Unit 5 Logic of Algebra

Unit 5: Logic ofAlgebra363m + 2s + 3h + 9 = 50502h + s = 1873m + 9 = 3h= ______ 7 = ______ 4 = ______ 49Transition to AlgebraTeacher Guide

The Transition to Algebra materials are being developed at Education Development Center, Inc. inNewton, MA. More information on this research and development project is available at ttalgebra.edc.org.Copyright © 2011 by Education Development Center, Inc.Pre-publication draft. Do not copy, quote, or cite without written permission.This material is based on work supported by the National Science Foundationunder Grant No. ESI-0917958. Opinions expressed are those of the authorsand not necessarily those of the Foundation.Development and Research Team:Cindy Carter, Tracy Cordner, Jeff Downin, Mary Fries, Paul Goldenberg, Mari Halladay, Susan Janssen,Jane M. Kang, Doreen Kilday, Jo Louie, June Mark, Deborah Spencer, Yu Yan XuPilot and Field Test Site Staff:Attleboro School District: Linda Ferreira, Jamie PlanteChelsea High School: Ralph Hannabury, Brittany Jordan, Jeanne Lynch-Galvin, Deborah Miller,Alex SomersLawrence High School: Yu Yan XuLowell High School: Jodi Ahern, Ornella Bascunan, Jeannine Durkin, Kevin Freeman, Samnang Hor,Wendy Jack, Patrick Morasse, Maureen Mulryan, Thy Oeur, Krisanne SantarpioMalden High School: Jason Asciola, Hava Daniels, Maryann Finn, Chris Giordano, Nick Lippman,Paul MarquesThe Rashi School: Cindy Carter

Unit 5: Logic of AlgebraTable of ContentsUnit 5 Mental Mathematics Activities .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiActivity 1: Who Am I? Number Bingo.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiActivity 2: Divisibility by 10, 5, and 2.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiActivity 3: Divisibility by 4 by Halving Twice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiActivity 4: Squares of Integers.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiActivity 5: Roots of Squares.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivActivity 6: Silent Collaborative Review.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivActivity 7: Factor Race.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vUnit 5: Logic of Algebra Teacher Guide .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Introduction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Lesson 1: Staying Balanced .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Additional Practice.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Lesson 2: Mobiles and Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Additional Practice.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Lesson 3: Keeping Track of Your Steps .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Additional Practice.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Lesson 4: Ordering Instructions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Additional Practice.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Lesson 5: The Last Instruction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Additional Practice.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Lesson 6: Solving Equations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Additional Practice.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Lesson 7: Solving with Squares .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Additional Practice.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Lesson 8: Solving with Systems.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Additional Practice.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Unit Additional Practice.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Unit 5 Resource Materials .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AMental Math Activity 1: Who Am I? Number Bingo Clue Cards. . . . . . . . . . . . . . . . . . . . . . . AMental Math Activity 7: Balanced Mobiles Sorting Cards.. . . . . . . . . . . . . . . . . . . . . . . . . . . . LProjection Page for Lesson 1: Balanced Mobiles Sorting Cards .. . . . . . . . . . . . . . . . . . . . . . MProjection Pages for Lesson 2: Four Corners Activity.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NProjection Page for Lesson 3: Lena and Jay’s Think-of-a-Number Tables. . . . . . . . . . . . . . . . RProjection Page for Lesson 4: Blank Think-of-a-Number Table. . . . . . . . . . . . . . . . . . . . . . . . SProjection Page for Lesson 5: Last Instruction Sorting Cards .. . . . . . . . . . . . . . . . . . . . . . . . . TProjection Page for Lesson 8: Solving with Systems Mobiles.. . . . . . . . . . . . . . . . . . . . . . . . . VSnapshot Check-in. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ZAnswer Key .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AAUnit Assessment .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ABAnswer Key .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AD

Unit 5 Mental Mathematics ActivitiesActivity 1: Who Am I? Number Bingo (2+ days)Materials:• Cut out the Clue Cards for Who-Am-I? Number Bingo on pages A-K in the Resource Materials.Setup:• All players write their own set of five 3-digit whole numbers (e.g., 472, 689, 253, 205, 100).• Shuffle the clue cards and stack them face down.Play:• Turn one clue over at a time, and read it aloud.• Each player checks all five numbers to see if the clue fits.• If the clue fits, the player crosses that number out.Ending the Game:• Any player who has crossed out all five numbers says “Bingo!”• The whole group reviews the clues to check the player(s). If the numbers are correct, the player(s) who calledBingo win. If there are errors, play continues until there is a winner.Introducing the Game for the First Time:• When playing for the first time with the whole class, read each clue, pause briefly, read it again, and pause oncemore for students to check their numbers. When students are (in subsequent days) playing this on their own inpairs or small groups, they may take turns reading a clue out loud.Play:• You (or students) can create clues with any content at all (squares, comparison of value or magnitude, multiplesor factors, fractions, decimals, sign, etc.)Activity 2: Divisibility by 10, 5, and 2 (1+ day)You say a number, and students tell you if the number is divisible by 10, 5, and/or 2.Begin by having students identify whether a number is divisible by 10, then by 5, then by 2 (do a few of each).Then switch to giving students a number and having them tell you which of 10, 5 and 2 it is divisible by.Teacher: I’ll say a number and you tellme whether it is divisible by 10.Teacher: 85. Students: No.Teacher: 20. Students: Yes.Teacher: 70. Students: Yes.Teacher: 320. Students: Yes.Teacher: 4320. Students: Yes.Teacher: 432. Students: No.Teacher: Now you tell mewhether it is divisible by 5.Teacher: 30. Students: Yes.Teacher: 52. Students: No.Teacher: 95. Students: Yes.Teacher: 19. Students: Yes.Teacher: 193. Students: No.Teacher: 1000. Students: Yes.You say a number and students tell you whether it is divisible by 10, 5, and/or 2.Teacher: 15. Students: 5.Teacher: 30. Students: 2, 5, and 10.Teacher: 44. Students: 2.Teacher: 8. Students: 2.Teacher: 7 Students: None.Teacher: 20. Students: 2, 5, and 10.Teacher: 28. Students: 2.Teacher: 505. Students: 5.Keep the pace lively. Even if notall players get the clue, moveon. That’s part of the game!Teacher: 25. Students: 5.Teacher: 50. Students: 2, 5, and 10.Teacher: 100. Students: 2, 5, and 10.Teacher: 20. Students: 2, 5, and 10.Teacher: 4. Students: 2.Teacher: 40. Students: 2, 5, and 10.Teacher: 17. Students: None.Teacher: 18. Students: 2.Use thisactivity onmore thanone day.Keep track of the cards you’veused so you can check.Teacher: Now tell me whetherit is divisible by 2.Teacher: 40. Students: Yes.Teacher: 41. Students: No.Teacher: 42. Students: Yes.Teacher: 11. Students: No.Teacher: 22. Students: Yes.Teacher: 44. Students: Yes.Prompts like 40, 18, 22,20, 24, 30, and 28, canbe fun to explore forother factors too.This exercise can be done in a particularly rapid-fire manner, as students may have little difficulty with dividingby 10, 5 or 2. However, if students struggle with divisibility by 10, 5 or 2, choose more examples for that number.iiUnit 5: Logic of Algebra

Unit 5 Mental Mathematics ActivitiesActivity 3: Divisibility by 4 by Halving Twice (1+ day)You say a number, and students tell you if the number is divisible by 4 by determining whether they can halvethe number twice. For example, for the number 68, students halve once to get 34, then realize that 34 is even, so canbe halved again, making 68 divisible by 4. For the number 50, students halve once to get 25, but then, since 25 isodd, conclude that 50 is not divisible by 4. Begin with two digit numbers.Teacher: 20. Students: Yes.Teacher: 30. Students: No.Teacher: 31. Students: No.Teacher: 24. Students: Yes.Teacher: 60. Students: Yes.Teacher: 82. Students: No.Teacher: 66. Students: No.Teacher: 36. Students: Yes.Teacher: 28. Students: Yes.Teacher: 26. Students: No.Teacher: 74. Students: No.Teacher: 58. Students: No.Teacher: 86. Students: No.Teacher: 90. Students: No.Teacher: 92. Students: Yes.Finish with a few exercises where you prompt with three-digit numbers (and higher). Note that since 100 is divisibleby 4, students only need to consider the last two digits to determine divisibility by 4.Teacher: 216. Students: Yes.Teacher: 316. Students: Yes.Teacher: 580. Students: Yes.Teacher: 642. Students: No.Teacher: 924. Students: Yes.Teacher: 521. Students: No.Teacher: 736. Students: Yes.Teacher: 154. Students: No.Activity 4: Squares of Integers (1+ day)Teacher: 5,632. Students: Yes.Teacher: 1,868. Students: Yes.Teacher: 3,870. Students: No.Teacher: 12,644. Students: Yes.Start by explaining in your own words, “I will say a number from 0 to twelve, and you will respond with the ‘square’of that number, that number times itself. So, for example, if I say ‘7’ you will answer ‘49’ because 49 is 7 x 7.”Identify three facts that students are struggling with (e.g., the squares of 7, 8, and 9 or the squares of 7, 9, and12). “Assign” one student to each of those three facts. “Assigning” a fact is saying something like “OK, for today,Brandon, you will be the ‘specialist’ for the number 9. Any time 9 comes up, you’re in charge, and you answer 81.”Those three students are now the specialists. Now, instead of asking “what is 8 x 8?,” ask “who is in charge of 8 x8?” or “who is in charge of 64?” or “who are today’s specialists?”(The point of asking who rather than the facts themselves is that the personalization of the facts–since they areassigned to particular individuals–makes them more memorable to the whole class–a more meaningful mnemonic.)Then, after it is clear that everyone knows who’s the specialist for what, begin the regular exercise, below.You say a number and students say the square of that number. Begin using only positive integers.Teacher: 4. Students: 16.Teacher: 5. Students: 25.Teacher: 9. Students: 81.Teacher: 0. Students: 0.Teacher: 2. Students: 4.Teacher: 3. Students: 9.Teacher: 1. Students: 1.Teacher: 10. Students: 100.Teacher: 5. Students: 25.Teacher: 12. Students: 144.Teacher: 7. Students: 49.Teacher: 11. Students: 121.Teacher: 8. Students: 64.Teacher: 0. Students: 0.Teacher: 11. Students: 121.Teacher: 3. Students: 9.Teacher: 4. Students: 16.Teacher: 7. Students: 49.When students are responding well, start including negative integers as prompts.Include, as a comment, something like “The square of negative 7 is the same as the square of 7. Who was in chargeof 7 today?” If students answer -49, add (with no lesson at this point), “Actually, the square is exactly the same. Justas 9 squared is, um, who’s in charge of that? Right, Maria, 81. Just as 9 squared is 81, when we square negative 9,we also get 81 (not negative 81).” You can refer back to the extended multiplication table in Unit 4 to show why.Teacher: -3. Students: 9.Teacher: -4. Students: 16.Teacher: -8. Students: 64.Teacher: 8. Students: 64.Teacher: -7. Students: 49.Teacher: -10. Students: 100.Teacher: -11. Students: 121.Teacher: -6. Students: 36.Teacher: -2. Students: 4.Teacher: -0. Students: 0.Teacher: -12. Students: 144.Teacher: -1. Students: 1.Teacher: -5. Students: 25.Teacher: -9. Students: 81.Teacher: -8. Students: 64.Teacher Guideiii

Unit 5 Mental Mathematics ActivitiesYou say a perfect square and studentssay the positive square root.Start with squares from 0 to 144.You may choose to keep the same“specialists” from the previous day toreinforce the mnemonic.Activity 5: Roots of SquaresTeacher: 25. Students: 5.Teacher: 100. Students: 10.Teacher: 9. Students: 3.Teacher: 49. Students: 7.Teacher: 16. Students: 4.Teacher: 1. Students: 1.Teacher: 81. Students: 9.Teacher: 4. Students: 2.Teacher: 36. Students: 6.Teacher: 64. Students: 8.Teacher: 0. Students: 0.Teacher: 121. Students: 11.If students start out by offering negatives as well, skip directly to the second part of this activity below.When students are responding well, start asking students for a second answer in addition to the positive answer.Teacher: 9. Students: 3.Teacher: What’s another number youcan square to get 9? Students: -3.Teacher: 36. Students: 6 and -6.Teacher: 25. Students: 5 and -5.Teacher: 64. Students: 8 and -8.Teacher: 49. Students: 7 and -7.Teacher: 81. Students: 9 and -9.Teacher: 121. Students: 11 and -11Teacher: 144. Students: 12 and -12.As an extra challenge, throw in numbers that aren’t perfect squares, too.Teacher: 100. Students: 10 and -10.Teacher: 90. Students: Not perfect.Teacher: 36. Students: 6 and -6.Teacher: 54. Students: Not perfect.Activity 6: Silent Collaborative Review (1 day)Draw or project the Silent Collaborative Review Table shown right andlocated on page L in the Teacher Resource Materials.Model what to do by silently looking over the table as if choosing a place towrite, and filling in one entry (not the first).Hand your marker to a student, and beckon, without words, for the student tofill in another entry.Hand out several more markers so a few students are up at once. As theyfinish, have them hand their markers to a classmate who hasn’t been up.Partway through the exercise, tell students “If you think something isincorrect, you may come and correct it.”Keep your words minimal to maintain the general silence.When the table is complete, point briefly at each entry, silently “asking” theclass to approve or correct it.Once all entries are correct, you can break the silence and begin the day’slesson.Teacher: 1. Students: 1 and -1.Teacher: 16. Students: 4 and -4.Teacher: 4. Students: 2 and -2.Teacher: 36. Students: 6 and -6.Teacher: 0. Students: 0.Teacher: 1. Students: 1 and -1.Teacher: 2. Students: Not perfect.Squares0 2 = √144 =1 2 = √121 =2 2 = √100 =3 2 = √81 =4 2 = √64 =5 2 = √49 =6 2 = √36 =7 2 = √25 =8 2 = √16 =9 2 = √9 =10 2 = √4 =11 2 = √1 =12 2 = √0 =Square RootsivUnit 5: Logic of Algebra

Unit 5 Mental Mathematics ActivitiesActivity 7: Factor Race (2+ days)Divide your board into 4-5 sections, one for each “team.” Have students stand in a line in their teams, facing theboard, but at least 6 feet away.When you say “Go!” the first two students in each team go the board. Each pair can talk with each other, but notwith the rest of their team. Give them a number (e.g. 30). As fast as they can, they write down all of the factors(e.g. 1, 2, 3, 5, 6, 10, 15, 30). After only 8-10 seconds, call “Time!” and call out the answers. Each team awardsthemselves one point for each factor they get correct. (You can adjust the scoring scheme to take away points forincorrect or missing factors, but it may get complicated.) If there are more than two people per team, the pair at theboard lines up at the back of their team’s line, and when you call “Go!” the next pair goes up.Keep the pace lively and competitive! Don’t wait for any team to get all the factors before calling “Time!”Teacher: 25. Students: 1, 5, 25. (You may want to give an extra point for “perfect square.”)Teacher: 8. Students: 1, 2, 4, 8.Teacher: 20. Students: 1, 2, 4, 5, 10, 20.Teacher: 17. Students: 1, 17. (You may want to give an extra point if they write “prime.”)Teacher: 45. Students: 1, 3, 5, 9, 15, 45.Teacher: 7. Students: 1, 7. (prime)Teacher: 14. Students: 1, 2, 7, 14Teacher: 28. Students: 1, 2, 4, 7, 14, 28.Teacher: 16. Students: 1, 2, 4, 8, 16. (perfect square)Teacher: 32. Students: 1, 2, 4, 8, 16, 32.Teacher: 64. Students: 1, 2, 4, 8, 16, 32, 64. (perfect square)Teacher: 6. Students: 1, 2, 3, 6.Teacher: 12. Students: 1, 2, 3, 4, 6, 12.Teacher: 24. Students: 1, 2, 3, 4, 6, 8, 12, 24.Teacher: 23. Students: 1, 23. (prime)Teacher: 21. Students: 1, 3, 7, 21.Teacher: 60. Students: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.Teacher: 9. Students: 1, 3, 9. (perfect square)Teacher: 18. Students: 1, 2, 3, 6, 9, 18.Teacher: 90. Students: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90.Teacher: 180. Students: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 45, 60, 90, 180.Teacher: 150. Students: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150.Teacher: 88. Students: 1, 2, 4, 8, 11, 22, 44, 88.Teacher: 15. Students: 1, 3, 5, 15.Teacher: 2. Students: 1, 2. (prime)Teacher Guidev

