(MP) NUMBER AND QUANTITY - Quantities

(N.Q) ALGEBRA - Seeing Structure in Expressions

(A.SSE) ALGEBRA - Creating Equations

(A.CED) ALGEBRA - Reasoning With Equations and Inequalities (A.REI) FUNCTIONS - Interpreting Linear and Exponential Functions (F.IF) FUNCTIONS - Building Linear or Exponential Functions (F.BF) FUNCTIONS - Linear and Exponential

(F.LE) GEOMETRY - Congruence

(G.CO) GEOMETRY - Expressing Geometric Properties With Equations (G.GPE) STATISTICS AND PROBABILITY - Interpreting Categorical and Quantitative Data (S.ID) HONORS: NUMBER AND QUANTITY - Vector and Matrix Quantities (N.VM)

Standard SI.MP.1

Make sense of problems and persevere in solving them. Explain the meaning of a problem and look for entry points to its solution. Analyze givens, constraints, relationships, and goals. Make conjectures about the form and meaning of the solution, plan a solution pathway, and continually monitor progress asking, "Does this make sense?" Consider analogous problems, make connections between multiple representations, identify the correspondence between different approaches, look for trends, and transform algebraic expressions to highlight meaningful mathematics. Check answers to problems using a different method.

Standard SI.MP.2

Reason abstractly and quantitatively. Make sense of the quantities and their relationships in problem situations. Translate between context and algebraic representations by contextualizing and decontextualizing quantitative relationships. This includes the ability to decontextualize a given situation, representing it algebraically and manipulating symbols fluently as well as the ability to contextualize algebraic representations to make sense of the problem.

Standard SI.MP.3

Construct viable arguments and critique the reasoning of others. Understand and use stated assumptions, definitions, and previously established results in constructing arguments. Make conjectures and build a logical progression of statements to explore the truth of their conjectures. Justify conclusions and communicate them to others. Respond to the arguments of others by listening, asking clarifying questions, and critiquing the reasoning of others.

Standard SI.MP.4

Model with mathematics. Apply mathematics to solve problems arising in everyday life, society, and the workplace. Make assumptions and approximations, identifying important quantities to construct a mathematical model. Routinely interpret mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Standard SI.MP.5

Use appropriate tools strategically. Consider the available tools and be sufficiently familiar with them to make sound decisions about when each tool might be helpful, recognizing both the insight to be gained as well as the limitations. Identify relevant external mathematical resources and use them to pose or solve problems. Use tools to explore and deepen their understanding of concepts.

Standard SI.MP.6

Attend to precision. Communicate precisely to others. Use explicit definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose. Specify units of measure and label axes to clarify the correspondence with quantities in a problem. Calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context.

Standard SI.MP.7

Look for and make use of structure. Look closely at mathematical relationships to identify the underlying structure by recognizing a simple structure within a more complicated structure. See complicated things, such as some algebraic expressions, as single objects or as being composed of several objects.For example, see 5 - 3(x - y)^{2}as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Standard SI.MP.8

Look for and express regularity in repeated reasoning. Notice if reasoning is repeated, and look for both generalizations and shortcuts. Evaluate the reasonableness of intermediate results by maintaining oversight of the process while attending to the details.

- NUMBER & QUANTITY - Quantities (N.Q) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Quantities (N.Q).

- Introduction to the Materials (Math 1)

Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 1: Sequences - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 1, Sequences is written as two intertwined learning cycles that begin by alternating from arithmetic sequence to geometric sequences, so students can compare and contrast features as they represent both types of sequences with tables, graphs, story contexts, diagrams, and equations. - Module 1: Sequences - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 1 Teacher Notes, Sequences is written as two intertwined learning cycles that begin by alternating from arithmetic sequence to geometric sequences, so students can compare and contrast features as they represent both types of sequences with tables, graphs, story contexts, diagrams, and equations. - Module 4: Equations & Inequalities - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 4, Equations and Inequalities, students use units in combinations to define new variables for use in modeling with equations and inequalities, and interpret expressions that are the result of combining units. - Module 4: Equations & Inequalities - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 4 Teacher Notes, Equations and Inequalities, students use units in combinations to define new variables for use in modeling with equations and inequalities, and interpret expressions that are the result of combining units.

- Accuracy of Carbon 14 Dating I

This task examines, from a mathematical and statistical point of view, how scientists measure the age of organic materials by measuring the ratio of Carbon 14 to Carbon 12. The focus here is on the statistical nature of such dating. - Accuracy of Carbon 14 Dating II

This Illustrative Mathematics task is a refinement of "Carbon 14 dating" which focuses on accuracy. While the mathematical part of this task is suitable for assessment, the context makes it more appropriate for instructional purposes. This type of question is very important in science and it also provides an opportunity to study the very subtle question of how errors behave when applying a function: in some cases the errors can be magnified while in others they are lessened. - Bus and Car

This Illustrative Mathematics task operates at two levels. In part it is a simple exploration of the relationship between speed, distance, and time. Part (c) requires understanding of the idea of average speed, and gives an opportunity to address the common confusion between average speed and the average of the speeds for the two segments of the trip. - Calories in a sports drink

This Illustrative Mathematics task involves the meaning of numbers found on labels. When the level of accuracy is not given we need to make assumptions based on how the information is reported. An unexpected surprise awaits in this case, however, as no reasonable interpretation of the level of accuracy makes sense of the information reported on the bottles in parts (b) and (c). Either a miscalculation has been made or the numbers have been rounded in a very odd way. - Dinosaur Bones

The purpose of this Illustrative Mathematics task is to illustrate through an absurd example the fact that in real life quantities are reported to a certain level of accuracy, and it does not make sense to treat them as having greater accuracy. - Felicia's Drive

This Illustrative Mathematics task provides students the opportunity to make use of units to find the gas need to make some sensible approximations. - Framing a House - student task

This task has students recreate house plans on graph paper and then determine how many linear feet of wall plate material will be needed. - Fuel Efficiency

Sadie has a cousin Nanette in Germany. Both families recently bought new cars and the two girls are comparing how fuel efficient the two cars are. Sadie tells Nanette that her family's car is getting 42 miles per gallon. Nanette has no idea how that compares to her famiy's car because in Germany mileage is measured differently. She tells Sadie that her family's car uses 6 liters per 100 km. Which car is more fuel efficient? - Giving raises

A small company wants to give raises to their 5 employees. They have $10,000 available to distribute. Imagine you are in charge of deciding how the raises should be determined. - Harvesting the Fields

This is a challenging Illustrative Mathematics task, suitable for extended work, and reaching into a deep understanding of units. The task requires students to exhibit MP1, Make sense of problems and persevere in solving them. - How Much is a Penny Worth?

The price of copper fluctuates. Between 2002 and 2011, there were times when its price was lower than $1.00 per pound and other times when its priace was higher than $4.00 per pound. Copper pennies minted between 1962 and 1982 are 95% copper and 5% zinc by weight, and each penny weighs 3.11 grams. At what price per pound of copper does such a penny contain exactly one cent worth of copper? - Ice Cream Van

The purpose of this Illustrative Mathematics task is to engage students, probably working in groups, in a substantial and open-ended modeling problem. Students will have to brainstorm or research several relevant quantities, and incorporate these values into their solutions. - Runners' World

This Illustrative Mathematics task provides students with an opportunity to engage in Standard for Mathematical Practice 6, attending to precision. It intentionally omits some relevant information. The incompleteness of the problem statement makes the task more amenable to having students do work in groups. - Selling Fuel Oil at a Loss

This Illustrative Mathematics task is a modeling problem which ties in to financial decisions faced routinely by businesses, namely the balance between maintaining inventory and raising short-term capital for investment or re-investment in developing the business. - Solar Radiation Model

The task is a seemingly straightforward modeling task that can lead to more involved tasks if the instructor expands on it. In this task, students also have to interpret the units of the input and output variables of the solar radiation function. - Traffic Jam

Last Sunday an accident caused a traffic jam 12 miles long on a straight stretch of a two lane freeway. How many vehicles do you think were in the traffic jam? Explain your thinking and show all calculations. - Weed killer

The principal purpose of this Illustrative Mathematics task is to explore a real-world application problem with algebra, working with units and maintaining reasonable levels of accuracy throughout.

- Access Ramp - Student Task

This task has students design an access ramp, which complies with the Americans with Disabilities Act (ADA) requirements and include pricing based on local costs. - Fences - student task

This task has students design a fence that meets the city ordinances and the client's specifications. - Framing a House - student task

This task has students recreate house plans on graph paper and then determine how many linear feet of wall plate material will be needed. - Lesson Starter: Solving Math Problems with a Team

Students will work in teams to solve mathematical problems; they listen to the reasoning of others and offer correction with supporting arguments; they modify their own arguments when corrected; they learn from mistakes and make repeated attempts at solving problems. - Storage Shed - student task

Students are going to build storage sheds as a fund raising project, but before they can start they must determine the best dimensions for the shed, make scale drawings and decide on how much to charge for each shed.

Standard N.Q.1

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.Standard N.Q.2

Define appropriate quantities for the purpose of descriptive modeling.Standard N.Q.3

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

- ALGEBRA - Seeing Structure in Expressions (A.SSE) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Seeing Structure in Expressions (A.SSE)

- Introduction to the Materials (Math 1)

Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 1: Sequences - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 1, Sequences is written as two intertwined learning cycles that begin by alternating from arithmetic sequence to geometric sequences, so students can compare and contrast features as they represent both types of sequences with tables, graphs, story contexts, diagrams, and equations. - Module 1: Sequences - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 1 Teacher Notes, Sequences is written as two intertwined learning cycles that begin by alternating from arithmetic sequence to geometric sequences, so students can compare and contrast features as they represent both types of sequences with tables, graphs, story contexts, diagrams, and equations. - Module 2: Linear & Exponential Functions - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 2, Linear and Exponential Functions, begins with a learning cycle that introduces contexts with continuous domains and defining linear functions as having a constant rate of change and exponential functions as having a constant ratio over equal intervals. - Module 2: Linear & Exponential Functions - Teacher Notes (Math 1)

Mathematics Vision Project, Secondary Math One Module 2 Teacher Notes, Linear and Exponential Functions, begins with a learning cycle that introduces contexts with continuous domains and defining linear functions as having a constant rate of change and exponential functions as having a constant ratio over equal intervals.

- Animal Populations

This Illustrative Mathematics task students have to interpret expressions involving two variables in the context of a real world situation. - Delivery Trucks

The task has students look for structure in algebraic expressions related to a context, and asks them to relate that structure to the context. In particular, it is worth emphasizing that the task requires no algebraic manipulation from the students. - Exponential Parameters

The task provides a reasonably straight-forward introduction to interpreting the parameters of an exponential function in terms of a modeling context. The task has students both generate an exponential expression from a contextual description, and in reverse, interpret parameters in a context from an algebraic expression. - Increasing or Decreasing? Variation 1

This Illustrative Mathematics task encourages students to reason quantitatively about the structure of the expression. - Increasing or Decreasing? Variation 2

The purpose of this task is to help students see manipulation of expressions as an activity undertaken for a purpose. The expression arises in physics as the reciprocal of the combined resistance of two resistors in parallel. However, the context is not explicitly considered here. - Kitchen Floor Tiles

The purpose of this task is to give students practice in reading, analyzing, and constructing algebraic expressions, attending to the relationship between the form of an expression and the context from which it arises. - Mixing Fertilizer

This Illustrative Mathematics task deals with a rational expression which is built up from operations arising naturally in a context: adding the volumes of the fertilizer and the water, and dividing the volume of the fertilizer by the resulting sum. - Modeling London's Population

The purpose of this task is to model the population data for London with a variety of different functions. In addition to the linear, quadratic, and exponential models, this task introduces an additional model, namely the logistic model. - Quadrupling Leads to Halving

This Illustrative Mathematics task provides students with an opportunity to see expressions as constructed out of a sequence of operations. - Throwing Horseshoes

This Illustrative Mathematics task requires students to identify expressions as sums or products and interpret each summand or factor.

- Adding and Subtracting Polynomials video

This video introduces and explains the topic. - Multiplying Polynomials video

This video introduces and explains the concept. - Polynomials video

This video introduces and explains the topic. - Special Products of Polynomials

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Special Products of Polynomials video

This video introduces and explains the concept.

Standard A.SSE.1

Interpret linear expressions and exponential expressions with integer exponents that represent a quantity in terms of its context.^{★}

- Interpret parts of an expression, such as terms, factors, and coefficients.
- Interpret complicated expressions by viewing one or more of their parts as a single entity.
For example, interpret P(1+r)^{n}as the product of P and a factor not depending on P.

