Resources by Strand
Strand: MATHEMATICAL PRACTICES (MP)
The Standards for Mathematical Practice in Secondary Mathematics III describe mathematical habits of mind that teachers should seek to develop in their students. Students become mathematically proficient in engaging with mathematical content and concepts as they learn, experience, and apply these skills and attitudes
(Standards MP.1-8).
Standard SIII.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 SIII.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 SIII.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 SIII.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 SIII.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 SIII.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 SIII.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 SIII.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.
Strand: NUMBER AND QUANTITY - The Complex Number System (N.CN)
Use complex numbers in polynomial identities and equations. Build on work with quadratic equations in Secondary Mathematics II (Standards N.CN.8–9).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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: Polynomial Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function. - Module 3: Polynomial Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function.
Standard N.CN.8
Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i).
Standard N.CN.9
Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Limit to polynomials with real coefficients.
Strand: ALGEBRA - Seeing Structures in Expressions (A.SSE)
Interpret the structure of expressions. Extend to polynomial and rational expressions (Standards A.SSE.1–2)
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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: Polynomial Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function. - Module 3: Polynomial Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function.
Instructional Tasks
Review Activitiy
Standard A.SSE.1
Interpret polynomial and rational expressions that represent a quantity in terms of its context.★
- Interpret parts of an expression, such as terms, factors, and coefficients.
- Interpret complex expressions by viewing one or more of their parts as a single entity. For example, examine the behavior of P(1+r/n)nt as n becomes large.
Standard A.SSE.2
Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).
Write expressions in equivalent forms to solve problems (Standard A.SSE.4).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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.
Instructional Tasks
- A Lifetime of Savings
The purpose of this instructional task is to give students an opportunity to construct and find the value of a geometric series (A-SSE.4) in a financial literacy context. The task assumes that students have already developed the formula for a geometric series themselves; having them recognize the need for this formula (and look up if necessary) allows them to engage in MP 5, Use appropriate tools strategically. The task also provides students with an opportunity to look for and express regularity in repeated reasoning (MP 8), as the solution shows. This task also asks students to interpret the variables in the future value formula in the context of the problem (A-SSE.1). - Cantor Set
The purpose of this task is to use finite geometric series to investigate an amazing mathematical object that might inspire students' curiosity. The Cantor Set is an example of a fractal. - Course of Antibiotics
This task presents a real world application of finite geometric series. The context can lead into several interesting follow-up questions and projects. Many drugs only become effective after the amount in the body builds up to a certain level. This can be modeled very well with geometric series. - Triangle Series
The purpose of this task is to emphasize the adjective "geometric" in the "geometric" series, namely, that the algebraic notion of a common ratio between terms corresponds to the geometric notion of a repeated similarity transformation. - YouTube Explosion
The purpose of this task is to have students derive the formula for the sum of a specific finite geometric series. In determining the total number of views the YouTube video has, students will first come across the sum of the terms of a geometric sequence. Without a formula, students will have to calculate this sum by adding each term individually. By the end of the task, the students will have come up with a formula that will help them find the sum much quicker than by rote calculation.
Standard A.SSE.4
Understand the formula for the sum of a series and use the formula to solve problems.
- Derive the formula for the sum of an arithmetic series.
- Derive the formula for the sum of a geometric series, and use the formula to solve problems. Extend to infinite geometric series. For example, calculate mortgage payments.★
Strand: ALGEBRA - Arithmetic With Polynomials and Rational Expressions (A.APR)
Perform arithmetic operations on polynomials, extending beyond the quadratic polynomials (Standard A.APR.1).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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: Polynomial Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function. - Module 3: Polynomial Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function.
Instructional Tasks
- Non-Negative Polynomials
The task helps foster student understanding of the analogy described in the standard -- "Understand that polynomials form a system analogous to the integers..." -- in addition to having the same arithmetic operations available, there are many other instances in which integers and polynomials share common properties. - Powers of 11
This task might be used as either practice with polynomial arithmetic or as an introduction to the binomial theorem, providing a process for raising binomials to powers without dredging through many repetitive applications of the distributive law.
Standard A.APR.1
Understand that all polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Understand the relationship between zeros and factors of polynomials (Standards A.APR.2–3)
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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: Polynomial Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function. - Module 3: Polynomial Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function.
Instructional Tasks
- Graphing from Factors I
The purpose of this task is to help students understand the relationship between the factors of a polynomial and the x-intercepts of the graph of the polynomial. By giving students two different polynomials with the same factors the task draws attention to the fact that both polynomials cross the x-axis at the same points. Students are then invited to reflect on why this is so by looking at the structure of the polynomials. - Graphing from Factors II
The purpose of this task is to give students an opportunity to see and use the structure of the factored form of a polynomial (MP7). - Graphing from Factors III
The task has students use the remainder theorem to deduce a linear factor of a cubic polynomial, and then to completely factor the polynomial. Students will need some procedure (e.g., synthetic or long division, or guess-and-check the coefficients) for determining the quadratic factor. Having the factored form permits students to deduce much about the structure of the graph. - Solving a Simple Cubic Equation
The purpose of this task is twofold. First, it prompts students to notice and explain a connection between the factored form of a polynomial and the location of its zeroes when graphed. Second, it highlights a complication that results from a seemingly innocent move that students might be tempted to make: "dividing both sides by x." - The Missing Coefficient
The purpose of this task is to emphasize the use of the Remainder Theorem (a discussion of which should obviously be considered as a prerequisite for the task) as a method for determining structure in polynomial in equations, and in this particular instance, as a replacement for division of polynomials. - Zeroes and factorization of a general polynomial
This task builds on ''Zeroes and factorization of a quadratic function'' parts I and II. The teacher may wish to recall the result from the first of these tasks, generalized to the polynomials of degree d considered here. - Zeroes and factorization of a non polynomial function
The level of the task is appropriate for assessment but since its intention is to provide extra depth to the standard A-APR.2 it is principally designed for instructional purposes only. - Zeroes and factorization of a quadratic polynomial I
For a polynomial function p, a real number r is a root of p if and only if p(x) is evenly divisible by xr. This fact leads to one of the important properties of polynomial functions: a polynomial of degree d can have at most d roots. This is the first of a sequence of problems aiming at showing this fact. - Zeroes and factorization of a quadratic polynomial II
This task is intended for instructional purposes to help students see more clearly the link between factorization of polynomials and zeroes of polynomial functions.
