Download Unit 2 The Number System: Adding and Subtracting Integers and

Unit 2 The Number System:
Adding and Subtracting Integers
and Fractions
Introduction
In this unit, students will add, subtract, and order positive and negative integers and fractions,
including mixed numbers. Students will also multiply integers and fractions by whole numbers,
using repeated addition.
Materials. We recommend that students always work in grid paper notebooks. Paper with 1/4inch grids works well in most lessons. Grid paper is very helpful when comparing multi-digit
numbers, drawing number lines, and performing operations with multi-digit numbers and using
standard algorithms. If students do not use grid paper notebooks in general, you will need to
have lots of grid paper on hand throughout the unit. If students who have difficulties in visual
organization will be working without grid paper, they should be taught to draw a grid before
starting to work on a problem.
Number line notation. Arrows are sometimes drawn on number lines to show that the number
line can continue in the direction indicated. In the lesson plans, when we ask teachers and
students to sketch number lines for the purpose of solving a problem, we recommend using the
arrows only when the mathematical information they convey is relevant to solving the problem.
That is never the case in this unit, since the problems students are asked to solve involve only
comparing and ordering or adding and subtracting numbers. Including only the relevant
information when drawing diagrams to solve problems is an important strategy used throughout
mathematics. Students should not be encouraged to show arrows on number lines in these
situations, but their work should not be marked as incorrect when they do.
In the AP Books, however, we follow the standard practice of showing the arrows in both
directions.
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-1
NS7-1
Integers
Pages 27–29
Standards: preparation for 7.NS.A.1
Goals:
Students will compare and order integers, including multi-digit integers.
Prior Knowledge Required:
Knows that greater numbers are more to the right on a number line
Understands that the value of each digit in a number is determined by its place
Vocabulary: integer, negative, opposite integer, positive, whole number
Materials:
BLM Number Lines from −6 to 6 (p. C-85)
Introduce integers. Draw the number line below leaving out the negative numbers. Point out
that if you start at 0, the next whole number up is 1, then 2, and so on. SAY: You can also start
at 1 on the other side of 0, but now the 1 is negative and has a minus sign in front. Mark −1
and −2, and have volunteers mark all the remaining numbers to −7.
Tell students that an integer is any number that is a positive whole number, zero, or a negative
whole number.
Ordering integers. Draw students’ attention to the number line above. ASK: Which number is
greater, 3 or 4? (4) How does the number line show this? (4 is to the right of 3) Which number is
greater, 2 or −5? (2) How can you tell? (because 2 appears to the right of −5 on the number
line). Have volunteers put the correct sign (< or >) between pairs of numbers:
1
−2
−3
0
−4
−5
(1 > −2, −3 < 0, and −4 > −5) Remind students that the bigger (wider) end of the sign always
faces the bigger (greater) number. Write on the board:
3 > 1 and 1 < 3
Exercises: Use a number line. Write < or >.
a) −5 6
b) 5
−6
c) −1
Answers: a) <, b) >, c) >, d) <
C-2
−4
d) −3
−2
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
Tell students that when they want to order many numbers, number lines make it easy. Write on
the board:
Order: −5, 4, −6, 7, −2, 0
Mark −5 with a dot on the number line and have a volunteer mark the rest of the numbers. Ask
another volunteer to order the numbers from least to greatest, using the number line. ASK: Why
was that so easy to do? (because they are in order from least to greatest when they are placed
from left to right on a number line) Give each student BLM Number Lines from −6 to 6.
Exercises: Use the number line to order each set of numbers from least to greatest.
a) 2, 3, −6
b) 6, −1, 1, 4, −3
Bonus: 5, 0, −3, 3, 6, −5
Answers: a) −6 < 2 < 3, b) −3 < −1 < 1 < 4 < 6, Bonus: −5 < −3 < 0 < 3 < 5 < 6
Introduce opposite integers. Tell students that two integers are called opposite integers when
they are the same distance from 0, but in opposite directions. Show students a number line from
BLM Number Lines from −6 to 6. Fold the number line through 0, with the outside showing
and demonstrate putting a hole through 5. Turn the number line around and point out that the
hole is also passing through −5. SAY: That’s because 5 and −5 are opposite integers—they are
the same distance from 0. ASK: What is the opposite of 5? (−5) What is the opposite of −3? (3)
Write 3 and −3 on the board. ASK: What is the same about how we write opposite integers?
(they have the same number part) What is different about how we write them? (whether the
minus sign is there)
Exercises: What is the opposite integer?
a) 8
b) −9
c) 10
Bonus: 23,809
Answers: a) −8, b) 9, c) −10, d) 7, Bonus: −23,809
d) −7
The special case of 0. ASK: How far from 0 is 0? (0 units) SAY: So the opposite of 0 must also
be 0 units away from 0. ASK: So what is the opposite of 0? (0) Point out that 0 is the only
number that is the same as its opposite.
Tell students that people sometimes write positive numbers with a “+” sign in front. Then you
can say that opposite numbers have the same number part, but opposite signs.
(MP.8) Comparing two negative integers by comparing their positive opposites first.
ASK: By looking at a number line, how can you tell whether one integer is greater than another?
(if it is to the right, it is greater) Draw on the board a number line from −6 to 6 (or have students
refer to the BLM).
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-3
Exercises: Use the number line to copy and complete the sentences.
a) 3 is to the ________ of 4, so 3 is _________ than 4.
−3 is to the _________ of −4, so −3 is ___________ than −4.
b) +5 is to the ________ of +2, so +5 is _________ than +2.
−5 is to the _________ of −2, so −5 is ___________ than −2.
Answers: a) left, less; right, greater; b) right, greater; left, less
SAY: If you know how two numbers compare, their opposites compare the opposite way.
Exercises: Write < or > in the box.
a) 4 7, so −4 −7.
b) +8 +6, so −8 −6.
c) +13 +9, so −13 −9.
Bonus: 5,000 8,000, so −5,000
Answers: a) <, >; b) >, <; c) >, <; Bonus: <, >
−8,000.
Comparing multi-digit numbers. SAY: To compare two multi-digit numbers, line up their place
values. Start by comparing the largest place values first. Write on the board:
2,134
38,476
35,402
2,173
35,391
6,807
Point out that you had to line up the digits from the right, so that the ones digits are aligned.
SAY: If you make sure the ones digits are lined up, and all the commas are lined up, then all the
digits in the same place value should be lined up. Then for the first pair, ASK: Are the thousands
digits the same? (yes) Are the hundreds digits the same? (yes) The tens digits? (no) Circle the
tens digits in both numbers and SAY: 2,173 has more tens, so 2,173 is greater than 2,134. It
doesn’t matter which number has more ones—the number with the greater tens digit is greater
overall if all the larger place values are the same.
Repeat for the next pair, but start by comparing the ten thousands digits (this time, the
thousands place is the first place they differ, so 38,476 > 35,391), then again for the next pair
(this time, they differ in the ten thousands place because 6,807 doesn’t have any ten thousands,
so 35,402 > 6,807):
2,134
38,476
35,402
2,173
35,391
6,807
Exercises: Copy the numbers onto grid paper and line up the place values.
Which number is greater?
a) 632 or 641
b) 2,741 or 850
c) 532,417 or 276,329
d) 912,401 or 91,401
e) 62,713 or 68,290
f) 8,322,407 or 8,322,415
Answers: a) 641; b) 2,741; c) 532,417; d) 912,401; e) 68,290; f) 8,322,415
C-4
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
(MP.3, MP.8) Comparing multi-digit negative integers. SAY: Remember: If you can compare
positive numbers, you can compare negative numbers, too. Write on the board:
1 < 4 so −1 > −4
SAY: Because 1 is less than 4, you know that −1 > −4, so just flip the sign.
Exercises: Write < or >.
a) +324 +351 so −324 −351
c) 93,412 86,270 so −93,412 −86,270
Answers: a) <, >; b) >, <; c) >, <; d) >, <
b) +516
d) 5,080
+89 so −516
−89
5,008 so −5,080 −5,008
SAY: Now imagine the opposite positive integers to decide how these negative integers
compare.
Exercises: Write < or >.
a) −4,312
−11,217
c) −3,222,814 −3,222,726
Answers: a) >; b) <; c) <; d) >
b) −45,009
−43,782
d) −4,321 −4,321,000
Comparing positive numbers to negative numbers. Write on the board:
2
−3
ASK: Which integer is greater? (2) How do you know? (because it is positive and the other
integer is negative; because it would be more to the right on a number line) Point out that 2 is
greater than 0, and −3 is less than 0, so 2 is greater than −3. Repeat with 742 and −5,603. Then
SAY: When two numbers have different signs, the positive number is always greater, so you
don’t even have to look at the number parts.