- ALGEBRA - Creating Equations (A.CED) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Creating Equations (A.CED). - ALGEBRA - Seeing Structure in Expressions (A.SSE) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Seeing Structure in Expressions (A.SSE)

- Introduction to the Materials (Math 1)

Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 2: Linear & Exponential Functions - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 2, Linear and Exponential Functions, begins with a learning cycle that introduces contexts with continuous domains and defining linear functions as having a constant rate of change and exponential functions as having a constant ratio over equal intervals. - Module 2: Linear & Exponential Functions - Teacher Notes (Math 1)

Mathematics Vision Project, Secondary Math One Module 2 Teacher Notes, Linear and Exponential Functions, begins with a learning cycle that introduces contexts with continuous domains and defining linear functions as having a constant rate of change and exponential functions as having a constant ratio over equal intervals. - Module 3: Features of Functions - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 3, Features of Functions, is the culminating functions module in Secondary Math I. In this module, students broaden their thinking about functions to relationships that are not either linear or exponential. - Module 3: Features of Functions - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 3 Teacher Notes, Features of Functions, is the culminating functions module in Secondary Math I. In this module, students broaden their thinking about functions to relationships that are not either linear or exponential. - Module 4: Equations & Inequalities - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 4, Equations and Inequalities, students use units in combinations to define new variables for use in modeling with equations and inequalities, and interpret expressions that are the result of combining units. - Module 4: Equations & Inequalities - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 4 Teacher Notes, Equations and Inequalities, students use units in combinations to define new variables for use in modeling with equations and inequalities, and interpret expressions that are the result of combining units. - Module 5: Systems of Equations & Inequalities - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 5, Systems of Equations and Inequalities, has two learning cycles, built around a common story context that is used throughout the module. The first learning cycle begins by making the representations, tables, graphs, equations, and diagrams, needed for the rest of the module available. - Module 5: Systems of Equations & Inequalities - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 5 Teacher Notes, Systems of Equations and Inequalities, has two learning cycles, built around a common story context that is used throughout the module. The first learning cycle begins by making the representations, tables, graphs, equations, and diagrams, needed for the rest of the module available.

- Basketball

This task provides a simple but interesting and realistic context in which students are led to set up a rational equation (and a rational inequality) in one variable, and then solve that equation/inequality for an unknown variable. - Harvesting the Fields

This is a challenging Illustrative Mathematics task, suitable for extended work, and reaching into a deep understanding of units. The task requires students to exhibit MP1, Make sense of problems and persevere in solving them. - Linear Equations in One Variable

This tutorial is designed to help students understand how to solve linear equations by using the addition, subtraction, multiplication and division properties of equalities. - Student Task: Best Buy Tickets

Susie is organizing the printing of tickets for a show. She has collected prices from several printers. The student's task is to use graphs and algebra to advise Susie on how to choose the best printer. - Student Task: Printing Tickets

Susie is organizing the printing of tickets for a show. She has collected prices from several printers. The student's task is to use graphs and algebra to advise Susie on how to choose the best printer. - Student Task: Skeleton Tower

In this task, students must work out a rule for calculating the total number of cubes needed to build towers of different heights.

- Applications of Quadratic Functions

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Common Math Formulas

A list of common mathematics formulas. - Compound Interest Simulator

This applet will allow students to investigate savings account earnings, credit card debt, and a stock market simulation. - Graphing Equations in Slope Intercept Form

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Graphing Equations in Slope Intercept Form Video

This is a video introduction to the topic. - Graphing Equations on the Cartesian Plane: Slope

This lesson plan is designed to teach students about the slope of lines. They are asked to determine the slope given a graph and given 2 points on the line, and compare the slopes of parallel and perpendicular lines. - Graphing Quadratic Functions video

This video introduces and explains the topic. - Intercepts of Linear Equations

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Intercepts of Linear Equations video

This video introduces the topic. - Linear Functions

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Linear Functions video

This video compares proportional and non-proportional linear functions. - Non-Linear Functions

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Parallel Lines

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Parallel Lines video

This video explains the concept. - Perpendicular Lines

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Perpendicular Lines video

This video introduces and explains perpendicular lines. - Point Slope Form and Standard Form of Linear Equations

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Point Slope Form and Standard Form of Linear Equations video

This a video explanation of the topic. - Polynomials

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Polynomials video

This video introduces and explains the topic. - Proportional Functions

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Proportional Functions video

This video introduces proportional functions. - Solving Linear Inequalities

Students are helped to understand how to solve inequalities in this tutorial. Simple and more advanced problems are provided for practice. - Solving Quadratic Equations Using the Quadratic Formula video

This video introduces and explains the topic. - Solving Systems of Equations by Graphing

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Solving Systems of Linear Equations by Graphing

This is a video introduction and explanation of the topic. - Solving for a Specific Variable video

This is an introductory video to formulas. - Stairway - Student Task

This task students to design a stairway for a custom home. They will need to gather information regarding design, safety, and the utility of staircases. - Translating Word Problems into Equations

Seven steps are presented on this site showing the student how to translate word problems into a solvable equation. - Writing Expressions and Equations video

How to write an equation using what we know to solve a problem we don't know. - Writing and Using Inequalities

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Writing and Using Inequalities video

This video introduces and explains the topic.

Standard A.CED.1

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and simple exponential functions.Standard A.CED.2

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.Standard A.CED.3

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.Standard A.CED.4

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.For example, rearrange Ohm's law V = IR to highlight resistance R.

- ALGEBRA - Reasoning With Equations and Inequalities (A.REI) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Reasoning with Equations and Inequalities (A.REI).

- Introduction to the Materials (Math 1)

Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 4: Equations & Inequalities - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 4, Equations and Inequalities, students use units in combinations to define new variables for use in modeling with equations and inequalities, and interpret expressions that are the result of combining units. - Module 4: Equations & Inequalities - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 4 Teacher Notes, Equations and Inequalities, students use units in combinations to define new variables for use in modeling with equations and inequalities, and interpret expressions that are the result of combining units.

- Complex Square Roots

This Illustrative Mathematics task is intended as an introduction to the algebra of the complex numbers, and also builds student's comfort and intuition with these numbers. - How does the solution change?

The equations in this task are intentionally very simple; the point of the task is not to test technique in solving equations, but to encourage students to reason about them. - Products and Reciprocals

The purpose of this task is to test student skill at converting verbal statements into two algebraic equations and then solving those equations - Reasoning with linear inequalities

This problem is intended to detect the ability of the student to identify errors in mathematical reasoning, and to help students see the process of solving a equation or inequality is a special kind of proof. - Same solutions?

The purpose of this task is to provide an opportunity for students to look for structure when comparing equations and to reason about their equivalence. - Zero Product Property 1

This task is the first in a series that leads students to understand and apply the zero product property to solving quadratic equations. The emphasis is on using the structure of a factorable expression in order to justify the steps in a solution (rather than memorizing steps without understanding). - Zero Product Property 2

This task is part of a series of tasks that lead students to understand and apply the zero product property to solving quadratic equations. The emphasis is on using the structure of a factorable expression to help find its solutions (rather than memorizing steps without understanding). - Zero Product Property 3

This task is part of a series of tasks that lead students to understand and apply the zero product property to solving quadratic equations. The emphasis is on using the structure of a factorable expression to help find its solutions (rather than memorizing steps without understanding). - Zero Product Property 4

This task is the fourth in a series of tasks that leads students to understand The Zero Product Property (ZPP) and apply it to solving quadratic equations. The emphasis is on using the structure of a factorable expression to justify the solution method (rather than memorizing steps without understanding)

- Absolute Value

This array of resources teaching absolute value includes warm-up problems, a video introducing the topic, worked examples, practice problems, and a review. - Applying Radical Equations

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Applying Radical Equations video

This video introduces and explains the topic. - Applying Rational Equations

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Applying Rational Equations video

This video introduces and explains the topic. - Graphing Quadratic Functions

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Graphing Quadratic Functions video

This video introduces and explains the topic. - Intercepts of Linear Equations

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Intercepts of Linear Equations video

This video introduces the topic. - Proportional Functions

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Proportional Functions video

This video introduces proportional functions. - Solve Quadratic Expressions by Factoring video

This video introduces and explains the concept. - Solving Quadratic Equations Using the Quadratic Formula video

This video introduces and explains the topic. - Solving Rational Expressions video

This video introduces and explains the topic. - Writing and Using Inequalities video

This video introduces and explains the topic.

Standard A.REI.1

Explain each step in solving a linear equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Students will solve exponential equations with logarithms in Secondary Mathematics III.

- ALGEBRA - Reasoning With Equations and Inequalities (A.REI) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Reasoning with Equations and Inequalities (A.REI).

- Introduction to the Materials (Math 1)

Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 1: Sequences - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 1, Sequences is written as two intertwined learning cycles that begin by alternating from arithmetic sequence to geometric sequences, so students can compare and contrast features as they represent both types of sequences with tables, graphs, story contexts, diagrams, and equations. - Module 1: Sequences - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 1 Teacher Notes, Sequences is written as two intertwined learning cycles that begin by alternating from arithmetic sequence to geometric sequences, so students can compare and contrast features as they represent both types of sequences with tables, graphs, story contexts, diagrams, and equations. - Module 4: Equations & Inequalities - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 4, Equations and Inequalities, students use units in combinations to define new variables for use in modeling with equations and inequalities, and interpret expressions that are the result of combining units. - Module 4: Equations & Inequalities - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 4 Teacher Notes, Equations and Inequalities, students use units in combinations to define new variables for use in modeling with equations and inequalities, and interpret expressions that are the result of combining units. - Secondary I Textbook

Secondary I Textbook is composed of modules that are aligned with the Utah Core State Standards for Mathematics. Each lesson begins with a worthwhile task that has been designed to develop mathematical understanding, solidify that understanding, or allow for practice of the new concepts, while focusing on the mathematical goals of the chosen learning cycle.

- Reasoning with linear inequalities

This problem is intended to detect the ability of the student to identify errors in mathematical reasoning, and to help students see the process of solving a equation or inequality is a special kind of proof.

- IXL Game: Solve Two-Step Linear Equations

This game is designed to help the student understand how to solve two-step linear equations. This is just one of many online games that supports the Utah Math core. Note: The IXL site requires subscription for unlimited use. - Solving Linear Inequalities

Students are helped to understand how to solve inequalities in this tutorial. Simple and more advanced problems are provided for practice. - Solving Rational Expressions video

This video introduces and explains the topic. - Solving Systems of Linear Equations by Substitution video

This is a video introduction and explanation of the topic. - Writing and Using Inequalities

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Writing, Solving and Graphing Inequalities in One Variable

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review.

Standard A.REI.3

Solve equations and inequalities in one variable.

- Solve one-variable equations and literal equations to highlight a variable of interest.
- Solve compound inequalities in one variable, including absolute value inequalities.
- Solve simple exponential equations that rely only on application of the laws of exponents (limit solving exponential equations to those that can be solved without logarithms).
For example, 5^{x}= 125 or 2^{x}= 1/16.

- ALGEBRA - Reasoning With Equations and Inequalities (A.REI) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Reasoning with Equations and Inequalities (A.REI).

- Introduction to the Materials (Math 1)

Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 5: Systems of Equations & Inequalities - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 5, Systems of Equations and Inequalities, has two learning cycles, built around a common story context that is used throughout the module. The first learning cycle begins by making the representations, tables, graphs, equations, and diagrams, needed for the rest of the module available. - Module 5: Systems of Equations & Inequalities - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 5 Teacher Notes, Systems of Equations and Inequalities, has two learning cycles, built around a common story context that is used throughout the module. The first learning cycle begins by making the representations, tables, graphs, equations, and diagrams, needed for the rest of the module available.

- Accurately weighing pennies I

This task asks students to solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. - Accurately weighing pennies II

This task is a somewhat more complicated version of ''Accurately weighing pennies I'' as a third equation is needed in order to solve part (a) explicitly. - Cash Box

The purpose of this task is to gives students an opportunity to engage in Mathematical Practice #3 Construct viable arguments and critique the reasoning of others. This task gives a teacher the opportunity to ask students not only for a specific answer of whether the dollar came from in the cash box or not, but for students to construct an argument as to how they came to their solution. - Estimating a Solution via Graphs

The purpose of this task is to give students an opportunity use quantitative and graphical reasoning to detect an error in a solution. - Find A System

The purpose of this task is to encourage students to think critically about both the algebraic and graphical interpretation of systems of linear equations. They are expected to take what they know about solving systems of linear equations, and then reverse the usual process. - Pairs of Whole Numbers

This task addresses solving systems of linear equations exactly, and provides a simple example of a system with three equations and three unknowns. - Quinoa Pasta 2

This task is a variant of 8.EE Quinoa Pasta 1, where all the relevant information is given as part of the task statements and the students are asked to set up a system of equations. - Quinoa Pasta 3

This task is an example of a mathematical modeling problem (SMP 4) and it also illustrates SMP 1 (Making sense of a problem). This task is a variant of 8.EE Quinoa Pasta 1 and A-REI.6 Quinoa Pasta 2. - Solving Two Equations in Two Unknowns

The goal of this task is to help students see the validity of the elimination method for solving systems of two equations in two unknowns.