- 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.
Standard A.APR.2
Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
Standard A.APR.3
Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
Use polynomial identities to solve problems (Standards A.APR.4–5)
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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: Polynomial Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function. - Module 3: Polynomial Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function.
Instructional Tasks
- Powers of 11
This task might be used as either practice with polynomial arithmetic or as an introduction to the binomial theorem, providing a process for raising binomials to powers without dredging through many repetitive applications of the distributive law. - Trina's Triangles
This task is a fleshing-out of the example suggested in A-APR.4 of the Common Core document, using the polynomial identity (x2+y2)2=(x2y2)2+(2xy)2 to generate Pythagorean triples.
Standard A.APR.4
Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.
Standard A.APR.5
Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers. For example, with coefficients determined by Pascal’s Triangle.
Rewrite rational expressions (Standards A.APR.6–7).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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: Rational Expressions and Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 4, Rational Expressions and Functions, students work with the fractions that are ratios of polynomials, rational expressions and functions. - Module 4: Rational Expressions and Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 4, Rational Expressions and Functions, students work with the fractions that are ratios of polynomials, rational expressions and functions.
Instructional Tasks
- Combined Fuel Efficiency
The primary purpose of this problem is to rewrite simple rational expressions in different forms to exhibit different aspects of the expression, in the context of a relevant real-world context (the fuel efficiency of of a car). - Egyptian Fractions II
The purpose of this task is for students rewrite a simple rational expression and study the arithmetic of these expressions. Egyptian fractions provide an interesting context, both historically and mathematically, for students to use rational expressions.
- Adding and Subtracting Rational Expressions
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. - Adding and Subtracting Rational Expressions video
This video introduces and explains the topic. - Interpreting Algebraic Expressions
This lesson unit is intended to help educators assess how well students are able to translate between words, symbols, tables, and area representations of algebraic expressions - Multiplying and Dividing Rational Expressions
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. - Multiplying and Dividing Rational Expressions video
This video introduces and explains the topic. - Polynomials video
The video introduces and explains the topic. - Simplifying Rational Expressions
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.APR.6
Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
Standard A.APR.7
Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
Strand: ALGEBRA - Creating Equations (A.CED)
Create equations that describe numbers or relationships, using all available types of functions to create such equations (Standards A.CED.1–4).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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: Polynomial Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function. - Module 3: Polynomial Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function. - Module 4: Rational Expressions and Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 4, Rational Expressions and Functions, students work with the fractions that are ratios of polynomials, rational expressions and functions. - Module 4: Rational Expressions and Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 4, Rational Expressions and Functions, students work with the fractions that are ratios of polynomials, rational expressions and functions.
Instructional Tasks
- 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. - Bernardo and Sylvia Play a Game
This task presents a simple but mathematically interesting game whose solution is a challenging exercise in creating and reasoning with algebraic inequalities. The core of the task involves converting a verbal statement into a mathematical inequality in a context in which the inequality is not obviously presented, and then repeatedly using the inequality to deduce information about the structure of the game. - Buying a Car
The emphasis in this task is not on complex solution procedures. Rather, the progression of equations, from two that involve different values of the sales tax, to one that involves the sales tax as a parameter, is designed to foster the habit of looking for regularity in solution procedures, so that students don't approach every equation as a new problem but learn to notice familiar types. - Clea on an Escalator
This task has students create equations to model a physical scenario, and then reason with those equations to come up with a solution. - Dimes and Quarters
This task does not actually require that the student solve the system but that they recognize the pairs of linear equations in two variables that would be used to solve the system. - Equations and Formulas
This task asks students to use inverse operations to solve the equations for the unknown variable, or for the designated variable if there is more than one. - 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. - Global Positioning System I
This question examines the algebraic equations for three different spheres. The intersections of each pair of spheres are then studied, both using the equations and thinking about the geometry of the spheres. - Growing coffee
This task is designed to make students think about the meaning of the quantities presented in the context and choose which ones are appropriate for the two different constraints presented. In particular, note that the purpose of the task is to have students generate the constraint equations for each part (though the problem statements avoid using this particular terminology), and not to have students solve said equations. - How Much Folate?
This task a could be used as an introduction to writing and graphing linear inequalities. Part (a) includes significant scaffolding to support the introduction of the ideas. Part (b) demonstrates that, in some situations, writing down all possible combinations is not feasible. - Introduction to Polynomials - College Fund
This task could serve as an introduction to polynomials or as an application after students are familiar with this type of function. - Optimization Problems: Boomerangs
This lesson unit is intended to help educators assess how well students are able to interpret a situation and represent the constraints and variables mathematically, select appropriate mathematical methods to use, explore the effects of systematically varying the constraints, and interpret and evaluate the data generated and identify the optimum case, checking it for confirmation.
- Paper Folding
This is a very open-ended task designed for students to develop some of the basic ideas surrounding exponential growth. - Paying the rent
This simple conceptual task focuses on what it means for a number to be a solution to an equation, rather than on the process of solving equations. - Planes and wheat
This is a simple exercise in creating equations from a situation with many variables. By giving three different scenarios, the problem requires students to keep going back to the definitions of the variables, thus emphasizing the importance of defining variables when you write an equation. - Regular Tessellations of the plane
This task examines the ways in which the plane can be covered by regular polygons in a very strict arrangement called a regular tessellation. The goal of the task is to use algebra in order to understand which tessellations of the plane with regular polygons are possible. - Rewriting equations
The goal of this task is to manipulate equations in order to solve for a specified variable. - Silver Rectangle
This task provides a geometric context for working with ratios and algebraic equations. Students will create and then solve an algebraic equation describing a remarkable shape, the silver rectangle. - Sum of angles in a polygon
This problem provides students with an opportunity to discover algebraic structure in a geometric context. More specifically, the student will need to divide up the given polygons into triangles and then use the fact that the sum of the angles in each triangle is 180. - Throwing a Ball
Although this task is quite straightforward, it has a couple of aspects designed to encourage students to attend to the structure of the equation and the meaning of the variables in it. - Uranium 238
The goal of this task is to represent an exponential relationship by an equation and identify, using knowledge of the context and the structure of the equation, possible graphs for the equation. - Writing constraints
The purpose of this task is to give students practice writing a constraint equation for a given context.
Standard A.CED.1
Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and 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 nonviable options in a modeling context. For example, maximizing the volume of a box for a given surface area while drawing attention to the practical domain.
