## Course Standards

## General Course Information and Notes

### Version Description

This course is targeted for students who are not yet "college ready" in mathematics or simply need some additional instruction in content to prepare them for success in college level mathematics. This course incorporates the Florida Standards for Mathematical Practices as well as the following Florida Standards for Mathematical Content: Expressions and Equations, The Number System, Functions, Algebra, Geometry, Number and Quantity, Statistics and Probability, and the Florida Standards for High School Modeling. The standards align with the Mathematics Postsecondary Readiness Competencies deemed necessary for entry-level college courses.

**English Language Development (ELD) Standards Special Notes Section:**

Teachers are required to provide listening, speaking, reading and writing instruction that allows English Language Learners (ELL) to communicate information, ideas and concepts for academic success in the content area of Mathematics. For the given level of English language proficiency and with visual, graphic, or interactive support, students will interact with grade level words, expressions, sentences and discourse to process or produce language necessary for academic success. The ELD standard should specify a relevant content area concept or topic of study chosen by curriculum developers and teachers which maximize an ELL's need for communication and social skills. To access an ELL supporting document which delineates performance definitions and descriptors, please clock on the following link: http://www.cpalms.org/uploads/docs/standards/eld/MA.pdf.

For additional information on the development and implementation of the ELD standards, please contact the Bureau of Student Achievement through Language Acquisition at sala@fldoe.org.

**Additional Instructional Resources:**

A.V.E. for Success Collection is provided by the Florida Association of School Administrators: http://www.fasa.net/4DCGI/cms/review.html?Action=CMS_Document&DocID=139. Please be aware that these resources have not been reviewed by CPALMS and there may be a charge for the use of some of them in this collection.

### General Information

**Course Number:**1200700

**Course Path:**

**Abbreviated Title:**MATH COLL READINESS

**Number of Credits:**One (1) credit

**Course Length:**Year (Y)

**Course Attributes:**

- Class Size Core Required
- Highly Qualified Teacher (HQT) Required
- Florida Standards Course

**Course Type:**Core Academic Course

**Course Level:**2

**Course Status:**Course Approved

**Grade Level(s):**9,10,11,12,30,31

**Graduation Requirement:**Mathematics

## Educator Certifications

## Student Resources

## Original Student Tutorials

Learn how to solve rational linear and quadratic equations using cross multiplication in this interactive tutorial.

Type: Original Student Tutorial

Learn to construct a function to model a linear relationship between two quantities and determine the slope and y-intercept given two points that represent the function with this interactive tutorial.

Type: Original Student Tutorial

Learn how to solve and graph compound inequalities and determine if solutions are viable in part 2 of this interactive tutorial series.

Click **HERE** to open Part 1.

Type: Original Student Tutorial

Learn how to write equations in two variables in this interactive tutorial.

Type: Original Student Tutorial

Learn how to solve and graph one variable inequalities, including compound inequalities, in part 1 of this interactive tutorial series.

Click **HERE** to open Part 2.

Type: Original Student Tutorial

Learn how reflections of a function are created and tied to the value of *k* in the mapping of *f*(*x*) to -1*f*(*x*) in this interactive tutorial.

Type: Original Student Tutorial

Explore translations of functions on a graph that are caused by *k* in this interactive tutorial. GeoGebra and interactive practice items are used to investigate linear, quadratic, and exponential functions and their graphs, and the effect of a translation on a table of values.

Type: Original Student Tutorial

Practice solving and checking two-step equations with rational numbers in this interactive tutorial.

This is part 2 of the two-part series on two-step equations. **Click HERE to open Part 1.**

Type: Original Student Tutorial

Professor E. Qual will teach you how to solve and check two-step equations in this interactive tutorial.

This is part 1 of a two-part series about solving 2-step equations. **Click HERE to open Part 2.**

Type: Original Student Tutorial

Use models to solve balance problems on a space station in this interactive, math and science tutorial.

Type: Original Student Tutorial

Learn how to find the point on a directed line segment that partitions it into a given ratio in this interactive tutorial.

Type: Original Student Tutorial

Learn what slope is in mathematics and how to calculate it on a graph and with the slope formula in this interactive tutorial.

Type: Original Student Tutorial

Quadratic functions can be used to model real-world phenomena. Key features of quadratic functions such as maximum values and zeros can often reveal important qualities of these phenomena. By the end of this tutorial, you should be able to find the zeros of a quadratic function and interpret their meaning in real-world contexts.

Type: Original Student Tutorial

Learn how to add and subtract polynomials in this online tutorial. You will learn how to combine like terms and then use the distribute property to subtract polynomials.

This is part 2 of a two-part lesson. Click below to open part 1.

Type: Original Student Tutorial

Learn how to identify monomials and polynomials and determine their degree in this interactive tutorial.

This is part 1 in a two-part series. **Click HERE to open Part 2**.

Type: Original Student Tutorial

Learn how to write the equation of a linear trend line when fitted to bivariate data in a scatterplot in this online tutorial.

Type: Original Student Tutorial

Use long division to rewrite a rational expression of the form *a*(*x*) divided by *b*(*x*) in the form *q*(*x*) plus the quantity *r*(*x*) divided by *b*(*x*), where *a*(*x*), *b*(*x*), *q*(*x*), and *r*(*x*) are polynomials.

Type: Original Student Tutorial

Learn to define, calculate, and interpret marginal frequencies, joint frequencies, and conditional frequencies in the context of the data with this interactive tutorial.

Type: Original Student Tutorial

Learn to complete the square of a quadratic expression and identify the maximum or minimum value of the quadratic function it defines. In this interactive tutorial, you'll also interpret the meaning of the maximum and minimum of a quadratic function in a real world context.

Type: Original Student Tutorial

Learn to rewrite products involving radicals and rational exponents using properties of exponents in this interactive tutorial.

Type: Original Student Tutorial

Learn how to explain the meaning of additive inverse, identify the additive inverse of a given rational number, and justify your answer on a number line.

Type: Original Student Tutorial

Learn how to calculate and interpret an average rate of change over a specific interval on a graph.

Type: Original Student Tutorial

Learn how to explain the steps used to solve a simple equation and provide reasons to support those steps with this interactive tutorial.

Type: Original Student Tutorial

Learn to determine the number of possible solutions for a linear equation with this interactive tutorial.

Type: Original Student Tutorial

Explain why the x-coordinate of the point of intersection of two functions is the solution of the equation f(x) = g(x).

Type: Original Student Tutorial

The graph of a quadratic equation is called a parabola [puh-ra-bow-luh]. The key features we will focus on in this tutorial are the vertex (a maximum or minimum extreme) and the direction of its opening. You will learn how to examine a quadratic equation written in vertex form in order to distinguish each of these key features.

Type: Original Student Tutorial

Learn to construct linear functions from tables that contain sets of data that relate to each other in special ways as you complete this interactive tutorial.

Type: Original Student Tutorial

Use mathematical properties to explain why a negative factor times a negative factor equals a positive product… instead of just quoting a rule with this interactive tutorial.

Type: Original Student Tutorial

Write linear inequalities for different money situations in this interactive tutorial.

Type: Original Student Tutorial

## Educational Games

Challenge yourself with this Prodigi game to see if you can answer questions about points and lines in graphs. Practice using slope-intercept form of a line. Try the "Teach Me" button to prepare yourself. When you are ready, play Prodigi! Be sure to use the review function at the end for any incorrect answers! Have fun!

Type: Educational Game

In this challenge game, you will be simplifying fractional expressions with exponents. Use the "Teach Me" button to review content before the challenge. During the challenge you get one free solve and two hints! After the challenge, review the problems as needed. Try again to get all challenge questions right! Question sets vary with each game, so feel free to play the game multiple times as needed! Good luck!

Type: Educational Game

In this challenge game, you will be solving inequalities and working with graphs of inequalities. Use the "Teach Me" button to review content before the challenge. During the challenge you get one free solve and two hints! After the challenge, review the problems as needed. Try again to get all challenge questions right! Question sets vary with each game, so feel free to play the game multiple times as needed! Good luck!

Type: Educational Game

This interactive game has 4 categories: adding integers, subtracting integers, multiplying integers, and dividing integers. Students can play individually or in teams.

Type: Educational Game

This addition game encourages some logical analysis as well as addition skills. This particular circle game uses positive and negative integers. There is only one way to combine all the given numbers so that every circle sums to zero.

(source: NLVM grade 6-8 "Circle 0")

Type: Educational Game

In this timed activity, students solve linear equations (one- and two-step) or quadratic equations of varying difficulty depending on the initial conditions they select. This activity allows students to practice solving equations while the activity records their score, so they can track their progress. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Educational Game

In this activity, two students play a simulated game of Connect Four, but in order to place a piece on the board, they must correctly solve an algebraic equation. This activity allows students to practice solving equations of varying difficulty: one-step, two-step, or quadratic equations and using the distributive property if desired. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.

Type: Educational Game

This virtual manipulative provides students with practice adding positive and negative integers. Students are given an addition problem and using one-to-one correspondence, the student is able to see what happens when adding negative integers. The addition problems can be computer generated or teacher generated and there is a free play mode which allows the student to practice with the chips and become familiar with the process of moving the chips around the page and creating a visual representation of an addition problem with integers.

Type: Educational Game

## Educational Software / Tools

This Excel spreadsheet allows the educator to input data into a two way frequency table and have the resulting relative frequency charts calculated automatically on the second sheet. This resource will assist the educator in checking student calculations on student-generated data quickly and easily.

Steps to add data: All data is input on the first spreadsheet; all tables are calculated on the second spreadsheet

- Modify column and row headings to match your data.
- Input joint frequency data.
- Click the second tab at the bottom of the window to see the automatic calculations.

Type: Educational Software / Tool

In this activity, students solve arithmetic problems involving whole numbers, integers, addition, subtraction, multiplication, and division. This activity allows students to track their progress in learning how to perform arithmetic on whole numbers and integers. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Educational Software / Tool

## Perspectives Video: Experts

Jump to it and learn more about how quadratic equations are used in robot navigation problem solving!

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Expert

The tide is high! How can we statistically prove there is a relationship between the tides on the Gulf Coast and in a fresh water spring 20 miles from each other?

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Expert

Statistical analysis played an essential role in using microgravity sensors to determine location of caves in Wakulla County.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Expert

It's important to stay inside the lines of your project constraints to finish in time and under budget. This NASA systems engineer explains how constraints can actually promote creativity and help him solve problems!

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Listen in as a computing enthusiast describes how hexadecimal notation is used to express big numbers in just a little space.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

Get fired up as you learn more about ceramic glaze recipes and mathematical units.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

## Problem-Solving Tasks

In this online problem-solving challenge, students apply algebraic reasoning to determine the "costs" of individual types of faces from sums of frowns, smiles, and neutral faces. This page provides three pictorial problems involving solving systems of equations along with tips for thinking through the problem, the solution, and other similar problems.

Type: Problem-Solving Task

Students explore the structure of the operation *s*/(v*n*). This question provides students with an opportunity to see expressions as constructed out of a sequence of operations: first taking the square root of *n*, then dividing the result of that operation into *s*.

Type: Problem-Solving Task

This problem solving task asks students to make deductions about what kind of music students like by examining a table with data.

Type: Problem-Solving Task

The purpose of this task is to assess ability to interpret the slope and intercept of the least squares regression line in context.

Type: Problem-Solving Task

This problem solving task asks students to examine the relationship between shops and crimes by using a correlation coefficient.

Type: Problem-Solving Task

This problem solving task gives students the opportunity to prove a fact about quadrilaterals: that if we join the midpoints of an arbitrary quadrilateral to form a new quadrilateral, then the new quadrilateral is a parallelogram, even if the original quadrilateral was not.

Type: Problem-Solving Task

This problem solving task asks students to find the area of a triangle by using unit squares and line segments.

Type: Problem-Solving Task

This problem solving task challenges students to use ideas about linear functions in order to determine when certain angles are right angles.

