Goals and objectives Topic overview Resources Print resources …pvhs.fms.k12.nm.us/teachers/malonzo/08EF690A-00757… · · 2015-07-26Goals and objectives Topic overview Resources

Advice for Instruction | 3. Foundations of algebra | Prepare instruction

Goals and objectives Topic overview Resources Print resources

Goals and objectives

In the topic Foundations of algebra, studentsinvestigate the use of variables to represent unknownsand to generalize relationships. They also reviewimportant graphing skills.

In this topic, students will:

Use variables to represent unknownsCreate, interpret, and evaluate algebraicexpressionsUse the distributive propertyUse variables to generalize input-output relationshipsPlot points on a graphMake graphs from tablesInterpret the meaning of points on a graph

The Staying Sharp problems in Topic 3 are organized around several key ideas:

Problems 1 and 2 (Practicing algebra skills and concepts): Finding numbers that fit twoconditions (in a “square box problem” structure) and finding (or using) a function rule for aninput-output table

1.

Problems 3 and 4 (Preparing for upcoming lessons): Continuing sequences and translatingamong multiple representations

2.

Problems 5 and 6 (Focus skill: Operations with signed numbers): Practicing operations onsigned integers, articulating the rules for them, and explaining misconceptions in theirapplication

3.

Topic overview

Lesson3.1:

Students use variables and expressions to generalize arithmetic processes. Theyexplore the commutative and associative properties of both addition andmultiplication.

Lesson3.2:

Students review the distributive property of addition over multiplication. Theyinvestigate this property by looking at magic number puzzles as well as an arearepresentation of the property.

Agile Mind http://tng.agilemind.com/LMS/com.agilemind.common.Print/Print.html

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Lesson3.3:

Students continue to explore ideas related to variables and algebraic expressions.They evaluate and describe the practical meaning of expressions in context.

Lesson3.4:

Students create and evaluate algebraic expressions as they investigate areas andperimeters of rectangular regions.

Lesson3.5:

Students investigate input-output relationships. They generate algebraic rules todescribe these relationships and use input-output tables and rules to solvereal-world problems.

Lesson3.6:

Students use variables to describe input-output relationships. They state input-output rules in various ways and apply rules to problem situations.

Lesson3.7:

Students review graphing skills.

Lesson3.8:

Students review their earlier work with expressions and with signed numbers. Theyalso complete an online assessment to gauge their understanding of important ideasin the topic.

Lesson3.9:

Students learn about creating accurate graphs and conduct critical analyses ofgraphs to identify and correct common graphing errors.

Lesson3.10:

Students take the end-of-unit assessment to demonstrate their understanding of keyconcepts and skills in the unit. Then they use what they have learned about scalinggraphs to adjust the viewing window of a graphing calculator to view a set of data.

Resources

Computer with projection device and Internet connectionGraphing calculatorsWhiteboards, dry erase markers, and erasersPocket folders, one for each pair of students

Print resources

Unit 1 Topic 3 Student Activity Book (updated 4/29/14)Unit 1 Topic 3 Student Activity Book answer key (updated 4/29/14)Card sort (Lesson 3.4)The search is on! (Lesson 3.7)Expressions game (Lesson 3.8)


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Advice for Instruction | 3. Foundations of algebra | Deliver instruction

Agile Mind materials Overview of the day Lesson activities

Lesson materials

Foundations of algebra:

Lesson 3.1 “Variables, expressions, and properties”Student Activity BookStudent whiteboards, markers, and erasers

Lesson preview

Suggestedtime Activity Goals

10 min. Opener Investigate patterns in arithmetic

30 min. Core activity Use variables and expressions to understand themathematics of magic number puzzles

10 min. Process homework Learn from reviewing the homework due today

25 min. Consolidation activity Explore expressions and algebraic properties used inmagic number puzzles

5 min. Wrap up and introducehomework

Reflect on the day’s lesson and understand tonight’shomework assignment

Lesson activities

OPENER (10 minutes)

Students each complete an arithmetic puzzle by following directions, then compare the result withtheir partners to find the pattern. The Opener provides an entry point for students to explore andreview ideas related to variables, expressions, and algebraic properties (including the distributiveproperty) in today’s lesson.

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Students complete the puzzle and then compare their answers with their partner. (If donecorrectly, the final result will be a three-digit number: the first two digits correspond to the


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student’s age, and the final digit corresponds to the number of siblings the student has.) Thenstudents discuss and record why they think the puzzle works. Note that the puzzle “works”only as long as the number of siblings is 9 or less. [SAB, questions 1-3]Quickly debrief, making sure students found the pattern. Do not spend too much time onstudent responses about why the puzzle works. Students should just have a sense that algebracan be used to explain why the puzzle works. (The puzzle is not “magic,” despite its name.)Tell students that they will use magic number puzzles to investigate some importantalgebraic ideas in the next activity.Classroom strategy. If some students have figured out why the puzzle works, you might askthem to write up an explanation and submit it to you. You might also tell students that youwill leave this question as an open-ended challenge, encouraging them to think about it andwork on it on their own.

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Preview the activities and learning goals for the day’s lesson.

CORE ACTIVITY (30 minutes)

Students solve magic number puzzles to develop algebraic thinking; define and formalize theconcepts of variable, algebraic expression, and algebraic equation; and explore and test algebraicfield properties, including the distributive property.

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Students solve magic number puzzles for different starting numbers.

Have students work individually on questions 1 and 2. [SAB, questions 1 and 2] When theyfinish, have them discuss the questions with their partners. Then briefly discuss thesequestions as a class. Students should verify that the ending number for question 1 will alwaysbe 2. Their work on question 2 should confirm that the ending number is always 4 greaterthan the starting number.

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Students are introduced to key vocabulary and then identify correct magic number puzzles,informally exploring algebraic properties, order of operations, and equivalent expressions.

Page 4: Use this page to formalize the algebraic term variable. (Students will use a mathjournal in tonight’s homework assignment to organize their understanding of the algebraicterms presented in today’s lesson.)Page 5: Use the animation on this page to demonstrate to students how the letter variable ncan be used to represent the outcome of the magic number puzzle they completed inquestion 2. Before playing the animation, ask students to record (on whiteboards) whathappens to the variable in each step; then use the animation to check, discussing anydifferences between students’ expressions and the ones on the screen.

Step 4: Pause the animation when it shows(3n + 12) ÷ 3. Ask: Why are there parentheses around the 3n + 12?Continue animation and ask: What happened to get from (3n + 12) to n + 4? Why isn’t itn + 12?