Unit 5: Logic of AlgebraWhere’s the Logic?Algebra makes logical sense in the same way that mobiles make sense and number lines makesense and multiplying makes sense. Now is the time to connect them all together. In this Unit,you will revisit mobiles and you’ll see a lot of equations - it may look like materials from anyother algebra class. At every step, you can make sense of what you’re doing. You’ll find thatcomplicated problems can be thought of in simpler ways, and the tools you’ve been using so farcan really help you with algebra.Lessons in this Unit:1: Staying Balanced2: Mobiles and Algebra3: Keeping Track of Your Steps4: Ordering Instructions5: The Last Instruction6: Solving Equations7: Solving with Squares8: Solving with SystemsMysteryGrid 7-8-9 Puzzle56, x 72, x7 9 824, + 63, x8 7 9You will solvepuzzles like these.MysteryGrid 2-7-9 Puzzle63, x 14, x9 2 718, + 7, –7 9 219 29= ______ 5 = ______ 716 – a+ 527Think of a number.Subtract it from 16.Divide by 7.9872 7 9Add 52.Algebraic Habits of Mind: Using StructureYou will see that, sometimes, algebra problems can look exactly like mobile problems, and you can use thesame strategies you used for mobiles to simplify and reorganize an equation.Sometimes, an algebra equation looks complicated, but is really just a lot of layers of instructions, like inThink-of-a-Number puzzles. If you can figure out what the instructions are, you can start to unravel themuntil you arrive at the number in the bucket.And sometimes, you see numbers that you recognize, but they’re mixed in with expressions that look toocomplicated. You’ll learn that sometimes it’s good to leave those expressions as a complicated chunk todeal with later, and you’ll learn how to use the numbers you recognize first.And through these structures, you’ll start to see the logic in algebra.Copyright © 2011 by Education Development Center, Inc. 1

Unit 5 Teacher GuideUnit 5: Logic of Algebra builds facilitywith manipulating expressions and equationsthoughtfully, logically, and accurately.Students will use the common-sense logicthey have developed–particularly in theThink-of-a-Number tricks and mobilepuzzles–to make sense of the syntacticmanipulations of algebra (the standard“solving rules”). Primary emphasis isplaced on using logic and understanding theunderlying structure of algebra problems.Lessons 1 and 2 relate mobiles to algebra.Students have encountered mobiles since Unit 1.Their work in Unit 5 will make the connectionbetween mobile pictures and algebraic equationsexplicit.Lessons 3, 4, and 5 explicitly connect theinstructions of a Think-of-a-Number trick tolayers of an algebraic expression. Studentswill see the structure behind complicatedalgebraic expressions using order of operationsand will use ideas of “undoing” that structuresystematically to solve the equations.Lessons 6 and 7 extend the ideas of Lessons 3, 4, and 5 butwithout using the Think-of-a-Number table. In Lesson 5,students identify the last instruction of a Think-of-a-Numbertrick; in Lesson 6, students isolate that step and treat the rest ofan expression as a single “chunk” to solve equations. Lesson 7extends that idea to solving equations with squared expressions.Lesson 8 explores systems of equations through mobiles. Thislesson is not intended to teach methods of manipulating systemsof equations, but rather lets students explore the way thatsystems of equations give more information about variables, andshows that sometimes these equations (or mobiles) have to beused together to solve for a variable.Algebraic Habits of Mind in Unit 5:• Using Structure: Students will see the structuralrelationship between mobiles and equations. Studentswill see that the structure of an algebraic expressioncan be made simpler by identifying just the last step ofa series of instructions and treating everything else as a“chunk” of information.• Using Tools Strategically: Mobiles are useful forvisualizing what kinds of manipulations maintainbalance, but the metaphor becomes weak for negativenumbers and awkward for fractions and large numbers.Think-of-a-Number tricks are used to analyze thestructure of a complex expression, but are not as usefulwhen a problem requires complicated expressions onboth sides of the equation. We use these tools to helpus see the logic behind algebra, but ultimately, algebraitself is the most versatile tool we have.• Clear Communication: Students will experience theneed to be precise not only with the vocabulary theyuse to describe algebraic steps, but also with the orderin which they present these steps.Learning Goals:• Solve equations using properties of operations and the logic of preserving equality.• Translate between verbal instructions and algebraic expressions.• Articulate the “common sense” behind rules of algebraic manipulations.• Develop mathematical language related to calculations and equations.Many students experience mathematics as aset of rules to follow and get answers. Withstudents who have had limited success withmathematics, it is especially tempting to teachthat way since it often feels like the least riskyway to help them succeed on tests and pushingfor the logic requires more time and effort andmay even be beyond what these students canachieve. Focusing on rules is faster in the shortrun–it does help with tests that follow soon afterinstruction–but without constant maintenance itis not long-lasting, and strugglers are typicallybetter at the “common-sense” logic than atremembering and appropriately applying rulesthey don’t understand.Unit 5: Logic of Algebra TG - 1

Lesson 1: Staying BalancedWhy This Lesson?This lesson identifies moves that preserve balance and are therefore allowed in algebra.Why This Way?Using mobiles encourages students to extend mobile logic to algebraic expressions to understand allowablealgebraic manipulations.. Using mobiles and symbols together helps students consider algebraic steps moregenerally and not only as ways to make an equation simpler towards the goal of solving for a variable.Mental Mathematics:(5 min) Activity 1: Who Am I? Number BingoBalanced Mobiles Sorting Cards:As a Class(10 min) Before class, photocopy page M fromthe Resource Materials. Provide one copy forevery group of 2-3 students. Pre-cut the cardsif possible.Sample: Card AIf this mobile balances.. does this? YES or NOStudents will sort these cards into piles for YES and NO. Ifyou wish, you may have them tape or glue their cards onto a sheet of paper divided into a YES and NO section. Asgroups work, refrain from correcting individual cards. Instead, give hints like: “Three of your group’s cards are inthe correct category.” As you circulate, ask questions of both correct and incorrect cards:• “How did you know for sure that this mobile also balances?”• “Why isn’t this mobile guaranteed to balance?”After 5-7 minutes, bring the class together, even if they aren’t finished.Write the answers on the board (Yes: A, B, D, F, H; No: C, E, G). Usethe remaining time to have a student-centered discussion about whysome mobiles stay balanced and others don’t. Try to limit your role asmuch as possible to recording student responses.Leave this language on the board for students to refer to as they workon their student pages. If students don’t volunteer answers, or someanswers are incomplete, that’s ok for now. You will have a chance torevisit this question at the end of the lesson.Answers for Card-Sorting Activity:A: Yes, both sides were doubled (or, since 2 hearts = pentagon + moon, both sides increased by the same weight).B: Yes, a leaf and bucket were removed from both sides.C: No, a circle and star were exchanged, but they aren’t necessarily equivalent weights.D: Yes, the sides of the mobiles and the shapes were rearranged, but equal quantities still hang on both sides.E: No, a clover was moved from the left to the right side, so the right side is now heavier.F: Yes, a pentagon was removed from both sides.G: No, a moon was added to the left and a circle was added to the right, and they aren’t necessarily equal weights.H: Yes, a diamond weight was added to both sides.It is important for all students toexperience success, but also importantto move activities along at aninteresting pace. Even fun activitiesbecome boring when they drag. Usethis card sorting activity to whetstudents’ curiosity, and use the studentpages to provide further support.TG - 2Unit 5: Logic of Algebra

5-1 Staying BalancedIn each problem, the first mobile is balanced. Use it to figure out whether that means thesecond mobile for sure has to balance. Circle your response and explain how you knowyour answer is right.12The only movesallowed on a mobileare moves that keep themobile balanced.If this mobile balances...does this? YES or NOExplain:An equal weight (a circle) was addedto both sides.If this mobile balances... does this? YES or NOBoth sides were halved. (Or,Explain:since a clover = 2 hexagons,an equal quantity was subtractedfrom both sides.)34If this mobile balances...does this? YES or NOIf this mobile balances...does this? YES or NOExplain:Both a diamond and a bucket weresubtracted from both sides.Explain: The pentagon weight wasmoved from the left to the right, sothe right is now heavier.Translate each mobile into algebra. Let = m, = s, and = h.5 6 7____________ 2m + s = ____________ 2h + 3s ____________ h + m + h = _________________ 2s + 2m + s _________ 5s = _______________m + 2h + mOR 3s + 2mOR 2m + 2h8 Which of the following moves would keep the mobile balanced? Circle all that apply.A Add a pentagon to both sides. D Switch the leaf and circle.B Add 5 leaves to both sides. E Add a circle to the right side.CMove all the pentagons to theright side.FRemove one pentagon fromboth sides.2 Unit 5: Logic of Algebra

Algebraic Habits of Mind:Communicating Clearly:Encourage students to reason throughthese mobiles together. Students needthe experience of describing theseideas out loud in order to clarifyexactly what moves are allowed (e.g.“you group the diamonds togetherand group the circles together, butonly on the same side of the mobile”).Let students use their own internallogic, intuition, and understanding todescribe what they are doing.Additional Discussion Questions:• If 4b = 10c, then 3b = 9c. Is this true? (No) Explain.• If 4b = 10c, then 2b = 5c. Is this true? (Yes) Explain.• Why is “doubling” a legitimate move to make on a mobile?If 3 = + can you really say 6 = 2 + 2 ?Aren’t you adding 3 stars to the left side and a circle and abucket to the right side? (Yes, but 3 stars is the same quantityas a circle and bucket.)• A student asks: “Is 3x + 8 + 2x the same thing as 8 + 6x?”How would you explain the student’s mistake? (Think of amobile and realize that you’re not adding 3x twice (that’smultiplication), but that combining 3x’s and 2x’s gives 5x’s.It is correct to say 8 + 6x or 6x + 8.)Student Problem Solving:Small Groups (20 min) Encourage students to work through the problems in the Student Book together. Asstudents work, help them refine the language they use to describe the moves happening on themobiles and in the equations. Judiciously ask:• “How did you know for sure that this mobile also balances?”• “Why isn’t this mobile guaranteed to balance?”• “Could this mobile balance? Ah, but only if... So it’s not guaranteed to balance.”Summarizing Discussion:As a Class(10 min) Share answers from the Discuss and Write What You Think box. This discussion shouldfollow from the conversation from the beginning of class. Students have had a chance to practicedescribing the moves as they’ve solved problems, so now is a good time to bring the class togetherto clarify that, for example, crossing out the samesymbols on both sides of the mobiles is equivalent tosubtracting their symbols in an equation. Emphasize thatthis is why algebra steps work they way they do: allsteps in solving an algebra problem preserve equality.Teacher Notes:You’ll notice much more Additional Practice in Unit5. Be strategic in how you use these problems. Inthis unit, you’ll notice that most problem types aregrouped together in the Additional Practice section.This was done on purpose to allow you to easilyassign certain problems to groups of students. EachAdditional Practice section also includes reviewfrom previous units and fun puzzles. If students areworking ahead, let them loose on these puzzles!Looking ahead: Lesson 3 usesThink-of-a-Number tricks. Use theThink-of-a-Number problems in theAdditional Practice in Lessons 1and/or 2 to help prepare them.TG - 3Unit 5: Logic of Algebra

Translate each mobile into algebra or draw the mobile for the equation provided. Explain the change from the first to thesecond mobile and equation. Let = m, = s, and = h.910_________ m = _________ 2h ________ 3m = __________ 2h + 2mHow did the mobile and the equation change?2m was added to both sides._________ 3s = _________ h + m _________ h + m = 3sHow did the mobile and the equation change?The sides of the equation (andmobile) were switched.1112_________ 2h = _________ 2s + m 4h = ___________ 4s + 2m _________ m+s+m = _________ 2h+s+m _________ s + 2m = _________ s + 3mHow did the mobile and the equation change?Both sides were doubled. (Or, since2h = 2s + m, an equivalent amountwas added both both sides.)_________________ 2s + m + 2s = _________________s + m + h + mHow did the mobile and the equation change?The order of the variableschanged, but the same quantitiesstay on both sides.13 This mobile balances. Translate it into an equation and circle all of the equations that must also be balanced.m =s =h =Discuss & Write What You ThinkAB2s + m + 2s = s + m + h + m4s + m = s + 2m + hLesson 1: Staying Balanced 3CDEm + 4s = 2m + s + h5s + m = 2m + h3s = h + mHint: Youshould havecircled fouranswers here.And in #8.Encourage students to keep expressing theseideas in words. Listen for the following:What moves are you allowed to make on a mobile or equation? What moves are you not allowed to make?Allowed: Adding/Subtracting the same thing to both sides, Switching theentire left and right sides, Changing the order of variables within one side,Doubling/Halving/Tripling/etc. both sides, Substituting equal quantities.Not Allowed: Moving one piece from one side to the other, Adding/Subtractingunequal amounts

Additional Practice ProblemsSelect problems that will help you learn. Do some problems now. Do some later.In each problem, the first mobile is balanced. Use it to figure out whether that means the second mobile for sure has tobalance. Circle your response and explain how you know your answer is right.ABIf this mobile balances...does this? YES or NOIf this mobile balances...does this? YES or NOExplain:One heart was added to the left, buttwo hearts were added to the right.Explain:Two drops and a leaf weresubtracted from both sides.Translate each mobile into algebra or draw the mobile for the equation provided. Explain the change from the first to thesecond mobile and equation in words. Let = m, = s, and = h.CD_______ 2h = ___________ 2m + 2s_________ h = _________ m + s _________ 2s = _________ h ___________ 2s + 2m = h _________ + 2mHow did the mobile and the equation change?Both sides were halved. (Or, sinceh=m+s, the same quantity was subtractedfrom both sides.)How did the mobile and the equation change?2m was added to both sides.EF_________ s + m = _________ 3h _________ 3h = _________ m + s m+s+2m _________ = _________ s + 2h _________ 3m = _________ 2hHow did the mobile and the equation change?The sides were switched.How did the mobile and the equation change?One star was removed from bothsides.4 Unit 5: Logic of Algebra

