Download Unit 2: Geography of the Number Line

Unit 2: Geography of the
Number Line
Put A, B, and C in order on the number line:
Geography?
Clue 1: A is positive.
Clue 2: B is negative.
Clue 3: A < C
A number line shows the position,
order, and space between numbers, kind
of like a map for numbers.
I
I
0
B
I
I
A
C
If n – 5 = 13, what is n?
13
14
I
I
n–5 n–4
I
I
18
I
I
n
Dear Student,
Every real number lives on the number line, no matter how huge
or how small, including the most cumbersome fractions and the most
complicated decimals. Every number lives close to a lot of other numbers,
and between every two numbers are infinitely many other numbers. Because
the number line captures all of this in one image, it is a valuable tool used by
mathematicians to picture relationships among numbers.
Your work in this unit will involve finding where positive and negative
numbers live and learning how to use the number line to picture addition,
subtraction, and even algebra.
The Authors
I
-15
I
-12
I
-9
I
-6
I
-3
I
0
I
3
I
6
I
9
I
12
I
15
-9 ≤ x < 6
t
Who Am I?
• I am odd.
1
• u>t
• I am less than 30.
• The sum of my digits is 10.
I
18
I
21
I
24
10
10
u
10
9
10
2
3
4
+
5
6
7
8
In each box, put the sum of the numbers that point to it.
Unit 2 Introduction
Teacher Guide
1
T1
The Transition to Algebra materials are being developed at Education Development Center, Inc. in Waltham, MA.
More information on this research and development project is available at ttalgebra.edc.org.
Copyright © 2013 by Education Development Center, Inc.
Pre-publication draft. Do not copy, quote, or cite without written permission.
All rights reserved. No part of this book may be reproduced in any form
or by any electronic or mechanical means except for classroom use only.
You may not distribute these materials outside the classroom.
The use of these materials is limited to the 2013-14 school year.
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 K. 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 2: Geography of the Number Line
This unit presents the number line as a tool for reasoning about integers and the relationships between integers,
including order and distance. The number line is also used as a tool for making sense of the operations of
addition and subtraction, first with numbers and then generalized to variables.
Lesson 1: Placing Integers . . . . . . . . . . . . . . . . . T12
Lesson 2: Operations with Integers. . . . . . . . . . . T20
Lesson 3: Checkers and Who Am I? Puzzles. . . T28
Snapshot Check-in. . . . . . . . . . . . . . . . . . . . T73
Lesson 4: Distance and Inequalities. . . . . . . . . . T36
Lesson 5: Geography of Addition and Subtraction.T43
Lesson 6: Algebra on the Number Line . . . . . . . T50
Unit Assessment. . . . . . . . . . . . . . . . . . . . . . T76
Mental Mathematics
• Distances, Adding, and Subtracting
Exploration
• Toothpick Rows. . . . . . . . . . . . . . . . . . . . . . . T4
• Color Towers 2 . . . . . . . . . . . . . . . . . . . . . . . T8
Related Activity
• Who Am I? Number Bingo. . . . . . . . . . . . . T59
Number Line
Using these Materials
In order to make sense of arithmetic operations without
reverting to rules, which may seem arbitrary or be
misapplied, students must be comfortable with the
geography of the number line. Students must learn that
numbers have distinctive “personalities” due to their
location and order on the number line, and they can make
sense of the results of operations performed on these
numbers by considering the relationship between them.
Explorations are a key component of the
curriculum as a way of teaching students how to
think mathematically. There are two Explorations in
Unit 2; both are located at the beginning of the unit
in the Student Book and Teacher Guide. Since the
Explorations are not tied to any particular lesson,
you may choose when in the unit to use them.
The Related Activity for Unit 2 is a bingo game.
The instructions are located on page T59 in the Teacher
Guide and on page 35 in the Student Book. The game
uses a set of cards which are found in the Teacher
Guide on page T60.
Wh
er
The number line is presented as a tool for reasoning
about numbers because it builds students’ familiarity
with the magnitude of numbers, distance between
numbers, and order of numbers. Students will revisit
the number line in Units 3 and 6. In Unit 3, MicroGeography of the Number Line, students will see that
decimals and fractions follow the same logic of integers on a zoomed in number line. In Unit 6, Geography of the
Plane, students will see that two orthogonal number lines form the axes of a coordinate plane.
I?
Where Am I? Number Puzzles
m
a
e
Where Am I? number puzzles support flexible thinking with the number line as
students use clues to find the possible location of numbers. Two forms of the puzzles
are shown here:
Clue: I am less than 60.
My name is n.
Where am I? Use symbols: n < 60
Show everywhere I could be.
I
T2
I
0
I
20
I
40
II
60
I
Clue: I am 20 units away from -4.
Where am I?
I
-24
I I
-4 0
I
16
Unit 2: Geography of the Number Line
Note that both Where Am I? puzzles have more than one solution. In the first puzzle, students learn how to mark
inequalities on the number line to indicate infinite solutions. In the second puzzle, students use the number line to
determine the two possible solutions.
Students connect these ideas to algebra in problems like these:
Clue: I am 9 units away from p.
Where am I?
p – 9
I
p + -9
I
p
If a + 4 = 1, what is a?
p – -9
I
p + 9
I
-3 -2
I
a
I
-1
I
0
I
1
I
a+4
Who Am I? Number Puzzles
In solving these puzzles, students develop their own methods for
coordinating multiple “clues” that deal with the properties of an
unknown number and its digits. These puzzles strengthen logical
reasoning as well as mathematical language comprehension and
translation.
Who Am I?
• The sum of my digits is 16.
• My tens digit is greater than
my units digit.
t
u
9
7
Algebraic Habits of Mind
Using Tools Strategically: The number line is a tool that can help make sense of a calculation. The focus of this unit
is not just on how to use the number line but also on developing the mathematical thinking promoted by picturing
a number line. Students learn to look at problems and make sense of them by turning to a tool that helps locate and
compare numbers. Students learn to see addition and subtraction as more than an action to be done to two numbers;
they are part of a larger system in which numbers are related to each other. Adding and subtracting positive and
negative numbers is no longer about error-prone algorithms and mis-remembered rules, but about predicting where
the answer will fall and making sense of problems using distance and order on the number line.
Puzzling and Persevering: Where Am I? puzzles ask students to make sense of clues to locate values on the number
line. Because puzzles may have multiple (often infinitely many) solutions, students must figure out if they have
found all possible solutions. Who Am I? puzzles help students build working memory and the ability to coordinate
multiple pieces of information and draw logical conclusions. Students must find a way to start the problem and
evaluate possible solutions against multiple criteria.
Communicating Clearly: Students continue to practice the use of mathematical language in Thinking Out Loud
dialogues. Describing what they see on a number line helps students make sense of each problem and solidify their
reasoning.
Learning Goals:
• Order and place positive and negative numbers on the number line.
• Make and use observations about sums and differences on the number line.
• Use the number line to determine the distance between two numbers.
• Represent inequalities on the number line.
• Reason abstractly about number placement and operations using variables.
• Translate verbal and symbolic puzzle clues.
Teacher Guide
T3
Exploration: Toothpick Rows
Purpose
This Exploration leads students toward building an algebraic expression. Students first observe a numerical
pattern, experience how the numerical pattern is applied to more numbers, and then describe the pattern
of calculation using an algebraic expression. The pattern can be explained visually (using geometry) and
physically (using manipulatives).
Exploration at a Glance
Preparation: Each group needs approximately 30 toothpicks or other short sticks (straws, crayons, etc.).
Mental Mathematics: (5 min) See pages XX-XX of the TTA Mental Mathematics book.
Launch: (5 min)
• Quickly make sure students understand the task.
Student Exploration and Discussion: (35 min)
• Provide time for students to explore on their own.
• Discuss algebraic expressions that students may write for the number of toothpicks.
Unit 2 Related Game: Who Am I? Number Bingo (See Teacher Guide page T59 and Student Book page 35)
Launch the Activity:
The activity doesn’t take much introduction. You may choose to draw or
demonstrate making 4 squares with 13 toothpicks so that students are clear
about what they are counting. Then have them start the Exploration.
4 squares
What if...
What if students try to build the row of 10 or 20 squares to answer problems 4 and 5? Some may try to
build all 20 squares. Let them do it (although they’ll probably run out of toothpicks). Students will probably get
tired of building squares before they reach 100 squares. That’s the point – direct exploration is great, but you
do need more structure and organization when you start to generalize. Ask: “Can you describe what’s going on
without having to build every single square?”
What if students can only express the relationship “add 3 to the number of toothpicks before it” and so
can’t write an algebraic expression? First of all, acknowledge that the recursive definition (“you keep adding
3 toothpicks”) is exactly right and mathematically important (computer programmers often use this kind of
thinking). For students who only see the recursive nature of the pattern, ask “Now how do you use that to find
the number of toothpicks in 10 squares?”
Discussing Student Findings:
Invite students to share their written descriptions for problem 7. The Answer Key includes four different possible
descriptions. All of them are mathematically interesting:
• “It’s one more than 3 times the number of squares” may come from a student who only looked at the numbers,
or from a student who considered the first toothpick to start the chain, and saw the first square as adding 3
T4
Unit 2: Geography of the Number Line
toothpicks to that initial toothpick.
Teacher Tips
• “You add 3 toothpicks to the number you had
last time” may at first seem incomplete, but
At this point in the year, students may not generate an
is a nice recursive explanation of the pattern,
algebraic expression on their own. The process outlined in
and touches upon the main point of the
this Exploration is called “Guess-Check-Generalize” and
pattern having a constant rate of change.
will be revisited many times over the year. The important
• “You need four for the first square, but only
part of this Exploration is that they have the expreince of
three extra for each square after that” uses the
describing repeated reasoning. As they find the number of
language of considering the initial value (4),
toothpicks it takes to make 5, 6, 10, 20, and 100 squares,
then the constant rate of change (3).
make sure they see that the answer is not as important as
• “4 for each square minus 1 for each square
being able to describe the process for finding it.
after the first one” may come from a student
who understands needing to multiply the
number of squares by 4 since a square has four sides, and then modifying the squares based on their shared
sides.
Use the shared verbal descriptions to have a discussion about ways to translate the verbal descriptions into
algebraic expressions. Depending on your class, you may have a productive discussion by asking students to
show how they translated their verbal descriptions into algebraic expressions. Or you may assume the role of the
“native speaker” in the class to demonstrate how to translate each description into algebra. Show students, for
example, how 3n + 1 is directly related to the description “3 times the number of squares, then add 1.” Then have
a discussion about how different interpretations of the pattern can lead to expressions that can all be shown to be
equivalent.
Teacher Tips
Students may also have other descriptions and corresponding algebraic expressions. For example, a student
may have reasoned “To make 20 squares, I add 16 squares to the original 4. Each of those 16 squares takes 3
toothpicks. So 20 squares takes 13 + 16(3) toothpicks. In the same way, s squares would take 13 + (s – 4)(3)
toothpicks.” Note also that the recursive expression wasn’t provided in the Answer Key, but should certainly
be accepted as a correct answer: f(0) = 1; f(n) = f(n – 1) + 3 for n ≥ 1.
Further Exploration:
In problems 9-11, students consider the problem in reverse: given the number of toothpicks, how many squares
in a row can be built? Each of these questions ask about a number of toothpicks for which you can build a
row of squares without having any toothpicks left over. You may also modify this question to ask about other
numbers of toothpicks with the additional question: “How many toothpicks will be left over?” Or you might
ask students to generalize a process for finding the number of squares given t toothpicks (and finding how
many toothpicks are left over in that case).
Problem 12 of the Further Exploration asks students extend the reasoning from the original Exploration to a
slightly different shape. Encourage students to describe the relationship between problem 12 and the original
Exploration.
Exploration: Toothpick Rows
T5
Exploration: Toothpick Rows
Make a row of squares with toothpicks as in the picture below.
How many toothpicks does it take to build a row of 4 squares?
1
13 toothpicks
4 squares
2
4
5
6
2
T6
Find out how many toothpicks are necessary for
rows of various sizes. Record them in this table.
Squares
Toothpicks
1
4
2
7
3
10
4
13
5
16
6
19
3
Describe a relationship between the number of squares
and the number of toothpicks.
Responses will vary. At this stage,
either of these interpretations is
fine:
Every time you add a square, you
add three toothpicks.
or
The number of toothpicks is 1 more
than three times the number of
squares.
How many toothpicks does it take to build a row of 10 squares? Describe how you can figure this out
without building the whole row.
It takes 31 toothpicks. Responses will vary.
Students may continue the pattern of adding 3 toothpicks each for 7, 8,
9, and 10 squares. Or students may reason that to make 10 squares, you
have to build 4 more squares, needing 12 more toothpicks. Or students
may figure out that to make the row, you can place 1 toothpick, then add
3 toothpicks for every square, needing 3(10) + 1 toothpicks.
How many toothpicks does it take to build a row of 20 squares? Describe how you can figure this out
without building the row of squares.
It takes 61 toothpicks. Responses will vary.
Students may continue the pattern of adding 3 toothpicks. Or students may
reason that it takes 31 toothpicks to make 10 squares so to build 10 more
squares requires 30 more toothpicks. Or students may figure out that the
row takes 3(20) + 1 toothpicks (after the first toothpick, each square is
made with 3 more toothpicks. Or students may see that doubling the number
of squares uses one less than double the number of toothpicks.
How many toothpicks does it take to build a row of 100 squares? Describe how you figured this out.
It takes 301 toothpicks. Responses will vary. Students may reason that
building a row uses one toothpick plus 3 toothpicks per square, so 100
toothpicks uses 3(100) + 1 toothpicks. Or students may add 240 toothpicks
(3 toothpicks for 80 squares) to the result for 20 squares.
Unit 2: Geography of the Number Line
Unit 2: Geography of the Number Line
7
8
If someone tells you the number of squares they want to make, how would you describe how many toothpicks
they will need? Responses will vary. Possible responses include:
“It’s one more than 3 times the number of squares” or
“You add 3 toothpicks to the number you had last time.” or
“You need four for the first square, but only three extra for each
square after that.” or
“4 for each square minus 1 for each square after the first one.”
How many toothpicks does it take to build a row of s squares?
Responses will vary. Students might provide a verbal description or an
algebraic expression. “Multiply the number of squares by 3 and add 1” or
3s + 1 or
4 + 3(s – 1) or
4s – (s – 1)
Further Exploration
9
If you build a row with 25 toothpicks, then how many squares will it have?
25 toothpicks will build a row of 8 squares.
10
If you build a row with 40 toothpicks, then how many squares will it have? Describe how you figured this out.
40 toothpicks will build a row of 13 squares. Responses will vary. Put 1
toothpick down, then the remaining 39 toothpicks will make 13 squares.
Subtract 1 from 40, then divide by 3.
11
If you build a row with 100 toothpicks, then how many squares will it have? Describe how you figured this out.
100 toothpicks will build a row of 33 squares. Responses will vary. Put 1
toothpick down, then the remaining 99 toothpicks will make 33 squares.
Subtract 1 from 100, then divide by 3. Or, use the response to problem 10
and reason that the 60 additional toothpicks will make 20 more squares.
12
Suppose you build a row of rectangles as shown below. How many toothpicks does it take to build a
row of r rectangles?
5 rectangles
Exploration: Toothpick Rows
Exploration: Toothpick Rows
It takes 4r + 2 toothpicks to build a row
of r rectangles.
Other ways to write this expression include:
6 + 4(r – 1) or
6s – 2(s – 1)
“Multiply the number of rectangles by 4 and
add 2.”
3
T7
Exploration: Color Towers 2
Purpose
Students explore a variation of the Color Towers Exploration from Unit 1. In Unit 1, students looked for all the
possible arrangements for a tower of a certain height built with two colors, for which exactly two of the blocks
are one color and the rest are another color. In this Exploration, students are told to build as many towers of a
certain height as possible from two colors. Students are led through a systematic process for organizing their
answers as an example of how to use an organized solution to describe and make predictions about general
patterns.