Note to teachers. The animation on page 5 foreshadows the idea of the distributive property,


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which will be taught later in this topic. In panel 4 of the animation, the magic number puzzlemoves from the expression (3n + 12) ÷ 3 to the expression n + 4. Students have hadexperience with this magic number puzzle from the opener and from the previous onlinepages and already know the end result of the puzzle. They will be able to grasp thismovement from step 3 to step 4 without the formal introduction of the distributive property.Page 6: Have students work with their partners on question 3 in the activity book. They areasked to use the variable n to represent the outcome of two magic number puzzles. [SAB,question 3] After sufficient time has passed, ask students to share their resulting expressionsfor the tables in question 3. Then, use the checks on this page to verify students’ thinking.Page 7: Use this page to define algebraic expression. Emphasize that all the terms in theresults column are algebraic expressions, even single terms like n and 3n.Page 8: Use this page to formalize the definition of an algebraic equation. Some questions toask to check for understanding:

Can anyone tell us what a variable is in their own words?What about an algebraic expression?How is an expression different from an equation?Who can provide an example of an algebraic expression?Who can provide an example of a numeric equation? [Sometimes we call these “numbersentences.”] How about an algebraic equation?

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Students are introduced to additional vocabulary. They will investigate the associative andcommutative properties of addition and multiplication by looking at several shape equations.

Page 9: Pose the question on the page to students. Ask:What number could the square represent that would make the equation true?Are there other numbers that would make the equation true?Why do you think the equation will be true no matter what number you use?

Page 10: Ask students to complete the activity in their activity book. [SAB, question 4] Usethis page to verify students’ responses to question 4. Ask students to explain why they thinkeach equation is true or false. If students think the equation is false, ask them to provide anexample that supports their response.Page 11: Use this page to formalize the definitions for the commutative properties ofmultiplication and addition.Page 12: This page formalizes the definitions for the associative properties of multiplicationand addition.

PROCESS HOMEWORK (10 minutes)

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Students process the homework due today: Homework 2.5 and Staying Sharp 2.5.

CONSOLIDATION ACTIVITY (25 minutes)

Students practice working with expressions. They will build expressions from a starting number n ina magic number puzzle and translate verbal descriptions into expressions. They will also use whatthey learned about the commutative and associative properties to identify correct magic numberpuzzles and justify their selections.


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This page cues students to work with their partner on the Consolidation activity.To debrief questions 1 and 2, select students to share their magic number expressions. Payparticular attention to order of operations and make sure students have correctly applied thedistributive property. Since the distributive property will be covered in the next lesson, youmay want to have students use numerical examples to see whether their expressions matchthe verbal descriptions.Classroom strategy. To debrief question 3, have the groups do the third puzzle on thewhiteboard. Have them hold up their whiteboards and have the class scan the answers. Askstudents to discuss their reasoning as a class. You can ask questions like:

Some of you said that Alisha was right, and some of you said that Brianna was right.Who would like to explain why you think [Alisha’s answer] is the correct answer?Who would like to explain why you think [Brianna’s answer] is correct?Now that you’ve heard your classmates’ explanations, which answer do you think iscorrect?

WRAP UP AND INTRODUCE HOMEWORK (5 minutes)

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Homework 3.1The homework problems are modeled after the problems in today’s lesson. Additionally,students use a math journal to reflect upon and organize their understanding of the mathconcepts variable, algebraic expression, equation, commutative property, andassociative property. Staying Sharp 3.1The main concepts and skills students will review in these problems are:

Finding the sum and product of two given numbers (in a “square box” structure)1.Writing an algebraic expression to describe a sequence of operations (“magic number”)2.Finding subsequent terms of a given arithmetic sequence3.Translating from a graphical representation to a table4.Generating examples of addition and subtraction of signed integers to find patterns5.Writing statements articulating structure in addition and subtraction of signed integers6.


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Lesson materials


Lesson 3.2 “The distributive property”Student Activity BookStudent whiteboards, markers and erasers

Lesson preview


10 min. Opener Investigate patterns in arithmetic

30 min. Core activity Learn about a property used in arithmetic and algebra


25 min. Consolidation activity Investigate an area representation for the distributiveproperty



Lesson activities

OPENER (10 minutes)

Students compare two magic number puzzles to determine which resulting magic numberexpression is correct. This activity is seeding the idea for students’ investigation of the distributiveproperty of multiplication over addition.

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Ask students to complete the opener in the activity book. If you notice that students arestruggling with determining which result is correct, encourage them to use numericalexamples as they have done in past lessons.To debrief the opener, ask students to share their explanations and examples. Do not delvetoo deeply into the distributive property since this is the topic of the day’s lesson. However,


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you may want to seed the idea by asking:In step 2, what are you adding 3 to?In step 3, what are you multiplying by 4?How do you know what you are multiplying by 4?Does the order of the steps have an impact on what you multiply by 4?

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Students review the distributive property by investigating two different ways to calculate theanswer to a verbal problem. The distributive property of multiplication over addition and oversubtraction is formally defined with both numeric equations and algebraic equations. In the nextlesson a concrete model involving the area of rectangles will be used to represent the distributiveproperty.

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Page 3: Call on various students to read parts of the online text, pausing to check for studentunderstanding and to add clarification. Have students briefly discuss the question with theirpartners. You may want to ask students to try substituting a number for n to show that 4(n +3) ≠ 4n + 3.Page 4: This page provides a concrete example of the distributive property of multiplicationover addition. Use this page to demonstrate how the distributive property works in areal-world situation.Page 5: The distributive property of multiplication over addition is formally defined. Havestudents work with their partners on questions 1-4. [SAB, questions 1-4]. As you circulatearound the room, check for understanding and offer assistance to student pairs as needed.After about 10 minutes, bring the class together for a brief processing of the activity. Call ondifferent student pairs to share their solutions.Page 6: Ask students what they think should go into the blank at the top of the page. Studentssolidify their understanding by using the distributive property to write equivalent expressionsand by creating their own illustration of this property. [SAB, questions 5-7] Encouragestudents who finish early to begin thinking about the final “mental math” question on thispage.Use the reveals on this page to allow students to check their work on problems 5 and 6. Todebrief question 7, solicit student illustrations and ask for other students to explain why theyare or are not examples of the distributive property. Close by debriefing the “mental math”question.


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Discuss question 3 from the homework as a class. Students have had multiple opportunities totranslate verbal descriptions into expressions; however, they may still struggle with verballydescribing what is happening at each step in the puzzle given the expression.


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Students are introduced to area models for the distributive property. They explain and create theirown models of products of increasing complexity, including the product of a constant and abinomial.

This lesson focuses on using the distributive property to simplify a product and then furthersimplify the expression if possible. Students need to simplify a product to recognize equivalentalgebraic expressions in the upcoming topics. Later in the course, students will revisit area modelsto multiply binomials and learn about factoring.

The work with area models here is limited to a(bx + c), where a, b, and c are all positive integers.Although it is possible to create models for the case when a, b, and c are negative, in this coursestudents are asked to first model easy cases, and later extend what they’ve learned to deal withcases that are not as obvious to model. The problems in the second part of this activity can also bemodeled with algebra tiles.