- Linear Functions

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Linear Functions video

This video compares proportional and non-proportional linear functions. - Rate Problems video

This video introduces the explains the topic. - Solving Rational Expressions video

This video introduces and explains the topic. - Solving Systems of Equations by Graphing

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Solving Systems of Linear Equations by Elimination video

This video introduces and explains the concept. - Solving Systems of Linear Equations by Graphing

This is a video introduction and explanation of the topic.

Standard A.REI.5

Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.Standard A.REI.6

Solve systems of linear equations exactly and approximately (numerically, algebraically, graphically), focusing on pairs of linear equations in two variables.

- ALGEBRA - Reasoning With Equations and Inequalities (A.REI) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Reasoning with Equations and Inequalities (A.REI).

- Introduction to the Materials (Math 1)

Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 3: Features of Functions - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 3, Features of Functions, is the culminating functions module in Secondary Math I. In this module, students broaden their thinking about functions to relationships that are not either linear or exponential. - Module 3: Features of Functions - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 3 Teacher Notes, Features of Functions, is the culminating functions module in Secondary Math I. In this module, students broaden their thinking about functions to relationships that are not either linear or exponential. - Module 5: Systems of Equations & Inequalities - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 5, Systems of Equations and Inequalities, has two learning cycles, built around a common story context that is used throughout the module. The first learning cycle begins by making the representations, tables, graphs, equations, and diagrams, needed for the rest of the module available. - Module 5: Systems of Equations & Inequalities - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 5 Teacher Notes, Systems of Equations and Inequalities, has two learning cycles, built around a common story context that is used throughout the module. The first learning cycle begins by making the representations, tables, graphs, equations, and diagrams, needed for the rest of the module available.

- A Linear and Quadratic System

The purpose of this task is to give students the opportunity to make connections between equations and the geometry of their graphs. They must read information from the graph (such as the vertical intercept of the quadratic graph or the slope of the linear one), use that information to construct and solve an equation, then interpret their solution in terms of the graph. - Collinear points

This task leads students through a series of problems which illustrate a crucial interplay between algebra (e.g., being solutions to equations) and geometry (e.g., being points on a curve). - Fishing Adventures 3

This task is the last in a series of three tasks that use inequalities in the same context at increasing complexity in 6th grade, 7th grade and in HS algebra. Students write and solve inequalities, and represent the solutions graphically. - Ideal Gas Law

The goal of this task is to interpret the graph of a rational function and use the graph to approximate when the function takes a given value. - Population and Food Supply

In this task students construct and compare linear and exponential functions and find where the two functions intersect. - Solution Sets

The purpose of this task is to give students a chance to go beyond the typical problem and make the connections between points in the coordinate plane and solutions to inequalities and equations. - Two Squares are Equal

This classroom task is meant to elicit a variety of different methods of solving a quadratic equation.

- Graphing Equations in Slope Intercept Form

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Graphing Equations in Slope Intercept Form Video

This is a video introduction to the topic. - Graphing Quadratic Functions

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Graphing Quadratic Functions video

This video introduces and explains the topic. - Graphing Systems of Inequalities

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Graphing Systems of Inequalities video

This video introduces and explains the topic. - Intercepts of Linear Equations

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Intercepts of Linear Equations video

This video introduces the topic. - Linear Functions

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Linear Functions video

This video compares proportional and non-proportional linear functions. - Linear Inequalities

This lesson contains an activity that allows students to use a coordinate plane to create linear inequalities. - Non-Linear Functions

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Non-linear Functions video

This video introduces non-linear functions. - Parallel Lines

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Parallel Lines video

This video explains the concept. - Perpendicular Lines

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Perpendicular Lines video

This video introduces and explains perpendicular lines. - Point Slope Form and Standard Form of Linear Equations

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Point Slope Form and Standard Form of Linear Equations video

This a video explanation of the topic. - Proportional Functions

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Proportional Functions video

This video introduces proportional functions. - Representing Inequalities Graphically

This lesson unit is intended to help educators assess how well students are able to use linear inequalities to create a set of solutions. - Solving Quadratic Equations Using the Quadratic Formula video

This video introduces and explains the topic. - Solving Rational Expressions video

This video introduces and explains the topic. - Solving Systems of Equations by Graphing

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Solving Systems of Linear Equations by Graphing

This is a video introduction and explanation of the topic. - Solving and Graphing Linear Inequalities in Two Variables

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Solving and Graphing Linear Inequalities in Two Variables video

This video introduces and explains the concept. - Stairway - Student Task

This task students to design a stairway for a custom home. They will need to gather information regarding design, safety, and the utility of staircases. - Student Task: Cubic Graph

In this task, students will look at the properties of a cubic equation.

Standard A.REI.10

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).Standard A.REI.11

Explain why thex-coordinates of the points where the graphs of the equationsy = f(x)andy = g(x)intersect are the solutions of the equationf(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases wheref(x)and/org(x)are linear and exponential functions.^{★}Standard A.REI.12

Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

- ALGEBRA - Seeing Structure in Expressions (A.SSE) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Seeing Structure in Expressions (A.SSE) - FUNCTIONS - Interpreting Linear and Exponential Functions (F.IF) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I Interpreting Linear and Exponential Functions (F.IF).

- Introduction to the Materials (Math 1)

Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 2: Linear & Exponential Functions - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 2, Linear and Exponential Functions, begins with a learning cycle that introduces contexts with continuous domains and defining linear functions as having a constant rate of change and exponential functions as having a constant ratio over equal intervals. - Module 2: Linear & Exponential Functions - Teacher Notes (Math 1)

Mathematics Vision Project, Secondary Math One Module 2 Teacher Notes, Linear and Exponential Functions, begins with a learning cycle that introduces contexts with continuous domains and defining linear functions as having a constant rate of change and exponential functions as having a constant ratio over equal intervals. - Module 3: Features of Functions - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 3, Features of Functions, is the culminating functions module in Secondary Math I. In this module, students broaden their thinking about functions to relationships that are not either linear or exponential. - Module 3: Features of Functions - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 3 Teacher Notes, Features of Functions, is the culminating functions module in Secondary Math I. In this module, students broaden their thinking about functions to relationships that are not either linear or exponential.

- Cell Phones

This simple task assesses whether students can interpret function notation. The four parts of the task provide a logical progression of exercises for advancing understanding of function notation and how to interpret it in terms of a given context. - Do two points always determine a linear function?

This problem allows the student to think geometrically about lines and then relate this geometry to linear functions. Or the student can work algebraically with equations in order to find the explicit equation of the line through two points (when that line is not vertical). - Domains

The purpose of this task to help students think about an expression for a function as built up out of simple operations on the variable, and understand the domain in terms of values for which each operation is invalid (e.g., dividing by zero or taking the square root of a negative number). - Finding the domain

The purpose of this task is to introduce the idea of the domain of a function by linking it to the evaluation of an expression defining the function. - Interpreting the Graph

Students will use the graph (for example, by marking specific points) to illustrate the statements in (a) and (d). If possible, label the coordinates of any points you draw. - Linear Functions

The applet in this lesson allows students to manipulate variables and see the changes in the graphed line. - Pizza Place Promotion

Students will use a function that models a relationship between two quantities to figure out how a pizza restaurant's promotion that prices pizza based on a function of time causes the cost to fluctuate. - Points on a Graph

This task is designed to get at a common student confusion between the independent and dependent variables. This confusion often arises in situations like (b), where students are asked to solve an equation involving a function, and confuse that operation with evaluating the function. - Random Walk II

This task follows up on ''The Random Walk,'' looking in closer detail at what outcomes are possible. These problems form a bridge between work on functions and work on probability. The task is better suited for instruction than for assessment as it provides students with a non standard setting in which to interpret the meaning of functions. Students should carry out the process of flipping a coin and modeling this Random Walk in order to develop a sense of the process before analyzing it mathematically. - Snake on a Plane

This task has students approach a function via both a recursive and an algebraic definition, in the context of a famous game of antiquity that they may have encountered in a more modern form. - The Customers

The purpose of this task is to introduce or reinforce the concept of a function, especially in a context where the function is not given by an explicit algebraic representation. Further, the last part of the task emphasizes the significance of one variable being a function of another variable in an immediately relevant real-life context. Instructors might prepare themselves for variations on the problems that the students might wander into (e.g., whether one person could have two home phone numbers) and how such variants affect the correct responses. - The Parking Lot

The purpose of this task is to investigate the meaning of the definition of function in a real-world context where the question of whether there is more than one output for a given input arises naturally. In more advanced courses this task could be used to investigate the question of whether a function has an inverse. - The Random Walk

This task requires interpreting a function in a non-standard context. While the domain and range of this function are both numbers, the way in which the function is determined is not via a formula but by a (pre-determined) sequence of coin flips. In addition, the task provides an opportunity to compute some probabilities in a discrete situation. The task could be used to segue the discussion from functions to probability, in particular the early standards in the S-CP domain. - Using Function Notation I

This task deals with a student error that may occur while students are completing F-IF Average Cost. - Using Function Notation II

The purpose of the task is to explicitly identify a common error made by many students, when they make use of the "identity" f(x+h)=f(x)+f(h). The task has students find a single explicit example for which the identity is false, but it is worth emphasizing that in fact the identity fails for the vast majority of functions. - Vertical Line Test

This interactive applet asks the student to connect points on a plane in order to build a function and then test it to see if it's valid. - Yam in the Oven

The purpose of this task is to give students practice interpreting statements using function notation. It can be used as a diagnostic if students seem to be having trouble with function notation, for example interpreting f(x) as the product of f and x. - Your Father

This is a simple task touching on two key points of functions. First, there is the idea that not all functions have real numbers as domain and range values. Second, the task addresses the issue of when a function admits an inverse, and the process of "restricting the domain" in order to achieve an invertible function.

- Algebra Why and When video

This video explains why and when algebra is needed instead of arithmetic functions. - Derivate

Students may use the applet in this lesson to graph a function and a tangent line and view its equation. - Domain and Range

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Function Flyer

The applet on this site allows the students to manipulate the graph of a function by changing the value of exponents, coefficients and constants. - Inductive Patterns

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Inductive Patterns video

This video explains patterns and how we can use math with patterns. - Inductive Reasoning

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Inductive Reasoning video

This video introduces and explains the topic. - Representing Functions and Relations

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Representing Functions and Relations video

Explains how algebra can be used to describe, represent and predict relations. - Representing Patterns

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Representing Patterns video

This video introduces tables and graphs as representations of patterns. - Sequencer

By using this applet students are able to create sequences by changing the values of starting numbers, multipliers, and add-ons.

Standard F.IF.1

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. Iffis a function andxis an element of its domain, thenf(x)denotes the output offcorresponding to the inputx. The graph offis the graph of the equationy = f(x).Standard F.IF.2

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.Standard F.IF.3

Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Emphasize arithmetic and geometric sequences as examples of linear and exponential functions.For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

- FUNCTIONS - Interpreting Linear and Exponential Functions (F.IF) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I Interpreting Linear and Exponential Functions (F.IF).

- Introduction to the Materials (Math 1)

Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 3: Features of Functions - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 3, Features of Functions, is the culminating functions module in Secondary Math I. In this module, students broaden their thinking about functions to relationships that are not either linear or exponential. - Module 3: Features of Functions - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 3 Teacher Notes, Features of Functions, is the culminating functions module in Secondary Math I. In this module, students broaden their thinking about functions to relationships that are not either linear or exponential.

- 1,000 is half of 2,000

This real-life modeling task could serve as a summative exercise in which many aspects of students' knowledge of functions are put to work. - As the Wheel Turns

In this task, students use trigonometric functions to model the movement of a point around a wheel and, in the case of part (c), through space (F-TF.5). Students also interpret features of graphs in terms of the given real-world context (F-IF.4). - Average Cost

For a function that models a relationship between two quantities, students will interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. - Containers

The purpose of the task is to help students think about how two quantities vary together in a context where the rate of change is not given explicitly but is derived from the context. - From the flight deck

This task is designed to help students learns how to Interpret functions that arise in applications in terms of the context. - Functions and the Vertical Line Test

The vertical line test for functions is the focus of this lesson plan. - Hoisting the Flag 1

In this task, students are given a scenario and different graphs that could describe the relationship of the quantities in the situation, the height of a flag as a function of time. - Hoisting the Flag 2

In this task, students are given a scenario and different graphs that could describe the relationship of the quantities in the situation, the height of a flag as a function of time. - How is the Weather?