GWhich of the following moves would keep the mobile balanced? Circle all that apply.i Add 3 circles to the right side. iv Subtract a circle from both sides.ii Add 2 clovers ( ) to both sides. v Cross out both hexagons.iiiMove the shapes so all 5 circlesare on the left side and bothhexagons are on the right side.viSubtract 3 circles from the leftside.HWhich of the following moves would keep the mobile balanced? Circle all that apply.i Add 4 water drops to both sides. iv Add a bucket to both sides.ii Remove a bucket from both sides. v Switch the leaf and the circle.iiiMove all the buckets to the left sideand the water drop to the right side.viSwitch the leaf and the waterdrop.IWhich of the following equations also must be balanced? Circle all that apply.ih + s + s + h = m + m + hiv2m + h = h + 2s + hii2s + h = 2mv3s + 2h = 2m + h + sJFill in the blanks in this Think-of-a-Number Trick.iii6h + 6s = 6m + 3hvi3h = 2m + 2sAnother hint:G: 3 answersH: 4 answersI: 5 answersInstructions Pictures Result Jacob Mali KaylaThink of a number.b 4 7 3Add 5. b + 5 9 12 8Multiply by 2. 2b + 10 18 24 16Subtract 4. 2b + 6 14 20 12KMysteryGrid 1-2-3-4 PuzzleLMysteryGrid 1-3-4-5 PuzzleMMysteryGrid 2-3-7-8 Puzzle8, x 11, + 6, +13 4 24, + 4, ÷ 1, –17, + 56, x 8, +3 1 4 5 8 7 2 32 4 1 320, x 12, +75 3 1 4 2 8 3 7412, x 4, x23 12, –18, x 4, ÷1 4 5 3 7 3 8 23, ÷312 415, x1, –4 5 3 1 3 2 7 8Lesson 1: Staying Balanced5

Lesson 2: Mobiles and AlgebraWhy This Lesson?This lesson connects the process of solving a mobile to the process of solving an equation.Why This Way?Mobiles are a tool for getting an intuitive understanding of the logic behind solving equations, and thislesson makes the point of using mobiles to explore equations that students typically find daunting (especiallyequations with variables on both sides or like terms that are split up). Students are encouraged to understandand justify different approaches for solving equations using mobile logic to help them understand steps theymay be asked to perform in algebra.Mental Mathematics:(5 min) Activity 2: Divisibility by 10, 5, and 2Four Corners:As a Class(15 min) Project the mobile on page N of the Resource Materials (shownto the right).Ask: What are some ways to start solving this mobile? You may hear:• Cross out any 2 hearts on the left side and the 2 hearts on the right side. (*)• Cross out 6 on the left side and turn the 16 on the right side into a 10. (*)• Cross out 6 on the left side and turn the 14 on the right side into 8. (*)• Combine the hearts on the left side as 5 hearts + 6 on the left side.• Combine the numbers on the right side as 30 + 2 hearts on the right side. (*)Ask: What moves are you not allowed to make?• Lump all seven hearts together.• Combine all the numbers together.• Add 3 extra hearts to the right to “even out” the hearts.• Cross out 6 on the left side and then both turn the 16 into a 10 and also turnthe 14 into an 8.The four starred (*) possible steps above are provided as pages O and P in theResource Materials. Print out a copy of these pages, cut on the dotted lines, andtape the four options to the four corners of your classroom. Tell everyone to picktheir favorite first step and that when you say “Go,” they should run (well, walk)to the corner of the step they chose. Remind them that all are valid steps and thatthis is simply a matter of preference.You can modify the activityso that the students are splitinto four groups, and eachgroup gets a different “Step1” to start with. Putting theoptions in the four cornersjust gives students morechoice and gives studentsa different experience ofworking together.Once students get to the four corners, give them a few moments to confer and solve the mobile, using their first step.You may choose to print out four copies of the mobile to help them do this, and page Q in the Resource Materials isprovided for this purpose.After 1-2 minutes, get everyone’s attention. Using your discretion, you may ask students stay in their corners andpay attention, or you may ask students to return to their seats. First, make sure that all students understand how totranslate the mobile into algebra: h + 6 + 4h = 14 + 2h + 16.The answer is h = 8(Continued on page TG-7)61416TG - 6Unit 5: Logic of Algebra

5-2 Mobiles and AlgebraSolve both the mobile and the equation, and check thatyou get the same answer.The only steps that may be performed to an algebraequation are steps that keep the equation balanced.1Find c...by using this mobile.2Find p...by using this mobile.9 + 4c = 10 + c + 11Encourage variousmethods.One possibility:(subtract 9)4c = 1 + c + 11(cross out c)3c = 12c = 49 1011p + 39 + p = 5p + 12Encourage variousmethods.One possibility:(cross out 2p)39 = 3p + 12(subtract 12)27 = 3p9 = p3912Find h...by drawing and using this mobile.3 4Find m...by drawing and using this mobile.h + 5 + 2h = 13 + hEncourage variousmethods.One possibility:(cross out h)5 + 2h = 13(subtract 5)2h = 8h = 4515 + 2m = 2m + 9 + m13Encourage variousmethods.One possibility:(cross out 2m)15 = 9 + m(subtract 9)6 = m1595 Which of the following would work as a first step to solving this equation (use s = )? Circle all that apply.ASubtract s from both sides.91227BSubtract 9 from both sides.C Add 12 and 27 so the right side becomes 39 + s.DMove all the stars to the left and all the numbers to the right to get 5s = 48.ECombine the stars on the left side to get 4s + 9 = 12 + 27 + s.Choose one of the first steps you circled, and useit as your first step to find s:Find a friend who chose a different first step andcompare your methods. Record their process here:Encourage various methods. You may also choose to have severalstudents communicate their process on the board.Ultimately, 3s = 30, so s = 106 Unit 5: Logic of Algebra

Ask one representative from each group to say theirsteps for solving the mobile, in order. While they dothis, you will show the same steps using algebra. (Bythe way, if you had students who were “too cool” toactually go to one of the four corners, they can be thescribes at the board for this part.) As students sharetheir steps, keep asking students: “How would wewrite that step using algebra?” Keep allfour solutions on the board, if possible.h + 6 + 4h = 14 + 2h + 16“Cross out 2 hearts on both sides”Show students that subtracting 2hfrom the left side can happen byh + 6 + 4h = 14 + 2h + 16-h -h -2hOR6 + 3h = 14 + 16h + 6 + 4h = 14 + 2h + 16-2h -2hh + 6 + 2h = 14 + 16Student Problem Solving:Small Groupsh + 4h = 8 + 2h + 10(15 min) Havestudents work in theStudent Book in small groups.Encourage students to use different methods ofsolving the mobiles. Don’t let students get stuckin the trap of thinking that there is only onecorrect method. Emphasize the logic – solvingequations requires the mobile to stay balanced.Ask: “What move makes sense to make?”Summarizing Discussion:As a Class(10 min) Ask for studentanswers to the purple Discuss& Write What You Think box.Keep in mind that explanations are hard towrite, but continue to insist that students writedown their thoughts to develop their skills. Theability to explain a mistake is useful in figuringout why certain algebraic steps are incorrect.h + 6 + 4h = 14 + 2h + 16“Cross out 6 on both sides”The goal is for students to understand that every step ona mobile can be represented with algebra, and why wewrite the steps the way we do. For example, on a mobile,it may make intuitive sense to just “cross out” two starson each side, but, in algebra, we don’t just cross out theletter h twice, we represent the process with subtraction.Show students that subtracting 6can happen byh + 6 + 4h = 14 + 2h + 16OR-6 -6h + 4h = 8 + 2h + 16h + 6 + 4h = 14 + 2h + 16-6 -6h + 4h = 14 + 2h + 10BUT NOT (Why not?)h + 6 + 4h = 14 + 2h + 16-6 -6 -6h + 6 + 4h = 14 + 2h + 16Show students the logicof combining like terms.After students, for example,subtract 6 from the left side,most students will naturallyjust say there are 5 heartsleft. Your job is to show thatwhat they see as obvious alsoworks in algebra as the stepcommonly called “combininglike terms.” Remember,the primary emphasis is onusing logic. Your job is tohelp make the connection toalgebra explicit.Additional DiscussionQuestions:• You ask a friend forhelp in solving Problem 8, and he or she says to you “Ilearned that when you have an equation like this (h + 25+ h = 3h + 2h +1) you have to combine like terms first.”• What does your friend mean by “combine liketerms”?• Your friend thinks that the first step has to becombining like terms. Explain to your friendanother first step he or she can take and why it alsoworks.• You may have heard an algebra teacher say “Whateveryou do to one side of an equation, you have to do tothe other.” You can say: “That’s not true! In Problem1, I can add 10 and 11 on the right side to get 21without doing anything to the left side.” Your teachermight respond: “You’re right. What I meant to saywas......” What did your teacher mean by his or her firststatement?TG - 7Unit 5: Logic of Algebra

6 Which of the following would work as a first step to solving this equation? Circle all that apply.5b + 2 = 2b + 16 + bFind b, too.One possibility:Subtract 3b:2b + 2 = 16Subtract 2:2b = 14b = 7 b = ______ 7Discuss & Write What You ThinkABCDEFOn the right side, turn 2b + b into 3b.Subtract 2 from both sides.Subtract 3b from both sides.Combine all the b’s and all the numbers to get 8b = 18.Algebraic Habits of Mind: Using Tools StrategicallySubtract b from the right and add it to the left side to get 6b + 2 = 2b + 16.Subtract b from the left side and move it to the right to get the same numberof b’s on both sides: 4b + 2 = 4b + 16.For problem 6, explain why... Emphasize clear communication. Listen for:choice d won’t balance: choice e won’t balance: choice f won’t balance:You can’t combine If you subtract b fromb’s on oppositethe right, you have tosides. Moving 2 subtract b from theto the right side left, too. The sides onlydoesn’t necessarilystay balanced by doingbalance moving the same thing to both3b to the left.sides.Draw a mobileif it helps yourexplanation.It’s not about justbalancing the b’s.Moving b to theright is the same asadding b to the right,but you subtracted bfrom the left.Mobiles are useful tools for visualizing balance. They help make sense of algebra, since solving algebraproblems always requires the equation to stay balanced.Mobiles are not always useful, however. For instance, how would you represent -5 on a mobile? Or -x?Or 1 y? Or -6.27s? You get the idea.2You can’t draw a mobile for every algebra equation. But every equation balances. And anything you do toan equation has to keep the balance.Find s...by drawing and using this mobile.7 8Find h...by drawing and using this mobile.10 + 4s = 8 + s + 11Encourage variousmethods.One possibility:(cross out 10)4s = 8 + s + 1(subtract s)3s = 9s = 310 811h + 25 + h = 3h + 2h + 1Encourage variousmethods.One possibility:(cross out 2h)25 = 3h + 1(subtract 1)24 = 3h8 = h251Lesson 2: Mobiles and Algebra7

Additional Practice ProblemsSelect problems that will help you learn. Do some problems now. Do some later.AFind c...by using this mobile.BFind h...by using this mobile.c + 8 + 4c = c + 13 + c + 13Encourage variousmethods.One possibility:(cross out 2c)3c + 8 = 13 + 13(subtract 8)3c = 5 + 133c = 18c = 68 13137 + h + 2h = 5 + 4hEncourage variousmethods.One possibility:(cross out 3h)7 = 5 + h(subtract 5)2 = h75CFind s...by drawing and using this mobile.DFind m...by drawing and using this mobile.15 + 2s + 5 = s + 2 + 3sEncourage variousmethods.One possibility:(subtract 2)13 + 2s + 5 = 4s(subtract 2s)13 + 5 = 2s18 = 2s9 = s15528 + m + 31 = 2m + 6 + 2mEncourage variousmethods.One possibility:(cross out m)8+31 = 2m + 6 + m(subtract 6)2 + 31 = 2m + m33 = 3m11 = m8316EFWhich of the following would work as a first step to solving this equation? Circle all that apply.i Subtract 7 from both sides. Find p, too.ii Add 19 + 7 + 18 = 44.One possibility:Subtract 18:1918 iii Subtract 18 from both sides. p + 1 + 7 = 3p7iv Subtract a pentagon from both Subtract p:sides.8 = 2pp = 4Which of the following would work as a first step to solving this equation? Circle all that apply.10 + 3d + 2 = 6 + 5di Add 10 + 2 = 12 on the left. Find d, too.iiiiiivSubtract 3d from both sides.Subtract 6 from both sides.Subtract 2 from both sides.One possibility:Subtract 3d:10 + 2 = 6 + 2dSubtract 6:6 = 2dd = 3.p = ______ 4d = ______ 38 Unit 5: Logic of Algebra

GFill in the blanks in this Think-of-a-Number Trick.Instructions Pictures Result Luis Jess RajThink of a number.Multiply by 3.b 7 -2 33b 21 -6 9Add 5. 3b + 5 26-1 14Multiply by 2. 6b + 10 52 -2 28Subtract 8. 6b + 2 44 -10 20I am q. Where am I? ______ 5.28I am s. Where am I? ______ 5.335I am v. Where am I? ______ 5.375HI I I I I I I I I I I I I I I I5.31 5.4I am r. Where am I? ______ 5.3I am t. Where am I? ______ 5.35Use an area model to multiply these expressions.I 100 ____ 20 ____ 1J ____ a ____ 3____ 30 3000 600 30____ 8 8a 24____ 4 400 80 4____ -b -ab -3b121 • 34 = _________ 4114(a + 3)(8 – b) = 8a - ab + 24 - 3bK____ 2c ____ 3 ____ d ____ y ____ -7____ c 2c 2 3c____ 9 18c 27L____ dd 2dy -7d____ 4 4d 4y -28M(c + 9)(2c + 3) = 2c 2 + 21c + 27(d + 4)(d + y – 7) = d 2 + dy - 3d + 4y - 28MysteryGrid 1-2-3-4 Puzzle6, x 12, x1 2 3 4NMysteryGrid 3-4-5-6 Puzzle11, + 36, x5 6 3 4OMysteryGrid 5-6-7-8 Puzzle1, – 3, –6 7 8 5316, x 3, +41 24 15, x 10, +456 340, x 30, x 42, x856 78, x 3234 118, x 30, x634 55635, x 48, x7 8415, +2 3320, x45 656, x785 6Lesson 2: Mobiles and Algebra9

Lesson 3: Keeping Track of Your StepsWhy This Lesson?Students will see that complicated algebraic expressions can be seen as a series of individual steps.Why This Way?Algebra students are frequently taught to “undo” steps in order to solve equations. For this to make sense,students must first see that a complicated algebraic expression is actually made up of simpler elements. Thislesson uses the familiar Think-of-a-Number trick format to show how expressions can be built by adding onelayer of complexity at a time. This logic will also be developed through Lessons 4 and 5 to support the logic ofsolving equations in Lesson 6.Mental Mathematics:(5 min) Activity 3: Divisibility by 4 by HalvingTwiceThinking out Loud:As a Class(10 min) Project the top half ofpage 10. If you like, a blank copy isprovided for you as page R in theResource Materials. Students should observe thatLena and Jay are both correct, and their bucketpictures show that they get equivalent results.Have students read the Thinking out Loud dialogueas a class. Make sure students see how 3(b + 4) – 10is not only a way to record the result, but also toencode the instructions as well. Make sure students also answer the question in the Pausing to Think box and work tounderstand the main points of the dialogue.Student Problem Solving:Small Groups(15 min) Have students work in the Student Book insmall groups.Summarizing Discussion:As a Class(10 min) There are several ideas introduced in theseproblems that will be addressed in later lessons. Inparticular, students will see more problems whichaddress the language used with subtraction and division, and studentswill see “chunking” as a way to solve equations. Students have beenrewriting complicated expressions from the previous step withoutworrying about what the expression is (thereby treating theexpression as a “chunk”). Depending on what students are ready for,you may wish to explore some of these topics by using the discussionquestions in the purple boxes to the right and on page TG-11.Algebraic Habits of Mind: Using StructureAs students fill out these Think-of-a-Number tables,they are learning about the structure of algebraicexpressions. In particular, they are learning that newinstructions (or steps) do not affect previous steps;they simply follow in sequence. This experiencesupports “chunking” in Lesson 6, because they’veseen that larger pieces of information can be groupedand considered together (as a “chunk”) withoutworrying about the details of what is contained in thatchunk.Discussion Questions:• When we talk about subtraction,we frequently use the word“from.” If someone is havingtrouble figuring out the differencebetween “subtract 4 from 10” and“subtract 10 from 4,” how wouldyou help explain it to them?• When we talk about division,we usually use the word “by.” Ifsomeone is having trouble figuringout the difference between “divide6 by 3” and “divide 3 by 6,” howwould you help explain it to them?TG - 10Unit 5: Logic of Algebra

