Cluster 1: Create equations that describe numbers or relationships. (Algebra 1 - Major Cluster) (Algebra 2 - Supporting Cluster)Archived

A variety of graph paper types for printing, including Cartesian, polar, engineering, isometric, logarithmic, hexagonal, probability, and Smith chart.

Type: Educational Software / Tool

Students are asked to write absolute value inequalities to represent the relationship among values described in word problems.

Type: Formative Assessment

Students are asked to solve a set of absolute value equations.

Type: Formative Assessment

Students are asked to solve a set of absolute value inequalities.

Type: Formative Assessment

Students are asked to solve a set of absolute value equations.

Type: Formative Assessment

Students are given the slope formula and the slope-intercept equation and are asked to solve for specific variables.

Type: Formative Assessment

Students are given three literal equations, each involving three variables and either addition or subtraction, and are asked to solve each equation for a specific variable.

Type: Formative Assessment

Students are given three literal equations, each involving three variables and either multiplication or division, and are asked to solve each equation for a specific variable.

Type: Formative Assessment

Students are asked to write a function that represents an annual loss of 3 percent each year.

Type: Formative Assessment

Students are asked to write and solve an equation that models a given problem.

Type: Formative Assessment

Students are asked to write and solve an equation that models an exponential relationship between two variables.

Type: Formative Assessment

Students are asked to represent constraints using inequalities given in a problem context.

Type: Formative Assessment

Students are asked to model a problem involving constraints using inequalities.

Type: Formative Assessment

Students are asked to sketch a graph that depicts the exponential decline in the population of fir trees in a forest.

Type: Formative Assessment

Students are asked to graph a function in two variables given in context.

Type: Formative Assessment

Students are asked to write an equation in two variables given a verbal description of the relationship among the variables.

Type: Formative Assessment

Students are asked to solve the formula for the surface area of a cube for e, the length of an edge of the cube.

Type: Formative Assessment

Students are asked to graph an equation in two variables given in context.

Type: Formative Assessment

Students are given a literal equation involving four variables and are asked to solve for the variable in the quadratic term.

Type: Formative Assessment

Students are asked to write an equation in three variables from a verbal description.

Note: This task may assess skills that exceed the general expectation for this mathematical concept at this grade level.  The task is intended for students who have demonstrated mastery within the scope of instruction who may be ready for a more rigorous extensions of the content. As with all materials, ensure to gauge the readiness of students or adapt according to students needs prior to administration.

Type: Formative Assessment

Students are asked to write an equation in two variables from a verbal description.

Type: Formative Assessment

Students are asked to determine the constraint on a profit equation and to interpret solutions as being viable or not in the context of the problem.

Type: Formative Assessment

Students are asked to write and solve an equation that models a given problem.

Type: Formative Assessment

Students are given a real world context and asked to model the situation by writing and then solving a multistep inequality.

Type: Formative Assessment

Students investigate the amount of space that could be saved by flattening cardboard boxes. The analysis includes linear graphs and regression analysis along with discussions of slope and a direct variation phenomenon.

Type: Lesson Plan

Students will create an outline of a room and write equations of the lines that contain the sides of the room. This lesson provides an opportunity to review and reinforce writing equations of lines (including horizontal and vertical lines) and to apply the relationship between the slopes of parallel and perpendicular lines.

Type: Lesson Plan

This will be a lesson designed to introduce students to the concept of 9.81 m/s² as a sort of clock that can be used for solving all kinematics equations where a = g.

Type: Lesson Plan

This lesson is on motion of objects. Students will learn what factors affect the speed of an object through experimentation with gumballs rolling down an incline. The students will collect data through experimenting, create graphs from the data, interpret the slope of the graphs and create equations of lines from data points and the graph. They will understand the relationship of speed and velocity and be able to relate the velocity formula to the slope intercept form of the equation of a line.

Type: Lesson Plan

This lesson unit is intended to help you assess how well students can formulate and solve problems using algebra and, in particular, to identify and help students who have difficulties solving a problem using two linear equations with two variables and interpreting the meaning of algebraic expressions.

Type: Lesson Plan

This lesson unit is intended to help you assess how well students are able to use the Pythagorean theorem to derive the equation of a circle and translate between the geometric features of circles and their equations.

Type: Lesson Plan

This is a great lab activity that allows students to develop a true understanding of slope as a rate of change. Students are active and involved and must use higher order thinking skills in order to answer questions. Students work through an activity, measuring heights of cups that are stacked. Students them determine a "rate of change - slope". Students are then asked to put this into slope-intercept form. The important part here is in their determining the y-intercept of the equation. Students then take this further and finally attempt to create a linear inequality to determine how many cups, stacked vertically, will fit under a table.

