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Unit 5: Logic of Algebra 9 = 50 3m + 2s + 3h + 36 = 18 2h + s 50 7 3m + 9 = 3h 9 7 = ______ 4 = ______ 4 = ______ Transition to Algebra Teacher Guide The Transition to Algebra materials are being developed at Education Development Center, Inc . in Newton, 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 Foundation under Grant No . ESI-0917958 . Opinions expressed are those of the authors and 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 Xu Pilot and Field Test Site Staff: Attleboro School District: Linda Ferreira, Jamie Plante Chelsea High School: Ralph Hannabury, Brittany Jordan, Jeanne Lynch-Galvin, Deborah Miller, Alex Somers Lawrence High School: Yu Yan Xu Lowell High School: Jodi Ahern, Ornella Bascunan, Jeannine Durkin, Kevin Freeman, Samnang Hor, Wendy Jack, Patrick Morasse, Maureen Mulryan, Thy Oeur, Krisanne Santarpio Malden High School: Jason Asciola, Hava Daniels, Maryann Finn, Chris Giordano, Nick Lippman, Paul Marques The Rashi School: Cindy Carter Unit 5: Logic of Algebra Table of Contents Unit 5 Mental Mathematics Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Activity 1: Who Am I? Number Bingo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Activity 2: Divisibility by 10, 5, and 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Activity 3: Divisibility by 4 by Halving Twice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Activity 4: Squares of Integers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Activity 5: Roots of Squares. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Activity 6: Silent Collaborative Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Activity 7: Factor Race. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Unit 5: Logic of Algebra Teacher Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Lesson 1: Staying Balanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Additional Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Lesson 2: Mobiles and Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Additional Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Lesson 3: Keeping Track of Your Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Additional Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Lesson 4: Ordering Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Additional Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Lesson 5: The Last Instruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Additional Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Lesson 6: Solving Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Additional Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Lesson 7: Solving with Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Additional Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Lesson 8: Solving with Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Additional Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Unit Additional Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Unit 5 Resource Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Mental Math Activity 1: Who Am I? Number Bingo Clue Cards. . . . . . . . . . . . . . . . . . . . . . . . A Mental Math Activity 7: Balanced Mobiles Sorting Cards. . . . . . . . . . . . . . . . . . . . . . . . . . . . . L Projection Page for Lesson 1: Balanced Mobiles Sorting Cards . . . . . . . . . . . . . . . . . . . . . . . M Projection Pages for Lesson 2: Four Corners Activity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N Projection Page for Lesson 3: Lena and Jay’s Think-of-a-Number Tables. . . . . . . . . . . . . . . . . R Projection Page for Lesson 4: Blank Think-of-a-Number Table. . . . . . . . . . . . . . . . . . . . . . . . S Projection Page for Lesson 5: Last Instruction Sorting Cards . . . . . . . . . . . . . . . . . . . . . . . . . . T Projection Page for Lesson 8: Solving with Systems Mobiles. . . . . . . . . . . . . . . . . . . . . . . . . . V Snapshot Check-in. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Z Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AA Unit Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AB Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AD Unit 5 Mental Mathematics Activities Activity 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. Use this Setup: activity on • All players write their own set of five 3-digit whole numbers (e.g., 472, 689, 253, 205, 100). more than • Shuffle the clue cards and stack them face down. t no one day. lively. Even if Play: Keep the pace the clue, move • Turn one clue over at a time, and read it aloud. all players get game! • Each player checks all five numbers to see if the clue fits. on. That’s part of the • If the clue fits, the player crosses that number out. Keep track of the cards you’ve Ending the Game: used so you can check. • 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 called Bingo 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 once more for students to check their numbers. When students are (in subsequent days) playing this on their own in pairs 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, multiples or 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 tell me 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 me whether 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. Teacher: Now tell me whether it 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. 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. 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. Prompts like 40, 18, 22, 20, 24, 30, and 28, can be fun to explore for other factors too. This exercise can be done in a particularly rapid-fire manner, as students may have little difficulty with dividing by 10, 5 or 2. However, if students struggle with divisibility by 10, 5 or 2, choose more examples for that number. ii Unit 5: Logic of Algebra Unit 5 Mental Mathematics Activities Activity 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 halve the number twice. For example, for the number 68, students halve once to get 34, then realize that 34 is even, so can be halved again, making 68 divisible by 4. For the number 50, students halve once to get 25, but then, since 25 is odd, conclude that 50 is not divisible by 4. Begin with two digit numbers. Teacher: 20. Teacher: 30. Teacher: 31. Teacher: 24. Teacher: 60. Students: Yes. Students: No. Students: No. Students: Yes. Students: Yes. Teacher: 82. Teacher: 66. Teacher: 36. Teacher: 28. Teacher: 26. Students: No. Students: No. Students: Yes. Students: Yes. Students: No. Teacher: 74. Teacher: 58. Teacher: 86. Teacher: 90. Teacher: 92. Students: No. Students: No. Students: No. Students: No. Students: Yes. Finish with a few exercises where you prompt with three-digit numbers (and higher). Note that since 100 is divisible by 4, students only need to consider the last two digits to determine divisibility by 4. Teacher: 216. Teacher: 316. Teacher: 580. Teacher: 642. Students: Yes. Students: Yes. Students: Yes. Students: No. Teacher: 924. Teacher: 521. Teacher: 736. Teacher: 154. Students: Yes. Students: No. Students: Yes. Students: No. Teacher: 5,632. Students: Yes. Teacher: 1,868. Students: Yes. Teacher: 3,870. Students: No. Teacher: 12,644. Students: Yes. Activity 4: Squares of Integers (1+ day) 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, and 12). “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 x 8?” 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 are assigned 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 charge of 7 today?” If students answer -49, add (with no lesson at this point), “Actually, the square is exactly the same. Just as 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 Guide 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. iii Unit 5 Mental Mathematics Activities Activity 5: Roots of Squares You say a perfect square and students say the positive square root. Start with squares from 0 to 144. You may choose to keep the same “specialists” from the previous day to reinforce the mnemonic. Teacher: 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 you can 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 -11 Teacher: 144. Students: 12 and -12. 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. 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. Teacher: 1. Students: 1 and -1. Teacher: 2. Students: Not perfect. Activity 6: Silent Collaborative Review (1 day) Draw or project the Silent Collaborative Review Table shown right and located on page L in the Teacher Resource Materials. Model what to do by silently looking over the table as if choosing a place to write, and filling in one entry (not the first). Hand your marker to a student, and beckon, without words, for the student to fill in another entry. Hand out several more markers so a few students are up at once. As they finish, have them hand their markers to a classmate who hasn’t been up. Partway through the exercise, tell students “If you think something is incorrect, 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” the class to approve or correct it. Once all entries are correct, you can break the silence and begin the day’s lesson. iv Squares Square Roots 2 0 = √144 = 2 1 = √121 = 22 = √100 = 2 3 = √81 = 2 4 = √64 = 2 5 = √49 = 62 = √36 = 2 7 = √25 = 8 = √16 = 9 = √9 = 102 = √4 = 2 11 = √1 = 12 = √0 = 2 2 2 Unit 5: Logic of Algebra Unit 5 Mental Mathematics Activities Activity 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 the board, 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 not with 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 awards themselves one point for each factor they get correct. (You can adjust the scoring scheme to take away points for incorrect or missing factors, but it may get complicated.) If there are more than two people per team, the pair at the board 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, 14 Teacher: 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 Guide v Unit 5: Logic of Algebra Where’s the Logic? Algebra makes logical sense in the same way that mobiles make sense and number lines make sense 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 any other algebra class . At every step, you can make sense of what you’re doing . You’ll find that complicated problems can be thought of in simpler ways, and the tools you’ve been using so far can really help you with algebra . Lessons in this Unit: 1: Staying Balanced 2: Mobiles and Algebra 3: Keeping Track of Your Steps 4: Ordering Instructions 5: The Last Instruction 6: Solving Equations 7: Solving with Squares 8: Solving with Systems MysteryGrid 7-8-9 Puzzle 56, x 7 72, x 9 8 24, + 63, x e ill solv You w se . ike the l s e l z puz MysteryGrid 263, x 9 8 7 9 7 9 8 7 2 29 19 14, x 2 18, + 9 7 7-9 Puzzle 7 7, – 5 = ______ 7 = ______ 16 – a + 52 7 Think of a number. Subtract it from 16. 2 Divide by 7. 9 Add 52. Algebraic Habits of Mind: Using Structure You will see that, sometimes, algebra problems can look exactly like mobile problems, and you can use the same 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 in Think-of-a-Number puzzles . If you can figure out what the instructions are, you can start to unravel them until 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 too complicated . You’ll learn that sometimes it’s good to leave those expressions as a complicated chunk to deal 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 Guide Unit 5: Logic of Algebra builds facility with manipulating expressions and equations thoughtfully, logically, and accurately. Students will use the common-sense logic they have developed–particularly in the Think-of-a-Number tricks and mobile puzzles–to make sense of the syntactic manipulations of algebra (the standard “solving rules”). Primary emphasis is placed on using logic and understanding the underlying 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 connection between mobile pictures and algebraic equations explicit. Lessons 3, 4, and 5 explicitly connect the instructions of a Think-of-a-Number trick to layers of an algebraic expression. Students will see the structure behind complicated algebraic expressions using order of operations and will use ideas of “undoing” that structure systematically to solve the equations. Algebraic Habits of Mind in Unit 5: • Using Structure: Students will see the structural relationship between mobiles and equations. Students will see that the structure of an algebraic expression can be made simpler by identifying just the last step of a series of instructions and treating everything else as a “chunk” of information. • Using Tools Strategically: Mobiles are useful for visualizing what kinds of manipulations maintain balance, but the metaphor becomes weak for negative numbers and awkward for fractions and large numbers. Think-of-a-Number tricks are used to analyze the structure of a complex expression, but are not as useful when a problem requires complicated expressions on both sides of the equation. We use these tools to help us see the logic behind algebra, but ultimately, algebra itself is the most versatile tool we have. • Clear Communication: Students will experience the need to be precise not only with the vocabulary they use to describe algebraic steps, but also with the order in which they present these steps. Lessons 6 and 7 extend the ideas of Lessons 3, 4, and 5 but without using the Think-of-a-Number table. In Lesson 5, students identify the last instruction of a Think-of-a-Number trick; in Lesson 6, students isolate that step and treat the rest of an expression as a single “chunk” to solve equations. Lesson 7 extends that idea to solving equations with squared expressions. Lesson 8 explores systems of equations through mobiles. This lesson is not intended to teach methods of manipulating systems of equations, but rather lets students explore the way that systems of equations give more information about variables, and shows that sometimes these equations (or mobiles) have to be used together to solve for a variable. Many students experience mathematics as a set of rules to follow and get answers. With students who have had limited success with mathematics, it is especially tempting to teach that way since it often feels like the least risky way to help them succeed on tests and pushing for the logic requires more time and effort and may even be beyond what these students can achieve. Focusing on rules is faster in the short run–it does help with tests that follow soon after instruction–but without constant maintenance it is not long-lasting, and strugglers are typically better at the “common-sense” logic than at remembering and appropriately applying rules they don’t understand. 