Type: Problem-Solving Task

In this task, students use trigonometric functions to model the movement of a point around a wheel and, through space. Students also interpret features of graphs in terms of the given real-world context.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

In order to engage this task meaningfully, students must be aware of the convention that va for a positive number a refers to the positive square root of a. 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.

Type: Problem-Solving Task

This task 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. The brand of pasta is quite commonly available at supermarkets or health food stores such as Whole Foods and even at Amazon.com. The box has the nutritional label and a reference to the website where the students can find other information about the ingredients

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

This task addresses A-REI.3.6, solving systems of linear equations exactly, and provides a simple example of a system with three equations and three unknown. Two (of many) methods for solving the system are presented. The first takes the given information to make three equations in three unknowns which can then be solved via algebraic manipulation to find the three numbers. The second solution is more clever, creating a single equation in three unknowns from the given information. This equation is then combined with the given information about the sums of pairs of numbers to deduce what the third number is. In reality, this solution is not simpler than the first: rather it sets up a slightly different set of equations which can be readily solved (the key being to take the sum of the three equations in the first solution). It provides a good opportunity for the instructor to show different methods for solving the same system of linear equations.

Type: Problem-Solving Task

The purpose of this task is to continue a crucial strand of algebraic reasoning begun at the middle school level (e.g, 6.EE.5). By asking students to reason about solutions without explicitly solving them, we get at the heart of understanding what an equation is and what it means for a number to be a solution to an equation. The equations are intentionally very simple; the point of the task is not to test technique in solving equations, but to encourage students to reason about them.

Type: Problem-Solving Task

In this task students use verbal descriptions to construct and compare linear and exponential functions and to find where the two functions intersect (F-LE.2, F-LE.3, A-REI.11).

Type: Problem-Solving Task

This task provides an exploration of a quadratic equation by descriptive, numerical, graphical, and algebraic techniques. Based on its real-world applicability, teachers could use the task as a way to introduce and motivate algebraic techniques like completing the square, en route to a derivation of the quadratic formula.

Type: Problem-Solving Task

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. This aspect of the task is illustrative of mathematical practice standard MP4 (Model with mathematics), and crucial as the system has an integer solution for both situations, that is, whether or not we include the dollar on the floor in the cash box or not.

Type: Problem-Solving Task

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.) Part (b) is new to this task, as with only two types of pennies the weight of the collection determines how many pennies of each type are in the collection. This is no longer the case with three different weights but in this particular case, a collection of 50 is too small to show any ambiguity. This is part of the reason for part (c) of the question where the weight alone no longer determines which type of pennies are in the roll. This shows how important levels of accuracy in measurement are as the answer to part (b) could be different if we were to measure on a scale which is only accurate to the nearest tenth of a gram instead of to the nearest hundredth of a gram.

Type: Problem-Solving Task

Solving this problem with algebra requires factoring a particular cubic equation (the difference of two cubes) as well as a quadratic equation. An alternative solution using prime numbers and arithmetic is presented.

Type: Problem-Solving Task

This classroom task is meant to elicit a variety of different methods of solving a quadratic equation (A-REI.4). Some are straightforward (for example, expanding the square on the right and rearranging the equation so that we can use the quadratic formula); some are simple but clever (reasoning from the fact that x and (2x - 9) have the same square); some use tools (using a graphing calculator to graph the functions f(x) = x^2 and g(x) = (2x-90)^2 and looking for values of x at which the two functions intersect). Some solution methods will work on an arbitrary quadratic equation, while others (such as the last three) may have difficulty or fail if the quadratic equation is not given in a particular form, or if the solutions are not rational numbers.

Type: Problem-Solving Task

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. The main work in parts (a) and (b) is in setting up the equation(s) appropriately. Question (c) is more subtle and it requires thinking carefully about the accuracy available in a particular measurement (weight). The first two parts of this task could be used for instructional or assessment purposes while the third part should strictly be implemented for instructional purposes.

Type: Problem-Solving Task

The purpose of this task is to provide an opportunity for students to reason about equivalence of equations. The instruction to give reasons that do not depend on solving the equation is intended to focus attention on the transformation of equations as a deductive step.

Type: Problem-Solving Task

This problem-solving task challenges students to find all quadratic functions described by given equation and coordinates, and describe how the graphs of those functions are related to one another.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

This task addresses a common misconception about function notation.

Type: Problem-Solving Task

This task could be used for assessment or for practice. It allows 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, students are asked to determine which function has the greatest maximum and the greatest non-negative root.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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

Type: Problem-Solving Task

This task asks students to find the average, write an equation, find the domain, and create a graph of the cost of producing DVDs.

Type: Problem-Solving Task

The problem presents a context where a quadratic function arises. Careful analysis, including graphing of the function, is closely related to the context. The student will gain valuable experience applying the quadratic formula and the exercise also gives a possible implementation of completing the square.

Type: Problem-Solving Task

The goal of this task is to give students a context to investigate large numbers and measurements. Students need to fluently convert units with very large numbers in order to successfully complete this task. The total number of pennies minted either in a single year or for the last century is phenomenally large and difficult to grasp. One way to assess how large this number is would be to consider how far all of these pennies would reach if we were able to stack them one on top of another: this is another phenomenally large number but just how large may well come as a surprise.

Type: Problem-Solving Task

In this task students investigate and ultimately prove the validity of the method of generating Pythagorean Triples that involves the polynomial identity (x^{2}+y^{2})^{2}=(x^{2}−y^{2})^{2}+(2xy)^{2}.

Type: Problem-Solving Task

In this task, students are asked to determine the unit price of a product under two different circumstances. They are also asked to generalize the cost of producing *x* items in each case.

Type: Problem-Solving Task

The principal purpose of the task is to explore a real-world application problem with algebra, working with units and maintaining reasonable levels of accuracy throughout. Students are asked to determine which product will be the most economical to meet the requirements given in the problem.

Type: Problem-Solving Task

This task asks students to consider functions in regard to temperatures in a high school gym.

Type: Problem-Solving Task

The purpose of this task is to introduce or reinforce the concept of a function, especially in a context where the function is not given by an explicit algebraic representation. Further, the last part of the task emphasizes the significance of one variable being a function of another variable in an immediately relevant real-life context.

Type: Problem-Solving Task

In this task students are given graphs of quantities related to weather. The purpose of the task is to show that graphs are more than a collection of coordinate points; they can tell a story about the variables that are involved, and together they can paint a very complete picture of a situation, in this case the weather. Features in one graph, like maximum and minimum points, correspond to features in another graph. For example, on a rainy day, the solar radiation is very low, and the cumulative rainfall graph is increasing with a large slope.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

This problem is a simple de-contextualized version of F-IF Your Father and F-IF Parking Lot. It also provides a natural context where the absolute value function arises as, in part (b), solving for x in terms of y means taking the square root of x^2 which is |x|.This task assumes students have an understanding of the relationship between functions and equations.

Type: Problem-Solving Task

This deceptively simple task asks students to find the domain and range of a function from a given context. The function is linear and if simply looked at from a formulaic point of view, students might find the formula for the line and say that the domain and range are all real numbers.

Type: Problem-Solving Task

This problem introduces a logistic growth model in the concrete settings of estimating the population of the U.S. The model gives a surprisingly accurate estimate and this should be contrasted with linear and exponential models.

Type: Problem-Solving Task

This task is for instructional purposes only and students should already be familiar with some specific examples of logistic growth functions. The goal of this task is to have students appreciate how different constants influence the shape of a graph.

Type: Problem-Solving Task

This task can be used as a quick assessment to see if students can make sense of a graph in the context of a real world situation. Students also have to pay attention to the scale on the vertical axis to find the correct match. The first and third graphs look very similar at first glance, but the function values are very different since the scales on the vertical axes are very different. The task could also be used to generate a group discussion on interpreting functions given by graphs.

Type: Problem-Solving Task

In this task, students will use inverse operations to solve the equations for the unknown variable or for the designated variable if there is more than one.

Type: Problem-Solving Task

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 (i.e., "constraining" said values). In particular, it is worth differentiating the role of constraint equations from more functional equations, e.g., formulas to convert from degrees Celsius to degree Fahrenheit. The task has students interpret the context and choose variables to represent the quantities, which are governed by the constraint equation and the fact that they are non-negative (allowing us to restrict the graphs to points in the first quadrant only).

The four parts are independent and can be used as separate tasks.

Type: Problem-Solving Task

The purpose of this task is to help students learn to read information about a function from its graph, by asking them to show the part of the graph that exhibits a certain property of the function. The task could be used to further instruction on understanding functions or as an assessment tool, with the caveat that it requires some amount of creativity to decide how to best illustrate some of the statements.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Students are asked to write and solve an inequality to determine the number of people that can safely rent a boat.

Type: Problem-Solving Task

The student is asked to write and solve an inequality to match the context.

Type: Problem-Solving Task

The purpose of this task is meant to reinforce students' understanding of rational numbers as points on the number line and to provide them with a visual way of understanding that the sum of a number and its additive inverse (usually called its "opposite") is zero.

Type: Problem-Solving Task

In this task, students answer a question about the difference between two temperatures that are negative numbers.

Type: Problem-Solving Task

The purpose of this task is to help solidify students' understanding of signed numbers as points on a number line and to understand the geometric interpretation of adding and subtracting signed numbers. There is a subtle distinction in the Florida Standards between a fraction and a rational number. Fractions are always positive, and when thinking of the symbol ab as a fraction, it is possible to interpret it as a equal-sized pieces where b pieces make one whole.

Type: Problem-Solving Task

The student is asked to complete a long division which results in a repeating decimal, and then use multiplication to "check" their answer. The purpose of the task is to have students reflect on the meaning of repeating decimal representation through approximation.

Type: Problem-Solving Task

Students are given a word problem that can be solved by using a pair of linear equations. 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.

Type: Problem-Solving Task

This task examines the ways in which the plane can be covered by regular polygons in a very strict arrangement called a regular tessellation. These tessellations are studied here using algebra, which enters the picture via the formula for the measure of the interior angles of a regular polygon (which should therefore be introduced or reviewed before beginning the task). The goal of the task is to use algebra in order to understand which tessellations of the plane with regular polygons are possible.

Type: Problem-Solving Task

The purpose of this task is to give students an opportunity to solve a multi-step ratio problem that can be approached in many ways. This can be done by making a table, which helps illustrate the pattern of taxi rates for different distances traveled and with a little persistence leads to a solution which uses arithmetic. It is also possible to calculate a unit rate (dollars per mile) and use this to find the distance directly without making a table.

Type: Problem-Solving Task

This task provides an approximation, and definition, of *e*, in the context of more and more frequent compounding of interest in a bank account. The approach is computational.

Type: Problem-Solving Task

This task aims for students to understand the quadratic formula in a geometric way in terms of the graph of a quadratic function.

Type: Problem-Solving Task

This task is intended for instruction and to motivate "Building a general quadratic function." This task assumes that the students are familiar with the process of completing the square.

Type: Problem-Solving Task

This task asks the student to graph and compare two proportional relationships and interpret the unit rate as the slope of the graph.

Type: Problem-Solving Task

In this example, students will answer questions about unit price of coffee, make a graph of the information, and explain the meaning of slope in the given context.

Type: Problem-Solving Task

In this example, students use properties of rational exponents and other algebraic concepts to compare and verify the relative size of two real numbers that involve decimal exponents.

Type: Problem-Solving Task

In this example, students are given the graph of two functions and are asked to sketch the graph of the function that is their sum. The intent is that students develop a conceptual understanding of function addition.

Type: Problem-Solving Task

The primary purpose of this task is to elicit common misconceptions that arise when students try to model situations with linear functions. This task, being multiple choice, could also serve as a quick assessment to gauge a class' understanding of modeling with linear functions.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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

At a higher level, the task addresses MAFS.912.N-Q.1.3, since realistically neither the car nor the bus is going to travel at exactly the same speed from beginning to end of each segment; there is time traveling through traffic in cities, and even on the autobahn the speed is not constant. Thus students must make judgments about the level of accuracy with which to report the result.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

The problem requires students to not only convert miles to kilometers and gallons to liters but they also have to deal with the added complication of finding the reciprocal at some point.