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The distributive property is demonstrated with an area model in preparation for the activity.

Before playing the animation, remind students about their use of the candy store model. Tellthem that in this activity they will use a model called an area model that is particularlyuseful because it provides a concrete representation of the distributive property.

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Students complete the activity with their partners, creating area models in their activity books.

Before students start the work in their activity books, access students’ prior learning byreminding them of how the area of a rectangle is related to its length and its width.Page 9: Ask students to answer the questions on the page in their activity books. [SAB,questions 1-3]Debrief questions 1 and 2 with the entire class. Ask several students to present their models,clearly identifying the factors and products. Even though Carla’s diagram is labeled, theyshould articulate that the 5 rows of 14 squares represent 5 ⋅ 14, while the two smallerrectangles represent 5 ⋅ 10 and 5 ⋅ 4.Page 10: Discuss ways to represent the distributive property with expressions containingvariables. [SAB, question 4] Debrief question 4 with the class by comparing and contrastinghow each model represents x + 8, etc. Students may notice that Marty’s model shows moredetail, but Jim’s model is more efficient and easier to draw. Be sure to discuss how torepresent x. It is important that students understand that the length of x is not known. Theycannot conclude anything about the value of x by comparing the length of x to the length of 8or 4.Page 11: Give students time to complete the activity in their activity book. [SAB, questions5-7] As you circulate, ask selected students to draw their models on whiteboards, for useduring the debriefing.For question 5, be sure students understand they are to write the product as well as draw thearea model. Questions to ask struggling students include:

How can you represent x? How can you represent x + 4?


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What does 3(x + 4) mean? How could you show that with the rectangles? Look back atMarty’s and Jim’s models. How did they show a product

Question 7 is similar to the magic number puzzle problems. Students must represent thenumber 101 as (100 + 1) and the number 99 as (100 — 1) to see that these examples are bothapplications of the distributive property.


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Homework 3.2First, students use what they have learned about the distributive property to construct areamodels and write equivalent expressions. Then they complete a math journal to organizetheir understanding of the distributive property.Staying Sharp 3.2The main concepts and skills students will review in these problems are:

Finding sums and products (in a "square box" structure)1.Writing an algebraic expression to describe a sequence of operations2.Finding subsequent terms of a given sequence3.Using an input-output rule to generate a table of values4.Generating examples of multiplication of signed integers to find patterns5.Writing statements articulating structure in multiplication of signed integers6.


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Lesson materials


Lesson 3.3 “Working with algebraic expressions”Student Activity BookStudent whiteboards, markers, and erasers

Lesson preview


10 min. Opener Find values in shape equations by substituting numbers

30 min. Core activity Use variables to create meaningful algebraicexpressions and evaluate them


25 min. Consolidation activity Investigate an area representation for the distributiveproperty



Lesson activities

OPENER (10 minutes)

Students solve familiar shape equation problems in preparation for evaluating algebraicexpressions. In the process, they review mathematical proficiencies, including the application ofprocedural skills (applying exponents, and working with negative numbers and grouping symbols)and reasoning skills (determining whether two similar-looking shape expressions—one with groupingsymbols and one without—are equivalent).

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Give students several minutes to work on the opener; then debrief it as a class. [SAB,questions 1-2] In debriefing, ensure that students understand that both shapes and letters


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can be used as variables.Give students explicit feedback about their execution of the opener routine—coming intoclass, getting their notebooks and Student Activity Books, and starting the Opener withoutteacher prompting.

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In the context of a party scenario, students combine variables to construct meaningful algebraicexpressions and evaluate algebraic expressions for given values of variables.

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Page 3: Read the introduction to the party scenario as a class, pausing to check forunderstanding and to clarify. Subscripts are likely to be new for students; be sure theyunderstand that a letter with a subscript is a variable that represents a single quantity.Classroom strategy. You can ask students to paraphrase the meaning of particular sections ofthe introduction.Page 4: Use this page to demonstrate how algebraic expressions can be used to modelpractical situations. Be sure that students understand

the fundamental task of the activity: creating algebraic expressions by combiningvariables from the list with each other and with basic operations;the concept of a meaningful algebraic expression and the idea that not all expressionswill have practical meaning;how to summarize, in words, the practical meaning of an algebraic expression.

Page 5: The introduction on this page emphasizes how to write a description of an algebraicexpression so that it has practical meaning in the context of the situation. After reading theintroduction as a class, have students work on questions 1 and 2 with their partners. [SAB,questions 1-2] Circulate around the room asking questions that check for understanding,clarify thinking, and, where appropriate, push students’ thinking. Use the reveals on this pageto debrief question 1.Page 6: Use the animation to demonstrate the mathematics involved in evaluating anexpression. Remind students that, when investigating a problem, it does not make sense toevaluate an expression that has no practical meaning. Have students work with partners tocomplete the activity. [SAB, questions 3-5] Debrief questions 3-4 by allowing students toshare their responses. To debrief question 5, ask a few student pairs to share the descriptionsthey created, and then have the rest of the students try to determine the summary of theaccuracy of these descriptions .

Classroom strategy. There are several options for debriefing the activity, especially forstudents who finish early. You could have students place answers on their whiteboards forspecific problems. Alternatively, you could assign partner groups, especially those that finishearly, a specific problem to present on their whiteboards. For individuals or partners whofinish early, challenge them to create more complex or creative expressions.


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Review question 1 in the homework assignment. When asking students to justify why oneresult is correct and another is not, encourage students to use correct mathematicalterminology. Students should be able to identify the distributive property as part of theirjustification, especially when noticing that both results to question 1c are correct.


Students continue to investigate meaningful expressions in a new problem context. This activity issimilar to the Core activity.

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This page initiates a stop and work period. Read the introduction to the school dance scenarioas a class, pausing to check for understanding and to clarify. [SAB, questions 1-6]When debriefing the activity, pay particular attention to question 5. Students should be ableto use their answers from questions 1 and 4 to deduce the practical meaning of theexpression in this question.Ask a few student pairs to share the expressions they created, and then have the rest of thestudents try to determine the summary of the expression’s meaning.


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Homework 3.3The homework problems provide additional practice for working with meaningful expressions.Students continue this investigation of meaningful expressions in a new problem situationinvolving a class field trip.Staying Sharp 3.3The main concepts and skills students will review in these problems are:

Finding numbers that yield a given sum and product (in a “square box” structure)1.Using a linear rule to generate an input-output table2.Finding a term of an arithmetic sequence from a pictorial description3.Translating from a table to a graphical representation4.Rewriting subtraction as addition of the opposite (additive inverse)5.Identifying numbers that sum to positive integers, negative integers, or zero6.