5-3 Keeping Track of Your Steps1Lena filled in a Think-of-a-Number table this way.Instructions ResultJay filled in the same Think-of-a-Number table this way.Instructions ResultThink of a number.bThink of a number.bAdd 4. b + 4Multiply by 3. 3b + 12Subtract 10. 3b + 2Add 4. b + 4Multiply by 3. 3(b + 4)Subtract 10. 3(b + 4) – 10Draw Lena and Jay’s bucket pictures.Instructions Lena’s Picture Jay’s PictureThink of a number.Add 4.Multiply by 3.Subtract 10.Thinking out LoudLena: I guess we got the same answer. But I like my final result better. It’s simpler!Michael: I like Lena’s final result better, too. 3b + 2 is definitely simpler than 3(b + 4) – 10. Why didyou write it that way, Jay?Jay: Actually, I like Lena’s final result, too. But I was trying to make my final result look like aThink-of-a-Number trick.Lena: What do you mean? Why doesn’t mine look like a Think-of-a-Number trick?Jay: Well, it does... just not the one in the problem. Look at my equation. Looking at 3(b + 4) – 10,you can see the instructions, can’t you? You can see that you start with b, then you first add 4,then multiply by 3, and then subtract 10.Michael: Oh, I see! And Lena took the same steps to get to her final result, but you can’t tell what thosesteps were. If you just saw her final result, you could have thought the instructions were...Pausing to ThinkFinish Michael’s thought. If all you saw was 3b + 2, what might you say werethe Think-of-a-Number trick instructions?Think of a number, multiply by 3, then add 2.10 Unit 5: Logic of Algebra

Teacher Notes:Additional Discussion Questions:• If you just look at the last line of any of theseThink-of-a-Number tables, can you tell what thefirst instruction must have been? (locate the variableand use order of operations or just see what’s“closest”)• When do you need to worry about usingparentheses? (explain to students that a divisionbar automatically groups the numerator anddenominator, so parentheses are implied there)The Additional Practice pages includeproblems and puzzles from Units 2, 3,and 4. They are useful review materialsfor students who like to work aheadin the Student Book. They may alsobe useful to help fill gaps for studentswho missed portions of previous units.Let us know how you are using theseAdditional Practice problems.TG - 11Unit 5: Logic of Algebra

Fill in the missing information on the following Think-of-a-Number tricks.2 Instructions Result3 Instructions ResultThink of a number.cThink of a number.bMultiply by 2.2cAdd 4.b + 4Subtract 3.2c – 3Multiply by 3.3( b + 4 )Multiply by 5.Add 16.5(2c – 3)5(2c – 3) + 16Divide by 2.Subtract 9 from theresult.3( b + 4 )23(b + 4)2- 94 Instructions Result5 Instructions ResultThinkof a number.Subtract 2.Divide by 3.Add 25.Multiply by 8.8(kk – 2k –23k – 23k – 23+ 25+ 25)Think of a number. wDouble that.Subtract that resultfrom 9. 9 -Multiply that by 3(use parentheses).Add 8.2w2w3(9 - 2w)3(9 - 2w) + 8“Subtract9” wouldbe 2w – 9.“Subtractfrom 9”means9 – 2w.6 This mobile This mobileHow many heartsbalances. balances, too.will make thisbalance?How many heartswill make thisbalance?If = 8,then what arethe values ofand ?==4687 Instructions ResultInstructions ResultSubtract 1.Divide the result10( m3 + 11) – 1 2(y - 6) + 5by 3.3Think of a number.mThink of a number.yDivide by 3.Add 11.m3m3 + 11Subtract 6.Multiply by 2.y - 62(y - 6)Multiply by 10.Add 5.10( m3 + 11)2(y - 6) + 5Lesson 3: Keeping Track of Your Steps11

Additional Practice ProblemsSelect problems that will help you learn. Do some problems now. Do some later.Fill in the missing information on the following Think-of-a-Number tricks.A Instructions ResultB Instructions ResultThink of a number.nThink of a number.cMultiply by 9.9nMultiply by 14.14cAdd 1. 9n + 1Add 8.Multiply by 4.4(9n + 1)Multiply the resultby 5.4(9n + 1) + 2Subtract 2 from theAdd 2.result.14c + 85(14c + 8)5(14c + 8) – 2CInstructions ResultDInstructions ResultThink of a number.dThink of a number.hSubtract 2.d – 2Add 5.h + 5Multiply by 13.13(d - 2)Multiply the resultby 2.2(h + 5)Add 6.13(d – 2) + 6Add 13.2(h + 5) + 13Divide by 15.13(d – 2) + 615Divide by 9.2(h + 5) + 139EInstructions ResultFInstructions ResultThink of a number.Add 4.Divide 20 by the result.Subtract 1.Multiply by 6.pp + 420p + 420p + 4 – 16(20p + 4 – 1)Divide 20 bythe resultmeans theresult fromthe step abovegoes on thebottom.Think of a number.Divide 40 by theresult.Subtract 21.Multiply the resultby 3.Add 81.k40k40k - 213( 40k - 21)3( 40k- 21) + 81GPlace these numbers on the number line below: 1.5 1.05 0.5 0.05347854881680.05 0.51.05 1.5I 11 1 1 I 1 1 1 1 I0 3 7 1 515 24 8 8 48 168812 Unit 5: Logic of Algebra158

Illustrate each problem on a number line. Then say only whether the answer will be positive or negative.H 551 – 569 is _______________ NegativeI 5.28 –5.8 is _______________ NegativeJ -2.3 – -1.8 is _______________Negative569Markthe 0I0I551Don’t actuallysubtract these.I I0 5.28II-2.3 0KNWho Am I?h t u• I am even.• u = h4 8 4• I am greater than 200.• My hundreds digit is aperfect square.• t = u + h• My tens digit is twice my hundreds digit.I am exactly halfway between 8.2 and 8.7. Who am I? _______ 8.45I I8.2 8.7L-5a ________ w ____ 8 ____ -a-5aw-40a-5a(w + 8 – a) = -5aw - 40a + 5a 2MMysteryGrid 2-3-4-5 Puzzle12, x 10, + 8, x3 4 5 210, x525a 2212, x 10, +33 44 5OI am exactly halfway between 2.89 and 2.9. Who am I? ________ 2.89520, x452 3I I2.89 2.9P32 42= ______ 2 = ______ 4 = _______ 7QWho Am I?h t u• I’m not an even number.• Two of my digits are 6 2 1even.• My units digit is half mytens digit.• All three of my digits are different.• My hundreds digit is twice the sum ofmy units digit and my tens digit.R____ 2n____ 7p____ -6____ n ____ -3p2n 2 -6np7np -21p 2-6n 18p(n – 3p)(2n + 7p – 6) =2n 2 + np - 21p 2 - 6n + 18pLesson 3: Keeping Track of Your Steps13

Lesson 4: Ordering InstructionsWhy This Lesson?Students will see how the order of instructions is reflected in an algebraic expression.Why This Way?This lesson, like Lesson 3, seeks to support students’ logical reasoning about algebraic expressions. Studentswill see how the structure and meaning of an expression can vary widely depending on how the same set ofinstructions is ordered. Also, students will see that it is possible to discern the order of instructions by startingat the variable and “working outward.” This lesson seeks to reinforce the idea of order of operations in thecontext of algebraic expressions.Mental Mathematics:(5 min) Activity 4: Squares of IntegersMix It Up:As a Class(10 min) Have the following fourinstructions on the board: Multiply by5, Add 8, Subtract 1, Divide by 4.Write them horizontally so they don’t look like they’realready ordered in a Think-of-a-Number table. Dividethe class up into 8 groups (at most; you may havefewer). Give each group a blank Think-of-a-Numbertable (provided as page S in the Resource Materials). Each group willuse the instructions in a different order. You may decide how best tocommunicate this to groups. It’s best to alternate add(A)/subtract(S)with multiply(M)/divide(D), so the eight groups would be MADS,MSDA, DAMS, DSMA, AMSD, ADSM, SMAD, and SDAM. Youmight just write this on post-it notes and hand one to each group. Itshould be clear to all groups that they’re all using the same fourinstructions and it’s only the order that’s changing. If you have fewergroups, you don’t have to use all the permutations.Give each group two minutes to fill in their table. Then tell allthe groups to pass their paper to a different group. They nowget 30 seconds to check the original group’s work. They shouldfix any mistakes and initial the paper to say they’ve checked it.After 30 seconds, tell all the groups to pass their papers to yetanother group. They will also check the original group’s work,fix mistakes, and initial the paper. You may keep doing this, ifyou wish.Then collect all of the papers, shuffle them, and take one ofthe final expressions from one of the groups and write it on theboard.(Continued on page TG-15)Algebraic Habits of Mind: Using StructureAgain, students are learning how to organizeinformation to their advantage. Students are learninghow to look at complicated expressions and see themanageable pieces that are embedded within. Helpstudents focus their attention so that they when theysee a complicated equation, they are able to seeentry points they can use to make sense of it.Instructions ResultThink of a number.Answers for Mix It Up Activity:MADS: 5n + 8 – 14MSDA: 5n – 14+ 8DAMS: 5( n4 + 8) – 1DSMA: 5( n4 – 1) + 8AMSD:5(n + 8) –14ADSM: 5( n + 84 – 1)5(n – 1) + 8SMAD: 4SDAM: 5( n – 14 + 8)nMultiply by 5 (M)Add 8 (A)Subtract 1 (S)Divide by 4 (D)TG - 14Unit 5: Logic of Algebra

5-4 Ordering InstructionsArrange the steps below to create a trick that produces the final result. Fill in the table and the intermediate steps.Add 14. Divide by 7. Multiply by 11. Subtract 5.1Instructions Result2Instructions ResultThink of a number.wThink of a number.bMultiply by 11. 11w Add 14.b + 14Subtract 5. 11w - 5Multiply by 11.Divide by 7.11w - 57Subtract 5.Add 14.11w – 57+ 14Divide by 7.11(b + 14)11(b + 14) - 511(b + 14) – 57Divide by 8. Subtract 22. Add 13. Multiply by 9.3Instructions Result4Instructions ResultThink of a number.Add 13.nn + 13Think of a number.Divide by 8.cc8Divide by 8.n + 13c8Subtract 22. 8 - 22Subtract 22. n + 13cMultiply by 9.8- 22 9(8 - 22)Multiply by 9. 9( n + 138– 22)Add 13.9( c8 – 22) + 13Match each algebraic expression on the left with a set of instructions on the right.54n + 1824n2 + 1862(n + 18)7844(n + 18)2BADCABCDThink of a number.Multiply by 4.Divide by 2.Add 18.Think of a number.Multiply by 4.Add 18.Divide by 2.Think of a number.Add 18.Multiply by 4.Divide by 2.None of the above10 5( a + 10 )6 BA Think of a number.Multiply by 5.Add 10.Divide by 6.5(a + 10)116 C B Think of a number.Add 10.Divide by 6.5a126 + 10 Multiply by 5.DC Think of a number.Add 10.Multiply by 5.13 5a + 10Divide by 6.6 AD None of the above14 Unit 5: Logic of Algebra

Together, the class should figure out what all theinstructions were, in order. You may also want to havepost-its (or something larger) so that it’s easy to reorderthe steps, but it’s not necessary. Do this for 3 or4 of the final expressions. Ask questions like:• How do you know which step comes first? Howdo you know which step comes last?• How can you use parentheses to help you figureout the order of the steps?Student Problem Solving:Small GroupsSnapshot Check-in:(15 min) Have students work in theStudent Book in small groups.On Their Own (15 min) Use the last 15 minutes ofclass to assess student understandingof the ideas presented so far in theunit with the Snapshot Check-in included in theResource Materials on page Z. Use studentperformance on this assessment to help selectadditional problems, if necessary, before moving on.Teacher Notes:Additional Discussion Questions:• What are the steps that lead to an expression like2x? (Recall that multiplication and division are at5the “same level” in the order of operations. Thiscan be thought of as multiply by 2, then divide by5, or vice versa.)• Build this expression: Think of a number, add 3,then subtract 1, then add 8 (n + 3 – 1 + 8). Whathappens when you switch the order to add 3, add8, subtract 1? (n + 3 + 8 – 1, which is equivalent.Recall addition and subtraction are also at the“same level” in the order of operations. Butbeware! “Add 3, then subtract from 10” is notthe same as “Subtract from 10, then add 3.” Thesame caution applies to “dividing a number by”and multiplying.)• Explain how you know that “Add 3, then subtractfrom 10” and “Subtract from 10, then add 3”are not the same. Use numerical examples andalgebraic expressions.• Explain how you know that “Multiply by 5,then divide 10 by the result” and “Divide 10by a number, than multiply by 5” are not thesame. Use numerical examples and algebraicexpressions.TG - 15Unit 5: Logic of Algebra

For each algebraic expression, write the instructions step-by-step, in the right order.15 10 – n2– 19166(b – 9)3175(3 – 10q)Think of a number.Subtract it from 10.We’re notsubtracting10. We’resubtractingfrom 10.Think of a number.Subtract 9.Think of a number.Multiply it by ten.We’re notsubtracting3. We’resubtractingfrom 3.Divide the result by 2.Subtract 19 from theMultiply by 6.(OR firstSubtract thedivide byresult from 3.3, thenmultiplyresult. Divide by 3. by 6.) Multiply the result by 5.Match each algebraic expression on the left with a set of instructions on the right.18192021-4(k + 7) – 1410-4( k + 710-4k + 710-4( k + 7) – 14107 – 4k10– 14)– 14DABCABCThink of a number.Add 7.Divide by 10.Subtract 14.Multiply by -4.Think of a number.Multiply by -4.Add 7.Divide by 10.Subtract 14.You canuse Btwice.Think of a number.Divide by 10.Add 7.Multiply by -4.Subtract 14.None of the above22 – 14-8(m – 3)27DB23242526-8(m – 3) + 125-8m – 3512 –5+ 128(m – 3)5-3 – 8m + 125+ 12ACBDBABCDThink of a number.Subtract 3.Multiply by -8.Add 12.Divide by 5.Think of a number.Subtract 3.Multiply by -8.Divide by 5.Add 12.Use Btwice!Think of a number.Multiply by -8.Subtract 3.Divide by 5.Add 12.None of the above28MysteryGrid 4-5-6-7 Puzzle28, x 30, x7 4 5 624, x 35, x 42, x411, +56 76 711, + 20, x4 55 6 7 429 MysteryGrid 1-2-3 Puzzle6, x 18, x2, x3 2132 11233031Who Am I?• All of my digitsare odd.• h + t + u = 23• u – h = 4• t • h = 45Who Am I?• None of mydigits are odd.• h = 2u• t = 2u• u + t + h = 10h t u5 9 9h t u4 4 2Lesson 4: Ordering Instructions15