Type: Problem-Solving Task

This task asks students to calculate the cost of materials to make a penny, utilizing rates of grams of copper.

Type: Problem-Solving Task

There are many different ways to write exponential expressions that describe the same quantity, in this task the amount of a radioactive substance after *t* years. Depending on what aspect of the context we need to investigate, one expression of the quantity may be more useful than another. This task contrasts the usefulness of four equivalent expressions. Students first have to confirm that the given expressions for the radioactive substance are equivalent. Then they have to explain the significance of each expression in the context of the situation.

Type: Problem-Solving Task

Students are asked to use units to determine if the given statement is valid.

Type: Problem-Solving Task

This is a challenging task, suitable for extended work, and reaching into a deep understanding of units. Students are given a scenario and asked to determine the number of people required to complete the amount of work in the time described. The task requires students to exhibit MAFS.K12.MP.1.1, Make sense of problems and persevere in solving them. An algebraic solution is possible but complicated; a numerical solution is both simpler and more sophisticated, requiring skilled use of units and quantitative reasoning. Thus the task aligns with either MAFS.912.A-CED.1.1 or MAFS.912.N-Q.1.1, depending on the approach.

Type: Problem-Solving Task

This task is intended for instructional purposes so that students can become familiar and confident with using a calculator and understanding what it can and cannot do. This task gives an opportunity to work on the notion of place value (in parts [b] and [c]) and also to understand part of an argument for why the square root of 2 is not a rational number.

Type: Problem-Solving Task

Students manipulate a given equation to find specified information.

Type: Problem-Solving Task

Students solve problems tracking the balance of a checking account used only to pay 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.

Type: Problem-Solving Task

Students extrapolate the list price of a car given a total amount paid in states with different tax rates. 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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

In this resource, students refer to given information which defines 5 variables in the context of real world government expenses. They are then asked to write equations based upon specific known values for some of the variables. The emphasis is on setting up, rather than solving, the equations.

Type: Problem-Solving Task

This task asks students to write expressions for various problems involving distance per units of volume.

Type: Problem-Solving Task

The context of this example is the spread of a flu virus on campus and the related sale of tissue boxes sold. Students interpret the composite function and determine values simply by using the tables of values.

Type: Problem-Solving Task

In this example, fuel efficiency of a car can be analyzed by using rational expressions and operations with rational expressions.

Type: Problem-Solving Task

In this resource, a method of deriving the quadratic formula from a theoretical standpoint is demonstrated. This task is for instructional purposes only and builds on "Building an explicit quadratic function."

Type: Problem-Solving Task

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

Type: Problem-Solving Task

By definition, the square root of a number *n* is the number you square to get *n*. The purpose of this task is to have students use the meaning of a square root to find a decimal approximation of a square root of a non-square integer. Students may need guidance in thinking about how to approach the task.

Type: Problem-Solving Task

This task compares the usefulness of different forms of a quadratic expression. Students have to choose which form most easily provides information about the maximum value, the zeros and the vertical intercept of a quadratic expression in the context of a real world situation. Rather than just manipulating one form into the other, students can make sense out of the structure of the expressions.

(From *Algebra: Form and Function*, McCallum et al., Wiley 2010)

Type: Problem-Solving Task

The purpose of this task is to help students see manipulation of expressions as an activity undertaken for a purpose.

Variation 1 of this task presents a related more complex expression already in the correct form to answer the question.

The expression arises in physics as the reciprocal of the combined resistance of two resistors in parallel. However, the context is not explicitly considered here.

Type: Problem-Solving Task

MAFS.8.NS.1.1 requires students to "convert a decimal expansion which repeats eventually into a rational number." Despite this choice of wording, the numbers in this task are rational numbers regardless of choice of representation. For example, 0.333¯ and 13 are two different ways of representing the same number.

Type: Problem-Solving Task

This task would be especially well-suited for instructional purposes. Students will benefit from a class discussion about the slope, y-intercept, x-intercept, and implications of the restricted domain for interpreting more precisely what the equation is modeling.

Type: Problem-Solving Task

This task illustrates the process of rearranging the terms of an expression to reveal different aspects about the quantity it represents, precisely the language being used in standard MAFS.912.A-SSE.2.3. Students are provided with an expression giving the temperature of a container at a time *t*, and have to use simple inequalities (e.g., that 2^{t}>0 for all *t*) to reduce the complexity of an expression to a form where bounds on the temperature of a container of ice cream are made apparent.

Type: Problem-Solving Task

The task assumes that students are able to express a given repeating decimal as a fraction. Teachers looking for a task to fill in this background knowledge could consider the related task "8.NS Converting Decimal Representations of Rational Numbers to Fraction Representations."

Type: Problem-Solving Task

When students plot irrational numbers on the number line, it helps reinforce the idea that they fit into a number system that includes the more familiar integer and rational numbers. This is a good time for teachers to start using the term "real number line" to emphasize the fact that the number system represented by the number line is the real numbers. When students begin to study complex numbers in high school, they will encounter numbers that are not on the real number line (and are, in fact, on a "number plane"). This task could be used for assessment, or if elaborated a bit, could be used in an instructional setting.

Type: Problem-Solving Task

Students explore and manipulate expressions based on the following statement:

A function f defined for -a < x < a is even if f(-x)=f(x) and is odd if f(-x)=-f(x) when -a < x < a. In this task we assume f is defined on such an interval, which might be the full real line (i.e., a=8).

Type: Problem-Solving Task

Students compare graphs of different quadratic functions, then produce equations of their own to satisfy given conditions.

This exploration can be done in class near the beginning of a unit on graphing parabolas. Students need to be familiar with intercepts, and need to know what the vertex is. It is effective after students have graphed parabolas in vertex form (y=a(x–h)^{2}+k), but have not yet explored graphing other forms.

Type: Problem-Solving Task

This is a standard problem phrased in a non-standard way. Rather than asking students to perform an operation, expanding, it expects them to choose the operation for themselves in response to a question about structure. Students must understand the need to transform the factored form of the quadratic expression (a product of sums) into a sum of products in order to easily see *a*, the coefficient of the x^{2} term; *k*, the leading coefficient of the *x* term; and *n*, the constant term.

Type: Problem-Solving Task

Students are asked to interpret the effect on the value of an expression given a change in value of one of the variables.

Type: Problem-Solving Task

Students are given a pair of numbers. They are asked to determine which is larger, and then justify their answer. The numbers involved are rational numbers and common irrational numbers, such π and square roots. This task can be used to either build or assess initial understandings related to rational approximations of irrational numbers.

Type: Problem-Solving Task

Students examine and answer questions related to a scenario similar to a "mixture" problem involving two different mixtures of fertilizer. In this example, students determine and then compare expressions that correspond to concentrations of various mixtures. Ultimately, students generalize the problem and verify conclusions using algebraic rather than numerical expressions.

Type: Problem-Solving Task

Students are asked to interpret expressions and equations within the context of the amounts of caramels and truffles in a box of candy.

Type: Problem-Solving Task

This problem asks students to consider algebraic expressions calculating the number of floor tiles in given patterns. The purpose of this task is to give students practice in reading, analyzing, and constructing algebraic expressions, attending to the relationship between the form of an expression and the context from which it arises. The context here is intentionally thin; the point is not to provide a practical application to kitchen floors, but to give a framework that imbues the expressions with an external meaning.

Type: Problem-Solving Task

The goal of this task is to develop an understanding of rational exponents (MAFS.912.N-RN.1.1); however, it also raises important issues about distinguishing between linear and exponential behavior (MAFS.912.F-LE.1.1c) and it requires students to create an equation to model a context (MAFS.912.A-CED.1.2).

Type: Problem-Solving Task

This resource describes a simple scenario which can be represented by the use of variables. Students are asked to examine several variable expressions, interpret their meaning, and describe what quantities they each represent in the given context.

Type: Problem-Solving Task

This resource poses the question, "how many vehicles might be involved in a traffic jam 12 miles long?"

This task, while involving relatively simple arithmetic, promps students to practice modeling (MP4), work with units and conversion (N-Q.1), and develop a new unit (N-Q.2). Students will also consider the appropriate level of accuracy to use in their conclusions (N-Q.3).

Type: Problem-Solving Task

In this task students interpret the relative size of variable expressions involving two variables in the context of a real world situation. All given expressions can be interpreted as quantities that one might study when looking at two animal populations.

Type: Problem-Solving Task

This resource involves simplifying algebraic expressions that involve complex numbers and various algebraic operations.

Type: Problem-Solving Task

This task has students experiment with the operations of addition and multiplication, as they relate to the notions of rationality and irrationality.

Type: Problem-Solving Task

This task provides the opportunity for students to reason about graphs, slopes, and rates without having a scale on the axes or an equation to represent the graphs. Students who prefer to work with specific numbers can write in scales on the axes to help them get started.

Type: Problem-Solving Task

The purpose of this task is to identify the structure in the two algebraic expressions by interpreting them in terms of a geometric context. Students will have likely seen this type of process before, so the principal source of challenge in this task is to encourage a multitude and variety of approaches, both in terms of the geometric argument and in terms of the algebraic manipulation.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

This task provides students the opportunity to make use of units to find the gas needed (MAFS.912.N-Q.1.1). It also requires them to make some sensible approximations (e.g., 2.92 gallons is not a good answer to part (a)) and to recognize that Felicia's situation requires her to round up. Various answers to (a) are possible, depending on how much students think is a safe amount for Felicia to have left in the tank when she arrives at the gas station. The key point is for them to explain their choices. This task provides an opportunity for students to practice MAFS.K12.MP.2.1: Reason abstractly and quantitatively, and MAFS.K12.MP.3.1: Construct viable arguments and critique the reasoning of others.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

This problem solving task examines, in a graphical setting, the impact of adding a scalar, multiplying by a scalar, and making a linear substitution of variables on the graph of the function *f*. This resource also includes standards alignment commentary and annotated solutions.

Type: Problem-Solving Task

In this problem-solving task students are challenged to apply their understanding of linear relationships to determine the amount of chicken and steak needed for a barbecue, which will include creating an equation, sketching a graph, and interpreting both. This resource also includes standards alignment commentary and annotated solutions.

Type: Problem-Solving Task

This task asks students to reason about the relative costs per pound of two fruits without actually knowing what the costs are. Students who find this difficult may add a scale to the graph and reason about the meanings of the ordered pairs. Comparing the two approaches in a class discussion can be a profitable way to help students make sense of slope.

Type: Problem-Solving Task

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Type: Problem-Solving Task

In this problem-solving task students are challenged to apply their understanding of linear relationships to determine the amount of chicken and steak needed for a barbecue, which will include creating an equation, sketching a graph, and interpreting both. This resource also includes standards alignment commentary and annotated solutions.

Type: Problem-Solving Task

This problem-solving task asks students to find a linear function that models something in the real world. After finding the equation of the linear relationship between the depth of the water and the distance across the channel, students have to verbalize the meaning of the slope and intercept of the line in the context of this situation. Commentary on standards alignment and illustrated solutions are also included.

Type: Problem-Solving Task

This task asks the student to understand the relationship between slope and changes in *x*- and *y*-values of a linear function.

Type: Problem-Solving Task

This activity challenges students to recognize the relationship between slope and the difference in *x-* and *y-*values of a linear function. Help students solidify their understanding of linear functions and push them to be more fluent in their reasoning about slope and y-intercepts. This task has also produced a reasonable starting place for discussing point-slope form of a linear equation.

Type: Problem-Solving Task

The student is asked to perform operations with numbers expressed in scientific notation to decide whether 7% of Americans really do eat at Giantburger every day.

Type: Problem-Solving Task

This is an instructional task meant to generate a conversation around the meaning of negative integer exponents. While it may be unfamiliar to some students, it is good for them to learn the convention that negative time is simply any time before t=0.

Type: Problem-Solving Task

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).