This task can be used as a quick assessment to see if students can make sense of a graph in the context of a real world situation. The task could also be used to generate a group discussion on interpreting functions given by graphs. - Influenza Epidemic

The principal purpose of this task is to probe students' ability to correlate symbolic statements about a function using function notation with a graph of the function, and to interpret their answers in terms of the quantities between which the function describes a relationship. - Lake Sonoma

This task asks students to describe features of a graph. It provides an opportunity to introduce (or use) mathematical terminology that makes communication easier and more precise, such as: periodic behavior, maxima, minima, outliers, increasing, decreasing, slope. - Laptop Battery Charge 2

This task uses a situation that is familiar to students to solve a problem they probably have all encountered before: How long will it take until an electronic device has a fully charged battery? A linear model can be used to solve this problem. The task combines ideas from statistics, functions and modeling. It is a nice combination of ideas in different domains in the high school curriculum. - Linear Functions

The applet in this lesson allows students to manipulate variables and see the changes in the graphed line. - Logistic Growth Model, Abstract Version

This task is for instructional purposes only and students should already be familiar with some specific examples of logistic growth functions such as that given in ''Logistic growth model, concrete case.'' - Logistic Growth Model, Explicit Version

This problem introduces a logistic growth model in the concrete setting of estimating the population of the U.S. - Mathemafish Population

In this problem, students use given data points to calculate the average rate of change of a function over a specific interval, foreshadowing the idea of limits and derivatives to students. - Model air plane acrobatics

This task could serve as an introduction to periodic functions and as a lead-in to sinusoidal functions. By visualizing the height of a plane that is moving along the circumference of a circle several times, students get the idea that output values of the height functions will repeat themselves after each complete revolution. They also connect the situation with key features on the graph, for example they interpret the midline and amplitude of the function as the height of the center of the circle and its radius. - Modeling London's Population

The purpose of this task is to model the population data for London with a variety of different functions. In addition to the linear, quadratic, and exponential models, this task introduces an additional model, namely the logistic model. - Oakland Coliseum

This deceptively simple task asks students to find the domain and range of a function from a given context. The function is linear and if simply looked at from a formulaic point of view, students might find the formula for the line and say that the domain and range are all real numbers. However, in the context of this problem, this answer does not make sense, as the context requires that all input and output values are non-negative integers, and imposes additional restrictions. - Pizza Place Promotion

Students will use a function that models a relationship between two quantities to figure out how a pizza restaurant's promotion that prices pizza based on a function of time causes the cost to fluctuate. - Playing Catch

This task gives the graph of the height of a ball over time and asks for a story that could be represented by this graph. The graph is the mathematical representation of a situation and features of the graph correspond to specific moments in the story the graph tells. The purpose of the task is to get away from plotting graphs by focusing on coordinate points and instead looking at the bigger picture a qualitative view. - Solar Radiation Model

The task is a seemingly straightforward modeling task that can lead to more involved tasks if the instructor expands on it. In this task, students also have to interpret the units of the input and output variables of the solar radiation function. - Student Task: Interpreting Functions

This task consists of a set of 2 short questions. - Telling a Story With Graphs

In this task students are given graphs of quantities related to weather. The purpose of the task is to show that graphs are more than a collection of coordinate points, that they can tell a story about the variables that are involved and together they can paint a very complete picture of a situation, in this case the weather. - Temperature Change

This task gives an easy context to introduce the idea of average rate of change. This problem could be done as a Think-Pair-Share activity. After posing the question, students can decide what they think and why and then discuss their answer with their neighbor. - The Aquarium

The purpose of this task is to connect graphs with real life situations. Graphs tell a story. Specific features of a graph connect to specific features of a story. A point on a graph captures a specific instant in the story. - The Canoe Trip, Variation 1

The purpose of this task is to give students practice constructing functions that represent a quantity of interest in a context, and then interpreting features of the function in the light of that context. It can be used as either an assessment or a teaching task. - The Canoe Trip, Variation 2

The primary purpose of this task is to lead students to a numerical and graphical understanding of the behavior of a rational function near a vertical asymptote, in terms of the expression defining the function. The canoe context focuses attention on the variables as numbers, rather than as abstract symbols. - The High School Gym

In this task, students will calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. - The Restaurant

The purpose of this task is to get students thinking about the domain and range of a function representing a particular context. - The story of a flight

This task uses data from an actual flight computer. - Warming and Cooling

This task is meant to be a straight-forward assessment task of graph reading and interpreting skills. This task helps reinforce the idea that when a variable represents time, t=0 is chosen as an arbitrary point in time and positive times are interpreted as times that happen after that. - Words - Tables - Graphs

The purpose of the task is to show that graphs can tell a story about the variables that are involved.

- Coordinates and the Cartesian Plane

This lesson helps students understand functions and the domain and range of a set of data points. - Derivate

Students may use the applet in this lesson to graph a function and a tangent line and view its equation. - Domain and Range video

This video introduces the concepts of domain and range. - Function Flyer

The applet on this site allows the students to manipulate the graph of a function by changing the value of exponents, coefficients and constants. - Graphing Calculator

A free online graphing calculator. - Graphing Quadratic Functions video

This video introduces and explains the topic. - Graphing Stories

The purpose of this task is to have students represent each indicated relationship of a given variable vs. time graphically with special attention to representing key features of increasing and decreasing intervals, maximums and minimums, intercepts, and constant and variable rates of change. - Graphit

With this interactive applet students are able to create graphs of functions and sets of ordered pairs on the same coordinate plane. - IXL Game: Linear Functions: Standard Form

This game will help the student understand linear functions, specifically the standard form by finding x- and y-intercepts. This is just one of many online games that supports the Utah Math core. Note: The IXL site requires subscription for unlimited use. - More Complicated Functions: Introduction to Linear Functions

This lesson is designed to reinforce the concept of linear functions and ask students to write functions using English, tables and algebra. - Multi-Function Data Flyer

The applet in this lesson allows students to plot ordered pairs and then change the values in order to observe the effects of those changes. - Polynomials video

The video introduces and explains the topic. - Possible or Not

Students can look at graphed functions from real-life examples and determine whether the graph makes sense or not in this activity. - Proportional Functions

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Rate of Change and Slope

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Rate of Change and Slope video

This video introduces the concepts. - Reading Graphs

Through this lesson students will understand how to graph functions. - Solving Quadratic Equations Using the Quadratic Formula video

This video introduces and explains the topic.

Standard F.IF.4

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; and end behavior.^{★}Standard F.IF.5

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.^{★}Standard F.IF.6

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.^{★}

- FUNCTIONS - Interpreting Linear and Exponential Functions (F.IF) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I Interpreting Linear and Exponential Functions (F.IF).

- Introduction to the Materials (Math 1)

Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 2: Linear & Exponential Functions - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 2, Linear and Exponential Functions, begins with a learning cycle that introduces contexts with continuous domains and defining linear functions as having a constant rate of change and exponential functions as having a constant ratio over equal intervals. - Module 2: Linear & Exponential Functions - Teacher Notes (Math 1)

Mathematics Vision Project, Secondary Math One Module 2 Teacher Notes, Linear and Exponential Functions, begins with a learning cycle that introduces contexts with continuous domains and defining linear functions as having a constant rate of change and exponential functions as having a constant ratio over equal intervals. - Module 3: Features of Functions - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 3, Features of Functions, is the culminating functions module in Secondary Math I. In this module, students broaden their thinking about functions to relationships that are not either linear or exponential. - Module 3: Features of Functions - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 3 Teacher Notes, Features of Functions, is the culminating functions module in Secondary Math I. In this module, students broaden their thinking about functions to relationships that are not either linear or exponential. - Module 8: Connecting Algebra & Geometry - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 8, Connecting Algebra and Geometry, students use the Pythagorean Theorem to find the distance between two points and to derive the distance formula. - Module 8: Connecting Algebra & Geometry - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 8 Teacher Notes, Connecting Algebra and Geometry, students use the Pythagorean Theorem to find the distance between two points and to derive the distance formula.

- Analyzing Graphs

This task could be used as a review problem or as an assessment problem after many different types of functions have been discussed. Since the different parameters of the functions are not given explicitly, the focus is not just on graphing specific functions but rather students have to focus on how values of parameters are reflected in a graph. - Bank Account Balance

The purpose of this task is to study an example of a function which varies discretely over time. - Exponential Kiss

The purpose of this task is twofold: first using technology to study the behavior of some exponential and logarithmic graphs and secondly to manipulate some explicit logarithmic and exponential expressions. - Graphing Rational Functions

This task starts with an exploration of the graphs of two functions whose expressions look very similar but whose graphs behave completely differently. - Graphs of Power Functions

This task requires students to recognize the graphs of different (positive) powers of x. - Graphs of Quadratic Functions

This exploration can be done in class near the beginning of a unit on graphing parabolas. Students need to be familiar with intercepts, and need to know what the vertex is. It is effective after students have graphed parabolas in vertex form, but have not yet explored graphing other forms. - Identifying Exponential Functions

The task is an introduction to the graphing of exponential functions. - Identifying graphs of functions

The goal of this task is to get students to focus on the shape of the graph of an equation and how this changes depending on the sign of the exponent and on whether the exponential is in the numerator or denominator. - Linear Functions

The applet in this lesson allows students to manipulate variables and see the changes in the graphed line. - Modeling London's Population

The purpose of this task is to model the population data for London with a variety of different functions. In addition to the linear, quadratic, and exponential models, this task introduces an additional model, namely the logistic model.

- Derivate

Students may use the applet in this lesson to graph a function and a tangent line and view its equation. - Function Flyer

The applet on this site allows the students to manipulate the graph of a function by changing the value of exponents, coefficients and constants. - Graphing Calculator

A free online graphing calculator. - Graphit

With this interactive applet students are able to create graphs of functions and sets of ordered pairs on the same coordinate plane. - More Complicated Functions: Introduction to Linear Functions

This lesson is designed to reinforce the concept of linear functions and ask students to write functions using English, tables and algebra. - Multi-Function Data Flyer

The applet in this lesson allows students to plot ordered pairs and then change the values in order to observe the effects of those changes. - Proportional Functions

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Reading Graphs

Through this lesson students will understand how to graph functions. - Representing Functions and Relations

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Representing Functions and Relations video

Explains how algebra can be used to describe, represent and predict relations. - Representing Polynomials

This lesson unit is intended to help educators assess how well students are able to translate between graphs and algebraic representations of polynomials. - Running Time

This task provides an application of polynomials in computing. This purpose of this task is to serve as an introduction, and motivation, for the study of end behavior of polynomials, content specifically addresses in standard F-IF.C7c. - Student Task: Sorting Functions

Students are given four graphs, four equations, four tables, and four rules. Their task is to match each graph with an equation, a table and a rule.

Standard F.IF.7

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.^{★}

- Graph linear functions and show intercepts.
- Graph exponential functions, showing intercepts and end behavior.
Standard F.IF.9

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).For example, compare the growth of two linear functions, or two exponential functions such as y=3^{n}and y=100•2^{n}.

- ALGEBRA - Seeing Structure in Expressions (A.SSE) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Seeing Structure in Expressions (A.SSE) - FUNCTIONS - Building Linear or Exponential Functions (F.BF) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Building Linear or Exponential Functions (F.BF)

- Introduction to the Materials (Math 1)

Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 1: Sequences - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 1, Sequences is written as two intertwined learning cycles that begin by alternating from arithmetic sequence to geometric sequences, so students can compare and contrast features as they represent both types of sequences with tables, graphs, story contexts, diagrams, and equations. - Module 1: Sequences - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 1 Teacher Notes, Sequences is written as two intertwined learning cycles that begin by alternating from arithmetic sequence to geometric sequences, so students can compare and contrast features as they represent both types of sequences with tables, graphs, story contexts, diagrams, and equations. - Module 2: Linear & Exponential Functions - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 2, Linear and Exponential Functions, begins with a learning cycle that introduces contexts with continuous domains and defining linear functions as having a constant rate of change and exponential functions as having a constant ratio over equal intervals. - Module 2: Linear & Exponential Functions - Teacher Notes (Math 1)

Mathematics Vision Project, Secondary Math One Module 2 Teacher Notes, Linear and Exponential Functions, begins with a learning cycle that introduces contexts with continuous domains and defining linear functions as having a constant rate of change and exponential functions as having a constant ratio over equal intervals. - Module 3: Features of Functions - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 3, Features of Functions, is the culminating functions module in Secondary Math I. In this module, students broaden their thinking about functions to relationships that are not either linear or exponential. - Module 8: Connecting Algebra & Geometry - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 8, Connecting Algebra and Geometry, students use the Pythagorean Theorem to find the distance between two points and to derive the distance formula. - Module 8: Connecting Algebra & Geometry - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 8 Teacher Notes, Connecting Algebra and Geometry, students use the Pythagorean Theorem to find the distance between two points and to derive the distance formula.

