A-CED, A-REI Introduction to Polynomials - College Fund

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: When Marcus started high school, his grandmother opened a college savings account. On the first day of each school year she deposited money into the ac...

Material Type: Activity/Lab

Author: Illustrative Mathematics

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.

Material Type: Activity/Lab

Author: Illustrative Mathematics

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. In order to reinforce this aspect of the problem, the variables have not been given names that remind the student of what they stand for. The emphasis here is on setting up equations, not solving them.

Material Type: Activity/Lab

Author: Illustrative Mathematics

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_.

Material Type: Activity/Lab

Author: Illustrative Mathematics

Accurately Weighing Pennies II

This task is a somewhat more complicated version of "Accurately weighing pennies I'' as a third equation is needed in order to solve part (a) explicitly. Instead, students have to combine the algebraic techniques with some additional problem-solving (numerical reasoning, informed guess-and-check, etc.)

Material Type: Activity/Lab

Author: Illustrative Mathematics

Accurately Weighing Pennies I

This problem involves solving a system of algebraic equations from a context: depending how the problem is interpreted, there may be one equation or two.

Material Type: Activity/Lab

Author: Illustrative Mathematics

A-CED Cash Box

The given solutions for this task involve the creation and solving of a system of two equations and two unknowns, with the caveat that the context of the problem implies that we are interested only in non-negative integer solutions. Indeed, in the first solution, we must also restrict our attention to the case that one of the variables is further even.

Material Type: Activity/Lab

Author: Illustrative Mathematics

A-REI Estimating a Solution via Graphs

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: Jason and Arianna are working on solving the equations \begin{align} 6x + 17y &= 100\\ 5x + 9y &= 86. \end{align} Rounding their answer to the nearest ...

Material Type: Activity/Lab

Author: Illustrative Mathematics

A-REI Find A System

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: Without graphing, construct a system of two linear equations where $(-2,3)$ is a solution to the first equation but not to the second equation, and whe...

Material Type: Activity/Lab

Author: Illustrative Mathematics

A-REI Pairs of Whole Numbers

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?...

Material Type: Activity/Lab

Author: Illustrative Mathematics

Quinoa Pasta 3

This tasks is an example of a mathematical modeling problem (SMP 4) and it also illustrates SMP 1 (Making sense of a problem). Students are only told that there are two ingredients in the pasta and they have a picture of the box. It might even be better to just show the picture of the box, or to bring in the box and ask the students to pose the question themselves.

Material Type: Activity/Lab

Author: Illustrative Mathematics

Quinoa Pasta 2

This task has some aspects of a mathematical modeling problem (SMP 4) and it also illustrates SMP 1 (Making sense of a problem). Students are given all the relevant information on the nutritional labels, but they have to figure out how to use this information. They have to come up with the idea that they can set up two equations in two unknowns to solve the problem.

Material Type: Activity/Lab

Author: Illustrative Mathematics

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.

Material Type: Activity/Lab

Author: Illustrative Mathematics

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. This is an important step in the process of solving systems.

Material Type: Activity/Lab

Author: Illustrative Mathematics

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.

Material Type: Activity/Lab

Author: Illustrative Mathematics

A-CED How Much Folate?

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: Sara's doctor tells her she needs between 400 and 800 milligrams of folate per day, with part coming from her diet and part coming from a multi-vitamin...

Material Type: Activity/Lab

Author: Illustrative Mathematics

Writing Constraints

The purpose of this task is to give students practice writing a constraint equation for a given context. Instruction accompanying this task should introduce the notion of a constraint equation as an equation governing the possible values of the variables in question.

Material Type: Activity/Lab

Author: Illustrative Mathematics

A-CED Products and Reciprocals

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: The product of two positive numbers is 9. The reciprocal of one of these numbers is 4 times the reciprocal of the other number. What is the sum of the ...

Material Type: Activity/Lab

Author: Illustrative Mathematics

Zero Product Property 1

This task is the first in a series that leads students to understand and apply the zero product property to solving quadratic equations. The emphasis is on using the structure of a factorable expression in order to justify the steps in a solution (rather than memorizing steps without understanding). Teachers should feel free to skip any tasks in the series that students have already mastered.

Material Type: Activity/Lab

Zero Product Property 2

This task is part of a series of tasks that lead students to understand and apply the zero product property to solving quadratic equations. The emphasis is on using the structure of a factorable expression to help find its solutions (rather than memorizing steps without understanding). In this particular task, we are trying to get students to prove the zero product property, which is the lynchpin in understanding how to solve quadratic equations by factoring. In tasks that follow in this series, students will apply this property to solving quadratic equations and justifying their solutions.

Material Type: Activity/Lab