Type: Lesson Plan

This is a model-eliciting activity where students have been asked by a new website, CollegeReview.com, to come up with a system to rank various colleges based on five categories; tuition cost, social life, athletics, education, city population and starting salary upon graduation.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

Type: Lesson Plan

This lesson is designed to help students develop strategies for solving optimization problems. Such problems typically involve scenarios where limited resources must be used to greatest effect, as in, for example, the allocation of time and materials to maximize profit.

Type: Lesson Plan

This lesson is designed to be part of a sequence of lessons. It follows resource 52648 "BIOSCOPES Summer Institute 2013 - Forces" and precedes resource 52957 "BIOSCOPES Summer Institute 2013 - Thermal Energy." This lesson uses a predict, observe, and explain approach along with inquiry based activities to enhance student understanding of the conservation of energy.

Type: Lesson Plan

The focus of this MEA is oil spills and their effect on the environment. In this activity, students from a fictitious class are studying about the effects of an oil spill on marine ecosystems and have performed an experiment in which they were asked to try to rid a teaspoon of corn oil from a baking pan filled with two liters of water as thoroughly as possible in a limited timeframe and with limited resources. By examining, analyzing, and evaluating experimental data related to resource usage, disposal, and labor costs, students must face the tradeoffs that are involved in trying to preserve an ecosystem when time, money, and resources are limited.

Type: Lesson Plan

The Alternative Fuel Systems MEA provides students with an engineering problem in which they must develop a procedure to decide the appropriate course for an automobile manufacturer to take given a set of constraints. The main focus of the MEA is to apply the concepts of work and energy to a business model.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

Type: Lesson Plan

Students use systems of equations and inequalities to solve real world budgeting problems involving two variables.

Type: Lesson Plan

The topic of this MEA is work and power. Students will be assigned the task of hiring employees to complete a given task. In order to make a decision as to which candidates to hire, the students initially must calculate the required work. The power each potential employee is capable of, the days they are available to work, the percentage of work-shifts they have missed over the past 12 months, and the hourly pay rate each worker commands will be provided to assist in the decision process. Full- and/or part-time positions are available. Through data analysis, the students will need to evaluate which factors are most significant in the hiring process. For instance, some groups may prioritize speed of work, while others prioritize cost or availability/dependability.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

Type: Lesson Plan

In this lesson, students will apply their knowledge to model a real-world linear situation in a variety of ways. They will analyze a situation in which 2 candles burn at different rates. They will create a table of values, determine a linear equation, and graph each to determine if and when the candles will ever be the same height. They will also determine the domain and range of their functions and determine whether there are constraints on their functions.

Type: Lesson Plan

Students will explore how percentages, proportions, and solving for unknowns are used in important jobs. This interactive activity will open their minds and address the question, "When is this ever used in real life?"

Type: Lesson Plan

Piles of Paper is a student activity that demonstrates linear and exponential growth using heights of flat and folded paper. Data tables are created and then algebraic models are developed. Real world types of linear and exponential growth are also introduced.

Type: Lesson Plan

This is a short unit plan that covers position/time and velocity/time graphs. Students are provided with new material on both topics, will have practice worksheets, and group activities to develop an understanding of motion graphs.

Type: Lesson Plan

In this lesson, students learn how to use the graph of a system of linear inequalities to determine the feasible region. Students practice solving word problems to find the optimal solution that maximizes profits. Students will use the free application, GeoGebra (see download link under Suggested Technology) to help them create different graphs and to determine the feasible or non-feasible solutions.

Type: Lesson Plan

Students will work in pairs and compose three different linear equations in slope intercept form. They will discover and describe how different values for the slope and y-intercept affect the graph. After graphing lines on graph paper, they will do a physical activity involving graphing.

Type: Lesson Plan

Students will be guided through the investigation of y = mx+b. Through this lesson, students will be able to determine whether lines are parallel, perpendicular, or neither by looking at the graph and the equation.

Type: Lesson Plan

Students will research various types of amusement park rides and use their findings to design a feasible ride of their own. They will summarize their findings and present their ride design to the class. Each student will then write a persuasive letter to a local amusement park describing the reasons their ride design is the best.

Type: Lesson Plan

The purpose of this lesson is to have students practice manipulation of literal equations to solve for the variable of interest. A literal equation is an equation that has more than variable (letter).

Type: Lesson Plan

Students write and solve linear equations from real-life situations. 

Type: Lesson Plan

In this Model Eliciting Activity, MEA, students create a plan for a movie theater to stay in business. Data is provided for students to determine the best film to show, and then based on that decision, create a model of ideal sales. Students will create equations and graph them to visually represent the relationships.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

Type: Lesson Plan

Learn how to create systems of linear equations to represent contextual situations in this interactive tutorial.

This part 6 in a 7-part series. Click below to explore the other tutorials in the series. 