- 1,000 is half of 2,000

This real-life modeling task could serve as a summative exercise in which many aspects of students' knowledge of functions are put to work. - Building an Explicit Quadratic Function by Composition

This task is intended for instruction and to motivate the task Building a General Quadratic Function. This task assumes that the students are familiar with the process of completing the square. - Compounding with a 100% Interest Rate

This task provides an approximation, and definition, of e, in the context of more and more frequent compounding of interest in a bank account. This task is preliminary to F-LE Compounding Interest with a 5% Interest Rate which further develops the relationship between e and compound interest. - Compounding with a 5% Interest Rate

This task develops reasoning behind the general formula for balances under continuously compounded interest. While this task itself specifically addresses the standard (F-BF), building functions from a context, a auxiliary purpose is to introduce and motivate the number e, which plays a significant role in the (F-LE) domain of tasks. - Crude Oil and Gas Mileage

In this task students are asked to write expressions about the relation to the price of oil and gas mileage. - Exponential Parameters

The task provides a reasonably straight-forward introduction to interpreting the parameters of an exponential function in terms of a modeling context. The task has students both generate an exponential expression from a contextual description, and in reverse, interpret parameters in a context from an algebraic expression. - Flu on Campus

The purpose of this problem is to have students compose functions using tables of values only. Students are asked to consider the meaning of the composition of functions to solidify the concept that the domain of g contains the range of f. - Graphs and Functions

This lesson plan is designed to help the student understand how to plot functions on the Cartesian plane and how the graphing of functions leads to lines and parabolas. - Graphs of Compositions

This task addresses an important issue about inverse functions. In this case the function f is the inverse of the function g but g is not the inverse of f unless the domain of f is restricted. - Kimi and Jordan

In the middle grades, students have lots of experience analyzing and comparing linear functions using graphs, tables, symbolic expressions, and verbal descriptions. In this task, students may choose a representation that suits them and then reason from within that representation. When used in instruction, this task provides opportunities to compare representations and to make connections among them. - Lake Algae

The purpose of this task is to introduce students to exponential growth. While the context presents a classic example of exponential growth, it approaches it from a non-standard point of view. Instead of giving a starting value and asking for subsequent values, it gives an end value and asks about what happened in the past. - Skeleton Tower

This problem is a quadratic function example. - Student Task: Best Buy Tickets

Susie is organizing the printing of tickets for a show. She has collected prices from several printers. The student's task is to use graphs and algebra to advise Susie on how to choose the best printer. - Student Task: Printing Tickets

Susie is organizing the printing of tickets for a show. She has collected prices from several printers. The student's task is to use graphs and algebra to advise Susie on how to choose the best printer. - Student Task: Sidewalk Patterns

In this task, students will look for rules which let you work out how many blocks of different colors are needed to make different sized patterns. - Student Task: Skeleton Tower

In this task, students must work out a rule for calculating the total number of cubes needed to build towers of different heights. - Student Task: Table Tiling

In this task, students must work out how many whole, half and quarter tiles tiles are needed to cover the tops of tables of different sizes. - Sum of Functions

The intent of this problem is to have students think about how function addition works on a fundamental level, so formulas have been omitted on purpose. Although this problem does not ask students to "write a function that describes a relationship between two quantities", it can provide students with understandings preparatory for F.BF.1b. - Summer Intern

Students are given the following task and asked to write an expression. "You have been hired for a summer internship at a marine life aquarium. Part of your job is diluting brine for the saltwater fish tanks. The brine is composed of water and sea salt, and the salt concentration is 15.8% by mass, meaning that in any amount of brine the mass of salt is 15.8% of the total mass." - Susita's Account

This task asks students to determine a recursive process from a context. Students who study computer programming will make regular use of recursive processes. - Temperature Conversions

Unit conversion problems provide a rich source of examples both for composition of functions (when several successive conversions are required) and inverses (units can always be converted in either of two directions). - The Canoe Trip, Variation 1

The purpose of this task is to give students practice constructing functions that represent a quantity of interest in a context, and then interpreting features of the function in the light of that context. It can be used as either an assessment or a teaching task. - The Canoe Trip, Variation 2

The primary purpose of this task is to lead students to a numerical and graphical understanding of the behavior of a rational function near a vertical asymptote, in terms of the expression defining the function. The canoe context focuses attention on the variables as numbers, rather than as abstract symbols.

- Applications of Quadratic Functions video

This video introduces and explains the topic. - Function Flyer

The applet on this site allows the students to manipulate the graph of a function by changing the value of exponents, coefficients and constants. - Function Matching

In this student interactive, from Illuminations, students demonstrate their understanding of function expressions by matching a function graph to a generated graph. Choose from several function types or select random and let the computer choose. - Graphit

With this interactive applet students are able to create graphs of functions and sets of ordered pairs on the same coordinate plane. - Inductive Patterns

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Inductive Reasoning

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Inductive Reasoning video

This video introduces and explains the topic. - Multi-Function Data Flyer

The applet in this lesson allows students to plot ordered pairs and then change the values in order to observe the effects of those changes. - Polynomials

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Polynomials video

This video introduces and explains the topic. - Representing Patterns

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Representing Patterns video

This video introduces tables and graphs as representations of patterns. - Student Task: Patchwork

Kate makes patchwork cushions using right triangles made from squares of material. In this task, students must investigate number patterns and to find a rule, or a formula, that will help Kate figure out the number of squares she needs for cushions of different sizes. - Student Task: Sidewalk Stones

In this task, students will look for rules which let them work out how many blocks of different colors are needed to make different sized patterns. - Writing Expressions and Equations video

How to write an equation using what we know to solve a problem we don't know.

Standard F.BF.1

Write a function that describes a relationship between two quantities.^{★}

- Determine an explicit expression, a recursive process, or steps for calculation from a context.
- Combine standard function types using arithmetic operations.
For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.Standard F.BF.2

Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Limit to linear and exponential functions. Connect arithmetic sequences to linear functions and geometric sequences to exponential functions.^{★}

- ALGEBRA - Seeing Structure in Expressions (A.SSE) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Seeing Structure in Expressions (A.SSE) - FUNCTIONS - Building Linear or Exponential Functions (F.BF) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Building Linear or Exponential Functions (F.BF)

- Introduction to the Materials (Math 1)

Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 8: Connecting Algebra & Geometry - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 8, Connecting Algebra and Geometry, students use the Pythagorean Theorem to find the distance between two points and to derive the distance formula. - Module 8: Connecting Algebra & Geometry - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 8 Teacher Notes, Connecting Algebra and Geometry, students use the Pythagorean Theorem to find the distance between two points and to derive the distance formula.

- Building a General Quadratic Function

This task is for instructional purposes only and builds on ''Building an explicit quadratic function.'' First, it is vital that students have worked through ''Building an explicit quadratic function'' before undertaking this task. - Building a quadratic function from f(x)=x2

This is the first of a series of tasks aiming at understanding the quadratic formula in a geometric way in terms of the graph of a quadratic function. Here the student works with an explicit function and studies the impact of scaling and linear change of variables. - Building an Explicit Quadratic Function by Composition

This task is intended for instruction and to motivate the task Building a General Quadratic Function. This task assumes that the students are familiar with the process of completing the square. - Exploring Sinusoidal Functions

This task serves as an introduction to the family of sinusoidal functions. It uses a desmos applet to let students explore the effect of changing the parameters in y=Asin(B(xâh))+k on the graph of the function. - Identifying Even and Odd Functions

This task includes an experimental GeoGebra worksheet, with the intent that instructors might use it to more interactively demonstrate the relevant content material. - Identifying Quadratic Functions (Vertex Form)

This task has students explore the relationship between the three parameters a, h, and k in the equation f(x)=a(xh)2+k and the resulting graph. - Medieval Archer

This task addresses the first part of standard F-BF.3: âIdentify the effect on the graph of replacing f(x) by f(x)+k, kf(x), f(kx), and f(x+k) for specific values of k (both positive and negative).â Here, students are required to understand the effect of replacing x with x+k, but this task can also be modified to test or teach function-building skills involving f(x)+k, kf(x), and f(kx) in a similar manner.

- Function Flyer

The applet on this site allows the students to manipulate the graph of a function by changing the value of exponents, coefficients and constants. - Generalizing Patterns: Table Tiles

This lesson unit is intended to help educators assess how well students are able to identify linear and quadratic relationships in a realistic context: the number of tiles of different types that are needed for a range of square tabletops. - Graphit

With this interactive applet students are able to create graphs of functions and sets of ordered pairs on the same coordinate plane. - Multi-Function Data Flyer

The applet in this lesson allows students to plot ordered pairs and then change the values in order to observe the effects of those changes. - Representing Polynomials

This lesson unit is intended to help educators assess how well students are able to translate between graphs and algebraic representations of polynomials. - Sample Assessment Task: Quadratic transformation

This is a two-part assessment task. Part A asks students to interpret the symbolic representation of a transformation on a function. Part B is about connecting the representations of a function. Use the navigation at the upper right of this page to access the task. - Tidal Waves (pdf)

Students analyze a problem faced by the captain of a shipping vessel. Students may use a range of functions to model the situation and reflect on their usefulness. Because trigonometric functions can be useful, this task would be particularly appropriate for students who have had an introduction to graphing sine and cosine functions. - Transforming the graph of a function

Like "Building functions: concrete case'' this task examines, in a graphical setting, the impact of adding a scalar, multiplying by a scalar, and making a linear substitution of variables on the graph of a function f. The setting here is abstract as there is no formula for the function f. The focus is therefore on understanding the geometric impact of these three operations.

Standard F.BF.3

Identify the effect on the graph of replacingf(x) by f(x) + k, for specific values ofk(both positive and negative); find the value ofkgiven the graphs. Relate the vertical translation of a linear function to itsy-intercept. Experiment with cases and illustrate an explanation of the effects on the graph using technology.

- ALGEBRA - Seeing Structure in Expressions (A.SSE) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Seeing Structure in Expressions (A.SSE) - FUNCTIONS - Linear and Exponential (F.LE) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Linear and Exponential (F.LE).

- Introduction to the Materials (Math 1)

Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 1: Sequences - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 1, Sequences is written as two intertwined learning cycles that begin by alternating from arithmetic sequence to geometric sequences, so students can compare and contrast features as they represent both types of sequences with tables, graphs, story contexts, diagrams, and equations. - Module 1: Sequences - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 1 Teacher Notes, Sequences is written as two intertwined learning cycles that begin by alternating from arithmetic sequence to geometric sequences, so students can compare and contrast features as they represent both types of sequences with tables, graphs, story contexts, diagrams, and equations. - Module 2: Linear & Exponential Functions - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 2, Linear and Exponential Functions, begins with a learning cycle that introduces contexts with continuous domains and defining linear functions as having a constant rate of change and exponential functions as having a constant ratio over equal intervals. - Module 2: Linear & Exponential Functions - Teacher Notes (Math 1)

Mathematics Vision Project, Secondary Math One Module 2 Teacher Notes, Linear and Exponential Functions, begins with a learning cycle that introduces contexts with continuous domains and defining linear functions as having a constant rate of change and exponential functions as having a constant ratio over equal intervals.

- Algae Blooms

The problem statement describes a changing algae population as reported by the Maryland Department of Natural Resources. In part (a), students are expected to build an exponential function modeling algae concentration from the description given of the relationship between concentrations in cells/ml and days of rapid growth (F-LE.2). The intent of part (b) is for students to gain an appreciation for the exponential growth exhibited despite an apparently modest growth rate of 1 cell division per day. - Allergy medication

The purpose of the task is to help students become accustomed to evaluating exponential functions at non-integer inputs and interpreting the values. - Basketball Bounces, Assessment Variation 1

This task asks students to analyze a set of data from a physical context, choose a model that reasonably fits the data, and use the model to answer questions about the physical context. - Basketball Bounces, Assessment Variation 2

This task asks students to analyze a set of data from a physical context, choose a model that reasonably fits the data, and use the model to answer questions about the physical context. - Basketball Rebounds

This task involves a fairly straightforward decaying exponential. Filling out the table and developing the general formula is complicated only by the need to work with a fraction that requires decisions about rounding and precision. - Boiling Water

This task examines linear models for the boiling point of water as a function of elevation. Two sets of data are provided and each is modeled quite well by a linear function. - Boom Town

The purpose of this task is to give students experience working with simple exponential models in situations where they must evaluate and interpret them at non-integer inputs. - Carbon 14 Dating, Variation 2

This exploratory task requires the student to use this property of exponential functions in order to estimate how much Carbon 14 remains in a preserved plant after different amounts of time. - Carbon 14 dating in practice II

This problem introduces the method used by scientists to date certain organic material. It is based not on the amount of the Carbon 14 isotope remaining in the sample but rather on the ratio of Carbon 14 to Carbon 12. This ratio decreases, hypothetically, at a constant exponential rate as soon as the organic material has ceased to absorb Carbon 14, that is, as soon as it dies. - Choosing an appropriate growth model

The goal of this task is to examine some population data from a modeling perspective. Because large urban centers and their growth are governed by many complex factors, we cannot expect a simple model (linear, quadratic, or exponential) to give accurate values or predictions over large stretches of time. Deciding on an appropriate model is a delicate process requiring careful analysis. - Comparing Exponentials

This task gives students an opportunity to work with exponential functions in a real world context involving continuously compounded interest. They will study how the base of the exponential function impacts its growth rate and use logarithms to solve exponential equations. - Comparing Graphs of Functions

The goal of this task is to use appropriate tools to compare graphs of several functions. In addition, students are asked to study the structure of the different expressions to explain why these functions grow as they do. - Decaying Dice

This task provides concrete experience with exponential decay. It is intended for students who know what exponential functions are, but may not have much experience with them, perhaps in an Algebra 1 course. - Dido and the Foundation of Carthage

The goal of this task is to interpret the mathematics behind a famous story from ancient mythology, giving rise to linear and quadratic expressions which model the story. - Do two points always determine a linear function II?

