Download Unit 5: Logic of Algebra - Pittsburgh Public Schools

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 .