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Lesson materials


Lesson 3.4 "More about expressions"Student Activity BookStudent whiteboards, markers, and erasersCards for Card sort activity (Card_sort.pdf)Scissors (one per student pair)Tape (one per student pair)

Lesson preview


10 min. Opener Review how to calculate the area and perimeter of arectangular region

30 min. Core activity Evaluate expressions for perimeter and create expressionsthat represent perimeters by collecting like terms


25 min. Consolidation activity Match verbal expressions to algebraic expressions andevaluate algebraic expressions



Lesson activities

OPENER (10 minutes)

Students review finding area and perimeter of a rectangular region in preparation for exploringperimeters of rectangles with variable dimensions.

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Give students several minutes to work on the Opener; then debrief it as a class. [SAB,questions 1-2] Listen for different approaches to finding the perimeters, such as simple


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addition or doubling and then adding. Emphasize the appropriate units for perimeter andarea, and ask students to provide rationale for the units, e.g., Why is perimeter given infeet, and area given in square feet?

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In this activity, students find perimeters and areas of rectangles with side lengths containingvariables. While investigating perimeters, students will use the distributive property and connect itto the idea of collecting like terms. Students will continue to work with the distributive property asthey determine expressions to represent areas of rectangles.

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Page 3: This page sets up the context of the activity. When students understand the situation,have them work with their partners to determine the dimensions and perimeter and area of aroom given different values of x. [SAB, question 1]Page 4: Have students work with their partners to find an algebraic expression that can beused to represent the perimeters of the rectangular rooms. [SAB, questions 2-4] Use theanimation to verify their work. The animation introduces the concept of like terms.Page 5: Use this page to extend students’ work finding an expression for the perimeter of thegymnasium to considering equivalent algebraic expressions. Remind students of thecommutative property of addition and the distributive property of multiplication overaddition. Using these two algebraic properties, students can show that 2x + 10 + x + 2x + 10 +x and 2(x) + 2(2x + 10) are equivalent.Page 6: This animation asks students to explore the area of different rooms at the recreationcenter. Introduce students to the new task. Panel 1 of the animation uses the closet toremind students that exponents can be used to denote the repeated multiplication of a

variable and to introduce students to the concept of x2. After showing and discussing panel 1of the animation, have students work with their partners to find the area of the other threerooms. Use the remaining panels to allow students to verify their work. [SAB, questions 5-6]


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As a class, discuss question 3 in the homework assignment. Ask several students to share theirexpressions and explain why the expressions match the descriptions of the practical meaning.


In this activity, students translate verbal descriptions into algebraic expressions, evaluateexpressions, and analyze the results of the activity.

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To prepare for the card sort, conduct a brief whiteboard activity to combat possibleconfusion or misconceptions. Call out an expression from the list below and have studentswrite a matching algebraic (or, if appropriate, numeric) expression on their whiteboard. Havestudents hold up their whiteboards after each question so that you can gauge understandingand remaining sources of error or confusion.

Add four to sixAdd four to any numberSubtract four from sixSubtract four from any numberSubtract any number from fourSubtract any number from itselfMultiply four by six

Multiply four by any numberDivide four by sixDivide any number by sixDivide any number into sixMultiply any number by four, then add sixAdd six to any number, then multiply by four

Introduce the card sort activity by pointing out that translating verbal descriptions intoalgebraic expressions is a critical skill in being able to use algebra to solve real-worldproblems. Tell students they will be working with verbal descriptions of mathematicaloperations without a context.Distribute one set of cards to each pair of students. Read through the Part I activitydirections. Underscore that students will be matching and taping together matching cards (anumber card with a letter card) and that there will be extra cards. [SAB, Part I]As students work, circulate around the room, checking for understanding. Pay attention todifficulties that students may be having with expression descriptions that have multiple partsand which may require grouping symbols. Also watch for expressions for which the correctrepresentation of the commutative property is involved; for example, some students maymatch “3 less than a number” with 3 – x instead of x - 3.Bring the class together for a discussion of Part I. Call on different partners to share cardmatches. Ask questions like:

Which card did you match with Card A? Did other groups have something different?Who can explain to us why you think your match is correct?

Below is a key for the card match:1 -P

2 -G

3- I

4 -J

5 -C

6 -B

7 -A

8 -D

9 -F

10 -N

11 -E

12 -H

If time permits, read the directions for Part II and give students time to work with theirpartners to complete it. [SAB, Part II, 1-3]Discuss Part II as a whole class. You might use whiteboards for students to write the way theyordered the cards. You can use the information on the whiteboards to help you direct thewhole-class discussion. Ask questions like:

I noticed that some of you had G before B and that others had B before G. Who canexplain which one should come first?

Here are the answers to the ordering of the cards (from high to low). Note there is a tie whenx = -1, meaning that these letters or cards could be reversed on some students’ answersheets:1. When substituting 10:

CARD VALUE

2 - G 100

6 - B 65

7 - A 53

2. When substituting -1:CARD VALUE

11 - E 4

5 - C 2

2 - G 1


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4 - J 47

5 - C 35

9 - F 25

8 - D 15

1 - P 7

10 - N 1.4

12 - H 1

3 - I -1

11 - E -7

12 - H 1

10 - N -0.8

7 - A -2

3 - I -3.2

1 - P -4

4 - J -8

9 - F -8

6 - B -10

8 - D -18

Be sure students articulate their written answers for question 3. To underscore the meaningsof variable and variability, explore the idea that different orderings occur depending on whatvalue is substituted into the expression. Make sure students understand that as the inputschange, the outputs change.


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Homework 3.4The homework problems provide additional practice for working with expressions thatrepresent perimeters and areas of rectangles. The following skills are addressed: (1)evaluating expressions; (2) creating expressions to represent perimeters; (3) creatingexpressions to represent areas; (4) collecting like terms; and (5) using the distributiveproperty.Staying Sharp 3.4The main concepts and skills students will review in these problems are:

Finding numbers that yield a given sum and product (in a “square box” structure)1.Writing a linear function rule for a given input-output table2.Creating a graphical representation of an arithmetic sequence from a pictorialdescription

3.

Generating examples related to addition and subtraction of the additive identity orinverse

4.

Articulating patterns related to the additive identity or inverse5.


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Lesson materials


Lesson 3.5 “Using expressions to represent relationships”Student Activity BookStudent whiteboards, markers, and erasers

Lesson preview


10 min. Opener Use variables to represent the relationships betweenstarting and ending numbers in magic number puzzles

30 min. Core activity Find a rule for an important type of mathematicalrelationship


25 min. Consolidation activity Apply input-output rules to problem situations



Lesson activities

OPENER (10 minutes)

Students turn magic number expressions into magic number rules, setting the stage for thedevelopment of input-output machines and rules, the main focus of today’s lesson.

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Debrief the activity by asking students how they determined what the ending number wouldbe for a starting number n. [SAB, questions 1-2] Emphasize that, since there is a rule thatconnects each starting number to the corresponding ending number, there is a special type ofrelationship between the starting number and the ending number. If students know the rule,they can find the ending number for any starting number. And if this special type of


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relationship exists, then they can figure out the rule if given a set (or table) of starting andending numbers.