Exercises: Write < or >.
a) 23 −517
b) −62 +417
Answers: a) >, b) <, c) >, d) <
c) +3,416
−23,416
d) −278,914
6,341
SAY: For these questions, you might have to look at the number part.
Exercises: Write < or >.
a) 34 27
b) −34
27
c) 34
−27
e) 34,217 −819
f) −34,217 819
g) −348,921
Answers: a) >, b) <, c) >, d) <, e) >, f) <, g) >
−356,821
d) −34
−27
SAY: For these, if you need to compare the number part, remember that fractions with the same
denominator are easy to compare, and fractions with different denominators can be changed to
have the same denominator.
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-5
Exercises: Write < or >.
3 2
3
1
b) 
a) 

4
4
5 5
Answers: a) <, b) <, c) >, d) <
c)
1
2

2
5
d) 
1
2

2
5
Extensions
(MP.6) 1. What symbol can be put in the box to make the statement true?
34,207 >
110,642
Answer: a negative sign (−) or a decimal point
(MP.1, MP.3) 2. Try to put a single digit in each box to make the statement true. Which one is
not possible?
a) 5 < 4
b) 8 7 < 04
c) 9 7 < 04
Sample answers: a) 52 < 54, 59 < 64; b) 837 < 904, c) not possible, because the first number
is at least 907, and the second number is at most 904.
3. Put the same digit in each box.
a) 8 < 5
b)
37 < 2
Answers: a) 9, b) 1, c) 3
5
c)
95 < 42
<4
1
4. (MP.7, MP.1) a) How many 6-digit positive integers are greater than 999,985?
b) How many 6-digit negative integers are less than −999,985?
Answers: a) 999,999 − 999,985 = 14, b) 14, because the 6-digit negative integers that are less
than −999,985 are exactly the opposites of the 6-digit positive integers that are greater than
999,985
C-6
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
NS7-2
Integers in the Real World
Pages 30–31
Standards: 7.NS.A.1a
Goals:
Students will understand that integers can describe quantities in opposite directions from a
chosen point or quantities that can cancel each other out.
Students will relate integer comparisons to the real-world contexts.
Prior Knowledge Required:
Knows that an integer is a positive whole number, 0, or a negative whole number
Can compare and order integers
Vocabulary: cancel, credit, debit, gain, integer, loss, negative, opposite direction, opposite
integer, opposite value, positive
Using integers to describe quantities in opposite directions from a chosen point. Tell
students that integers are really convenient for describing quantities in opposite directions from
a chosen point. SAY: When talking about how high up a location is, locations above sea level
are positive and locations below sea level are negative. Sea level is considered to be exactly 0.
Write on the board:
3 m above sea level is +3 m
3 m below sea level is −3 m
Exercises: Write an integer to represent each quantity. Include the unit.
a) A location 7 m above than sea level
b) A location 5 ft below sea level
c) A location 4 mi below sea level
d) A location 3 km above sea level
Answers: a) +7 m, b) −5 ft, c) −4 mi, d) +3 km
SAY: Temperatures warmer than 0°F are positive and temperatures colder than 0°F are
negative.
Exercises: Write an integer to represent each quantity. Include the unit.
a) a temperature 12 degrees warmer than 0°F
b) a temperature 8 degrees colder than 0°F
c) a temperature 9 degrees colder than 0°F
Answers: a) +12°F, b) −8°F, c) −9°F
Relating integer comparisons to the temperatures they represent. ASK: Do greater integers
represent warmer or colder temperatures? (warmer) ASK: Which temperature is warmer, +40°F
or −50°F? (+40°F) Which integer is greater, +40 or −50? (+40)
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-7
Exercises: Write “warmer” or “colder.” Then write > or <.
a) 3°F is ______________ than −2°F and 3
−2.
b) −8°F is ______________ than −5°F and −8
−5.
c) −3°F is ______________ than −7°F and −3
−7.
Answers: a) warmer, >; b) colder, <; c) warmer, >
(MP.4) Integers and time zones. Tell students that time zones are also represented using
integers, with the time in London, England, considered to be 0. Ask students if they have ever
wanted to watch a live event on television but couldn’t because it was airing in the middle of the
night. Point out that in some parts of the world, it is the middle of the night right now! If you have
friends or relatives in other parts of the world, you have to be careful about what time you call
them. Tell students which time zone they are in (e.g., −5, −6, or −7) and explain that the minus
sign tells them they are behind London’s time, and the number tells them by how many hours.
Exercises: Write the integer to describe the time zone. Include the unit.
a) four hours ahead of London
b) two hours behind London
c) three hours behind London
d) one hour ahead of London
Answers: a) +4, b) −2, c) −3, d) +1
(MP.6) SAY: Integers are used in many real-world contexts. It is important to include the units,
because many things are measured using different units. Write on the board:
Distance above/below sea level:
+8 inches
+8 miles
SAY: Both of these locations can be represented by the integer +8, but they are very different
locations. ASK: Which one is higher up? (+8 miles) SAY: 8 miles above sea level is much higher
than 8 inches above sea level.
ASK: What units can temperature be measured in? (degrees Fahrenheit or degrees Celsius) If
students haven’t heard about Celsius, point out that it is a temperature scale used in many parts
of the world.
Exercises: Write an integer to represent each quantity. Include the unit.
a) a temperature 8 degrees warmer than 0°F
b) a temperature 5 degrees colder than 0°C
c) a location 6 m below sea level
d) a location 700 ft above sea level
e) a time zone 3 hours ahead of London
f) a time zone 7 hours behind London
Answers: a) +8°F, b) −5°C, c) −6 m, d) +700 ft, e) +3 hours, f) −7 hours
Relating integer comparisons to the real-world situation they represent. Write on the board:
−5°F
C-8
+2°F
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
ASK: Which temperature is warmer? (+2°F) Repeat for −5°C and +2°C. (+2°C is warmer
than −5°C) SAY: You don’t even have to know how warm or cold the temperatures are or
anything about the units being used. If the same units are used for both temperatures, you can
tell which one is warmer. SAY: You can make comparisons using the integer comparisons in
many real-world contexts.
Exercises: Which location is higher up?
a) +5 feet or −4 feet
b) −3 meters or −8 meters
c) −6 inches or −2 inches
Bonus: −5 feet or −2 yards (Hint: Be careful about the units.)
Answers: a) +5 feet, b) −3 meters, c) −2 inches, Bonus: −5 feet
Exercises: Which time zone is ahead?
a) A. +6 hours
b) A. −4 hours
c) A. −3 hours
B. +7 hours
B. −5 hours
B. +1 hour
Bonus: For each part above, if you live in Time Zone A and want to visit Time Zone B, do you
need to move your clock ahead or back?
Answers: a) B, b) A, c) B, Bonus: a) ahead, b) behind, c) ahead
Using integers to describe opposite values that cancel each other out. Tell students that
integers are also used to describe opposite values in the sense that positive values cancel out
negative values. Write on the board:
+$3 = a gain of $3
−$3 = a loss of $3
ASK: If you gained $3 then lost $3, what can you say about your situation? (it’s the same as if I
didn’t gain or lose anything) Point out that the two actions cancel each other out, so they are
opposite actions. SAY: This happens when you gain or lose yards in football, too, or in any sport
where you get a point and the other team gets a point—the result brings neither team closer to
winning, so they cancel each other out. Write on the board:
+1 = my team gets a point
−1 = the other team gets a point
(MP.6) Exercises: Write an integer to represent each action. Include the unit.
a) a gain of 3 yards in football
b) a loss of 25¢
c) a loss of 8 yards in football
d) a gain of $18
e) your team gets 3 points
f) the other team gets 4 points
Answers: a) +3 yards, b) −25¢, c) −8 yards, d) +$18, e) +3 points, f) −4 points
SAY: A bank statement shows a credit when you add money to your account and a debit when
you take money out of your account. Write on the board:
credit = +
debit = −
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-9
Exercises: Write the integer that represents the action. Include the unit.
a) a debit of $5
b) a credit of $3
c) a credit of $6
d) a debit of 85¢
Answers: a) −$5, b) +$3, c) +$6, d) −85¢
Explain that a negative balance in a bank account means that you owe the bank money. Then
tell students that four people have different bank account balances. Write on the board:
Anne +$4
Bob +$7
Cathy −$3
Dan −$5
Have volunteers show the locations of each balance on the number line. ASK: Whose bank
account balance is least? (Dan) Whose is greatest? (Bob) Whose balance would you most like
to have? (Bob’s) Whose would you least like to have? (Dan’s)
Exercises: What is better in the real-world context?
a) bank account balances: +$4 or −$5
b) gains and losses: −$2 or −$1
c) gains and losses in football: −2 yards or −5 yards
d) points for and against: +6 or +7
e) points for and against: +2 or −5
Answers: a) +$4, b) −$1, c) −2 yards, d) +7 points, e) +2 points
Extensions
1. Project idea. Investigate the history and current use of temperature scales. Examples of
questions students might ask:
● What are the three main scales used today?