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. Unit 5: Logic of Algebra TG - 1 Lesson 1: Staying Balanced Why 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 allowable algebraic manipulations.. Using mobiles and symbols together helps students consider algebraic steps more generally 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 Bingo Sample: Card A Balanced Mobiles Sorting Cards: As a Class (10 min) Before class, photocopy page M from the Resource Materials. Provide one copy for every group of 2-3 students. Pre-cut the cards if possible. If this mobile balances.. does this? YES or NO Students will sort these cards into piles for YES and NO. If you wish, you may have them tape or glue their cards onto a sheet of paper divided into a YES and NO section. As groups work, refrain from correcting individual cards. Instead, give hints like: “Three of your group’s cards are in the 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). Use the remaining time to have a student-centered discussion about why some mobiles stay balanced and others don’t. Try to limit your role as much as possible to recording student responses. Leave this language on the board for students to refer to as they work on their student pages. If students don’t volunteer answers, or some answers are incomplete, that’s ok for now. You will have a chance to revisit this question at the end of the lesson. It is important for all students to experience success, but also important to move activities along at an interesting pace. Even fun activities become boring when they drag. Use this card sorting activity to whet students’ curiosity, and use the student pages to provide further support. 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. - 2 TG Unit 5: Logic of Algebra 5-1 Staying Balanced The only moves obile allowed on a m keep the at are moves th . mobile balanced In each problem, the first mobile is balanced . Use it to figure out whether that means the second mobile for sure has to balance . Circle your response and explain how you know your answer is right . 1 2 If this mobile balances . . . does this? YES or NO Explain: 3 Both sides were halved. (Or, since a clover = 2 hexagons, an equal quantity was subtracted from both sides.) 4 If this mobile balances . . . does this? YES or NO Explain: Both a diamond and a bucket were subtracted from both sides. = m, Translate each mobile into algebra . Let 5 = s, and If this mobile balances . . . does this? YES or NO Explain: The pentagon weight was moved from the left to the right, so the right is now heavier. = h . 6 2h + 3s 2m + s = ____________ ____________ 2 does this? YES or NO Explain: An equal weight (a circle) was added to both sides. 8 If this mobile balances . . . 7 2s + 2m + s h + m + h = _________________ ____________ 5s = _______________ m + 2h + m _________ OR 3s + 2m Which of the following moves would keep the mobile balanced? Circle all that apply . OR 2m + 2h 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 . C Move all the pentagons to the right side . F Remove one pentagon from both sides . Unit 5: Logic of Algebra Algebraic Habits of Mind: Communicating Clearly: Encourage students to reason through these mobiles together. Students need the experience of describing these ideas out loud in order to clarify exactly what moves are allowed (e.g. “you group the diamonds together and group the circles together, but only on the same side of the mobile”). Let students use their own internal logic, intuition, and understanding to describe 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 a bucket to the right side? (Yes, but 3 stars is the same quantity as 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 a mobile and realize that you’re not adding 3x twice (that’s multiplication), 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. As students work, help them refine the language they use to describe the moves happening on the mobiles 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 should follow from the conversation from the beginning of class. Students have had a chance to practice describing the moves as they’ve solved problems, so now is a good time to bring the class together to clarify that, for example, crossing out the same symbols on both sides of the mobiles is equivalent to subtracting their symbols in an equation. Emphasize that You’ll notice much more Additional Practice in Unit this is why algebra steps work they way they do: all 5. Be strategic in how you use these problems. In steps in solving an algebra problem preserve equality. this unit, you’ll notice that most problem types are grouped together in the Additional Practice section. Teacher Notes: This was done on purpose to allow you to easily assign certain problems to groups of students. Each Additional Practice section also includes review from previous units and fun puzzles. If students are working ahead, let them loose on these puzzles! Looking ahead: Lesson 3 uses Think-of-a-Number tricks. Use the Think-of-a-Number problems in the Additional Practice in Lessons 1 and/or 2 to help prepare them. - 3 TG Unit 5: Logic of Algebra Translate each mobile into algebra or draw the mobile for the equation provided . Explain the change from the first to the second mobile and equation . Let = m, = s, and = h . 10 9 2h m _________ = _________ 3m = __________ 2h + 2m ________ 3s = _________ h + m _________ h + m = 3s _________ How did the mobile and the equation change? How did the mobile and the equation change? 2m was added to both sides. The sides of the equation (and mobile) were switched. 11 12 2h = _________ 2s + m _________ 4s + 2m 4h = ___________ How did the mobile and the equation change? Both sides were doubled. (Or, since 2h = 2s + m, an equivalent amount was added both both sides.) 13 m+s+m = _________ 2h+s+m _________ s + 2m = _________ s + 3m _________ How did the mobile and the equation change? The order of the variables changed, but the same quantities stay on both sides. This mobile balances . Translate it into an equation and circle all of the equations that must also be balanced . m= s= h= _________________ 2s + m + 2s = _________________ s + m + h + m C A 2s + m + 2s = s + m + h + m D B 4s + m = s + 2m + h E m + 4s = 2m + s + h Hint: You 5s + m = 2m + h should have circled four 3s = h + m answers here . And in #8 . Discuss & Write What You Think Encourage students to keep expressing these ideas 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 the entire 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/Subtracting unequal amounts Lesson 1: Staying Balanced 3 Additional Practice Problems Select 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 to balance . Circle your response and explain how you know your answer is right . B A If this mobile balances . . . does this? YES or NO Explain: One heart was added to the left, but two hearts were added to the right. If this mobile balances . . . does this? YES or NO Explain: Two drops and a leaf were subtracted from both sides. Translate each mobile into algebra or draw the mobile for the equation provided . Explain the change from the first to the second mobile and equation in words . Let = m, = s, and = h . D C 2h = ___________ 2m + 2s _______ h m + s _________ = _________ How did the mobile and the equation change? Both sides were halved. (Or, since h=m+s, the same quantity was subtracted from both sides.) 4 2s + 2m = h_________ + 2m ___________ How did the mobile and the equation change? 2m was added to both sides. F E s + m = _________ 3h _________ 2s = _________ h _________ 3h = _________ m + s _________ m+s+2m s + 2h _________ = _________ 3m = _________ 2h _________ How did the mobile and the equation change? How did the mobile and the equation change? The sides were switched. One star was removed from both sides. Unit 5: Logic of Algebra G H I J Which 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 . iii Move the shapes so all 5 circles are on the left side and both hexagons are on the right side . vi Subtract 3 circles from the left side . Which 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 . iii Move all the buckets to the left side and the water drop to the right side . vi Switch the leaf and the water drop . Which of the following equations also must be balanced? Circle all that apply . i h+s+s+h=m+m+h iv 2m + h = h + 2s + h ii 2s + h = 2m v 3s + 2h = 2m + h + s iii 6h + 6s = 6m + 3h vi 3h = 2m + 2s Fill in the blanks in this Think-of-a-Number Trick . Instructions Pictures Result Jacob Mali Kayla b 4 7 3 b + 5 9 12 8 Multiply by 2 . 2b + 10 18 24 16 Subtract 4 . 2b + 6 14 20 12 Think of a number . Add 5 . K Another hint: G: 3 answers H: 4 answers I: 5 answers MysteryGrid 1-2-3-4 Puzzle 8, x 1 2 4 3, ÷ 3 11, + 3 4 4 1 12, x 2 1 Lesson 1: Staying Balanced 3 2 6, + 2 3 4, x 1 4 L MysteryGrid 1-3-4-5 Puzzle 4, + 3 20, x 5 1 4 1 12, + 4, ÷ 4 3 1 4 5 15, x 5 3 1, – 5 4 2, – 3 1 M MysteryGrid 2-3-7-8 Puzzle 17, + 8 2 7 3 56, x 7 8 18, x 3 2 8, + 2 3 4, ÷ 8 1, – 7 3 7 7 2 8 5 Lesson 2: Mobiles and Algebra Why 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 this lesson makes the point of using mobiles to explore equations that students typically find daunting (especially equations with variables on both sides or like terms that are split up). Students are encouraged to understand and justify different approaches for solving equations using mobile logic to help them understand steps they may be asked to perform in algebra. Mental Mathematics: (5 min) Activity 2: Divisibility by 10, 5, and 2 Four Corners: As a Class (15 min) Project the mobile on page N of the Resource Materials (shown to 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. (*) 14 6 16 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 turn the 14 into an 8. The four starred (*) possible steps above are provided as pages O and P in the Resource Materials. Print out a copy of these pages, cut on the dotted lines, and tape the four options to the four corners of your classroom. Tell everyone to pick their 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 that this is simply a matter of preference. You can modify the activity so that the students are split into four groups, and each group gets a different “Step 1” to start with. Putting the options in the four corners just gives students more choice and gives students a different experience of working 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 is provided for this purpose. After 1-2 minutes, get everyone’s attention. Using your discretion, you may ask students stay in their corners and pay attention, or you may ask students to return to their seats. First, make sure that all students understand how to translate the mobile into algebra: h + 6 + 4h = 14 + 2h + 16. The answer is h = 8 (Continued on page TG-7) - 6 TG Unit 5: Logic of Algebra 5-2 Mobiles and Algebra Solve both the mobile and the equation, and check that you get the same answer . 1 Find c . . . by using this mobile . 9 + 4c = 10 + c + 11 Find h... 9 10 11 by drawing and using this mobile . h + 5 + 2h = 13 + h by using this mobile . Encourage various methods. One possibility: (cross out 2p) 39 = 3p + 12 (subtract 12) 27 = 3p 9 = p Find m... 4 12 by drawing and using this mobile . 