Math - Secondary Math I

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Reason quantitatively and use units to solve problems. Working with quantities and the relationships between them provides grounding for work with expressions, equations, and functions (Standards N.Q.1–3).

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.

Interpret the structure of expressions (Standard A.SSE.1).

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.

Create equations that describe numbers or relationships. Limit these to linear equations and inequalities, and exponential equations. In the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs (Standards A.CED.1–4).

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.

Understand solving equations as a process of reasoning and explain the reasoning (Standard A.REI.1)

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)

Solve equations and inequalities in one variable (Standard A.REI.3).

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.

Solve systems of equations. Build on student experiences graphing and solving systems of linear equations from middle school. Include cases where the two equations describe the same line - yielding infinitely many solutions - and cases where two equations describe parallel lines - yielding no solution; connect to GPE.5, which requires students to prove the slope criteria for parallel lines (Standards A.REI.5-6)

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.

Represent and solve equations and inequalities graphically (Standards A.REI.10-12).

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.

Understand the concept of a linear or exponential function and use function notation. Recognize arithmetic and geometric sequences as examples of linear and exponential functions (Standards F.IF.1–3).

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.

nterpret linear or exponential functions that arise in applications in terms of a context (Standards F.IF.4–6).

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.

Analyze linear or exponential functions using different representations (Standards F.IF.7, 9).

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.

Build a linear or exponential function that models a relationship between two quantities (Standards F.BF.1–2).

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.

Build new functions from existing functions (Standard F.BF.3).

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.

Construct and compare linear and exponential models and solve problems (Standards F.LE.1– 3).

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)=aâbx 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.

Interpret expressions for functions in terms of the situation they model. (Standard F.LE.5).

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.

Experiment with transformations in the plane. Build on student experience with rigid motions from earlier grades (Standards G.CO.1–5).

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.

Understand congruence in terms of rigid motions. Rigid motions are at the foundation of the definition of congruence. Reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems (Standards G.CO.6–8).

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.

Make geometric constructions (Standards G.CO.12–13).

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.

Use coordinates to prove simple geometric theorems algebraically (Standards G.GPE.4–5, 7).

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.

Summarize, represent, and interpret data on a single count or measurement variable (Standards S.ID.1–3).

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.

Summarize, represent, and interpret data on two categorical and quantitative variables (Standard S.ID.6).

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.

Interpret linear models building on students’ work with linear relationships, and introduce the correlation coefficient (Standards S.ID.7–9).

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.

Represent and model with vector quantities (Standards N.VM.1–3). Perform operations on vectors (Standards N.VM.4–5)

Perform operations on matrices and use matrices in applications (Standards N.VM.6–13).

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Math - Secondary Math I