Type: Problem-Solving Task

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

Type: Problem-Solving Task

This task asks students to use proportions of mass and volume to create ideal brine for saltwater fish tanks. It also asks students to compare graphs.

Type: Problem-Solving Task

This problem is a quadratic function example. The other tasks in this set illustrate MAFS.912.F.BF.1.1.a in the context of linear, exponential, and rational functions.

Type: Problem-Solving Task

The task addresses the first part of standard MAFS.912.F-BF.2.3: "Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative)."

Type: Problem-Solving Task

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. If desired, instructors could also use this task to touch on such solutions by finding and interpreting solutions to the system of equations created in parts (a) and (b).

Type: Problem-Solving Task

The purpose of this task is to introduce students to exponential growth. While the context presents a classic example of exponential growth, it approaches it from a non-standard point of view.

Type: Problem-Solving Task

The primary purpose of this task is to lead students to a numerical and graphical understanding of the behavior of a rational function near a vertical asymptote, in terms of the expression defining the function.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

Students are asked to solve an inequality in order to answer a real-world question.

Type: Problem-Solving Task

This task asks students to determine whether a the set of given functions is odd, even, or neither.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

## Student Center Activities

Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.

Type: Student Center Activity

This video will demonstrate how to multiply polynomials.

Type: Student Center Activity

## Tutorials

You will learn in this video how to solve Quadratic Equations using the Quadratic Formula.

Type: Tutorial

You will learn how the parent function for a quadratic function is affected when f(x) = x^{2}.

Type: Tutorial

You will learn int his video how to solve the Quadratic Equation by Completing the Square.

Type: Tutorial

This video will demonstrate how to simplify square roots involving variables.

Type: Tutorial

This video is an example of solving systems by elimination where the system has infinite solutions.

Type: Tutorial

This video shows how to solve a system of equations through simple elimination.

Type: Tutorial

This video explains how to identify systems of equations without a solution.

Type: Tutorial

This video will demonstrate how to solve radical equations with additional practice problems.

Type: Tutorial

In this tutorial, students will look at input and output values of quadratic functions to help them understand why the graph of a second degree polynomial curves.

Type: Tutorial

This video tutorial shows students: the standard form of a polynomial, how to identify polynomials, how to determine the degree of a polynomial, how to add and subtract polynomials, and how to represent the area of a shape as an addition or subtraction of polynomials.

Type: Tutorial

This video shows how to solve systems by elimination.

Type: Tutorial

This tutorial will help the students to understand the function notation such as f(x), which can be thought as another way of representing the y-value in a function, especially when graphing. The y-axis is even labeled as the f(x) axis, when graphing.

Type: Tutorial

In this video tutorial students will learn how to solve quadratic equations by square roots.

Type: Tutorial

This video is an introduction to the elimination method of solving a system of equations.

Type: Tutorial

This example demonstrates solving a system of equations algebraically and graphically.

Type: Tutorial

This video demonstrates a system of equations with no solution.

Type: Tutorial

This video shows how to solve a system of equations using the substitution method.

Type: Tutorial

This video demonstrates solving a system of equations word problem using elimination.

Type: Tutorial

In this tutorial, students will learn how to solve and graph a system of equations.

Type: Tutorial

This tutorial shows students how to solve and graph a system of equations. Students will see how to sketch their solution after solving the system of equations.

Type: Tutorial

This tutorial shows how to solve a system of equations by graphing. Students will see what a no solution system of equations looks like in a graph.

Type: Tutorial

This tutorial shows how to solve a system of equations using substitution.

Type: Tutorial

In this video, you will practice finding the slope of a line from data in a table, and interpret what the slope means in the problem.

Type: Tutorial

In this video, you will use a linear graph to determine the y-intercept (starting point) and slope (rate of change), as well as interpret what these mean in the given scenario.

Type: Tutorial

In this video, you will practice writing the slope-intercept form for a line, given a table of x and y values.

Type: Tutorial

This video shows some examples that test your understanding of what happens when positive and negative numbers are multiplied and divided.

Type: Tutorial

Given two points on a line, you will find the slope and the y-intercept. You will then write the equation of the line in slope-intercept form.

Type: Tutorial

Given the slope of a line and a point on the line, you will write the equation of the line in slope-intercept form.

Type: Tutorial

Many real world problems involve involve percentages. This lecture shows how algebra is used in solving problems of percent change and profit-and-loss.

Type: Tutorial

Students will learn how to determine an equation by checking solutions. Students will be given a table and 4 linear equations and they will have to determine which equation created the table.

Type: Tutorial

Students will learn how to solve a consecutive integer problem. Checking the solution will be left to the student.

Type: Tutorial

In this video, you will practice writing the equations of lines in slope-intercept form from graphs. You will then practice graphing lines from equations in slope-intercept form.

Type: Tutorial

In this video, you will learn about Rene Descartes, and how he bridged the gap between algebra and geometry.

Type: Tutorial

This tuptorial shows students how to set up and solve an age word problem. The tutorial also shows how tp check your work using substitution.

Type: Tutorial

In this video, you will practice comparing an irrational number to a percent. First you will try it without a calculator. Then you will check your answer using a calculator.

Type: Tutorial

In this video, you will learn how to approximate a square root to the hundredths place.

Type: Tutorial

In this video, you will practice approximating square roots of numbers that are not perfect squares. You will find the perfect square below and above to approximate the value of the square root between two whole numbers.

Type: Tutorial

In this tutorial, you will practice classifying numbers as whole numbers, integers, rational numbers, and irrational numbers.

Type: Tutorial

This video discusses exponent properties involving products.

Type: Tutorial

This video models how to use the Quotient of Powers property.

Type: Tutorial

Students will learn the difference between rational and irrational numbers.

Type: Tutorial

This video demonstrates multiplying in scientific notation.

Type: Tutorial

This example demonstrates mathematical operations with scientific notation used to solve a word problem.

Type: Tutorial

This tutorial shows students the rule for negative exponents. Students will see, using variables, the pattern for negative exponents.

Type: Tutorial

This video demonstrates a scientific notation word problem involving division.

Type: Tutorial

This is an example showing how to simplify an expression into scientific notation.

Type: Tutorial

Students will learn how to convert difficult repeating decimals to fractions.

Type: Tutorial

This tutorial shows students how to convert basic repeating decimals to fractions.

Type: Tutorial

In this tutorial, students will learn about negative exponents. An emphasis is placed on multiplying by the reciprocal of a number.

Type: Tutorial

Students will learn how to convert a fraction into a repeating decimal. Students should know how to use long division before starting this tutorial.

Type: Tutorial

This video explains how to subtract polynomials with multiple variables and reinforces how to distribute a negative number.

Type: Tutorial

This video demonstrates how to factor a linear expression by taking a common factor.

Type: Tutorial

This video shows how to construct and solve a basic linear equation to solve a word problem.

Type: Tutorial

In this video, you will practice changing a fraction into decimal form.

Type: Tutorial

This video covers squaring a binomial with two variables. Students will be given the area of a square.

Type: Tutorial

You will learn how multiplication and division problems give us a positive or negative answer depending on whether there are an even or odd number of negative integers used in the problem.

Type: Tutorial

This video teaches about combining like terms in linear equations.

Type: Tutorial

In this tutorial, you will simplify expressions involving positive and negative fractions.

Type: Tutorial

In this tutorial, you will see how to simplify complex fractions.

Type: Tutorial

This video demonstrates adding and subtracting decimals in the context of an overdrawn checking account.

Type: Tutorial

Students will solve the inequality and graph the solution.

Type: Tutorial

It's helpful to represent an equation on a graph where we plot at least 2 points to show the relationship between the dependent and independent variables. Watch and we'll show you.

Type: Tutorial

Given a graph, we will be able to find the equation it represents.

Type: Tutorial

This video tutorial discusses how to create a system of equations.

Type: Tutorial

In this tutorial, you will evaluate fractions involving negative numbers and variables to determine if expressions are equivalent.

Type: Tutorial

In this tutorial, you will see how to divide fractions involving negative integers.

Type: Tutorial

In this tutorial you will practice multiplying and dividing fractions involving negative numbers.

Type: Tutorial

In this tutorial, you will learn rules for multiplying positive and negative integers.

Type: Tutorial

In this tutorial you will learn how to divide with negative integers.

Type: Tutorial

In this tutorial you will use the repeated addition model of multiplication to help you understand why multiplying negative numbers results in a positive answer.

Type: Tutorial

In this tutorial, you will use the distributive property to understand why the product of two negative numbers is positive.

Type: Tutorial

This 8 minute video will show step-by-step directions for using the elimination method to solve a system of linear equations.

Type: Tutorial

This video provides a real-world scenario and step-by-step instructions to constructing equations using two variables. Possible follow-up videos include *Plotting System of Equations - Yoga Plan, Solving System of Equations with Substitution - Yoga Plan, and Solving System of Equations with Elimination - Yoga Plan.*

Type: Tutorial

Practice substituting positive and negative values for variables.

Type: Tutorial

In this video, we will find the absolute value as distance between rational numbers.

Type: Tutorial

This video uses the number line to find unknown values in subtraction statements with negative numbers.

Type: Tutorial

This video asks you to select the model that matches the given expression.

Type: Tutorial

Use a number line to solve a word problem that includes a negative number.

Type: Tutorial

In this video, we figure out the temperature in Fairbanks, Alaska by adding and subtracting integers.

Type: Tutorial

This tutorial will help you to explore slopes of lines and see how slope is represented on the x-y axes.

Type: Tutorial

This tutorial will help the students to identify the vertex of a parabola from the equation, and then graph the parabola.

Type: Tutorial

This tutorial will help the learners to graph the equation of the quadratic function using the coordinates of the vertex of a parabola adn its x- intercepts.

Type: Tutorial

Evaluating Expressions with Two Variables

Type: Tutorial

This tutorial will help you to learn about the exponential functions by graphing various equations representing exponential growth and decay.

Type: Tutorial

This video demonstrates how to add and subtract negative fractions with unlike denominators.

Type: Tutorial

This video demonstrates use of a number line and absolute value to add negative numbers.

Type: Tutorial

This video demonstrates use of a number line to add numbers with positive and negative signs.

Type: Tutorial

Find out why subtracting a negative number is the same as adding the absolute value of that number.

Type: Tutorial

In this example we have a formula for converting Celsius temperature to Fahrenheit. Let's substitute the variable with a value (Celsius temp) to get the degrees in Fahrenheit. Great problem to practice with us!

Type: Tutorial

Learn how to evaluate an expression with variables using a technique called substitution (or "plugging in").

Type: Tutorial

This video demonstrates adding and subtracting integers using a number line.

Type: Tutorial

This video discusses multiplication and division of inequalities with negative numbers to solve the inequality.

Type: Tutorial

This tutorial reviews the concept of exponents and powers and includes how to evaluate powers with negative signs.

Type: Tutorial

Great question. In algebra, we do indeed avoid using the multiplication sign. We'll explain it for you here.

Type: Tutorial

Our focus here is understanding that a variable is just a letter or symbol (usually a lower case letter) that can represent different values in an expression. We got this. Just watch.

Type: Tutorial

This tutorial demonstrates how to use the power of a power property with both numerals and variables.

Type: Tutorial

This tutorial will help you understand the concept of absolute value. Take the quiz after the lesson to practice!

Type: Tutorial

This tutorial will help you to solve one-step equations using multiplication and division. For practice, take the quiz after the lesson!

Type: Tutorial

This tutorial demonstrates the number line method of multiplying integers. You will encounter four different combinations when multiplying integers: (1) Positive times positive, (2) Positive times negative, (3) Negative times negative, (4) Negative times positive. The lesson is available in video format, and there is a quiz for practice.

Type: Tutorial

This short video uses both an equation and a visual model to explain why the same steps must be used on both sides of the equation when solving for the value of a variable.

Type: Tutorial

This lecture shows how algebra is used to solve problems involving mixtures of solutions of different concentrations.

Type: Tutorial

If a term raised to a power is enclosed in parentheses and then raised to another power, this expression can be simplified using the rules of multiplying exponents.