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Students are introduced to the important mathematical idea of input-output machines and writerules to describe the relationship (or pattern) among inputs and outputs in a set of values. Theywill also begin to develop a strategy for finding the rule that represents an input-outputrelationship.

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Students view an input-output machine in an animation. They use a rule to determine the outputfor a given input.

Page 3: Tell students they will be building off of the ideas in the Opener. Read the framinginformation at the top of the page. Before you play the animation, tell students to focus onwhat is happening to the input numbers—how they are being “transformed” by the machineinto output values.Panel 1: Discuss with students that they have investigated many input-output relationships inmagic number puzzles. However, they used the idea of starting number and ending numberinstead of input and output.Panel 2: Ask students how they think the rule “Output = 3(Input) + 1” can be used to find theoutput that goes with an input of -2. Allow students to share a few ideas, but do not verify.The next panel of the animation will demonstrate this idea.Panel 3: Use this panel to demonstrate how the input-output rule can be used to find outputsthat correspond to given inputs. Ask:

Is this what you expected to happen?Can you describe in words how the machine turned the input into the output?

Give students a few minutes to fill out the rest of the table in their activity books. [SAB,question 1]Panel 4: Use this panel to allow students to verify their work on question 1.

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Students are given a table of input-output values and are asked to determine the rule.

Tell students that sometimes they will be given input and output values and will need tofigure out the rule. Ask students to fill out the table in their activity book. [SAB, question 2]Once students have had time to complete the table in their activity book, use the puzzle toallow students to verify their work. Ask:

Is there a rule that you know will not produce the table?How do you know that that rule will not work? [Students may identify one or more inputvalues that, when substituted into the rule, will not produce the desired output.]Is there a rule that you think will produce the values in the table?Does this rule work for all of the values in the table?

Make sure students understand that, in order for a rule to be correct, it must work for all of


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the input-output pairs in the table.Tell students to work on question 3 in their activity book with their partner . Then havestudents share which rule works for the table. Ask students to explain how they know thatthis rule works and the other rules do not work. [SAB, question 3]

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Students develop a strategy for writing a rule for a set of input-output values.

Read the framing information at the top of the page. Ask students to generate a list of rulesthat would work for the first row of values in the input-output table. [SAB, question 4]Use the first panel of the animation to verify students’ rules. If students came up with rulesthat are not listed in the animation, write them on the board.Play the second panel of the animation. Ask students to look at the second row of values inthe input-output table. Tell students that, in order for an input-output rule to represent therelationship, it must work with all of the input-output pairs in the table. Ask:

Are there any rules in our list that do not work for the second pair of input-outputvalues?How can you determine whether a rule works for the second pair of input-outputvalues?

Use the third panel of the animation to show the elimination of rules that do not work for thesecond pair of input-output values. Once the animation has played, remind students that therule that they are looking for must work for all of the values in the table. Ask:

How do you determine which of the remaining rules is correct? [SAB, question 5]Classroom strategy. After the first correct rule is discovered and checked, challenge studentsto find other correct rules that look different from the first rule. As new rules are proposed,ask students to determine whether or not the new rule works for all of the input-outputvalues in the table. This foreshadows work with equivalent expressions in the next unit.

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Students work with partners to write rules for different sets of input-output values.

Allow students time to work with their partners on the remaining questions in the activity.[SAB, questions 6-8] As you circulate, watch for whether the rules students come up withwork for all rows of the input-output table. A common error is to find a rule that works foronly one row—usually the first one—of the input-output table. Here are some questions to askas you visit with student pairs:

Can you explain to me why your rule works?Does your rule work for all rows of the table?

Review the activity as a class. Consider using whiteboards to process the activity, assigningdifferent groups different problems to present. As part of processing the activity, makeexplicit that input-output rules connect to the theme of problem-solving with patterns. Alsopoint out that input-output relationships constitute an important pattern relationship inalgebra—one that will be studied a great deal this year.Be sure that students have gained a solid foundation of the basics of input-output machinesfrom this activity before moving on. If necessary, you can add on a whiteboard activity inwhich you place an input-output table on the board and ask each student to write a rule onhis/her small whiteboard. You can do several rounds of this, as needed.The big picture. Input-output machines set the stage for the development of function


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relationships. The idea of a function will be introduced and formalized in Unit 2. This unituses the more informal terminology of “input-output tables” and “input-output machines” sothat students can concentrate on building/reviewing important foundational understandings.


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Discuss question 4 as a class. Students should recognize that both methods for findingperimeter are valid and that they can identify the step in which the error occurred thatcaused the results to be different.


Students continue to work with input-output relationships, finding rules and stating them in severalways. They explore cases where multiple rules can describe the relationship and cases where norule exists.

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Tell students that the activity is similar to the input-output activity they completed earlier inthe lesson, but this set of problems challenges them to consider some different types ofproblems and push their understanding into new areas.Have students work on the activity with their partners. As they work, circulate around theroom to monitor progress. [SAB, questions 1-5]Briefly process the activity. If possible, choose various questions and have two student pairswho solved the problem differently present their solutions.


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Homework 3.5Tonight’s homework problems are modeled closely on today’s class activities and aredesigned to give students practice with input-output relationships, including finding rules andusing input-output tables and rules as tools to solve problems.Staying Sharp 3.5The main concepts and skills students will review in these problems are:

Finding numbers that yield a given sum and product (in a “square box” structure)1.Using a linear rule to generate an input-output table2.Writing a linear function rule of an arithmetic sequence from a pictorial description3.Translating from a table to a verbal representation4.Generating products of positive integers with positive, zero, or negative integers5.Articulating statements related to the multiplication of signed integers6.


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Lesson materials


Lesson 3.6 “More input-output relationships”Student Activity BookStudent whiteboards, markers, and erasers

Lesson preview


10 min. Opener Find input and output values that meet a givencondition

30 min. Core activity Use variables to describe input-output relationships


25 min. Consolidation activity State input-output rules in different ways and applythem to problem situations



Lesson activities

OPENER (10 minutes)

Students are given a verbal description of an input-output relationship and determine input-outputpairs that fit the scenario. This scenario will be used in the lesson to transition students to usingvariables to represent unknown quantities in an input-output relationship.

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To debrief the opener, ask several students to share some of their input-output pairs. Askstudents what they think is the input and what they think is the output in the problemsituation. Once students have identified Cassandra’s age as the input variable, ask studentshow they determined Cassandra’s mother’s age after they chose an age for Cassandra.


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Students continue to investigate input-output relationships and begin transitioning to writing input-output rules using variables.