● How did human body temperature play a role in the Fahrenheit scale?
● How were 0°C and 100°C chosen?
● How was 0 kelvin chosen?
● How was the value of each degree chosen in each scale?
● In which order were the three scales invented?
● Were temperatures warmer than 0 ever chosen to be negative?
(MP.4) 2. Tell students that a golfer’s score is recorded in integers. Par is the number of shots,
or strokes, that a golfer should aim for to get the ball in each hole. Scores are determined by
how they compare to par. For example, someone who took 7 fewer shots than par to complete 9
holes would have their score recorded as −7. Someone who took 7 more shots than par would
have their score recorded as +7. In golf, a lower score (“below par”) is better because it means
fewer shots were needed.
a) Kami needed 3 more shots than par. How would she record her score?
b) Roy’s score is +2 and Sharon’s score is −1. Whose score is better?
c) Jon’s score is −3 and Eva’s score is −5. Whose score is better?
Answers: a) +3, b) Sharon’s, c) Eva’s
C-10
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
3. A temperature increases from −2°F to +5°F. How much did it increase by? Sketch a number
line to show your answer.
Answer: 7°F
4. The temperature was +3°F. The temperature decreased 5°, then increased 4°, then
decreased 2°. What was the final temperature? Sketch a number line to show your answer.
Answer: 0°F
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-11
NS7-3
Adding Gains and Losses
Pages 32–33
Standards: 7.NS.A.1d
Goals:
Students will add gains and losses to determine an overall outcome.
Students will cancel amounts that can be canceled before adding.
Prior Knowledge Required:
Can represent gains and losses using integers
Vocabulary: cancel, gain, loss
Deciding whether a gain and a loss represent an overall gain or loss. Tell students that
when playing games such as Monopoly, they can win and lose money. SAY: You had a good
day if you gained more than you lost, and a bad day if you lost more than you gained. Write on
the board:
gain = +
loss = −
Exercises: Was it a good day or a bad day?
a) + 3 − 5
b) − 4 + 2
c) − 8 + 12
d) + 6 − 3
Answers: a) bad, b) bad, c) good, d) good, e) bad
e) − 5 + 4
Review how to use integers to represent gains and losses. Tell students that you will use +1
to represent an overall gain of $1 and −1 to represent an overall loss of $1. Write various
integers (e.g., +3, −4, −2, +8) on the board and have volunteers tell you what each represents.
(a gain of $3, a loss of $4, a loss of $2, and a gain of $8)
Exercises: Write the integer to represent the gain or loss.
a) a gain of $6
b) a loss of $7
c) a gain of $12
d) a loss of $11
Answers: a) +6, b) −7, c) +12, d) −11
Determining the overall result of a gain and a loss. Write on the board:
+ 7 − 4 = ____
− 4 + 4 = ____
Referring to the first example, ASK: Was more gained or lost? (more was gained than lost) How
much more? ($3) Write “+3” in the blank. SAY: So, overall, $3 was gained. Referring to the
second example, ASK: Was more gained or lost? (the same was gained as lost) Overall, what
was the result? (nothing was gained or lost overall) Write “0” in the blank.
C-12
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
Exercises: Write an integer to show how much was gained or lost overall.
a) + 7 − 8
b) − 5 + 8
c) − 6 + 2
d) + 3 − 1
Answers: a) −1, b) +3, c) −4, d) +2
Determining the overall result of two gains or two losses. Write on the board:
− 5 − 3 = _____
+ 2 + 2 = _____
Referring to the first example, SAY: This time there are two losses. How much was lost overall?
($8) Write “−8” in the blank. Referring to the second example, SAY: This time there are two
gains. ASK: How much was gained overall? ($4) Write “+4” in the blank.
Exercises: Write an integer to show how much was gained or lost overall.
a) − 2 − 5
b) + 7 + 1
c) − 3 – 3
d) + 4 + 5
e) + 6 − 3
f) − 7 + 7
Answers: a) −7, b) +8, c) −6, d) +9, e) +3, f) 0
Adding more than two gains and losses. Write on the board:
+3−4−5
SAY: Now there was a gain, then two losses. Cover the “− 5” and ASK: What do the first two
actions result in? (−1) Write on the board:
+ 3 − 4 − 5 = − 1 − 5 = ______
ASK: What is the overall gain or loss? (a loss of $6) How do I write that as an integer? (−6) Fill
in the blank.
An easier way to add more than two gains and losses. Tell students that there is another
way to add more than two gains and losses. Write on the board:
+3−4−5=
SAY: You can add all the gains, then all the losses. There was one gain of 3 (circle it) and there
were two losses. SAY: Circling the gains makes it easier to spot the gains and the losses. ASK:
What did the losses total? (9) Write on the board:
+ 3 − 4 − 5 = + 3 − 9 = ______
ASK: What was the overall gain or loss? (a loss of $6) Write “−6” in the blank. ASK: Was that
the same answer we got the other way? (yes) Write on the board:
+2−5−4+8−3
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-13
ASK: What was the total of all the gains? ($10) What was the total of all the losses? ($12) Write
on the board:
+ 10 − 12 = _______
ASK: What was the overall result? (a loss of $2)
Exercises: Add the gains and losses.
a) − 2 + 5 − 1
b) + 3 − 4 + 5
c) + 7 − 5 − 6
d) − 3 + 5 + 1
Answers: a) + 5 − 3 = +2, b) + 8 − 4 = +4, c) + 7 − 11 = −4, d) + 6 − 3 = +3
(MP.1) Bonus: Do the same problems but, this time, add the first two numbers, then add the
third number. Make sure you get the same answers.
Exercises: How much was gained or lost overall? Hint: Circle the gains.
a) − 5 + 4 + 3 + 1 − 2 − 6
b) − 1 − 2 − 3 − 4
c) − 3 − 2 + 6 + 5 + 1 − 7 − 2
d) + 3 + 4 − 1 − 1 − 2
Bonus: Find an easy way to add: − 1 + 2 − 3 + 4 − 5 + 6 − 7 + 8 − 9 + 10 = ______
Answers: a) + 8 − 13 = −5, b) −10, c) + 12 − 14 = −2, d) + 7 − 4 = +3, Bonus: +5 because each
of the five pairs (− 1 + 2, − 3 + 4, − 5 + 6, − 7 + 8, and − 9 + 10) add to +1
Adding integers. Tell students that by adding gains and losses, they’ve been adding integers.
Write on the board:
Add +3 and −4
+ 3 − 4 = _____
SAY: Positive three can represent a gain of $3 and negative four can represent a loss of $4.
ASK: What is the result of both? (a loss of $1) Write “−1” in the blank. SAY: You can think of
adding +3 and −4 as gaining 3 and losing 4, or adding 3 and subtracting 4.
Canceling before adding. Write on the board:
− 3 + 6 + 3 = + 9 − 3 = +6
SAY: There was $9 gained and $3 lost, so $6 gained overall. But there is an easier way to do
this problem. Write on the board:
− 3 + 3 + 6 = ______
SAY: Instead of adding all the gains first, you can add the ones that cancel first. Because I know
that gaining $3 cancels out losing $3, I can cancel them out:
− 3 + 3 + 6 = ______
SAY: If I lost $3, gained $3, then gained 6 more dollars, then I know that overall I gained $6
because I don’t even have to count the gaining and losing of $3. ASK: Do I get the same answer
C-14
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
as before? (yes) Why does that make sense? (because it doesn’t matter what order you have
your gains and losses in, gaining and losing the same amounts will give the same result)
Exercises: Cancel as much as you can before adding.
a) − 9 + 9 − 4
b) + 7 + 3 − 7
c) − 8 + 2 − 2
d) + 6 − 4 − 6
e) − 100 + 77 + 100
f) − 15 + 43 − 43
Bonus: − 5 + 3 + 2 − 3 − 9 + 5
Answers: a) −4, b) +3, c) −8, d) −4, e) +77, f) −15, Bonus: + 2 − 9 = −7
Canceling positive and negative pairs that add to 10. Write on the board:
+7−5+2−5+3
Tell students that pairs that add to 10 are easy to find. ASK: Do you see two gains that add to a
gain of $10? (+ 7 and + 3) Circle them. Do you see two losses that add to a loss of $10? (− 5
and − 5) Circle them in a different color.