Exploration at a Glance
Preparation: Each group needs about 30 blocks: 15 of one color, and 15 of another color. You might use
LEGOs, stacking cubes, or even just squares of paper.
Mental Mathematics: (5 min) See pages XX-XX of the TTA Mental Mathematics book.
Launch: (10 min)
• Pose the question, “Using two colors, how many towers can we make that are 2 stories tall?”
Student Exploration and Discussion: (30 min)
• Provide time for students to explore on their own.
• Discuss students’ predictions and organization strategies.
Unit 2 Related Game: Who Am I? Number Bingo (See Teacher Guide page T59 and Student Book page 35)
Launch the Activity:
Before students turn to page 4 in the Student Book, show them that you have two different colors of blocks and ask,
“Using these two colors, how many different towers can we make that are 2 stories tall?” The four possible towers
are shown at the top of the Student Book page.
What if...
What if students don’t understand the organizational scheme in problem 2? First, examine the students’
response to problem 1 to understand how they have organized their towers and help them see how they can craft
an alternate explanation for the fact that the number of possible towers has doubled. It may also help students to
compare the organizational scheme to putting words in alphabetical order. This comparison might help students
to see that this way of organizing the tower possibilities will capture all possibilities, and will help to order
them.
Discussing Student Findings:
Invite students to share their prediction and their reasoning from problem 4. The prediction is less important than the
reasoning. Listen for students who have made sense of the doubling pattern. You may hear responses like:
• “There are 8 possible 3-story towers. Now imagine adding a blue block to the bottom of all 8 towers. That’s
8 different 4-story towers. Now replace that blue block with a green block. That gives 8 more 4-story towers.
None of those towers are the same, because we know the top three blocks all made different combinations, and
T8
Unit 2: Geography of the Number Line
that last block will either be blue or green. So there are 16 possible 4-story towers.”
• “I listed them out in a tree diagram. The first block is either blue or green. Then you can add either a blue or
green block to each of those, so the second level has 4 possibilities. Then draw 2 branches out from all of those,
and you get your 8 3-story towers. Finally, draw 2 branches out from each of those possibilities, and you get 16
possible 4-story towers.”
Then discuss problem 5 by asking students to describe how they organized their responses. Make sure every student
has an idea for how to write an answer (whether or not they are successful at writing all 16 possible arrangements).
Further Exploration:
Problem 6 asks students to extend the pattern to 5 blocks. If students have made sense of the doubling pattern,
then they just need to double their response from problem 5. Some students may also make the connection that
they can find the number of possible arrangements for the 5-story tower by finding 25.
Problem 7 gives students a sense of how quickly the number of possible arrangements grows, while also asking
students to extend the pattern in the same way.
In problem 8, students may approach the problem by building towers, by listing possible arrangments, or by
making a hypothesis about three-color towers by extending their reasoning for two-color towers. Encourage
students to make and test further hypotheses about how the number of colors and the number of stories affects
the number of possible arrangements of towers in general.
Exploration: Color Towers 2
T9
Exploration: Color Towers 2
You have two piles of blocks, one blue and one green.
Here are all four ways you can arrange these colors to
make a tower
that is exactly
B
B
G
G
2 blocks tall:
B
1
G
B
G
How many different ways can you arrange two colors to make a tower that is exactly 3 blocks tall?
This space is for experimenting.
You may not need all the towers.
Organize your solution in a sensible way here.
B
B
B
B
G
G
G
G
B
B
G
G
B
B
G
G
B
G
B
G
B
G
B
G
This is only one of many possible ways to organize the solutions.
One possible way to explain the growth of this pattern is to examine how adding a block changes the problem.
For example, here’s a way to think about how adding one block changed the 2-story tower into a 3-story tower.
2
4
T10
These towers have These towers
BG
have the _______
the BB 2-story
tower at the top. 2-story tower at
the top.
B
B
B
B
B
B
G
G
B
G
B
G
These towers
These towers have ________________
have the GG
________________
GB 2-story
the ____________
2-story tower
________________
tower at the
________________
at the top.
top.
________________
________________
G
G
G
G
B
B
G
G
B
G
B
G
Unit 2: Geography of the Number Line
Unit 2: Geography of the Number Line
3
Explain the connection between the number of 2-story towers and the number of 3-story towers.
4
Using two colors, there are ______
ways to build a 2-story tower.
(how many?)
adding a block
Every 3-story tower can be built from a 2-story tower by _________________.
2
There are only ______
possible colors that the third block can be. That means
there are twice as many possible 3-story towers as there are 2-story towers.
8
So there are ______
possible ways to build a 3-story tower using two colors.
4
Make a prediction. How many different 4-story towers do you think you can make
with two colors? Describe the reasoning you used to make your prediction.
Responses will vary. Students may extend the reasoning
by realizing that since the fourth block can only be one
of two colors, there are 16 different 4-story towers.
5
Check your prediction by showing all the different ways you can arrange two colors to
make a tower that is exactly 4 blocks tall? Organize your solution in a sensible way.
This is only one of many possible ways to
organize the solutions:
BBBB
BBBG
BBGB
BBGG
BGBB
BGBG
BGGB
BGGG
GBBB
GBBG
GBGB
GBGG
GGBB
GGBG
GGGB
GGGG
Every 4-story
tower can be built
from a 3-story
tower by...
You can draw
your own towers,
or you can use
abbreviations like
BGGB to show
the order of the
colors in a tower.
Further Exploration
6
How many different ways can you arrange two colors to make a tower that is exactly 5 blocks tall?
32 ways
7
Using two colors, there are 1024 ways to build a tower that is 10 blocks tall. How many different ways can you
arrange two colors to make a tower that is exactly 11 blocks tall?
1024 × 2 = 2048 ways
8
What if you have three colors (blue, green, and yellow)? How many different 2-story towers you can make
with three colors?
Like
3 × 3 = 9 ways
Y
G
B
G
Exploration: Color Towers 2
Exploration: Color Towers 2
and so on...
BB
BG
BY
GB
GG
GY
YB
YG
YY
5
T11
Lesson 1: Placing Integers
Purpose
In this lesson, students place positive and negative integers on the number line and examine and complete
number lines drawn at different scales. Students also start to interpret inequality symbols along the number
line. This lesson introduces the unit as students use the number line to visually compare distance and order of
numbers. Students will see similar problems in Unit 3, Micro-Geography of the Number Line, as they make
sense of decimals and fractions on the number line.
Lesson at a Glance
1
Preparation: Prepare 36 sticky notes for the Launch. Write the following numbers on sticky notes: -8002, -55,
1
1
1
15, -33.5, 0, 142, 652, 20, 304.5, 1000, 204, 295.5, 250, -552, 152, 743, -555.5, -200, 3902, 13, 11.5, -1, 23.5,
1
1
1
31, 50, -82, 9.5, 80, -80, 162, 9, -21, -610, -8.5, 112, and 14.5.
Mental Mathematics: (5 min) See pages XX-XX of the TTA Mental Mathematics book.
Launch: (15 min) Thinking Out Loud Dialogue and Zooming In on the Number Line
• Perform the Thinking Out Loud dialogue as a class.
• Hand out the prepared sticky notes and ask students to place them on a number line
marked with a scale of 200 (shown below).
• Repeat the sticky note exercise on a number line marked with a scale of 20.
As a Class
Student Problem Solving and Discussion: (25 min)
• Allow students to work through the rest of the Important Stuff and explore additional problems.
• Discuss strategies for using given information to fill in number lines.
• Discuss using clues with inequalities to determine the order of values on the number line.
In Groups
As a Class
Unit 2 Related Game: Who Am I? Number Bingo (See Teacher Guide page T59 and Student Book page 35)
Launch – Thinking Out Loud Dialogue and Zooming in on the Number Line:
Write the sequence from problem 1 on the board: 11, 8, 5, 2, ____. Ask three students to read through the dialogue at
the board. Make sure they draw a number line and mark on it when prompted. Call on several students to share their
thoughts about Lena’s interpretation of the problem using the Pausing to Think box.
Then, draw a long line on the board, mark 0 and 1000, and ask students to help draw the rest of the number line from
-1000 to 1000, referring back to the dialogue to talk about the placement of negative numbers. Draw the number line
with a hash mark every 200 units, as shown below. Draw attention to the symmetry of the marks across 0.
Give each student one numbered sticky note, and ask them to place it where it goes on the number line. Some
students may get two or more sticky notes to place. Having three or more students up at once helps move this
activity along. Resist the urge to correct placement while students are working.
I
-1000
-800 1
2I
-800
I
-600
I
-400
I
-200
15
-55 I 20
0
I
200
I
400
I
600
I
800
I
1000
Some numbers will be too crowded to place properly. Making a “pile” there is correct – at that scale, all of those
numbers are correctly placed in about the same location.
T12
Unit 2: Geography of the Number Line
Once everyone is done, quickly read off the
other numbers, skipping over the “mess” for the
moment, and have students decide whether these
others are placed correctly. Ask students roughly
what numbers the “mess” is between.
Remark that you’ll zoom in on that interval to
unpack the “mess.” Draw a new number line
from -100 to 100 with a hash mark every 20.
Make it about as long as the original. Collect
only the sticky notes from the original “mess.”
Distribute these for students to place on the new
“zoomed-in” number line.
What if...
What if students incorrectly place negatives, such as
placing -610 to the right of -600? Offer that “just as 610 is
between 600 and 700, -610 is between -600 and -700.” Let
them suggest a new place for the number.
What if students misread numbers with decimals as
whole numbers, such as placing 23.5 at 235? Suggest that
“twenty-three point five is just a little more than twentythree, not up in the two hundreds...” and let them suggest
a different place to put it. Such errors don’t need extended
instruction at this point. Students will naturally gain more
skill as the unit progresses.
As before, when students finish, have them
briefly check the placement of the numbers that
are not clustered in the mess. If you feel it would
help, you may acknowledge the new mess, and zoom in once more on a -10 to 10 number line to unpack the mess.
Student Problem Solving and Discussion:
Allow time for students to work together in pairs or small groups on the Important Stuff. Listen for strategies
students use to fill in the spaces in problems 7-12, especially if you hear students using methods other than counting
or trial-and-error, such as subtraction and then division to find the scale. Problems 17-20 are different from the
previous problems and from the launch. This type of problem will return throughout the unit. Students may need
support to remember how to read the inequality symbols.
Supporting Classroom Discussion
Consider using these additional questions with individuals, small groups, or the whole class to stimulate
productive mathematical communication:
• Which number did you fill in first in problem 6? Some students aren’t in the habit of looking around for
a way to enter into a problem; instead, they may try to fill in the spaces in problem 6 by working from left
to right, using trial and error. Have students share the insight of recognizing how the marked -2 and -4 give
valuable information.
• Which number did you fill in first in problems 7, 8, and 9? Have students share how they found the
number halfway between 0 and the other marked number and how they used this to fill in the rest.
• Which number(s) did you fill in first in problem 10, 11, and 12? The explanation for filling in these
spaces is a bit more involved. Some students may have used some form of trial and error, but others may
be able to explain how they thought of cutting distances in half to get the scale without guessing.
• How did you use the clues in problem 18 to mark A, B, and C on the number line? Invite students to
share their process. Use this opportunity to remind students that B < A is read “B is less than A.”
• How did you use the clues in problem 19 to mark A, B, and C on the number line? In this case, to
solve the problem, a student has to dive in and mark at least one of the letters on the number line to be able
to mark the location of the other letters relative to the first. If students notice that 0 is not marked, ask them
“Does your picture change if I tell you that A is negative? Or if B is positive?”
• How did you use the clues in problem 20 to put Adam, Brianna, and Chris in order? Invite students to
share their methods, then point out how the same logic can describe a strategy for solving problem 19.
Lesson 1: Placing Integers
T13
Connections to Algebra
Teacher Tips
Thinking about the relationship between
numbers is an essential part of algebraic
thinking. Consider an expression like m + 4, or a
relationship like h = m + 4. For students to make
sense of such an expression or relationship,
they have to make sense of h being 4 more
than the number m. A student who doesn’t
have the mental image of a distance of 4 on the
number line may find it difficult to understand
how h = m + 4 is a relationship between two
quantities and will only be able to approach
these problems with procedures, not intuition
and logic.
The thinking students do in problems 13-16 will help
students think about distances later in the unit and will
serve as a framework for thinking about decimals and
fractions in Unit 3.
Notes:
For now, students will use a number’s distance to
multiples of 10 to find the total distance between -43
and -92.
2
I
I
-92 -90
7
I
I
-50
-43
In Unit 3, students will find distances with decimals
and fractions.
I
0.08
3.92
I
3
85
T14
40
6
I
4
I
0.4
10
2
5
1
5
7
I
10.4
I
I
I
9
16
16 5
1
Unit 2: Geography of the Number Line
2-1 Placing Integers
Important Stuff
1
Try imagining a number line.
-1
Find a pattern and write the number that comes next: 11, 8, 5, 2, _____
Thinking Out Loud
Michael , Lena, and Jay are working on problem 1.
Michael: Is it a negative or a fraction? Maybe it’s a decimal. Something’s got to change...
Lena:
Yeah, something’s going to change. The numbers go down by 3, so will it be 2 minus 3?
Pausing to Think
What does Lena mean by “the numbers go down by 3”?
I pictured it on a number line. (Jay draws a number line and marks 11, 8, 5, and then 2.)
Jay:
I
I
I
I
I
I
I
2
I
I
I
I
5
I
I
8
I
11
Then I counted down 3 more and got to...
Michael: Oh, I know! Look: 1, 0, -1. We subtracted three so many times we went below zero!
(Michael fills in a few more numbers on the number line to show how he got to -1.)
I
2
I
I
I
I
-1
0
1
2
I
I
I
I
I
I
5
I
I
8
I
11
Finish labeling this number line to help you answer the problems below.
I
I
I
I
I
-11
-10
-9
-8
-7
I
-6
I
I
I
I
I
-5
-4
-3
-2
-1
I
0
3
-3 , _____
-5 , _____
-7
Find three more numbers for this pattern: 5, 3, 1, -1, _____
4
-10
Find a pattern and write the number that comes next: 2, -2, -6, _____
5
-1 _____
2
Complete this pattern in a sensible way: -10, -7, -4, _____,
6
This number line is marked using a different scale. Finish labeling it in a consistent way.
I
-22
I
-20
6
Lesson 1: Placing Integers
I
I
I
I
I
I
I
-18
-16
-14
-12
-10
-8
-6
I
1
I
I
2
3
Often, we use regularly
spaced markings on a
number line (every half,
every one, or every five or
ten) depending on the scale.
I
-4
I
-2
I
I
I
0
2
4
I
6
Unit 2: Geography of the Number Line
T15
These number lines all have regular markings. Watch for different scales, and finish labeling the marks on these lines.
7
9
11
I
-10
I
0
I
I
-16
I
0
I
I
20
10
-12
I
-8
-4
I
I
I
25
50
75
I
I
0
8
30
I
I
0
10
I
100
12
I
I
6
3
I
-15
I
I
-28
-32
I
9
12
I
I
0
5
I
-20
I
I
-5
-10
I
I
I
-24
-16
Using a number line, you establish the order of the numbers (from least to greatest) and can mark distances.
For example, this one diagram shows that the number
• 53 comes between 50 and 60 (that’s 50 < 53 < 60)
• 53 is 3 more than 50 (that’s 53 = 50 + 3)
• and 53 is 7 less than 60 (53 = 60 – 7)
3
7
I
I
I
50
53
60
Place each target number between its closest multiples of 10. Show how far away it is from each multiple of 10.
The results should look like the diagram above. Problem 13 is partially done for you.
