Students participate in an icebreaker activity, finding a classmate whose card contains an expression equivalent to the expression on their own card. The resulting student pairs will be partners for this unit. Students spend time exploring the digital course. They learn new symbols for multiplication and detect possible errors in evaluating numeric expressions. The class discusses and decides upon norms for math class.Key ConceptsStudents evaluate numerical expressions and identify equivalent expressions. They explore why the order of operations affects calculation results and how to use parentheses to clearly describe the order of the operations.Goals and Learning ObjectivesEvaluate numerical expressions.Understand the reason for the order of operations and how to use parentheses in numerical expressions.Use the basic features of the application.Create and understand the classroom norms.Use mathematical reasoning to justify an answer.PreparationPrint out the Expressions Icebreaker cards. Select the number of pairs of Partner 1 and Partner 2 cards needed for your class. Shuffle the cards before distributing to students.Write on the board or chart paper: Find a classmate whose card has an expression that is equivalent to the expression on your card.Choose a hand signal or phrase for common activities, such as putting technology away and focusing on the teacher.

Students are introduced to classroom routines and expectations, and complete a full mathematics lesson. The class discusses how to clearly present work to classmates. Partner work is modeled, and partners then work to match numerical expressions to corresponding word descriptions. Students read and discuss a summary of the math in the lesson, and then write a reflection about their thoughts.Key ConceptsStudents match a numerical expression to its corresponding description in words. Students interpret parentheses and brackets in numerical expressions and they construct viable arguments and critique the reasoning of others. Students learn to use the exponent 2 to represent squaring.Goals and Learning ObjectivesDescribe the classroom routines and expectations.Consider how to present work clearly to classmates.Collaborate with a partner.Critique a partner’s reasoning.Connect a numerical expression to its corresponding word description.Learn to use an exponent of 2 to represent squaring.

Students are introduced to classroom routines and expectations, and complete a full mathematics lesson. The class discusses how to clearly present work to classmates. Partner work is modeled, and partners then work to match numerical expressions to corresponding word descriptions. Students read and discuss a summary of the math in the lesson, and then write a reflection about their thoughts.Key ConceptsStudents match a numerical expression to its corresponding description in words. Students interpret parentheses and brackets in numerical expressions and they construct viable arguments and critique the reasoning of others. Students learn to use the exponent 2 to represent squaring.Goals and Learning ObjectivesDescribe the classroom routines and expectations.Consider how to present work clearly to classmates.Collaborate with a partner.Critique a partner’s reasoning.Connect a numerical expression to its corresponding word description.Learn to use an exponent of 2 to represent squaring.

The class reviews the properties of operations. The use of “ask myself” questions to make sense of problems and persevere is modeled. Students review things to do when they feel stuck on a problem. Finally, students use the properties of operations to evaluate expressions.Key ConceptsStudents use the properties of operations to justify whether two expressions are equivalent.Goals and Learning ObjectivesTo start to work on a problem, make sense of the problem by using “ask myself” questions.Persevere in solving a problem even when feeling stuck.Use the properties of operations to evaluate expressions.

Students discuss as a class the important ways that listeners contribute to mathematical discussions during Ways of Thinking presentations. Students then use the properties of operations to find the value of each fruit used in equations.Key ConceptsStudents use the properties of operations to find the value of each fruit used in different equations. By considering several equations, students can match each of the 10 fruits to the whole numbers 0 through 9. This work helps students see why representing unknown numbers with letters is useful.Goals and Learning ObjectivesContribute as listeners during the Ways of Thinking discussion.Identify the whole numbers that make an equation true.Use the properties of operations, when appropriate, to justify which whole numbers represent unknown values.

Students work in a whole-class setting, independently, and with partners to design and implement a problem-solving plan based on the mathematical concepts of rates and multiple representations (e.g., tables, equations, and graphs). They analyze a rule of thumb and use this relationship to calculate the distance in miles from a viewer's vantage point to lightning.Key ConceptsThroughout this unit, students are encouraged to apply the mathematical concepts they have learned over the course of this year to new settings. Help students develop and refine these problem-solving skills:Creating a problem-solving plan and implementing the plan systematicallyPersevering through challenging problems to find solutionsRecalling prior knowledge and applying that knowledge to new situationsMaking connections between previous learning and real-world problemsCommunicating their approaches with precision and articulating why their strategies and solutions are reasonableCreating efficacy and confidence in solving challenging problems in the real worldGoals and Learning ObjectivesCreate and implement a problem-solving plan.Organize and interpret data presented in a problem situation.Analyze the relationship between two variables.Create a rate table to organize data and make predictions.Apply the relationship between the variables to write a mathematical formula and use the formula to solve problems.Create a graph to display proportional relationships, and use this graph to make predictions.Articulate strategies, thought processes, and approaches to solving a problem, and defend why the solution is reasonable.