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This page expands upon the work students did in the Opener. Use this page to transitionstudents from the statement of the problem to an input-output rule using the terms input andoutput. Ask:

What is the input value in this scenario?What is the output value in this scenario?How would you write a rule using the terms input and output? (Students should be ableto come up with the rule: Output = 2(Input) + 9.)How would you write this rule using the variable C to represent Cassandra’s age andthe variable M to represent her mother's age? [SAB, question 1]

Give students time to answer the remaining questions associated with this scenario. [SAB,questions 2-4] Ask:

Is this what you expected the rule to look like?How could you use your rule to find Cassandra’s mother’s age if you knew thatCassandra was 16 years old?

To debrief this activity, draw students’ attention to the process column in the table. Ask:What do you notice is staying the same in each row of the process column?How do these terms relate to the rule that you wrote in question 1?What is changing in each row of the process column?How does this term relate the rule you wrote in question 1?What do you notice about these terms and the terms in the first column of the table?

While debriefing, remind students that variables are used to represent quantities thatchange. Also, discuss with students the importance of defining variables in a problemscenario. Tell students that sometimes they may want to use different letters to representdifferent variables. For example, in this problem scenario, they used C to representCassandra’s age and M to represent her mother’s age. Make the connection between theinput values and the variable C and the output values and the variable M.

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Ask students to work with their partner to complete the activity in their Student ActivityBooks. [SAB, questions 5-7] Then, debrief the activity and allow students to check theirthinking.


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Debrief questions 2 and 3 in the homework assignment. Ask students to share the rules they


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wrote for question 3 and how they knew these rules worked for the table. Remind studentsthat in order for a rule to work, it must work for all input-output pairs in the table.


Students practice writing input-output rules given a table of values or a problem scenario.

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Page 6: Tell students that they will be working with their partner to write more input-outputrules to generalize input-output relationships. [SAB, questions 1-4]

Page 7: Use this page to debrief question 1. Make sure students understand that in order for arule to work, it must work for all input-output pairs in the table.Page 8: Use this page to debrief question 2. Remind students that their rules may not lookexactly like the rules shown in the process columns, and help students make sense ofequivalent expressions. Students may need additional support in verifying whether their rulesare equivalent to the rules shown.To debrief the third and fourth questions, create an input -output table on the board and askstudents to share some of their table values and the rule that they wrote to model therelationship. Students should be able to explain how the function rule relates to the problemsituation, including how they decided to define their variables.


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Homework 3.6The problems in tonight’s homework are similar to the problems in the day’s lesson. Studentscontinue to practice writing input-output rules to model relationships in a table of values orproblem situation.Staying Sharp 3.6The main concepts and skills students will review in these problems are:

Writing an algebraic expression to describe a sequence of operations1.Finding a rule for an arithmetic sequence given a pictorial representation2.Translating from a rule to tabular and graphical representations3.Adding, subtracting, multiplying, and dividing signed integers and variables4.Finding the sum a product of integers (in a “square box” structure)5.


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Lesson materials


Lesson 3.7 “Representing relationships with graphs”Student Activity BookThe search is on!.pdf

Lesson preview


10 min. Opener Represent a relationship in multiple ways

25 min. Core activity Plot points on graph


25 min. Consolidation activity Interpret the meaning of points on a graph



Lesson activities

OPENER (10 minutes)

Students review plotting points in the context of representing data from input-output tables.

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As students complete the activity, monitor to ensure they are transferring the data fromtable to graph and from graph to table correctly. Debrief the activity by asking differentstudents to share their solutions with the class. Be sure each student explains the process fortransferring from table to graph and from graph to table.

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Students review coordinate plane skills by playing the game “The search is on!” in pairs. The“Dispatcher” plots a point representing a missing hiker on his/her coordinate grid. The “Pilot” triesto guess the location of the point. To play this game, you may want to download “The search ison!.pdf” document from the Lesson materials section to print and make extra copies of the gamemaster in case they are needed.

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Students review characteristics of the coordinate plane in preparation for playing the game.

Ask students what they remember about plotting points on the coordinate plane. Discussbriefly; this is an opportunity to activate students’ prior knowledge. Then, use the animationto explain how the coordinate plane is divided into four quadrants and how ordered pairs areplotted.

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Students play the game “The search is on!”

Use the grid on this page to introduce the game and let students practice finding coordinatesto prepare for the game. Tell students:

I’m going to tell you two numbers. The first number tells you how far east or west ofcenter the hiker is located. The second number tells you how far north or south ofcenter the hiker is located.

Call out pairs of coordinates (e.g., “The hiker is located at (2,7).”) Each time ask a student tocome up and point/mark where the hiker is located. Use all four quadrants and continue untilyou’re sure students have the idea.Now play a practice game with the class. Pass out three copies of the handout titled Thesearch is on!.pdf to each student. Record the location of the hiker on a small whiteboardthat you do not show the students. Say:

I have found the hiker. Who has a first guess of where he is located?As you play the practice game, record the guesses and clues on a projected grid and in theGuess column of the table. Students should also record the guesses and clues on their activitysheets. If the location is not guessed correctly, give a clue: “No, go (east/west) and(north/south)”. Be sure to give the east/west direction before the north/south direction toreinforce that the first coordinate tells the east/west (horizontal) direction. Record your cluein the Clue column of the table. Continue soliciting guesses and giving hints until the classlocates the hiker. As you play, you might need to ask whether the guesses are consistent withthe clues.Tell students they will now play the game with their partner using the recording sheets youhanded out to them. They will take turns being the Pilot and Dispatcher in each game. Theyshould arrange themselves so that they cannot see each other’s grids; e.g., they could place abinder or book between them to shield the view. They should play four games (two asDispatcher and two as Pilot) if possible.Classroom strategy. This partner activity provides an opportunity to extend thecommunication work students began in topic 1. To extend this conversation, start byreferencing the “Characteristics of Good Mathematical Communication” poster studentscreated in topic 1. Ask:


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How do the actions recorded on this poster apply to actions needed in this game?Which of these characteristics was most important as the speaker? Which was mostimportant as the listener or audience member?What is one speaker characteristic and one listener characteristic that you will chooseto apply as you play the game?

To emphasize how these skills extend beyond audience members, you can add the title“Listener” to the audience member column.Classroom strategy. For students who are ready for an additional challenge, allow them toplace the hiker’s location between gridlines, such as five-tenths or three-fourths.After completing four games, students should discuss the reflection questions in the StudentActivity Book and record their responses. [SAB, questions 1-5]Discuss students’ responses to the reflection questions. For question 5, elicit students’strategies and probe why they think those strategies are the most efficient, thencompare/contrast strategies. (Students might come to the realization that the best place tostart is (0, 0), the center of grid, because regardless of what the next clue is, you only haveone-quarter of the area to search.)


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Students process the homework due today:Homework 3.6 and Staying Sharp 3.6.

Debrief questions 4 and 5 in the homework assignment. Ask students to share the rule theywrote for question 4, part b, and how they used this rule to answer question 5. Discuss withstudents the idea that Sean cannot mow part of a lawn, so by mowing 16 lawns, he willactually make more than $190. However, if he only mows 15 lawns, he will not have enoughmoney to buy the festival ticket.