13
4
Target: -36
6
14
1
Target: 71
9
I
I
I
I
I
I
-40
-36
-30
70
71
80
Mark number lines from least to greatest.
15
Target: -18 2
I
8
I
16
-20 -18
Target: 117
I
I
-10
110
7
3
I
I
117
120
Use the clues to mark one possible location each for A, B, and C so they are in the correct order.
17
Clue 1: A is negative.
Clue 2: B is positive.
Clue 3: C is greater than B.
I
A
I I
0 B
I
T16
Clue 1: A is negative.
Clue 2: B < A
Clue 3: C is not negative.
I
C
Since all you know is that A
is negative, you may pick any
negative place.
Lesson 1: Placing Integers
18
B
20
I
A
I
0
I
C
19
Clue 1: A > B
Clue 2: A > C
Clue 3: C < B
I
C
I
B
I
A
Adam is older than Brianna; Adam is older than Chris; Chris is younger
than Brianna. List them in order from youngest to oldest.
Youngest: Chris; Middle: Brianna; Oldest: Adam
7
Unit 2: Geography of the Number Line
Stuff to Make You Think
Finish labeling the marks on these lines in a consistent way.
21
23
I
-15
I
0
I
15
I
I
140
160
I
180
I
I
30
45
I
22
I
220
200
I
I
I
28
35
I
-18
24
42
I
49
I
56
I
I
I
-14
-10
-6
I
-2
Place each target number between its closest multiples of 10. Then zoom out and place the integer between its closest
multiples of 100. Write in the distances. Problems 25 and 26 are partially done for you.
25
2
Target: 362
Between 10’s
I
8
I
I
360 362
Between 100’s
370
62
38
I
300
26
I I
360
362
I
370
6
Target: -124
Between 10’s
Between 100’s
-200
Target: 413
3
Between 10’s
I
410
Between 100’s
13
400
Lesson 1: Placing Integers
-124
I
I
-120
-124
I
-100
7
I
I
413
420
87
I
I
410 420
413
8
I
-120
24
I
-130
I
I
76
I
I
4
I
-130
27
I
400
I
500
Unit 2: Geography of the Number Line
T17
28
Target: -981
1
9
Between 10’s
I
I
-981 -980
-990
Between 100’s
81
19
I
-990 -980
-981
-1000
29
I I
I
I
I
-900
What number is 4 away from -60 and 6 away
from -70?
30
-64
31
-11
32
48
8
= ______
What number is 9 away from -20 and is greater
than -20?
4
= ______
3
= ______
48
7
= ______
2
= ______
6
= ______
Tough Stuff
A, B, C, D, and E are numbers. Use the clues to mark them in the correct order on the number line.
33
Clue 1: C is negative.
Clue 2: B < C
Clue 3: A < E < D
Clue 4: A is positive.
I
B
35
I
C
34
I
0
I
A
I
E
I
D
Clue 1: A < C < B
Clue 2: E > B
Clue 3: C < D
Clue 4: B > D
I
A
I
C
I
D
I
B
I
E
Make Your Own: On this number line, mark the positions of A, B, C, D, and E in any order you like.
Then write a set of clues about the order you
picked.
Responses will vary.
Lesson 1: Placing Integers
T18
9
Unit 2: Geography of the Number Line
Additional Practice
A
Finish labeling this number line in a consistent way.
I
I
I
-6
-4
-2
I
0
I
I
I
I
2
4
6
8
I
10
I
I
I
I
I
12
14
16
18
20
B
4 , _____
0 , _____,
-4 _____,
-8 _____
-12
Find five more numbers that continue this pattern: 20, 16, 12, 8, _____
C
6 , 4, _____
2 , _____
0 , _____
-2 , -4 , _____
-6
Fill in numbers that continue this pattern: 12, 10 , 8, _____
Finish labeling the marks on these lines in a consistent way.
D
I
I
0
-6
F
H
I
-11
I
6
I
I
-9
-10
I
12
I
18
I
I
-8
-7
I
0
7
I
I
-120
-90
E
G
I
I
14
I
-60
I
I
21
28
I
I
0
-30
This number line has yet another scale. Finish labeling it to help you answer the problems below.
I
-45
I
I
I
I
I
I
I
I
-40
-35
-30
-25
-20
-15
-10
-5
I
0
I
I
10
5
I
I
I
15
20
25
I
-5 , -15, _____
-25 , _____
-35 , -45
Fill in numbers that continue this pattern in a consistent way: 15, 5, _____
J
15 , 0, _____
-15 , _____
-30 , -45
Fill in numbers that continue this pattern: 30, _____
Place each target number between its closest multiples of 10. Show how far away it is from each one.
K
M
8
Target: 68
2
6
Target: -24
L
I
I
I
I
I
60
68
70
-30
1
Target: 41
9
I
I
I
40 41
-20
7
3
I
I
50
I
-24
Target: -73
N
4
-80
-73
I
-70
Finish labeling these number lines in a consistent way.
O
I
-500
P
I
I
-450 -400
I
I
I
I
-350 -300 -250 -200
I
I
I
I
I
-44
-40
-36
-32
-28
10
Lesson 1: Placing Integers
I
-24
I
I
I
I
I
50
-150
-100
-50
0
I
I
I
I
I
-20
-16
-12
-8
-4
I
0
I
I
I
100
150
200
I
I
4
8
I
12
Unit 2: Geography of the Number Line
T19
Lesson 2: Operations with Integers
Purpose
Students examine addition and subtraction in a way that uses the number line and that also supports algebraic
thinking by placing the focus on the quantity of the sum or difference rather than on the calculation used to find
them. This allows students to focus on the logic of the operations and how the operations affect the values of
the results.
Lesson at a Glance
Preparation: Prepare two identical sets of sticky notes each with the numbers 38, 39, 80, 81,
13, 15, 76, 78, 129, 130, 246, and 247.
As a Class
Mental Mathematics: (5 min) See pages XX-XX of the TTA Mental Mathematics book.
Launch: (15 min) Balance
In Groups
Student Problem Solving and Discussion: (25 min)
• Students should work through the Important Stuff and explore additional problems.
• Use the discussion questions in the Student Book to give students the opportunity to share
their observations.
• Discuss how turning to the number line can be a strategy for making sense of addition and
subtraction problems with numbers or algebraic expressions.
As a Class
Unit 2 Related Game: Who Am I? Number Bingo (See Teacher Guide page T59 and Student Book page 35)
Launch – Balance:
Draw two empty mobile beams on the board. Ask for two volunteers (or two teams of two students each) to come
forward to compete in a quick game. You will hand each volunteer (or each team) four sticky
notes, and their job is to hang all four sticky notes on the mobile so that the mobile balances.
Give the first two volunteers (or teams) the numbers 38, 39, 80, and 81. You will likely
38
39
see teams show the correct grouping (38 with 81 and 39 with 80) or a common incorrect
81
80
grouping that groups 38 with 80 and 39 with 81. Ask each team to briefly explain how they
came up with their answer. Ask both teams, even if they are both correct or both incorrect.
If only one team got the correct answer, ask them to explain
first, in order to give the other team the opportunity to see
Teacher Tips
and correct their response.
Unfortunately, speed in mathematics is often
Listen for ideas like: “I put 80 and 81 on different strings
taken as a sign of ability. Speed is not the point
because if they were on the same string, that side would
of this game. The only reason this game is
be too heavy,” “39 goes with 80 because 39 is heavier than
designed as a competition is because, in this
38 and 80 is ligher than 81,” “Once I put the 80’s and 30’s
case, the speed factor will likely push students
together, I added the ones digits and saw that 1 + 8 is the
away from calculating a sum, and towards
same as 0 + 9.”
looking for interesting features of the numbers,
Call up a second set of volunteers (or teams) to do the
such as relative magnitude.
same with the numbers 13, 15, 76, and 78. Again, ask for
T20
Unit 2: Geography of the Number Line
brief explanations. Repeat with the numbers 129, 130, 246, and 247. If students are getting a lot out of the activity,
consider making more sticky notes of a similar nature and play more rounds at the end of class, after students have
worked on the problems in the Student Book.
Finally, draw a number line from 4 to 14 on the board. Draw two arrows coming up from 9 and leading to a box, and
ask students to add 9 + 9, and put 18 in the box.
18
+
4
5
6
7
8
9
10
11
12
13
14
Then draw two more arrows coming up from 8 and 10, respectively, leading to another box. Ask students for the
sum, and place it in the box.
18
18
4
5
6
7
8
+
9
10
11
12
13
14
If students seem to understand how the picture works, ask them to turn to Lesson 2-2 in their Student Books, where
they can complete problem 1.
Student Problem Solving and Discussion:
Allow time for students to work together in pairs or small groups on the Important Stuff. Use problems 14 and
15 in the Discuss & Write What You Think box to start discussion about ways to think flexibly about addition and
subtraction by using a visual representation of the sum or difference instead of performing a calculation.
Supporting Classroom Discussion
These questions ask students to think about how they might use visual models for addition and subtraction:
• Out of the pictures in problems 5, 6, and 7, which picture of addition do you like the most? Why?
• If you are asked to picture the sum 49 + 81, what picture comes to mind? How does having the visual
image help you find the sum? If a student’s answer for the picture that comes to mind is different from
the picture they like the most, this may be a sign that the student is using these visual tools strategically.
• If you want to draw a number line picture for the difference 91 – 59, what would you draw? How
would you use the number line to come up with a difference that is easier to find? Students might see
that 91 – 59 = 90 – 58 or 92 – 60, and students may express preference for one form over another.
Algebraic Habits of Mind: Using Tools Strategically
You don’t want students to think of using the number line as a “new” or “right” way of adding and subtracting.
But at the same time, it is helpful to help students develop the strategy of using a visual tool to aid them,
especially with mental calculations. Mentally calculating 39 + 41 requires more working memory than picturing
39 + 41 as a sum that “surrounds” 40 + 40, and having that visual image can increase accuracy.
Unfortunately, students—especially those who struggle—tend to latch onto particular methods as the “best
way” to solve a problem instead of thinking about what will work best in each situation. Communicate to
students that any tool should be used strategically, and a large part of thinking mathematically is to judge
whether a tool is useful for a given situation. Students who work to make sense of problems in this way will
find that they are more confident about solutions.
Lesson 2: Operations with Integers
T21
Connections to Algebra
In Lesson 6, students will use algebra to show that the addition pattern on the number line shown in problem 1
works the same way for any integer at the center of the arrows. In this lesson, students work first with numbers
in order to gain familiarity with the pattern and to begin to form ideas for how to justify the pattern before using
algebra.
Even though the materials don’t make the connection explicit, throughout this lesson students are using the
associative, commutative, and distributive properties. Here are some examples.
Find a.
59 + 23 = 60 + a
59 + 23 = (59 + 1) + a
59 + 23 = 59 + (1 + a)
59 + (1 + 22) = 59 + (1 + a)
So a = 22
Find b.
90 – 51 = 87 – b
90 – 51 = (90 – 3) – b
90 – 51 = 90 – 3 – b
90 – 51 = 90 – (3 + b)
90 – (3 + 48) = 90 – (3 + b)
So b = 48
Again, students don’t formally write down this process and would likely find it difficult to understand these
steps as they are written here. But the reasoning students use through their own intuition leads them through
each equivalence, and it is this algebraic thinking that is the goal of the lesson.
Notes:
T22
Unit 2: Geography of the Number Line
2-2 Operations with Integers
Important Stuff
1
18
In each box, put the
sum of the numbers
that point to it.
18
18
18
18
18
4
5
6
7
+
8
9
10
11
12
13
14
Try to figure out whether these mobiles are balanced without doing the addition on either side. If they are balanced, fill
in = in the expression below. If they are not balanced, circle the heavier side, and fill in the expression with < or > .
2
63
?
37
63 + 37
5
3
If you think
first, you
can do these
without
calculating.
65
37
<
65 + 37
n
28
30
Think first,
and you can
do this with
almost no
calculation
at all!
43 + 28 = n + 30
n=
7
74
51 + 74
Find the weight of n that makes
the mobile balance.
43
51
41
?
=
4
50
123
75
94
50 + 75
123 + 94
?
120
95
>
120 + 95
Discuss & Write What You Think
6
Luis drew this picture to make sense of problem 5.
Explain how he can
use it to find n.
43
n
28
30
Responses will vary. Sample: The total row
lengths are equal since 43 + 28 is equal to
n + 30. Since 30 is two larger than 28, the
picture shows that n must be two smaller
than 43. So n is 41.
Ayana drew this picture to make sense of problem 5. Explain how she can use it to find n.
this
equals
that
28
30
Lesson 2: Operations with Integers
Lesson 2: Operations with Integers
+
n
43
Responses will vary. Sample:
The number line picture shows
that since 30 is 2 more than
28, the number n must be 2
less than 43 in order for the
sums to be equal. So n is 41.
11
T23
The green line shows another number line relationship: it connects numbers whose difference is 4.
Here, the green line connects 1 and 5 because 5 – 1 = 4.
There are more possible
Draw three more lines that connect numbers that differ by 4.
8
lines than are drawn here.
5 – 1 = 4
I
-3
I
-2
I
-1
I
0
I
1
I
2
I
3
I
4
I
5
I
6
I
-90
I
-80
I
-70
I
-60
I
-50
I
-40
I
-30
I
-20
I
-10
I
0
Figure out the value of the variable using the number line, not by subtracting.
10
52 – 19 = a – 20
a=
11
I
I
52
I
I
b 37
34
I
9
I
10
I
11
53
I
80
I
83
I
10
I
20
I
30
I
40
I
50
Algebraic Habits of Mind:
Using Tools Strategically
Using the number line is not a
“new way” to add and subtract
numbers. It is just another
way to think about sums and
differences and make sense
of them. One way to use the
number line strategically is to
487 – 290 = c – 300
c=
I
20
Number lines aren’t always precise about exact
spacing, but the order always has to be correct.
83 – 37 = 80 – b
b=
12
53
I
19
I
8
There are more possible
lines than are drawn here.
50 – 0 = 50
Here’s a line between numbers whose difference is 50.
Draw three more lines that connect numbers that differ by 50.
9
I
7
I
I
290 300
497
I
487
I
look for a relationship between
c
the order of numbers and the
result you get when you add or
13
8000 – d = 7998 – 179
d=
181
I
179
I
d
I
I
subtract them.
7998 8000
Discuss & Write What You Think
14
If we’re told that 77 + m = 80 + 43,
explain how you can find m without
adding 80 + 43 first.
Responses will vary. Sample:
The two sides balance. 80 is
3 heavier than 77, so, to
make up for it, m must be 3
heavier than 43. So m must
be 46.
12
T24
15
If we know that 94 – 75 = 92 – x, explain
how to find x without ever figuring out
what 94 – 75 is.
Responses will vary. Sample: The
distance between 94 and 75 is
the same as the distance between
92 and x. Since 92 is 2 less
than 94, x must be 2 less than
75, so x is 73.
Unit 2: Geography of the Number Line
Unit 2: Geography of the Number Line
Stuff to Make You Think
16
Clue: A + B = C + D. Mark the location of D.
I
A
18
19
I
C
I
D
17
Clue: A – B = C – D. Mark the location of D.
I
B
I
B
I
I
A
D
Find the missing values by considering how the numbers on each side of the equation
balance each other, not by adding the numbers in the problem.
72 + a = 73 + 49
50
a = __________
84 + 66 = b + 85
65
b = __________
257 + 4528 = 255 + c
c = __________
I
C
Consider drawing
a picture to go
with the problem.
4530
Make up two similar addition problems and give them to a classmate to solve.
Responses will vary.
20
21
Find the missing values without subtracting the numbers in the problem.
164 – 72 = d – 73
165
d = __________
87 – e = 90 – 59
62
e = __________
9249 – 381 = f – 380
9250
f = __________
This time, a drawing mi
ght
look like this:
Make up two similar subtraction problems and give them to a classmate to solve.