Rate

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Solve problems involving all four operations with rational numbers.
Understand quantity as a number used with a unit of measurement.
Solve problems involving quantities such as distances, intervals of time, liquid volumes, masses of objects, and money, and with the units of measurement for these quantities.
Understand that a ratio is a comparison of two quantities.
Write ratios for problem situations.
Make and interpret tables, graphs, and diagrams.
Write and solve equations to represent problem situations.

Lesson Flow

In this unit, students will explore the concept of rate in a variety of contexts: beats per minute, unit prices, fuel efficiency of a car, population density, speed, and conversion factors. Students will write and refine their own definition for rate and then use it to recognize rates in different situations. Students will learn that every rate is paired with an inverse rate that is a measure of the same relationship. Students will figure out the logic of how units are used with rates. Then students will represent quantitative relationships involving rates, using tables, graphs, double number lines, and formulas, and they will see how to create one such representation when given another.

In this lesson, students define rate. After coming up with a preliminary definition on their own, students identify situations that describe rates and situations that do not.Students determine what is common among rate situations and then revise their definitions of rate based on these observations. Students present and discuss their work and together create a class definition. They compare the class definition of rate with the Glossary definition and revise the class definition as needed.Key ConceptsA good definition of rate has to be precise, yet general enough to be useful in a variety of situations. For example, the statement “a rate compares two quantities” is true, but it is so general that it is not helpful. The statement “speed is a rate” is true, but it is not useful in determining whether unit price or population density are rates.A good definition of rate needs to state that a rate is a single quantity, expressed with a unit of the form A per B, and derived from a comparison by division of two measures of a single situation.Goals and Learning ObjectivesGain a deeper understanding of rate by developing, refining, testing, and then refining again a definition of rate.Use a definition of rate to determine the kinds of situations that are rate situations and to recognize rates in new and different situations.Understand the importance of precision in communicating mathematical concepts.

Gallery OverviewAllow students who have a clear understanding of the content thus far in the unit to work on Gallery problems of their choosing. You can then use this time to provide additional help to students who need review of the unit's concepts or to assist students who may have fallen behind on work.Gallery DescriptionsCreate Your Own RateStudents create their own rate problems, given three quantities that must all be used in the problems or the answers.Paper Clip ChallengeStudents think about rate in the context of setting a record for making a paperclip chain.The Speed of Light Students must determine the speed of light so they can figure out how long it will take a light beam from Earth to reach the Moon (assuming it would make it there). They conduct research and perform calculations.Tire WeightStudents connect area and a rate they may not be familiar with, tire pressure, to indirectly weigh a car. They find and add areas and do a simple rate calculation. Please note this problem requires adult supervision for the process of measuring the car tires. If no adult supervision is available, you can provide students with measurements to work with inside the classroom. Do not allow students to work with a car without permission from the owner and adult supervision.Planting Wildflowers Students apply area and length concepts (square miles, acres, and feet) to rectangles, choose and carry out appropriate area conversions, and show each step of their solutions. While specific solution paths will vary, all students who show good conceptualization will make at least one area conversion and show understanding about area even when dimensions and units change. This task allows several different correct solution paths.Train Track Students use information about laying railroad ties for the Union Pacific Railroad. These rates are different from those used elsewhere in the unit, asking how many rails per gang of workers, how long it takes to lay one mile of track, and how many spikes are needed for a mile of track.HeartbeatsStudents will investigate and compare the heartbeats of different animals and their own heartbeat.FoghornStudents use the relationships among seconds, minutes, and hours to find equivalent rates. Each step requires students to express an equivalent rate in terms of these different units of time. In any strong response, students use conversion factors and the given rate to find equivalent rates.

In this lesson, students represent quantitative relationships involving rates using tables, graphs, double number lines, and formulas. Students will understand how to create one such representation when given another representation.Key ConceptsQuantitative relationships involving rates can be represented using tables, graphs, double number lines, and formulas. One such representation can be used to create another representation. Two rates can describe each situation: the rate and its inverse. For the water pump situation, there are two related formulas: a formula for finding the quantity of water pumped for any amount of time, and a formula for finding the amount of time for any quantity of water.Goals and Learning ObjectivesUnderstand that tables, graphs, double number lines, and formulas can be used to represent the same situation.Compare the different representations within a situation and the same representation across similar situations.Understand each representation and how to find the rate in each one.