Standard A.CED.4
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange the compound interest formula to solve for t: A = P(1+ r/n)nt.
Strand: ALGEBRA - Reasoning with Equations and Inequalities (A.REI)
Understand solving equations as a process of reasoning and explain the reasoning (Standard A.REI.2)
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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: Rational Expressions and Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 4, Rational Expressions and Functions, students work with the fractions that are ratios of polynomials, rational expressions and functions. - Module 4: Rational Expressions and Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 4, Rational Expressions and Functions, students work with the fractions that are ratios of polynomials, rational expressions and functions.
Instructional Tasks
- An Extraneous Solution
The goal of this task is to examine how extraneous solutions can arise when solving rational equations. - 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. - Canoe Trip
The goal of this task is to set up and solve an equation involving a simple rational expression. - Radical Equations
The purpose of the task is to show students a situation where squaring both sides of an equation can result in an equation with more solutions than the original one. - Who wins the Race?
The goal of this task is to solve an equation involving a radical and then verify whether the solutions of the resulting quadratic equation are relevant.
- 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. - 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. - Solving 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. - Solving Radical Equations video
This video introduces and explains the topic. - Solving 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. - Solving Rational Expressions video
This video introduces and explains the topic.
Standard A.REI.2
Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
Represent and solve equations and inequalities graphically (Standard A.REI.11).
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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.
Instructional Tasks
- Introduction to Polynomials - College Fund
This task could serve as an introduction to polynomials or as an application after students are familiar with this type of function. - Optimization Problems: Boomerangs
This lesson unit is intended to help educators assess how well students are able to interpret a situation and represent the constraints and variables mathematically, select appropriate mathematical methods to use, explore the effects of systematically varying the constraints, and interpret and evaluate the data generated and identify the optimum case, checking it for confirmation.
- 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.
Standard A.REI.11
Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(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 where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★
Strand: FUNCTIONS - Interpreting Functions (F.IF)
Interpret functions that arise in applications in terms of a context (Standards F.IF.4–6).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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: Polynomial Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function. - Module 3: Polynomial Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function. - Module 4: Rational Expressions and Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 4, Rational Expressions and Functions, students work with the fractions that are ratios of polynomials, rational expressions and functions. - Module 4: Rational Expressions and Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 4, Rational Expressions and Functions, students work with the fractions that are ratios of polynomials, rational expressions and functions. - Module 6: Modeling Periodic Behavior - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 6, Modeling Periodic Behavior. In this module students use a Ferris wheel as a context for constructing conceptual understanding of circular trigonometry. They begin by calculating heights on the Ferris wheel, progress to calculating the heights at a given time on the Ferris wheel, and then, graphing the heights to show a sine function. - Module 6: Modeling Periodic Behavior - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 6, Modeling Periodic Behavior. In this module students use a Ferris wheel as a context for constructing conceptual understanding of circular trigonometry. They begin by calculating heights on the Ferris wheel, progress to calculating the heights at a given time on the Ferris wheel, and then, graphing the heights to show a sine function. - Module 7: Trig. Functions, Equations & Identities - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 7, Trigonometric Functions, Equations, and Identities, students work with more trigonometric graphs, beginning with the familiar Ferris wheel context. - Module 7: Trig. Functions, Equations & Identities - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 7, Trigonometric Functions, Equations, and Identities, students work with more trigonometric graphs, beginning with the familiar Ferris wheel context.
Review Activitiy
- Polynomials video
The video introduces and explains the topic. - Simplifying Rational Expressions
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 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; end behavior; and periodicity.★
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.★
Analyze functions using different representations (Standards F.IF.7–9).
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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: Logarithmic Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 2, Logarithmic Functions, picks up where Module 1 leaves off. Students begin to understand logarithms by drawing upon their experiences with inverses and exponential functions to evaluate, approximate, and order logarithmic expressions such log2 8 and log2 20. - Module 2: Logarithmic Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 2, Logarithmic Functions, picks up where Module 1 leaves off. Students begin to understand logarithms by drawing upon their experiences with inverses and exponential functions to evaluate, approximate, and order logarithmic expressions such log2 8 and log2 20. - Module 3: Polynomial Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function. - Module 3: Polynomial Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function. - Module 4: Rational Expressions and Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 4, Rational Expressions and Functions, students work with the fractions that are ratios of polynomials, rational expressions and functions. - Module 4: Rational Expressions and Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 4, Rational Expressions and Functions, students work with the fractions that are ratios of polynomials, rational expressions and functions.
Instructional Tasks
- Throwing Baseballs
This task could be used for assessment or for practice. It allows the students to compare characteristics of two quadratic functions that are each represented differently, one as the graph of a quadratic function and one written out algebraically. Specifically, we are asking the students to determine which function has the greatest maximum and the greatest non-negative root. - Which Function?
The task addresses knowledge related to interpreting forms of functions derived by factoring or completing the square. It requires students to pay special attention to the information provided by the way the equation is represented as well as the sign of the leading coefficient, which is not written out explicitly, and then to connect this information to the important features of the graph.
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 square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Compare and contrast square root, cubed root, and step functions with all other functions.
- Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
- Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
- Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
Standard F.IF.8
Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
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, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
Strand: FUNCTIONS - Building Functions (F.BF)
Build a function that models a relationship between two quantities. Develop models for more complex or sophisticated situations (Standards F.BF.1).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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: Functions and Their Inverses - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 1, Functions and Their Inverses, reviews the features of linear, exponential, and quadratic functions and general inverse relationships. The idea that the inputs and outputs are reversed in inverse functions is reinforced in the module using tables, graphs, equations, and story context. - Module 1: Functions and Their Inverses - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 1, Functions and Their Inverses, reviews the features of linear, exponential, and quadratic functions and general inverse relationships. The idea that the inputs and outputs are reversed in inverse functions is reinforced in the module using tables, graphs, equations, and story context. - Module 3: Polynomial Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function. - Module 3: Polynomial Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function. - Module 8: Modeling With Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 8, Modeling with Functions, focuses on a big idea of functions: Functions can be combined together (using basic operations or composed) to make new functions, usually retaining some of the features of both functions. - Module 8: Modeling With Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 8, Modeling with Functions, focuses on a big idea of functions: Functions can be combined together (using basic operations or composed) to make new functions, usually retaining some of the features of both functions.