Responses will vary.
Lesson 2: Operations with Integers
Lesson 2: Operations with Integers
13
T25
22
23
42
3
= ______
7
= ______
60
9
= ______
5
= ______
2
= ______
3
= ______
Tough Stuff
24
There’s a way you can find the sum of all the integers from 1 to 10 without doing all the adding!
The two pictures suggest slightly different
ways to figure out the sum without adding all
the numbers.
Find the sum using whichever picture you find
more convenient and explain how the picture
shows a shorter way of adding
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10.
10
10
10
10
+
1
2
3
4
5
6
7
8
9
10
11
11
11
11
11
1
25
2
3
4
5
+
6
7
8
9
10
Responses will vary. Sample:
Both pictures show that the
sum of the integers from 1 to
10 is 55.
Picture 1: 4(10) + 10 + 5 = 55
Picture 2: 5(11) = 55
Both pictures take advantage
of the fact that you can add
the ten numbers in any order,
so we can choose pairs of
numbers that give the same
sum to reduce the amount of
information to keep track of.
Imagine or draw a similar picture to find the sum of all the integers from 1 to 20.
Adjust the pictures to show 1 to 20:
Picture 1: 9(20) + 20 + 10 = 210
Picture 2: 10(21) = 210
Both show that the sum of the
integers from 1 to 20 is 210.
14
T26
Unit 2: Geography of the Number Line
Unit 2: Geography of the Number Line
Additional Practice
Try to figure out whether these mobiles are balanced without doing the addition on either side. If they are balanced,
write = in the box. If they are not balanced, circle the heavier side and fill in the box with < or >.
A
71
?
99
71 + 99
B
=
70
213
100
36
70 + 100
213 + 36
?
<
C
38
58
215
93
38 + 215
?
60
90
58 + 93
>
60 + 90
Fill in the blank spaces in each picture so that each one shows a way to picture 63 + 89 = n + 90.
D
G
E
63
n
89
90
63
89
n
90
F
this
equals
that
63
n
62
All three pictures show that n = _______
+
89
90
Find the weight of the variable that will make mobile balance.
H
I
J
79
m
812
p
1000
b
41
40
97
100
77
998
79 + 41 = m + 40
812 + 97 = p + 100
1000 + 77 = b + 998
80
m = _______
809
p = _______
79
b = _______
Find the value of each variable by drawing on the number line, not by subtracting.
K
63 – 29 = c – 30
I
29
64
c = _______
L
200 – n = 199 – 85
2000 – w = 2003 – 349
346
w = _______
Additional Practice
Lesson 2: Operations with Integers
I
63
I
I
85 n
86
n = _______
M
I
30
I
w
I
349
I
c
I
I
199 200
I
I
2000 2003
15
T27
Lesson 3: Checkers and Who Am I? Puzzles
Purpose
By using black and red checkers to represent positive and negative integers, students make sense of adding and
subtracting positive and negative numbers. Students are also introduced to Who Am I? puzzles in which they
interpret and use multiple clues to determine a mystery number.
Lesson at a Glance
Preparation:
• Prepare red and black checkers. Have at least 5-10 of each color to use at the board (either physical or
virtual pieces that the whole class can see). You may also find it helpful to have extra checkers for students
to use at their desks as they solve problems.
• Prepare to project the three-digit Who Am I? puzzle on page T31.
• Photocopy the Snapshot Check-in on page T72.
Mental Mathematics: (5 min) See pages XX-XX of the TTA Mental Mathematics book.
As a Class
Launch: (15 min) Who Am I? Puzzle and Checkers Introduction
Student Problem Solving: (15 min) Students should work through the Important Stuff and
explore additional problems.
Reflection and Assessment: (10 min) Snapshot Check-in
In Groups
As a Class
Unit 2 Related Game: Who Am I? Number Bingo (See Teacher Guide page T59 and Student Book page 35)
Launch – Who Am I? Puzzle and Checkers Introduction:
Tell students they are about to solve a puzzle together in which they figure out a three-digit mystery number. Project
the puzzle on page T31 or write these clues so all students
can see them.
Who Am I?
h
t
u
• I am even.
• My digits are all different.
• I am greater than 319.
• My hundreds digit is less than 7.
• u = 1 + h
• My tens digit is my largest digit.
• My hundreds digit is my only odd digit.
• My units digit is one more than my hundreds digit.
• The sum of all three of my digits is 19.
• My units digit is not 4.
T28
(The answer is 586)
Algebraic Habits of Mind: Puzzling and
Persevering
Persevering means hanging in there even when the
going is tough. But problem solving isn’t just about
tackling the tough stuff. Good problem solvers
often start by looking for the easiest clues or ideas
to use. Then when they’ve used up the easy clues,
they see what else they can do.
Teacher Tips
Explain that just like letters are used to spell
words, digits are used to “spell” numbers. We’re
just looking for the individual digits that spell the
number.
Unit 2: Geography of the Number Line
Let the spirit be playful: they’re detectives solving
a mystery. The clues are intentionally not labeled
because they should not be considered in order. If
students suggest it, label the clues (a-j) so they are
easier to refer to. Let students examine the clues
quietly on their own for 30 seconds, then let them
start to offer ideas about how to solve the puzzle. As
the class works together, set a rule: You may only
share one new idea on your turn. For example, if
a clue says “My units digit is a perfect square,” a
student can say, “That tells me the units digit is a 0,
1, 4, or 9.” They cannot go on to say “... and also,
since another clue says ‘I am odd,’ the units digit
must be 1 or 9.” That comment must be left for
another student! This can be difficult, but the goal is
to get all your students involved and contributing.
Algebraic Habits of Mind: Puzzling and
Persevering
Who Am I? puzzles are great for encouraging students
to find entry points into problems and for helping them
organize information. They encourage students to move
away from asking “What am I supposed to do here?”
towards a more sophisticated question: “What can I do?”
When helping students, don’t lead them down a certain
path. Just keep asking “What do you know?” Then help
them keep track of what they’ve figured out. If you have
helpers in your classroom or you hear students helping
each other, encourage them to take the same approach.
This type of social problem solving also forces students to keep track of what they know. If they’re going to build on
what they already know, they have to keep track of what’s been said. When students have ideas, ask how to record
those ideas. If students are really having trouble, you may suggest writing the digits 0-9 underneath each box and
crossing out digits as they are eliminated. But go with student suggestions first since this isn’t always the best way to
keep track of the information and so that students can be thinking about how best to organize their thoughts.
After solving the Who Am I? puzzle but before students begin solving problems in the Student Book, use checkers to
familiarize them with the “game” context of the problems. Black checkers are worth +1 and red checkers are worth
-1. In each case, set up the situation or perform the action, and ask the students to find the total score.
• Start with 4 black checkers. (Score: 4)
-1
1
• Put in 2 red checkers (Score: 2)
• Put in 3 more red checkers (Score: -1)
• Remove 1 black checker (Score: -2)
Continue to put in and remove checkers until students seem to understand the idea of finding the score.
Student Problem Solving:
Give students time to work together in pairs or small groups on the Important Stuff. Use problems 23 and 24 to
check that students have made sense of how to use the checkers to talk about adding and subtracting positive and
negative numbers.
Supporting Classroom Discussion
As students are solving problems, check for student understanding with these questions:
• How does putting in (adding) a negative number affect the score?
• How does removing (subtracting) a positive number affect the score?
• How does removing (subtracting) a negative number affect the score?
• Why is removing a negative number like putting in a positive number?
• If a score is already negative, what happens if you put in (add) a negative value?
• If a score is already negative, what happens if you remove (subtract) a negative value?
Lesson 3: Checkers and Who Am I? Puzzles
T29
Connections to Algebra
m – 5
m – -5
In Lesson 5, students will revisit the idea of adding and
subtracting positive and negative values but as represented
I
I
I
m + -5
m
m + 5 on a number line centered on a variable. When they
encounter this general picture of adding or subtracting
values to end up to the right or left of the original value, remind them of their work with putting in and
removing black and red checkers so that students continue to work with the image of checkers and use the
language and the context to help them make sense of calculations.
Who Am I? puzzles are also connected to algebra. Each set of clues is a system of equations and inequalities
that has a unique solution, and students use deductive reasoning skills to find the solution. Students will
encounter clues like u = t (from problem 30) and have to understand that the clue shows a relationship between
two unknown digits. The ability to coordinate multiple clues and think flexibly about how to use the clues is a
way to train a student to think algebraically.
Student Reflections and Snapshot Check-in:
Ask students to reflect on their learning by responding to the following prompts:
• What are some things you’ve learned so far in this unit?
• What questions do you still have?
Assess student understanding of the ideas presented so far in the unit with the Snapshot Check-in (on page T72). Use
student performance on this assessment to guide students to solve targeted additional practice problems from this
or prior lessons as necessary.
Notes:
T30
Unit 2: Geography of the Number Line
Who Am I? Puzzle
Who Am I?
h
t
u
•I am even.
•My digits are all different.
•I am greater than 319.
•My hundreds digit is less than 7.
•u = 1 + h
•My tens digit is my largest digit.
•My hundreds digit is my only odd digit.
•My units digit is one more than my
hundreds digit.
•The sum of all three of my digits is 19.
•My units digit is not 4.
Lesson 3: Checkers and Who Am I? Puzzles
T31
2-3 Checkers and Who Am I? Puzzles
Important Stuff
1
For the following problems, a black checker is worth 1 point and a red checker is worth -1 point.
The total score is found by adding the value of all the checkers.
For example, in
-1
, the total score is -3.
What score is represented in each case?
1
2
-4
5
4
3
7
If the score is negative, are there
more black or red checkers in
the pile?
6
Red
1
If the score is positive, which
color is ahead (has more checkers
in the pile), black or red?
-6
If the number of black and red
checkers in the pile is the same,
what is the score?
7
Black
0
Putting checkers in the pile has the effect of adding a positive (black) or negative (red) number to the overall score.
Write a calculation for each scenario.
8
Black is ahead by 5. Find the new
score if you put in 3 red checkers.
9
2
5 + (-3) = _____
11
Red is ahead by 6.
Put in 3 red checkers.
12
Red is ahead by 1. Put in 3 red checkers.
What’s the score?
Red’s lead goes up, so
red is ahead by 4. The
score is -4.
16
10
Red is ahead by 2.
Put in 4 black checkers.
-2 + 4 = 2
15
Black is ahead by 1.
Put in 4 red checkers.
1 + (-4) = -3
-2
-5 + 3 = _____
-6 + (-6) = -12
14
Red is ahead by 5.
Put in 3 black checkers.
13
Red is ahead by 3.
Put in 4 red checkers.
-3 + (-4) = -7
Red is ahead by 1. Take out 3 black checkers.
What’s the score?
Red is already ahead, and black
takes away checkers, so red’s
lead goes up further. Red is now
ahead by 4. The score is -4.
In problems 14 and 15, putting in 3 red checkers has the same effect as removing 3 black checkers. Why do these
actions have the same effect on which color is ahead and by how many?
Responses will vary. For example: Red is ahead by 1, and adds three red
checkers, so red’s lead goes up. Removing three black checkers also means
that red’s lead goes up, so it has the exact same effect. In both cases, the
amount of change is 3.
16
T32
Unit 2: Geography of the Number Line
Unit 2: Geography of the Number Line
Removing checkers from the pile has the effect of subtracting a positive (black) or negative (red) number from the
overall score. Write a calculation for each scenario.
17
Black is ahead by 5.
Remove 2 black checkers.
Black is ahead by 5.
Remove 2 red checkers.
18
3
5 – 2 = _____
20
Red is ahead by 3
Remove 6 black checkers.
7
5 – (-2) = _____
Red is ahead by 7.
Remove 3 red checkers.
21
-3 – 6 = -9
23
25
27
29
31
-7 – (-3) = -4
Use checkers to make sense of the
calculation -15 + -4.
Responses will vary.
Sample:
In this situation, red
is ahead by 15, then
we put in 4 more red
checkers. The final
score is -19.
24
u
8
8
Who Am I?
• I am odd.
• The sum of my digits is 2.
t
u
1
1
Who Am I?
• I am even.
• The sum of my digits is 4.
• My units digits is not 2.
t
u
Who Am I?
• Both of my digits are odd.
• My tens digit is 6 less than
my units digit.
• I am less than 20.
Lesson 3: Checkers and Who Am I? Puzzles
Lesson 3: Checkers and Who Am I? Puzzles
26
0
2
4
6
8
28
30
4 0
t
u
1
7
Red is ahead by 6.
Remove 1 black checker.
-6 – 1 = -7
22
Black is ahead by 1.
Remove 5 black checkers.
1 – 5 = -4
Use checkers to make sense of the
calculation 5 – (-7).
Responses will vary. Sample:
In this situation, black is ahead
by 5, then we remove 7 red
checkers. The final score is 12.
If it helps,
draw checkers
and describe
your picture
to explain the
calculation.
t
Who Am I?
• I am even.
• The sum of my digits is 16.
19
32
t
u
Who Am I?
• I am even.
3 6
• I am a perfect square.
• My units digit is twice my tens digit.
Who Am I?
• I am a multiple of 10.
• I am between 42 and 52.
Who Am I?
• I am odd.
• u=t
• The sum of my digits is 6.
t
u
2 0
t
u
3
3
t
u
Who Am I?
• I am a perfect square.
8 1
• The sum of my digits is 9.
• My units digit is less than my tens digit.
17
T33
Stuff to Make You Think
Use
+ for
and
– for
Imagine a bucket contains a pile of checkers. The picture of the bucket represents the total value of
the checkers inside. If there are more black checkers, the number is positive; if there are more red
checkers, the number is negative. Either way, the bucket stands for the unknown number.
33
Words
Checker Description
Think of a number.
Bucket
Add -5.
Add 5 red
Subtract -3.
Remove 3 red
Subtract your
original number.
Remove bucket
34
Words
Checker Description
Think of a number.
Bucket
Subtract 4.
Pictures
Jay
Eva
Abbreviation
7
11
b
– –
– – –
2
6
b - 5
–
–
5
9
b - 2
-2
-2
-2
Ben
Carla
Abbreviation
20
7
b
16
3
b - 4
–
–
Pictures
– –
– –
Add 4 red
know.
You can’t reach into the bucket because you can’t change what you don’t
Adding 4 red has the same effect as removing 4 black.
35
Words
Checker Description
Think of a number.
Bucket
Pictures
Lena
Mali
Abbreviation
10
21
b
Subtract 2.
Add 2 red
–
–
8
19
b - 2
Add -3.
Add 3 red
– –
– – –
5
16
b - 5
–
9
20
b - 1
+ + +
– + +
14
25
b + 4
4
4
4
Subtract -4.
Add 5.
Subtract your
original number.
Remove 4 red
Add 5 black
Remove bucket
+ +
+ +
Tough Stuff
36
18
T34
t
Who Am I?
• I am even.
3
• My units digit is 3t – 1.
• I am closer to 30 than I am to 0.
u
8
37
Who Am I?
• 2t is 5 more than u.
• u2 is one less than 50.
• 2u is eight more than t.
t
u
6
7
Unit 2: Geography of the Number Line
Unit 2: Geography of the Number Line
Additional Practice
A black checker is worth 1 and a red checker is worth -1. The total score is found by adding the value of
all the checkers.
1
-1
What score is represented in each case?
A
B
C
-1
D
3
-2
4
Putting checkers into the pile has the effect of adding a positive (black) or negative (red) number to the overall score.
Write a calculation for each scenario.
E
Black is ahead by 4.
Put in 6 red checkers.
F
-2
4 + (-6) = _____
Red is ahead by 3.
Put in 8 black checkers.
G
-3 + 8 = 5
Red is ahead by 10.
Put in 2 black checkers.