Additional Practice ProblemsSelect problems that will help you learn. Do some problems now. Do some later.For each algebraic expression, write the instructions step-by-step, in the right order.A5(c – 4) + 18Bd – 163+ 30 C -3( x8 + 9)Think of a number.Think of a number.Think of a number.Subtract 4.Subtract 16.Divide by 8.Multiply by 5.Add 18.Divide by 3.Add 30 to the result.Add 9.Multiply the resultby -3.Match each algebraic expression on the left with a set of instructions on the right.D5(w – 8)i+ 6Think of a number.17 ivSubtract 8.Multiply by 5.I 11( p + 328– 2)E 5( w17 – 8) + 6 iiAdd 6.Divide by 17.J11(p + 32) – 28ii Think of a number.Divide by 17.FSubtract 8.5( w – 811p + 32K – 217 + 6) iii Multiply by 5.8Add 6.Giii Think of a number.5(w – 8) + 6Subtract 8.17 iDivide by 17.Add 6.L 11( p8 + 32) – 2HMultiply by 5.5w – 8+ 611(p + 32)Miv None of the above– 217 iv8iiiiviiviiiiiiiiivThink of a number.Multiply by 11.Add 32.Divide by 8.Subtract 2.Think of a number.Add 32.Multiply by 11.Divide by 8.Subtract 2.Think of a number.Add 32.Divide by 8.Subtract 2.Multiply by 11.None of the aboveN Who Am I?• My tens digit is odd.• I am a multiple of 7.• t + 1 = u• u is not less than t.O MysteryGrid 4-5-6 Puzzlet u24, x 15, +5 6 4 5 66 420, x5PMysteryGrid 1-2-3 Puzzle10, +18, x1 2313230, x56423116 Unit 5: Logic of Algebra

For each algebraic expression, write the instructions step-by-step, in the right order.Q5 – y16+ 4R193a + 2S-4(6 – h) + 15Think of a number.Subtract itfrom 5.We’re notsubtracting5. We’resubtractingfrom 5.Think of a number.Multiply by 3.Think of a number.Subtract theresult from 6.We’re notsubtracting6.Divide by 16.Add 2.Multiply the result by -4.Add 4.Divide 19 bythe result.We’re notdividingby 19.Add 15 to the result.Fill in the blanks on the number line illustrations and solve.12.3 – 8.6 = ________ 3.7TU45.1 – 17.8 = ________ 27.3____ 0.3 8 0.3 ____ 0.7 2 10 5 0.1I I I II I I I I I____ 3.7 ____ 4 12 12.3 27.3 ____ ____ 28 ____ 30 ____ 40 45 45.1V-52 – -4.7 = ________ -47.3 W 18 2 7 – 26 7 = ________ 15 37____ 40.7I I I I I I I-52 ____ -48 -47.3 ________ 15 32____ 16 18 ____ 18747____ 2277X____ 7y____ -9Y____ 6 ____ -y42y -7y 2-54 9y(7y – 9)(6 – y) = -7y 2 + 51y - 54____ x ____ 8ZMysteryGrid 1-2-3-4-5 Puzzle40, x 12, x 3, ÷5 2 4 1 320, x 10, x4 1 3 5 26, x 8, x3 5 2 4 1____ x x 2 8x____ 3 3x 24248, +11, – 9, +3 5( ____ x + ____ 3 ) ( ____ x + ____ 8 ) = x 2 + 11x + 24or vice versa (x + 8)(x + 3)Lesson 4: Ordering Instructions1352417

Lesson 5: The Last InstructionWhy This Lesson?This lesson further trains students to see only the last instruction and treat everything else in an algebraicexpression as a “chunk” of information so they can focus on “Where to start?” or on “What do undo first?”when solving an equation. This lesson also allows students to practice interpreting some tricky language.Why This Way?This lesson serves as the bridge between Think-of-a-Number tricks and solving equations. We use “inkspills” to cover pats of equations, allowing students to focus only on the most recent instruction (or algebraicstep) performed. This allows students to come to an intuitive understanding of “working backwards,” astrategy used in solving algebraic equations. For example, if you know the last instruction given was “add 6”and you ended up at 50, you can figure out (without any formal process) that before the last instruction youhad 44, regardless of how “messy” the rest of the equation is. Students will use this logic to solve equationsin Lesson 6.Mental Mathematics:(5 min) Activity 5: Roots of SquaresLast Instruction Sorting Activity:As a Class(15 min) Provide a set of cards (page T in the ResourceMaterials) for each group of 2-3 students. To save time, youmay consider cutting the cards ahead of time.Students will sort the 18 cards into 6 piles, according to their lastinstruction. Depending on your students, you may choose to tell themwhat the six categories are, or you can let them figure the categoriesout themselves. (You may not even want to tell them there are sixcategories.)You can also differentiate the activity by providing more or fewercards to different groups. For example, you may wish to givestruggling students only cards A-H to begin with. Students who arefinished early may get blank cards where they have to make an extraexample for each category.After about 5 minutes, bring the class together and check answers. You maychoose to project page U of the Resource Materials to aid the discussion. Saythat “add 3 to the result” always looks the same. It’s either + 3 (like incards E and G) or 3 + (like in card J). Start to use the language that represents a “chunk” of information,but no matter what’s in the chunk, ultimately, this is a problem where something is being added to 3. Draw a similarpicture for the rest of the categories: “subtract 4 from the result” ( – 4), “subtract theresult from 7” (7 – ), “multiply the result by 4” (4 ), “divide 5 by the result” (5/), and “divide the result by 3” ( / 3).Student Problem Solving:Small Groups(15 min) Have students work in the Student Book in small groups.Sample: Card A4(3b – 5)Answers to Sorting Activity:“Add 3 to the result”: E, G, J“Subtract 4 from the result”: B, F, O“Subtract the result from 7”: I, N, Q“Multiply the result by 4”: A, D, P“Divide 5 by the result”: L, M, R“Divide the result by 3”: C, H, KSample: Card A on page U4(3b – 5)Look ahead toLesson 6 to see how“chunking” will helpstudents make senseof solving equations.TG - 18Unit 5: Logic of Algebra

5-5 The Last InstructionThese Think-of-a-Number puzzles have seen better days! Can you still fill in the LAST instruction?1 Instructions Result2 Instructions ResultThink of a number.dThink of a number.z3(d + 16)Subtract 9. – 912Multiply by -5.-5( 7z – 936– 14)Write the LAST instruction of each Think-of-a-Number trick. Use the words “the result” to refer to the rest of theexpression.4x + 3113 48 – 3hDivide the result by 11.Subtract theresult5 6from 8.9(m + 2)14p + 3Multiply the result by 9.Divide 14 by the result.Subtract 6 from the2(9r – 4) – 6result.7 8k3 + 4Add 4 to the result.3 + 5(y – 1)Add 3 to the result.9 102( a – 65+ 4)Multiply the result by 2.114 – k – 32Subtract the resultfrom 4.125(p + 2) – 13Divide the result by 3.13Make up an expression with thevariable n and with at least 4 differentnumbers, where the LAST instructionwould be “Add 23 to the result.”Answers will vary.Encourage creativeexpressions with bothx + 23 and 23 + xforms.Check youranswerswith yourclassmates.1415Who Am I?• t + u = 10• My tens digit is four morethan my units digit.Who Am I?• My tens digit is 3 less thanmy units digit.• u + t = 7t u7 3t u2 518 Unit 5: Logic of Algebra

The tricky language in Problems 16-37is tricky for everyone, from strugglingstudents to experienced mathematicians!These problems never really become“easy.” Rather, teach students thatwhenever they see this language, theyshould always take a second look at theproblem, and take the extra time to makesense of the words before moving on.Summarizing Discussion:As a Class(10 min) Share answers fromthe Discuss and Write WhatYou Think box. Also use thistime to discuss some of the trickiervocabulary in the lesson using some of thediscussion questions to the right.Teacher Notes:Discussion Questions:• A friend is having trouble figuring out how to interpret“subtract a from 4” versus “subtract 4 from a.” Howwould you help explain it to them?• A friend is having trouble figuring out how to interpret“divide n by 5” versus “divide 5 by n.” How would youhelp explain it to them?• “Less than” is sometimes used to refer to subtraction andsometimes used to refer to an inequality. How do youknow which is appropriate? (Suggest using numbers tohelp make sense of the vocabulary: “4 less than 5” versus“4 is less than 5” versus “5 less than 4”)• Why don’t we have these same issues with languageabout addition and multiplication?• Explain how you know which instruction is the lastinstruction in a Think-of-a-Number table.TG - 19Unit 5: Logic of Algebra

Match each algebraic expression on the left with a set of instructions on the right.16171819208 – aa – 8a + 8a < 88 < aDABCEABCDE8 less than a8 more than aa is less than 8a less than 88 is less than a212223247 – b 2(7 – b) 2b 2 – 7(b – 7) 2CDBAABCDSubtract 7 from band then square theresult.Square b, thensubtract 7.Square b, thensubtract the resultfrom 7.Subtract b from7, then square theresult.25262728292n – 10n – 1010 – 2n10 – nn – 20Write an algebraic expression to match each sentence, using n for the unknown number.3436CBEADFour less than twicesome number n.ABCDESubtract n from 10.Subtract 10 from n.Subtract 10 fromtwice n.Subtract 10 twicefrom n.Subtract twice nfrom 10.Four minus twicesome number n._______________2n - 4_______________4 - 2n1_______________2 n - 3 _______________n 2 - 1Subtract 3 from halfsome number n.37 One less than the squareof some number n.353031323324d + 524d + 5d24 + 5d + 524DABCABCDDivide 24 by d,then add 5 to theresult.Divide d by 24,then add 5 to theresult.Add 5 to d, thendivide by 24.Add 5 to d, thendivide 24 by theresult.Algebraic Habits of Mind:Communicating ClearlyThe details of language areimportant because talking aboutmath will also help you thinkabout and make sense of math.3839Sovann’s age is twice that of his sister.If Sovann is 16, how old is his sister?Sovann’s sister is 8.Pengfei’s age is twice that of his brother.If his brother is 16, then how old is Pengfei?Pengfei is 32.Discuss & Write What You ThinkWait a second. Did you get the same answer or differentanswers for problems 38 and 39? Which is it? Convince askeptic.Different answers.Listen for ideas like “Sovann is older thanhis sister, and Pengfei is older than hisbrother.” Perhaps students will use algebra:S=2(sister) or P=2(brother)Lesson 5: The Last Instruction19

Additional Practice ProblemsSelect problems that will help you learn. Do some problems now. Do some later.Write the LAST instruction of each Think-of-a-Number trick. Use the words “the result” to refer to the rest of theexpression.B407t – 6p – 110Divide 40 by the result.What are youdividing by?Divide the result by 10.AC5h – 8Subtract 8 from theresult.5 + 9(a – 6) Add 5 to the result.DMultiply the result by 3.3( n E2 + 4) 12 +64 – xAdd 12 to the result.FSubtract 3 from the8(2k – 1) – 3result.G 7 – 4mSubtract the resultfrom 7.H5 + 3(y – 4)11Divide the result by 11.I16 – a + 315Subtract the resultfrom 16.Match each algebraic expression on the left with a set of instructions on the right.JKLMN2(5 – b)2b – 5(5 – b) 25 – 2b2(b – 5)iviiviiiiiiiiiiivvSubtract 2b from 5.Subtract 5 from 2b.OPy18 + 3018y + 30Add 30 to y, thendivide the result by18.Divide y by 18,then add 30 to theresult.Subtract 5 from b,y + 30then double the result. Qiiii18Divide 18 by y,then add 30 to theSubtract b from 5,result.then double the result. R18ivy + 30 iiiAdd 30 to y, thenSubtract b from 5,divide 18 by thethen square the result.result.iiiviiiWrite an algebraic expression to match each sentence, using n for the unknown number.S Think of a number n, divideit by 8, then add 2.T Think of a number n,multiply it by -4, then addU Think of a number n,subtract 3 from it, then12 to the result.multiply the result by -5.n_______________ 8 + 2 _______________-4n + 12 _______________-5(n - 3)20 Unit 5: Logic of Algebra

VAnother Balancing ActThis mobilebalances.This mobilebalances, too.How many hearts will make this balance?How many stars will make this balance?If = 20, then what are the values of and ?==4040WWho Am I?• My units digit is four morethan my tens digit.• u > t• The product of my digits is45.X Who Am I?t5u9• My tens digit is 2 less thanmy units digit.• t + 2 = u• Neither of my digits is odd.• u + t = 10t4u6Write an algebraic expression to match each sentence, using n for the unknown number.Y Think of a number n, square it,then subtract it from 10.Z Think of a number n, divide 9by it, then square the result.AA Think of a number, n, subtract2 from it, then divide by 9.10 - n 2 ( 9 n ) 2 n - 2_______________ _______________ _______________ 9Illustrate each problem on a number line. Then say only whether the answer will be positive or negative.BB 30.2 – 31 is _______________ NegativeCC -8.03 – -8 is _______________ NegativeDD 2.18 – 2.3 is _______________Negative31EEMarkthe 0I0I30.2MysteryGrid 1-3-5-7 Puzzle3, x 15, +15, +49, x 5, ÷5, x 3Don’t actuallysubtract.FFMysteryGrid 2-4-6-8 Puzzle12, x 12, +24, x 2, ÷ 12, x16, xI-8.03I02, – 6, +GGMysteryGrid 1-2-3-4 Puzzle3, ÷ 8, x1 5 3 7 2 6 4 8 1 3 2 46, x 16, x 33 7 1 5 4 8 2 6 2 4 3 15 3 7 1 6 4 8 2 3 1 4 27 1 5 3 8 2 6 4 4 2 1 37, +I0I2.186, xLesson 5: The Last Instruction21

Lesson 6: Solving EquationsWhy This Lesson?Students will learn the ‘chunking’ method for solving equations which builds on the logic of undoing Thinkof-a-Numbertricks to support understanding of solving algebraic equations step-by-step.Why This Way?It is easy for students to forget the meaning behind solving equations. This method seeks to reinforce thenumerical thinking behind solving equations. If 3 + “a chunk” = 26, then the “chunk” must be 23. Studentsare more than capable of that numerical reasoning, and should use it to make sense of the process of solvingequations. By emphasizing the numerical relationship shown in an equation rather than making studentsapply steps to “both sides,” students can see that algebra is a process that they can make sense of, and thatalgebra is much more connected than they thought to the arithmetic that they’ve learned.Mental Mathematics:(5 min) Activity 6: Silent Collaborative ReviewThinking out Loud:As a Class(15 min) Have students read theThinking out Loud dialogue as a class.Follow the discussion with a briefdiscussion using the suggested questions in the purple boxto the right.Then reinforce the ideas by modeling the thinking again.Write all four of the following problems up across theboard (leave space underneath to solve each of them) sothat students can see the structural similarity of all fourproblems:11 – 3x = 5 11 – (y – 4) = 5 11 – (a + 20) = 5 11 – (Stress the importance of thinking about the logicof algebra. Don’t make this lesson about reteachingor pre-teaching algebra concepts.Discuss the Dialogue:• Michael just changed the equation from 11 –2x = 5 to 2x = 6. Is that allowed? Are theseequations the same? Are they both going togive the same answer?• How can Michael check his work?24n + 1 ) = 5Then go through solving all four problems, emphasizing the reasoning used in the Thinking out Loud dialogue.Keep saying “I know 11 – 6 = 5 so this ‘chunk’ must be 6.” For the last problem, help students see that they just haveto “unwrap” an extra layer before they get to the answer, but that they can still identify the last instruction used, andso can still use their numerical reasoning (“I know that 24/4 = 6, so that ‘chunk’ must be 4”).2411 – 3x = 5 11 – (y – 4) = 5 11 – (a + 20) = 5 11 – (n + 1 ) = 5243x = 6 y - 4 = 6 a + 20 = 6= 6n + 1x = 2 y = 10 a = -14 n + 1 = 4, so n = 3“Chunking” is not just another rule for students toapply. The problems in this lesson were specificallychosen to demonstrate how chunking can make solvingequations easier. Chunking is certainly not the onlyway to solve equations, and may not always be themost strategic way to solve an equation.Checking work is an important part of themathematical process. Model the answer-checkingprocess for students with both correct andincorrect answers so that students understand thatchecking work is not just for finding mistakes, butalso for validating correct answers.TG - 22Unit 5: Logic of Algebra