This task is designed as a follow-up to the task F-LE Do Two Points Always Determine a Linear Function? Linear equations and linear functions are closely related, and there advantages and disadvantages to viewing a given problem through each of these points of view. This task is not intended for assessment purposes: rather it is intended to show the depth of the standard F-LE.2 and its relationship to other important concepts of the middle school and high school curriculum, including ratio, algebra, and geometry. - Do two points always determine a linear function?

This problem allows the student to think geometrically about lines and then relate this geometry to linear functions. Or the student can work algebraically with equations in order to find the explicit equation of the line through two points (when that line is not vertical). - Do two points always determine an exponential function?

This task asks students to construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). - Equal Differences over Equal Intervals 1

Students prove that linear functions grow by equal differences over equal intervals. They will prove this for equal intervals of length one unit, and note that in this case the equal differences have the same value as the slope. - Equal Differences over Equal Intervals 2

Linear functions grow by equal differences over equal intervals. In this task students prove the property in general (for equal intervals of any length). - Equal Factors over Equal Intervals

Examples in this task is designed to help students become familiar with this language "successive quotient". Depending on the students's prior exposure to exponential functions and their growth rates, instructors may wish to encourage students to repeat part (b) for a variety of exponential functions and step sizes before proceeding to the most general algebraic setting in part (c). - Exponential Functions

This task requires students to use the fact that the value of an exponential function f(x)=aâ bx increases by a multiplicative factor of b when x increases by one. It intentionally omits specific values for c and d in order to encourage students to use this fact instead of computing the point of intersection, (p,q), and then computing function values to answer the question. This task is preparatory for standard F.LE.1a. - Exponential Parameters

The task provides a reasonably straight-forward introduction to interpreting the parameters of an exponential function in terms of a modeling context. The task has students both generate an exponential expression from a contextual description, and in reverse, interpret parameters in a context from an algebraic expression. - Exponential growth versus linear growth I

The purpose of this task it to have students discover how (and how quickly) an exponentially increasing quantity eventually surpasses a linearly increasing quantity. - Exponential growth versus linear growth II

The purpose of this task it to have students discover how (and how quickly) an exponentially increasing quantity eventually surpasses a linearly increasing quantity. - Exponential growth versus polynomial growth

This problem shows that an exponential function takes larger values than a cubic polynomial function provided the input is sufficiently large. - Extending the Definitions of Exponents, Variation 2

The goal of this task is to develop an understanding of why rational exponents are defined as they are (N-RN.1), however it also raises important issues about distinguishing between linear and exponential behavior (F-LE.1c) and it requires students to create an equation to model a context (A-CED.2) - Finding Linear and Exponential Models

The goal of this task is to present students with real world and mathematical situations which can be modeled with linear, exponential, or other familiar functions. In each case, the scenario is presented and students must decide which model is appropriate. - Finding Parabolas through Two Points

In this task students are asked to find all quadratic functions described by given equations. - Functions and the Vertical Line Test

The vertical line test for functions is the focus of this lesson plan. - Graphs and Functions

This lesson plan is designed to help the student understand how to plot functions on the Cartesian plane and how the graphing of functions leads to lines and parabolas. - Identifying Exponential Functions

The task is an introduction to the graphing of exponential functions. - Identifying Functions

This task emphasizes the expectation that students know linear functions grow by constant differences over equal intervals and exponential functions grow by constant factors over equal intervals. - Illegal Fish

This task asks students to interpret the relevant parameters in terms of the real-world context and describe exponential growth. - In The Billions and Linear Modeling

This problem assumes students have completed several preliminary tasks about the fact that linear functions change by equal differences over equal intervals. - In the Billions and Exponential Modeling

This problem provides an opportunity to experiment with modeling real data. - Interesting Interest Rates

Given two bank interest rate scenarios, students will compare returns, write an expression for a balance, and create a table of values for the balances. - Introduction to Functions

This lesson introduces students to functions and how they are represented as rules and data tables. They also learn about dependent and independent variables. - Linear Functions

This task requires students to use the fact that on the graph of the linear function h(x)=ax+b, the y-coordinate increases by a when x increases by one. Specific values for a and b were left out intentionally to encourage students to use the above fact as opposed to computing the point of intersection, (p,q), and then computing respective function values to answer the question. - Linear or exponential?

This task gives a variet of real-life contexts which could be modeled by a linear or exponential function. The key distinguishing feature between the two is whether the change by equal factors over equal intervals (exponential functions), or by a constant increase per unit interval (linear functions). The task could either be used as an assessment problem on this distinction, or used as an introduction to the differences between these very important classes of functions. - Moore's Law and Computers

The goal of this task is to construct and use an exponential model to approximate hard disk storage capacity on personal computers. - Paper Folding

This is a very open-ended task designed for students to develop some of the basic ideas surrounding exponential growth. - Population and Food Supply

In this task students construct and compare linear and exponential functions and find where the two functions intersect. - Predicting the Past

The purpose of this instructional task is to provide an opportunity for students to use and interpret the meaning of a negative exponent in a functional relationship. - Rising Gas Prices Compounding and Inflation

The purpose of this task is to give students an opportunity to explore various aspects of exponential models (e.g., distinguishing between constant absolute growth and constant relative growth, solving equations using logarithms, applying compound interest formulas) in the context of a real world problem with ties to developing financial literacy skills. - Rumors

This problem is an exponential function example. - Sample Assessment Task: Golf Balls in Water

This sample task requires that students model approximately linear data with a linear function. To do so, they must use contextual information and construct representations of linear relationships as well as interpret parameters of the linear models in the context. Use the navigation at the upper right of this page to access the task. - Sandia Aerial Tram

Students are asked to write an equation for a function (linear, quadratic, or exponential) that models the relationship between the elevation of the tram and the number of minutes into the ride. - Snail Invasion

The purpose of this task is to give students experience modeling a real-world example of exponential growth, in a context that provides a vivid illustration of the power of exponential growth, for example the cost of inaction for a year. - Solving Problems with Linear and Exponential Models

The goal of this task is to provide examples of exponential and linear functions modeling different real world phenomena. Students will create the appropriate model and then use it to solve linear and exponential equations. - Student Task: Table Tiling

In this task, students must work out how many whole, half and quarter tiles tiles are needed to cover the tops of tables of different sizes. - Temperatures in degrees Fahrenheit and Celsius

Temperature conversions provide a rich source of linear functions which are encountered not only in science but also in our every day lives when we travel abroad. The first part of this task provides an opportunity to construct a linear function given two input-output pairs. The second part investigates the inverse of a linear function while the third part requires reasoning about quantities and/or solving a linear equation. - Triangular Numbers

The goal of this task is to work on producing a quadratic equation from an arithmetic context. - Two Points Determine an Exponential Function I

Given the graph of a function students must find the value of 2 variables. - Two Points Determine an Exponential Function II

Given the graph of a function students must find the value of 2 variables. - US Population 1790-1860

This task focuses on the fact that exponential functions are characterized by equal successive quotients over equal intervals. This task can be used alongside F-LE Equal Factors over Equal Intervals. - US Population 1982-1988

This task provides a preliminary investigation of mathematical modeling using linear functions. In particular, students are asked to make predictions using a linear model without ever writing down an equation for a line. As such, the task could be used to motivate or introduce the observation that linear functions are precisely those that change by constant differences over equal intervals. - Valuable Quarter

Successful work on this task involves modeling a bank account balance with an exponential function and then solving an exponential equation arising from the given information. This can be done either by extracting a root or taking a logarithm: either method will require a calculator in order to evaluate the expressions. Students will also need to be familiar with the context of annual interest and of compounding interest. - What functions do two graph points determine?

Given two points on a plane, students will demonstrate an understanding of unique linear function, unique exponential function, and quadratic function.

- Exploring Linear Functions: Representational Relationships

This lesson plan helps students better understand linear functions by allowing them to manipulate values and get a visual representation of the result. - Lesson Starter: Populated Communities

Students will use statistics and probability knowledge, as well as critical thinking skills, to solve problems. - Modeling: Having Kittens

This lesson unit is intended to help you assess how well students are able to interpret a situation and represent the constraints and variables mathematically, select appropriate mathematical methods to use, make sensible estimates and assumptions and investigate an exponentially increasing sequence. - More Complicated Functions: Introduction to Linear Functions

This lesson is designed to reinforce the concept of linear functions and ask students to write functions using English, tables and algebra. - Perpendicular Lines video

This video introduces and explains perpendicular lines. - Sequencer

By using this applet students are able to create sequences by changing the values of starting numbers, multipliers, and add-ons.

Standard F.LE.1

Distinguish between situations that can be modeled with linear functions and with exponential functions.

- Prove that linear functions grow by equal differences over equal intervals; exponential functions grow by equal factors over equal intervals.
- Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
- Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
Standard F.LE.2

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).Standard F.LE.3

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly.

- FUNCTIONS - Linear and Exponential (F.LE) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Linear and Exponential (F.LE).

- Introduction to the Materials (Math 1)

Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 1: Sequences - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 1, Sequences is written as two intertwined learning cycles that begin by alternating from arithmetic sequence to geometric sequences, so students can compare and contrast features as they represent both types of sequences with tables, graphs, story contexts, diagrams, and equations. - Module 1: Sequences - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 1 Teacher Notes, Sequences is written as two intertwined learning cycles that begin by alternating from arithmetic sequence to geometric sequences, so students can compare and contrast features as they represent both types of sequences with tables, graphs, story contexts, diagrams, and equations. - Module 2: Linear & Exponential Functions - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 2, Linear and Exponential Functions, begins with a learning cycle that introduces contexts with continuous domains and defining linear functions as having a constant rate of change and exponential functions as having a constant ratio over equal intervals. - Module 2: Linear & Exponential Functions - Teacher Notes (Math 1)

Mathematics Vision Project, Secondary Math One Module 2 Teacher Notes, Linear and Exponential Functions, begins with a learning cycle that introduces contexts with continuous domains and defining linear functions as having a constant rate of change and exponential functions as having a constant ratio over equal intervals.

- Carbon 14 dating in practice I

In the task ''Carbon 14 Dating'' the amount of Carbon 14 in a preserved plant is studied as time passes after the plant has died. In practice, however, scientists wish to determine when the plant died and, as this task shows, this is not possible with a simple measurement of the amount of Carbon 14 remaining in the preserved plant. - DDT-cay

The purpose of this task is for students to encounter negative exponents in a natural way in the course of learning about exponential functions. - Illegal Fish

This task asks students to interpret the relevant parameters in terms of the real-world context and describe exponential growth. - Mixing Candies

This task assumes students are familiar with mixing problems. This approach brings out different issues than simply asking students to solve a mixing problem, which they can often set up using patterns rather than thinking about the meaning of each part of the equations. - Newton's Law of Cooling

The coffee cooling experiment in this task is a popular example of an exponential model with immediate appeal. The model is realistic and provides a good context for students to practice work with exponential equations. - Profit of a company, assessment variation

The primary purpose of this task is to assess students' knowledge of certain aspects of the mathematics described in the High School domain A-SSE: Seeing Structure in Expressions. - Sample Assessment Task: Golf Balls in Water

This sample task requires that students model approximately linear data with a linear function. To do so, they must use contextual information and construct representations of linear relationships as well as interpret parameters of the linear models in the context. Use the navigation at the upper right of this page to access the task. - Saturating Exponential

The context here is a familiar one: a cold beverage warms once it is taken out of the refrigerator. Rather than giving the explicit function governing this warmth, a graph is presented along with the general form of the function. Students must then interpret the graph in order to understand more specific details regarding the function. - Taxi!

This simple conceptual problem does not require algebraic manipulation, but requires students to articulate the reasoning behind each statement. - US Population 1982-1988

This task provides a preliminary investigation of mathematical modeling using linear functions. In particular, students are asked to make predictions using a linear model without ever writing down an equation for a line. As such, the task could be used to motivate or introduce the observation that linear functions are precisely those that change by constant differences over equal intervals.