-10 + 2 = -8
Removing checkers from the pile has the effect of subtracting a positive (black) or negative (red) number from the
overall score. Write a calculation for each scenario.
H
Black is ahead by 6.
Remove 2 red checkers.
I
8
6 – (-2) = _____
K
Red is ahead by 8.
Remove 4 black checkers.
-8 – 4 = -12
Use checkers to explain how to make sense of the
calculation -9 + -5.
L
In this situation, red is ahead
by 9, then we put in 5 more red
checkers. The final score is -14.
M
O
Q
t
u
Who Am I?
• Both of my digits are odd.
3 5
• Neither of my digits are
perfect squares.
• I am one less than a perfect square.
t
u
Who Am I?
• The sum of my digits is 9.
4 5
• My tens digit is even.
• My tens digit and units digit differ by 1.
t
Who Am I?
• Both of my digits are even.
8
• My digits add to 10.
• My tens digit is 6 more than u.
Additional Practice
Lesson 3: Checkers and Who Am I? Puzzles
u
2
J
Red is ahead by 4.
Remove 5 red checkers.
-4 – (-5) = 1
Use checkers to explain how to make sense of the
calculation -5 – (-3).
In this situation, red is ahead
by 5, then we remove 3 red
checkers. The final score is -2.
N
P
R
Who Am I?
• The sum of my digits is 14.
• My digits are not the same.
• Both my digits are odd.
• I am less than 90.
t
u
5 9
t
u
9
9
t
Who Am I?
• My digits are the same.
7
• The sum of my digits is
greater than 12 and less than 16.
u
Who Am I?
• The sum of my digits is 18.
7
19
T35
Lesson 4: Distance and Inequalities
Purpose
In Lesson 4, students return their focus to the geography of the number line. They use the number line to find
and compare distances and learn how to represent an inequality statement on a number line. In both types of
problems, students must consider the order and placement of positive and negative numbers as they continue to
solidify their mental framework for distance between numbers.
Lesson at a Glance
Mental Mathematics: (5 min) See pages XX-XX of the TTA Mental Mathematics book.
Launch: (15 min) Thinking Out Loud Dialogue and Showing Inequalities
• Read and work through the Thinking Out Loud dialogue together as a class.
• Ask students to think of a number less than 10, and find a way to represent all the possible
numbers less than 10 on a number line.
As a Class
In Groups
Student Problem Solving and Discussion: (25 min)
• Allow time for students to work on the problems in the Student Book.
• Discuss a way to represent given sets of numbers on the number line.
As a Class
Unit 2 Related Game: Who Am I? Number Bingo (See Teacher Guide page T59 and Student Book page 35)
Launch – Thinking Out Loud Dialogue and Showing Inequalities:
Pose problem 1 to the class: Which is longer, the distance from -26 to 59 or the distance from -27 to 62? Ask
three students to read through the dialogue at the board. Make sure they draw a number line and mark on it when
prompted. Call on other students to share their reseponses to the problems in the Pausing to Think boxes.
Check for understanding by asking students to compare the distance between -13 and 92 to the distance between 13
and 92. Ask students to explain their response, listening for explanations that use visual comparison like “-13 must
be farther from 92 because -13 is on the other side of 0.”
Then draw a number line on the board without any markings. Call up a stream of students to come up and each mark
a number that is less than 10. For now, don’t specify how students should mark the numbers. They may just make
hash marks or draw dots on numbers. You may end up with a picture that looks something like this:
I
-3
I
1
I
2
I
4
I
5
I
6
I
7
I
9
After a while, ask, “Just from looking at this picture, could you tell what rule we’re using to make the picture?”
Listen for answers that provide reasons for why the picture is insufficient: “No, we haven’t marked all the numbers
less than 10,” “No, these numbers are also all less than 20,” “You can’t even tell which numbers are important,” etc.
Keep asking for more students to come up and continue to mark numbers that are less than 10. Depending on the
points raised in the previous conversation, you might find that more students are using dots to mark their points. If
students are only filling in integers, keep asking for students to fill in more and more numbers (without extending the
number line) until students start to mark non-integer numbers. Now, the number line might look more like this:
T36
Unit 2: Geography of the Number Line
I
-3
I
-2
I
-1
I
0
I
1
I
2
2.5
I
3
I
4
I
5
I
6
I
7
I
8
8.5
I
9
9.9
Again, ask, “Is this enough information to show anyone that we were only marking numbers less than 10?” Again,
listen for responses like “These numbers are also all less than 11 or 20 or 400.”
Depending on the conversation, you might respond, “To show that we don’t want numbers that are more than 10,
we could use a different mark. Actually, mathematicians have a way to show that we don’t want numbers: an open
circle.” You might show some examples. Ask, “What about the number 10?” Mark 10 with an open circle.
2.5
8.5 9.9
12.2
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
-3 -2 -1
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16
Ask again, “Now does this picture show clearly that we only want numbers less than 10?” Probe with questions like
“What abot 9.95? Does this picture tell us about 9.95?” Or “What about 6.5, or any other number that isn’t marked?
Can we assume that those numbers should have been marked, even though they aren’t?”
Finally, show students how to indicate inequalities on the number line, while describing the logic of the notation.
Emphasize to students that this notation isn’t just about “coloring” or “shading” one side versus the other. The dark
line we use to indicate inequalities is actually a way of showing an infinite number of points. This logic will return
when students produce graphs on a coordinate grid. Graphs may also represent an infinite number of points. The
only specific points that we have to distinguish are at the ends of the line we draw. Off to the left, we use an arrow to
show that any point to the left of our drawing is also a solution. At 10, we use an open circle to indicate that, while
10 is not less than 10, 10 is our boundary.
I
-3
I
-2
I
-1
I
0
I
1
I
2
I
3
I
4
I
5
I
6
I
7
I
8
I
9
I
10
I
11
I
12
I
13
I
14
I
15
I
16
Ask, one final time, “Now does the picture show clearly that we only want numbers less than 10?” Ask, “Could it
possibly mean anything else?” Review with students that this means if p is less than 10, that means p can be located
anywhere on the dark line. The point p is a specific point, and the line indicates all the possible locations of p.
Therefore, when we use the notation p < 10, we associate it with this picture on the number line. Let students voice
concerns or issues and be convinced of how to interpret this notation.
Supporting Classroom Discussion
Use these questions to check student understanding of the logic of how to indicate solutions on the number line.
• How could we show all the numbers that are not 5?
I
I
I
I
I
5
• How could we show all the positive numbers? Ask: Is 0 positive?
I
I
I
0
I
I
• How could we show all the numbers that are not positive?
I
I
I
0
I
I
I
I
0
1
• How could we show all the numbers that are less than 5 units away from 0?
I
2
I
-5
I
-5
I
3
I
0
I
0
I
4
I
5
I
5
• How could we show all the numbers that are 2-point-something?
• How could we show all the numbers that are more than 5 units away from 0?
Lesson 4: Distance and Inequalities
T37
Connections to Algebra
When considering specific numbers, inequalities may not seem to have much significance. 8 < 10 is a true
statement, but the specific relationship doesn’t seem very useful, since there are other ways of relating 8 and 10
that are much more informative.
Students might gain some new insights into the nature of a variable by thinking about and discussing the
meaning of a statement like b > -2. In this case, it’s possible that b stands for a particular number and we have
a clue that b > -2, which limits the possibilities for the identity of that particular number. Or it could be that b is
not a specific number but describes a quantity that may vary. In that case, b > -2 may be a description for how
b may vary. Students don’t need to be able to examine the nature of a variable in such depth, but having a clear
way to describe variables to students will likely help students understand more about algebra.
Teacher Tips
Whenever students ask for help in drawing inequalities on a number line, go through the process of the launch.
If students are graphing n ≥ 5, ask them to name and plot numbers that are greater than or equal to 5 with closed
circles and numbers that are less than 5 with open circles. Have them plot more and more and more points until
it is possible to see the line that would be drawn to represent n ≥ 5. Repeating this process will help students
understand that lines on a graph indicate the location of infinitely many points.
Some students may have learned rules like “‘less than’ means shade to the left.” Unfortunately, this procedurebased rule not only obscures understanding of why a person would shade anything at all, but it’s also wrong
when representing an inequality like 5 ≤ n. Continue to emphasize logic and sense-making to combat incorrect
rules and shortcuts that students may already have.
Notes:
T38
Unit 2: Geography of the Number Line
2-4 Distance and Inequalities
Important Stuff
1
(See dialogue for explanation)
Which is longer, the distance from -26 to 59 or the distance from -27 to 62? Explain how you know.
-27 and
62
Thinking Out Loud
Michael , Lena, and Jay are working on problem 1.
Lena:
Huh... are we supposed to subtract? Like 59 minus -26 is... um...
Michael: Wait. If we’re comparing distances, I want to picture it. Let’s draw a number line. Here, I’ll start it:
I
I
-27 -26
Lena:
I
I
I
0
59
62
The way you drew it, the distance between 59 and 62 looks like about 20!
Michael: Yeah, fine, but the only thing that matters is the order.
Pausing to Think
Place the numbers -26 and -27 in the blank spaces.
Jay:
So, we’re comparing from -26 to 59 (points to both numbers) and from -27 to 62 (points to both).
Lena:
-27 and 62
Ok, so the longer distance is between ___________________
Michael: That’s the answer... But how do we explain it?
Lena:
What is there to explain?! We can see which one is longer!
Jay:
-27 and 62 because of
Yeah, so what did we see? The distance has to be longer between _______________
the order of the numbers on the number line. We can see that...
Pausing
to Think
2
Which way should Jay end his sentence? Choose one.
A
... the distance from -26 to 59 fits inside the distance from -27 to 62.
B
... the distance from -27 to 62 fits inside the distance from -26 to 59.
Algebraic Habits of Mind: Using Tools
Strategically
You overhear two classmates talking.
Jing:
Eva:
a
What’s the distance between -20 and 70?
I think it’s 50...
No, it can’t be. I’m picturing the number
line, and the distance has to be bigger.
Show -20 and 70 on the number line.
I
-20
b
(write your answer again here)
I
0
What’s the distance between them?
20
Lesson 4: Distance and Inequalities
The number line is a tool to show the positions,
order, and space between real numbers. Use number
lines to make sense of problems that ask about
comparing numbers (Greater? Less? In between?)
I
70
90
and how far apart they are (How much bigger? What
do we have to add? What’s the distance? How did
it change?). Drawing a number line may help you
organize a problem and see how you might solve it.
Unit 2: Geography of the Number Line
T39
Show the numbers on the number line, and find the distance between them.
3
25 and 100
I
0
5
-25 and -100
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-100
75
I
75
25
4
-25 and 100
I
I
I
-25 0
100
6
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25 and -100
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0
-100
m
ea
“Think of a number.”
7
The number you think of may be at
any one of these specific points.
A thick line with arrows shows that any
point within that section is acceptable.
8
“Think of a number greater than 3.”
I
I
3
An open circle ( ) means the number is
not allowed.
Is the number 3 allowed?
No
“Think of a number greater than or
equal to 3.”
I
3
This number line shows n ≥ 3. The
number 3 is still the boundary point, but
now it is allowed.
Lesson 4: Distance and Inequalities
T40
9
nt
Hi
25
I?
Clue: I am less than 300.
My name is p.
Where am I? Use symbols:
Show everywhere I could be.
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I
0
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I
I
I
p < 300
I
I
I
100 200 300
Clue: I am greater than -3.
My name is y.
Where am I? Use symbols:
y > -3
Show everywhere I could be.
-3
-2
I
-1
I
0
Clue: I am greater than 28.
My name is c.
Where am I? Use symbols:
Show everywhere I could be.
I
10
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Wh
er
Inequality symbols compare numbers. Inequality sentences can
be shown visually as sections of the number line.
I
100
125
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I
-25 0
I
3
A thick line (and the arrow) shows which
numbers are allowed. If the number is
“n,” then this number line shows n > 3.
The number 3 is not “greater than 3” so
it’s not allowed.
125
I
I
28
I
30
I
1
c > 28
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32
I
34
Clue: I am less than or equal to -20.
My name is n.
Where am I? Use symbols:
n ≤ -20
Show everywhere I could be.
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11
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I
I
I
I
-40 -30 -20 -10
0
Clue: I am greater than or equal to -12.
My name is k.
Where am I? Use symbols:
k ≥ -12
Show everywhere I could be.
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-12 -9
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-6
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-3
I
0
21
Unit 2: Geography of the Number Line
Stuff to Make You Think
Clue 1: I am less than 4.
Clue 2: I am greater than or equal to -1.
a
b
c
d
I?
Wh
er
12
m
ea
Place 10 filled circles ( ) on points that are true for both clues.
Place 10 open circles ( ) on points that are false for at least one of the clues.
Is the number 4 allowed? Put the correct circle ( or ) on 4.
Is the number -1 allowed? Put or on -1.
I
-3
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-2
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-1
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0
I
1
The exact placement of circles will vary.
I
2
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3
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4
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5
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6
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7
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8
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9
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10
For problem 12, if you cover all the filled circles ( ) with a dark line and get rid of all open circles ( ) except for
the boundary point, you should get a number line like the one below. We keep the filled and open circles on -1
and 4 because they are boundary points. They tell you if you should count the number on the boundary or not.
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-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
If the number line
shows where b is,
write b in symbols:
13
Clue 1: I am less than or equal to 0.
Clue 2: I am greater than -3.
My name is k.
Where am I? Use symbols:
-3 < k ≤
Show everywhere I could be.
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15
I
-3
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I
-2
“b is between -1 and 4. It’s greater than or equal to -1
and less than 4.”
-1 ≤ b < 4
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0
-1
14
0
I
1
Clue 1: I am greater than or equal to -18.
Clue 2: I am less than or equal to -9.
My name is x.
Where am I? Use symbols:
-18 ≤ x ≤ -9
Show everywhere I could be.
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I
I
-18 -15
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-12
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-9
Clue 1: I am greater than -20.
Clue 2: I am less than or equal to 10.
My name is c.
Where am I? Use symbols:
-20 < c ≤
Show everywhere I could be.
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I
-40 -30 -20 -10
16
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0
10
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10
I am somewhere in here. Write my two clues.
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4
-6 -4 -2 0
2
I am greater or equal to -6
Clue 1: _________________________________
Clue 2: _________________________________
I am less than 2
My name is n.
Where am I? Use symbols:
-6 ≤ n < 2
I
-6
Tough Stuff
Match each clue with the correct number line.
17
Clue: I am positive.
A
18
Clue: I am not positive.
D
19
Clue: I am negative.
B
20
Clue: I am not negative.
C
22
Lesson 4: Distance and Inequalities
A
I
0
B
I
0
C
I
0
D
I
0
Unit 2: Geography of the Number Line
T41
Additional Practice
Show the numbers on the number line and find the distance between them.
A
35 and 100
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I
0
C
25 and -300
E
402 and 100
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0
G
35
100
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I
-10 0
-70
-74 and 10
84
I
F
302
I
100
-55 and -40
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15
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402
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I
0
-55 -40
H
-88 to 70
Which is longer: the distance from -142 to -22 or
from -142 to 22?
-142 to 22
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I
-88 -85
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0
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I
68 70
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-142
m
ea
I
Hiroshi: “I remember that these mea
n ‘or equal to’
because that line underneath looks like
part of
an equal sign. So, x ≥ -4 means x cou
ld equal
-4 or any larger number, like -3 or 1
or 9...”
Clue: I am greater than 120.
My name is w.
Where am I? Use symbols:
Show everywhere I could be.
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L
I
100
w > 120
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I
110 120
I
Clue: I am greater than or equal to 16.
My name is m.
Where am I? Use symbols:
m ≥ 16
Show everywhere I could be.
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-4
Additional Practice
T42
I
90
K
I
0
I
4
I
8
I
12
I
16
I
54
I
58
y < 58
I
I
Clue: I am less than or equal to -20.