5-6 Solving EquationsThinking out LoudMichael: Here’s a Think-of-a-Number puzzle: Think of a number, multiply by 2, and subtract yourresult from 11. I got 5. Can you figure out my original number?Lena: Well, I like to think of it as an equation.(Lena writes:) 11 – 2x = 5Then we solve for x.Pausing to ThinkUse Lena’s equation to try to find Michael’s original number.Jay: When I look at that equation, I don’t just focus on the x. Instead, I look at the bigger picture.That equation is really just saying “11 minus something equals 5.” And we know that answer!Michael: Well, 11 minus 6 is 5. How does that help?Jay: Watch. Let me write down what we know under the equation.(Jay writes:) 11 – 2x = 511 – 6 = 5See? They’re almost the same thing.The only thing that’s different is that...Lena: 2x is 6. Oh! I get it! 2x is 6! So I can write:2x = 6Michael: And if 2x = 6, then x must be 3.And that was my original number!Algebraic Habits of Mind: ChunkingLooking at the bigger picture can help youmake sense of a complicated equation. Even ifthe equation has lots of steps and numbers andsymbols, if you can see that11 – something = 5,at least you know that “something” must equal 6.Solve each equation.120 – a = 2What is ?182That’s20 – 3p= 220 – 3p = 2 3 20 – (a + 10) = 23p = ____ 18a + 10 = 18p = ____ 6a = 8430= 35306b + 1 = 3 305y = 3What is ?10b + 1 = ____ 10b = ____ 95y = 10y = 222 Unit 5: Logic of Algebra

Student Problem Solving:Small Groups(15 min) Have students workin the Student Book in smallgroups.Summarizing Discussion:As a Class(10 min) Discuss howstudents might check theiranswers. Remind them thatthese problems can all be thought of asThink-of-a-Number problems, but now theyare finding the original number. Encouragestudents to practice checking their answers outloud, using the same language they would usewith Think-of-a-Number tricks. You maychoose to start by checking Problems 9 or 12, where the language is already written.For Problem 1, for example, a student might say, “I started with the number 4. First add 1, to get 5, then subtract theresult from 16, and that gives 11. So 4 was the right answer.”Teacher Notes:Additional Discussion Questions:• In an equation like Problem 7, the left side of theequation (16 – (p + 1)) can be thought of as a set ofinstructions in a Think-of-a-Number trick. What does theright side (11) represent? (The final number). What doesthe answer (p = 4) represent? (The original number).• Your friend looks at the expression 3x + 5 and says that53x is the “chunk.” Why isn’t he or she correct?• In a problem like 5 + 3h – 2 = 12, where’s the chunk?Is it 5 + 3h? Is it 3h – 2? (It could be either one).Remember, chunking is only one tool in making sense ofan equation like this.TG - 23Unit 5: Logic of Algebra

Solve these equations. Show your process.16 – (p + 1) = 11That’s just16 – = 11.So what’s p?7 8p + 1 = ____ 5p = ____ 445y + 3 = 9y + 3 = ____ 5y = ____ 29Think of a number, subtract 8, multiply by 6, andadd 5. Imani followed the instructions and got 29as her final result. What number did she think of?6(n - 8) + 5 = 296(n - 8) = 24n - 8 = 4n = 12Students maynot write thisas an equation.That’s ok ifthey use logic.102 + 3(y – 1) = 233(y - 1) = 21y - 1 = 7y = 811242n – 10 = 32n - 10 = 82n = 18n = 912Think of a number, multiply by 3, add 10, thendouble the result. Jacob followed the instructionsand got 62 as his final result. What number did hethink of?2(3n + 10) = 623n + 10 = 313n = 21n = 7Students maynot write thisas an equation.That’s ok ifthey use logic.15 –21a + 3 = 12 3h + 55= 713 1421a + 3 = 3 3h + 5 = 35a + 3 = 7a = 43h = 30h = 101548 __Find the total weight16 42= 10= ______ 4= ______ 2 = 6 = ______ 5Lesson 6: Solving Equations23

Additional Practice ProblemsSelect problems that will help you learn. Do some problems now. Do some later.Solve these equations. Show your process.A10 – 3x = 4B5 + (m – 2) = 203x = ____ 6m - 2 = 15x = ____ 2m = 17C25a + 3 = 5Dp + 46= 10a + 3 = ____ 5p + 4 = 60a = ____ 2p = 56E20 – (y + 6) = 10F1002b – 6 = 25y + 6 = 102b - 6 = ____ 4y = 42b = ____ 10b = ____ 5G7(10 – k) + 4 = 117(10 - k) = 710 - k = 1k = 9H 8 – 10 h = 310h= 5h = 2I1 + 2(3c + 5) = 472(3c + 5) = 463c + 5 = 233c = 18c = 6J 9 –1210 – 8y = 31210 - 8y = 610 - 8y = 28y = 8y = 124 Unit 5: Logic of Algebra

K64L36= 10 = ______ 3 = ______ 1 = ______ 8 = ______ 3= 2MPlace these numbers on the number line below: 0.5 0.25 0.75150.25 0.5 0.75I 1 1 1 I11 1 1 I0 1234 15555253545Write an algebraic expression to match each sentence, using n for the unknown number.Subtract a number n from 15. 15 minus a number n. 15 less than a number n.N O P_______________ 15 - n _______________ 15 - n _______________ n - 15Where am I?QI am exactly halfway between 27 and 43. Who am I? _____ 35I I27 43RI am exactly halfway between -0.02 and 0.08. Who am I? _____ 0.03I I-0.02 0.08S MysteryGrid 1-2-3-4 PuzzleT MysteryGrid 3-6-9 Puzzle2, – 1 12, x2 1 3 4412, x324, x 1, –31 2U18, x 15, +3 6 954, x 15, +____ a ____ 99 3 62 4 1____ a a 2 9a6 9 36, x____ 7 7a 631 4 2 3( ____ a + ____ 7 ) ( ____ a + ____ 9 ) = a 2 + 16a + 63or vice versa (a + 9)(a + 7)Lesson 6: Solving Equations25

Lesson 7: Solving with SquaresWhy This Lesson?We extend the idea of chunking to equations with more than one solution, in this case, with perfect squares.Students will also use logic to determine the identity of mystery numbers by solving simultaneous equations.Why This Way?By introducing quadratic equations this way, we allow students to extend what they have learned aboutlinear equations to quadratic equations without learning about a new class of equations. Secret Code (orMystery) Numbers are fun, intuitive puzzles that support the use of properties (such as the additive andmultiplicative identities) as well as the important idea that multiple variables can require the simultaneoususe of multiple equations to solve.Mental Mathematics:(5 min) Activity 7: Factor RaceThinking out Loud:As a Class(5 min) Have students read the Thinkingout Loud dialogue together as a class.Use the Pausing to Think box to modelthe process of answer-checking. You may say, forexample, “Start with 11. Subtract 5 to get 6, then square itto get 36. Start again, but with -1. Subtract 5 to get -6,then square it to get 36. Both answers work.” Follow thediscussion with a brief discussion using the suggested question in the purple box above.Student Problem Solving:Small Groups (25 min) Have students work in theStudent Book in small groups. Makesure students get to spend at least 10minutes puzzling through problems 7-15. Studentsshould tackle these problems in the order they arewritten, since the ideas build upon each other.Summarizing Discussion:As a Class(10 min) Convene students at the endof class to discuss problem 15. IfDiscuss the Dialogue:• It’s easy to see why in n 2 = 36, n is both thepositive and negative versions of 6. But 11and -1 are not just positive/negative versionsof each other. What is going on here?Encourage students to work together on the Secret CodeNumber problems (7-15). As students work, circulateand ask:• How do you know your answers work?• Are you sure there are no other possible answers?Ask students about both correct and incorrect answers.Even students with correct answers need practicejustifying their answers and developing convincingarguments.students have finished this problem, have them explain their process, or if students have not finishedthe problem, work on the problem together as a class. Students should take turns volunteering their observations.Facilitate the conversation so that every student gets a chance to think and speak, and encourage the class tocontribute in a way that makes it feel like they solved the puzzle collectively. You may want to point out connectionsbetween the fragments in problem 15 and the previous problems. Encourage students to discuss these connections. Inparticular, fragment A connects to problem 7, fragment B connects to problem 8, and fragments D and E are relatedto problems 10 and 14. If students are stuck, you may choose to use one of these connections to get them started, butin general, try to offer as few hints as possible.TG - 26Unit 5: Logic of Algebra

5-7 Solving with SquaresThinking out LoudMichael: If n 2 = 36, then there are two possible numbers that n can be: either 6 or -6. But what happensLena:when something more complicated is squared?I saw a problem like that the other day. It was... oh yes.(Lena writes:) (p – 5) 2 = 36Michael: Something is still squared, but this time it’s p – 5. But that would mean that p – 5 could be -6Jay:or 6. How would we write that?Why not with two equations? Write what you just said:(Jay writes:) p – 5 = 6 or p – 5 = -6Michael: And so we get two possible answers for p.Lena:Pausing toThinkIt must either be 11 or -1... So which one is it?Both! They both work in the equation.Show that both 11 and -1 work in theequation (p – 5) 2 = 36.Algebraic Habits of Mind:ChunkingWe’re chunking again! But this time,2something = 36,so you know that “something” mustequal -6 or 6!Jay:There are two answers! Just like there are two answers for n 2 = 36. That makes sense. Theonly two numbers that work in this equation are 11 and -1.Solve the following. Show your steps.That’sSo what’s c?2= 64.1 (c + 3) 2 = 642 (y – 1) 2 = 49c + 3 = ____ 8 OR c + 3 = ____ -8y - 1 = 7 OR y - 1 = -7c = ____ 5 OR c = ____ -11y = 8 OR y = -63 (10 – n) 2 = 814 (4x + 2) 2 = 10010 - n = 9 OR 10 - n = -9n = 1 OR n = 194x + 2 = 10 OR 4x + 2 = -104x = 8 OR 4x = -12x = 2 OR x = -35 2 (h + 3) 2 = 50(h + 3) 2 = ____ 256 Think of a number, subtract 5, then square theresult. If you end up with 16, what are the only twonumbers you could have started with?h + 3 = ____ 5 OR h + 3 = ____ -5(n - 5) 2 = 16h = ____ 2 OR h = ____ -8n - 5 = 4 OR n - 5 = -4n = 9 OR n = 126 Unit 5: Logic of Algebra

Teacher Notes:Additional Discussion Questions:• Can an equation with a squared variable ever have justone solution? (Yes, for example in the case of n 2 = 0 oran equation like (n – 10) 2 = 0)• For Problem 7, how do you know that 0 and 1 are theonly two numbers that work? (You may ask students toexplain their thinking about Problems 8-14 in the sameway.)TG - 27Unit 5: Logic of Algebra

Secret Code Numbers: All the shapes represent one-digit numbers, 0 through 9.7What are the only two numbers that could8 What is the only number that could be ifbe if • = ?+ = ?0 • 0 = 0= _________ 0 or 10 + 0 = 0= ____ 01 • 1 = 1Consideronly thedigits 0–9.91113What are the numbers that could be if• = ? Then what could be?0 • 0 = 01 • 1 = 12 • 2 = 43 • 3 = 9What could , , and be if all the shapes aredifferent numbers?• =+ =3 • 3 = 93 + 3 = 6What could , , and be if all the shapes aredifferent numbers?• =+ =+ + =2 • 3 = 63 + 3 = 62 + 2 + 2 = 6= _______________0, 1, 2, 3= ________________0, 1, 4, 9= _____ 3= _____ 9= _____ 61012The numbershave to workfor all theequations ina problem.14Let’s look at two equations at once. What couldand be?0 • 0 = 0• = 0 + 0 = 0= _________ 0 or 2+ =2 • 2 = 4= _________ 0 or 42 + 2 = 4What could be? Don’t try to find . (Why not?)• =+ =0 • 0 = 00 + == _____ 0can beany number.What could , , and be if all the shapes aredifferent numbers?= _____ 6 • == _____ 1• == _____ 2 = _____ 2+ == _____ 3 = _____ 41 • 2 = 22 • 2 = 42 + 2 = 415You found several fragments of pottery—maybe a secret code, or from a space ship oran ancient ruin, or just from some countrywhose language you don’t know. From otherfragments, you already figured out that the “•”and “+” and “=” mean exactly what they meanin your language, and it turns out that the othersymbols are digits, all different numbers from0 through 9. But what digits?!Fragment AFragment DFragment BFragment EFragment C= 1 = 0 = 2 = 3 = 4Lesson 7: Solving with Squares27

Additional Practice ProblemsSelect problems that will help you learn. Do some problems now. Do some later.Solve the following. Show your steps.A (a + 10) 2 = 144B (w – 8) 2 = 81a + 10 = ____ 12 OR a + 10 = ____ -12w - 8 = 9 OR w - 8 = -9a = ____ 2 OR a = -22 ____w = 17 OR w = -1C(13 – b) 2 = 25D(2n – 1) 2 = 8113 - b = 5 OR 13 - b = -5b = 8 OR b = 182n - 1 = 9 OR 2n - 1 = -92n = 10 OR 2n = -8n = 5 OR n = -4E (h – 14) 2 + 3= 28F 20 – (m + 1) 2 = 4(h - 14) 2 = ____ 25h - 14 = 5 OR h - 14 = -5h = 19 OR h = 9(m + 1) 2 = ____ 16m + 1 = ____ 4 OR m + 1 = ____ -4m = ____ 3 OR m = ____ -5G5(12 – x) 2 = 45H(b + 9) 22= 8(12 - x) 2 = 912 - x = 3 OR 12 - x = -3x = 9 OR x = 15(b + 9) 2 = 16b + 9 = 4 OR b + 9 = -4b = -5 OR b = -13IYou have found thesecalculations carved on a pieceof rock in your backyard. Youknow that “•”, “+”, and “=”are exactly like our algebraand that the other symbols areall different digits (integers0-9). What are the digits?• =+ =+ =+ =• == ______ 1= ______ 2= ______ 3= ______ 6= ______ 928 Unit 5: Logic of Algebra

Solve the following. Show your steps!J 16 – (c + 8) 2 = 15K 4(10 – k) 2 – 3 = 13L(c + 8) 2 = 1c + 8 = 1 OR c + 8 = -1c = -7 OR c = -9Fill in the blanks in this Think-of-a-Number Trick.4(10 - k) 2 = 16(10 - k) 2 = 410 - k = 2 OR 10 - k = -2k = 8 OR k = 12Instructions Result Jacob Mali KaylaThink of a number.n6 4 1Add 6.n + 6 12 10 7Multiply by 2. 2(n + 6) 24 20 14Subtract 4. 2(n + 6) - 4 20 16 10Fill in the blanks.M 13 – 15.6 = ________ -2.6N 74 – 6.3 = ________ 67.7____ 0.6 2 ____ 13____ 0.3 2 4I I III I II-2.6 ____ ____ -2 013 67.7 ____ ____ 68 ____ 70 74O____ 4m ____ -7 -3m ________ m____ 54m 220m-7m-35-3m 2-15m(m + 5)(4m – 7 – 3m) = m 2 - 2m - 35P____ x ____ 9____ x x 2 9x____ 4 4x 36( ____ x + ____ 4 ) ( ____ x + ____ 9 ) = x 2 + 13x + 36or vice versa (x + 9)(x + 4)Q48 __R50= 12 = ______ 4 = ______ 6 = ______ 4 = ______ 7= 1Lesson 7: Solving with Squares29