Review Activitiy
- 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. - Inductive Reasoning video
This video introduces and explains the topic. - Modeling: Rolling Cups
This lesson unit is intended to help educators assess how well students are able to choose appropriate mathematics to solve a non-routine problem, generate useful data by systematically controlling variables, and develop experimental and analytical models of a physical situation. - Solving Rational Expressions video
This video introduces and explains the topic.
Standard F.BF.1
Write a function that describes a relationship between two quantities.★
- 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.
Build new functions from existing functions (Standards F.BF.3–4).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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: Functions and Their Inverses - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 1, Functions and Their Inverses, reviews the features of linear, exponential, and quadratic functions and general inverse relationships. The idea that the inputs and outputs are reversed in inverse functions is reinforced in the module using tables, graphs, equations, and story context. - Module 1: Functions and Their Inverses - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 1, Functions and Their Inverses, reviews the features of linear, exponential, and quadratic functions and general inverse relationships. The idea that the inputs and outputs are reversed in inverse functions is reinforced in the module using tables, graphs, equations, and story context. - Module 2: Logarithmic Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 2, Logarithmic Functions, picks up where Module 1 leaves off. Students begin to understand logarithms by drawing upon their experiences with inverses and exponential functions to evaluate, approximate, and order logarithmic expressions such log2 8 and log2 20. - Module 2: Logarithmic Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 2, Logarithmic Functions, picks up where Module 1 leaves off. Students begin to understand logarithms by drawing upon their experiences with inverses and exponential functions to evaluate, approximate, and order logarithmic expressions such log2 8 and log2 20. - Module 3: Polynomial Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function. - Module 3: Polynomial Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function. - Module 4: Rational Expressions and Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 4, Rational Expressions and Functions, students work with the fractions that are ratios of polynomials, rational expressions and functions. - Module 4: Rational Expressions and Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 4, Rational Expressions and Functions, students work with the fractions that are ratios of polynomials, rational expressions and functions. - Module 6: Modeling Periodic Behavior - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 6, Modeling Periodic Behavior. In this module students use a Ferris wheel as a context for constructing conceptual understanding of circular trigonometry. They begin by calculating heights on the Ferris wheel, progress to calculating the heights at a given time on the Ferris wheel, and then, graphing the heights to show a sine function. - Module 6: Modeling Periodic Behavior - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 6, Modeling Periodic Behavior. In this module students use a Ferris wheel as a context for constructing conceptual understanding of circular trigonometry. They begin by calculating heights on the Ferris wheel, progress to calculating the heights at a given time on the Ferris wheel, and then, graphing the heights to show a sine function. - Module 7: Trig. Functions, Equations & Identities - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 7, Trigonometric Functions, Equations, and Identities, students work with more trigonometric graphs, beginning with the familiar Ferris wheel context. - Module 7: Trig. Functions, Equations & Identities - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 7, Trigonometric Functions, Equations, and Identities, students work with more trigonometric graphs, beginning with the familiar Ferris wheel context. - Module 8: Modeling With Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 8, Modeling with Functions, focuses on a big idea of functions: Functions can be combined together (using basic operations or composed) to make new functions, usually retaining some of the features of both functions. - Module 8: Modeling With Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 8, Modeling with Functions, focuses on a big idea of functions: Functions can be combined together (using basic operations or composed) to make new functions, usually retaining some of the features of both functions.
Instructional Tasks
- Exponentials and Logarithms II
This task and its companion, F-BF Exponentials and Logarithms I, is designed to help students gain facility with properties of exponential and logarithm functions resulting from the fact that they are inverses. - Invertible or Not?
This task illustrates several components of standard F-BF.B.4.c: Find Inverse Functions. Here, instead of presenting two functions and asking the students to decide which one is invertible, students are asked to complete a table of input-output pairs for the functions in such a way that one of the functions is invertible and the other one is not. - Latitude
This task requires students to use data to generate understanding of an invertible function. Some brief notes: First, the table has data ordered by percentage, not latitude, so students will have to reorder the data in order to generate a graph of N(â). Second, students are asked to interpret statements about inverse functions, for which an understanding of the quantities' units is particularly helpful. - Parabolas and Inverse Functions
This task assumes students have an understanding of the relationship between functions and equations. Using this knowledge, the students are prompted to try to solve equations in order to find the inverse of a function given in equation form: when no such solution is possible, this means that the function does not have an inverse. - Rainfall
In this task students are asked to analyze a function and its inverse when the function is given as a table of values. In addition to finding values of the inverse function from the table, they also have to explain why the given function is invertible. - 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). - 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. - US Households
The purpose of this task is to construct and use inverse functions to model a a real-life context. Students choose a linear function to model the given data, and then use the inverse function to interpolate a data point.
- 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. - 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. - Applications of Quadratic Functions video
This video introduces and explains the topic. - 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. - 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. - 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. - Modeling: Rolling Cups
This lesson unit is intended to help educators assess how well students are able to choose appropriate mathematics to solve a non-routine problem, generate useful data by systematically controlling variables, and develop experimental and analytical models of a physical situation. - 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. - Solve Quadratic Expressions by Factoring video
This video introduces and explains the concept. - Solving Absolute Value 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. - Solving Absolute Value Equations video
This video explains how to solve absolute value expressions. - Solving Equations video
Answers the questions "what are equations?" and "how do we solve them?" - Solving Quadratic Equations Using the Quadratic Formula video
This video introduces and explains the topic. - Solving Quadratic Equations by Completing the Square
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 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. - Solving Radical Equations video
This video introduces and explains the topic. - Solving 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. - Solving Rational Expressions video
This video introduces and explains the topic. - Systems of Non-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. - Systems of Non-Linear Equations video
This video introduces and explains the topic.
Standard F.BF.3
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Note the effect of multiple transformations on a single function and the common effect of each transformation across function types. Include functions defined only by a graph. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
Standard F.BF.4
Find inverse functions.
- Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1.
Strand: FUNCTIONS - Linear, Quadratic, and Exponential Models (F.LE)
Construct and compare linear, quadratic, and exponential models and solve problems (Standards F.LE.3–4).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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: Logarithmic Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 2, Logarithmic Functions, picks up where Module 1 leaves off. Students begin to understand logarithms by drawing upon their experiences with inverses and exponential functions to evaluate, approximate, and order logarithmic expressions such log2 8 and log2 20. - Module 2: Logarithmic Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 2, Logarithmic Functions, picks up where Module 1 leaves off. Students begin to understand logarithms by drawing upon their experiences with inverses and exponential functions to evaluate, approximate, and order logarithmic expressions such log2 8 and log2 20. - Module 3: Polynomial Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function. - Module 3: Polynomial Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function.