+7−5+2−5+3
ASK: After we cancel the +10 and the −10, what’s left? (+2) SAY: Every time you add +10
and −10, you can cancel them.
Exercises: Cancel pairs that add to +10 with pairs that add to −10. Then add the integers.
a) + 4 + 5 − 1 − 9 + 6
b) − 3 − 5 + 2 − 5 + 8
c) − 7 − 2 + 4 + 5 − 8 + 6
d) + 5 + 6 − 9 − 7 + 5 + 8 − 1 + 4 − 3
SAY: If there is a 10 or −10 in the sum, it automatically adds to 10 or −10.
e) + 8 − 5 − 6 + 7 − 4 + 3 + 10 − 5
Bonus: Cancel pairs that add to +100 with pairs that add to −100. Then add the integers.
f) + 70 − 60 + 30 − 20 − 40
g) − 72 + 35 + 65 + 75 − 28
Answers: a) +5, b) −3, c) −2, d) +8, e) +8, Bonus: f) −20, g) +75
Extensions
(MP.7) 1. Add amounts that almost cancel first.
a) − 100 + 72 + 99
b) − 703 − 704 + 705
Answers: a) − 1 + 72 = +71, b) − 703 + 1 = −702 or + 2 − 704 = −702
(MP.7) 2. Look for shortcut ways to add the gains and losses.
a) − 4 − 5 − 6 + 7 + 8 + 9
b) − 8 + 17 − 4 − 9 + 4
c) + 1 − 3 + 5 − 7 + 9 − 11
Sample solutions: a) − 6 + 7 + 8 = 1 + 8 = 9; b) − 8 − 9 + 17 = 0 and − 4 + 4 = 0, so the total is
0; c) − 2 − 2 − 2 = −6
(MP.7, MP.1) 3. Add.
a) 1 − 2 + 3 − 4 + 5 − 6 + … + 99 − 100
b) 1 − 2 + 3 − 4 + 5 − 6 + … + 999
Answers: a) −50, b) +500
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-15
NS7-4
Opposite Actions and Opposite Integers
Pages 34–35
Standards: 7.NS.A.1a, 7.NS.A.1b
Goals:
Students will describe situations in which opposite quantities combine to make 0.
Students will use pictures to add integers and will understand that an integer and its opposite
add to 0.
Prior Knowledge Required:
Knows that integers can be used to describe values that can cancel each other
Can add gains and losses
Materials:
10 two-color counters (see Extension)
Vocabulary: bank account, bank balance, bank statement, cancel, credit, debit, electric
charge, electron, gain, loss, opposite actions, opposite integers, proton
Introduce opposite actions. Tell students that actions that cancel each other out are called
opposite actions. Then SAY: If you gain any amount of money and lose the same amount of
money, the result is as though nothing happened, so gaining and losing the same amount of
money are opposite actions.
Exercises: Write the integers that represent the amounts that cancel.
a) a gain of $4 and a loss of $4
b) a loss of $17 and a gain of $17
_____
and ______
_____ and ______
c) a loss of $6 and a gain of $6
d) a gain of $9 and a loss of $9
_____
and ______
_____ and ______
Answers: a) +4 and −4, b) −17 and +17, c) −6 and +6, d) +9 and −9
Exercises: Write the amount that cancels the given amount. Then write both amounts as
integers.
a) a loss of $6 cancels a _________________________
_______ and ________
b) a gain of 7 yards cancels a _____________________
_______ and ________
Answers: a) gain of $6, −6 and +6; b) loss of 7 yards, +7 and −7
ASK: What can you say about integers that represent amounts that cancel? (they are opposite
integers)
C-16
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
Using integers to describe positive and negative electric charges. ASK: What do you use
electricity for? (sample answers: turning on lights, watching television, and so on) Tell students
that protons and electrons are what create electricity. Protons have a positive electric charge
and electrons have a negative electric charge. Tell students that a proton has a charge of +1
and an electron has a charge of −1. Write on the board:
proton = +1
electron = −1
NOTE: Protons are located in an atom’s nucleus, or center, and electrons orbit around that
center.
Exercises: What is the electric charge?
a) 3 protons
b) 2 electrons
c) 4 protons
d) 5 electrons
Bonus: 1,700 electrons
Answers: a) +3, b) −2, c) +4, d) −5, Bonus: −1,700
Representing protons and electrons with pictures. Show students how you will represent a
proton and an electron.
proton:
electron:
Tell students that protons and electrons actually look nothing like what you are drawing—they
are just symbols to represent what they mean, not what they look like.
Exercises: What is the electric charge?
a)
b)
c)
d)
Answers: a) +2, b) −3, c) +6, d) −4
Electrons and protons cancel each other out. Tell students that protons and electrons cancel
each other out when they are near each other. Draw two protons on the board. SAY: This object
has a charge of +2. Then draw one electron with the two protons. SAY: The electron cancels
out one of the protons. It’s as though neither of them is there. Show their removal as below.
ASK: Now what is the electric charge? (+1) Point out that because there are more protons than
electrons, the electric charge is positive.
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-17
Exercises: Circle the amounts that cancel to find the total electric charge.
a) + 4 − 2 = _____
b) + 3 − 4 = _____
c) + 4 − 4 = _____
d) − 5 + 2 = _____
Now draw the picture yourself.
e) + 3 − 6
f) − 4 + 5
g) − 2 + 3
Answers: a) +2, b) −1, c) 0, d) −3, e) −3, f) +1, g) +1, h) −3
h) + 1 − 4
Point out that when there are more electrons than protons, the charge is negative. Then SAY:
Protons and electrons have opposite electric charges that cancel each other out. That’s why it is
convenient to use integers to represent them.
SAY: You can also add two negative charges and get a stronger negative charge, and you can
add two positive charges and get a stronger positive charge. Write on the board:
−2−1
So − 2 − 1 = −3.
Exercises: Draw a picture to find the total electric charge.
a) − 5 − 3
b) + 3 + 2
c) − 2 − 2
Answers: a) −8, b) +5, c) −4, d) +6
d) + 5 + 1
SAY: What you’ve just done is added integers. This time, instead of thinking of the integers as
representing gains and losses, you’ve thought of them as representing positive and negative
charges. Write on the board:
− 5 − 3 = −8, so (−5) + (−3) = −8
Using pictures to represent debits and credits. Tell students that you can show debits and
credits the same way you show protons and electrons. Write on the board:
________________ =
_________________ =
ASK: Does anyone remember which one adds money to your account, a debit or a credit?
(credit) Write in the first blank “credit of $1” and in the second blank “debit of $1.”
Exercises: Draw a picture to find the resulting bank balance.
a) a credit of $4 and a debit of $1
b) a debit of $3 and a credit of $7
c) a credit of $3 and a debit of $5
d) a debit of $2 and a credit of $2
Answers: a) $3, b) $4, c) −$2, d) $0
Amounts that cancel add to 0. ASK: Which balance ended up as 0? (part d) Why was that the
case? (because the credit and debit were the same dollar amounts) SAY: A credit of any
amount cancels out a debit of the same amount.
C-18
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
Write on the board:
A credit of $5 followed by a debit of $5 results in no change.
(+5)
+
(−5)
=
0
SAY: We can show the situation with an addition of integers. Have a volunteer do the same
thing for the following situation:
A loss of $3 followed by a gain of $3 results in no gain or loss.
(the integer addition is (−3) + (+3) = 0)
Exercises: Write an integer addition to show the situation.
a) Ken added $5 to his bank account, then took out $5 from his bank account. The balance was
the same as when he started.
b) Amy’s football team lost 6 yards, then gained 6 yards. The team ended up where it started.
c) Ivan walked 8 steps north (+), then 8 steps south (−). He ended up where he started.
d) Liz’s team scored 5 points, then the other team scored 5 points. They ended up tied.
Answers: a) (+5) + (−5) = 0, b) (−6) + (+6) = 0, c) (+8) + (−8) = 0, d) (+5) + (−5) = 0
Point out that amounts that cancel add to 0.
Exercises:
1. Write the missing integer.
a) (+3) + ____ = 0
b) (−5) + _____ = 0 c) (+7) + _____ = 0
Answers: a) (−3), b) (+5), c) (−7), d) (+794)
2. Write the missing sign (+ or −) in the box.
a) + 5
5=0
b) − 9
9=0
d)
2−2=0
e) (+7) + ( 7) = 0
Answers: a) −, b) +, c) −, d) +, e) −, f) +
c)
f) (
d) (−794) + _____ = 0
4+4=0
6) + (−6) = 0
Extensions
(MP.1, MP.3) a) Take 10 two-color counters and toss them. The counters that land on yellow
represent protons and the counters that land on red represent electrons. Determine the resulting
electric charge by pairing up protons with electrons and seeing what’s left. Record the results of
various tosses. Alternatively, instead of tossing counters, students can select any two integers
whose positive parts add to 10, draw them as charges, then find the resulting electric charge.
b) Investigate the following question: Can the resulting electric charge ever be an odd number?