Students interpret the relative placement of points on graphs that contain axis labels but not axisscales, prioritizing conceptual understanding of graphing. The activity builds the skills of studentswho do not have solid experience with reading graphs, but it is also appropriate for students withmore experience, because of the level of conceptual understanding that is required to answer thequestions.

Classroom strategy. In preparation for supporting the students, take some time to work throughthe activity yourself, writing down the answers to the questions, thinking about particularmisconceptions that may be surfaced by a particular question or set of questions, and identifyingchallenges that students might encounter.

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Students are introduced to the concept of “specific evidence” in preparation for the activity.

Before students begin work in their activity books, discuss what is meant by “specificevidence,” as shown on these pages. It may be helpful to tell students to describe theirevidence as if they are writing to someone who is just learning how to make and read graphs.

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Students work with partners to interpret graphs describing characteristics of airplanes.

Have students work with their partners to complete the activity in the Student Activity Book.[SAB, questions 1-3] Check in with each team and ask questions that require students toverbalize their thinking. This activity uses the idea repeatedly of the “story of the graph.”Consider the example of the graph that shows maximum number of passengers versus lengthof airplane. This graph tells the story: the longer the plane, the more passengers that can becarried. (Note that there may be other correct ways to capture the story of a graph;however, there are correct and incorrect answers/interpretations for these questions.) Youmay choose to ask:

Questions that are simple checks for understanding (e.g., Can you explain youranswer to this question to me?);Questions that help students clarify their thinking (e.g., What does it mean if a pointis higher on the graph for this situation?);Questions that extend student understanding (e.g., What would it mean to have apoint far up on the y-axis on this graph? [For example, on the graph of fuel efficiencyversus weight of airplane, a point far up the y-axis would constitute an impossiblesituation: a highly fuel efficient plane, but one that does not have any weight.]

To debrief question 1, ask students to share their answers and evidence. When solicitingresponses from students, project the graphs on the online page and ask students to point outtheir “specific evidence.”To debrief this part of the activity, ask several students to write their statement from eitherquestion 3d or 3e. Then have the class discuss whether the statement is true or false. If thestatement is false, ask students how they would rewrite the statement in order to make itcorrect.


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Homework 3.7The homework problems provide additional practice with creating and interpreting graphs.The following skills are addressed: (1) plotting points on a coordinate grid; and (2) answeringquestions using information in graphs.Staying Sharp 3.7The main concepts and skills students will review in these problems are:

Finding numbers that yield a given sum and product (in a “square box” structure)1.Writing a rule from an input-output table2.Finding subsequent terms of a given sequence following a quadratic rule3.Translating from a verbal representation to a graphical representation4.Generating quotients of positive integers with positive, zero, or negative integers5.Articulating statements related to the division of signed integers6.


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Lesson materials


Lesson 3.8 “Revisiting expressions”Guided assessmentStudent Activity BookExpression and value cards (one set per pair of students)Envelopes or baggies for storing the cardsExpressions gameScore sheet transparencyCalculators

Lesson preview


10 min. Opener Review signed number arithmetic as you evaluateexpressions

25 min. Core activity Continue to solidify your understanding of evaluatingexpressions


30 min. Online assessment Assess your understanding of key ideas and skills fromthis unit



Lesson activities

OPENER (10 minutes)

Students evaluate an expression, reviewing signed number arithmetic as well as the concept ofmultiple rules for the same equation.

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Debrief this activity by allowing students to share their answers. Problem 3 would beespecially interesting to review. Many students will choose integer values for a and b;however, you may want to point out that by opening up to all rational values, they have manymore options!

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Students play a game in which they determine possible values for given algebraic expressions; thenthey answer questions reflecting on the challenges of the game. You will need to download thedocument “Expressions game.pdf” from the Lesson materials section to provide copies of theexpression and value cards. A copy of the score sheet is also included in this document.

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Classroom strategy. For each possible value, ask students to use a student whiteboard to tellyou whether they think each value is a possible value of the expression. Ask students to writeeither “yes” or “no” on the whiteboard and hold it up in the air so that you can see theiranswer. This allows you to make a formative assessment of student understanding andaddress misconceptions before students play the Expressions game with their partner.Use the puzzle to allow students to check their thinking. [SAB, question 1] For each numbergiven, select a student to explain whether the number is a possible value of the expression.

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Read the text at the top of the page. Remind students that, while playing the game, theyshould not let the form of the algebraic expression influence what they think the possiblevalues of the expression are.Introduce the game and its goals as described in the Student Activity Book. Students shouldcut out the expression and value cards, keeping them in separate piles.To illustrate how the game is played, model a few turns with the class. Make a transparencyof the score sheet so that you can model how to record moves and keep scores. Studentsshould play along with you.Students should play two games and then answer question 3 in the Student Activity Book.[SAB, questions 2-3] As students play the game, circulate to ensure that they are makingcorrect judgments about possible values for given expressions. To debrief question 3, brieflydiscuss parts a-d.Classroom strategy. Part c of question 3 provides an opportunity to respect, inspect, andlearn from common errors. Consider polling the class about the most challengingexpression/values combinations—or gather this information as you circulate while studentswork on the question— to focus the discussion appropriately.


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You may want to discuss question 5 as a class. In the discussion, press students forjustifications and listen to make sure they are interpreting the graph correctly.

ONLINE ASSESSMENT (30 minutes)

Online page 6 and Guided assessment, pages 1-10

Students complete the online Guided assessment for this topic. You will need to reserve acomputer lab for this activity. Completion of the items in the Guided assessment will provide youand your students with formative assessment data related to student understanding of the keyideas in this topic, and will help students prepare for the upcoming end-of-unit assessment. Toview and analyze data from this assessment, you will need to create an assignment for your class,and then view the assignment report.


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Summarize the lesson and introduce tonight’s homework following the established routine.

Homework 3.8Students evaluate expressions and create graphs from data, making decisions about scalingand labeling.Staying Sharp 3.8The main concepts and skills students will review in these problems are:

Finding numbers that yield a given sum and product (in a “square box” structure)1.Using a quadratic rule to generate an input-output table2.Finding subsequent terms of a given sequence following a linear rule3.Translating from a graphical representation to a verbal representation4.Generating products of negative integers with positive, zero, or negative integers5.Articulating statements related to the multiplication of signed integers6.


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Lesson materials


Lesson 3.9 “Making graphs”Student Activity Book

Lesson preview


10 min. Opener Learn how scaling can be used to create misleadinggraphs

30 min. Core activity Learn how to scale graphs to accurately represent data


25 min. Review online assessment Analyze and learn from performance on the onlineassessments



Lesson activities

OPENER (10 minutes)

Students compare different graphs representing the same data.