Instructional Tasks
- 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. - Bacteria Populations
This task provides a real world context for interpreting and solving exponential equations. There are two solutions provided for part (a). The first solution demonstrates how to deduce the conclusion by thinking in terms of the functions and their rates of change. The second approach illustrates a rigorous algebraic demonstration that the two populations can never be equal. - Carbon 14 dating
The task requires the student to use logarithms to solve an exponential equation in the realistic context of carbon dating, important in archaeology and geology, among other places. - 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. - 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. - Graphene
This task provides a real world context for examining the incredible power of exponential growth/decay. - 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. - 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.
- Applying Rational Equations video
This video introduces and explains the topic. - 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.
- Solving Rational Expressions video
This video introduces and explains the topic.
Standard F.LE.3
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quanitity increasing linearly, quadratically, or (more generally) as a polynomial function.
Standard F.LE.4
For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. Include the relationship between properties of logarithms and properties of exponents, such as the connection between the properties of exponents and the basic logarithm property that log xy = log x + log y.
Interpret expressions for functions in terms of the situation it models. Introduce f(x) = ex as a model for continuous growth (Standard F.LE.5).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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.
Review Activitiy
- 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.
Standard F.LE.5
Interpret the parameters in a linear, quadratic, or exponential function in terms of a context.
Strand: FUNCTIONS - Trigonometric Functions (F.TF)
Extend the domain of trigonometric functions using the unit circle (Standards F.TF.1–3).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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: Modeling Periodic Behavior - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 6, Modeling Periodic Behavior. In this module students use a Ferris wheel as a context for constructing conceptual understanding of circular trigonometry. They begin by calculating heights on the Ferris wheel, progress to calculating the heights at a given time on the Ferris wheel, and then, graphing the heights to show a sine function. - Module 6: Modeling Periodic Behavior - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 6, Modeling Periodic Behavior. In this module students use a Ferris wheel as a context for constructing conceptual understanding of circular trigonometry. They begin by calculating heights on the Ferris wheel, progress to calculating the heights at a given time on the Ferris wheel, and then, graphing the heights to show a sine function. - Module 7: Trig. Functions, Equations & Identities - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 7, Trigonometric Functions, Equations, and Identities, students work with more trigonometric graphs, beginning with the familiar Ferris wheel context. - Module 7: Trig. Functions, Equations & Identities - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 7, Trigonometric Functions, Equations, and Identities, students work with more trigonometric graphs, beginning with the familiar Ferris wheel context.
Instructional Tasks
- Bicycle Wheel
The purpose of this task is to introduce radian measure for angles in a situation where it arises naturally. - Coordinates of Points on a Circle
The purpose of this task it to use geometry and algebra in order to understand the behavior of the trigonometric function f(x)=sinx+cosx. The task has been stated in an open ended fashion as there are natural solutions using geometry, or using the trigonometric identity sin2x=2sinxcosx, or algebraically solving a system of equations. - Equilateral triangles and trigonometric functions
The purpose of this task is to apply knowledge about triangles to calculate the sine and cosine of 30 and 60 degrees. - 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. - Foxes and Rabbits 2
The example of rabbits and foxes was introduced to illustrate two functions of time given in a table. We are now in a position to actually model the data given previously with trigonometric functions and investigate the behavior of this predator-prey situation. - Foxes and Rabbits 3
The example of rabbits and foxes was introduced to illustrate two functions of time given in a table. The same situation was used in F-TF Foxes and Rabbits 2 to find trigonometric functions modeling the data in the table. The previous situation was somewhat unrealistic since we were able to find functions that fit the data perfectly. In this task, on the other hand, we do some legitimate modelling, in that we come up with functions that approximate the data well, but do not perfectly match, the given data. - Properties of Trigonometric Functions
The goal of this task is to use the unit circle and rigid transformations in order to establish some fundamental trigonometric function identities. - Special Triangles 1
Using known facts about the unit circle and isosceles triangles together with the Pythagorean Theorem, we can derive the sine and cosine of special angles, in this case of Ï/4. This task can be done as a mini lecture soliciting responses from the students, or as a challenge problem for students to ponder and discuss. - Special Triangles 2
Using known facts about the unit circle and isosceles triangles together with the Pythagorean Theorem, we can derive the sine and cosine of special angels, in this case of Ï/6. This task can be done as a mini lecture soliciting responses from the students, or as a challenge problem for students to ponder and discuss. - Trig Functions and the Unit Circle
The purpose of this task is to help students make the connection between the graphs of sint and cost and the x and y coordinates of points moving around the unit circle. Students have to match coordinates of points on the graph with coordinates and angles in the diagram of the unit circle. - 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. - Trigonometric functions for arbitrary angles
The purpose of this task is to examine trigonometric functions for obtuse angles. The values sinx and cosx are defined for acute angles by referring to a right triangle one of whose acute angles measures x. For an obtuse angle, no such triangle exists and so an alternate definition is required. Prior to working on this task, students should have experience working with trigonometric functions and how they relate to the unit circle. - What exactly is a radian?
Radians are often mysterious to students, yet they are a very straight forward way to measure an angle by relating the measure of the angle to the length of the arc on the unit circle it subtends. This task is not designed to discover the definition of radian, rather it allows students to make meaning out of the definition.
- Representing Trigonometric Functions - Ferris Wheel
This lesson unit is intended to help educators assess how well students are able to model a periodic situation, the height of a person on a Ferris wheel, using trigonometric functions, and interpret the constants a, b, c in the formula h = a + b cos ct in terms of the physical situation, where h is the height of the person above the ground and t is the elapsed time.
Standard F.TF.1
Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
Standard F.TF.2
Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
Standard F.TF.3
Use special triangles to determine geometrically the values of sine, cosine, tangent for Π/3, Π/4 and Π6, and use the unit circle to express the values of sine, cosine, and tangent for Π - x, Π + x, and 2Π - x in terms of their values for x, where x is any real number.