Lesson 8: Solving with SystemsWhy This Lesson?Students will see that systems of equations are used to establish more than one relationship betweenvariables. They can see these relationships displayed in mobiles that give additional information, such as thetotal weight of the mobile, or in sets of mobiles that can be used together.Why This Way?Solving the mobiles in this lesson requires the use of many algebraic ideas, particularly substitution.Students continue to use intuition to explore the logic of algebra. While many of these problems are actuallysystems of equations, and students do translate them to algebra, students do not focus on solving systemsalgebraically; instead, they start to develop intuition about working with multiple equations and substitution.Mental Mathematics:(5 min) Activity 1: Who Am I? Number Bingo OR 7: Factor RaceMobiles and Equations:As a Class(15 min) Project the first mobile and set of equationsshown on the right (provided as page V in the ResourceMaterials). Four of the equations (A, C, E, F) can beconstructed from the mobiles and two (B, D) cannot. Students shouldwork together as a class to figure out which two equations cannot bewritten using the mobiles. Students may also solve the mobiles, if thatmakes sense for your class (s = 10 and c = 5).Project the second set of mobiles (on page W in the ResourceMaterials). This time, the class will work together to write manydifferent equations. Possible equations include:• p + 2m = 27• 2p = m + p• p = m(Solution: p = 9 and m = 9)Mobile 1 from page VUses =c =Mobile 2 from page W70Which of these equations can youwrite using the mobile above?A 2s + 3c = 35B 5s = 4cC 3s + c = 2s + 3cD 3s + c = 70E 4c + 5s = 70F s = 2cUse p = and m =If there is time, you may project the third mobile (on page X in theResource Materials) and again work together as a class to write manydifferent equations. Possible equations include:• 3h = 4b• 3h + b + 4b = 54• 3h + 5b = 54(Solution: h = 8 and b = 6)The goal is for students to understand that systems of equations areused when there is more than one relationship between variables.27Mobile 3 from page XUse h = and b =54If students desire a challenge, consider asking students to writeequations for the optional multi-tiered mobile on page Y of theResource Materials.TG - 30Unit 5: Logic of Algebra

5-8 Solving with Systems1= ______ 330= ______ 6Solve the mobile. You can use theequations you wrote if you would like,but you don’t have to.Let c stand for and let p stand for .There are many equations that you can write using the mobile.Write two different equations:_____________________________________3c + p = 2p + c_____________________________________3c + p = 15 2p + c = 15Write two more true equations that other classmates wrote:_____________________________________4c + 2p = 30 2c = pShare yourequationsand yourstrategies forsolving._____________________________________Students may also write equations from partlysolved mobiles, too, like: 5p = 30 or 5c = 15, etc.2Write two different equations using these two mobiles. Let p stand for , let s stand for , and let h stand for .Equations:s _____________________________________+ 3h = p= 30 = ______ 6 = ______ 8Solve the mobiles, too.s _____________________________________+ 3h = 5sStudents may make the observation thatp = 5s or maybe that 3h = 4s, etc.Students may also use p = 30 to writeequations (s + 3h = 30, for example).Write two different equations using these twomobiles. Let h stand for and let c stand for .3 4Write two different equations using this mobile.Let b stand for and let m stand for .5619 29= ______ 5 = ______ 7Solve the mobiles, too.Equations:= ______ 8 = ______ 8Solve the mobiles, too.Equations:_____________________________________h + 2c = 19 _____________________________________2b + m = b + 2m b = m_____________________________________2c + 3h = 29Other possibilities: 2h = 10, 2c = 14,etc._____________________________________3b + 4m = 56 7b = 56Other possibilities: 7m = 56,2b + m = 24, b + 2m = 24, etc.30 Unit 5: Logic of Algebra

Student Problem Solving:Small Groups(15 min) Have students work in theStudent Book in small groups.Summarizing Discussion:As a Class(10 min) Depending on how studentshave done, the conversation may go inseveral directions. Problem 6 would beinteresting to discuss with students, since the equationsmay be represented with two separate mobiles orcombined into one mobile with the total weight displayed (the solutionpresented in the answer key). If most students have drawn two mobiles,you may tell students that you can actually use just one mobile torepresent both equations, and wait for students to suggest a way to dothis. If they need a hint, you may ask them to remember that both sidesof the mobiles need to balance each other out.You may also choose to discuss some of the solving techniques ingreater detail. For example, to solve the mobiles in problems 1-4,students need to use the skill of substitution in one way or another.You may wish to make the connections between the substitution on themobile to the algebraic substitution more clear. For example, in problem1, students need to realize that one pentagon = two circles in order tosolve the mobile. You may choose to show how the equality p = 2c maybe used algebraically by replacing p with 2c, and help students make the connection.Teacher Notes:Additional Discussion Questions:• When did you need two mobiles to solve forthe variables in these problems? When couldyou solve with only one mobile?• A student said, “The more shapes there are, themore mobiles I need to solve it.” What wouldyou say to that student?• Look at the mobile in Problem 4. How do youincorporate the middle weight into an equation?Answers to optional multi-tiered mobileon page Y of the Resource Materials:Possible equations:• 3h + 2d = d + 5s• 3h + 2s = h + 2s + d• 2h = d• 4h + 4s + 3d = 56• 3h + 3d + 5s = 56• 7h + 6d + 9s = 112• And many more!Solution: h = 4, s = 4, and d = 8TG - 31Unit 5: Logic of Algebra

5 More Mystery Numbers!You found these calculations painted on an old piece of leather.You know that “•” (“times”), “+” and “=” are exactly like our algebra.You’ve also figured out that the other symbols are all different digits.You have to look attwo or more of theseequations at a time tofigure out anything.=2 = 8 = 3 = 4 = 6Use the two equations to draw one or twomobiles relating h ( ) and s ( ).6 7Use the two equations to draw one or twomobiles relating b ( ) and c ( ).You mayuse oneor bothmobiles.20Equations:2h + s = 105h = 10Of course,studentsmay drawtwo mobiles.Equations:b + c = 112b + 3c = 2411 24= ______ 2 = ______ 6= ______ 9 = ______ 2Solve the mobiles, too.Solve the mobiles, too.8 Who Am I?9 Who Am I?• I’m odd.h t u• I’m odd.h t u• h + t + u = 3 • 3 • 3• My units digit is not greaterthan my tens digit.9 9 9 • h + t + u = 6• My units digit > my tensdigit.3 0 3• My hundreds digit is not less than my tensdigit.• t > 8• h > t• t + u = h• The product of my three digits is 0.Lesson 8: Solving with Systems31

Additional Practice ProblemsSelect problems that will help you learn. Do some problems now. Do some later.ALet d = and let m = .28There are many equations that you can write using the mobile.Write two different equations:_____________________________________4d + m = d + 2m_____________________________________4d + m = 14 d + 2m = 14= ______ 2 = ______ 6Solve the mobile, too.5d + 3m = 28 m = 3d 7d = 14Students may have other equations.BWrite two different equations using these twomobiles. Let h = and let s = .CWrite two different equations using this mobile.Let b stand for and let c stand for .211348= ______ 9 = ______ 4Solve the mobiles, too.Equations:_____________________________________h + 3s = 21_____________________________________s + h = 13Other possibilities: 2s = 8, etc.= ______ 6 = ______ 5Solve the mobiles, too.Equations:_____________________________________3b = 2c + 8_____________________________________2c = 4 + bOther possibilities: 3b = 12 + b,2b = 12, etc.DYou found this puzzle in a very olde algebra book.Because it’s algebra, these letters can stand for any numbers at all.Different letters can even stand for the same number, unless you’re told otherwise.In this problem, no mystery number happens to be negative,but at least one of them is greater than 10.What can you figure out?A . E = EB . B = BC + B = A . AD + D = AD . D = AB > EA = 4 B = 1 C = 15 D = 2 E = 032 Unit 5: Logic of Algebra

Match each algebraic expression on the left with a set of instructions on the right.EFGH5(m + 4) – 83I5m + 43i Think of a number.iiAdd 4.Divide by 3.Subtract 8.JKh – 9h < 9iiviiih less than 9.Subtract 9 fromMultiply by 5.3 + 4) – 8 ivM 9 – h iiv 9 minus twice h.– 8) i Multiply by 5.ii Think of a number.Add 4.L 2h – 9 iiiiii 9 less than twice h.5( m + 435( m5(m + 4)3– 8– 8iviiiiiiivSubtract 8.Divide by 3.Think of a number.Multiply by 5.Add 4.Divide by 3.Subtract 8.None of the aboveNP9 – 2h____ x x 2iv____ x 88xvh is less than 9.ORWho Am I?• All of my digits are even.• My units digit is theproduct of my tens digitand my hundreds digit.h2t4u8• My hundreds digit is less than my tensdigit.____ 3y ____ -x____ x 3xy -x 2____ y 3y 2 -xyQ9 9x 72( ____ x + ____ 9 ) ( ____ x + ____ 8 ) = x 2 + 17x + 72or vice versa (x + 8)(x + 9)____ 7b____ 2c____ 5c -8b ____35bc10c 2 -56b 2-16bc(7b + 2c)(5c – 8b) =19bc - 56b 2 + 10c 2____ z 3yz -xz(x + y + z)(3y – x) =2xy - x 2 + 3y 2 + 3yz - xzSMysteryGrid 1-3-5-7 PuzzleTMysteryGrid 2-4-6-8 PuzzleUMysteryGrid 6-7-8-9 Puzzle21, x 7, x 15, +2, – 48, x3 1 7 5 4 2 8 61, – 1, –6 7 8 925, x16, x 3, ÷ 32, x7 5 1 3 8 6 2 415, + 54, x 13, +896 72, –24, x1 3 5 7 2 4 6 878 63, x89 612, + 3, ÷2, – 2, ÷5 7 3 1 6 8 4 215, + 89 6 7 8Lesson 8: Solving with Systems33

Review for Unit AssessmentTeacher Notes:ERROR! Problem 13 should havespecified that all the shapes represent onedigitnumbers, as they did in Lesson 5-7.Students may also offer other answers,such as = 16, = 4, and = 8,or = 25, = 5, and = 10, etc.TG - 34Unit 5: Logic of Algebra

Unit Additional Practice ProblemsUse these questions to help you prepare for the Unit Assessment.1 2 Find s ( )... by drawing and using this mobile.If this mobile balances...does this? YES or NOExplain:The same quantity (leaf and diamond)were removed from both sides.2s + 10 + 6 = 2 + 4sVarious methods.One possibility:(take away 2)2s + 8 + 6 = 4s(remove 2s)8 + 6 = 2s14 = 2s7 = s10623 This mobile balances. Translate it into an equation and circle all of the equations that must also be balanced.m =s =_________________ 3m + s = _________________2s + mA3m + s = m + 2sCD2m + s = 2s2m = sB4m = 3sE3m = m + sMatch each algebraic expression on the left with a set of instructions on the right.45678139(bA Think of a number.2 – 3) + 7 CSubtract 3.9 y – 5 AA Subtract 5 from y.Divide by 2.Add 7.10 5 – 2y CB 5 minus y.9(b – 3) + 7Multiply by 9.2 BB Think of a number.11 2y – 5 DC Subtract twice ySubtract 3.from 5.9b – 3Multiply by 9.+ 72 D Add 7.12 5 – y BD 5 less than twice y.Divide by 2.9( b – 32+ 7) AC Think of a number.Divide by 2.Subtract 314 InstructionsResultMultiply by 9.Add 7.9(b – 3)Think of a number.n2+ 7 D D None of the abovenDivide by 2.2What could , , and be if all the shapes aredifferent numbers?Add 8.n2 + 8• =+ =3 • 3 = 93 + 3 = 6= _____ 9= _____ 3= _____ 6Multiply by 10.Subtract the resultfrom 15.10( n2 + 8)15 - 10( n2 + 8)34 Unit 5: Logic of Algebra

Solve these equations. Show your process!1520 – (h + 5) = 816483x = 8h + 5 = 123x = 6h = 7x = 2175(p – 1) + 15 = 3018185(p - 1) = 15m - 10 = 3p - 1 = 3m – 10 = 6 4a + 5m = 13p = 419(a + 5) 2 = 49 20 (n – 8) 2 = 100a + 5 = 7 OR a + 5 = -7a = 2 OR a = -12n - 8 = 10 OR n - 8 = -10n = 18 OR n = -2212436Let c = and let p = .Write two equations:_____________________________________5c + p = c + 2p_____________________________________5c + p = 18 c + 2p = 186c + 3p = 36 p = 4c= ______ 2 = ______ 8 5c = c + pSolve the mobile, too. And others!Your neighbor also foundcalculations carved on pieces ofrock in his backyard. You want tohelp him find out what they mean.You know that “•”, “+”, and“=” are exactly like our algebra.You’ve also figured out that theother symbols are all differentdigits (integers 0-9). Can you helpyour neighbor find the digits?• =+ =+ =+ =• =22– 1Write the3instructionsin order.Think of a number.Multiply by 4.Add 5.Divide by 3.Subtract 1.= ______ 1= ______ 2= ______ 6= ______ 3= ______ 5Additional Practice 35

Mental Mathematics Activity 1: Who-Am-I? Number Bingo Clue Cards1 The tens digit is 3 times theones digit.2 The tens digit is twice theones digit.3 The ones digit is twice thehundreds digit.4 The ones digit is 3 times thetens digit.5 The hundreds digit is 3 lessthan the ones digit.6 The hundreds digit is 3 lessthan one of the other digits.7 The number is one less thana multiple of 10.8 One of the digits is 3 lessthan anther digit.9 The ones digit is twice thetens digit.10 The tens digit is 2 greaterthan one of the other digits.11 The hundreds digit is 2greater than the ones digit.12 The ones digit is less thantwice the hundreds digit.13 The tens digit is 2 greaterthan one of the other digits.14 The hundreds digit is 2greater than one of the otherdigits.Teacher GuideA

Mental Mathematics Activity 1: Who-Am-I? Number Bingo Clue Cards15 The tens digit is 2 greaterthan the hundreds digit.16 The hundreds digit is 3 lessthan the ones digit.17 The hundreds digit is greaterthan the ones digit.18 The tens digit is 3 times theones digit.19 The number is two more thana multiple of 5.20 The tens digit is twice thehundreds digit.21 The number is four less thana multiple of 10.22 The ones digit is twice thetens digit.23 The hundreds digit is 3 lessthan the tens digit.24 The hundreds digit is greaterthan twice the ones digit.25 The ones digit is less thantwice the hundreds digit.26 The tens digit is 3 times theones digit.27 The tens digit is more thantwice the hundreds digit.28 The hundreds digit is 3 lessthan one of the other digits.BUnit 5: Logic of Algebra

Mental Mathematics Activity 1: Who-Am-I? Number Bingo Clue Cards29 The hundreds digit is 3 lessthan one of the other digits.30 The hundreds digit is twicethe ones digit.31 The tens digit is twice theones digit.32 Tens digit < ones digit.33 The hundreds digit is 2greater than one of the otherdigits.34 The ones digit is 3 times thetens digit.35 Sum of all the digits is 9, 12,15, or 18.36 The ones digit is greater thanthe hundreds digit.37 The hundreds digit is twicethe ones digit.38 The hundreds digit is twicethe tens digit.39 The tens digit is twice thehundreds digit.40 One of the digits is 3 lessthan another digit.41 One of the digits is 3 lessthan another digit.42 The tens digit is greater thanthe hundreds digit.Teacher GuideC

Mental Mathematics Activity 1: Who-Am-I? Number Bingo Clue Cards43 The ones digit is 3 times thetens digit.44 Ones digit < tens digit.45 The hundreds digit is twicethe tens digit.46 The ones digit is twice thetens digit.47 The hundreds digit is greaterthan twice the ones digit.48 The ones digit is less thantwice the hundreds digit.49 The sum of all the digits is amultiple of 3.50 The tens digit is twice theones digit.51 The hundreds digit is 3 timesthe tens digit.52 The hundreds digit is 3 lessthan the ones digit.53 The tens digit times the onesdigit is 0.54 The hundreds digit is greaterthan the tens digit..55 The hundreds digit times theones digit is a multiple of 3.56 The tens digit is twice thehundreds digit.DUnit 5: Logic of Algebra