13 5 Encourage various methods. One possibility: (cross out 2m) 15 = 9 + m (subtract 9) 6 = m Which of the following would work as a first step to solving this equation (use s = 12 9 39 15 + 2m = 2m + 9 + m Encourage various methods. One possibility: (cross out h) 5 + 2h = 13 (subtract 5) 2h = 8 h = 4 5 Find p... 2 p + 39 + p = 5p + 12 Encourage various methods. One possibility: (subtract 9) 4c = 1 + c + 11 (cross out c) 3c = 12 c = 4 3 ra be performed to an algeb The only steps that may d . ce ep the equation balan equation are steps that ke 27 15 9 )? Circle all that apply . A Subtract s from both sides . B Subtract 9 from both sides . C Add 12 and 27 so the right side becomes 39 + s . D Move all the stars to the left and all the numbers to the right to get 5s = 48 . E Combine the stars on the left side to get 4s + 9 = 12 + 27 + s. Choose one of the first steps you circled, and use it as your first step to find s: Find a friend who chose a different first step and compare your methods . Record their process here: Encourage various methods. You may also choose to have several students communicate their process on the board. Ultimately, 3s = 30, so s = 10 6 Unit 5: Logic of Algebra Ask one representative from each group to say their The goal is for students to understand that every step on steps for solving the mobile, in order. While they do a mobile can be represented with algebra, and why we this, you will show the same steps using algebra. (By write the steps the way we do. For example, on a mobile, the way, if you had students who were “too cool” to it may make intuitive sense to just “cross out” two stars actually go to one of the four corners, they can be the on each side, but, in algebra, we don’t just cross out the scribes at the board for this part.) As students share letter h twice, we represent the process with subtraction. their steps, keep asking students: “How would we write that step using algebra?” Keep all h + 6 + 4h = 14 + 2h + 16 h + 6 + 4h = 14 + 2h + 16 four solutions on the board, if possible. “Cross out 6 on both sides” Show students the logic h + 6 + 4h = 14 + 2h + 16 of combining like terms. Show students that subtracting 6 “Cross out 2 hearts on both sides” After students, for example, can happen by subtract 6 from the left side, Show students that subtracting 2h h + 6 + 4h = 14 + 2h + 16 most students will naturally from the left side can happen by -6 -6 just say there are 5 hearts h + 6 + 4h = 14 + 2h + 16 left. Your job is to show that h + 4h = 8 + 2h + 16 -h -h -2h what they see as obvious also OR works in algebra as the step 6 + 3h = 14 + 16 h + 6 + 4h = 14 + 2h + 16 commonly called “combining OR like terms.” Remember, -6 -6 h + 6 + 4h = 14 + 2h + 16 the primary emphasis is on h + 4h = 14 + 2h + 10 using logic. Your job is to -2h -2h BUT NOT (Why not?) help make the connection to h + 6 + 2h = 14 + 16 algebra explicit. h + 6 + 4h = 14 + 2h + 16 Student Problem Solving: Small Groups -6 Summarizing Discussion: (10 min) Ask for student answers to the purple Discuss & Write What You Think box. Keep in mind that explanations are hard to write, but continue to insist that students write down their thoughts to develop their skills. The ability to explain a mistake is useful in figuring out why certain algebraic steps are incorrect. - 7 TG -6 Additional Discussion Questions: • You ask a friend for help in solving Problem 8, and he or she says to you “I learned 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 like terms”? • Your friend thinks that the first step has to be combining like terms. Explain to your friend another first step he or she can take and why it also works. • You may have heard an algebra teacher say “Whatever you do to one side of an equation, you have to do to the other.” You can say: “That’s not true! In Problem 1, I can add 10 and 11 on the right side to get 21 without doing anything to the left side.” Your teacher might respond: “You’re right. What I meant to say was......” What did your teacher mean by his or her first statement? h + 4h = 8 + 2h + 10 (15 min) Have students work in the Student Book in small groups. Encourage students to use different methods of solving the mobiles. Don’t let students get stuck in the trap of thinking that there is only one correct method. Emphasize the logic – solving equations requires the mobile to stay balanced. Ask: “What move makes sense to make?” As a Class -6 Unit 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 + b Find b, too . One possibility: Subtract 3b: 2b + 2 = 16 Subtract 2: 2b = 14 b = 7 7 b = ______ A On the right side, turn 2b + b into 3b . B Subtract 2 from both sides . C Subtract 3b from both sides . D Combine all the b’s and all the numbers to get 8b = 18 . E Subtract b from the right and add it to the left side to get 6b + 2 = 2b + 16 . F Subtract b from the left side and move it to the right to get the same number of b’s on both sides: 4b + 2 = 4b + 16 . Discuss & Write What You Think For problem 6, explain why . . . Emphasize clear communication. Listen for: choice d won’t balance: choice e won’t balance: You can’t combine b’s on opposite sides. Moving 2 to the right side doesn’t necessarily balance moving 3b to the left. If you subtract b from the right, you have to subtract b from the left, too. The sides only stay balanced by doing the same thing to both sides. Draw a mo bile if it helps y our explanatio n . choice f won’t balance: It’s not about just balancing the b’s. Moving b to the right is the same as adding b to the right, but you subtracted b from the left. Algebraic Habits of Mind: Using Tools Strategically Mobiles are useful tools for visualizing balance . They help make sense of algebra, since solving algebra problems 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 . 2 You can’t draw a mobile for every algebra equation . But every equation balances . And anything you do to an equation has to keep the balance . 7 Find s... by drawing and using this mobile . 10 + 4s = 8 + s + 11 Encourage various methods. One possibility: (cross out 10) 4s = 8 + s + 1 (subtract s) 3s = 9 s = 3 Lesson 2: Mobiles and Algebra 8 Find h... by drawing and using this mobile . h + 25 + h = 3h + 2h + 1 10 8 11 Encourage various methods. One possibility: (cross out 2h) 25 = 3h + 1 (subtract 1) 24 = 3h 8 = h 25 1 7 Additional Practice Problems Select problems that will help you learn . Do some problems now . Do some later . A Find c... by using this mobile . c + 8 + 4c = c + 13 + c + 13 Encourage various methods. One possibility: (cross out 2c) 3c + 8 = 13 + 13 (subtract 8) 3c = 5 + 13 3c = 18 c = 6 C Find s... 13 8 13 15 + 2s + 5 = s + 2 + 3s E Encourage various methods. One possibility: (cross out 3h) 7 = 5 + h (subtract 5) 2 = h D Find m... 7 5 by drawing and using this mobile . 8 + m + 31 = 2m + 6 + 2m 15 2 5 18 7 Encourage various methods. One possibility: (cross out m) 8+31 = 2m + 6 + m (subtract 6) 2 + 31 = 2m + m 33 = 3m 11 = m 8 31 6 i Subtract 7 from both sides . ii Add 19 + 7 + 18 = 44 . iii Subtract 18 from both sides . iv Subtract a pentagon from both sides . Find p, too . One possibility: Subtract 18: p + 1 + 7 = 3p Subtract p: 8 = 2p 4 p = ______ p = 4 Which of the following would work as a first step to solving this equation? Circle all that apply . 10 + 3d + 2 = 6 + 5d 8 by using this mobile . Which of the following would work as a first step to solving this equation? Circle all that apply . 19 F Find h... 7 + h + 2h = 5 + 4h by drawing and using this mobile . Encourage various methods. One possibility: (subtract 2) 13 + 2s + 5 = 4s (subtract 2s) 13 + 5 = 2s 18 = 2s 9 = s B i Add 10 + 2 = 12 on the left . ii Subtract 3d from both sides . iii Subtract 6 from both sides . iv Subtract 2 from both sides . Find d, too . One possibility: Subtract 3d: 10 + 2 = 6 + 2d Subtract 6: 6 = 2d 3 d = ______ d = 3. Unit 5: Logic of Algebra G Fill in the blanks in this Think-of-a-Number Trick . Instructions Pictures Result Luis Jess Raj b 7 -2 3 3b 21 -6 9 Add 5 . 3b + 5 26 -1 14 Multiply by 2 . 6b + 10 52 -2 28 Subtract 8 . 6b + 2 44 -10 20 Think of a number . Multiply by 3 . 5.335 I am s . Where am I? ______ 5.28 I am q . Where am I? ______ I H I I I 5.31 I I I I I I 5.3 I am r . Where am I? ______ I I 5.375 I am v . Where am I? ______ I I I 5.35 I am t . Where am I? ______ I 5.4 Use an area model to multiply these expressions . 100 30 ____ 20 ____ 1 ____ 3000 600 30 8 ____ 4 ____ 400 80 4 -b ____ I J 4114 121 • 34 = _________ c ____ 3 ____ 2c 2 3c d ____ 9 ____ 18c 27 4 ____ L (c + 9)(2c + 3) = 2c 2 + 21c + 27 M MysteryGrid 1-2-3-4 Puzzle 6, x 1 3 8, x 2 4 2 16, x 3 12, x 3 4 1 3 4 1 5, + 2 Lesson 2: Mobiles and Algebra 3 ____ 8a 24 -ab -3b (a + 3)(8 – b) = 8a - ab + 24 - 3b 2c ____ K a ____ 4 3, + 2 1 3 N d ____ y ____ -7 ____ d2 dy -7d 4d 4y -28 (d + 4)(d + y – 7) = d 2 + dy - 3d + 4y - 28 MysteryGrid 3-4-5-6 Puzzle 11, + 5 4 4 18, x 6 3 6 15, x 5 3 20, x 4 36, x 3 10, + 6 4 5 4 3 30, x 5 6 O MysteryGrid 5-6-7-8 Puzzle 1, – 6 40, x 8 5 56, x 7 7 30, x 5 6 8 3, – 8 42, x 6 35, x 7 5 5 7 48, x 8 6 9 Lesson 3: Keeping Track of Your Steps Why 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. This lesson uses the familiar Think-of-a-Number trick format to show how expressions can be built by adding one layer of complexity at a time. This logic will also be developed through Lessons 4 and 5 to support the logic of solving equations in Lesson 6. Mental Mathematics: (5 min) Activity 3: Divisibility by 4 by Halving Twice Thinking out Loud: As a Class (10 min) Project the top half of page 10. If you like, a blank copy is provided for you as page R in the Resource Materials. Students should observe that Lena and Jay are both correct, and their bucket pictures show that they get equivalent results. Algebraic Habits of Mind: Using Structure As students fill out these Think-of-a-Number tables, they are learning about the structure of algebraic expressions. In particular, they are learning that new instructions (or steps) do not affect previous steps; they simply follow in sequence. This experience supports “chunking” in Lesson 6, because they’ve seen that larger pieces of information can be grouped and considered together (as a “chunk”) without worrying about the details of what is contained in that chunk. Have students read the Thinking out Loud dialogue as a class. Make sure students see how 3(b + 4) – 10 is not only a way to record the result, but also to encode the instructions as well. Make sure students also answer the question in the Pausing to Think box and work to understand the main points of the dialogue. Student Problem Solving: Small Groups (15 min) Have students work in the Student Book in small groups. Summarizing Discussion: As a Class (10 min) There are several ideas introduced in these problems that will be addressed in later lessons. In particular, students will see more problems which address the language used with subtraction and division, and students will see “chunking” as a way to solve equations. Students have been rewriting complicated expressions from the previous step without worrying about what the expression is (thereby treating the expression as a “chunk”). Depending on what students are ready for, you may wish to explore some of these topics by using the discussion questions in the purple boxes to the right and on page TG-11. - 10 TG Discussion Questions: • When we talk about subtraction, we frequently use the word “from.” If someone is having trouble figuring out the difference between “subtract 4 from 10” and “subtract 10 from 4,” how would you help explain it to them? • When we talk about division, we usually use the word “by.” If someone is having trouble figuring out the difference between “divide 6 by 3” and “divide 3 by 6,” how would you help explain it to them? Unit 5: Logic of Algebra 5-3 Keeping Track of Your Steps 1 Lena filled in a Think-of-a-Number table this way . Instructions Think of a number . Result Instructions b Result b Think of a number . b + 4 Add 4 . Jay filled in the same Think-of-a-Number table this way . b + 4 Add 4 . Multiply by 3 . 3b + 12 Multiply by 3 . Subtract 10 . 3b + 2 Subtract 10 . 3(b + 4) 3(b + 4) – 10 Draw Lena and Jay’s bucket pictures . Instructions Lena’s Picture Jay’s Picture Think of a number . Add 4 . Multiply by 3 . Subtract 10 . Thinking out Loud Lena: 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 did you 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 a Think-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 those steps were . If you just saw her final result, you could have thought the instructions were . . . Pausing to Think Finish Michael’s thought . If all you saw was 3b + 2, what might you say were the 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 these Think-of-a-Number tables, can you tell what the first instruction must have been? (locate the variable and use order of operations or just see what’s “closest”) • When do you need to worry about using parentheses? (explain to students that a division bar automatically groups the numerator and denominator, so parentheses are implied there) The Additional Practice pages include problems and puzzles from Units 2, 3, and 4. They are useful review materials for students who like to work ahead in the Student Book. They may also be useful to help fill gaps for students who missed portions of previous units. Let us know how you are using these Additional Practice problems. - 11 TG Unit 5: Logic of Algebra Fill in the missing information on the following Think-of-a-Number tricks . 2 4 Instructions Result Instructions Think of a number. c Think of a number . Multiply by 2. 2c Add 4 . Subtract 3. 2c – 3 Multiply by 5. 5(2c – 3) Add 16. 5(2c – 3) + 16 Instructions Divide by 2 . Subtract 9 from the result . 5 Subtract 2. k–2 Double that . Divide by 3. k –2 3 Add 25. k–2 + 25 3 Subtract that result from 9 . Multiply that by 3 (use parentheses) . 3( b + 4 ) 2 3(b + 4) 2 Result w Think of a number . k–2 + 25) 3 This mobile balances, too . 3( b + 4 ) Instructions k This mobile balances . b b + 4 Think of a number. 8( Result Multiply by 3 . Result Multiply by 8. 6 3 Add 8 . How many hearts will make this balance? 2w 9 - 2w - 9 “Subtract 9” would be 2w – 9 . “Subtract from 9” means 9 – 2w . 3(9 - 2w) 3(9 - 2w) + 8 How many hearts will make this balance? If = 8, then what are the values of and ? = 4 = 6 7 Instructions Think of a number. Divide by 3. Result Think of a number . m 3 Subtract 6 . m + 11 3 Multiply by 10. 10( m + 11) 3 Lesson 3: Keeping Track of Your Steps Instructions m Add 11. Subtract 1. 8 10( m + 11) – 1 3 Multiply by 2 . Result y y - 6 2(y - 6) Add 5 . 2(y - 6) + 5 Divide the result by 3 . 