My name is a.
Where am I? Use symbols:
a ≤ -20
Show everywhere I could be.
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M
I
50
22
I?
Clue: I am less than 58.
My name is y.
Where am I? Use symbols:
Show everywhere I could be.
I
46
I
Wh
er
g
nd being LOUD > soft or a river bein
Ayana: “I see these and think of sou
s BIG > small. So, 5 > 3.”
WIDE > narrow. The symbol itself goe
≥ and ≤
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-22 0
> and <
J
I I
0 10
-74
Which is longer: the distance from -85 to 68 or
from -88 to 70?
I
60
I
I I
0 25
-300
-10 and -70
I
D
325
I
B
65
I
I
-20 -15
I
-10
I
-5
Clue: I am greater than -21.
My name is u.
Where am I? Use symbols:
Show everywhere I could be.
I
I
-21
I
-14
I
-7
I
0
u > -21
I
0
I
23
Unit 2: Geography of the Number Line
Lesson 5: Geography of Addition and Subtraction
Purpose
Students use what they have learned about position and order on the number line to solve Where Am I? puzzles
with distance. Then, students use that visual image to make sense of addition and subtraction and see that there
is a logic to the structure of addition and subtraction on the number line. The way that addition and subtraction
affects position on the number line is not arbitrary, but follows a predictable pattern, which students will learn
to use in their calculations.
Lesson at a Glance
Preparation: The launch suggests the use of individual whiteboards, though the same work can
be done on paper.
Mental Mathematics: (5 min) See pages XX-XX of the TTA Mental Mathematics book.
As a Class
Launch: (10 min) Finding Distances
Student Problem Solving and Thinking Out Loud Dialogue: (30 min)
• Students work through the first page on finding distances.
• When students have generally started to reach problem 12, bring the class together to work
through the Thinking Out Loud dialogue.
• Use the language in the Thinking Out Loud dialogue to go through problems 13-16 and
discuss.
In Groups
As a Class
Unit 2 Related Game: Who Am I? Number Bingo (See Teacher Guide page T59 and Student Book page 35)
Launch – Finding Distances:
Ask students to draw a number line that helps them find the
distance between 78 and 303. Each student should sketch their
own number line on an individual whiteboard (or piece of
paper). Acknowledge that subtraction is also a way to find the
distance, but in this case, subtraction is a bit onerous to do on
paper and quite difficult to do mentally. This may help convince
students that it is useful to explore an alternate strategy for
finding distance.
Students will likely draw one of two diagrams, both of which
can be useful. A student may justify drawing a number line that
spans only from 78 to 303 as its endpoints because we only
need to consider the space between the two numbers. Another
student might mark “0” on their number line and mark 78 and
303 relative to 0. This visual model makes sense especially
for a student who strongly identifies distance with finding the
difference between two numbers, since it shows that 303 is at a
greater distance from 0 than 78 is from 0.
Lesson 5: Geography of Addition and Subtraction
Teacher Tips
In Unit 2, we have seen number lines where
regular markings were used to establish
a scale (see Lesson 1) and other number
lines where the scale was distorted either
to emphasize the order of numbers or as a
practical matter to provide space to mark
relative distances between numbers even if
the drawing was not to scale.
Part of learning how to think mathematically
about interpreting diagrams and using the
number line strategically is learning to
switch between different representations and
to distinguish important from unimportant
features of a diagram. Similarly, when
students produce their own diagrams, they
have a chance to practice figuring out how
to make a diagram that helps them.
T43
In either case, once the student has a visual representation for the distance between 78 and 303, ask, “What is
a distance you know is too large to be the answer?” (Possible responses: 1000, 303, 300, 250). Ask, “What is a
distance you know is too small to be the answer?” (Possible responses: 1, 10, 100, 200)
Depending on how the conversation goes, this may be a good way to start to talk about drawing “jumps” to identify
large chunks of distances between 78 and 303. If a student says that 200 is a distance that is too small, ask “How do
200
you know?” They might say that 200 is the distance between 100 and 300, and
we can see the distance between 78 and 303 is greater. Show students how to
78
100
300 303
indicate this observation on a number line.
Then give students time to generate the rest of their picture.
Share interesting student responses with the class. Point out similarities between different methods: students will
likely have identified 80 or 300 as useful reference points. Here are two possible responses you may see.
2 20
78 80 100
200
3
300 303
25
75 78
200
100
300 303
Both of these responses use ideas from earlier in the unit. In Lesson 1, students located numbers relative to the
nearest multiple of 10, in Lesson 2, they noticed that the distance between 78 and 303 is the same as the distance
from 75 to 300, and their Mental Mathematics activities have supported finding distances to 10 and 100.
Student Problem Solving and Thinking Out Loud Dialogue:
Students should work in pairs or small groups on problems 1-11. When students have generally started to reach
problem 12, gather the class and ask three volunteers to read the Thinking Out Loud dialogue at the board. It is
important that they recreate the picture on the board as they act. To give more students the opportunity to act and to
further reinforce the vocabulary of “adding the opposite,” ask a second group of students to perform the dialogue
again. As students consider the question in the Pausing to Think box, refer back to what students did with checkers
in Lesson 3. Subtracting -5 is like removing 5 red checkers from a pile of checkers where the score is m.
Students will use the ideas from the dialogue to fill in the number lines in problems 13-16, making sense of how
the generic diagram relating addition and subtraction from a variable (problems 13 and 14) relates to the a diagram
showing addition and subtraction from a number (problems 15 and 16).
What if...
What if students don’t know how to add and subtract positive and negative numbers? This difficulty is
not uncommon. Emphasize to students that the point is not to “know how,” but to be able to make sense of
the calculation. Refrain from reducing this process into a set of rules. It is tempting to say something like “In
subtraction, when the signs are the same, you subtract, and when the signs are different, you add.” For most
students, they already have a set of rules in their heads which are confusing (“In 2–5, the signs are different, so
I add?”). Keep pointing students back to the number line image so that they use that tool to make sense of the
numbers instead of just applying rules without thinking about what’s going on.
What if students don’t want to draw “jumps” on the number line and instead find distances by
subtracting using the standard algorithm? Students may feel frustrated if they feel like they are being made
to learn a new procedure for a problem they already feel like they know how to do. The message is never
“Don’t use what you know.” The subtraction algorithm is very useful to know. But when learning algebra,
these strategies become much less useful. There’s no way to apply the subtraction algorithm to x – 45. The
number line is a tool that is useful beyond arithmetic calculations to algebraic reasoning.
T44
Unit 2: Geography of the Number Line
Supporting Classroom Discussion
Use these additional questions to discuss ideas from this lesson.
• If someone asks you to find -14 – 3, several possible answers might come to mind. Describe a number
line image that will help you be confident you have the correct answer. Ask several students the same
question with different examples.
• Describe your results in problems 15 and 16 using the black and red checkers from Lesson 3. What
is the connection between the number line and the checkers? Recall that putting in checkers is like
adding and removing checkers is like subtracting. For example, it makes sense that 3 – -12 is to the right
of 3 on the number line because subtracting -12 is like removing 12 red checkers, which increases black’s
score.
Connections to Algebra
Algebraic thinking often requires students to think “inside out” and “backwards.” In problems 1-4, students
find distances between two numbers using strategies demonstrated in the launch. Then, in problems 5-11,
students must switch their thinking. They are now given a distance, and they need to figure out a way to use the
distance as a measurement along the number line. Finally, problems 12-16 use a similar image of distance as a
general image of addition and subtraction of positive and negative numbers.
In order to develop algebraic thinking, it’s important that students understand that all the problems are
connected, and they should be encouraged to make sense of new problems based on their understanding of
previous problems.
Notes:
Lesson 5: Geography of Addition and Subtraction
T45
2-5 Geography of Addition and Subtraction
Important Stuff
Show the numbers on the number line and find the distance between them.
1
71 and 100
2
Hint: Think of
70 to 100 first.
Then adjust.
3
I
I
0
71
29
I
I
0
-40
I
235
I I
-5 0
-240
Where am I? Show all possible solutions and fill in all of the blanks.
Clue: I am 14 units away from 40.
Where am I?
I
26
6
-7
-23
-19
24
T46
I
0
I
3
I
0
I
9
Clue: I am 5 units away from m.
Where am I?
Mark 0 to
help
you imagin
e the
Clue: I am 40 units away from 3.
negative si
Where am I?
de.
I
0
I
43
I
-93
11
I
3
Clue: I am 83 units away from -10.
Where am I?
I
37
-4 0
54
-37
I
I
I?
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I
13
I
7
I
I
50
7
Clue: I am 15 units away from -4.
Where am I?
I
12
I
40
Clue: The distance between me and 7 is 30.
Where am I?
I
10
I
30
Clue: I am 10 units away from 3.
Where am I?
I
8
m
ea
Wh
er
5
60
-5 and -240
I
125
I
I
0
-18
4
I
78
I
100
125 and -40
165
-18 and 60
I
I
-10 0
73
Clue: I am 20 units away from -33.
Where am I?
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I
11
-53
I
m
I
-33
I
-13
I
0
See dialogue to
learn how to label
this number line.
Unit 2: Geography of the Number Line
Unit 2: Geography of the Number Line
Thinking Out Loud
Michael, Lena, and Jay are working on problem 12.
Michael: I know how to draw the arrows (draws picture),
I
but then what? How do we label the answers?
I
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m
Well, this number on the right (she points), is five more than m, so that’s m + 5 (labels m + 5).
Lena:
I
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I
m
m + 5
Michael: So, can we write both answers with addition? What would we add to get to the number on the left?
What about -5? Adding -5 is adding the opposite of 5, so m + -5 is on the opposite side. (Jay labels
Jay:
m + -5.)
I
I
m + -5
I
m
m + 5
Why don’t we just use subtraction? Jay, you labeled this number (she points to m + -5), but we
Lena:
m – 5
could also call it m – 5 because it’s five less than m.
I
I
m + -5
Jay:
I
m
Sure. It’s both. Hey, do you think we can use subtraction to name the number on the right?
m + 5
Michael: Let’s use the idea you had, Jay. We want to subtract the opposite of 5, right? So we subtract -5.
Multiple Choice: What could Michael say next to explain what he means?
Pausing
to Think
A
That means m – -5 ends up in the same place as m + 5.
B
That means subtracting -5 is like adding 5.
C
Subtracting a negative number does the opposite of subtraction, which is addition.
D
All of the above.
m – 5
Then label m – -5 on the number line.
13
Clue: I am 12 units away from c
Where am I?
c - 12
I
c + -12
15
3 + 12
Plot:
3 - 12
I
3 + -12
I
c
3 + -12
I
3
14
I
l
n+8
c - - 12
n - 8
c + 12
n + -8
I
3 – 12
3 – - 12
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16
Plot:
-10 –
3 - - 12
-10 - 8
3 + 12
-10 + -8
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Lesson 5: Geography of Addition and Subtraction
Lesson 5: Geography of Addition and Subtraction
I
m + -5
Plot:
I
I
m
n + -8
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n
-8
m – -5
-10 – 8
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-10
m + 5
n–
n– 8
8
n - -8
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n + 8
-10 + -8
-10 +
8
-10 - - 8
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-10 + 8
25
T47
Stuff to Make You Think
Find the temperatures for the rest of the week.
Monday
Tuesday
Wednesday
Thursday
Friday
12º
27 º
25 º
35 º
15 º
Temperature
+15º
change:
–2º
–20º
Monday
Tuesday
Wednesday
Thursday
Friday
5º
20 º
18º
28 º
8º
+10º
Tuesday’s is higher. The
temp. change from Tues.
to Wed. is -2 º , so to go
back to Tuesday’s temp, we
add 2 º .
–20º
☀
I
40º
20º
I
20º
0º
I
☀
0º
-20º
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Today the temperature is -20º, and
tomorrow’s temperature is expected to be
15º colder. Find tomorrow’s temperature.
40º
-20º
-40º
I
It got warmer by 35 º .
22
I
If today’s temperature is 30º and
yesterday was -5º, find the temperature
change. Did it get warmer (+) or cooler
(–)?
Is Tuesday’s temperature higher or
lower than Wednesday’s temperature?
Explain why your answer makes sense.
I
–2º
20
A temperature of -14 º
indicates a cold day. A
temperature change of -14 º
means it got 14 º cooler,
whether the weather is
actually hot or cold.
I
Temperature
+15º
change:
21
+10º
Explain how “a temperature of -14º” is
different from “a temperature change
of -14º.”
I
19
18
I
17
Tomorrow’s temperature
will be -35 º .
❄
23
-40º
❄
Today’s temperature is 30º.
It is 12º warmer than it was yesterday.
What was yesterday’s temperature?
24
Yesterday’s temperature was 18 º .
Today’s temperature is -4º.
It is 10º warmer than it was yesterday.
What was yesterday’s temperature?
Yesterday’s temperature was -14 º .
Tough Stuff
25
Points A, B, C, and D are on the same line. The distance between A and B is 1, the distance between B and C is 2,
and the distance between C and D is 4. What can be the distance between A and D? List all the possible cases.
Hint: The points can be in any order.
The distance between A and D can be 1, 3, 5, or 7. For example:
I
I
I
D
A
B
I
I
I
I
I
C
C
A
B
I
I
I
I
I
D
B
A
C
I
I
I
I
I
I
D
A
B
I
(Each number line can be drawn in the backwards orientation.)
26
T48
I
C
I
I
I
I
D
Unit 2: Geography of the Number Line
Unit 2: Geography of the Number Line
Additional Practice
Where am I? Show all possible solutions.
A
Clue: I am 9 units away from 40.
Where am I?
I
0
C
31
I
k + -7
1–7
Plot:
1 - 7
I
1 + -7
K
L
I
I
-20
1 + -7
0
I
1
-38
b - 11
k + 7
b + -11
I
J
Plot:
l
-6 +
-6 - 11
1 + 7
-6 + -11
I
t
u
Who Am I?
• The sum of my digits is 5.
2 3
• My tens digit is not odd.
• My tens digit and units digit differ by 1.
t
u
Who Am I?
• Both of my digits are even.
4 8
• My units digit is twice my
tens digit.
• My units digit is not a perfect square.
Lesson 5: Geography of Addition and Subtraction
-14
I
1 - -7
Additional Practice
I
1
b+1
Plot:
k - -7
1+7
I
-2 0
I
H
I
30
Clue: I am 24 units away from -14.
Where am I?
I
-7
1 – -7
I
-34
F
I
13
Clue: I am 32 units away from -2.
Where am I?
26
I
I
k
D
I
Clue: I am 7 units away from k.
Where am I?
k - 7
I
49
Clue: I am 13 units away from -20.
Where am I?
-33
G
-13
I
0 6
I
I
0
I
I
-14
Clue: I am 13 units away from 0.
Where am I?
I
Clue: The distance between me and 6 is 20.
Where am I?
Mark 0
I
E
I
40
I
B
I
M
b–1
1
I
b
11
I
I
10
0
- 1
b– 1
b + -11
b - - 11
I
b + 11
- 1 -6 + -11
-6 – 1
I
-6
-6 – 11
-6 - - 11
I
-6 + 11
52
6
= ______
=1
10
= ______
27
T49
Lesson 6: Algebra on the Number Line
Purpose
In this lesson, students re-encounter a diagram of addition on the number line from Lesson 2. They use the
diagram and generalize the result by considering what happens when the same structure is used on a number
line that is centered on any value c. The Thinking Out Loud dialogue conveys the idea that this process of
making generalizations about how numbers work is an important part of algebra. Students use variables on the
number line to solve simple algebraic equations and figure out what a statement like “n + 8 is negative” tells
you about n.
Lesson at a Glance
Mental Mathematics: (5 min) See pages XX-XX of the TTA Mental Mathematics book.