Model periodic phenomena with trigonometric functions (Standards F.TF.5–7).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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: Modeling Periodic Behavior - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 6, Modeling Periodic Behavior. In this module students use a Ferris wheel as a context for constructing conceptual understanding of circular trigonometry. They begin by calculating heights on the Ferris wheel, progress to calculating the heights at a given time on the Ferris wheel, and then, graphing the heights to show a sine function. - Module 6: Modeling Periodic Behavior - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 6, Modeling Periodic Behavior. In this module students use a Ferris wheel as a context for constructing conceptual understanding of circular trigonometry. They begin by calculating heights on the Ferris wheel, progress to calculating the heights at a given time on the Ferris wheel, and then, graphing the heights to show a sine function. - Module 7: Trig. Functions, Equations & Identities - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 7, Trigonometric Functions, Equations, and Identities, students work with more trigonometric graphs, beginning with the familiar Ferris wheel context. - Module 7: Trig. Functions, Equations & Identities - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 7, Trigonometric Functions, Equations, and Identities, students work with more trigonometric graphs, beginning with the familiar Ferris wheel context.
Instructional Tasks
- 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). - Foxes and Rabbits 2
The example of rabbits and foxes was introduced to illustrate two functions of time given in a table. We are now in a position to actually model the data given previously with trigonometric functions and investigate the behavior of this predator-prey situation. - Foxes and Rabbits 3
The example of rabbits and foxes was introduced to illustrate two functions of time given in a table. The same situation was used in F-TF Foxes and Rabbits 2 to find trigonometric functions modeling the data in the table. The previous situation was somewhat unrealistic since we were able to find functions that fit the data perfectly. In this task, on the other hand, we do some legitimate modelling, in that we come up with functions that approximate the data well, but do not perfectly match, the given data.
- Representing Trigonometric Functions - Ferris Wheel
This lesson unit is intended to help educators assess how well students are able to model a periodic situation, the height of a person on a Ferris wheel, using trigonometric functions, and interpret the constants a, b, c in the formula h = a + b cos ct in terms of the physical situation, where h is the height of the person above the ground and t is the elapsed time. - 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.
Standard F.TF.5
Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.★
Standard F.TF.7
Use inverse functions to solve trignometric equations that arise in modeling context; evaluate the solutions using technology and interpret them in terms of context. Limit solutions to a given interval. ★
Strand: GEOMETRY - Similarity, Right Triangles, and Trigonometry (G.SRT)
Apply trigonometry to general triangles. With respect to the general case of the Laws of Sines and Cosines, the definitions of sine and cosine must be extended to obtuse angles (Standards G.SRT.9–11).
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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: Modeling with Geometry - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 5, Geometric Modeling, begins with students visualizing two-dimensional cross sections of three-dimensional objects and solids of rotation. They learn to approximate the volume of an irregular solid by decomposing it into cylinders, frustrums, and cones with volumes that can be easily calculated. - Module 5: Modeling with Geometry - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 5, Geometric Modeling, begins with students visualizing two-dimensional cross sections of three-dimensional objects and solids of rotation. They learn to approximate the volume of an irregular solid by decomposing it into cylinders, frustrums, and cones with volumes that can be easily calculated.
Standard G.SRT.9
Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
Standard G.SRT.10
Prove the Laws of Sines and Cosines and use them to solve problems.
Standard G.SRT.11
Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
Strand: GEOMETRY - Geometric Measurement and Dimension (G.GMD)
Visualize relationships between two-dimensional and three-dimensional objects (Standards G.MD.4).
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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: Modeling with Geometry - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 5, Geometric Modeling, begins with students visualizing two-dimensional cross sections of three-dimensional objects and solids of rotation. They learn to approximate the volume of an irregular solid by decomposing it into cylinders, frustrums, and cones with volumes that can be easily calculated. - Module 5: Modeling with Geometry - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 5, Geometric Modeling, begins with students visualizing two-dimensional cross sections of three-dimensional objects and solids of rotation. They learn to approximate the volume of an irregular solid by decomposing it into cylinders, frustrums, and cones with volumes that can be easily calculated.
Instructional Tasks
- Global Positioning System II
Reflective of the modernness of the technology involved, this is a challenging geometric modelling task in which students discover from scratch the geometric principles underlying the software used by GPS systems. - Tennis Balls in a Can
This task is inspired by the derivation of the volume formula for the sphere.
- 2D Representations of 3D Objects
This lesson unit is intended to help educators assess how well students are able to visualize two-dimensional cross-sections of representations of three-dimensional objects. - Modeling: Rolling Cups
This lesson unit is intended to help educators assess how well students are able to choose appropriate mathematics to solve a non-routine problem, generate useful data by systematically controlling variables, and develop experimental and analytical models of a physical situation. - Student Task: Propane Tanks
In this task, students must work out how much you need to change the radius of a propane gas tank in order to double its capacity.
Standard G.GMD.4
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
Strand: GEOMETRY - Modeling With Geometry (G.MG)
Apply geometric concepts in modeling situations (Standards G.MG.1–3).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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: Modeling with Geometry - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 5, Geometric Modeling, begins with students visualizing two-dimensional cross sections of three-dimensional objects and solids of rotation. They learn to approximate the volume of an irregular solid by decomposing it into cylinders, frustrums, and cones with volumes that can be easily calculated. - Module 5: Modeling with Geometry - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 5, Geometric Modeling, begins with students visualizing two-dimensional cross sections of three-dimensional objects and solids of rotation. They learn to approximate the volume of an irregular solid by decomposing it into cylinders, frustrums, and cones with volumes that can be easily calculated.
Instructional Tasks
- A Ton of Snow
The goal of this task is to examine a mathematical statement about the mass of snow, hopefully providing some stimulating thought to go along with the very arduous and demanding physical exercise. - Archimedes and the King's crown
This problem combines the ideas of ratio and proportion within the context of density of matter. - Eratosthenes and the circumference of the earth
This task is designed for the student to apply geometric concepts in modeling situations. - 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. - Global Positioning System II
Reflective of the modernness of the technology involved, this is a challenging geometric modelling task in which students discover from scratch the geometric principles underlying the software used by GPS systems. - Hexagonal Pattern of Beehives
The goal of this task is to use geometry study the structure of beehives. - How far is the horizon?
The purpose of this modeling task is to have students use mathematics to answer a question in a real-world context using mathematical tools that should be very familiar to them. The task gets at particular aspects of the modeling process, namely, it requires them to make reasonable assumptions and find information that is not provided in the task statement. - How many cells are in the human body?
The purpose of this task is for students to apply the concepts of mass, volume, and density in a real-world context. - How many leaves on a tree?