2(y - 6) + 5 3 11 Additional Practice Problems Select 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 C Instructions Result Think of a number. n Think of a number . Multiply by 9. 9n Multiply by 14 . Add 1. 9n + 1 Multiply by 4. 4(9n + 1) Add 2. 4(9n + 1) + 2 Instructions 14c 14c + 8 Multiply the result by 5 . Subtract 2 from the result . 5(14c + 8) – 2 Instructions Result D 5(14c + 8) d Subtract 2. d–2 Multiply by 13. 13(d - 2) Add 6. 13(d – 2) + 6 Add 13 . 13(d – 2) + 6 15 2(h + 5) + 13 Divide by 9 . 2(h + 5) + 13 9 Instructions Divide 20 by the result. Subtract 1. Multiply by 6. 0 Multiply the result by 2 . p Divide 20 by lt p + 4 the resu th means e 20 result from p + 4 the step above goes on the 20 p + 4 – 1 bottom . 1.05 I 1 1 1 3 7 1 4 8 8 8 2(h + 5) Instructions F 20 6( p + 4 – 1) 0.5 1 h + 5 Add 5 . Place these numbers on the number line below: 1 .5 0.05 I1 h Think of a number . Result Add 4. 12 c Think of a number. Think of a number. G Result Add 8. Result Divide by 15. E Instructions B Result k 40 k 40 k - 21 Think of a number . Divide 40 by the result . Subtract 21 . Multiply the result by 3 . 3( 3( Add 81 . 1 .05 1 5 4 0 .5 1.5 1 0 .05 3 4 1 I 15 2 8 16 8 7 8 5 4 40 - 21) k 40 - 21) + 81 k 8 8 16 8 15 8 Unit 5: Logic of Algebra Illustrate each problem on a number line . Then say only whether the answer will be positive or negative . H Negative 551 – 569 is _______________ 569 I I 551 0 Mark the 0 K I Negative 5 .28 –5 .8 is _______________ I Who Am I? h t u • I am even . 4 8 4 • u=h • I am greater than 200 . • My hundreds digit is a perfect square . • t=u+h • My tens digit is twice my hundreds digit . Negative -2 .3 – -1 .8 is _______________ I 0 Don’t actually subtract these . J I 5.28 -2.3 w ____ 8 ____ -a ____ -5a ____ -5aw -40a 5a 2 L M MysteryGrid 2-3-4-5 Puzzle 10, x I 8.2 8.7 I 2.89 5 2 20, x 2.895 I am exactly halfway between 2 .89 and 2 .9 . Who am I? ________ I P 3 8.45 I am exactly halfway between 8 .2 and 8 .7 . Who am I? _______ I O 0 -5a(w + 8 – a) = -5aw - 40a + 5a 2 12, x N I 4 4 2 12, x 10, + 5 8, x 2 3 3 4 5 2 4 10, + 5 3 2.9 32 Q 42 2 = ______ 4 = ______ 7 = _______ R n ____ -3p ____ 2n ____ 2n 2 -6np 7p ____ 7np -21p 2 -6n 18p -6 ____ Lesson 3: Keeping Track of Your Steps Who Am I? h t u • I’m not an even number . 6 2 1 • Two of my digits are even . • My units digit is half my tens digit . • All three of my digits are different . • My hundreds digit is twice the sum of my units digit and my tens digit . (n – 3p)(2n + 7p – 6) = 2n 2 + np - 21p 2 - 6n + 18p 13 Lesson 4: Ordering Instructions Why 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. Students will see how the structure and meaning of an expression can vary widely depending on how the same set of instructions is ordered. Also, students will see that it is possible to discern the order of instructions by starting at the variable and “working outward.” This lesson seeks to reinforce the idea of order of operations in the context of algebraic expressions. Mental Mathematics: (5 min) Activity 4: Squares of Integers Mix It Up: As a Class Algebraic Habits of Mind: Using Structure Again, students are learning how to organize information to their advantage. Students are learning how to look at complicated expressions and see the manageable pieces that are embedded within. Help students focus their attention so that they when they see a complicated equation, they are able to see entry points they can use to make sense of it. (10 min) Have the following four instructions on the board: Multiply by 5, Add 8, Subtract 1, Divide by 4. Write them horizontally so they don’t look like they’re already ordered in a Think-of-a-Number table. Divide the class up into 8 groups (at most; you may have Instructions Result fewer). Give each group a blank Think-of-a-Number table (provided as page S in the Resource Materials). Each group will Think of a number. n use the instructions in a different order. You may decide how best to communicate 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. You might just write this on post-it notes and hand one to each group. It should be clear to all groups that they’re all using the same four instructions and it’s only the order that’s changing. If you have fewer groups, you don’t have to use all the permutations. Answers for Mix It Up Activity: Give each group two minutes to fill in their table. Then tell all 5n + 8 – 1 MADS: 4 the groups to pass their paper to a different group. They now Multiply by 5 (M) 5n –1 +8 get 30 seconds to check the original group’s work. They should MSDA: Add 8 (A) 4 fix any mistakes and initial the paper to say they’ve checked it. n DAMS: 5( 4 + 8) – 1 Subtract 1 (S) After 30 seconds, tell all the groups to pass their papers to yet n another group. They will also check the original group’s work, Divide by 4 (D) DSMA: 5( 4 – 1) + 8 fix mistakes, and initial the paper. You may keep doing this, if 5(n + 8) –1 AMSD: you wish. 4 n+8 ADSM: 5( 4 – 1) Then collect all of the papers, shuffle them, and take one of the final expressions from one of the groups and write it on the 5(n – 1) + 8 SMAD: 4 board. n–1 SDAM: 5( 4 + 8) (Continued on page TG-15) - 14 TG Unit 5: Logic of Algebra 5-4 Ordering Instructions Arrange the steps below to create a trick that produces the final result . Fill in the table and the intermediate steps . Add 14 . 1 Divide by 7 . Instructions Result 2 w Think of a number. Subtract 5 . Instructions Result b Think of a number. Multiply by 11. 11w Add 14. b + 14 Subtract 5. 11w - 5 Multiply by 11. 11(b + 14) Subtract 5. 11(b + 14) - 5 11w - 5 7 Divide by 7. 11w – 5 + 14 7 Add 14. Divide by 8 . 3 Multiply by 11 . Subtract 22 . Instructions Result Add 13 . 4 n Think of a number. Subtract 22. Multiply by 9. 9( Instructions Result c c 8 Divide by 8. n + 13 8 n + 13 - 22 8 Divide by 8. Multiply by 9 . Think of a number. n + 13 Add 13. 11(b + 14) – 5 7 Divide by 7. c - 22 8 c 9( - 22) 8 Subtract 22. Multiply by 9. n + 13 – 22) 8 c 9( 8 – 22) + 13 Add 13. Match each algebraic expression on the left with a set of instructions on the right . 14 5 4n + 18 2 B A 10 5( a + 10 ) 6 B A A B 11 C B 7 2(n + 18) 4 D Think of a number . Multiply by 4 . Add 18 . Divide by 2 . 5(a + 10) 6 Think of a number . Multiply by 5 . Add 10 . Divide by 6 . 6 4n + 18 2 Think of a number . Multiply by 4 . Divide by 2 . Add 18 . C 5a + 10 6 D C 8 4(n + 18) 2 C Think of a number . Add 18 . Multiply by 4 . Divide by 2 . 12 Think of a number . Add 10 . Divide by 6 . Multiply by 5 . 5a + 10 6 A Think of a number . Add 10 . Multiply by 5 . Divide by 6 . D None of the above D None of the above 13 Unit 5: Logic of Algebra Together, the class should figure out what all the instructions were, in order. You may also want to have post-its (or something larger) so that it’s easy to reorder the steps, but it’s not necessary. Do this for 3 or 4 of the final expressions. Ask questions like: • How do you know which step comes first? How do you know which step comes last? • How can you use parentheses to help you figure out the order of the steps? Student Problem Solving: Small Groups (15 min) Have students work in the Student Book in small groups. Snapshot Check-in: (15 min) Use the last 15 minutes of class to assess student understanding of the ideas presented so far in the unit with the Snapshot Check-in included in the Resource Materials on page Z. Use student performance on this assessment to help select additional problems, if necessary, before moving on. On Their Own Teacher Notes: - 15 TG Additional Discussion Questions: • What are the steps that lead to an expression like 2x ? (Recall that multiplication and division are at 5 the “same level” in the order of operations. This can be thought of as multiply by 2, then divide by 5, or vice versa.) • Build this expression: Think of a number, add 3, then subtract 1, then add 8 (n + 3 – 1 + 8). What happens when you switch the order to add 3, add 8, subtract 1? (n + 3 + 8 – 1, which is equivalent. Recall addition and subtraction are also at the “same level” in the order of operations. But beware! “Add 3, then subtract from 10” is not the same as “Subtract from 10, then add 3.” The same caution applies to “dividing a number by” and multiplying.) • Explain how you know that “Add 3, then subtract from 10” and “Subtract from 10, then add 3” are not the same. Use numerical examples and algebraic expressions. • Explain how you know that “Multiply by 5, then divide 10 by the result” and “Divide 10 by a number, than multiply by 5” are not the same. Use numerical examples and algebraic expressions. Unit 5: Logic of Algebra For each algebraic expression, write the instructions step-by-step, in the right order . 15 10 – n – 19 2 16 Think of a number. We’re not subtracting 10 . We’re Subtract it from 10. subtracting from 10 . 6(b – 9) 3 17 5(3 – 10q) Th ink of a number. Think of a number. Subtract 9. Multiply it by ten. We’re not Divide the result by 2. Multiply by 6. Subtract 19 from the result. Divide by 3. (OR first divide by 3, then multiply by 6.) subtracting 3 . We’re subtracting from 3 . Subtract the result from 3. Multiply the result by 5. Match each algebraic expression on the left with a set of instructions on the right . 18 -4(k + 7) – 14 10 19 -4( k + 7 – 14) A 10 20 21 22 28 -4k + 7 – 14 10 A D B B C -4( k + 7) – 14 C 10 7 – 4k – 14 10 B D Think of a number . Add 7 . Divide by 10 . Subtract 14 . Multiply by -4 . Think of a number . Multiply by -4 . You can Add 7 . Divide by 10 . use B Subtract 14 . twice . Think of a number . Divide by 10 . Add 7 . Multiply by -4 . Subtract 14 . None of the above 23 -8(m – 3) + 12 A 5 24 -8m – 3 + 12 5 C 25 12 – 8(m – 3) 5 B 26 -3 – 8m + 12 5 27 -8(m – 3) + 12 B 5 MysteryGrid 4-5-6-7 Puzzle 28, x 7 24, x 4 6 11, + 5 4 35, x 5 7 6 30, x 5 42, x 6 11, + 4 7 Lesson 4: Ordering Instructions 30 6 7 20, x 5 4 29 MysteryGrid 1-2-3 Puzzle 6, x 3 2, x 2 18, x 1 1 3 2 2 1 3 31 A Think of a number . Subtract 3 . Multiply by -8 . Add 12 . Divide by 5 . B Think of a number . Subtract 3 . Multiply by -8 . Divide by 5 . Use B ! Add 12 . twice C Think of a number . Multiply by -8 . Subtract 3 . Divide by 5 . Add 12 . None of the above D D Who Am I? • All of my digits are odd . • h + t + u = 23 • u–h=4 • t • h = 45 Who Am I? • None of my digits are odd . • h = 2u • t = 2u • u + t + h = 10 h t u 5 9 9 h t u 4 4 2 15 Additional Practice Problems Select 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 . A 5(c – 4) + 18 d – 16 + 30 3 B C -3( x + 9) 8 Think of a number. Think of a number. Think of a number. Subtract 4. Subtract 16. Divide by 8. Multiply by 5. Divide by 3. Add 9. Add 30 to the result. Multiply the result by -3. Add 18. Match each algebraic expression on the left with a set of instructions on the right . D 5(w – 8) + 6 17 E 5( w – 8) + 6 ii 17 F w–8 5( 17 + 6) G 5(w – 8) + 6 17 H 5w – 8 + 6 17 N iv i ii iii iii i iv iv Who Am I? • My tens digit is odd . • I am a multiple of 7 . • t+1=u • u is not less than t . Think of a number . Subtract 8 . Multiply by 5 . Add 6 . Divide by 17 . Think of a number . Divide by 17 . Subtract 8 . Multiply by 5 . Add 6 . Think of a number . Subtract 8 . Divide by 17 . Add 6 . Multiply by 5 . None of the above O t u 5 6 11( J 11(p + 32) – 2 iv 8 K 11p + 32 – 2 8 L 11( M 11(p + 32) – 2 8 24, x 4 p + 32) – 2 iv 8 6 5 15, + 5 4 6 i Think of a number . Multiply by 11 . Add 32 . Divide by 8 . Subtract 2 . ii Think of a number . Add 32 . Multiply by 11 . Divide by 8 . Subtract 2 . iii Think of a number . Add 32 . Divide by 8 . Subtract 2 . Multiply by 11 . None of the above i ii MysteryGrid 4-5-6 Puzzle 30, x 16 p + 32 8 – 2) iii I 6 20, x 5 4 iv P MysteryGrid 1-2-3 Puzzle 10, + 1 2 3 3 1 2 2 3 1 18, x Unit 5: Logic of Algebra For each algebraic expression, write the instructions step-by-step, in the right order . Q 5–y +4 16 R Think of a number. We’re not subtracting 5 . We’re subtracting from 5 . Subtract it from 5. 19 3a + 2 Think of a number. Think of a number. Multiply Subtract the result from 6. by 3. Divide by 16. Add 2. Add 4. Divide 19 by the result. 3.7 12 .3 – 8 .6 = ________ 0.3 ____ We’re not dividing by 19 . I U 8 I 3.7 ____ 4 ____ V 12 0.7 2 ____ I I I 2 I X 6 ____ -y ____ 7y ____ 42y -7y -9 ____ -54 9y 2 (7y – 9)(6 – y) = -7y 2 + 51y - 54 x ____ 8 ____ x ____ x2 8x 3 ____ 3x 24 x + ____ 3 ) ( ____ x + ____ 8 ) = x2 + 11x + 24 ( ____ or vice versa (x + 8)(x + 3) Lesson 4: Ordering Instructions I 0.1 I I I 40 45 45.1 ____ 3 6 4 7 -48 -47.3 ____ ____ 5 15 7 18 7 – 2 7 = ________ 0.7 I 10 27.3 28 ____ 30 ____ ____ W -52 Y I 12.3 -47.3 -52 – -4 .7 = ________ 4 ____ Add 15 to the result. 