As a Class
Launch: (15 min) Addition Patterns and Thinking Out Loud Dialogue
Student Problem Solving and Discussion: (25 min)
• Give students time to work through the remaining problems.
• Discuss further what it means to have variables on the number line.
• Discuss ways to think about the values of variables when they are on a number line.
In Groups
As a Class
Unit 2 Related Game: Who Am I? Number Bingo (See Teacher Guide page T59 and Student Book page 35)
Launch – Addition Patterns and Thinking Out Loud Dialogue:
Give students time to work on problems 1-3 in pairs or small groups. After students have worked on and discussed
problem 3, go through that problem together as a class. Make sure students understand the equations in the boxes—
both how to construct them and what they mean. Make sure students also understand the connection that “adding the
middle number to itself,” c + c, is equivalent to “twice the middle number,” 2c.
Then invite two students forward to act out the Thinking Out Loud dialogue for the class. Check for student
understanding with the Pausing to Think box.
Student Problem Solving and Discussion:
Allow time for students to work together in small groups through the remaining problems. Problems 7-14 continue
the idea from Lesson 5 of connecting the image of a number line with a variable (with a and a + 3) to a number line
with numbers. Problems 15-17 ask students to think more flexibly about variables on the number line, particularly in
determining whether those variables are positive or negative.
Teacher Tips
The ideas of problems 16-17 can be discussed in many different ways. The example in the Student Book shows
a more formal approach to the problem, but you may also observe students having productive discussions by
considering specific counterexamples. For example, for the statement “If a is a negative number, then a + 100
is always positive” encourage students to consider specific negative numbers (-1, -5, -20). Often, students will
suggest checking numbers like -300 or -1000. You might ask questions like “How many examples would we
have to try before we can declare the answer is ‘true’?” “What about ‘false’?”
T50
Unit 2: Geography of the Number Line
Supporting Classroom Discussion
Use these questions to give students the opportunity to talk about number lines with variables.
• We can draw a picture of adding 10 to any number. In fact, we can draw the general picture for
what it looks like to add 10, subtract 10, add -10, and subtract -10 from any number.
10
x – 10
x + -10
10
x + 10
x – -10
x
• Can we tell from this picture where 0 goes? No.
• If 0 is located over to the right of everything in the picture, what does that tell us? Everything in
the picture is negative.
• If 0 is exactly halfway between x and x – 10, what does that tell us? Allow for a longer
discussion. Some students will just conclude that x – 10 is negative (which is true). Some students
might figure out all the values exactly, by figuring out that x = 5, so x – 10 = -5 and x + 10 = 15.
Connections to Algebra
Students may already have experience solving equations like x + 7 = 10. In fact, they’ve already seen similar
equations in Unit 1. Problems 7-14 aren’t intended to teach “the way” for solving these equations. The
emphasis is again on making sense of calculations by considering relative positions on the number line. If
students see a problem like x + 7 = -10, they can (and probably should) think of “subtracting 7 from both
sides.” The number line image supports this calculation, and also provides extra support by confirming that
since x lies to the left of x + 7 on the number line, the result of -17 is reasonable.
-17
-10
x
x+7
Student Reflections and Unit Assessment:
Before conducting a summative assessment of the unit, ask students to reflect on their learning by having them
consider the following:
• What are some things you learned in this unit?
• What questions do you still have?
Use this feedback to help students select additional practice problems to help them prepare for the Unit
Assessment.
Lesson 6: Algebra on the Number Line
T51
2-6 Algebra on the Number Line
Important Stuff
1
Find the sums for this number line centered on -2.
-5
3
-4
-3
Fill out this number line so that it is centered on 35.
2
-4
70
-4
70
-4
70
-4
70
+
-2
-1
0
+
1
32
33
34
35
36
37
38
Problems 1 and 2 used specific numbers. Now use algebra to show what happens with any number in the center.
Call the center c.
c - 3 + c + 3 = 2c
c - 2 + c + 2 = 2c
c – 1 + c + 1 = 2c
c + c = 2c
+
c-3
c-1
c–2
c
c+1
Thinking Out Loud
c+2
c+3
c + 1 is the number that is 1 more than c.
Michael: Why use algebra? I’d rather just use numbers.
Jay:
What do you mean?
Michael: Why did we even do problem 3? We filled in enough of these number lines that I knew what to
expect. Add the middle number to itself, and you get an answer. If you add the numbers one more
and one less than the middle number, you get the same answer.
Jay:
Ha! You just used algebra!
Michael: No, I didn’t. I was talking about the numbers and what happens with the number problems.
Jay:
But that’s all algebra does, too. It just abbreviates everything a little more. Algebra describes
the same thing you described in words, and the variable is there to show that we can expect any
number to behave the same way.
Write the algebraic statements from problem 3 that go with Michael’s words.
Pausing
to Think
28
T52
4
“Add the middle number to
itself, and you get an answer”
c + c = 2c
5
“If you add the numbers one more and one less than
the middle number, you get the same answer.”
c - 1 + c + 1 = 2c
Unit 2: Geography of the Number Line
Unit 2: Geography of the Number Line
Plot these values and label them
on the number line below.
I
I
m-8 m - 7
7
-1
0
I
a
I
If a + 3 = 51, what is a?
I
a
I
I
I
m-3
I
I
c–2
1
I
2
I
a+3
10
I
I
I
m–
m+2
I
m
I
I
I
m+2
13
13
a =
I
I
This number line shows both of these statements to be true.
n is negative.
n + 8 is negative.
What are some possible values for n? Name at least five values.
I
I
I
m+8
If a + 3 = 2, what is a + 5?
If b + 10 = 98, what is b?
88
-12
I
a+3
98
I
b
14
I
b + 10
If m + 6 = -20, what is m?
-26
-20
m
m + 6
I
I
n
n – 8
m+4
11
3
I
I
c
I
m – -5
a + 5 = 4
(2 more than a + 3)
-1
If n – 8 = -5, what is n?
-5
m- - 5
I
8
If a + 3 = -12, what is a?
I
a
m + -5
7
Stuff to Make You Think
a + 3 = 2
I
-15
51
I
a+3
If c – 2 = 11, what is c?
11
15
I
I
48
12
I
m+4
m–3
If you know that a + 3 = 2, you can use a number line to find a.
I
9
m+-5
m–8
l
6
I
n
I
I
n+8
I
0
For example, n can’t be -3. Why not?
3
Responses will vary as long as n < -8, such as -10, -8.01, -200 4 , etc.
Answer True or False for each statement below. If the statement is true, show how you know. If the statement is false,
provide a specific example using numbers, algebra, or a picture to show why it’s not always true.
16
If a is a negative number,
False
then a + 100 is always positive. _______________
This picture shows
that it’s possible
to draw a number
line that makes the
statement false.
I
a
I
a + 100
I
0
17
a + 100 can be negative
if a < -100
For example, if a = -300,
then a + 100 is negative.
Lesson 6: Algebra on the Number Line
Lesson 6: Algebra on the Number Line
If m is a positive number,
False
then m – 8 is always negative. _______________
Responses
I
I
I
will vary.
0
m – 8
m
m – 8 can be positive if
m > 8
For example, if m = 10,
then m – 8 is positive.
29
T53
Stuff to Make You Think
18
19
On this number line, 0 and y are labeled. Label the other spaces.
I
I
I
I
I
-6y
-5y
-4y
-3y
-2y
-y
I
y
I
I
I
I
I
2y
3y
4y
5y
6y
In both problems 18 and 19, y and -y
are “opposites” of each other. And 2y
and -2y are also opposites. Opposite
numbers are the same distance from
0, but on opposite sides of 0.
I
I
I
4y
3y
2y
5
17
-11
-8
63
22
-19
-105
-14.5
0
y
-5
-17
11
8
-63
-22
19
105
14.5
0
-y
I
y
I
0
I
-y
I
I
-2y -3y
5n
Show where each of these belongs on the number line.
I
I
I
k
2k
22
I
0
On this number line, 0 and y are labeled, and y is a negative number.
Label the other spaces.
20
21
I
-2n
I
I
0
Plot and label these values on the number line.
I
I
4r
2r
I
r
I
I
n
2t
I
0
I
t
-r
I
-k
2k
I
4r
I
-k
3t
I
2t
-2k
-2n
I
I
5n
-2k
-r
2t
I
3t
Discuss & Write What You Think
23
At first, this number line looks wrong, because we know that negative numbers live to the left of 0 on the
number line. But because x is a variable, it is possible for -x to be a positive number. Use examples to
explain this number line to a person seeing it for the first time.
I
x
30
T54
I
0
I
-x
Responses will vary, but encourage students
to use examples (If x=-10, we would mark x
to the left of 0...) and use the idea of -x as
the “opposite” of x, and not the same thing
as a “negative number.”
Unit 2: Geography of the Number Line
Unit 2: Geography of the Number Line
Use the clues to mark one possible location each for A and B.
24
Clue 1: A is positive
Clue 2: B = 2A
Clue 3: B > A
25
I
0
Clue 1: A is negative
Clue 2: B = 2A
Clue 3: B < A
I
I
I
I
A
B
B
A
I
0
Discuss & Write What You Think
26
Problems 24 and 25 show that the answer to “Which is greater, x or 2x?” is “It depends.”
What does it depend on?
It depends on whether x is positive or negative. When x is positive,
2x is greater than x. When x is negative, 2x is less than x.
27
Is it possible for “a number” and “twice the same number” to be the same? In other words, can n = 2n?
If so, when? If not, why is this impossible?
Yes, it is possible, in only one place. When n = 0, 2n = 0.
28
Is it possible for “a number” and “two more than the same number” to be the same? In other words,
can y = y + 2? If so, when? If not, why is this impossible?
No, this is impossible. “Two more than y” will always be to the right of
y on the number line, no matter what y is. They can never be the
same number.
Tough Stuff
When you are taking a long test, you want to use methods that save you time. That always means understanding what
the test problem is asking for. It also means understanding what the problem is not asking for, to avoid unnecessary
work. For each problem, find each answer, but do no more work than is really needed.
29
If a + 20 = 12, then a + 21 =
30
If 6 – b = 0, then 10 – b =
31
If 150 – c = 17120, then 160 – c =
A
32
A
16
A
17270
B
13
B
10
B
17030
C
12
C
6
C
17130
D
11
D
4
D
17110
E
8
E
0
E
310
Lesson 6: Algebra on the Number Line
Lesson 6: Algebra on the Number Line
31
T55
Additional Practice
On this number line, a is labeled. Plot:
a–
a + 6 a + -6
6
a– 6
a - 6
a + -6
C
I
I
I
I
a + 6
I
-7+s
h+ - 8
I
Show where each of these belongs on
the number line.
I
On this number line, and h + 12 are labeled. Plot:
h + 4 h + -4 h + -8 h – -8
a - -6
I
a
I
B
I
s+9
I
I
s + -1
s–4
l
A
I
I
s
s+–1
s–4
I
I
I
h
h+ - 4
I
I
I
I
I
h+4
-7 + s
6+s
h- - 8
s – -8
s– 2
I
I
s–-2
I
h + 12
I
I
I
s – - 8 s+9
6+s
Use the number line to find the value of the variable.
G
If n + 2 = 9, what is n?
7
I
J
I
n
I
10
If y – 3 = 49, what is y?
49
I
y–3
K
I
I
If h + 2 = -13, what is h?
52
-15
I
y
I
h
L
I
I
h+2
-23
-3
m
m + 20
I
I
c
c - 10
-13
If m + 20 = -3, what is m?
-8
I
I
w
I
If c – 10 = -18, what is c?
-18
41
I
w – 10
N
9
I
n+2
If w – 10 = 31, what is w?
31
M
H
I
2
1
-10
-11
44
83
-26
-9
-5
-6
p
8
7
-4
-5
50
89
-20
-3
1
0
p+6
8
9
-7
-6
56
40
-17
-30
2
3
x
5
6
-10
-9
53
37
-20
-33
-1
0
x–3
Draw a number line which shows both of these statements to be true.
n is negative.
n + 8 is positive.
What are some possible values for n? Name at least five values.
I
n
I
0
I
n + 8
1
Responses will vary as long as -8 < n < 0, such as -1, -7.9, -2 3 , etc.
32
T56
Unit 2: Geography of the Number Line
Unit 2: Geography of the Number Line
Unit 2 Additional Practice
Use these pages to prepare for the unit exam.
Finish labeling the marks on these lines in a consistent way.
I
-2
I
0
I
I
-9
-6
I
-3
I
I
I
I
I
-33
-30
-27
-24
-21
1
3
5
I
I
4
2
I
2
6
I
I
3
0
4
I
-18
I
-30
I
-25
I
I
I
-20
-15
-10
I
-300
I
I
I
-200
-100
I
0
I
I
I
I
I
-15
-12
-9
-6
-3
I
0
100
I
I
3
6
I
9
Use the clues to mark one possible location each for A, B, and C so they are in the correct order.
6
Clue 1: A < B
Clue 2: B is negative.
Clue 3: C > 0
I
I
A B
7
Clue 1: B < A < C
Clue 2: B > 0
Clue 3: C > B
I
I
0
Clue 1: A > C
Clue 2: C < A
Clue 3: A < B
8
I I I I
0 BA C
C
I
I
I
C
A
B
Find the weight of the variable so that the mobile balances.
9
12
13
10
11
61
h
34
30
103
250
89
90
w
200
248
p
61 + 89 = h + 90
34 + w = 30 + 200
103 + 248 = 250 + p
60
h = _______
196
w = _______
101
p = _______
t
u
Who Am I?
• My tens digit is even.
8 3
• The sum of my digits is 11.
• My units digit is less than my tens digit.
• My tens digit is five more than u.
t
u
Who Am I?
• I am odd.
1 1
• I am less than 40.
• Both of my digits are perfect squares.
• t–u=0
Additional Practice
Unit Additional Practice
14
20
3
= ______
2
= ______
3
= ______
33
T57
Show the numbers on the number line and find the distance between them.
15
60 and -35
I
18
I
I
-2 0
20 and -90
100
Wh
er
19
Clue: I am less than or equal to 36.
My name is w.
Where am I? Use symbols:
w ≤ 36
Show everywhere I could be.
I
24
18
30
I
36
Mark 0
I
9
0
I
-61
I
I
I
b-8
I
I
I
I
I
I
-8
-3+b
I
I
b-2
I
I
b
I
2
4
6
I
I
8
I
16
Clue: I am 13 units away from -14.
Where am I?
I
-27
b–2
I
0
I
b+5
n > 6
Clue: I am less than or equal to -8.
My name is a.
Where am I? Use symbols:
a ≤ -8
Show everywhere I could be.
79
I
I
0
-2
I
Plot the numbers. Be sure to label them.
Some of them end up in the same place.
b+ - 8
I
I
23
20
Clue: I am greater than 6.
My name is n.
Where am I? Use symbols:
Show everywhere I could be.
I
Clue: I am 70 units away from 9.
Where am I?
I
24
I
l
22
I
21
I
I
0
-90
I?
I
12
I
0
110
I
I
-4
20
I
-43 -23
102
I
20
I
60
-2 and 100
m
ea
-23 and -43
I
I
0
-35
17
16
95
3+b
I
b–8
I
b+3
I
3+b
I
-14
b+3
I
-1
b+
I
b+5
-8
I
I
0
b–10
-3 + b
I
I
I
I
b- - 10
Use the number line to find the value of the variable.
25
If x + 5 = 3, what is x?
-2
I
x
34
T58
3
I
x+5
26
If a – 10 = -25, what is a?
-25
I
a - 10
-15
I
a
27
If h + 8 = -10, what is h?
-18
-10
h
h + 8
I
I
Unit 2: Geography of the Number Line
Unit 2: Geography of the Number Line
Game: Who Am I? Number Bingo
Purpose
Students are introduced to Who Am I? puzzles in Lesson 3. This bingo game is a fun way to introduce the
language of the many clues students may encounter in the Who Am I? puzzles.