Explain. (You can tell students that an integer is odd if the number part without the plus or minus
sign is odd.)
Answer: b) No, because you always pair up the counters, red with yellow. So what’s not paired
up of one color is 10 minus an even number, which always leaves an even number of counters.
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-19
NS7-5
Adding Integers on a Number Line
Pages 36–37
Standards: 7.NS.A.1b
Goals:
Students will add integers using a number line.
Students will interpret addition of integers when the integers are written in brackets.
Prior Knowledge Required:
Can represent integers on a number line
Can add whole numbers on a number line
Vocabulary: integer, negative, positive
Materials:
pre-cut arrows (made from Bristol board, see below for details)
BLM Number Lines from −6 to 6, 2 copies for each student (p. C-85)
Preparation. Using Bristol board, cut out eight arrows, one of each length, in 6-inch intervals: 6
inches, 12 inches, 18 inches, and 24 inches. Label the arrows, front and back, with the positive
integers from 1 to 4 showing arrows pointing right and the negative integers −1 to −4 showing
arrows pointing left. Keep them for a later class (NS7-11: Adding and Subtracting Fractions with
the Same Denominator) as well.
1
–1
2
3
4
–2
–3
–4
Using arrows to represent integers on number lines. Draw on the board a number line
from −4 to 4 with the numbers 6 inches apart, and show students how to represent 3 by starting
the “3” arrow at 0.
3
−4
−3
−2
−1
0
1
2
3
4
SAY: We can use the arrows starting at 0 to show the number. An arrow 3 units long will point at
3 if it points right. ASK: Where will the same arrow point if it points left? (at −3) Turn the arrow
around to show this (your arrow should have −3 written on the back). Ask volunteers to show
other integers using the arrows: −2, + 4, −1.
C-20
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
Using arrows to add integers on number lines. Write on the board:
(+3) + (−4)
Using the same number line as above, demonstrate how to use arrows to add 3 and −4. SAY:
You have to start at 0 and find where you end up after adding both numbers. So, after adding
the “3” arrow, you start the “−4” arrow where the “3” arrow finishes. Show this on the board:
–4
3
−4
−3
−2
−1
0
1
2
3
4
So (+3) + (−4) = ______
ASK: So what is positive three plus negative four? (negative one) Have volunteers use the
arrows and the same number line to add other pairs of numbers:
(−2) + (+3)
(+1) + (−3)
(−2) + (−1)
(+3) + (+1)
Answers: +1, −2, −3, + 4. Now provide students with BLM Number Lines from −6 to 6 to do
the exercises below (give each student two copies double-sided). Point out that when using a
number line to add, students don’t need to draw the arrows on the number line because the fact
that the number line can continue isn’t needed to solve the problem.
Exercises: Add using a number line.
a) (−4) + (+1)
b) (+3) + (−6)
c) (−4) + (−2)
d) (−3) + (+2)
e) (+2) + (−3)
Bonus: (−5) + (+3) + (−2)
Answers: a) −3, b) −3, c) −6, d) −1, e) −1, Bonus: −4
(MP.1) Tell students that in parts d) and e) they are adding the same two numbers, so they
should get the same answers.
Bonus: Use other pairs of numbers to check that adding the same numbers gets the same
answer, no matter which order you add them in.
SAY: You can also add integers by thinking of them as gains and losses.
(MP.1) Exercise: Do the previous exercises without using a number line. Make sure you get all
the same answers.
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-21
(MP.7) A shortcut way to add on a number line. Tell students that there is a shortcut way to
add on a number line. ASK: What can you do instead of drawing the first arrow? (rather than the
first arrow starting at 0 and ending at the first number, you can just start at the first number and
draw only the arrow for the second number)
Exercises: Use a number line to add. Draw only one arrow.
a) (−3) + (+5)
b) (+4) + (−5)
c) (−5) + (−1)
Bonus: Draw only two arrows to add (−3) + (+7) + (−5).
Answers: a) +2, b) −1, c) −6, Bonus: −1
A different notation is sometimes used for adding integers. Tell students that they can add
integers by writing the integers without brackets, the same as when adding gains and losses.
Write on the board:
(+7) + (−4) = + 7 − 4
SAY: Adding +7 is just like adding a gain of 7, and adding −4 is just like adding a loss of 4. It’s
easy to change the notation to not have brackets. Write on the board:
+(+) = +
+(−) = −
Exercises: Write the addition without brackets.
a) (−3) + (+4)
b) (+5) + (−6)
c) (+5) + (+1)
d) (−3) + (−4)
Answers: a) − 3 + 4, b) + 5 − 6, c) + 5 + 1, d) − 3 − 4
Students will need BLM Number Lines from −6 to 6 for the exercises below.
(MP.1) Exercises: Add the integers in two ways by thinking of them as gains and losses, then
by using a number line. Make sure you get the same answer both times.
a) (−5) + (+3)
b) (+6) + (−3)
c) (−4) + (+6)
d) (−2) + (−3)
Answers: a) −2, b) +3, c) +2, d) −5
Extensions
1. a) Extend the pattern using BLM Number Lines from −6 to 6.
i) +4, +3, +2, _____, _____, ______
ii) +5, +3, +1, _____, _____, ______
iii) −6, −3, 0, _____, ______
Answers: i) +1, 0, −1; ii) −1, −3, −5; iii) +3, +6
C-22
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
b) Use the pattern to add.
i)
(+5) + (+3)
+8
ii)
(+3) + (−7)
(+5) + (+2)
+7
(+2) + (−7)
(+5) + (+1)
+6
(+1) + (−7)
(+5) + (0)
+5
(0) + (−7)
(+5) + (−1)
_____ + (−7)
5 + _ __
_
_ + (−7)
5+_
_
_ + (−7)
_
Answers: i)
(+5) + (+3)
+8
(+5) + (+2)
+7
(+5) + (+1)
+6
(+5) + (0)
+5
(+5) + (−1)
+4
5 + _(−2)__
+3
5 + _(−3)_
+2
ii)
–1
–1
–1
–1
–1
–1
(+3) + (−7)
−4
(+2) + (−7)
−5
(+1) + (−7)
−6
(0) + (−7)
−7
_(−1)_ + (−7)
−8
_(−2)_ + (−7)
−9
_(−3)_ + (−7)
−10
–1
–1
–1
–1
–1
–1
2. Solve the puzzle by placing the same integer in each shape.
a)
+
+
= −6
b)
+
+
= −30
Answers: a) −2, b) −10
(MP.1) 3. In the square, the integers in each row and column and the two diagonals (these
include the center box) add up to +6. Find the missing integers.
–1
−2
+5
Answers:
+3
+4
–1
−2
+2
+6
+5
0
+1
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-23
NS7-6
Using Pictures to Subtract Integers
Pages 38–39
Standards: 7.NS.A.1c, 7.NS.A.1d
Goals:
Students will use pictures to subtract integers.
Prior Knowledge Required:
Can use pictures to represent integers
Vocabulary: positive, negative, integer
Materials:
10 “+” cards and 10 “−” cards
BLM Number Lines from −6 to 6 (p. C-85, see Extension 2)
Preparation. Create 10 cards with a “+” sign in a circle, and 10 cards with a “−” sign in a circle,
that you can tape to the board and remove easily. Tell students that the positive sign in a circle
can represent a gain of $1 or a positive charge. The negative sign in a circle can represent a
loss of $1 or a negative charge.
Using pictures to subtract positives from positives and negatives from negatives. Tape to
the board five “+” cards. ASK: What integer does this represent? (+5) SAY: In the same way you
can add integers by adding positives and negatives, you can subtract integers by taking some
away. Now take away two of the cards and ASK: Now what integer does this represent? (+3)
Write on the board:
(+5) − (+2) = +3
Then repeat with five “−” cards, again taking away two cards. Have a volunteer write the
equation on the board:
(−5) − (−2) = −3
Then draw on the board:
For each picture, ASK: What integer does this picture represent? (the first picture represents +3
and the second picture represents −4) Then show this on the board:
C-24
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
ASK: When I take away +2 from +3, what do I get? (+1) Have a volunteer write the equation
under the picture:
(+3) − (+2) = +1
When I take away −1 from −4, what am I left with? (−3) Have a volunteer write the equation
under the picture:
(−4) − (−1) = −3
SAY: When you subtract a number that has a positive or negative sign in front, you always have
to put brackets around the number you are subtracting so that you don’t have to write the
subtracting symbol right next to the positive or negative sign. But you don’t always have to put
brackets around the first number. Write on the board:
−4 − (−1) = −3
Exercises: Write a subtraction for the picture.
a)
b)
Answers: a) 5 − 3 = 3 or +5 − (+3) = +2, b) −6 − (−2) = −4
Representing an integer with both positives and negatives. Draw on the board:
ASK: What number does this represent? (−3) Then add a positive and a negative as shown
below and ASK: Now what number is represented? (still −3) SAY: The positive and negative
that I added cancel each other (circle them), so we are back where we started.