In this lesson, students use their knowledge of rates, graphs of rates, and formulas to solve problems.Key ConceptsThe formula for a rate is a mathematical way of writing a rule for computing a value. Rate formulas describe a constant relationship between two quantities. Each point on the graph of a rate shows a pair of related values. A graph of a constant rate is a straight line.Goals for Learning ObjectivesUncover any partial understandings and misconceptions students have about rate, graphs of rates, and formulas.Develop a more robust understanding of rate.Help identify which Gallery problems students should work on.

In this lesson, students write formulas to represent different rate relationships.Key ConceptsA formula is a mathematical way of writing a rule for computing a value.Formulas, like c = 2.50w or d = 20g, describe the relationship between quantities.The formula c = 2.50w describes the relationship between a cost and a quantity that costs $2.50 per unit of weight. Here, w stands for any weight, and c stands for the cost of w pounds at $2.50 per pound.The formula d = 20g describes the relationship between the distance, d, and the number of gallons of gas, g, for a car that gets 20 miles per gallon.Goals and Learning ObjectivesUse equations with two variables to express relationships between quantities that vary together.

In this lesson, students first watch three racers racing against each other. The race is shown on a track and represented on a graph. Students then change the speed, distance, and time to create a race with different results. They graph the new race and compare their graph to the original race graph.Key ConceptsA rate situation can be represented by a graph. Each point on a graph represents a pair of values. In today's situation, each point represents an amount of time and the distance a racer traveled in that amount of time. Time is usually plotted on the horizontal axis. The farther right a point is from the origin, the more time has passed from the start. Distance is usually plotted on the vertical axis. The higher up a point is from the origin, the farther the snail has traveled from the start. A graph of a constant speed is a straight line. Steeper lines show faster speeds.Goals and Learning ObjectivesUnderstand that a graph can be a visual representation of an actual rate situation.Plot pairs of related values on a graph.Use graphs to develop an understanding of rates.

In this lesson, students watch a video of a runner and express his speed as a rate in meters per second. Students then use the rate to determine how long it takes the runner to go any distance.Key ConceptsSpeed is a rate that is expressed as distance traveled per unit of time. Miles per hour, laps per minute, and meters per second are all examples of units for speed. The measures of speed, distance, and time are all related. The relationship can be expressed in three ways: d = rt, r = dt, t = dr.Goals and Learning ObjectivesExplore speed as a rate that measures the relationship between two aspects of a situation: distance and time.In comparing distance, speed, and time, understand how to use any two of these measures to find the third measure.

Students use their knowledge of rates to solve problems.Key ConceptsGiven any two values in a rate situation, you can find the third value.These three equations are equivalent, and they all describe rate relationships:y = rx,  r = yx,  x = yrAt the beginning of this lesson (or for homework), students will revise their work on the pre-assessment Self Check. Their revised work will provide data that you and your students can use to reassess students' understanding of rate. You can use this information to clear up any remaining misconceptions and to help students integrate their learning from the past several days into a deeper and more coherent whole.The work students do in this lesson and in revising their pre-assessments will help you and your students decide how to help them during the Gallery. In this lesson, students will reveal the depth and clarity of their understanding of rate.Students whose understanding of rate is still delicate should get extra help during the Gallery.Students who feel that they have a robust understanding of rate may choose from any of the problem-solving or deeper mathematics problems in the Gallery.Goals and Learning ObjectivesUncover any partial understandings and misconceptions about rate.Develop a more robust understanding of rate.Identify which Gallery problems to work on.

Students watch a video in which two students discuss the problem of how to compare fuel efficiency. Students then analyze the work of the two students as they use rates to determine fuel efficiency in two different ways.Key ConceptsFuel efficiency is a rate. Fuel efficiency can be expressed in miles per gallon (mpg). This rate is useful for determining how far a vehicle can travel using any number of gallons of gas. Fuel efficiency can also be expressed in gallons per mile (gpm). This rate is useful for determining how many gallons of gas a vehicle uses to travel any number of miles.The rates miles per gallon and gallons per mile are inverse rates—they both describe the same relationship. For example, the rates 20 miles per gallon and 0.05 gallon per mile both describe the relationship between 300 miles and 15 gallons. The greater the rate in miles per gallon, the better the fuel efficiency. The smaller the rate in gallons per mile, the better the fuel efficiency.SWD: Some students with disabilities will benefit from a preview of the goals in each lesson. This will help to highlight for students the critical features and/or concepts and will help them to pay close attention to salient information.Goals and Learning ObjectivesExplore rate in the context of fuel efficiency.Express fuel efficiency as the rate miles per gallon (mpg) and as its inverse, gallons per mile (gpm).Use the rate miles per gallon to find the number of miles a car can travel on a number of gallons of gas.Use the rate gallons per mile to find the number of gallons of gas used for a number of miles driven.