Timing: This activity is intended to be done in short 5-10 min sessions throughout Unit 2.
Preparation: Print and cut out a set of Clue Cards (on pages T60- T70), preferably on cardstock or laminated.
Students will need extra paper to write their numbers.
Game Suggestions:
• Game instructions are on page 35 of the Student Book.
• 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. Once students have experienced the game, increase the pace of the
game. Even if not all players get the clue, move on to the next one. That’s part of the game!
• Make sure you keep track of the cards you’ve used in a round, so that the class can check to see whether the
player is correct. In order to speed up the checking process, you may also ask students to write down “Clue
120” next to the number when they’ve crossed it off their list. This has the potential to confuse, however, since
students will be evaluating their set of numbers against the clue, and will also have to keep track of the clue
number. This option may be better used when students are playing in pairs or small groups and can see each
card, rather than when playing with the whole class.
Variations:
• Play with the whole class, with the teacher or a volunteer drawing and reading the clues.
• Give students their own stack of cards, or split up the deck, and have students play in small groups of 4-5
students.
• Reverse the game: Hand out five cards to each student. Call out a series of three-digit numbers. If the card
describes the number, the player puts it down. The first player to put all of their cards down calls “Bingo!”
• Split up the deck, giving each pair of students three cards, drawn randomly. Their challenge is to make up
as many numbers as possible for which all three clues are true. If they can show that two or more clues are
mutually exclusive (so that no numbers are possible), they can trade in their cards for a new set.
Game Extension:
Students may make up their own clue cards. They may write clues to add to the existing set about square numbers,
prime numbers, factors, or make up clues which use more symbols, such as “h + t = 2u.” Or, students might make
up an entirely new set of cards to use with:
• Four digit numbers: Clues include references to the thousand’s digit, or “k.”
• Fractions: Clues refer to the numerator and denomintaor.
• Decimals: Clues refer to the tenths digit, or “d.”
Game: Who Am I? Number Bingo
T59
Who-Am-I? Number Bingo Clue Cards
T60
1
The tens digit is 3 times the
units digit.
2
The tens digit is twice the
units digit.
3
The units digit is twice the
hundreds digit.
4
The units digit is 3 times the
tens digit.
5
The hundreds digit is 3 less
than the units 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 units 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 units 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.
Unit 2: Geography of the Number Line
Who-Am-I? Number Bingo Clue Cards
15
The tens digit is 2 greater
than the hundreds digit.
16
The hundreds digit is 3 less
than the units digit.
17
The hundreds digit is greater
than the units digit.
18
The tens digit is 3 times the
units 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 units 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 units digit.
25
The units digit is less than
twice the hundreds digit.
26
The tens digit is 3 times the
units 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.
Game: Who Am I? Number Bingo
T61
Who-Am-I? Number Bingo Clue Cards
T62
29
The hundreds digit is 3 less
than one of the other digits.
30
The hundreds digit is twice
the units digit.
31
The tens digit is twice the
units digit.
32
Tens digit < units digit.
33
The hundreds digit is 2
greater than one of the other
digits.
34
The units digit is 3 times the
tens digit.
35
Sum of all the digits is 9, 12,
15, or 18.
36
The units digit is greater than
the hundreds digit.
37
The hundreds digit is twice
the units 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 more
than another digit.
42
The tens digit is greater than
the hundreds digit.
Unit 2: Geography of the Number Line
Who-Am-I? Number Bingo Clue Cards
43
The units digit is 3 times the
tens digit.
45
44
Units digit < tens digit.
The hundreds digit is twice
the tens digit.
46
The units digit is twice the
tens digit.
47
The hundreds digit is greater
than twice the units digit.
48
The units 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
units digit.
51
The hundreds digit is 3 times
the tens digit.
52
The hundreds digit is 3 less
than the units digit.
53
The tens digit times the units
digit is 0.
54
The hundreds digit is greater
than the tens digit..
55
The hundreds digit times the
units digit is a multiple of 3.
56
The tens digit is twice the
hundreds digit.
Game: Who Am I? Number Bingo
T63
Who-Am-I? Number Bingo Clue Cards
T64
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 units
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 units digit.
64
The units digit is twice the
hundreds digit.
65
The units digit times the tens
digit is a multiple of 10.
66
The hundreds digit is 2
greater than the units 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 units
digit is a multiple of 10.
70
The hundreds digit is greater
than twice the units digit.
Unit 2: Geography of the Number Line
Who-Am-I? Number Bingo Clue Cards
71
The tens digit times the units
digit is less than 8.
72
The units digit plus the
hundreds digit < 8.
73
The units 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
units digit is less than 8.
76
The hundreds digit times the
units 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
units digit is a multiple of 3.
80
The product of all the digits
is less than 8.
81
Sum of the units digit and the
hundreds digit is a multiple
of 3.
82
The units digit times the tens
digit is a multiple of 10.
83
The product of all the digits
is less than 8.
84
The hundreds digit times the
tens digit is a multiple of 3.
Game: Who Am I? Number Bingo
T65
Who-Am-I? Number Bingo Clue Cards
T66
85
All three digits are the same.
86
The tens digit times the units
digit is a multiple of 10.
87
The number is four less than
a multiple of 10.
88
The tens digit times the units
digit is 0.
89
Units digit < 5.
90
The sum of all the digits is a
multiple of 5.
91
Hundreds digit > 6.
92
The tens digit times the units
digit is a multiple of 3.
93
At least two digits are the
same.
94
Sum of the tens digit and the
units digit is a multiple of 3.
95
Two of the digits are less
than 5.
96
The hundreds digit is 3 times
the units digit.
97
The number is a multiple of
10.
98
The tens digit times the units
digit is less than 8.
Unit 2: Geography of the Number Line
Who-Am-I? Number Bingo Clue Cards
99
Units 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.
100 Sum
of the tens digit and the
units 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 units digit
and the hundreds digit is a
multiple of 3.
107 Two
108 The
109 The
110 The
111 The
112 The
hundreds digit is greater
than the units digit.
number is two more than
a multiple of 5.
Game: Who Am I? Number Bingo
tens digit is more than
twice the hundreds digit.
tens digit is twice the
hundreds digit.
hundreds digit is greater
than the tens digit.
T67
Who-Am-I? Number Bingo Clue Cards
113 None
115 The
of the digits are even.
tens digit is 4.
117 Tens
digit < 3.
119 Hundreds
121 The
tens digit is 9.
123 The
hundreds digit is 9.
125 Tens
T68
digit > 2.
digit is 7, 8, or 9.
114 The
tens digit is 2 greater
than the hundreds digit.
116 The
units digit is twice the
hundreds digit.
118 The
hundreds digit is 2
greater than one of the other
digits.
120 The
units digit is less than
twice the hundreds digit.
122 The
units digit is greater than
the hundreds digit.
124 The
hundreds digit is twice
the tens digit.
126 The
hundreds digit is 2
greater than the units digit.
Unit 2: Geography of the Number Line
Who-Am-I? Number Bingo Clue Cards
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.
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
139 The
140 The
of all the digits is 9, 12,
15, or 18.
tens digit is more than
twice the hundreds digit.
units digit is twice the
hundreds digit.
tens digit is 2 greater
than the hundreds digit.
Game: Who Am I? Number Bingo
digit < units digit.
of the digits is 3 less
than another digit.
hundreds digit is twice
the units digit.
tens digit is 3 times the
units digit.
hundreds digit is greater
than twice the ones digit.
tens digit is twice the
units digit.
T69
Who-Am-I? Number Bingo Clue Cards
141 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 2 greater
than one of the other digits.
144 Units
145 The
146 The
147 The
148 The
149 The
150 The
hundreds digit is twice
the units digit.
hundreds digit is 2
greater than the units digit.
T70
142 The
digit < tens digit.
hundreds digit is 3 less
than the units digit.
units digit is twice the
tens digit.
hundreds digit is 3 less
than one of the other digits.
Unit 2: Geography of the Number Line
Game: Who Am I? Number Bingo
Instructions:
472
689
255
205
100
• Every player writes down their own set of five 3-digit numbers.
• Shuffle the clue cards and stack them face down.
• Turn one clue over at a time and read it aloud.
99
Units d
s are odd.
e of the digit
101 Non
igit > 6
.
120 The
units digit is less than
twice the hundreds digit.
• Each player checks all five of their numbers to see if the clue fits. If the clue fits, the player
can cross that number out.
472
689
255
205
100
Clue120
Clue99
Clue120
• Any player who has crossed out all five numbers says “Bingo!”
• Check the numbers against the clues. If the person was correct, they win. If there are errors,
continue to play until there is a winner.
Game: Who Am I? Number Bingo
Game: Who Am I? Number Bingo
35
T71
Snapshot Check-in
Name:
Finish labeling the marks on these lines in a consistent way.
1
I
3
I
-30
I
I
I
I
I
2
I
I
4
I
I
2
I
I
I
I
I
I
-20
I
I
0
4
Find three numbers that continue this pattern: 12, 6, 0, _____ , _____ , _____
5
Is this mobile balanced or is one side heavier than the other?
Explain your reasoning for without doing the addition on either side.
68
?
I
I
I
0
I
I
9
I
70
21
20
Find the weight of the variable so that the mobile balances.
6
T72
7
39
h
52
50
89
90
w
100
39 + 89 = h + 90
52 + w = 50 + 100
h = _______
w = _______
Unit 2: Geography of the Number Line
Snapshot Check-in
Name: Answer Key
Finish labeling the marks on these lines in a consistent way.
1
3
4
5
I
I
I
I
2
I
4
I
I
-4
-2
0
-40
-30
I
-30
I
I
I
I
I
I
I
I
I
-27
-24
-21
-18
-15
-12
-9
-6
-3
2
I
-20
I
0
I
I
0
-10
I
I
3
6
I
9
I
12
-6 , _____
-12 , _____
-18
Find three numbers that continue this pattern: 12, 6, 0, _____
Is this mobile balanced or is one side heavier than the other?
The right side is heavier.
Explain your reasoning for without doing the addition on either side.
68
Responses will vary. Sample responses include:
?
70
21
20
20 is 1 less than 21 but 70 is 2 more than 68 so the
right side is heavier by 1.
The mobile would be balanced if the left side were 69 and
21 (or 68 and 22 or another reasonable combination),
but since 68 is less than 69, the left side is lighter.
Find the weight of the variable so that the mobile balances.
6
7
39
h
52
50
89
90
w
100
39 + 89 = h + 90
52 + w = 50 + 100
38
h = _______
98
w = _______
Snapshot Check-in
T73
Unit Assessment
Name:
Finish labeling the marks on these lines in a consistent way.
1
I
I
0
I
I
6
I
2
I
-18
I
-16
I
I
I
3
I
I
I
-8
I
I
0
4
I
I
0
I
I
I
60
5
I
I
I
I
I
I
-30
I
I
I
I
I
I
0
I
I
I
15
Use the clues to mark one possible location each for A, B, and C so they are in the correct order.
6
Clue 1: A > B
Clue 2: B > C
Clue 3: C is positive.
7
Clue 1: A > B
Clue 2: C < B
Clue 3: C < A
I
0
Find the weight of the variable so that the mobile balances.
8
10
T74
9
48
n
100
98
77
49
c
38
48 + 77 = n + 49
100 + c = 98 + 38
n = _______
c = _______
t
u
Who Am I?
• I am odd.
• My tens digit is one less than
my units digit.
• None of my digits is a perfect square.
• I am greater than 40.
11
t
u
Who Am I?
• My digits are both odd.
• My tens digit is a perfect
square.
• My units digit is two more than my tens
digit.
Unit 2: Geography of the Number Line
Show the numbers on the number line and find the distance between them.
12
-10 and 110
13
-20 and -70
I
0
Clue: I am greater than or equal to 15.
My name is h.
Where am I? Use symbols:
Show everywhere I could be.
I
15
I
0
I
I
I
I
I
60
Clue: I am less than -2.
My name is p.
Where am I? Use symbols:
Show everywhere I could be.
I
I
Clue: I am less than or equal to 40.
My name is y.
Where am I? Use symbols:
Show everywhere I could be.
I
17
I
5
16
I
I
m
ea
I
I
1
I
I
3
I?
Wh
er
14
I
0
I
80
Clue: I am 80 units away from -2.
Where am I?
18
Mark 0
Clue: I am 30 units away from -31.
Where am I?
I
-2
19
Plot the numbers. Be sure to label them.
Some of them end up in the same place.
I
I
I
I
m–6
I
I
I
l
m–6
I
m–2
I
I
m
m + -5
I
m – -3
m+6
I
I
I
m–
I
8
m+3
I
I
m+
I
-1
I
Use the number line to find the value of the variable.
20
If k + 10 = 2, what is k?
I
k
Unit Assessment
I
k + 10
21
If a – 3 = -14, what is a?
I
I
a
T75
Unit Assessment
Name: Answer Key
Finish labeling the marks on these lines in a consistent way.
I
1
3
5
-3
I
0
I
I
6
I
I
-16
-12
I
I
I
I
I
-55
-50
-45
-40
-35
3
I
-8
I
2
9
I
I
0
-4
I
-18
I
-16
I
I
0
4
-20
I
-30
I
I
I
I
I
-25
-20
-15
-10
-5
I
I
I
-14
-12
-10
I
I
20
40
I
60
I
0
I
I
5
10
I
15
Use the clues to mark one possible location each for A, B, and C so they are in the correct order.
6
Clue 1: A > B
Clue 2: B > C
Clue 3: C is positive.
I I
0 C
7
Clue 1: A > B
Clue 2: C < B
Clue 3: C < A
I
I
I
I
I
B
A
C
B
A
Find the weight of the variable so that the mobile balances.
8
10
T76
9
48
n
100
98
77
49
c
38
48 + 77 = n + 49
100 + c = 98 + 38
76
n = _______
36
c = _______
t
u
Who Am I?
• I am odd.
6 7
• My tens digit is one less than
my units digit.
• None of my digits is a perfect square.
• I am greater than 40.
11
t
u
Who Am I?
• My digits are both odd.
1 3
• My tens digit is a perfect
square.
• My units digit is two more than my tens
digit.
Unit 2: Geography of the Number Line
Show the numbers on the number line and find the distance between them.
12
-10 and 110
13
120
I
I
I
-10 0
15
I
0
I
5
I
10
I
I
16
I
15
40
I
60
50
I
I
m–8
I
I
70
I
1
0
p ≤ -2
I
I
3
2
I?
m
ea
18
Mark 0
Clue: I am 30 units away from -31.
Where am I?
I
78
I
I
-1
I
m+-5
m–6
I
-2
Plot the numbers. Be sure to label them.
Some of them end up in the same place.
I
Clue: I am less than -2.
My name is p.
Where am I? Use symbols:
Show everywhere I could be.
I
80
I I
-2 0
-82
I
0
-20
I
Clue: I am 80 units away from -2.
Where am I?
I
19
I
I
-70
Clue: I am less than or equal to 40.
My name is y.
Where am I? Use symbols:
y ≤ 40
Show everywhere I could be.
I
17
110
Clue: I am greater than or equal to 15.
My name is h.
Where am I? Use symbols:
h ≥ 15
Show everywhere I could be.
I
I
50
Wh
er
14
-20 and -70
I
I
I
-61
l
m–6
I
m–2
m+-1
m–2
I
I
m
m + -5
I
-31
m – -3
m+6
I
I
m–-3
I
m+3
I
I
-1
m–
I
8
0
m+3
I
m+6
I
m+
I
-1
I
Use the number line to find the value of the variable.
20
If k + 10 = 2, what is k?
-8
I
k
Unit Assessment
2
I
k + 10
21
If a – 3 = -14, what is a?
-14
I
a – 3
-11
I
a
T77
Transition to Algebra
40
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.