Type: Tutorial

Any expression consisting of multiplied and divide terms can be enclosed in parentheses and raised to a power. This can then be simplified using the rules for multiplying exponents.

Type: Tutorial

When solving a system of linear equations in x and y with a single solution, we get a unique pair of values for x and y. But what happens when try to solve a system with no solutions or an infinite number of solutions?

Type: Tutorial

Parallel lines have the same slope and no points in common. However, it is not always obvious whether two equations describe parallel lines or the same line.

Type: Tutorial

Perpendicular lines have slopes which are negative reciprocals of each other, but why?

Type: Tutorial

Systems of two linear equations in two variables can have a single solution, no solutions, or an infinite number of solutions. This video gives a great description of inconsistent, dependent, and independent systems. A consistent independent system of equations will have one solution. A consistent dependent system of equations will have infinite number of solutions, and an inconsistent system of equations will have no solution. This tutorial also provides information on how to distinguish a given system of linear equations as inconsistent, independent, or dependent system by looking at the slope and intercept.

Type: Tutorial

A graph in Cartesian coordinates may represent a function or may only represent a binary relation. The "vertical line test" is a visual way to determine whether or not a graph represents a function.

Type: Tutorial

Systems of two equations in x and y can be solved by adding the equations to create a new equation with one variable eliminated. This new equation can then be solved to find the value of the remaining variable. That value is then substituted into either equation to find the value of other variable.

Type: Tutorial

A system of two equations in x and y can be solved by rearranging one equation to represent x in terms of y, and "substituting" this expression for x in the other equation. This creates an equation with only y which can then be solved to find y's value. This value can then be substituted into either equation to find the value of x.

Type: Tutorial

Scientific notation is used to conveniently write numbers that require many digits in their representations. How to convert between standard and scientific notation is explained in this tutorial.

Type: Tutorial

The first fractions used by ancient civilizations were "unit fractions." Later, numerators other than one were added, creating "vulgar fractions" which became our modern fractions. Together, fractions and integers form the "rational numbers."

Type: Tutorial

When number systems were expanded to include negative numbers, rules had to be formulated so that multiplication would be consistent regardless of the sign of the operands.

Type: Tutorial

A look behind the fundamental properties of the most basic arithmetic operation, addition

Type: Tutorial

The video tutorial discusses about two typical polynomial multiplications. First, squaring a binomial and second, product of a sum and difference.

Type: Tutorial

This tutorial will help the learners practice division of polynomials. Students will recognize that dividing polynomials is similar to simplifying fractions.

Type: Tutorial

This tutorial will help the learners practice multiplication of polynomials. Learners will understand that when they multiply expressions with more than two terms, they need to make sure each term in the first expression multiplies every term in the second expression.

Type: Tutorial

Binomials are the polynomials with two terms. This tutorial will help the students learn about the multiplication of binomials. In multiplication, we need to make sure that each term in the first set of parenthesis multiplies each term in the second set.

Type: Tutorial

In this tutorial, students will learn how to add and subtract polynomials functions using horizontal and vertical methods. In a horizontal format, like terms should be grouped together using the commutative property. In vertical format, terms should be listed by ascending degree with like terms placed below each other.

Type: Tutorial

In this tutorial students learn how to identify a polynomial, how to find the degree of a polynomial, and how to write a polynomial in standard format.

Type: Tutorial

This tutorial will help the learner to understand the concept of subtracting the positive and negative integers with the help of a number line. Learners can also take a quiz after the concept is internalized.

Type: Tutorial

Students will be able to see examples of addition of integers while watching a short video, and practice adding integers using an online quiz.

Type: Tutorial

This lesson introduces students to linear equations in one variable, shows how to solve them using addition, subtraction, multiplication, and division properties of equalities, and allows students to determine if a value is a solution, if there are infinitely many solutions, or no solution at all. The site contains an explanation of equations and linear equations, how to solve equations in general, and a strategy for solving linear equations. The lesson also explains contradiction (an equation with no solution) and identity (an equation with infinite solutions). There are five practice problems at the end for students to test their knowledge with links to answers and explanations of how those answers were found. Additional resources are also referenced.

Type: Tutorial

Upon completing this lesson, the student should be able to use the addition, subtraction, multiplication, and division properties of equality to solve linear inequalities, write the answer to an inequality using interval notation and draw a graph to give a visual answer to an inequality problem.

The lesson begins with explanations of inequality signs and interval notation and then moves on to demonstrate addition/subtraction and multiplication/division properties of equality. The site demonstrates a strategy for solving linear inequalities and presents three problems for students to practice what they have learned.

There is also a link to a previous tutorial which covers solving linear equations of one variable for students who need the review.

Type: Tutorial

This video models solving equations in one variable with variables on both sides of the equal sign.

Type: Tutorial

This Khan Academy presentation models solving two-step equations with one variable.

Type: Tutorial

The video describes how to multiply fractions and state the answer in lowest terms.

Type: Tutorial

## Video/Audio/Animations

This video will demonstrate how to determine what is and is not a function.

Type: Video/Audio/Animation

This video will demonstrate how to solve a quadratic equation using square roots.

Type: Video/Audio/Animation

This video demonstrates how to determine if a relation is a function and how to identify the domain.

Type: Video/Audio/Animation

Based upon the definition of speed, linear equations can be created which allow us to solve problems involving constant speeds, time, and distance.

Type: Video/Audio/Animation

How do we create linear equations to solve real-world problems? The video explains the process.

Type: Video/Audio/Animation

Although the Greeks initially thought all numeric quantities could be represented by the ratio of two integers, i.e. rational numbers, we now know that not all numbers are rational. How do we know this?

Type: Video/Audio/Animation

Although the domain and codomain of functions can consist of any type of objects, the most common functions encountered in Algebra are real-valued functions of a real variable, whose domain and codomain are the set of real numbers, R.

Type: Video/Audio/Animation

Exponents are not only integers. They can also be fractions. Using the rules of exponents, we can see why a number raised to the power " one over n" is equivalent to the nth root of that number.

Type: Video/Audio/Animation

Exponents are not only integers and unit fractions. An exponent can be any rational number expressed as the quotient of two integers.

Type: Video/Audio/Animation

Radical expressions can often be simplified by moving factors which are perfect roots out from under the radical sign.

Type: Video/Audio/Animation

Mixture problems can involve mixtures of things other than liquids. This video shows how Algebra can be used to solve problems involving mixtures of different types of items.

Type: Video/Audio/Animation

When should a system of equations with multiple variables be used to solve an Algebra problem, instead of using a single equation with a single variable?

Type: Video/Audio/Animation

The points of intersection of two graphs represent common solutions to both equations. Finding these intersection points is an important tool in analyzing physical and mathematical systems.

Type: Video/Audio/Animation

This chapter presents a new look at the logic behind adding equations- the essential technique used when solving systems of equations by elimination.

Type: Video/Audio/Animation

Any fraction can be converted into an equivalent decimal number with a sequence of digits after the decimal point, which either repeats or terminates. The reason can be understood by close examination of the number line.

Type: Video/Audio/Animation

Integer exponents greater than one represent the number of copies of the base which are multiplied together. hat if the exponent is one, zero, or negative? Using the rules of adding and subtracting exponents, we can see what the meaning must be.

Type: Video/Audio/Animation

Exponential expressions with multiplied terms can be simplified using the rules for adding exponents.

Type: Video/Audio/Animation

Exponential expressions with divided terms can be simplified using the rules for subtracting exponents.

Type: Video/Audio/Animation

Exponential expressions with multiplied and divided terms can be simplified using the rules of adding and subtracting exponents.

Type: Video/Audio/Animation

Two sets which are often of primary interest when studying binary relations are the domain and range of the relation.

Type: Video/Audio/Animation

"Slope" is a fundamental concept in mathematics. Slope is often defined as " the rise over the run"....but why?

Type: Video/Audio/Animation

Th point-slope form of the equation for a line can describe any non-vertical line in the Cartesian plane, given the slope and the coordinates of a single point which lies on the line.

Type: Video/Audio/Animation

The two point form of the equation for a line can describe any non-vertical line in the Cartesian plane, given the coordinates of two points which lie on the line.

Type: Video/Audio/Animation

Linear equations can be used to solve many types of real-word problems. In this episode, the water depth of a pool is shown to be a linear function of time and an equation is developed to model its behavior. Unfortunately, ace Algebra student A. V. Geekman ends up in hot water anyway.

Type: Video/Audio/Animation

Literal equations are formulas for calculating the value of one unknown quantity from one or more known quantities. Variables in the formula are replaced by the actual or 'literal' values corresponding to a specific instance of the relationship.

Type: Video/Audio/Animation

This video shows how to determine which lines are parallel from a set of three different equations.

Type: Video/Audio/Animation

This video illustrates how to determine if the graphs of a given set of equations are parallel.

Type: Video/Audio/Animation

This video describes how to determine the equation of a line that is perpendicular to another line. All that is given initially the equation of a line and an ordered pair from the other line.

Type: Video/Audio/Animation

This video takes a look at rearranging a formula to highlight a quantity of interest.

Type: Video/Audio/Animation

This video demonstrates writing a function that represents a real-life scenario.

Type: Video/Audio/Animation

This video gives a more in-depth look at graphing quadratic functions than previously offered in Quadratic Functions 1.

Type: Video/Audio/Animation

Khan Academy video tutorial on graphing linear equations: "Algebra: Graphing Lines 1"

Type: Video/Audio/Animation

Khan Academy tutorial video that demonstrates with real-world data the use of Excel spreadsheet to fit a line to data and make predictions using that line.

Type: Video/Audio/Animation

This Khan Academy video tutorial introduces averages and algebra problems involving averages.

Type: Video/Audio/Animation

## Virtual Manipulatives

This resource will assess students' understanding of addition and subtraction of polynomials.

Type: Virtual Manipulative

This resource can be used to assess students' understanding of solving quadratic equation by taking the square root. A great resource to view prior to this is "Solving quadratic equations by square root' by Khan Academy.

Type: Virtual Manipulative

This virtual manipulative is intended to allow the student to practice multiplication of binomials. The student should understand how to use algebra tiles before using this tool.

Type: Virtual Manipulative

This manipulative will help students in understanding scatter plots which are particularly useful when investigating whether there is a relationship between two variables. Students could develop a systematic plan for collecting and entering data into the scatter plot manipulative and set appropriate ranges for the *x* and *y* scales.

Type: Virtual Manipulative

In this activity, students adjust slider bars which adjust the coefficients and constants of a linear function and examine how their changes affect the graph. The equation of the line can be in slope-intercept form or standard form. This activity allows students to explore linear equations, slopes, and y-intercepts and their visual representation on a graph. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

Using this virtual manipulative, students apply dimensional analysis (AKA factor-label method or unit-factor method) to solve unit conversion problems. There is also the opportunity to create your own unit conversion problems.

Type: Virtual Manipulative

In this activity, students plug values into the independent variable to see what the output is for that function. Then based on that information, they have to determine the coefficient (slope) and constant(y-intercept) for the linear function. This activity allows students to explore linear functions and what input values are useful in determining the linear function rule. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.

Type: Virtual Manipulative

This virtual manipulative guides the student in the use of color counters to model subtraction of integers.

Type: Virtual Manipulative

This resource provides linear functions in standard form and asks the user to graph it using intercepts on an interactive graph below the problem. Immediate feedback is provided, and for incorrect responses, each step of the solution is thoroughly modeled.

Type: Virtual Manipulative

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative

Using this virtual manipulative, students are able to graph a function and a set of ordered pairs on the same coordinate plane. The constants, coefficients, and exponents can be adjusted using slider bars, so the student can explore the affect on the graph as the function parameters are changed. Students can also examine the deviation of the data from the function. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

This is a graphing tool/activity for students to deepen their understanding of polynomial functions and their corresponding graphs. This tool is to be used in conjunction with a full lesson on graphing polynomial functions; it can be used either before an in depth lesson to prompt students to make inferences and connections between the coefficients in polynomial functions and their corresponding graphs, or as a practice tool after a lesson in graphing the polynomial functions.