Exercises: What number is represented? Hint: Circle amounts that cancel. What’s left?
a)
b)
c)
Answers: a) +3, b) +2, c) −3
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-25
Subtracting a negative number from a positive number. Write on the board:
+3 − (−1)
SAY: I want to take away one negative, but there aren’t any negatives to subtract. ASK: What
can I do? Have students work in pairs to try to find a solution. As students work, provide
guidance as needed. For example, you can PROMPT: How can I add a negative and keep the
value represented the same? (add a positive, too) Show doing so on the board. Then SAY: Now
I can take away −1. Cross out the minus sign.
ASK: I started with +3 and took away −1. What’s left? (+4) Now write on the board:
+3 − (−1) = +4
+2 − (−3) =
Ask a volunteer to draw a picture (or use the cards) of +2 that has 3 negatives.
ASK: If I remove the three negatives, what’s left? (+5) Write that answer on the board.
Exercises:
a) Draw a picture of +5 that has …
i) 1 negative
ii) 2 negatives
iii) 3 negatives
b) Use your pictures from part a) to subtract.
i) +5 − (−1)
ii) +5 − (−2)
iii) +5 − (−3)
(MP.8) c) Predict +5 − (−5).
Answers:
a) i)
ii)
iii)
iv) 4 negatives
iv) +5 − (−4)
iv)
b) i) 6, ii) 7, iii) 8, iv) 9
c) 10
Have volunteers explain their prediction for part c). (all the numbers are becoming 1 bigger)
C-26
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
Exercises: Add enough positives and negatives to the picture of the first number so that you
can subtract the second number.
a) +3 − (−1)
b) +1 − (−2)
c) +2 − (−2)
d) +1 − (−4)
Answers: a) 4, b) 3, c) 4, d) 5
Subtracting a positive number from a negative number. Write on the board:
−2 − (+3)
SAY: I have to add 3 positives to make enough to subtract. ASK: What else do I need to add if I
want to keep the values the same? (3 negatives) Demonstrate doing so:
−2 − (+3)
Have a volunteer do the subtraction by removing the 3 positives (if you used cards, the
volunteer can literally remove them):
−2 − (+3) = −5
Exercises: Draw a picture of the first number so that you can subtract the second number.
Then subtract.
a) −2 − (+1)
b) −1 − (+3)
c) −4 − (+2)
d) −3 − (+2)
(MP.1, MP.3) Bonus: How do your answers to parts c) and d) compare? Why does this make
sense?
Answers: a) −3, b) −4, c) −6, d) −5, Bonus: the answer to part d) is one more that the answer to
part c) because −3 is one more than −4
Subtracting when there is not enough to subtract—more cases. Write on the board:
−2 − (−3)
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-27
ASK: Are there enough negatives to subtract? (no) I already have two negatives: how many
more do I need? (1 more) If I add one negative, what else do I need to add? (one positive) Ask
a volunteer to do that and another volunteer to do the subtraction:
−2 − (−3)
−2 − (−3)
−2 − (−3)
−2 − (−3) = +1
Repeat with more examples for volunteers to solve: + 2 − (+4), −1 − (−3), + 2 − (+5). (−2, 2, −3)
Exercises: Change the picture of the first number so you can subtract the second number.
Then subtract.
a) −1 − (+2)
b) +2 − (−4)
c) −3 − (−4)
d) +1 − (+3)
Bonus: How do the answers to parts b) and c) compare? Why does this make sense?
Answers: a) −3, b) +6, c) +1, d) −2, Bonus: the answer to part c) is 5 less than the answer to
part b) because −3 is 5 less than +2
SAY: Now you need to draw the picture of the first number yourself. Add enough positives and
negatives to it so you can subtract the second number.
Exercises: Subtract.
a) +3 − (+5)
b) (−1) − (+2)
Answers: a) −2, b) −3, c) +4, d) +6
c) −2 − (−6)
d) (+3) − (−3)
Extensions
1. a) Complete the picture to show −3.
i)
ii)
iii)
b) Use the pictures you made in part a) to subtract.
i) −3 − (+1)
ii) −3 − (+2)
iii) −3 − (+3)
Answers:
a) i)
ii)
iii)
b) Removing the positives results in i) −4, ii) −5, iii) −6
NOTE: After students finish Extension 1, point out that another way of looking at what they did
is to solve missing addend problems.
i) +1 + _____ = −3
ii) +2 + ______ = −3
iii) +3 + ______ = −3
For example, they had to add 4 negatives to +1 to get −3. So, for integers it’s the same as for
whole numbers—subtracting is the same as finding the missing addend.
C-28
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
(MP.4) 2. Give students a copy of BLM Number Lines from −6 to 6. Demonstrate how they
can use the BLM to determine the time in one city when given the time in another city in a
different time zone. For example, tell students that it is 3:00 p.m. in the +4 time zone. Tell
students that you want to determine the time in the −5 time zone.
Point out that it is earlier to the left on the number line and later to the right on the number line.
Together, fill in the number line to the left to find the time in the −5 time zone.
So, it is 6 a.m. in the −5 time zone. Now display the time zones below.
Time Zones
Muscat, Oman
Helsinki, Finland
Rome, Italy
New York City, USA
Chicago, USA
+4
+2
+1
−5
−6
a) A sporting event is being held in Rome at 2 p.m. Abdul lives in Muscat. At what time should
he turn on his television to watch the event live?
b) An acting awards ceremony is being held in New York City at 7:30 p.m. Alexa lives in
Helsinki. At what time should she turn on her television to watch the event live?
c) Jin lives in Chicago. His friend Clara lives in Helsinki. Jin wants to call Clara when he gets
home from school at 4 p.m. Clara goes to bed at 10:30 p.m. Will she get the call before she
goes to bed?
Answers: a) 5 p.m., b) 2:30 a.m., c) no, it will be 12:00 midnight where Clara lives when Jin
calls
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-29
NS7-7
Subtracting Integers on a Number Line
Pages 40–42
Standards: 7.NS.A.1c
Goals:
Students will use a number line to subtract integers.
Students will recognize that the distance between two integers is the absolute value of their
difference.
Prior Knowledge Required:
Knows how to subtract by counting up
Vocabulary: absolute value, integer, negative, positive
Materials:
BLM Subtraction on a Number Line (p. C-86)
BLM Number Lines from −6 to 6 (p. C-85)
Review using a number line to subtract two ways. Write on the board:
5−3
−5
−4
−3
−2
−1
0
1
2
3
4
5
ASK: How would you use the number line to subtract 5 − 3? Have a volunteer show their work.
(most likely, students will start at 5 and move left 3 spaces to reach 2) Then let another
volunteer show a different way. If no one suggests it, remind students that they can count up
from 3 to 5. SAY: You can ask: How can you get from 3 to 5 on the number line? You need to
move 2 units right, so the answer is +2. Show this on the board as follows:
5 − 3 = +2
−5
−4
−3
−2
−1
0
1
2
3
4
5
SAY: In this case, the answer isn’t where you end up. This time, the answer is how you get from
3 to 5.
Now write on the board: 3 − 5. Have two volunteers each show a different way to solve this on a
number line. Make sure that both solutions come up.
C-30
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
Solution 1: Start at 3 and move left 5 spaces to reach −2.
−5
−4
−3
−2
−1
0
1
2
3
4
5
Solution 2: How do you get from 5 to 3? Move two places left, so 3 − 5 = −2 If needed, you can
PROMPT: To subtract 5 − 3, you asked how you can get from 3 to 5; what question would you
ask to subtract 3 − 5? (How can you get from 5 to 3?)
3 − 5 = −2
−5
−4
−3
−2
−1
0
1
2
3
4
5
SAY: Both ways of subtracting 3 − 5 are correct and they both get the answer −2. In this lesson,
we will use the second way to subtract integers.