- Possible or Not

Students can look at graphed functions from real-life examples and determine whether the graph makes sense or not in this activity. - Stairway - Student Task

This task students to design a stairway for a custom home. They will need to gather information regarding design, safety, and the utility of staircases.

Standard F.LE.5

Interpret the parameters in a linear or exponential function in terms of a context. Limit exponential functions to those of the formf(x) = b.^{x}+ k

- GEOMETRY - Congruence (G.CO) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Congruence (G.CO).

- Introduction to the Materials (Math 1)

Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 6: Transformations & Symmetry - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 6, Transformations and Symmetry, builds on students experiences with rigid motion in earlier grades to formalize the definitions of translation, rotation, and reflection. - Module 6: Transformations & Symmetry - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 6 Teacher Notes, Transformations and Symmetry, builds on students experiences with rigid motion in earlier grades to formalize the definitions of translation, rotation, and reflection. - Module 7: Congruence, Construction & Proof - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 7, Congruence, Construction, and Proof, begins by developing constructions as another tool to be used to reason about figures and to justify properties of shapes. Individual constructions are not taught for the sake of memorizing a series of steps, but rather to reason using known properties of shapes such as circles. - Module 7: Congruence, Construction & Proof - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 7 Teacher Notes, Congruence, Construction, and Proof, begins by developing constructions as another tool to be used to reason about figures and to justify properties of shapes. Individual constructions are not taught for the sake of memorizing a series of steps, but rather to reason using known properties of shapes such as circles.

- 3D Transmographer

This lesson contains an applet that allows students to explore translations, reflections, and rotations. - Constructions

This site provides both a video and step-by-step directions on how to complete a variety of constructions. - Defining Parallel Lines

The goal of this task is to critically analyze several possible definitions for parallel lines. - Defining Perpendicular Lines

The purpose of this task is to critically examine some different possible definitions of what it means for two lines to be perpendicular. - Defining Reflections

The goal of this task is to compare and contrast the visual intuition we have of reflections with their technical mathematical definition. - Defining Rotations

The goal of this task is to encourage students to be precise in their use of language when making mathematical definitions. - Dilations and Distances

The goal of this task is to study the impact of dilations on distances between points. - Fixed points of rigid motions

The purpose of this task is to use fixed points at a tool for studying and classifying rigid motions of the plane. - Horizontal Stretch of the Plane

The goal of this task is to compare a transformation of the plane (translation) which preserves distances and angles to a transformation of the plane (horizontal stretch) which does not preserve either distances or angles. - Identifying Rotations

The purpose of this task is to use the definition of rotations in order to find the center and angle of rotation given a triangle and its image under a rotation. - Identifying Translations

The purpose of this task is to study the impact of translations on triangles. - Origami regular octagon

The goal of this task is to study the geometry of reflections in the context of paper folding. - Reflected Triangles

This task is a reasonably straight-forward application of rigid motion geometry, with emphasis on ruler and straightedge constructions, and would be suitable for assessment purposes. - Seven Circles II

This task is intended primarily for instructional purposes. It provides a concrete geometric setting in which to study rigid transformations of the plane - Showing a triangle congruence: a particular case

This task provides experience working with transformations of the plane and also an abstract component analyzing the effects of the different transformations. - Showing a triangle congruence: the general case

The purpose of this task is to work with transformations to exhibit triangle congruences in a general setting. - Symmetries of a circle

This task asks students to examine lines of symmetry using the high school definition of reflections. - Symmetries of a quadrilateral I

This task provides an opportunity to examine the taxonomy of quadrilaterals from the point of view of rigid motions. - Symmetries of a quadrilateral II

This task examines quadrilaterals from the point of view of rigid motions and complements. - Symmetries of rectangles

This task examines the rigid motions which map a rectangle onto itself. - Taking a Spin (pdf)

Although students are often asked to find the angles of rotational symmetry for given regular polygons, in this task they are asked to find the regular polygons for a given angle of rotational symmetry, a reversal that yields some surprising results. This task would be most appropriate with students who have at least some experience in exploring rotational symmetry. - Tangent Lines and the Radius of a Circle

This task presents a foundational result in geometry, presented with deliberately sparse guidance in order to allow a wide variety of approaches. - Trigonometric Identities and Rigid Motions

The purpose of this task is to apply translations and reflections to the graphs of the equations f(x)=cosx and g(x)=sinx in order to derive some trigonometric identities. - Unit Squares and Triangles

This problem provides an opportunity for a rich application of coordinate geometry.

- Geometry in Tessellations

In this lesson students will learn about lines, angles, planes, and experiment with the area and perimeter of polygons. - Identifying Unknown Transformations

This applet allows the student to drag a shape and then observe the changes to its behavior. They then determine whether the alteration is due to reflection, a rotation, or a translation/slide transformation. - Tessellations: Geometry and Symmetry

Students can explore polygons, symmetry, and the geometric properties of tessellations in this lesson. - Translations, Reflections, and Rotations

Students are introduced to the concepts of translation, reflection and rotation in this lesson plan. - Visual Patterns in Tessellations

In this lesson students will learn about types of polygons and tessellation patterns around us.

Standard G.CO.1

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.Standard G.CO.2

Represent transformations in the plane using, for example, transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).Standard G.CO.3

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.Standard G.CO.4

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.Standard G.CO.5

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, for example, graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Point out the basis of rigid motions in geometric concepts, for example, translations move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle.

- GEOMETRY - Congruence (G.CO) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Congruence (G.CO).

- Introduction to the Materials (Math 1)

Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 6: Transformations & Symmetry - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 6, Transformations and Symmetry, builds on students experiences with rigid motion in earlier grades to formalize the definitions of translation, rotation, and reflection. - Module 6: Transformations & Symmetry - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 6 Teacher Notes, Transformations and Symmetry, builds on students experiences with rigid motion in earlier grades to formalize the definitions of translation, rotation, and reflection. - Module 7: Congruence, Construction & Proof - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 7, Congruence, Construction, and Proof, begins by developing constructions as another tool to be used to reason about figures and to justify properties of shapes. Individual constructions are not taught for the sake of memorizing a series of steps, but rather to reason using known properties of shapes such as circles. - Module 7: Congruence, Construction & Proof - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 7 Teacher Notes, Congruence, Construction, and Proof, begins by developing constructions as another tool to be used to reason about figures and to justify properties of shapes. Individual constructions are not taught for the sake of memorizing a series of steps, but rather to reason using known properties of shapes such as circles.

- 3D Transmographer

This lesson contains an applet that allows students to explore translations, reflections, and rotations. - Are the Triangles Congruent?

The purpose of this task is primarily assessment-oriented, asking students to demonstrate knowledge of how to determine the congruency of triangles. - Building a tile pattern by reflecting hexagons

This task applies reflections to a regular hexagon to construct a pattern of six hexagons enclosing a seventh: the focus of the task is on using the properties of reflections to deduce this seven hexagon pattern. - Building a tile pattern by reflecting octagons

This task applies reflections to a regular octagon to construct a pattern of four octagons enclosing a quadrilateral: the focus of the task is on using the properties of reflections to deduce that the quadrilateral is actually a square. - Congruence Criterion

The goal of this task is to establish the SSS congruence criterion using rigid motions. - Properties of Congruent Triangles

The goal of this task is to understand how congruence of triangles, defined in terms of rigid motions, relates to the corresponding sides and angles of these triangles. - Reflections and Equilateral Triangles

This activity is one in a series of tasks using rigid transformations of the plane to explore symmetries of classes of triangles, with this task in particular focusing on the class of equilaterial triangles. - Reflections and Equilateral Triangles II

This task examines some of the properties of reflections of the plane which preserve an equilateral triangle: these were introduced in ''Reflections and Isosceles Triangles'' and ''Reflection and Equilateral Triangles I''. - Reflections and Isosceles Triangles

This activity is one in a series of tasks using rigid transformations of the plane to explore symmetries of classes of triangles, with this task in particular focussing on the class of isosceles triangles. - When Does SSA Work to Determine Triangle Congruence?

The triangle congruence criteria, SSS, SAS, ASA, all require three pieces of information. It is interesting, however, that not all three pieces of information about sides and angles are sufficient to determine a triangle up to congruence. In this problem, we considered SSA. Also insufficient is AAA, which determines a triangle up to similarity. Unlike SSA, AAS is sufficient because two pairs of congruent angles force the third pair of angles to also be congruent. - Why Does ASA Work?

The two triangles in this problem share a side so that only one rigid transformation is required to exhibit the congruence between them. In general more transformations are required and the "Why does SSS work?'' and "Why does SAS work?'' problems show how this works. - Why does SAS work?

For these particular triangles, three reflections were necessary to express how to move from ABC to DEF. Sometimes, however, one reflection or two reflections will suffice. Since any rigid motion will take triangle ABC to a congruent triangle DEF, this shows the remarkable fact that any rigid motion of the plane can be expressed as one reflection, a composition of two reflections, or a composition of three reflections. - Why does SSS work?

This particular sequence of transformations which exhibits a congruency between triangles ABC and DEF used one translation, one rotation, and one reflection.

- Geometry in Tessellations

In this lesson students will learn about lines, angles, planes, and experiment with the area and perimeter of polygons. - Tessellations: Geometry and Symmetry

Students can explore polygons, symmetry, and the geometric properties of tessellations in this lesson. - Translations, Reflections, and Rotations

Students are introduced to the concepts of translation, reflection and rotation in this lesson plan. - Visual Patterns in Tessellations

In this lesson students will learn about types of polygons and tessellation patterns around us.

Standard G.CO.6

Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.Standard G.CO.7

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.Standard G.CO.8

Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

- GEOMETRY - Congruence (G.CO) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Congruence (G.CO).

- Introduction to the Materials (Math 1)

Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 7: Congruence, Construction & Proof - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 7, Congruence, Construction, and Proof, begins by developing constructions as another tool to be used to reason about figures and to justify properties of shapes. Individual constructions are not taught for the sake of memorizing a series of steps, but rather to reason using known properties of shapes such as circles. - Module 7: Congruence, Construction & Proof - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 7 Teacher Notes, Congruence, Construction, and Proof, begins by developing constructions as another tool to be used to reason about figures and to justify properties of shapes. Individual constructions are not taught for the sake of memorizing a series of steps, but rather to reason using known properties of shapes such as circles.

- Angle bisection and midpoints of line segments

This task provides a construction of the angle bisector of an angle by reducing it to the bisection of an angle to finding the midpoint of a line segment. - Bisecting an angle

This task provides the most famous construction to bisect a given angle. - Construction of perpendicular bisector

The construction of the perpendicular bisector of a line segment is one of the most common in plane geometry and it is undertaken here. - Geometry Construction Reference

Thirteen straightedge and compass constructions are described and illustrated. The original version, in Word format, can be downloaded and distributed. - Inscribing a hexagon in a circle

This task is primarily for instructive purposes but can be used for assessment as well. Parts (a) and (b) are good applications of geometric constructions using a compass and could be used for assessment purposes but the process is a bit long since there are six triangles which need to be constructed. - Inscribing a square in a circle

This task provides an opportunity for students to apply triangle congruence theorems in an explicit, interesting context. - Inscribing an equilateral triangle in a circle

This task implements many important ideas from geometry including trigonometric ratios, important facts about triangles, and reflections. As a result, it is recommended that this task be undertaken relatively late in the geometry curriculum. - Locating Warehouse

This task can be implemented in a variety of ways. For a class with previous exposure to the incenter or angle bisectors, part (a) could be a quick exercise in geometric constructions,. Alternatively, this could be part of a full introduction to angle bisectors, culminating in a full proof that the three angle bisectors are concurrent, an essentially complete proof of which is found in the solution below. - Origami equilateral triangle

The purpose of this task is to explore reflections in the context of paper folding. - Origami regular octagon

The goal of this task is to study the geometry of reflections in the context of paper folding. - Placing a Fire Hydrant

This task can be implemented in a variety of ways. For a class with previous exposure to properties of perpendicular bisectors, part (a) could be a quick exercise in geometric constructions, and an application of the result. Alternatively, this could be part of an introduction to perpendicular bisectors, culminating in a full proof that the three perpendicular bisectors are concurrent at the circumcenter of the triangle.

- Evaluating Conditions for Congruency

This lesson unit is intended to help educators assess how well students are able to work with concepts of congruency and similarity, including identifying corresponding sides and corresponding angles within and between triangles. They will also identify and understand the significance of a counter-example, and prove and evaluate proofs in a geometric context. - Inscribing and Circumscribing Right Triangles

This lesson unit is intended to help educators assess how well students are able to use geometric properties to solve problems. - Introduction to Constructions

Introduction to Euclidean Construction - tools and rules. - Patterns in Fractals

In this lesson students will be introduced to patterns, the terminology used in patterns, and practice finding patterns in the observable process of fractal generation.