Algebraic Reasoning

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Add, subtract, multiply, and divide rational numbers.
Evaluate expressions for a value of a variable.
Use the distributive property to generate equivalent expressions including combining like terms.
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true?
Write and solve equations of the form x+p=q and px=q for cases in which p, q, and x are non-negative rational numbers.
Understand and graph solutions to inequalities x<c or x>c.
Use equations, tables, and graphs to represent the relationship between two variables.
Relate fractions, decimals, and percents.
Solve percent problems included those involving percent of increase or percent of decrease.

Lesson Flow

This unit covers all of the Common Core State Standards for Expressions and Equations in Grade 7. Students extend what they learned in Grade 6 about evaluating expressions and using properties to write equivalent expressions. They write, evaluate, and simplify expressions that now contain both positive and negative rational numbers. They write algebraic expressions for problem situations and discuss how different equivalent expressions can be used to represent different ways of solving the same problem. They make connections between various forms of rational numbers. Students apply what they learned in Grade 6 about solving equations such as x+2=6 or 3x=12 to solving equations such as 3x+6=12 and 3(x−2)=12. Students solve these equations using formal algebraic methods. The numbers in these equations can now be rational numbers. They use estimation and mental math to estimate solutions. They learn how solving linear inequalities differs from solving linear equations and then they solve and graph linear inequalities such as −3x+4<12. Students use inequalities to solve real-world problems, solving the problem first by arithmetic and then by writing and solving an inequality. They see that the solution of the algebraic inequality may differ from the solution to the problem.

Students use algebraic expressions and equations to represent rules of thumb involving measurement. They use properties of operations and the relationships between fractions, decimals, and percents to write equivalent expressions.Key ConceptsExpressions and equations are different. An expression is a number, a variable, or a combination of numbers and variables. Some examples of expressions are:74x5a + b3(2m + 1)In Grade 7, the focus is on linear expressions. A linear expression is a sum of terms that are either rational numbers or a rational number times a variable (with an exponent of either 0 or 1). If an expression contains a variable, it is called an algebraic expression. To evaluate an expression, each variable is replaced with a given value.Equivalent expressions are expressions for which a given value can be substituted for each variable and the value of the expressions are the same.An equation is a statement that two expressions are equal. An equation can be true or false. To solve an equation, students find the value of the variable that makes the equation true.Students solve an equation that involves finding 10% of a number. They see that finding 10% of the number is the same as finding 0.1 of the number, or finding 110 of the number.Goals and Learning ObjectivesWrite expressions and equations to represent real-world situations.Evaluate expressions for given values of a variable.Use properties of operations to write equivalent expressions.Solve one-step equations.Check the solution to an equation.

Students explore the effects of wind on a plane's time and distance and represent these situations using algebraic expressions and equations. They use terms with positive, negative, and zero coefficients.Key ConceptsIn this lesson, students show what they remember from Grade 6 about writing expressions and solving one-step equations. They use what they learned earlier in Grade 7 about adding and subtracting integers. They extend these concepts to write and interpret an expression with a negative coefficient.Goals and Learning ObjectivesReview addition and subtraction of integers.Review the relationship between distance, time, and speed.Write an algebraic expression for distance in terms of time, t.Write a term with a negative coefficient.Review solving a one-step equation using the multiplication property of equality.

Students write and solve inequalities in order to solve two problems. One of the problems is a real-world problem that involves selling a house and paying the real estate agent a commission. The second problem involves the relationship of the lengths of the sides of a triangle.Key ConceptsIn this lesson, students again use algebraic inequalities to solve word problems, including real-world situations. Students represent a quantity with a variable, write an inequality to solve the problem, use the properties of inequality to solve the inequality, express the solution in words, and make sure that the solution makes sense.Students explore the relationships of the lengths of the sides of a triangle. They apply the knowledge that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side to solve for the lengths of sides of a triangle using inequalities. They solve the inequality for the length of the third side.Goals and Learning ObjectivesUse an algebraic inequality to solve problems, including real-world problems.Use the properties of inequalities to solve an inequality.