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In debriefing this activity, have students share their responses. Have them justify why theythink these graphs show the same thing or not. If students struggle with explaining why thegraphs are the same, ask them to find the coordinates of the points on each graph andcompare the points.

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Students review how to make graphs from tables; the values in the tables force students to makedecisions about minimum and maximum values and scales on the graphs.

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Students create graphs that they have to scale themselves. They work on their own, then checktheir work with their partners.

Each student should create a graph for each x-y table in the Student Activity Book. When bothpartners are finished, they can compare their graphs. If their graphs don’t match, they shoulddiscuss the graphs to determine if the different graphs are both correct, or to resolve thedifferences if possible. [SAB, questions 1-5]As students are making their graphs, circulate, checking for understanding and gatheringinformation about particular “trouble spots” to be highlighted during the whole classdebriefing. If students are struggling, ask questions. For example, if students are havingdifficulties scaling an axis:

What values do you need to graph? What is the maximum value? How many grid linesare there?Pick a number to try (as the interval on an axis). Does it work? Why or why not? If itdoesn’t work, what other number(s) might you try?

Look for students with correct graphs but different scales for graphs 1 and 2. You might askstudents to make transparencies or posters of their graphs for use in the class discussion.Focus the debriefing on locating points between gridlines and creating scales. The goal is tomake explicit for all students the thinking processes involved in these skills (question 3). Iftime permits, ask students who made correct graphs with different scales to present theirgraphs.

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Students work with their partners to find and correct errors in graphs.

The big picture. While multiple representations of quantitative relationships receive attention inthis unit, graphing is especially emphasized. Students come to high school with varyingproficiencies with graphing. Many students make the same errors, especially with scaling.Identifying and correcting errors and misconceptions has been found to be effective in helpingstudents move beyond these errors.

Have students work with their partners to analyze the graphs and find the errors. [SAB,questions 6-8] Here are some questions you might ask as you circulate:

What might be wrong with a graph?What have you checked so far?

Debrief the activity in a short whole-class discussion. Here is a summary of the errors in thegraphs and the misconceptions underlying them:

In Graph 1, the scaling on the x-axis is incorrect; the scale is sometimes 1 andsometimes 2.In Graph 2, the scaling is also incorrect. Here, the incorrect scaling results from placingnumbers on the axes according to the way the input-output values appear in the table,


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without regard to consistent spacing on the axes.In Graph 3, the x- and y-coordinates are reversed. For example, the point (1,10) isincorrectly plotted by going up 1 unit from the origin and then going right 10 units.

Classroom strategy. If pairs finish early, you might have them reproduce one of theircorrected graphs on a transparency or whiteboard and share their graphs as part of thewhole-class debriefing of this activity. Consider preparing transparencies of the student pagesfor the activity. This will help students make accurate graphs for their class presentations.


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Students process the homework due today:Homework 3.8 and Staying Sharp 3.8.

Discuss question 2c as a class and connect the decisions students made about plotting decimalvalues to the decisions they made in the lesson today about plotting values between gridlines.

REVIEW ONLINE ASSESSMENT (25 minutes)

Students analyze class performance on the Guided assessment.

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Display the class report for the Guided assessment questions. Have students identify theitems the class did well on and those the class did not do well on. Instruct students to writethe question numbers in the table provided in the activity book.Work as a class to identify the mathematics of problems the class did well on, and discussthese concepts and skills. Be prepared to project particular Guided assessment questions.Then identify and discuss the mathematics involved for problems on which the class did notperform well. Be prepared to project particular questions (perhaps two questions) from theGuided assessment. Next, allow students several minutes to work with their partners to re-dothese problems in their activity books, and then discuss as a class.Share with students their individual score reports. Tell students they should take some timeoutside of class to revisit problems from the Guided assessment, in particular those for whichthey did not get the correct answers.


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Homework 3.9Students review key concepts in preparation for the end-of-unit assessment. As part of theirreview, they complete the online More practice for this topic.Staying Sharp 3.9The main concepts and skills students will review in these problems are:

Finding numbers that yield a given sum and product (in a “square box” structure)1.Writing a linear function rule from an input-output table2.Finding subsequent terms of a given sequence following a quadratic rule3.Translating from a symbolic representation to a graphical representation4.


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Generating quotients involving positive, zero, or negative integers5.Articulating statements related to the division of signed integers6.


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Lesson materials


Lesson 3.10 “Checking for understanding”Student Activity BookGraphing calculatorsUnit 1 end-of-unit assessment: StudentUnit 1 end-of-unit assessment: Teacher

Lesson preview


10 min. Opener Reflect on effective effort and malleability of intelligencein preparation for assessment


45 min. End-of-unit assessment Demonstrate your understanding of importantmathematical skills and concepts from the unit

10 min. Consolidation activity Relate what you know about scaling graphs to graphsmade with a graphing calculator



Lesson activities

OPENER (10 minutes)

Students reflect on their thoughts and emotions going into their first end-of-unit exam.

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To debrief this activity, you may want to ask willing students to share their feelings as theyget ready to take the test. Remind students to be respectful of other students’ feelings. Ifthere are no students that are willing to share their thoughts and emotions, tell students toremember that a test is merely a way for them to show you and themselves what they know.


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END-OF-UNIT ASSESSMENT (45 minutes)


Students use what they have learned about scaling graphs to adjust the viewing window of agraphing calculator to view a set of data.

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Page 5: Display this page near the end of the assessment time. As students turn in theirend-of-unit assessment, ask students to read the online page and answer the question in theiractivity book. [SAB]Page 6: Once all students have had a few minutes to work on the graphing problem, debriefthe activity. Begin by asking students what they think Matthew should do to see his databetter. Use the online page to show students how adjusting the window on the graphingcalculator can give them a better view of a graph.Technology tip. If you have enough time after students have completed the assessment, youmight want to take this opportunity to teach them how to make scatterplots and set graphingwindows using a graphing calculator. Have students go through the steps of setting theoriginal window shown on page 5 and making the scatterplot. Then, have them adjust thewindow and view the new scatterplot. Connect the values of Xmin, Xmax, Xscl, Ymin, Ymax,and Yscl to the decisions that students have been making about scaling graphs they make byhand.


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Homework 3.10Students answer questions by interpreting a graph, then answer questions about a situationbased on a pattern.Staying Sharp 3.10The main concepts and skills students will review in these problems are:

Finding numbers that yield a given sum and product (in a “square box” structure)1.Writing a quadratic rule from an input-output table2.Explaining why a quadratic rule generates a given sequence3.Translating from a graphical representation to a symbolic representation4.Computing with signed integers while avoiding common errors5.Articulating statements related to patterns in operations with signed integers6.


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Goals and objectives Topic overview Resources Print resources …pvhs.fms.k12.nm.us/teachers/malonzo/08EF690A-00757… · · 2015-07-26Goals and objectives Topic overview Resources