Standard G.CO.12

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Emphasize the ability to formalize and defend how these constructions result in the desired objects.For example, copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.Standard G.CO.13

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Emphasize the ability to formalize and defend how these constructions result in the desired objects.

- GEOMETRY - Expressing Geometric Properties With Equations (G.GPE) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Expressing Geometric Properties with Equations (G.GPE).

- Introduction to the Materials (Math 1)

Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 6: Transformations & Symmetry - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 6, Transformations and Symmetry, builds on students experiences with rigid motion in earlier grades to formalize the definitions of translation, rotation, and reflection. - Module 6: Transformations & Symmetry - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 6 Teacher Notes, Transformations and Symmetry, builds on students experiences with rigid motion in earlier grades to formalize the definitions of translation, rotation, and reflection. - Module 8: Connecting Algebra & Geometry - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 8, Connecting Algebra and Geometry, students use the Pythagorean Theorem to find the distance between two points and to derive the distance formula. - Module 8: Connecting Algebra & Geometry - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 8 Teacher Notes, Connecting Algebra and Geometry, students use the Pythagorean Theorem to find the distance between two points and to derive the distance formula.

- A Midpoint Miracle

This classroom task gives students the opportunity to prove a surprising fact about quadrilaterals: that if we join the midpoints of an arbitrary quadrilateral to form a new quadrilateral, then the new quadrilateral is a parallelogram, even if the original quadrilateral was not. - Equal Area Triangles on the Same Base I

This task is an relatively easy application of the formula for the area of a triangle and use of parallel lines. - Equal Area Triangles on the Same Base II

This task has students apply their knowledge of parallel lines to solve a geometric problem on areas of triangles. - Slope Criterion for Perpendicular Lines

The goal of this task is to use similar triangles to establish the slope criterion for perpendicular lines. - Squares on a coordinate grid

The purpose of this task is to use the Pythagorean Theorem and knowledge about quadrilaterals in order to construct squares of different sizes on a coordinate grid. - Triangle Perimeters

The purpose of this task is to apply the Pythagorean theorem to calculate distances and areas. - Triangles inscribed in a circle

The goal of this task is to use ideas about linear functions in order to determine when certain angles are right angles. - Unit Squares and Triangles

This problem provides an opportunity for a rich application of coordinate geometry. - When are two lines perpendicular?

The goal of this task is to examine when two lines in the plane are perpendicular in terms of their slopes.

- GeoGebra

GeoGebra is dynamic online geometry software. Constructions can be made with points, vectors, segments, lines, polygons, conic sections, inequalities, implicit polynomials and functions. All of them can be changed dynamically afterwards. - Parallel Lines

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Parallel Lines video

This video explains the concept. - Perpendicular Lines

This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. - Perpendicular Lines video

This video introduces and explains perpendicular lines.

Standard G.GPE.4

Use coordinates to prove simple geometric theorems algebraically.For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).Standard G.GPE.5

Prove the slope criteria for parallel and perpendicular lines; use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).Standard G.GPE.7

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles; connect with The Pythagorean Theorem and the distance formula.^{★}

- STATISTICS - Interpreting Categorical and Quantitative Data (S.ID) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Interpreting Categorical and Quantitative Data (S.ID).

- Introduction to the Materials (Math 1)

Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 9: Modeling Data - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 9, Modeling Data, addresses representing data in dot plots, histograms, and box plots, and analyzing the data with appropriate summary statistics for center, shape, and spread and identifying the existence of extreme data points. They compare data sets to draw conclusions and justify arguments based upon story context. This work extends the experience that students had in grades 6-8 where they informally described both center and spread. - Module 9: Modeling Data - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 9 Teacher Notes, Modeling Data, addresses representing data in dot plots, histograms, and box plots, and analyzing the data with appropriate summary statistics for center, shape, and spread and identifying the existence of extreme data points. They compare data sets to draw conclusions and justify arguments based upon story context. This work extends the experience that students had in grades 6-8 where they informally described both center and spread.

- Box Plotter

This student interactive, from Illuminations, allows students to create a customized box plot as well as display pre-set box plots. - DASL

DASL (pronounced "dazzle") is an online library of datafiles and stories that illustrate the use of basic statistics methods. - Describing Data Sets with Outliers

The goal of this task is to look at the impact of outliers on two important statistical measures of center: the mean and the median. - Haircut Costs

This problem could be used as an introductory lesson to introduce group comparisons and to engage students in a question they may find amusing and interesting. More generally, the idea of the lesson could be used as a template for a project where students develop a questionnaire, sample students at their school and report on their findings. - Speed Trap

The purpose of this task is to allow students to demonstrate an ability to construct boxplots and to use boxplots as the basis for comparing distributions. - Understanding the Standard Deviation

The purpose of this task is to deepen student understanding of the standard deviation as a measure of variability in a data distribution. - Univariate and Bivariate Data

This lesson helps students understand these two types of data and choose the best type of graph or measure appropriate to each.

- Histogram Tool

This tool can be used to create a histogram for analyzing the distribution of a data set using data that you enter or using pre-loaded data that you select. - Mean and Median

In this applet from Illuminations, students investigate the mean, median, and box-and-whisker plot for a set of data that they create. The data set may contain up to 15 integers, each with a value from 0 to 100. - Measuring Variability Through Tracking Wildfires

Data about wildfires in the U.S. is the basis of this lesson. Students examine the data about active fires as well as historical data and use that data to find changes across decades. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Normal Distribution

Students will better understand normal distribution by changing the standard deviation in the applet of this lesson plan. - Representing Data 1: Using Frequency Graphs

This lesson unit is intended to help educators assess how well students are able to use frequency graphs to identify a range of measures and make sense of this data in a real-world context. - Representing Data 2: Using Box Plots

This lesson unit is intended to help educators assess how well students are able to interpret data using frequency graphs and box plots.

Standard S.ID.1

Represent data with plots on the real number line (dot plots, histograms, and box plots).Standard S.ID.2

Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.Standard S.ID.3

Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Calculate the weighted average of a distribution and interpret it as a measure of center.

- STATISTICS - Interpreting Categorical and Quantitative Data (S.ID) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Interpreting Categorical and Quantitative Data (S.ID).

- Introduction to the Materials (Math 1)

Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 9: Modeling Data - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 9, Modeling Data, addresses representing data in dot plots, histograms, and box plots, and analyzing the data with appropriate summary statistics for center, shape, and spread and identifying the existence of extreme data points. They compare data sets to draw conclusions and justify arguments based upon story context. This work extends the experience that students had in grades 6-8 where they informally described both center and spread. - Module 9: Modeling Data - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 9 Teacher Notes, Modeling Data, addresses representing data in dot plots, histograms, and box plots, and analyzing the data with appropriate summary statistics for center, shape, and spread and identifying the existence of extreme data points. They compare data sets to draw conclusions and justify arguments based upon story context. This work extends the experience that students had in grades 6-8 where they informally described both center and spread.

- Coffee and Crime

This task addresses many standards regarding the description and analysis of bivariate quantitative data, including regression and correlation. - DASL

DASL (pronounced "dazzle") is an online library of datafiles and stories that illustrate the use of basic statistics methods. - Musical Preferences

The basic idea is for students to demonstrate that they know what it means for two variables to be associated. - Olympic Men's 100-meter dash

The task asks students to identify when two quantitative variables show evidence of a linear association, and to describe the strength and direction of that association. - Restaurant Bill and Party Size

The purpose of this task is to assess student understanding of residuals and residual plots. - Support for a Longer School Day?

The purpose of this task is to provide students with an opportunity to calculate joint, marginal and relative frequencies using data in a two-way table. - Univariate and Bivariate Data

This lesson helps students understand these two types of data and choose the best type of graph or measure appropriate to each.

- Devising a Measure for Correlation

This lesson unit is intended to help educators assess how well students understand the notion of correlation. - Interpreting Statistics: A Case of Muddying the Waters

This lesson unit is intended to help educators assess how well students are able to interpret data and evaluate statistical summaries and critique someone else's interpretations of data and evaluations of statistical summaries. - Linear Regression and Correlation

In this lesson students will plot data onto a scatter plot and then determine the line of best fit for the data sets.

Standard S.ID.6

Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

- Fit a linear function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions, or choose a function suggested by the context. Emphasize linear and exponential models.
- Informally assess the fit of a function by plotting and analyzing residuals. Focus on situations for which linear models are appropriate.
- Fit a linear function for scatter plots that suggest a linear association.

- ALGEBRA - Seeing Structure in Expressions (A.SSE) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Seeing Structure in Expressions (A.SSE) - STATISTICS - Interpreting Categorical and Quantitative Data (S.ID) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Interpreting Categorical and Quantitative Data (S.ID).

- Introduction to the Materials (Math 1)

Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 9: Modeling Data - Student Edition (Math 1)

The Mathematics Vision Project, Secondary Math One Module 9, Modeling Data, addresses representing data in dot plots, histograms, and box plots, and analyzing the data with appropriate summary statistics for center, shape, and spread and identifying the existence of extreme data points. They compare data sets to draw conclusions and justify arguments based upon story context. This work extends the experience that students had in grades 6-8 where they informally described both center and spread. - Module 9: Modeling Data - Teacher Notes (Math 1)

The Mathematics Vision Project, Secondary Math One Module 9 Teacher Notes, Modeling Data, addresses representing data in dot plots, histograms, and box plots, and analyzing the data with appropriate summary statistics for center, shape, and spread and identifying the existence of extreme data points. They compare data sets to draw conclusions and justify arguments based upon story context. This work extends the experience that students had in grades 6-8 where they informally described both center and spread.

- Coffee and Crime

This task addresses many standards regarding the description and analysis of bivariate quantitative data, including regression and correlation. - DASL

DASL (pronounced "dazzle") is an online library of datafiles and stories that illustrate the use of basic statistics methods. - Golf and Divorce

This is a simple task addressing the distinction between correlation and causation. Students are given information indicating a correlation between two variables, and are asked to reason out whether or not a causation can be inferred. - High blood pressure

The purpose of this task is to assess understanding of how study design dictates whether a conclusion of causation is warranted. - Olympic Men's 100-meter dash

The task asks students to identify when two quantitative variables show evidence of a linear association, and to describe the strength and direction of that association. - Texting and Grades II

The purpose of this task is to assess ability to interpret the slope and intercept of the line of best fit in context. - Univariate and Bivariate Data

This lesson helps students understand these two types of data and choose the best type of graph or measure appropriate to each. - Used Subaru Foresters II

This problem could be used as a lesson or an assessment.

- Interpreting Statistics: A Case of Muddying the Waters

This lesson unit is intended to help educators assess how well students are able to interpret data and evaluate statistical summaries and critique someone else's interpretations of data and evaluations of statistical summaries. - Linear Regression and Correlation

In this lesson students will plot data onto a scatter plot and then determine the line of best fit for the data sets.

Standard S.ID.7

Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.Standard S.ID.8

Compute (using technology) and interpret the correlation coefficient of a linear fit.Standard S.ID.9

Distinguish between correlation and causation.

- HONORS - NUMBER AND QUANTITY: Vector and Matrix Quantities (N.VM) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I Honors - Vector and Matrix Quantities (N.VM)

HONORS - Standard N.VM.1

Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g.,v, |v|, ||v||,v).HONORS - Standard N.VM.2

Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.HONORS - Standard N.VM.3

Solve problems involving velocity and other quantities that can be represented by vectors.HONORS - Standard N.VM.4

Add and subtract vectors.

- Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
- Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
- Understand vector subtraction
v - wasv + (-w), where-wis the additive inverse ofw, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.HONORS - Standard N.VM.5

Multiply a vector by a scalar.

- Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as
c(vx , vy ) = (cvx , cry).- Compute the magnitude of a scalar multiple
cvusing ||cv|| = |c|v. Compute the direction ofcvknowing that when |c|v≠ 0, the direction ofcvis either alongv(forc>0) or againstvs(forc<0).

- HONORS - NUMBER AND QUANTITY: Vector and Matrix Quantities (N.VM) - Sec Math I Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I Honors - Vector and Matrix Quantities (N.VM)

HONORS - Standard N.VM.6

Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.HONORS - Standard N.VM.7

Multiply matrices by scalars to produce new matrices, e.g., as when all of the pay-offs in a game are doubled.HONORS - Standard N.VM.8

Add, subtract, and multiply matrices of appropriate dimensions.HONORS - Standard N.VM.9

Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.HONORS - Standard N.VM.10

Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.HONORS - Standard N.VM.11

Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.HONORS - Standard N.VM.12

Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.HONORS - Standard N.VM.13

Solve systems of linear equations up to three variables using matrix row reduction.

The Online Core Resource pages are a collaborative project between the Utah State Board of Education and the Utah Education Network. If you would like to recommend a high quality resource, contact Trish French (Elementary) or Lindsey Henderson (Secondary). If you find inaccuracies or broken links contact resources@uen.org.