27.3 45 .1 – 17 .8 = ________ 0.3 I not We’re ng cti subtra 6 . Multiply the result by -4. Fill in the blanks on the number line illustrations and solve . T -4(6 – h) + 15 S I 3 15 7 ____ 2 7 2 ____ I I 16 ____ I 2 18 7 ____ 18 MysteryGrid 1-2-3-4-5 Puzzle Z 40, x 5 12, x 2 4 20, x 4 1 6, x 3 2 3, ÷ 1 3 10, x 3 5 2 4 1 8, x 5 4 2 1, – 9, + 1 3 5 5 2 4 8, + 1 3 17 Lesson 5: The Last Instruction Why This Lesson? This lesson further trains students to see only the last instruction and treat everything else in an algebraic expression 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 “ink spills” to cover pats of equations, allowing students to focus only on the most recent instruction (or algebraic step) performed. This allows students to come to an intuitive understanding of “working backwards,” a strategy 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 you had 44, regardless of how “messy” the rest of the equation is. Students will use this logic to solve equations in Lesson 6. Mental Mathematics: (5 min) Activity 5: Roots of Squares Last Instruction Sorting Activity: Sample: Card A As a Class 4(3b – 5) (15 min) Provide a set of cards (page T in the Resource Materials) for each group of 2-3 students. To save time, you may consider cutting the cards ahead of time. Answers to Sorting Activity: Students will sort the 18 cards into 6 piles, according to their last “Add 3 to the result”: E, G, J instruction. Depending on your students, you may choose to tell them what the six categories are, or you can let them figure the categories “Subtract 4 from the result”: B, F, O out themselves. (You may not even want to tell them there are six “Subtract the result from 7”: I, N, Q categories.) “Multiply the result by 4”: A, D, P You can also differentiate the activity by providing more or fewer “Divide 5 by the result”: L, M, R cards to different groups. For example, you may wish to give struggling students only cards A-H to begin with. Students who are “Divide the result by 3”: C, H, K finished early may get blank cards where they have to make an extra example for each category. Sample: Card A on page U 4(3b – 5) After about 5 minutes, bring the class together and check answers. You may choose to project page U of the Resource Materials to aid the discussion. Say that “add 3 to the result” always looks the same. It’s either + 3 (like in cards 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 similar picture for the rest of the categories: “subtract 4 from the result” ( – 4), “subtract the result from 7” (7 – ), “multiply the result by 4” (4 ), “divide 5 by the result” (5/ Look ahead to ), and “divide the result by 3” ( / 3). Lesson 6 to see how Student Problem Solving: Small Groups - 18 TG (15 min) Have students work in the Student Book in small groups. “chunking” will help students make sense of solving equations. Unit 5: Logic of Algebra 5-5 The Last Instruction These Think-of-a-Number puzzles have seen better days! Can you still fill in the LAST instruction? Instructions 1 Result d Think of a number. Subtract 9. Instructions 2 Result z Think of a number. 3(d + 16) – 9 12 -5( 7z – 9 – 14) 36 Multiply by -5. Write the LAST instruction of each Think-of-a-Number trick . Use the words “the result” to refer to the rest of the expression . 3 4x + 3 11 Divide the result by 11. Subtract the result 5 8 – 3h 7 2(9r – 4) – 6 result. 9 3 + 5(y – 1) 11 13 Subtract 6 from the 4– k–3 2 Add 3 to the result. Subtract the result from 4. Make up an expression with the variable n and with at least 4 different numbers, where the LAST instruction would be “Add 23 to the result .” Answers will vary. Encourage creative expressions with both x + 23 and 23 + x forms. 18 from 8. 4 9(m + 2) 6 14 p+3 8 Divide 14 by the result. Add 4 to the result. 10 2( a – 6 + 4) Multiply the result by 2. 5 12 5(p + 2) – 1 Divide the result by 3. 3 14 Check your answers with your classmates . k +4 3 Multiply the result by 9. 15 Who Am I? • t + u = 10 • My tens digit is four more than my units digit . Who Am I? • My tens digit is 3 less than my units digit . • u+t=7 t u 7 3 t u 2 5 Unit 5: Logic of Algebra The tricky language in Problems 16-37 is tricky for everyone, from struggling students to experienced mathematicians! These problems never really become “easy.” Rather, teach students that whenever they see this language, they should always take a second look at the problem, and take the extra time to make sense of the words before moving on. Summarizing Discussion: As a Class (10 min) Share answers from the Discuss and Write What You Think box. Also use this time to discuss some of the trickier vocabulary in the lesson using some of the discussion questions to the right. Discussion Questions: • A friend is having trouble figuring out how to interpret “subtract a from 4” versus “subtract 4 from a.” How would 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 you help explain it to them? • “Less than” is sometimes used to refer to subtraction and sometimes used to refer to an inequality. How do you know which is appropriate? (Suggest using numbers to help 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 language about addition and multiplication? • Explain how you know which instruction is the last instruction in a Think-of-a-Number table. Teacher Notes: - 19 TG Unit 5: Logic of Algebra Match each algebraic expression on the left with a set of instructions on the right . 16 8–a D A 8 less than a 21 7 – b2 C A 17 a–8 A B 8 more than a 22 (7 – b)2 D B 18 a+8 B C a is less than 8 23 b2 – 7 B C C D 24 (b – 7)2 A D 19 a<8 a less than 8 20 8<a E E 8 is less than a 25 2n – 10 C A Subtract n from 10 . 30 24 d+5 D A 26 n – 10 B B Subtract 10 from n. 31 24 + 5 d A B 27 10 – 2n E C Subtract 10 from twice n. 32 d +5 24 B C A D 33 C D D d+5 24 E 28 29 10 – n n – 20 Subtract 10 twice from n. Subtract twice n from 10 . Subtract 7 from b and then square the result . Square b, then subtract 7 . Square b, then subtract the result from 7 . Subtract b from 7, then square the result . Divide 24 by d, then add 5 to the result . Divide d by 24, then add 5 to the result . Add 5 to d, then divide by 24 . Add 5 to d, then divide 24 by the result . Write an algebraic expression to match each sentence, using n for the unknown number . 34 Four less than twice 35 Four minus twice Algebraic some number n . some number n . 2n - 4 _______________ 36 Subtract 3 from half some number n . 1 4 - 2n _______________ 37 n - 3 2 _______________ 38 One less than the square of some number n . n - 1 _______________ Sovann’s age is twice that of his sister . If Sovann is 16, how old is his sister? Sovann’s sister is 8. 39 Pengfei’s age is twice that of his brother . If his brother is 16, then how old is Pengfei? Pengfei is 32. Lesson 5: The Last Instruction 2 Habits of Mind: Communicating Clearly The details of language are important because talking about math will also help you think about and make sense of math . Discuss & Write What You Think Wait a second . Did you get the same answer or different answers for problems 38 and 39? Which is it? Convince a skeptic . Different answers. Listen for ideas like “Sovann is older than his sister, and Pengfei is older than his brother.” Perhaps students will use algebra: S=2(sister) or P=2(brother) 19 Additional Practice Problems Select 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 the expression . 40 7t – 6 5h – 8 Divide the result by 10. C 5 + 9(a – 6) Add 5 to the result. Multiply the result by 3. E 12 + Divide 40 by the result. What are you dividing by? B D F H p–1 10 3( n + 4) 2 Subtract 8 from the result. A Subtract 3 from the 8(2k – 1) – 3 result. 5 + 3(y – 4) Divide the result by 11. 11 6 Add 12 to the result. 4–x Subtract the result from 7. G 7 – 4m I 16 – a + 3 from 16. 15 Subtract the result Match each algebraic expression on the left with a set of instructions on the right . J 2(5 – b) iv i Subtract 2b from 5 . O y + 30 18 ii i K 2b – 5 ii ii Subtract 5 from 2b. P 18 y + 30 iv ii L (5 – b)2 v iii Subtract 5 from b, then double the result . Q y + 30 18 i iii i iv R iii iv iii 18 + 30 y v M N 5 – 2b 2(b – 5) Subtract b from 5, then double the result . Subtract b from 5, then square the result . Add 30 to y, then divide the result by 18 . Divide y by 18, then add 30 to the result . Divide 18 by y, then add 30 to the result . Add 30 to y, then divide 18 by the result . Write an algebraic expression to match each sentence, using n for the unknown number . S Think of a number n, divide T Think of a number n, U Think of a number n, it by 8, then add 2 . multiply it by -4, then add subtract 3 from it, then 12 to the result . multiply the result by -5 . n + 2 _______________ 8 20 -4n + 12 _______________ -5(n - 3) _______________ Unit 5: Logic of Algebra Another Balancing Act V How many hearts will make this balance? This mobile balances . This mobile balances, too . How many stars will make this balance? If = 20, then what are the values of and ? = 40 = 40 W Who Am I? • My units digit is four more than my tens digit . • u>t • The product of my digits is 45 . t u 5 9 Who Am I? • My tens digit is 2 less than my units digit . • t+2=u • Neither of my digits is odd . • u + t = 10 X t u 4 6 Write an algebraic expression to match each sentence, using n for the unknown number . Y Think of a number n, square it, Z Think of a number n, divide 9 AA Think of a number, n, subtract then subtract it from 10 . by it, then square the result . 2 from it, then divide by 9 . 9 2 n _______________ n - 2 _______________ 9 ( ) 10 - n 2 _______________ Illustrate each problem on a number line . Then say only whether the answer will be positive or negative . Negative Negative CC -8 .03 – -8 is _______________ 30 .2 – 31 is _______________ Don’t actually 31 subtract . I I I I -8.03 0 30 .2 Mark 0 0 the BB EE MysteryGrid 1-3-5-7 Puzzle 3, x 1 3 15, + 5 7 15, + 5 49, x 7 3 5, x 1 3 1 7 5, ÷ 5 7 5 Lesson 5: The Last Instruction 1 3 3 FF MysteryGrid 2-4-6-8 Puzzle 12, x 2 6 24, x 2, ÷ 4 6 16, x 8 8 4 2 12, + 4 12, x 8 2 6 2, – 6, + 8 6 2 4 DD Negative 2 .18 – 2 .3 is _______________ I I 2.18 0 GG MysteryGrid 1-2-3-4 Puzzle 3, ÷ 1 6, x 2 3 7, + 4 3 16, x 4 8, x 3 2 4 3 1 1 4 2 1 6, x 2 3 21 Lesson 6: Solving Equations Why This Lesson? Students will learn the ‘chunking’ method for solving equations which builds on the logic of undoing Thinkof-a-Number tricks 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 the numerical thinking behind solving equations. If 3 + “a chunk” = 26, then the “chunk” must be 23. Students are more than capable of that numerical reasoning, and should use it to make sense of the process of solving equations. By emphasizing the numerical relationship shown in an equation rather than making students apply steps to “both sides,” students can see that algebra is a process that they can make sense of, and that algebra is much more connected than they thought to the arithmetic that they’ve learned. Mental Mathematics: (5 min) Activity 6: Silent Collaborative Review Thinking out Loud: As a Class (15 min) Have students read the Thinking out Loud dialogue as a class. Follow the discussion with a brief discussion using the suggested questions in the purple box to the right. Then reinforce the ideas by modeling the thinking again. Write all four of the following problems up across the board (leave space underneath to solve each of them) so that students can see the structural similarity of all four problems: Stress the importance of thinking about the logic of algebra. Don’t make this lesson about reteaching or pre-teaching algebra concepts. Discuss the Dialogue: • Michael just changed the equation from 11 – 2x = 5 to 2x = 6. Is that allowed? Are these equations the same? Are they both going to give the same answer? • How can Michael check his work? 24 11 – 3x = 5 11 – (y – 4) = 5 11 – (a + 20) = 5 11 – ( n + 1 ) = 5 Then 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 have to “unwrap” an extra layer before they get to the answer, but that they can still identify the last instruction used, and so can still use their numerical reasoning (“I know that 24/4 = 6, so that ‘chunk’ must be 4”). 11 – 3x = 5 11 – (y – 4) = 5 11 – (a + 20) = 5 3x = 6 y - 4 = 6 a + 20 = 6 11 – ( n + 1 ) = 5 24 = 6 x = 2 y = 10 a = -14 n + 1 = 4, so n = 3 “Chunking” is not just another rule for students to apply. The problems in this lesson were specifically chosen to demonstrate how chunking can make solving equations easier. Chunking is certainly not the only way to solve equations, and may not always be the most strategic way to solve an equation. - 22 TG 24 n + 1 Checking work is an important part of the mathematical process. Model the answer-checking process for students with both correct and incorrect answers so that students understand that checking work is not just for finding mistakes, but also for validating correct answers. Unit 5: Logic of Algebra 5-6 Solving Equations Thinking out Loud Michael: Here’s a Think-of-a-Number puzzle: Think of a number, multiply by 2, and subtract your result 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 = 5 Then we solve for x . Pausing to Think Use Lena’s equation to try to find Michael’s original number . When I look at that equation, I don’t just focus on the x . Instead, I look at the bigger picture . Jay: 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 = 5 Algebraic Habits of Mind: Chunking 11 – 6 = 5 Lena: See? They’re almost the same thing . Looking at the bigger picture can help you The only thing that’s different is that . . . make sense of a complicated equation . Even if 2x is 6 . Oh! I get it! 2x is 6! So I can write: the equation has lots of steps and numbers and symbols, if you can see that 2x = 6 11 – Michael: And if 2x = 6, then x must be 3 . 1 20 – a =2 What is 2 ? 30 22 5 ? 10 3 30 b+1=3 20 – (a + 10) = 2 a + 10 = 18 a = 8 6 p = ____ =3 What is That’s 20 – 3p = 2 18 3p = ____ 18 4 20 – 3p = 2 = 5, at least you know that “something” must equal 6 . And that was my original number! Solve each equation . something 6 30 5y = 3 10 b + 1 = ____ 5y = 10 9 b = ____ y = 2 Unit 5: Logic of Algebra Student Problem Solving: Small Groups (15 min) Have students work in the Student Book in small groups. Summarizing Discussion: As a Class Additional Discussion Questions: • In an equation like Problem 7, the left side of the equation (16 – (p + 1)) can be thought of as a set of instructions in a Think-of-a-Number trick. What does the right side (11) represent? (The final number). What does the answer (p = 4) represent? (The original number). 