• Part 1: Solving Systems of Linear Equations: Using Graphs
• Part 2: Solving Systems of Linear Equations: Substitution
• Part 3: Solving Systems of Linear Equations: Basic Elimination
• Part 4: Solving Systems of Linear Equations: Advanced Elimination
• Part 5: Solving Systems of Linear Equations: Connecting Algebraic Methods to Graphing
• Part 7: Solving Systems of Linear Equations: Word Problems (Coming soon)

Type: Original Student Tutorial

Learn how to solve rational linear and quadratic equations using cross multiplication in 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

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

Type: Original Student Tutorial

<p>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!</p>

Type: Perspectives Video: Expert

<p>Angela Dial discusses how she solves systems of equations to determine how the composition&nbsp;of ocean floor sediment has changed over 65 million years to help reveal more information&nbsp;regarding&nbsp;climate change.</p>

Type: Perspectives Video: Professional/Enthusiast

What happens when math models go wrong in forecasting hurricanes?

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

<p>Have a need for speed? Get out your spreadsheet! Race car drivers use algebraic formulas and spreadsheets to optimize car performance.</p>

Type: Perspectives Video: Professional/Enthusiast

<p>Brandon Reese, a PhD candidate in the FAMU-FSU College of Engineering, discusses the significance of both Bernoulli's equation and statistical analysis for the design of a "smart wing."</p>

Type: Perspectives Video: Professional/Enthusiast

<p>A science teacher demonstrates stepwise calculations involving multiple variables for designing robots with desired characteristics.</p>

Type: Perspectives Video: Teaching Idea

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 the mathematical practice of modeling with mathematics, and crucial as the system has an integer solution for both situations, that is, whether we include the dollar on the floor in the cash box or not.

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 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

This question examines the algebraic equations for three different spheres. The intersections of each pair of spheres are then studied, both using the equations and thinking about the geometry of the spheres.

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 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

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 , 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 provides a simple but interesting and realistic context in which students are led to set up a rational equation (and a rational inequality) in one variable, and then solve that equation/inequality for an unknown variable. It seems likely to be direct and relevant enough to be used for assessment purposes, either in part or in whole. Alternatively, this task could be used as a motivation for studying equations of this form in general, as while students might be able to solve the first part by trial and error, this becomes rather tedious for the later parts. Teachers might also find this task could be used to illustrate standard A-REI.A.1 if some more emphasis were placed on the reasoning behind the algebraic manipulations provided in the solutions.

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

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 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

This learning video is designed to develop critical thinking in students by encouraging them to work from basic principals to solve a puzzling mathematics problem that contains uncertainty. One class session of approximately 55 minutes is necessary for lesson completion. First-year simple algebra is all that is required for the lesson, and any high school student in a college-preparatory math class should be able to participate in this exercise. Materials for in-class activities include: a yard stick, a meter stick or a straight branch of a tree; a saw or equivalent to cut the stick; and a blackboard or equivalent. In this video lesson, during in-class sessions between video segments, students will learn among other things: 1) how to generate random numbers; 2) how to deal with probability; and 3) how to construct and draw portions of the X-Y plane that satisfy linear inequalities.

Type: Problem-Solving Task

This site shows students how to translate word problems into equations. It gives seven steps, from reading the problem carefully to checking the solution, to creating equations. The lesson moves on to a few simple exercises in which a natural language sentence is translated to an algebraic equation. It then moves on to more elaborate word problems which require students to identify the important data and follows the given seven steps to create and solve the equation. The more complex questions draw on student understanding of geometric formulae. There are six questions at the end for students to test their new knowledge of how to create and solve equations.

Type: Teaching Idea

This video demonstrates solving a word problem by creating a system of linear equations that represents the situation and solving them using elimination.

Type: Tutorial

Students will learn to solve a literal equation. 

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

Evaluating Expressions with Two Variables

Type: Tutorial

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

Type: Tutorial

In this example of evaluating expressions, we're dusting off some geometry. On top of that, it's a word problem. We're seeing how different concepts in math are layered on top of each other to create more interesting and complex problems to solve.

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

Type: Tutorial

Type: Unit/Lesson Sequence

This session on linear function and slope contains five parts, multiple problems and videos, and interactive activities geared to help students recognize and understand linear relationships, explore slope and dependent and independent variables in graphs of linear relationships, and develop an understanding of rates and how they are related to slopes and equations. Throughout the session, students use spreadsheets to complete the work, and are encouraged to think about the ways technology can aid in teaching and understanding. The solutions for all problems are given, and many allow students to have a hint or tip as they solve. There is even a homework assignment with four problems for students after they have finished all five parts of the session.

Type: Unit/Lesson Sequence

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

The 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

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 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

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

Type: Video/Audio/Animation

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

Type: Video/Audio/Animation

This resource provides guided practice for writing and graphing linear functions.

Type: Virtual Manipulative