This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree. - How many leaves on a tree? (Version 2)
Teachers who use this version of the task will need to bring tree leaves (or prepare a good sketch of a tree leaf) to class so that they can work on and discuss how to approximate the area of an irregular shape like a leaf. - How thick is a soda can? Variation I
This task's main goal is to provide a familiar context and a straightforward question which require a variety of tools to solve: modeling a situation with geometry, paying close attention to units, and converting units. - How thick is a soda can? Variation II
This is a variation of ''How thick is a soda can? Variation I'' which allows students to work independently and think about how they can determine how thick a soda can is. - Ice Cream Cone
This rich task is an excellent example of geometric concepts in a modeling situation and is accessible to all students. In this task, students will provide a sketch of a paper ice cream cone wrapper, use the sketch to develop a formula for the surface area of the wrapper, and estimate the maximum number of wrappers that could be cut from a rectangular piece of paper. - Indiana Jones and the Golden Statue
The goal of this task is to provide a introduction to the sometimes subtle use of density and units related to density, in a simple and fun context with minimal geometric complexity. - Paper Clip
This high level task is an example of applying geometric methods to solve design problems and satisfy physical constraints. - Regular Tessellations of the plane
This task examines the ways in which the plane can be covered by regular polygons in a very strict arrangement called a regular tessellation. The goal of the task is to use algebra in order to understand which tessellations of the plane with regular polygons are possible. - Running around a track I
This task uses geometry to find the perimeter of the track. - Running around a track II
The goal of this task is to model a familiar object, an Olympic track, using geometric shapes. - Satellite
This task is an example of applying geometric methods to solve design problems and satisfy physical constraints. - Solar Eclipse
Total solar eclipses are quite rare, so much so that they make the news when they do occur. This task explores some of the reasons why. - Solving Quadratic Equations: Cutting Corners
This lesson unit is intended to help educators assess how well students are able to solve quadratics in one variable. - Tennis Balls in a Can
This task is inspired by the derivation of the volume formula for the sphere. - The Lighthouse Problem
In addition to the purely geometric and trigonometric aspects of the task, this problem asks students to model phenomena on the surface of the earth. - Tilt of earth's axis and the four seasons
This task gives students a chance to relate their weather experiences with a simple geometric model which explains why the seasons occur. - Toilet Roll
The purpose of this task is to engage students in geometric modeling, and in particular to deduce algebraic relationships between variables stemming from geometric constraints. - Use Cavalieris Principle to Compare Aquarium Volumes
This task presents a context that leads students toward discovery of the formula for calculating the volume of a sphere.
- 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. - Image Tool
Students are able to explore scale by measuring angles in various images in this activity in order to better understand ratio and proportion. - Modeling: Rolling Cups
This lesson unit is intended to help educators assess how well students are able to choose appropriate mathematics to solve a non-routine problem, generate useful data by systematically controlling variables, and develop experimental and analytical models of a physical situation. - 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. - 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 G.MG.1
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).★
Standard G.MG.2
Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).★
Standard G.MG.3
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).★
Strand: STATISTICS - Interpreting Categorical and Quantitative Data (S.ID)
Summarize, represent, and interpret data on a single count or measurement variable. While students may have heard of the normal distribution, it is unlikely that they will have prior experience using it to make specific estimates. Build on students’ understanding of data distributions to help them see how the normal distribution uses area to make estimates of frequencies (which can be expressed as probabilities). Emphasize that only some data are well described by a normal distribution (Standard S.ID.4).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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: Statistics - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 9, Statistics, begins with a learning cycle where students construct the concept of normal distributions, understanding the effect of modifying either the mean or standard deviation. - Module 9: Statistics - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 9, Statistics, begins with a learning cycle where students construct the concept of normal distributions, understanding the effect of modifying either the mean or standard deviation.
Instructional Tasks
- 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. - Do You Fit In This Car?
This task requires students to use the normal distribution as a model for a data distribution. Students must use given means and standard deviations to approximate population percentages. - Musical Preferences
The basic idea is for students to demonstrate that they know what it means for two variables to be associated. - SAT Scores
This task is designed to help students use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. They should also recognize that there are data sets for which such a procedure is not appropriate, and use calculators, spreadsheets, and tables to estimate areas under the normal curve. - Should We Send Out a Certificate?
The purpose of this task is to have students complete normal distribution calculations and to use properties of normal distributions to draw conclusions. The task is designed to encourage students to communicate their findings in a narrative/report form in context not just simply as a computed number.
Standard S.ID.4
Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
Strand: STATISTICS - Making Inferences and Justifying Conclusions (S.IC)
Understand and evaluate random processes underlying statistical experiments (Standard S.IC.1).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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: Statistics - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 9, Statistics, begins with a learning cycle where students construct the concept of normal distributions, understanding the effect of modifying either the mean or standard deviation. - Module 9: Statistics - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 9, Statistics, begins with a learning cycle where students construct the concept of normal distributions, understanding the effect of modifying either the mean or standard deviation.
Instructional Tasks
- Fred's Flare Formula
This task is intended to engage students into considering how margin of error can be estimated from examining the results of repeated simple random sampling. - High blood pressure
The purpose of this task is to assess understanding of how study design dictates whether a conclusion of causation is warranted. - Musical Preferences
The basic idea is for students to demonstrate that they know what it means for two variables to be associated. - School Advisory Panel
This task is designed to help students understand statistics as a process for making inferences about population parameters based on a random sample from that population. - Strict Parents
This task is designed to help students understand statistics as a process for making inferences about population parameters based on a random sample from that population. - The Marble Jar
This task is designed as an instructional task to develop students understanding of how data from a random sample can be used to estimate a population proportion or percentage. - Types of Statistical Studies
The purpose of this task is to provide students with experience distinguishing between the various types of statistical studies and to understand the purpose of random selection in surveys and observational studies vs. random assignment to treatments in experiments. - Why Randomize?
This task is designed to help students understand statistics as a process for making inferences about population parameters based on a random sample from that population. - Words and Music II
The purpose of this task is to assess (1) ability to distinguish between an observational study and an experiment and (2) understanding of the role of random assignment to experimental groups in an experiment.