Type: Virtual Manipulative

In this online tool, students input a function to create a graph where the constants, coefficients, and exponents can be adjusted by slider bars. This tool allows students to explore graphs of functions and how adjusting the numbers in the function affect the graph. Using tabs at the top of the page you can also access supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

This is an online graphing utility that can be used to create box plots, bubble graphs, scatterplots, histograms, and stem-and-leaf plots.

Type: Virtual Manipulative

In this activity, students enter inputs into a function machine. Then, by examining the outputs, they must determine what function the machine is performing. This activity allows students to explore functions and what inputs are most useful for determining the function rule. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

With a mouse, students will drag data points (with their error bars) and watch the best-fit polynomial curve form instantly. Students can choose the type of fit: linear, quadratic, cubic, or quartic. Best fit or adjustable fit can be displayed.

Type: Virtual Manipulative

This interactive simulation investigates graphing linear and quadratic equations. Users are given the ability to define and change the coefficients and constants in order to observe resulting changes in the graph(s).

Type: Virtual Manipulative

This site provides a virtual balance on which the student can represent (and then solve) simple linear equations with integer answers. Conceptually, positive weights (unit-blocks and x-boxes) push the pans of the scale downward. Negative values are represented by balloons which can be attached to the pans of the scale. The student can then manipulate the weights to solve the equation while simultaneously seeing a visual display of these effects on the equation.

Type: Virtual Manipulative

The user drags batteries to create a circuit. The voltage of the batteries that are placed will be displayed on the voltmeter, and an equation will be displayed in a list on the right, giving an example of how positive and negative numbers work together.

Type: Virtual Manipulative

This is an interactive applet in which students or teachers can visualize how changes in the parameters of the exponential function, *y* = *a*(*b*) *x* + *c*, affect the shape of the graph.

Type: Virtual Manipulative

## Parent Resources

## Educational Games

This addition game encourages some logical analysis as well as addition skills. This particular circle game uses positive and negative integers. There is only one way to combine all the given numbers so that every circle sums to zero.

(source: NLVM grade 6-8 "Circle 0")

Type: Educational Game

This virtual manipulative provides students with practice adding positive and negative integers. Students are given an addition problem and using one-to-one correspondence, the student is able to see what happens when adding negative integers. The addition problems can be computer generated or teacher generated and there is a free play mode which allows the student to practice with the chips and become familiar with the process of moving the chips around the page and creating a visual representation of an addition problem with integers.

Type: Educational Game

## Perspectives Video: Expert

It's important to stay inside the lines of your project constraints to finish in time and under budget. This NASA systems engineer explains how constraints can actually promote creativity and help him solve problems!

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Listen in as a computing enthusiast describes how hexadecimal notation is used to express big numbers in just a little space.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

Get fired up as you learn more about ceramic glaze recipes and mathematical units.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

## Problem-Solving Tasks

In this online problem-solving challenge, students apply algebraic reasoning to determine the "costs" of individual types of faces from sums of frowns, smiles, and neutral faces. This page provides three pictorial problems involving solving systems of equations along with tips for thinking through the problem, the solution, and other similar problems.

Type: Problem-Solving Task

Students explore the structure of the operation *s*/(v*n*). This question provides students with an opportunity to see expressions as constructed out of a sequence of operations: first taking the square root of *n*, then dividing the result of that operation into *s*.

Type: Problem-Solving Task

This problem solving task asks students to make deductions about what kind of music students like by examining a table with data.

Type: Problem-Solving Task

The purpose of this task is to assess ability to interpret the slope and intercept of the least squares regression line in context.

Type: Problem-Solving Task

This problem solving task asks students to examine the relationship between shops and crimes by using a correlation coefficient.

Type: Problem-Solving Task

This problem solving task gives students the opportunity to prove a fact about quadrilaterals: that if we join the midpoints of an arbitrary quadrilateral to form a new quadrilateral, then the new quadrilateral is a parallelogram, even if the original quadrilateral was not.

Type: Problem-Solving Task

This problem solving task asks students to find the area of a triangle by using unit squares and line segments.

Type: Problem-Solving Task

This problem solving task challenges students to use ideas about linear functions in order to determine when certain angles are right angles.

Type: Problem-Solving Task

In this task, students use trigonometric functions to model the movement of a point around a wheel and, through space. Students also interpret features of graphs in terms of the given real-world context.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

In order to engage this task meaningfully, students must be aware of the convention that va for a positive number a refers to the positive square root of a. 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.

Type: Problem-Solving Task

This task 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. The brand of pasta is quite commonly available at supermarkets or health food stores such as Whole Foods and even at Amazon.com. The box has the nutritional label and a reference to the website where the students can find other information about the ingredients

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

This task addresses A-REI.3.6, solving systems of linear equations exactly, and provides a simple example of a system with three equations and three unknown. Two (of many) methods for solving the system are presented. The first takes the given information to make three equations in three unknowns which can then be solved via algebraic manipulation to find the three numbers. The second solution is more clever, creating a single equation in three unknowns from the given information. This equation is then combined with the given information about the sums of pairs of numbers to deduce what the third number is. In reality, this solution is not simpler than the first: rather it sets up a slightly different set of equations which can be readily solved (the key being to take the sum of the three equations in the first solution). It provides a good opportunity for the instructor to show different methods for solving the same system of linear equations.

Type: Problem-Solving Task

The purpose of this task is to continue a crucial strand of algebraic reasoning begun at the middle school level (e.g, 6.EE.5). By asking students to reason about solutions without explicitly solving them, we get at the heart of understanding what an equation is and what it means for a number to be a solution to an equation. The equations are intentionally very simple; the point of the task is not to test technique in solving equations, but to encourage students to reason about them.

Type: Problem-Solving Task

In this task students use verbal descriptions to construct and compare linear and exponential functions and to find where the two functions intersect (F-LE.2, F-LE.3, A-REI.11).

Type: Problem-Solving Task

This task provides an exploration of a quadratic equation by descriptive, numerical, graphical, and algebraic techniques. Based on its real-world applicability, teachers could use the task as a way to introduce and motivate algebraic techniques like completing the square, en route to a derivation of the quadratic formula.

Type: Problem-Solving Task

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. This aspect of the task is illustrative of mathematical practice standard MP4 (Model with mathematics), and crucial as the system has an integer solution for both situations, that is, whether or not we include the dollar on the floor in the cash box or not.

Type: Problem-Solving Task

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.) Part (b) is new to this task, as with only two types of pennies the weight of the collection determines how many pennies of each type are in the collection. This is no longer the case with three different weights but in this particular case, a collection of 50 is too small to show any ambiguity. This is part of the reason for part (c) of the question where the weight alone no longer determines which type of pennies are in the roll. This shows how important levels of accuracy in measurement are as the answer to part (b) could be different if we were to measure on a scale which is only accurate to the nearest tenth of a gram instead of to the nearest hundredth of a gram.

Type: Problem-Solving Task

Solving this problem with algebra requires factoring a particular cubic equation (the difference of two cubes) as well as a quadratic equation. An alternative solution using prime numbers and arithmetic is presented.

Type: Problem-Solving Task

This classroom task is meant to elicit a variety of different methods of solving a quadratic equation (A-REI.4). Some are straightforward (for example, expanding the square on the right and rearranging the equation so that we can use the quadratic formula); some are simple but clever (reasoning from the fact that x and (2x - 9) have the same square); some use tools (using a graphing calculator to graph the functions f(x) = x^2 and g(x) = (2x-90)^2 and looking for values of x at which the two functions intersect). Some solution methods will work on an arbitrary quadratic equation, while others (such as the last three) may have difficulty or fail if the quadratic equation is not given in a particular form, or if the solutions are not rational numbers.

Type: Problem-Solving Task

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. The main work in parts (a) and (b) is in setting up the equation(s) appropriately. Question (c) is more subtle and it requires thinking carefully about the accuracy available in a particular measurement (weight). The first two parts of this task could be used for instructional or assessment purposes while the third part should strictly be implemented for instructional purposes.

Type: Problem-Solving Task

The purpose of this task is to provide an opportunity for students to reason about equivalence of equations. The instruction to give reasons that do not depend on solving the equation is intended to focus attention on the transformation of equations as a deductive step.

Type: Problem-Solving Task

This problem-solving task challenges students to find all quadratic functions described by given equation and coordinates, and describe how the graphs of those functions are related to one another.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

This task addresses a common misconception about function notation.

Type: Problem-Solving Task

This task could be used for assessment or for practice. It allows 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, students are asked to determine which function has the greatest maximum and the greatest non-negative root.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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

Type: Problem-Solving Task

This task asks students to find the average, write an equation, find the domain, and create a graph of the cost of producing DVDs.

Type: Problem-Solving Task

The problem presents a context where a quadratic function arises. Careful analysis, including graphing of the function, is closely related to the context. The student will gain valuable experience applying the quadratic formula and the exercise also gives a possible implementation of completing the square.

Type: Problem-Solving Task

The goal of this task is to give students a context to investigate large numbers and measurements. Students need to fluently convert units with very large numbers in order to successfully complete this task. The total number of pennies minted either in a single year or for the last century is phenomenally large and difficult to grasp. One way to assess how large this number is would be to consider how far all of these pennies would reach if we were able to stack them one on top of another: this is another phenomenally large number but just how large may well come as a surprise.

Type: Problem-Solving Task

In this task students investigate and ultimately prove the validity of the method of generating Pythagorean Triples that involves the polynomial identity (x^{2}+y^{2})^{2}=(x^{2}−y^{2})^{2}+(2xy)^{2}.

Type: Problem-Solving Task

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^{°}.

Type: Problem-Solving Task

In this task, students are asked to determine the unit price of a product under two different circumstances. They are also asked to generalize the cost of producing *x* items in each case.

Type: Problem-Solving Task

This task does address some aspects of modeling as described in Florida Standard for Mathematical Practice 4. Also, students often think that time must always be the independent variable and so may need some help understanding that one chooses the independent and dependent variable based on the way one wants to view a situation.

Type: Problem-Solving Task

The principal purpose of the task is to explore a real-world application problem with algebra, working with units and maintaining reasonable levels of accuracy throughout. Students are asked to determine which product will be the most economical to meet the requirements given in the problem.

Type: Problem-Solving Task

This task asks students to consider functions in regard to temperatures in a high school gym.

Type: Problem-Solving Task

The purpose of this task is to introduce or reinforce the concept of a function, especially in a context where the function is not given by an explicit algebraic representation. Further, the last part of the task emphasizes the significance of one variable being a function of another variable in an immediately relevant real-life context.

Type: Problem-Solving Task

In this task students are given graphs of quantities related to weather. The purpose of the task is to show that graphs are more than a collection of coordinate points; they can tell a story about the variables that are involved, and together they can paint a very complete picture of a situation, in this case the weather. Features in one graph, like maximum and minimum points, correspond to features in another graph. For example, on a rainy day, the solar radiation is very low, and the cumulative rainfall graph is increasing with a large slope.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

This problem is a simple de-contextualized version of F-IF Your Father and F-IF Parking Lot. It also provides a natural context where the absolute value function arises as, in part (b), solving for x in terms of y means taking the square root of x^2 which is |x|.This task assumes students have an understanding of the relationship between functions and equations.

Type: Problem-Solving Task

This deceptively simple task asks students to find the domain and range of a function from a given context. The function is linear and if simply looked at from a formulaic point of view, students might find the formula for the line and say that the domain and range are all real numbers.

Type: Problem-Solving Task

This problem introduces a logistic growth model in the concrete settings of estimating the population of the U.S. The model gives a surprisingly accurate estimate and this should be contrasted with linear and exponential models.

Type: Problem-Solving Task

This task is for instructional purposes only and students should already be familiar with some specific examples of logistic growth functions. The goal of this task is to have students appreciate how different constants influence the shape of a graph.