3x + 5 • Your friend looks at the expression and says that 5 3x 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 of an equation like this. (10 min) Discuss how students might check their answers. Remind them that these problems can all be thought of as Think-of-a-Number problems, but now they are finding the original number. Encourage students to practice checking their answers out loud, using the same language they would use with Think-of-a-Number tricks. You may choose 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 the result from 16, and that gives 11. So 4 was the right answer.” Teacher Notes: - 23 TG Unit 5: Logic of Algebra Solve these equations . Show your process . That’s just 7 16 – (p + 1) = 11 16 – = 11 . So what’s p? 9 5 y + 3 = ____ 4 p = ____ 2 y = ____ Think of a number, subtract 8, multiply by 6, and add 5 . Imani followed the instructions and got 29 as her final result . What number did she think of? 10 - 8) + 5 = 29 Students may not write this - 8) = 24 as an equation. 8 = 4 12 That’s ok if they use logic. 24 2n – 10 = 3 2 + 3(y – 1) = 23 3(y - 1) = 21 y - 1 = 7 y = 8 12 2n - 10 = 8 Think of a number, multiply by 3, add 10, then double the result . Jacob followed the instructions and got 62 as his final result . What number did he think of? 2(3n + 10) = 62 3n + 10 = 31 3n = 21 n = 7 2n = 18 n = 9 13 45 y+3 =9 5 p + 1 = ____ 6(n 6(n n n = 11 8 21 15 – a + 3 = 12 14 Students may not write this as an equation. That’s ok if they use logic. 3h + 5 5 =7 3h + 5 = 35 21 = 3 a + 3 3h = 30 a + 3 = 7 h = 10 a = 4 __ 48 15 = 10 Lesson 6: Solving Equations Find the total weight 4 = ______ 42 16 2 = ______ =6 5 = ______ 23 Additional Practice Problems Select problems that will help you learn . Do some problems now . Do some later . Solve these equations . Show your process . A C E 10 – 3x = 4 B 5 + (m – 2) = 20 6 3x = ____ m - 2 = 15 2 x = ____ m = 17 25 a+3 =5 D p+4 6 = 10 5 a + 3 = ____ p + 4 = 60 2 a = ____ p = 56 20 – (y + 6) = 10 F 100 2b – 6 = 25 y + 6 = 10 4 2b - 6 = ____ y = 4 10 2b = ____ 5 b = ____ G 7(10 – k) + 4 = 11 H 7(10 - k) = 7 10 h 10 - k = 1 1 + 2(3c + 5) = 47 2(3c + 5) = 46 3c + 5 = 23 3c = 18 c = 6 24 = 5 h = 2 k = 9 I 10 8– h =3 J 12 9 – 10 – 8y = 3 12 = 6 10 - 8y 10 - 8y = 2 8y = 8 y = 1 Unit 5: Logic of Algebra K L 64 3 = ______ = 10 1 = ______ 8 = ______ Place these numbers on the number line below: M 0.25 1 1 I 0 36 0.5 I1 1 1 5 0 .5 2 5 0 .25 0 .75 3 = ______ 1 5 0.75 1 1 1 3 5 =2 2 5 3 5 4 5 I 1 4 5 Write an algebraic expression to match each sentence, using n for the unknown number . Subtract a number n from 15 . N 15 minus a number n . O 15 - n _______________ 15 - n _______________ Q I? 15 less than a number n . n - 15 _______________ 35 I am exactly halfway between 27 and 43 . Who am I? _____ I I 27 Wh er m ea P R 43 0.03 I am exactly halfway between -0 .02 and 0 .08 . Who am I? _____ I I -0.02 S 0.08 MysteryGrid 1-2-3-4 Puzzle 2, – 2 4 12, x 3 1 1 1 12, x 3 24, x 3 1 2 4 4 6, x Lesson 6: Solving Equations 2 T 18, x 4 3 1, – 2 1 3 MysteryGrid 3-6-9 Puzzle 9 54, x 15, + 3 6 9 3 a ____ 9 ____ 9 a ____ a2 9a 7 ____ 6 7a 63 U 6 15, + a + ____ 7 ) ( ____ a + ____ 9 ) = a2 + 16a + 63 ( ____ or vice versa (a + 9)(a + 7) 25 Lesson 7: Solving with Squares Why 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 about linear equations to quadratic equations without learning about a new class of equations. Secret Code (or Mystery) Numbers are fun, intuitive puzzles that support the use of properties (such as the additive and multiplicative identities) as well as the important idea that multiple variables can require the simultaneous use of multiple equations to solve. Mental Mathematics: (5 min) Activity 7: Factor Race Thinking out Loud: As a Class (5 min) Have students read the Thinking Discuss the Dialogue: out Loud dialogue together as a class. • It’s easy to see why in n2 = 36, n is both the Use the Pausing to Think box to model positive and negative versions of 6. But 11 the process of answer-checking. You may say, for and -1 are not just positive/negative versions example, “Start with 11. Subtract 5 to get 6, then square it of each other. What is going on here? to get 36. Start again, but with -1. Subtract 5 to get -6, then square it to get 36. Both answers work.” Follow the discussion with a brief discussion using the suggested question in the purple box above. Student Problem Solving: Small Groups (25 min) Have students work in the Student Book in small groups. Make sure students get to spend at least 10 minutes puzzling through problems 7-15. Students should tackle these problems in the order they are written, since the ideas build upon each other. Summarizing Discussion: As a Class Encourage students to work together on the Secret Code Number problems (7-15). As students work, circulate and 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 practice justifying their answers and developing convincing arguments. (10 min) Convene students at the end of class to discuss problem 15. If students have finished this problem, have them explain their process, or if students have not finished the 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 to contribute in a way that makes it feel like they solved the puzzle collectively. You may want to point out connections between the fragments in problem 15 and the previous problems. Encourage students to discuss these connections. In particular, fragment A connects to problem 7, fragment B connects to problem 8, and fragments D and E are related to problems 10 and 14. If students are stuck, you may choose to use one of these connections to get them started, but in general, try to offer as few hints as possible. - 26 TG Unit 5: Logic of Algebra 5-7 Solving with Squares Thinking out Loud Michael: If n2 = 36, then there are two possible numbers that n can be: either 6 or -6 . But what happens when something more complicated is squared? Lena: I saw a problem like that the other day . It was . . . oh yes . (Lena writes:) (p – 5)2 = 36 Michael: Something is still squared, but this time it’s p – 5 . But that would mean that p – 5 could be -6 or 6 . How would we write that? Jay: Why not with two equations? Write what you just said: (Jay writes:) p – 5 = 6 or p – 5 = -6 Algebraic Habits of Mind: Chunking It must either be 11 or -1 . . . So which one is it? We’re chunking again! But this time, 2 = 36, something Both! They both work in the equation . so you know that “something” must Michael: And so we get two possible answers for p . Lena: Pausing to Think Jay: equal -6 or 6! Show that both 11 and -1 work in the equation (p – 5)2 = 36 . There are two answers! Just like there are two answers for n2 = 36 . That makes sense . The only two numbers that work in this equation are 11 and -1 . Solve the following . Show your steps . 1 3 5 That’s So what’s c? (c + 3)2 = 64 2 = 64 . (y – 1)2 = 49 -8 8 OR c + 3 = ____ c + 3 = ____ y - 1 = 7 OR y - 1 = -7 5 c = ____ y = 8 OR -11 c = ____ (10 – n)2 = 81 4 10 - n = 9 OR 10 - n = -9 n = 1 n = 19 OR 2 (h + 3)2 = 50 5 OR h + 3 = ____ -5 h + 3 = ____ 2 h = ____ OR -8 h = ____ OR y = -6 (4x + 2)2 = 100 4x + 2 = 10 OR 4x + 2 = -10 6 25 (h + 3)2 = ____ 26 2 4x = 8 OR 4x = -12 x = 2 OR x = -3 Think of a number, subtract 5, then square the result . If you end up with 16, what are the only two numbers you could have started with? (n - 5)2 = 16 n - 5 = 4 n = 9 OR n - 5 = -4 OR n = 1 Unit 5: Logic of Algebra Teacher Notes: - 27 TG Additional Discussion Questions: • Can an equation with a squared variable ever have just one solution? (Yes, for example in the case of n2 = 0 or an equation like (n – 10)2 = 0) • For Problem 7, how do you know that 0 and 1 are the only two numbers that work? (You may ask students to explain their thinking about Problems 8-14 in the same way.) Unit 5: Logic of Algebra Secret Code Numbers: All the shapes represent one-digit numbers, 0 through 9 . 7 What are the only two numbers that • = ? be if 0 • 0 = 0 1 • 1 = 1 9 0 or 1 = _________ 3 = _____ • = + = What could , , and different numbers? 9 = _____ 6 = _____ + ers The numb ork have to w for all the in equations . a problem 14 3 = _____ 2 • 3 = 6 3 + 3 = 6 2 + 2 + 2 = 6 1 Lesson 7: Solving with Squares = 0 = + = • 0 • 0 = 0 0 + 0 = 0 can be any number. 0 • 0 = 0 0 + = What could , , and different numbers? = • = + = Fragment A be if all the shapes are 1 = _____ = _____ 2 = _____ 4 Fragment B 2 Fragment C Fragment E Fragment D = . (Why not?) 0 = _____ = • = _________ 0 or 4 be? Don’t try to find = + = _________ 0 or 2 2 • 2 = 4 2 + 2 = 4 1 • 2 = 2 2 • 2 = 4 2 + 2 = 4 You found several fragments of pottery— maybe a secret code, or from a space ship or an ancient ruin, or just from some country whose language you don’t know . From other fragments, you already figured out that the “•” and “+” and “=” mean exactly what they mean in your language, and it turns out that the other symbols are digits, all different numbers from 0 through 9 . But what digits?! = • What could 2 = _____ = 0 = ____ Let’s look at two equations at once . What could and be? 6 = _____ = + 12 be if all the shapes are = + 10 be if all the shapes are could be if 0 + 0 = 0 = ________________ 0, 1, 4, 9 , , and What could different numbers? • 15 Consider only the digits 0–9 . What is the only number that + = ? 0, 1, 2, 3 = _______________ 3 • 3 = 9 3 + 3 = 6 13 8 could be if What are the numbers that • = ? Then what could be? 0 • 0 = 0 1 • 1 = 1 2 • 2 = 4 3 • 3 = 9 11 could = 3 = 4 27 Additional Practice Problems Select problems that will help you learn . Do some problems now . Do some later . Solve the following . Show your steps . A C (a + 10)2 = 144 B 12 OR a + 10 = ____ -12 a + 10 = ____ w - 8 = 9 OR w - 8 = -9 2 a = ____ w = 17 OR a = -22 ____ (13 – b)2 = 25 D 13 - b = 5 OR 13 - b = -5 b = 8 E OR b = 18 (h – 14)2 + 3= 28 F OR 28 (2n – 1)2 = 81 2n - 1 = 9 OR 2n - 1 = -9 2n = 10 OR 2n = -8 n = 5 OR n = -4 20 – (m + 1)2 = 4 3 m = ____ h = 9 5(12 – x)2 = 45 OR -5 m = ____ (b + 9)2 =8 2 H (12 - x)2 = 9 I w = -1 4 OR m + 1 = ____ -4 m + 1 = ____ h - 14 = 5 OR h - 14 = -5 h = 19 OR 16 (m + 1)2 = ____ 25 (h - 14)2 = ____ G (w – 8)2 = 81 (b + 9)2 = 16 12 - x = 3 OR 12 - x = -3 b + 9 = 4 OR b + 9 = -4 x = 9 OR x = 15 b = -5 OR b = -13 You have found these calculations carved on a piece of rock in your backyard . You know that “•”, “+”, and “=” are exactly like our algebra and that the other symbols are all different digits (integers 0-9) . What are the digits? • + + + • = = = = = 1 = ______ 2 = ______ 3 = ______ 6 = ______ 9 = ______ Unit 5: Logic of Algebra Solve the following . Show your steps! J 16 – (c + 8)2 = 15 K 4(10 - k)2 = 16 2 (c + 8) = 1 L 4(10 – k)2 – 3 = 13 c + 8 = 1 OR c + 8 = -1 c = -7 OR c = -9 (10 - k)2 = 4 10 - k = 2 OR 10 - k = -2 k = 8 OR k = 12 Fill in the blanks in this Think-of-a-Number Trick . Instructions Think of a number . Add 6 . Multiply by 2 . Subtract 4 . Fill in the blanks . -2.6 M 13 – 15 .6 = ________ 0.6 2 ____ I I -2.6 ________ -2 Mali Kayla n 6 4 1 n + 6 12 10 7 2(n + 6) 24 20 14 2(n + 6) - 4 20 16 10 67.7 74 – 6 .3 = ________ 13 ____ I 0.3 ____ I 0 I -7 ____ -3m ____ m ____ 4m 2 -7m -3m 2 5 ____ 20m -35 -15m P ____ x ____ 4 (m + 5)(4m – 7 – 3m) = m 2 - 2m - 35 4 = ______ Lesson 7: Solving with Squares 6 = ______ 4 I I ____ x ____ 9 x2 9x 4x 36 I 74 4 ) ( ____ x + ____ 9 ) = x2 + 13x + 36 x + ____ ( ____ or vice versa (x + 9)(x + 4) R __ 48 2 67.7 68 ____ 70 ____ ____ 13 4m ____ = 12 Jacob N O Q Result 50 4 = ______ 7 = ______ =1 29 Lesson 8: Solving with Systems Why This Lesson? Students will see that systems of equations are used to establish more than one relationship between variables. They can see these relationships displayed in mobiles that give additional information, such as the total 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 actually systems of equations, and students do translate them to algebra, students do not focus on solving systems algebraically; 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 Race Mobiles and Equations: Mobile 1 from page V 70 Use s= c= As a Class (15 min) Project the first mobile and set of equations shown on the right (provided as page V in the Resource Materials). Four of the equations (A, C, E, F) can be constructed from the mobiles and two (B, D) cannot. Students should work together as a class to figure out which two equations cannot be written using the mobiles. Students may also solve the mobiles, if that makes sense for your class (s = 10 and c = 5). Project the second set of mobiles (on page W in the Resource Materials). This time, the class will work together to write many different equations. Possible equations include: • p + 2m = 27 • 2p = m + p • p = m Which of these equations can you write using the mobile above? A B C D E F 2s + 3c = 35 5s = 4c 3s + c = 2s + 3c 3s + c = 70 4c + 5s = 70 s = 2c Mobile 2 from page W Use p = and m = (Solution: p = 9 and m = 9) If there is time, you may project the third mobile (on page X in the Resource Materials) and again work together as a class to write many different equations. Possible equations include: • 3h = 4b • 3h + b + 4b = 54 • 3h + 5b = 54 (Solution: h = 8 and b = 6) 27 Mobile 3 from page X Use h = and b = 54 The goal is for students to understand that systems of equations are used when there is more than one relationship between variables. If students desire a challenge, consider asking students to write equations for the optional multi-tiered mobile on page Y of the Resource Materials. - 30 TG Unit 5: Logic of Algebra 5-8 Solving with Systems 1 Let c stand for 30 . There are many equations that you can write using the mobile . Write two different equations: Share your equations 3c + p = 2p + c and your _____________________________________ strategies for solving . 