- Fire!, Probability, and Chaos
A simulated forest fire is the starting place to help students understand probability and graphing and graph interpretation, mean, and variance. - 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. - 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. - Marbles
This activity helps the student understand randomness and probability by pulling marbles out of a bag. - Probability: Playing with Fire
This lesson is designed to help students understand possible outcomes of a probability experiment. - Replacement and Probability
This lesson will help students learn the difference between sampling with and without replacement. - Unexpected Answers
This lesson shows students some unexpected outcomes to probability problems and shows how fairness in games is not always as it seems.
Standard S.IC.1
Understand that statistics allows inferences to be made about population parameters based on a random sample from that population.
Standard S.IC.3
Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
Draw and justify conclusions from sample surveys, experiments, and observational studies. In earlier grades, students are introduced to different ways of collecting data and use graphical displays and summary statistics to make comparisons. These ideas are revisited with a focus on how the way in which data is collected determines the scope and nature of the conclusions that can be drawn from that data. The concept of statistical significance is developed informally through simulation as meaning a result that is unlikely to have occurred solely as a result of random selection in sampling or random assignment in an experiment. For S.IC.4, focus on the variability of results from experiments - that is, focus on statistics as a way of dealing with, not eliminating, inherent randomness (Standards S.IC.3-4, 6).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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.
Instructional Tasks
- Fred's Flare Formula
This task is intended to engage students into considering how margin of error can be estimated from examining the results of repeated simple random sampling. - High blood pressure
The purpose of this task is to assess understanding of how study design dictates whether a conclusion of causation is warranted. - Margin of Error for Estimating a Population Mean
The purpose of this task is to illustrate the development of margin of error when estimating a population mean. - Scratch 'n Win Blues
This task is intended to engage students in considering how margin of error can be estimated from examining the results of repeated simple random sampling. - The Marble Jar
This task is designed as an instructional task to develop students understanding of how data from a random sample can be used to estimate a population proportion or percentage. - Types of Statistical Studies
The purpose of this task is to provide students with experience distinguishing between the various types of statistical studies and to understand the purpose of random selection in surveys and observational studies vs. random assignment to treatments in experiments. - Words and Music II
The purpose of this task is to assess (1) ability to distinguish between an observational study and an experiment and (2) understanding of the role of random assignment to experimental groups in an experiment.
- Fire!, Probability, and Chaos
A simulated forest fire is the starting place to help students understand probability and graphing and graph interpretation, mean, and variance. - 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. - 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. - Probability: Playing with Fire
This lesson is designed to help students understand possible outcomes of a probability experiment. - Replacement and Probability
This lesson will help students learn the difference between sampling with and without replacement. - Unexpected Answers
This lesson shows students some unexpected outcomes to probability problems and shows how fairness in games is not always as it seems.
Standard S.IC.4
Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
Standard S.IC.6
Evaluate reports based on data.
HONORS - Strand: NUMBER AND QUANTITY - Complex Number System (N.CN)
Perform arithmetic operations with complex numbers (Standard N.CN.3)
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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.
HONORS - Standard N.CN.3
Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
Represent complex numbers and their operations on the complex plane (Standard N.CN.4–6).
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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.
HONORS - Standard N.CN.4
Represent complex numbers on the complex plane in rectangular form and polar form (including real and imaginary numbers), and explain why the rectangular form of a given complex number represents the same number.
HONORS - Standard N.CN.5
Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1 + √3 i)3 = 8 because (-1 + √3 i) has modulus 2 and argument 120°.
HONORS - Standard N.CN.6
Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
Use complex numbers in polynomial identities and equations (Standard N.CN.10).
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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.
HONORS - Standard N.CN.10
Multiply complex numbers in polar form and use DeMoivre's Theorem to find roots of complex numbers.
HONORS - Strand: FUNCTIONS - Interpreting Functions (F.IF)
Analyze functions using different representations (Standard F.IF.7, d and f).
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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.
HONORS - 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 rational functions, identifying zeros, asymptotes, and point discontinuities when suitable factorizations are available, and showing end behavior.
- Define a curve parametrically and draw its graph.
HONORS - Strand: FUNCTIONS - Building Functions (F.BF).
Build a function that models a relationship between two quantities (Standard F.BF.1.c).
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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.
HONORS - Standard F.BF.1
Write a function that describes a relationship between two quantities.
- Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
Build new functions from existing functions (Standards F.BF.4, b,c,d–5).
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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: Logarithmic Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 2, Logarithmic Functions, picks up where Module 1 leaves off. Students begin to understand logarithms by drawing upon their experiences with inverses and exponential functions to evaluate, approximate, and order logarithmic expressions such log2 8 and log2 20. - Module 2: Logarithmic Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 2, Logarithmic Functions, picks up where Module 1 leaves off. Students begin to understand logarithms by drawing upon their experiences with inverses and exponential functions to evaluate, approximate, and order logarithmic expressions such log2 8 and log2 20.
HONORS - Standard F.BF.4
Find inverse functions.
- Verify by composition that one function is the inverse of another.
- Read values of an inverse function from a graph or a table, given that the function has an inverse.
- Produce an invertible function from a non-invertible function by restricting the domain.
HONORS - Standard F.BF.5
Understand the inverse relationship between exponents and logarithms, and use this relationship to solve problems involving logarithms and exponents.
HONORS - Strand: FUNCTIONS - Trigonometric Functions (F.TF)
Extend the domain of trigonometric functions using the unit circle (Standard T.FT.4).
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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: Trig. Functions, Equations & Identities - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 7, Trigonometric Functions, Equations, and Identities, students work with more trigonometric graphs, beginning with the familiar Ferris wheel context. - Module 7: Trig. Functions, Equations & Identities - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 7, Trigonometric Functions, Equations, and Identities, students work with more trigonometric graphs, beginning with the familiar Ferris wheel context.
HONORS - Standard F.TF.4
Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
Model periodic phenomena with trigonometric functions (Standards T.FT.6–7).
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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.
HONORS - Standard F.TF.6
Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
HONORS - Standard F.TF.7
Use the inverse functions to solve trigonometric equations that arise in the modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
Prove and apply trigonometric identities (Standard T.FT.9).
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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.
HONORS - Standard F.TF.9
Prove the addition and subtraction formulas for sine, cosine, and tangent, and use them to solve problems.
HONORS - Strand: GEOMETRY - Geometric Measurement and Dimension (G.GMD)
Explain volume formulas and use them to solve problems (Standard G.GMD.2).
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three 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.
HONORS - Standard G.GMD.2
Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.
HONORS - Strand: STATISTICS AND PROBABILITY - Conditional Probability and the Rules of Probability (S.CP)
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.
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