The answer isn’t where you end up on the number line, but how you get from the second
number to the first number. You have to start at the second number and ask yourself: How do I
get to the first number? If you have to move right, the answer is positive, because you’re
subtracting a smaller number from a bigger number. Use the 5 − 3 example to illustrate. Then
SAY: But if you have to move left, then you’re subtracting a bigger number from a smaller
number, so the answer is negative. Use the 3 − 5 example to illustrate.
Give students BLM Subtraction on a Number Line.
Exercises: Do Question 1 on the BLM.
Answers: a) +4, b) −6, c) −1, d) +3
SAY: Where the arrow ends is where you start the subtraction.
Exercises: Do Question 2 on the BLM.
Answers: a) −1 − (−5) = +4, b) 2 − (−1) = +3, c) −5 − (−4) = −1, d) −1 − (+5) = −6
Students will need a copy of BLM Number Lines from −6 to 6 to do the exercises below.
SAY: Where you start the subtraction is where the arrow ends.
Exercises: Draw the arrow that shows the answer. For parts e) to h), do the subtraction.
a) +4 − (−1) = +5
b) −3 − (+2) = −5
c) −6 − (+3) = −9
d) −2 − (−5) = +3
e) −2 − (+4) = ____
f) −5 − (−2) = _____
g) +1 − (−1) = ____
h) +3 − (+6) = _____
Answers: a) from −1 to +4, b) from +2 to −3, c) from +3 to −6, d) from −5 to −2, e) from +4
to −2, −6; f) from −2 to −5, −3; g) from −1 to +1, +2, h) from +6 to +3, −3
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-31
Subtracting the same numbers in opposite orders gets opposite answers. Draw on the
board:
5 − 2 = _____
0
1
2
3
4
5
6
2
3
4
5
6
2 − 5 = _____
0
1
(MP.1, MP.2) Have volunteers draw the arrows for both subtractions, then give the answers.
ASK: How are the arrows the same? (they have the same length) How are the arrows different?
(the directions are different) How are the answers the same? (the number part is the same) How
are the answers different? (the sign is different) Point out that the length of the arrow, or how far
apart the numbers are, tells you the number part of the answer and the direction tells you the
sign.
SAY: The distance between the integers, or how far apart they are, is always a positive number.
But the subtraction is positive only if you subtract a smaller number from a bigger number.
Exercises: Which subtraction tells you how far apart the numbers are?
a) 3 − 5 or 5 − 3
b) +2 − (−1) or −1 − (+2)
c) −3 − (−5) or −5 − (−3)
Answers: a) 5 − 3, b) +2 − (−1), c) −3 − (−5)
Introduce absolute value notation. Remind students that the absolute value of a number is its
distance from 0. Tell students that because absolute value is used a lot, mathematicians have
created a notation for it so that you don’t have to keep writing out the words. Write on the board:
|−3| = 3
Read the equation as: “The absolute value of negative three is three.” Point out that the
absolute value is the number part without the sign.
Exercises: Write the number.
Answers: a) 9, b) 10, c) 12, d) 3.4, e) 3 1/2, f) 8.7
C-32
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
Subtraction and distance apart. Write on the board:
−3 − (+1) = _____
−4
−3
−2
−1
0
1
2
3
4
ASK: Is the answer positive or negative? (negative) SAY: The number I’m taking away (point
to +1) is bigger than the number I’m taking it from (point to −3), so the answer is negative. Write
the negative sign to begin the answer. ASK: What is the number part? (4) How do you know?
(the numbers are 4 units apart) SAY: The distance apart is always the number part of the
subtraction so, if you know the answer to the subtraction, then the absolute value is the distance
apart. Write on the board:
|−3 − (+1)| is the distance between the integers −3 and +1
Exercises: Subtract the second number from the first. Then take the absolute value to find the
distance between the integers.
a) −4 and +5
b) −3 and −6
c) −18 and −3
d) −100 and −124
Answers: a) |−9| = 9, b) |3| = 3, c) |−15| = 15, d) |24| = 24
(MP.4) Exercises: Find the distance apart. Show your answer with a subtraction equation.
a) How much warmer is +6°F than −4°F?
b) How much lower is −30 ft than +42 ft?
c) Mark lives 8 blocks north (+) of the school and Jenny lives 3 blocks south (−) of the school.
How far apart do they live?
d) Ravi lives 5 blocks south of the school and Sandy lives 12 blocks south of the school. How
far apart do they live?
Answers: a) |+6 − (−4)| = 10, so 10°F; b) |−30 − (+42)| = 72, so 72 ft; c) |+8 − (−3)| = 11, so 11
blocks; d) |−5 − (−12)| = 7, so 7 blocks apart
NOTE: There are two different ways to subtract on a number line. For example, to subtract
17 − 2, you can start at 17 and count 2 places left. The answer is where you end up. Or, you
can find the distance between 2 and 17 by counting how many steps you have to take to get
from 2 to 17. The answer is the number of steps. If you want to explore this further, students can
do Extension 1.
Extensions
1. Subtract 12 − 3 in two ways on a number line. Make sure you get the same answer both ways.
Answer: 9
(MP.1) 2. Show students the connection between the fact that 10 − 5 + 3 has the same answer
as 10 + 3 − 5 and the fact that addition is commutative: 10 − 5 + 3 is really just another way of
writing 10 + (−5) + 3 which, by commutativity, is 10 + 3 + (−5). Have students use the
commutativity of addition to make the calculation easier.
a) 17 + 4 − 7
b) 21 − 6 + 19
Answers: a) 17 − 7 + 4 = 10 + 4 = 14, b) 21 + 19 − 6 = 40 − 6 = 34
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-33
3. a) Find the distance between each pair of integers.
i) −3 and −7 are |−3 − (−7)| = | _____ | = _____ units apart.
+3 and +7 are |+3 − (+7)| = | _____ | = _____ units apart.
ii) −3 and +2 are |−3 − (+2)| = | _____ | = _____ units apart.
+3 and −2 are |+3 − (−2)| = | _____ | = _____ units apart.
iii) −4 and +6 are |−4 − (+6)| = | _____ | = _____ units apart.
+4 and −6 are |+4 − (−6)| = | _____ | = _____ units apart.
(MP.8) b) If you know the distance between two integers, how can you find the distance
between the two opposite integers?
Answers: a) i) +4, 4, −4, 4; ii) −5, 5, +5, 5; iii) −10, 10, +10, 10; b) they are the same
4. Which integer, +7 or −10, is farther from −2?
Answer: +7
(MP.1) 5. The distance between a positive integer and a negative integer is 5.
What might the integers be?
Bonus: List all possible solutions.
Answers: +4 and −1, +3 and −2, +2 and −3, +1 and −4
(MP.1) 6. The distance between two negative numbers is 5/7. Give one pair of numbers that work.
Sample answer: −1/7 and −6/7
7. The distance between 62 and a negative integer is 100. What is the negative integer?
Answer: −38
C-34
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
NS7-8
Patterns in Subtraction
Pages 43–44
Standards: 7.NS.A.1c, 7.NS.A.1d
Goals:
Students will understand subtraction of integers as adding the additive inverse.
Prior Knowledge Required:
Can subtract integers on a number line
Can use a picture to subtract integers
Can add integers
Vocabulary: integer, negative, opposite, positive
Materials:
10 “+” cards and 10 “−” cards
BLM Adding and Subtracting Integers on a Number Line (pp. C-87–88, see Extension 2)
Subtracting a number from 0 gets the opposite of the number. Draw on the board:
0−2
−3
−2
−1
0
1
2
3
−1
0
1
2
3
0 − (−3)
−3
−2
SAY: These subtractions start at 0, so the arrow ends at 0 and starts at the other number. Have
two volunteers draw the arrows and two more volunteers do the subtractions (0 − 2 = −2 and
0 − (−3) = +3). Then ask volunteers to predict:
0 − (+5)
0 − (−7)
0 − (+80)
0 − (−900)
PROMPTS: Is +5 greater than 0 or less than 0? (greater) So if the number I’m subtracting is
bigger than 0, is the difference positive or negative? (negative) SAY: Subtracting a bigger
number from a smaller number gets a negative answer. Repeat for the other subtractions (the
answers are −5, +7, −80, and +900). Point out that subtracting from 0 changes the sign, but
keeps the number part the same. Write on the board:
0 − (−3) = +3
0 − (+3) = −3
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-35
Exercises: Subtract from 0.
a) 0 − 4
b) 0 − (−4)
c) 0 − 18
Answers: a) −4, b) +4, c) −18, d) +43, e) −71
d) 0 − (−43)
e) 0 − (+71)
Have students show their answers to parts a) and b) on a number line (they can draw the
number line themselves or you can provide one for them).