Type: Problem-Solving Task

This task can be used as a quick assessment to see if students can make sense of a graph in the context of a real world situation. Students also have to pay attention to the scale on the vertical axis to find the correct match. The first and third graphs look very similar at first glance, but the function values are very different since the scales on the vertical axes are very different. The task could also be used to generate a group discussion on interpreting functions given by graphs.

Type: Problem-Solving Task

In this task, students will use inverse operations to solve the equations for the unknown variable or for the designated variable if there is more than one.

Type: Problem-Solving Task

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 (i.e., "constraining" said values). In particular, it is worth differentiating the role of constraint equations from more functional equations, e.g., formulas to convert from degrees Celsius to degree Fahrenheit. The task has students interpret the context and choose variables to represent the quantities, which are governed by the constraint equation and the fact that they are non-negative (allowing us to restrict the graphs to points in the first quadrant only).

The four parts are independent and can be used as separate tasks.

Type: Problem-Solving Task

The purpose of this task is to help students learn to read information about a function from its graph, by asking them to show the part of the graph that exhibits a certain property of the function. The task could be used to further instruction on understanding functions or as an assessment tool, with the caveat that it requires some amount of creativity to decide how to best illustrate some of the statements.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Students are asked to write and solve an inequality to determine the number of people that can safely rent a boat.

Type: Problem-Solving Task

The student is asked to write and solve an inequality to match the context.

Type: Problem-Solving Task

The purpose of this task is meant to reinforce students' understanding of rational numbers as points on the number line and to provide them with a visual way of understanding that the sum of a number and its additive inverse (usually called its "opposite") is zero.

Type: Problem-Solving Task

In this task, students answer a question about the difference between two temperatures that are negative numbers.

Type: Problem-Solving Task

The purpose of this task is to help solidify students' understanding of signed numbers as points on a number line and to understand the geometric interpretation of adding and subtracting signed numbers. There is a subtle distinction in the Florida Standards between a fraction and a rational number. Fractions are always positive, and when thinking of the symbol ab as a fraction, it is possible to interpret it as a equal-sized pieces where b pieces make one whole.

Type: Problem-Solving Task

The student is asked to complete a long division which results in a repeating decimal, and then use multiplication to "check" their answer. The purpose of the task is to have students reflect on the meaning of repeating decimal representation through approximation.

Type: Problem-Solving Task

Students are given a word problem that can be solved by using a pair of linear equations. 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.

Type: Problem-Solving Task

This task examines the ways in which the plane can be covered by regular polygons in a very strict arrangement called a regular tessellation. These tessellations are studied here using algebra, which enters the picture via the formula for the measure of the interior angles of a regular polygon (which should therefore be introduced or reviewed before beginning the task). The goal of the task is to use algebra in order to understand which tessellations of the plane with regular polygons are possible.

Type: Problem-Solving Task

The purpose of this task is to give students an opportunity to solve a multi-step ratio problem that can be approached in many ways. This can be done by making a table, which helps illustrate the pattern of taxi rates for different distances traveled and with a little persistence leads to a solution which uses arithmetic. It is also possible to calculate a unit rate (dollars per mile) and use this to find the distance directly without making a table.

Type: Problem-Solving Task

This task requires students to work with very large and small values expressed both in scientific notation and in decimal notation (standard form). In addition, students need to convert units of mass. The solution below converts the mass of humans into grams; however, we could just as easily converted the mass of ants into kilograms. Students are unable to go directly to a calculator without taking into account all of the considerations mentioned above. Even after converting units and decimals to scientific notation, students should be encouraged to use the structure of scientific notation to regroup the products by extending the properties of operations and then use the properties of exponents to more fluently perform the calculations involved rather than rely heavily on a calculator.

Type: Problem-Solving Task

The purpose of this task is to show how the ideas in the RP and EE domains in 6th and 7th grade mature in 8th grade. Parts (a)-(c) could easily be asked of 7th grade students. Part (a) asks students to do what is described in 7.RP.2.a, Part (b) asks students to do what is described in 7.RP.2.c, and Part (c) is the 7th grade extension of the work students do in MAFS.6.EE.3.9.

On the other hand, part (d) is 8th grade work. It is true that in 7th grade, "Students graph proportional relationships and understand the unit rate informally as a measure of the steepness of the related line, called the slope". However, in 8th grade students are ready to treat slopes more formally: 8.EE.5 says students should "graph proportional relationships, interpreting the unit rate as the slope of the graph" which is what they are asked to do in part (d).

Type: Problem-Solving Task

This task provides an approximation, and definition, of *e*, in the context of more and more frequent compounding of interest in a bank account. The approach is computational.

Type: Problem-Solving Task

This task aims for students to understand the quadratic formula in a geometric way in terms of the graph of a quadratic function.

Type: Problem-Solving Task

This task is intended for instruction and to motivate "Building a general quadratic function." This task assumes that the students are familiar with the process of completing the square.

Type: Problem-Solving Task

This task asks the student to graph and compare two proportional relationships and interpret the unit rate as the slope of the graph.

Type: Problem-Solving Task

In this example, students will answer questions about unit price of coffee, make a graph of the information, and explain the meaning of slope in the given context.

Type: Problem-Solving Task

In this example, students use properties of rational exponents and other algebraic concepts to compare and verify the relative size of two real numbers that involve decimal exponents.

Type: Problem-Solving Task

In this example, students are given the graph of two functions and are asked to sketch the graph of the function that is their sum. The intent is that students develop a conceptual understanding of function addition.

Type: Problem-Solving Task

This task could be put to good use in an instructional sequence designed to develop knowledge related to students' understanding of linear functions in contexts. Though students could work independently on the task, collaboration with peers is more likely to result in the exploration of a range of interpretations.

Type: Problem-Solving Task

The primary purpose of this task is to elicit common misconceptions that arise when students try to model situations with linear functions. This task, being multiple choice, could also serve as a quick assessment to gauge a class' understanding of modeling with linear functions.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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

At a higher level, the task addresses MAFS.912.N-Q.1.3, since realistically neither the car nor the bus is going to travel at exactly the same speed from beginning to end of each segment; there is time traveling through traffic in cities, and even on the autobahn the speed is not constant. Thus students must make judgments about the level of accuracy with which to report the result.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Type: Problem-Solving Task

The problem requires students to not only convert miles to kilometers and gallons to liters but they also have to deal with the added complication of finding the reciprocal at some point.

Type: Problem-Solving Task

This task asks students to calculate the cost of materials to make a penny, utilizing rates of grams of copper.

Type: Problem-Solving Task

There are many different ways to write exponential expressions that describe the same quantity, in this task the amount of a radioactive substance after *t* years. Depending on what aspect of the context we need to investigate, one expression of the quantity may be more useful than another. This task contrasts the usefulness of four equivalent expressions. Students first have to confirm that the given expressions for the radioactive substance are equivalent. Then they have to explain the significance of each expression in the context of the situation.

Type: Problem-Solving Task

Students are asked to use units to determine if the given statement is valid.

Type: Problem-Solving Task

This is a challenging task, suitable for extended work, and reaching into a deep understanding of units. Students are given a scenario and asked to determine the number of people required to complete the amount of work in the time described. The task requires students to exhibit MAFS.K12.MP.1.1, Make sense of problems and persevere in solving them. An algebraic solution is possible but complicated; a numerical solution is both simpler and more sophisticated, requiring skilled use of units and quantitative reasoning. Thus the task aligns with either MAFS.912.A-CED.1.1 or MAFS.912.N-Q.1.1, depending on the approach.

Type: Problem-Solving Task

This task is intended for instructional purposes so that students can become familiar and confident with using a calculator and understanding what it can and cannot do. This task gives an opportunity to work on the notion of place value (in parts [b] and [c]) and also to understand part of an argument for why the square root of 2 is not a rational number.

Type: Problem-Solving Task

Students manipulate a given equation to find specified information.

Type: Problem-Solving Task

Students solve problems tracking the balance of a checking account used only to pay 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.

Type: Problem-Solving Task

Students extrapolate the list price of a car given a total amount paid in states with different tax rates. 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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

In this resource, students refer to given information which defines 5 variables in the context of real world government expenses. They are then asked to write equations based upon specific known values for some of the variables. The emphasis is on setting up, rather than solving, the equations.

Type: Problem-Solving Task

This task asks students to write expressions for various problems involving distance per units of volume.

Type: Problem-Solving Task

The context of this example is the spread of a flu virus on campus and the related sale of tissue boxes sold. Students interpret the composite function and determine values simply by using the tables of values.

Type: Problem-Solving Task

In this example, fuel efficiency of a car can be analyzed by using rational expressions and operations with rational expressions.

Type: Problem-Solving Task

In this resource, a method of deriving the quadratic formula from a theoretical standpoint is demonstrated. This task is for instructional purposes only and builds on "Building an explicit quadratic function."

Type: Problem-Solving Task

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

Type: Problem-Solving Task

By definition, the square root of a number *n* is the number you square to get *n*. The purpose of this task is to have students use the meaning of a square root to find a decimal approximation of a square root of a non-square integer. Students may need guidance in thinking about how to approach the task.

Type: Problem-Solving Task

This task compares the usefulness of different forms of a quadratic expression. Students have to choose which form most easily provides information about the maximum value, the zeros and the vertical intercept of a quadratic expression in the context of a real world situation. Rather than just manipulating one form into the other, students can make sense out of the structure of the expressions.

(From *Algebra: Form and Function*, McCallum et al., Wiley 2010)

Type: Problem-Solving Task

The purpose of this task is to help students see manipulation of expressions as an activity undertaken for a purpose.

Variation 1 of this task presents a related more complex expression already in the correct form to answer the question.

The expression arises in physics as the reciprocal of the combined resistance of two resistors in parallel. However, the context is not explicitly considered here.

Type: Problem-Solving Task

MAFS.8.NS.1.1 requires students to "convert a decimal expansion which repeats eventually into a rational number." Despite this choice of wording, the numbers in this task are rational numbers regardless of choice of representation. For example, 0.333¯ and 13 are two different ways of representing the same number.

Type: Problem-Solving Task

This task would be especially well-suited for instructional purposes. Students will benefit from a class discussion about the slope, y-intercept, x-intercept, and implications of the restricted domain for interpreting more precisely what the equation is modeling.

Type: Problem-Solving Task

This task illustrates the process of rearranging the terms of an expression to reveal different aspects about the quantity it represents, precisely the language being used in standard MAFS.912.A-SSE.2.3. Students are provided with an expression giving the temperature of a container at a time *t*, and have to use simple inequalities (e.g., that 2^{t}>0 for all *t*) to reduce the complexity of an expression to a form where bounds on the temperature of a container of ice cream are made apparent.

Type: Problem-Solving Task

The task assumes that students are able to express a given repeating decimal as a fraction. Teachers looking for a task to fill in this background knowledge could consider the related task "8.NS Converting Decimal Representations of Rational Numbers to Fraction Representations."

Type: Problem-Solving Task

When students plot irrational numbers on the number line, it helps reinforce the idea that they fit into a number system that includes the more familiar integer and rational numbers. This is a good time for teachers to start using the term "real number line" to emphasize the fact that the number system represented by the number line is the real numbers. When students begin to study complex numbers in high school, they will encounter numbers that are not on the real number line (and are, in fact, on a "number plane"). This task could be used for assessment, or if elaborated a bit, could be used in an instructional setting.

Type: Problem-Solving Task

Students explore and manipulate expressions based on the following statement:

A function f defined for -a < x < a is even if f(-x)=f(x) and is odd if f(-x)=-f(x) when -a < x < a. In this task we assume f is defined on such an interval, which might be the full real line (i.e., a=8).

Type: Problem-Solving Task

Students compare graphs of different quadratic functions, then produce equations of their own to satisfy given conditions.

This exploration can be done in class near the beginning of a unit on graphing parabolas. Students need to be familiar with intercepts, and need to know what the vertex is. It is effective after students have graphed parabolas in vertex form (y=a(x–h)^{2}+k), but have not yet explored graphing other forms.