Mental Mathematics Activity 1: Who-Am-I? Number Bingo Clue Cards57 The sum of all the digits is amultiple of 5.58 The tens digit is 2 greaterthan the hundreds digit.59 The tens digit times the onesdigit is a multiple of 3.60 The hundreds digit is 3 lessthan the tens digit.61 The tens digit plus thehundreds digit is a multipleof 3.62 The hundreds digit is 2greater than one of the otherdigits.63 The hundreds digit is 3 timesthe ones digit.64 The ones digit is twice thehundreds digit.65 The ones digit times the tensdigit is a multiple of 10.66 The hundreds digit is 2greater than the ones digit.67 The hundreds digit times thetens digit is a multiple of 3.68 The tens digit is more thantwice the hundreds digit.69 The tens digit times the onesdigit is a multiple of 10.70 The hundreds digit is greaterthan twice the ones digit.Teacher GuideE

Mental Mathematics Activity 1: Who-Am-I? Number Bingo Clue Cards71 The tens digit times the onesdigit is less than 8.72 The ones digit plus thehundreds digit < 8.73 The ones digit plus thehundreds digit is less than 8.74 The sum of all the digits is amultiple of 3.75 Sum of the tens digit and theones digit is less than 8.76 The hundreds digit times theones digit is a multiple of 3.77 The product of all the digitsis a multiple of 3.78 The tens digit plus thehundreds digit is a multipleof 3.79 Sum of the tens digit and theones digit is a multiple of 3.80 The product of all the digitsis less than 8.81 Sum of the ones digit and thehundreds digit is a multipleof 3.82 The ones digit times the tensdigit is a multiple of 10.FUnit 5: Logic of Algebra

Mental Mathematics Activity 1: Who-Am-I? Number Bingo Clue Cards83 The product of all the digitsis less than 8.84 The hundreds digit times thetens digit is a multiple of 3.85 All three digits are the same.86 The tens digit times the onesdigit is a multiple of 10.87 The number is four less thana multiple of 10.88 The tens digit times the onedigit is 0.89 Ones digit < 5.90 The sum of all the digits is amultiple of 5.91 Hundreds digit > 6.92 The tens digit times the onesdigit is a multiple of 3.93 At least two digits are thesame.94 Sum of the tens digit and theones digit is a multiple of 3.95 Two of the digits are lessthan 5.96 The hundreds digit is 3 timesthe ones digit.Teacher GuideG

Mental Mathematics Activity 1: Who-Am-I? Number Bingo Clue Cards97 The number is a multiple of10.98 The tens digit times the onesdigit is less than 8.99 Ones digit > 6.100 Sum of the tens digit and theones digit is less than 8.101 None of the digits are odd.102 The hundreds digit is 3 timesthe tens digit.103 The number is a multiple of5.104 The product of all the digitsis a multiple of 3.105 Sum of all the digits is lessthan 12.106 Sum of all the ones digitand the hundreds digit is amultiple of 3.107 Two of the digits are greaterthan 5.108 The tens digit is more thantwice the hundreds digit.109 The hundreds digit is greaterthan the ones digit.110 The tens digit is twice thehundreds digit.HUnit 5: Logic of Algebra

Mental Mathematics Activity 1: Who-Am-I? Number Bingo Clue Cards111 The number is two more thana multiple of 5.112 The hundreds digit is greaterthan the tens digit.113 None of the digits are even.114 The tens digit is 2 greaterthan the hundreds digit.115 The tens digit is 4.116 The ones digit is twice thehundreds digit.117 Tens digit < 3.118 The hundreds digit is 2greater than one of the otherdigits.119 Hundreds digit > 2.120 The ones digit is less thantwice the hundreds digit.121 The tens digit is 9.122 The ones digit is greater thanthe hundreds digit.123 The hundreds digit is 9.124 The hundreds digit is twicethe tens digit.Teacher GuideI

Mental Mathematics Activity 1: Who-Am-I? Number Bingo Clue Cards125 Tens digit is 7, 8, or 9.126 The hundreds digit is 2greater than the ones digit.127 The number is four less thana multiple of 10.128 The hundreds digit is 3 lessthan the tens digit.129 The number is two more thana multiple of 5.130 Tens digit < ones digit.131 The number is one less thana multiple of 10.132 One of the digits is 3 lessthan another digit.133 Sum of all the digits is 9, 12,15, or 18.134 The hundreds digit is twicethe ones digit.135 The tens digit is more thantwice the hundreds digit.136 The tens digit is 3 times theones digit.137 The ones digit is twice thehundreds digit.138 The hundreds digit is greaterthan twice the ones digit.JUnit 5: Logic of Algebra

Mental Mathematics Activity 1: Who-Am-I? Number Bingo Clue Cards139 The tens digit is 2 greaterthan the hundreds digit.140 The tens digit is twice theones digit.141 The hundreds digit is 3 lessthan the tens digit.142 The tens digit is 2 greaterthan one of the other digits.143 The hundreds digit is greaterthan the tens digit.144 Ones digit < tens digit.145 The hundreds digit is twicethe tens digit.146 The hundreds digit is 3 lessthan the ones digit.147 The hundreds digit is twicethe ones digit.148 The ones digit is twice thetens digit.149 The hundreds digit is 2greater than the ones digit.150 The hundreds digit is 3 lessthan one of the other digits.Teacher GuideK

Mental Mathematics Activity 7: Silent Collaborative Review TableSquaresSquare Roots0 2 = √144 =1 2 = √121 =2 2 = √100 =3 2 = √81 =4 2 = √64 =5 2 = √49 =6 2 = √36 =7 2 = √25 =8 2 = √16 =9 2 = √9 =10 2 = √4 =11 2 = √1 =12 2 = √0 =LUnit 5: Logic of Algebra

Balanced Mobiles Sorting CardsABIf this mobile balances...does this? YES or NOIf this mobile balances...does this? YES or NOCDIf this mobile balances...does this? YES or NOIf this mobile balances...does this? YES or NOEFIf this mobile balances...does this? YES or NOIf this mobile balances...does this? YES or NOGHIf this mobile balances...does this? YES or NOIf this mobile balances...does this? YES or NOTeacher GuideM

Four Corners Mobile61416NUnit 5: Logic of Algebra

Four Corners - Corner OptionsStep 1: Cross out anytwo hearts on the leftside and both heartson the right side.Step 1: Cross out the6 on the left side andturn the 16 on theright side into 10.Teacher GuideO

Four Corners - Corner OptionsStep 1: Cross out the6 on the left side andturn the 14 on theright side into 8.Step 1: Add 14 and16 together and turnthe right side intotwo hearts and 30.PUnit 5: Logic of Algebra

Four Corners Mobile Solving61416Write directions for finding the value of the heart,step by step, starting with your group’s “Step 1”:____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________Teacher GuideQ

Lena & Jay’s Think-of-a-Number Tables (Blank)1Lena filled in a Think-of-a-Number table this way.Instructions ResultJay filled in the same Think-of-a-Number table this way.Instructions ResultThink of a number.bThink of a number.bAdd 4.Add 4.Multiply by 3.Multiply by 3.Subtract 10.Subtract 10.Draw Lena and Jay’s bucket pictures.Instructions Lena’s Picture Jay’s PictureThink of a number.Add 4.Multiply by 3.Subtract 10.RUnit 5: Logic of Algebra

Blank Think-of-a-Number TablesInstructions ResultThink of a number.nInstructions ResultThink of a number.nTeacher GuideS

Last Instruction Sorting CardsA4(3b – 5)Bb + 29– 4C4b + 53D4( b2 + 3)E4(b + 2) + 3F5(8b + 1) – 4Gb – 42+ 3H4(b – 7) + 53I7 – 3bJ3 + 5b – 7K7 – 4(b + 3)3L54b + 3M57 – 3bN7 –4(b + 3)3O7 – (b + 3)5– 4P4(3 + 7b + 52)Q7 – 4(b + 9)R54(b + 3) – 4TUnit 5: Logic of Algebra

Last Instruction Sorting Cards with ChunksA4(3b – 5)Bb + 29– 4C4b + 53D4( b2 + 3)E4(b + 2) + 3F5(8b + 1) – 4Gb – 42+ 3H4(b – 7) + 53I7 – 3bJ3 + 5b – 7K7 – 4(b + 3)3L54b + 3M57 – 3bN7 –4(b + 3)3O7 – (b + 3)5– 4P4(3 + 7b + 52)Q7 – 4(b + 9)R54(b + 3) – 4Teacher GuideU

Solving with Systems - 1Uses =c =70Which of these equations can youwrite using the mobile above?A 2s + 3c = 35B 5s = 4cC 3s + c = 2s + 3cD 3s + c = 70E 4c + 5s = 70F s = 2cVUnit 5: Logic of Algebra

Solving with Systems - 2Use p = and m =27Teacher GuideW

Solving with Systems - 3Use h = and b =54XUnit 5: Logic of Algebra

Solving with Systems - (Optional Challenge) Multi-Tiered MobileUse h = s = and d =112Teacher GuideY

Snapshot Check-inName:In each problem, the first mobile is balanced. Use it to figure out whether that means the second mobile for sure has tobalance. Circle your response and explain how you know your answer is right.12If this mobile balances...does this? YES or NOIf this mobile balances...does this? YES or NOExplain:Explain:Solve both the mobile and the equation, and check thatyou get the same answer.3Find c (2c + 5 + c = 9 + 2c)... by drawing and using this mobile.Fill in the missing information on the following Think-ofa-Numbertricks.4 Instructions ResultThink of a number.b4b4b + 1013(4b + 10)13(4b + 10) – 85Make up your own mobile problem. Include the solution.ZUnit 5: Logic of Algebra

Snapshot Check-inName:Answer KeyIn each problem, the first mobile is balanced. Use it to figure out whether that means the second mobile for sure has tobalance. Circle your response and explain how you know your answer is right.12If this mobile balances...does this? YES or NOIf this mobile balances...does this? YES or NOExplain: A bucket was taken from theleft but added to the right. The leftgot lighter and the right got heavier.Explain:An equal weight (one moon) wasremoved from both sides.Solve both the mobile and the equation, and check thatyou get the same answer.3Find c (2c + 5 + c = 9 + 2cEncourage variousmethods.One possibility:(remove 2c)5 + c = 9(subtract 5)c = 4)... by drawing and using this mobile.59Fill in the missing information on the following Think-ofa-Numbertricks.4 Instructions ResultThink of a number.Multiply by 4.Add 10.Multiply by 13.Subtract 8.b4b4b + 1013(4b + 10)13(4b + 10) – 85Make up your own mobile problem. Include the solution.Depending on your students, you may make this problem optional,or give bonus points. You can also provide more structure, orleave it open-ended, so that students can make a problem like#1-2 on this Snapshot (a “does this balance” problem), like #3 onthis Snapshot (a “solve this equation using the mobile” problem),or like the mobile problems from Unit 1 (find the values of theshapes). Encourage students to be creative, and make multitieredmobiles or multiple mobiles that work together (such ason the cover of the Unit 5 student book)Teacher GuideAA

Unit AssessmentName:1 2 Find m ( )... by drawing and using this mobile.4m + 2 + m = 10 + 2m + 7If this mobile balances...does this? YES or NOExplain:Match each algebraic expression on the left with a set of instructions on the right.34564( w + 824(w + 8) – 324(w + 8)24w + 82– 3)– 3– 3ABCDThink of a number.Add 8.Divide by 2.Subtract 3.Multiply by 4.Think of a number.Add 8.Multiply by 4.Divide by 2.Subtract 3.Think of a number.Multiply by 4.Add 8.Divide by 2.Subtract 3.None of the above7 12 – 2gA12 – g8 B2g – 129 Cg – 1210 D12 less than g.Subtract g from 12.Subtract 12 fromtwice g.Twice g less than 12.11InstructionsThink of a number.Resultn129( d4 – 13) + 8Think ofWrite the instructionsin order.Subtract it from 10.Divide by 7.Add 6 to it.Multiply by 3.ABUnit 5: Logic of Algebra

Solve these equations. Show your process.1310 + 6h = 34 14 30 – (a + 3) = 22153(y + 5) + 9 = 301640n – 2 = 417(k + 6) 2 = 64 18 (c – 3) 2 = 16- End of Test -OPTIONAL CHALLENGE!Only work on this problem after you have checked your work for all of the other problems.MysteryGrid 1-2-3-4-5-6 Puzzle4, x 1, – 180, x4 18, x 2, – 15, x11, + 6, x10, x 2, – 10, x18, x 4 4, x1, –Teacher GuideAC

Unit AssessmentName:Answer Key1 2 Find m ( )... by drawing and using this mobile.If this mobile balances...does this? YES or NOExplain:The same quantity (a heart) wereremoved from both sides.4m + 2 + m = 10 + 2m + 7Various methods.One possibility:(take away 2)5m = 8 + 2m + 7(remove 2m)3m = 8 + 73m = 15m = 52107Match each algebraic expression on the left with a set of instructions on the right.34564( w + 824(w + 8) – 324(w + 8)24w + 82– 3)– 3– 3ADBCABCDThink of a number.Add 8.Divide by 2.Subtract 3.Multiply by 4.Think of a number.Add 8.Multiply by 4.Divide by 2.Subtract 3.Think of a number.Multiply by 4.Add 8.Divide by 2.Subtract 3.None of the above12 – 2g7 A12 – g8 B2g – 129 Cg – 12DBCA10 D12 less than g.Subtract g from 12.Subtract 12 fromtwice g.Twice g less than 12.11InstructionsThink of a number.Subtract it from 10.Divide by 7.Add 6 to it.Resultn10 - n10 - n710 - n7 + 6129( d4 – 13) + 8Think ofDivide by 4.Subtract 13.Multiply by 9.Write the instructionsin order.a number.Multiply by 3.3( 10 - n7 + 6)Add 8.ADUnit 5: Logic of Algebra

Solve these equations. Show your process.1310 + 6h = 34 14 30 – (a + 3) = 226h = 24a + 3 = 8h = 4a = 5153(y + 5) + 9 = 301640n – 2 = 43(y + 5) = 21n - 2 = 10y + 5 = 7n = 12y = 217(k + 6) 2 = 64 18 (c – 3) 2 = 16k + 6 = 8 OR k + 6 = -8k = 2 OR k = -14c - 3 = 4 OR c - 3 = -4c = 7 OR c = -1- End of Test -OPTIONAL CHALLENGE!Only work on this problem after you have checked your work for all of the other problems.MysteryGrid 1-2-3-4-5-6 Puzzle4, x 1, – 180, x414 18, x 2, – 15, x11, + 6, x561 2 3 6 543 2 5 610, x 2, – 10, x18, x 4 4, x3652146415324231231, –5614Teacher GuideAE

Transition to AlgebraTransition to Algebra is an EDC project supported by theNational Science Foundation aimed at very quickly givingstudents the mathematical knowledge, skill, and confidenceto succeed in a standard first year algebra class.The familiar topic-oriented approach to mathematics isreplaced by a small number of key mathematical ideas andways of thinking: Algebraic Habits of Mind (puzzling,using structure, generalizing patterns, using tools, andcommunicating clearly). Conventional algebra topics arepart of the curriculum, but instead of the topics being thefocus of the lesson, they become contexts for exploringthese broadly-applicable problem-solving strategies.More information on this research and developmentproject is available at ttalgebra.edc.org.