3c + p = 15 2p + c = 15 _____________________________________ 3 = ______ Write two more true equations that other classmates wrote: 6 = ______ Solve the mobile . You can use the equations you wrote if you would like, but you don’t have to . 2 and let p stand for 4c + 2p = 30 2c = p _____________________________________ _____________________________________ Students may also write equations from partly solved mobiles, too, like: 5p = 30 or 5c = 15, etc. Write two different equations using these two mobiles . Let p stand for , let s stand for , and let h stand for . Equations: s_____________________________________ + 3h = p s_____________________________________ + 3h = 5s = 30 6 = ______ Students may make the observation that p = 5s or maybe that 3h = 4s, etc. Students may also use p = 30 to write equations (s + 3h = 30, for example). 8 = ______ Solve the mobiles, too . 3 Write two different equations using these two mobiles . Let h stand for and let c stand for 4 . Write two different equations using this mobile . Let b stand for and let m stand for . 56 29 19 5 = ______ 7 = ______ Solve the mobiles, too . Equations: Equations: h + 2c = 19 _____________________________________ 2b + m = b + 2m b = m _____________________________________ 2c + 3h = 29 _____________________________________ 3b + 4m = 56 7b = 56 _____________________________________ Other possibilities: 2h = 10, 2c = 14, etc. 30 8 8 = ______ = ______ Solve the mobiles, too . Other possibilities: 7m = 56, 2b + m = 24, b + 2m = 24, etc. Unit 5: Logic of Algebra Student Problem Solving: Small Groups (15 min) Have students work in the Student Book in small groups. Summarizing Discussion: As a Class Additional Discussion Questions: • When did you need two mobiles to solve for the variables in these problems? When could you solve with only one mobile? • A student said, “The more shapes there are, the more mobiles I need to solve it.” What would you say to that student? • Look at the mobile in Problem 4. How do you incorporate the middle weight into an equation? (10 min) Depending on how students have done, the conversation may go in several directions. Problem 6 would be interesting to discuss with students, since the equations may be represented with two separate mobiles or combined into one mobile with the total weight displayed (the solution presented in the answer key). If most students have drawn two mobiles, you may tell students that you can actually use just one mobile to represent both equations, and wait for students to suggest a way to do this. If they need a hint, you may ask them to remember that both sides of the mobiles need to balance each other out. Answers to optional multi-tiered mobile on 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! You may also choose to discuss some of the solving techniques in greater 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 the mobile to the algebraic substitution more clear. For example, in problem Solution: h = 4, s = 4, and d = 8 1, students need to realize that one pentagon = two circles in order to solve the mobile. You may choose to show how the equality p = 2c may be used algebraically by replacing p with 2c, and help students make the connection. Teacher Notes: - 31 TG Unit 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 at two or more of these equations at a time to figure out an ything . = 6 2 = 8 = 3 Use the two equations to draw one or two mobiles relating h ( ) and s ( ) . 7 Equations: You may use one or both mobiles . 20 4 = Use the two equations to draw one or two mobiles relating b ( ) and c ( ) . Equations: Of course, 2h + s = 10 students may draw 5h = 10 two mobiles. b + c = 11 2b + 3c = 24 11 2 = ______ 6 = ______ 9 = ______ Solve the mobiles, too . 8 Who Am I? h t u • I’m odd . • h+t+u=3•3•3 9 9 9 • My units digit is not greater than my tens digit . • My hundreds digit is not less than my tens digit . • t>8 Lesson 8: Solving with Systems 6 = 24 2 = ______ Solve the mobiles, too . 9 Who Am I? h t • I’m odd . 3 0 • h+t+u=6 • My units digit > my tens digit . • h>t • t+u=h • The product of my three digits is 0 . u 3 31 Additional Practice Problems Select problems that will help you learn . Do some problems now . Do some later . A Let d = 28 and let m = . There 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 = ______ 6 = ______ Solve the mobile, too . B 5d + 3m = 28 m = 3d 7d = 14 Students may have other equations. Write two different equations using these two mobiles . Let h = and let s = . C Write two different equations using this mobile . Let b stand for and let c stand for . 4 13 21 8 9 = ______ 4 = ______ 6 = ______ Solve the mobiles, too . Solve the mobiles, too . Equations: Equations: h + 3s = 21 _____________________________________ 3b = 2c + 8 _____________________________________ s + h = 13 _____________________________________ 2c = 4 + b _____________________________________ Other possibilities: 2s = 8, etc. D Other possibilities: 3b = 12 + b, 2b = 12, etc. A . E = E B . B = B C + B = A . A D+D=A D . D = A B>E You 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= 32 5 = ______ 4 B= 1 C= 15 D= 2 E= 0 Unit 5: Logic of Algebra Match each algebraic expression on the left with a set of instructions on the right . E 5(m + 4) – 8 3 F m+4 5( 3 – 8) i Think of a number . Add 4 . Multiply by 5 . Subtract 8 . Divide by 3 . ii m + 4) – 8 iv 3 G 5( H 5(m + 4) –8 3 I 5m + 4 – 8 3 O Think of a number . Add 4 . Divide by 3 . Subtract 8 . Multiply by 5 . i ii Think of a number . Multiply by 5 . Add 4 . Divide by 3 . Subtract 8 . None of the above iii iv iii iv 3y ____ -x ____ x ____ 3xy -x 2 y ____ 3y 2 -xy z ____ 3yz -xz h–9 ii i h less than 9 . K h<9 v ii Subtract 9 from h . L 2h – 9 iii iii 9 less than twice h . M 9–h i iv 9 minus twice h . N 9 – 2h iv v h is less than 9 . x ____ 8 x ____ x2 8x 9 9x 72 P Who Am I? h t u • All of my digits are even . 2 4 8 • My units digit is the product of my tens digit and my hundreds digit . • My hundreds digit is less than my tens digit . R J x + ____ 9 ) ( ____ x + ____ 8 ) = x2 + 17x + 72 ( ____ or vice versa (x + 8)(x + 9) 5c ____ Q -8b ____ 7b ____ 35bc -56b 2 2c ____ 10c 2 -16bc (7b + 2c)(5c – 8b) = 19bc - 56b 2 + 10c 2 (x + y + z)(3y – x) = 2xy - x 2 + 3y 2 + 3yz - xz S MysteryGrid 1-3-5-7 Puzzle 21, x 3 7 2, – 1 12, + 5 7, x 1 25, x 7 15, + 5 5 1 3 3 5 7 7 3, ÷ 3 Lesson 8: Solving with Systems 1 T MysteryGrid 2-4-6-8 Puzzle 2, – 4 16, x 8 2 2, – 6 2 3, ÷ 6 24, x 4 8 48, x 8 2 6 2, ÷ 4 6 32, x 4 8 2 U MysteryGrid 6-7-8-9 Puzzle 1, – 6 15, + 8 7 15, + 9 7 54, x 9 8 8 6 1, – 8 6 9 13, + 7 63, x 9 7 6 8 8 33 Review for Unit Assessment Teacher Notes: - 34 TG ERROR! Problem 13 should have specified that all the shapes represent onedigit numbers, as they did in Lesson 5-7. Students may also offer other answers, = 8, = 4, and such as = 16, = 10, etc. = 5, and = 25, or Unit 5: Logic of Algebra Unit Additional Practice Problems Use these questions to help you prepare for the Unit Assessment . 2 1 Find s ( ) . . . by drawing and using this mobile . 2s + 10 + 6 = 2 + 4s If this mobile balances . . . does this? YES or NO Explain: The same quantity (leaf and diamond) were removed from both sides. 3 Various methods. One possibility: (take away 2) 2s + 8 + 6 = 4s (remove 2s) 8 + 6 = 2s 14 = 2s 7 = s 2 10 6 This mobile balances . Translate it into an equation and circle all of the equations that must also be balanced . m= s= _________________ = _________________ 3m + s 2s + m C 2m + s = 2s A 3m + s = m + 2s D 2m = s B 4m = 3s E 3m = m + s Match each algebraic expression on the left with a set of instructions on the right . 4 5 9( b – 3) + 7 C 2 9(b – 3) + 7 2 6 9b – 3 + 7 2 7 b–3 9( 2 + 7) A 9(b – 3) +7 2 D 8 13 • = + = B D C What could , , and different numbers? 3 • 3 = 9 3 + 3 = 6 34 B A D Think of a number . Subtract 3 . Divide by 2 . Add 7 . Multiply by 9 . Think of a number . Subtract 3 . Multiply by 9 . Add 7 . Divide by 2 . Think of a number . Divide by 2 . Subtract 3 Multiply by 9 . Add 7 . None of the above be if all the shapes are = _____ 9 = _____ 3 = _____ 6 y–5 A A Subtract 5 from y . 10 5 – 2y C B 5 minus y. 11 2y – 5 D C Subtract twice y from 5 . 12 5–y B D 5 less than twice y . 9 14 Instructions Result Think of a number . n Divide by 2 . n 2 Add 8 . Multiply by 10 . Subtract the result from 15 . n 2 10( + 8 n 2 15 - 10( + 8) n 2 + 8) Unit 5: Logic of Algebra Solve these equations . Show your process! 15 17 20 – (h + 5) = 8 16 48 = 8 3x h + 5 = 12 3x = 6 h = 7 x = 2 5(p – 1) + 15 = 30 18 18 = 6 m – 10 5(p - 1) = 15 m - 10 = 3 p - 1 = 3 m = 13 p = 4 19 21 (a + 5)2 = 49 20 (n – 8)2 = 100 a + 5 = 7 OR a + 5 = -7 n - 8 = 10 OR n - 8 = -10 a = 2 OR a = -12 n = 18 OR n = -2 36 Let c = and let p = 22 . Write two equations: 5c + p = c + 2p _____________________________________ 5c + p = 18 c + 2p = 18 _____________________________________ = ______ 2 6c + 3p = 36 = ______ 8 Solve the mobile, too . 24 Your neighbor also found calculations carved on pieces of rock in his backyard . You want to help him find out what they mean . You know that “•”, “+”, and “=” are exactly like our algebra . You’ve also figured out that the other symbols are all different digits (integers 0-9) . Can you help your neighbor find the digits? Additional Practice p = 4c 5c = c + p And others! Write the instructions in order . 4a + 5 – 1 3 Think of a number. Multiply by 4. Add 5. Divide by 3. Subtract 1. • = 1 = ______ + = 2 = ______ + = + = 6 = ______ • = 3 = ______ 5 = ______ 35 Mental Mathematics Activity 1: Who-Am-I? Number Bingo Clue Cards 1 The tens digit is 3 times the ones digit. 2 The tens digit is twice the ones digit. 3 The ones digit is twice the hundreds digit. 4 The ones digit is 3 times the tens digit. 5 The hundreds digit is 3 less than the ones digit. 6 The hundreds digit is 3 less than one of the other digits. 7 The number is one less than a multiple of 10. 8 One of the digits is 3 less than anther digit. 9 The ones digit is twice the tens digit. 10 The tens digit is 2 greater than one of the other digits. 11 The hundreds digit is 2 greater than the ones digit. 12 The ones digit is less than twice the hundreds digit. 13 The tens digit is 2 greater than one of the other digits. 14 The hundreds digit is 2 greater than one of the other digits. Teacher Guide A Mental Mathematics Activity 1: Who-Am-I? Number Bingo Clue Cards B 15 The tens digit is 2 greater than the hundreds digit. 16 The hundreds digit is 3 less than the ones digit. 17 The hundreds digit is greater than the ones digit. 18 The tens digit is 3 times the ones digit. 19 The number is two more than a multiple of 5. 20 The tens digit is twice the hundreds digit. 21 The number is four less than a multiple of 10. 22 The ones digit is twice the tens digit. 23 The hundreds digit is 3 less than the tens digit. 24 The hundreds digit is greater than twice the ones digit. 25 The ones digit is less than twice the hundreds digit. 26 The tens digit is 3 times the ones digit. 27 The tens digit is more than twice the hundreds digit. 28 The hundreds digit is 3 less than one of the other digits. Unit 5: Logic of Algebra Mental Mathematics Activity 1: Who-Am-I? Number Bingo Clue Cards 29 The hundreds digit is 3 less than one of the other digits. 30 The hundreds digit is twice the ones digit. 31 The tens digit is twice the ones digit. 32 Tens digit < ones digit. 33 The hundreds digit is 2 greater than one of the other digits. 34 The ones digit is 3 times the tens digit. 35 Sum of all the digits is 9, 12, 15, or 18. 36 The ones digit is greater than the hundreds digit. 37 The hundreds digit is twice the ones digit. 38 The hundreds digit is twice the tens digit. 39 The tens digit is twice the hundreds digit. 40 One of the digits is 3 less than another digit. 41 One of the digits is 3 less than another digit. 42 The tens digit is greater than the hundreds digit. Teacher Guide C Mental Mathematics Activity 1: Who-Am-I? Number Bingo Clue Cards D 43 The ones digit is 3 times the tens digit. 45 44 Ones digit < tens digit. The hundreds digit is twice the tens digit. 46 The ones digit is twice the tens digit. 47 The hundreds digit is greater than twice the ones digit. 48 The ones digit is less than twice the hundreds digit. 49 The sum of all the digits is a multiple of 3. 50 The tens digit is twice the ones digit. 51 The hundreds digit is 3 times the tens digit. 52 The hundreds digit is 3 less than the ones digit. 53 The tens digit times the ones digit is 0. 54 The hundreds digit is greater than the tens digit.. 55 The hundreds digit times the ones digit is a multiple of 3. 56 The tens digit is twice the hundreds digit. Unit 5: Logic of Algebra Mental Mathematics Activity 1: Who-Am-I? Number Bingo Clue Cards 57 The sum of all the digits is a multiple of 5. 58 The tens digit is 2 greater than the hundreds digit. 59 The tens digit times the ones digit is a multiple of 3. 60 The hundreds digit is 3 less than the tens digit. 61 The tens digit plus the hundreds digit is a multiple of 3. 62 The hundreds digit is 2 greater than one of the other digits. 63 The hundreds digit is 3 times the ones digit. 64 The ones digit is twice the hundreds digit. 65 The ones digit times the tens digit is a multiple of 10. 66 The hundreds digit is 2 greater than the ones digit. 