Type: Problem-Solving Task

This is a standard problem phrased in a non-standard way. Rather than asking students to perform an operation, expanding, it expects them to choose the operation for themselves in response to a question about structure. Students must understand the need to transform the factored form of the quadratic expression (a product of sums) into a sum of products in order to easily see *a*, the coefficient of the x^{2} term; *k*, the leading coefficient of the *x* term; and *n*, the constant term.

Type: Problem-Solving Task

Students are asked to interpret the effect on the value of an expression given a change in value of one of the variables.

Type: Problem-Solving Task

In this task, students explore some important mathematical implications of using a calculator. Specifically, they experiment with the approximation of common irrational numbers such as pi (π) and the square root of 2 (√2) and discover how to properly use the calculator for best accuracy. Other related activities involve converting fractions to decimal form and a concrete example where rounding and then multiplying does not yield the same answer as multiplying and then rounding.

Type: Problem-Solving Task

Students are given a pair of numbers. They are asked to determine which is larger, and then justify their answer. The numbers involved are rational numbers and common irrational numbers, such π and square roots. This task can be used to either build or assess initial understandings related to rational approximations of irrational numbers.

Type: Problem-Solving Task

Students examine and answer questions related to a scenario similar to a "mixture" problem involving two different mixtures of fertilizer. In this example, students determine and then compare expressions that correspond to concentrations of various mixtures. Ultimately, students generalize the problem and verify conclusions using algebraic rather than numerical expressions.

Type: Problem-Solving Task

Students are asked to interpret expressions and equations within the context of the amounts of caramels and truffles in a box of candy.

Type: Problem-Solving Task

This problem asks students to consider algebraic expressions calculating the number of floor tiles in given patterns. The purpose of this task is to give students practice in reading, analyzing, and constructing algebraic expressions, attending to the relationship between the form of an expression and the context from which it arises. The context here is intentionally thin; the point is not to provide a practical application to kitchen floors, but to give a framework that imbues the expressions with an external meaning.

Type: Problem-Solving Task

The goal of this task is to develop an understanding of rational exponents (MAFS.912.N-RN.1.1); however, it also raises important issues about distinguishing between linear and exponential behavior (MAFS.912.F-LE.1.1c) and it requires students to create an equation to model a context (MAFS.912.A-CED.1.2).

Type: Problem-Solving Task

This resource describes a simple scenario which can be represented by the use of variables. Students are asked to examine several variable expressions, interpret their meaning, and describe what quantities they each represent in the given context.

Type: Problem-Solving Task

This resource poses the question, "how many vehicles might be involved in a traffic jam 12 miles long?"

This task, while involving relatively simple arithmetic, promps students to practice modeling (MP4), work with units and conversion (N-Q.1), and develop a new unit (N-Q.2). Students will also consider the appropriate level of accuracy to use in their conclusions (N-Q.3).

Type: Problem-Solving Task

In this task students interpret the relative size of variable expressions involving two variables in the context of a real world situation. All given expressions can be interpreted as quantities that one might study when looking at two animal populations.

Type: Problem-Solving Task

This resource involves simplifying algebraic expressions that involve complex numbers and various algebraic operations.

Type: Problem-Solving Task

This task has students experiment with the operations of addition and multiplication, as they relate to the notions of rationality and irrationality.

Type: Problem-Solving Task

This task provides the opportunity for students to reason about graphs, slopes, and rates without having a scale on the axes or an equation to represent the graphs. Students who prefer to work with specific numbers can write in scales on the axes to help them get started.

Type: Problem-Solving Task

The purpose of this task is to identify the structure in the two algebraic expressions by interpreting them in terms of a geometric context. Students will have likely seen this type of process before, so the principal source of challenge in this task is to encourage a multitude and variety of approaches, both in terms of the geometric argument and in terms of the algebraic manipulation.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

This task provides students the opportunity to make use of units to find the gas needed (MAFS.912.N-Q.1.1). It also requires them to make some sensible approximations (e.g., 2.92 gallons is not a good answer to part (a)) and to recognize that Felicia's situation requires her to round up. Various answers to (a) are possible, depending on how much students think is a safe amount for Felicia to have left in the tank when she arrives at the gas station. The key point is for them to explain their choices. This task provides an opportunity for students to practice MAFS.K12.MP.2.1: Reason abstractly and quantitatively, and MAFS.K12.MP.3.1: Construct viable arguments and critique the reasoning of others.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

This problem solving task examines, in a graphical setting, the impact of adding a scalar, multiplying by a scalar, and making a linear substitution of variables on the graph of the function *f*. This resource also includes standards alignment commentary and annotated solutions.

Type: Problem-Solving Task

In this problem-solving task students are challenged to apply their understanding of linear relationships to determine the amount of chicken and steak needed for a barbecue, which will include creating an equation, sketching a graph, and interpreting both. This resource also includes standards alignment commentary and annotated solutions.

Type: Problem-Solving Task

This task asks students to reason about the relative costs per pound of two fruits without actually knowing what the costs are. Students who find this difficult may add a scale to the graph and reason about the meanings of the ordered pairs. Comparing the two approaches in a class discussion can be a profitable way to help students make sense of slope.

Type: Problem-Solving Task

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Type: Problem-Solving Task

Type: Problem-Solving Task

This problem-solving task asks students to find a linear function that models something in the real world. After finding the equation of the linear relationship between the depth of the water and the distance across the channel, students have to verbalize the meaning of the slope and intercept of the line in the context of this situation. Commentary on standards alignment and illustrated solutions are also included.

Type: Problem-Solving Task

This task asks the student to understand the relationship between slope and changes in *x*- and *y*-values of a linear function.

Type: Problem-Solving Task

This activity challenges students to recognize the relationship between slope and the difference in *x-* and *y-*values of a linear function. Help students solidify their understanding of linear functions and push them to be more fluent in their reasoning about slope and y-intercepts. This task has also produced a reasonable starting place for discussing point-slope form of a linear equation.

Type: Problem-Solving Task

The student is asked to perform operations with numbers expressed in scientific notation to decide whether 7% of Americans really do eat at Giantburger every day.

Type: Problem-Solving Task

This is an instructional task meant to generate a conversation around the meaning of negative integer exponents. While it may be unfamiliar to some students, it is good for them to learn the convention that negative time is simply any time before t=0.

Type: Problem-Solving Task

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).

Type: Problem-Solving Task

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

Type: Problem-Solving Task

This task asks students to use proportions of mass and volume to create ideal brine for saltwater fish tanks. It also asks students to compare graphs.

Type: Problem-Solving Task

This problem is a quadratic function example. The other tasks in this set illustrate MAFS.912.F.BF.1.1.a in the context of linear, exponential, and rational functions.

Type: Problem-Solving Task

The task addresses the first part of standard MAFS.912.F-BF.2.3: "Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative)."

Type: Problem-Solving Task

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. If desired, instructors could also use this task to touch on such solutions by finding and interpreting solutions to the system of equations created in parts (a) and (b).

Type: Problem-Solving Task

The purpose of this task is to introduce students to exponential growth. While the context presents a classic example of exponential growth, it approaches it from a non-standard point of view.

Type: Problem-Solving Task

The primary purpose of this task is to lead students to a numerical and graphical understanding of the behavior of a rational function near a vertical asymptote, in terms of the expression defining the function.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

Students are asked to solve an inequality in order to answer a real-world question.

Type: Problem-Solving Task

This task asks students to determine whether a the set of given functions is odd, even, or neither.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

## Teaching Idea

This resource features two pairs of interactive graphs to help students explore rate of change and linear relationships. "Users can drag a slider on an interactive graph to modify a rate of change (cost per minute for phone use) and learn how modifications in that rate affect the linear graph displaying accumulation (the total cost of calls). In this first part, Constant Cost per Minute, the cost per minute for phone use remains constant over time. In the second part, Changing Cost per Minute, the cost per minute for phone use changes after the first sixty minutes of calls." (from NCTM's Illuminations)

Type: Teaching Idea

## Tutorials

This video tutorial shows students: the standard form of a polynomial, how to identify polynomials, how to determine the degree of a polynomial, how to add and subtract polynomials, and how to represent the area of a shape as an addition or subtraction of polynomials.

Type: Tutorial

This video discusses multiplication and division of inequalities with negative numbers to solve the inequality.

Type: Tutorial

This tutorial reviews the concept of exponents and powers and includes how to evaluate powers with negative signs.

Type: Tutorial

This tutorial demonstrates how to use the power of a power property with both numerals and variables.

Type: Tutorial

This tutorial will help you understand the concept of absolute value. Take the quiz after the lesson to practice!

Type: Tutorial

The video tutorial discusses about two typical polynomial multiplications. First, squaring a binomial and second, product of a sum and difference.

Type: Tutorial

This tutorial will help the learners practice division of polynomials. Students will recognize that dividing polynomials is similar to simplifying fractions.

Type: Tutorial

This tutorial will help the learners practice multiplication of polynomials. Learners will understand that when they multiply expressions with more than two terms, they need to make sure each term in the first expression multiplies every term in the second expression.

Type: Tutorial

Binomials are the polynomials with two terms. This tutorial will help the students learn about the multiplication of binomials. In multiplication, we need to make sure that each term in the first set of parenthesis multiplies each term in the second set.

Type: Tutorial

In this tutorial, students will learn how to add and subtract polynomials functions using horizontal and vertical methods. In a horizontal format, like terms should be grouped together using the commutative property. In vertical format, terms should be listed by ascending degree with like terms placed below each other.

Type: Tutorial

In this tutorial students learn how to identify a polynomial, how to find the degree of a polynomial, and how to write a polynomial in standard format.

Type: Tutorial

This tutorial will help the learner to understand the concept of subtracting the positive and negative integers with the help of a number line. Learners can also take a quiz after the concept is internalized.

Type: Tutorial

Students will be able to see examples of addition of integers while watching a short video, and practice adding integers using an online quiz.

Type: Tutorial

This video models solving equations in one variable with variables on both sides of the equal sign.

Type: Tutorial

The video describes how to multiply fractions and state the answer in lowest terms.

Type: Tutorial

This resource is a step-by-step tutorial on how to multiply polynomials.

Type: Tutorial

## Video/Audio/Animations

Khan Academy video tutorial on graphing linear equations: "Algebra: Graphing Lines 1"

Type: Video/Audio/Animation

Khan Academy tutorial video that demonstrates with real-world data the use of Excel spreadsheet to fit a line to data and make predictions using that line.

Type: Video/Audio/Animation

This Khan Academy video tutorial introduces averages and algebra problems involving averages.

Type: Video/Audio/Animation

## Virtual Manipulatives

This resource will assess students' understanding of addition and subtraction of polynomials.

Type: Virtual Manipulative

This resource can be used to assess students' understanding of solving quadratic equation by taking the square root. A great resource to view prior to this is "Solving quadratic equations by square root' by Khan Academy.

Type: Virtual Manipulative

This virtual manipulative is intended to allow the student to practice multiplication of binomials. The student should understand how to use algebra tiles before using this tool.

Type: Virtual Manipulative

This manipulative will help students in understanding scatter plots which are particularly useful when investigating whether there is a relationship between two variables. Students could develop a systematic plan for collecting and entering data into the scatter plot manipulative and set appropriate ranges for the *x* and *y* scales.

Type: Virtual Manipulative

Using this virtual manipulative, students apply dimensional analysis (AKA factor-label method or unit-factor method) to solve unit conversion problems. There is also the opportunity to create your own unit conversion problems.

Type: Virtual Manipulative

This virtual manipulative guides the student in the use of color counters to model subtraction of integers.

Type: Virtual Manipulative

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative

This site provides a virtual balance on which the student can represent (and then solve) simple linear equations with integer answers. Conceptually, positive weights (unit-blocks and x-boxes) push the pans of the scale downward. Negative values are represented by balloons which can be attached to the pans of the scale. The student can then manipulate the weights to solve the equation while simultaneously seeing a visual display of these effects on the equation.

Type: Virtual Manipulative

Section:Grades PreK to 12 Education Courses >Grade Group:Grades 9 to 12 and Adult Education Courses >Subject:Mathematics >SubSubject:Algebra >