67 The hundreds digit times the tens digit is a multiple of 3. 68 The tens digit is more than twice the hundreds digit. 69 The tens digit times the ones digit is a multiple of 10. 70 The hundreds digit is greater than twice the ones digit. Teacher Guide E Mental Mathematics Activity 1: Who-Am-I? Number Bingo Clue Cards F 71 The tens digit times the ones digit is less than 8. 72 The ones digit plus the hundreds digit < 8. 73 The ones digit plus the hundreds digit is less than 8. 74 The sum of all the digits is a multiple of 3. 75 Sum of the tens digit and the ones digit is less than 8. 76 The hundreds digit times the ones digit is a multiple of 3. 77 The product of all the digits is a multiple of 3. 78 The tens digit plus the hundreds digit is a multiple of 3. 79 Sum of the tens digit and the ones digit is a multiple of 3. 80 The product of all the digits is less than 8. 81 Sum of the ones digit and the hundreds digit is a multiple of 3. 82 The ones digit times the tens digit is a multiple of 10. Unit 5: Logic of Algebra Mental Mathematics Activity 1: Who-Am-I? Number Bingo Clue Cards 83 The product of all the digits is less than 8. 84 The hundreds digit times the tens digit is a multiple of 3. 85 All three digits are the same. 86 The tens digit times the ones digit is a multiple of 10. 87 The number is four less than a multiple of 10. 88 The tens digit times the one digit is 0. 89 Ones digit < 5. 90 The sum of all the digits is a multiple of 5. 91 Hundreds digit > 6. 92 The tens digit times the ones digit is a multiple of 3. 93 At least two digits are the same. 94 Sum of the tens digit and the ones digit is a multiple of 3. 95 Two of the digits are less than 5. 96 The hundreds digit is 3 times the ones digit. Teacher Guide G Mental Mathematics Activity 1: Who-Am-I? Number Bingo Clue Cards 97 The number is a multiple of 10. 99 Ones digit > 6. 101 None 103 The 5. of the digits are odd. number is a multiple of 105 Sum of all the digits is less than 12. of the digits are greater than 5. The tens digit times the ones digit is less than 8. 100 Sum of the tens digit and the ones digit is less than 8. 102 The hundreds digit is 3 times the tens digit. 104 The product of all the digits is a multiple of 3. 106 Sum of all the ones digit and the hundreds digit is a multiple of 3. 107 Two 108 The 109 The 110 The hundreds digit is greater than the ones digit. H 98 tens digit is more than twice the hundreds digit. tens digit is twice the hundreds digit. Unit 5: Logic of Algebra Mental Mathematics Activity 1: Who-Am-I? Number Bingo Clue Cards 111 The number is two more than a multiple of 5. 113 None 115 The of the digits are even. tens digit is 4. 117 Tens digit < 3. 119 Hundreds digit > 2. 121 The tens digit is 9. 123 The hundreds digit is 9. Teacher Guide 112 The hundreds digit is greater than the tens digit. 114 The tens digit is 2 greater than the hundreds digit. 116 The ones digit is twice the hundreds digit. 118 The hundreds digit is 2 greater than one of the other digits. 120 The ones digit is less than twice the hundreds digit. 122 The ones digit is greater than the hundreds digit. 124 The hundreds digit is twice the tens digit. I Mental Mathematics Activity 1: Who-Am-I? Number Bingo Clue Cards 125 Tens digit is 7, 8, or 9. 127 The number is four less than a multiple of 10. 129 The number is two more than a multiple of 5. number is one less than a multiple of 10. hundreds digit is 2 greater than the ones digit. 128 The hundreds digit is 3 less than the tens digit. 130 Tens 131 The 132 One 133 Sum 134 The 135 The 136 The 137 The 138 The of all the digits is 9, 12, 15, or 18. tens digit is more than twice the hundreds digit. ones digit is twice the hundreds digit. J 126 The digit < ones digit. of the digits is 3 less than another digit. hundreds digit is twice the ones digit. tens digit is 3 times the ones digit. hundreds digit is greater than twice the ones digit. Unit 5: Logic of Algebra Mental Mathematics Activity 1: Who-Am-I? Number Bingo Clue Cards 139 The tens digit is 2 greater than the hundreds digit. 140 The 141 The 142 The hundreds digit is 3 less than the tens digit. 143 The hundreds digit is greater than the tens digit. hundreds digit is twice the tens digit. tens digit is twice the ones digit. tens digit is 2 greater than one of the other digits. 144 Ones 145 The 146 The 147 The 148 The 149 The 150 The hundreds digit is twice the ones digit. hundreds digit is 2 greater than the ones digit. Teacher Guide digit < tens digit. hundreds digit is 3 less than the ones digit. ones digit is twice the tens digit. hundreds digit is 3 less than one of the other digits. K Mental Mathematics Activity 7: Silent Collaborative Review Table Squares L Square Roots 02 = √144 = 12 = √121 = 22 = √100 = 32 = √81 = 42 = √64 = 52 = √49 = 62 = √36 = 72 = √25 = 82 = √16 = 92 = √9 = 102 = √4 = 112 = √1 = 122 = √0 = Unit 5: Logic of Algebra Balanced Mobiles Sorting Cards A If this mobile balances... B does this? YES or NO C If this mobile balances... does this? YES or NO Teacher Guide If this mobile balances... does this? YES or NO F does this? YES or NO G If this mobile balances... does this? YES or NO D E If this mobile balances... If this mobile balances... If this mobile balances... does this? YES or NO H does this? YES or NO If this mobile balances... does this? YES or NO M Four Corners Mobile 14 6 16 N Unit 5: Logic of Algebra Four Corners - Corner Options Step 1: Cross out any two hearts on the left side and both hearts on the right side. Step 1: Cross out the 6 on the left side and turn the 16 on the right side into 10. Teacher Guide O Four Corners - Corner Options Step 1: Cross out the 6 on the left side and turn the 14 on the right side into 8. Step 1: Add 14 and 16 together and turn the right side into two hearts and 30. P Unit 5: Logic of Algebra Four Corners Mobile Solving 14 6 16 Write directions for finding the value of the heart, step by step, starting with your group’s “Step 1”: __________________________________________ __________________________________________ __________________________________________ __________________________________________ __________________________________________ __________________________________________ Teacher Guide Q Lena & Jay’s Think-of-a-Number Tables (Blank) 1 Lena filled in a Think-of-a-Number table this way. Instructions Think of a number. Result Jay filled in the same Think-of-a-Number table this way. Instructions b Result b Think of a number. Add 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 Picture Think of a number. Add 4. Multiply by 3. Subtract 10. R Unit 5: Logic of Algebra Blank Think-of-a-Number Tables Instructions Think of a number. Instructions Think of a number. Teacher Guide Result n Result n S Last Instruction Sorting Cards A C E G I K M O Q T 4(3b – 5) 4b + 5 3 4(b + 2) + 3 b–4 +3 2 7 – 3b 7 – 4(b + 3) 3 5 7 – 3b 7 – (b + 3) – 4 5 7 – 4(b + 9) B D F H J L N P R b+2 –4 9 4( b + 3) 2 5(8b + 1) – 4 4(b – 7) + 5 3 5 b–7 5 4b + 3 3+ 7 – 4(b + 3) 3 4(3 + 7b + 5 ) 2 5 4(b + 3) – 4 Unit 5: Logic of Algebra Last Instruction Sorting Cards with Chunks A 4(3b – 5) 4b + 5 3 C E 4(b + 2) + 3 b–4 +3 2 G I K M O 7 – 3b 7 – 4(b + 3) 3 5 7 – 3b 7 – (b + 3) – 4 5 Q Teacher Guide 7 – 4(b + 9) B D F H J L N P R b+2 –4 9 4( b + 3) 2 5(8b + 1) – 4 4(b – 7) + 5 3 5 b–7 5 4b + 3 3+ 7 – 4(b + 3) 3 4(3 + 7b + 5 ) 2 5 4(b + 3) – 4 U Solving with Systems - 1 70 Use s= c= Which of these equations can you write using the mobile above? A B V C D E F 2s + 3c = 35 5s = 4c 3s + c = 2s + 3c 3s + c = 70 4c + 5s = 70 s = 2c Unit 5: Logic of Algebra Solving with Systems - 2 Use p = and m = 27 Teacher Guide W Solving with Systems - 3 Use h = and b = 54 X Unit 5: Logic of Algebra Solving with Systems - (Optional Challenge) Multi-Tiered Mobile Use h = s= and d = 112 Teacher Guide Y Snapshot Check-in Name: In each problem, the first mobile is balanced. Use it to figure out whether that means the second mobile for sure has to balance. Circle your response and explain how you know your answer is right. 1 2 If this mobile balances... does this? YES or NO If this mobile balances... Explain: Solve both the mobile and the equation, and check that you get the same answer. 3 does this? YES or NO Explain: Fill in the missing information on the following Think-ofa-Number tricks. Find c ( )... by drawing and using this mobile. 2c + 5 + c = 9 + 2c 4 Instructions Think of a number. Result b 4b 4b + 10 13(4b + 10) 13(4b + 10) – 8 5 Z Make up your own mobile problem. Include the solution. Unit 5: Logic of Algebra Snapshot Check-in Name: Answer Key In each problem, the first mobile is balanced. Use it to figure out whether that means the second mobile for sure has to balance. Circle your response and explain how you know your answer is right. 1 2 If this mobile balances... does this? YES or NO If this mobile balances... Explain: A bucket was taken from the Explain: left but added to the right. The left got lighter and the right got heavier. Solve both the mobile and the equation, and check that you get the same answer. 3 An equal weight (one moon) was removed from both sides. Fill in the missing information on the following Think-ofa-Number tricks. Find c ( )... by drawing and using this mobile. 2c + 5 + c = 9 + 2c Encourage various methods. One possibility: (remove 2c) 5 + c = 9 (subtract 5) c = 4 5 does this? YES or NO 9 5 4 Instructions Result Think of a number. b Multiply by 4. 4b Add 10. 4b + 10 Multiply by 13. 13(4b + 10) Subtract 8. 13(4b + 10) – 8 Make 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, or leave it open-ended, so that students can make a problem like #1-2 on this Snapshot (a “does this balance” problem), like #3 on this Snapshot (a “solve this equation using the mobile” problem), or like the mobile problems from Unit 1 (find the values of the shapes). Encourage students to be creative, and make multitiered mobiles or multiple mobiles that work together (such as on the cover of the Unit 5 student book) Teacher Guide AA Unit Assessment Name: 2 1 Find m ( )... by drawing and using this mobile. 4m + 2 + m = 10 + 2m + 7 If this mobile balances... does this? YES or NO Explain: Match each algebraic expression on the left with a set of instructions on the right. 3 4( w + 8 – 3) 2 4 4(w + 8) – 3 2 5 6 4(w + 8) –3 2 4w + 8 –3 2 A Think of a number. Add 8. Divide by 2. Subtract 3. Multiply by 4. B Think of a number. Add 8. Multiply by 4. Divide by 2. Subtract 3. C D Think of a number. Multiply by 4. Add 8. Divide by 2. Subtract 3. None of the above 7 12 – 2g A 12 less than g. 8 12 – g B Subtract g from 12. 9 2g – 12 C Subtract 12 from twice g. g – 12 D Twice g less than 12. 10 12 11 Instructions Result Think of a number. n 9( d – 13) + 8 4 Write the instructions in order. Think of Subtract it from 10. Divide by 7. Add 6 to it. Multiply by 3. AB Unit 5: Logic of Algebra Solve these equations. Show your process. 13 10 + 6h = 34 14 30 – (a + 3) = 22 15 3(y + 5) + 9 = 30 16 40 = 4 n–2 17 (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 Puzzle 4, x 1, – 4 18, x 180, x 2, – 15, x 11, + 6, x 10, x 2, – 18, x 10, x 4 4, x 1, – Teacher Guide AC Unit Assessment Name: Answer Key 2 1 Find m ( )... by drawing and using this mobile. 4m + 2 + m = 10 + 2m + 7 If this mobile balances... does this? YES or NO Explain: The same quantity (a heart) were removed from both sides. Various methods. One possibility: (take away 2) 5m = 8 + 2m + 7 (remove 2m) 3m = 8 + 7 3m = 15 m = 5 10 2 7 Match each algebraic expression on the left with a set of instructions on the right. 3 4( w + 8 – 3) 2 A 4 4(w + 8) – 3 2 D 5 6 4(w + 8) –3 2 4w + 8 –3 2 B C A Think of a number. Add 8. Divide by 2. Subtract 3. Multiply by 4. B Think of a number. Add 8. Multiply by 4. Divide by 2. Subtract 3. C D Think of a number. Multiply by 4. Add 8. Divide by 2. Subtract 3. None of the above 7 12 – 2g D A 12 less than g. 8 12 – g B B Subtract g from 12. 9 2g – 12 C C Subtract 12 from twice g. g – 12 A D Twice g less than 12. 10 12 11 Instructions Result Think of a number. n Subtract it from 10. 10 - n 10 - n 7 Divide by 7. Add 6 to it. Multiply by 3. AD 10 - n 7 3( 10 - n 7 + 6 + 6) 9( d – 13) + 8 4 Write the instructions in order. Think of a number. Divide by 4. Subtract 13. Multiply by 9. Add 8. Unit 5: Logic of Algebra Solve these equations. Show your process. 13 15 10 + 6h = 34 14 30 – (a + 3) = 22 6h = 24 a + 3 = 8 h = 4 a = 5 3(y + 5) + 9 = 30 16 40 = 4 n–2 3(y + 5) = 21 n - 2 = 10 y + 5 = 7 n = 12 y = 2 17 (k + 6)2 = 64 18 (c – 3)2 = 16 k + 6 = 8 OR k + 6 = -8 c - 3 = 4 OR c - 3 = -4 k = 2 OR k = -14 c = 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 Puzzle 4, x 1, – 4 1 4 1 2 18, x 4 3 180, x 3 2, – 6 15, x 2 5 11, + 5 18, x 3 2 Teacher Guide 6 6, x 6 10, x 6 5 5 2 3 1 2, – 4 6 1, – 5 4 3 2 2 3 10, x 1 5 6 4 4 1 4, x 1 4 AE Transition to Algebra Transition to Algebra is an EDC project supported by the National Science Foundation aimed at very quickly giving students the mathematical knowledge, skill, and confidence to succeed in a standard first year algebra class . The familiar topic-oriented approach to mathematics is replaced by a small number of key mathematical ideas and ways of thinking: Algebraic Habits of Mind (puzzling, using structure, generalizing patterns, using tools, and communicating clearly) . Conventional algebra topics are part of the curriculum, but instead of the topics being the focus of the lesson, they become contexts for exploring these broadly-applicable problem-solving strategies . More information on this research and development project is available at ttalgebra .edc .org .