Subtracting a number is like adding its opposite. Write on the board, using cards for the
negatives:
−5 − (+2)
Have a volunteer show how to change the picture so that you can subtract +2. Then SAY: We
needed to add two positives and two negatives. Now we take away the two positives. Do so,
then SAY: It’s like all we did was add the two negatives. So subtracting two positives gets the
same answer as adding two negatives.
Exercises: Subtract +2 and add −2 to the same number. Make sure you get the same answer.
a) +3 − (+2) and +3 + (−2)
b) −3 − (+2) and −3 + (−2)
c) +5 − (+2) and +5 + (−2)
d) 0 − (+2) and 0 + (−2)
Answers: a) +1 and +1, b) −5 and −5, c) +3 and +3, d) −2 and −2
Write on the board:
+4 − (−1)
Have a volunteer show how to change the picture so you can subtract −1. Then SAY: We
needed to add one positive and one negative. Now we take away the one negative. Do so, then
SAY: It’s like all we did was add the one positive. So subtracting one negative gets the same
answer as adding one positive.
Exercises: Subtract −1 and add +1 to the same number. Make sure you get the same answer.
a) +3 − (−1) and +3 + (+1)
b) −3 − (−1) and −3 + (+1)
c) +5 − (−1) and +5 + (+1)
d) −2 − (−1) and −2 + (+1)
Answers: a) +4 and +4, b) −2 and −2, c) +6 and +6, d) −1 and −1
SAY: You can subtract any number by adding its opposite number.
Exercises: Fill in the blank so that both questions have the same answer. Then check your
answer by evaluating both expressions.
a) −5 − (−3) and −5 + _______
b) +5 − (+6) and +5 + _______
c) −3 − (−2) and −3 + _______
d) +3 − (+4) and +3 + _______
e) +4 − (+1) and +4 + _______
f) +2 − (−3) and +2 + _______
C-36
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
Answers: a) −5 − (−3) = −5 + (+3) = −2, b) +5 − (+6) = +5 + (−6) = −1,
c) −3 − (−2) = −3 + (+2) = −1, d) +3 − (+4) = +3 + (−4) = −1, e) +4 − (+1) = +4 + (−1) = +3,
f) +2 − (−3) = +2 + (+3) = +5
(MP.1) Demonstrate the connection between adding opposites and subtracting a number
from itself. Write on the board:
(−3) + (____) = 0
(−3) − (_____) = 0
ASK: What do I have to add to −3 to get 0? (+3) What do I have to subtract from −3 to get 0?
(−3) SAY: Any number added to its opposite gets 0, and any number subtracted from itself also
gets 0, so adding its opposite does the same thing as subtracting itself.
Exercises: Fill in the blanks.
a) (−5) + (____) = 0 and (−5) − (_____) = 0
b) (+4) + (____) = 0 and (+4) − (_____) = 0
c) (−9) + (____) = 0 and (−9) − (_____) = 0
Answers: a) +5, −5, b) −4, +4, c) +9, −9
Rewriting bracket notation without brackets. SAY: You can write brackets around both
numbers when writing addition or subtraction. Write on the board:
(+7) − (−8)
SAY: You can change it to notation without brackets by writing it as an addition of gains and
losses. Write on the board:
+7+8
SAY: Subtracting −8 is the same as adding +8 or just adding 8. Removing a loss is like adding a
gain. Then write on the board:
(−8) − (+5)
(−8) − (−5)
(+3) − (+7)
(−4) − (+3)
Ask volunteers to write the subtractions as additions without brackets. (−
8 − 5, − 8 + 5, + 3 − 7, − 4 − 3)
Exercises: Write the subtraction as an addition without brackets. Then evaluate.
a) (−3) − (−4)
b) (+3) − (−2)
c) (−4) − (+3)
d) (+2) − (+8)
Answers: a) − 3 + 4 = +1, b) + 3 + 2 = +5, c) − 4 − 3 = −7, d) + 2 − 8 = −6
SAY: Any addition or subtraction written in brackets can be written as an addition without
brackets. The rules for changing notation are:
+(+) = +
+(−) = −
−(+) = −
−(−) = +
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-37
Exercises: Write the expression as an addition without brackets. Then evaluate.
a) (−2) + (−7)
b) (−2) − (−7)
c) (−2) + (+7)
d) (−2) − (+7)
e) (+2) + (−7)
f) (+2) − (−7)
g) (+2) + (+7)
h) (+2) − (+7)
Answers: a) − 2 − 7 = −9, b) − 2 + 7 = +5, c) − 2 + 7 = +5, d) − 2 − 7 = −9, e) + 2 − 7 = −5,
f) + 2 + 7 = +9, g) + 2 + 7 = +9, h) + 2 − 7 = −5
Subtracting integers in real-world contexts. Write on the board:
What is the temperature now if …
a) the temperature was 40°F, then dropped 30°F?
b) the temperature was 40°F, then dropped 50°F?
c) the temperature was −40°F, then dropped 30°F?
Have a volunteer answer the first question. (10°F) ASK: How did you know to subtract?
(because “dropped 30°F” tells us it became less by 30°F) Have a volunteer answer the second
question and write a subtraction equation for it. (40 − 50 = −10) Repeat for the third question
(− 40 − 30 = −70) Point out that sometimes knowing how to solve a problem in a familiar
situation with positive numbers helps you to figure out how to solve the same type of problem in
a more unfamiliar situation, such as when there are negative numbers.
(MP.4) Exercises:
a) The temperature in California is 75°F and in New York is 40°F colder. What is the
temperature in New York?
b) The temperature at the South Pole in January can reach as high as 9°F. In July, it can get
100°F lower. What is the temperature in July?
c) Celsius is a temperature scale in which 0°C is the freezing point of water. When Sal adds
some salt to the water, the mixture freezes at a temperature 15°C colder. What is the freezing
temperature of the new mixture?
d) On January 15, 1972, in Loma, Montana, the temperature rose from −54°F to 49°F, all in one
day. How much of a temperature change was that?
e) On January 22, 1943, the temperature in Spearfish, South Dakota, rose from −4°F to 45°F in
only 2 minutes. How much of a temperature change was that?
Answers: a) 35°F, b) −91°F, c) −15°C, d) 49°F − (−54°F) = 49°F + 54°F = 103°F, e)
45°F − (−4°F) = 45°F + 4°F = 49°F
Extensions
1. (MP.7) a) Subtract the positive numbers. Then subtract the negative numbers by continuing
the pattern.
6 − 3 = _____
6 − 2 = _____
6 − 1 = _____
6 − 0 = _____
6 − (−1) = _____
6 − (−2) = _____
6 − (−3) = _____
C-38
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
b) Use a pattern to subtract (+5) − (−3).
Answers: a) 3, 4, 5, 6, 7, 8, 9; b) +8
2. Have students explore adding and subtracting integers on a number line by starting at the
first number and moving in the correct direction. See BLM Adding and Subtracting Integers
on a Number Line.
3. Write <, >, or =.
a) |−3|
2
b) |−1|
Answers: a) >, b) >, c) =, d) <
−1
c) |−5|
5
4. Write + or − to make the answer as large as possible: (+3)
Answer: (+3) − (−5) + (+8) − (−2) = 18
d) |−4|
(−5)
(+8)
|−17|
(−2)
5. Write 5 or 6 to make the answer as large as possible: − + (− ) − (− ) − (+ ) + (+ )
Answer: −5 + (−5) − (−6) − (+5) + (+6) = − 5 − 5 + 6 − 5 + 6 = −3. The correct strategy to use is
to insert 6 when the sign is positive and 5 when the sign is negative, using the
rules: +(+) = −(−) = + and −(+) = +(−) = −
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-39
NS7-9
Adding and Subtracting Multi-Digit Integers
Pages 45–47
Standards: 7.NS.A.1d
Goals:
Students will use place value to add and subtract multi-digit integers.
Prior Knowledge Required:
Can add and subtract integers
Knows how to subtract an integer by adding the opposite integer
Can add multi-digit numbers by lining up the place values
Vocabulary: absolute value, expanded form, hundreds, ones, place value, tens, whole number
Materials:
overhead projector
transparency of grid paper
BLM Balance Model (p. C-89, see Extension 1)
Using expanded form to add 3-digit numbers. Show how to add 452 + 273 using the
expanded form:
452
+ 273
______ hundreds + ______ tens + ______ ones
+ ______ hundreds + ______ tens + ______ ones
______ hundreds + ______ tens + ______ ones
______ hundreds + ______ tens + ______ ones
After regrouping:
Point out that students need to regroup until all place values have only a single digit.
Exercise: Add 869 + 237 using expanded form.
Bonus: Add 5,846 + 2,571 using expanded form.
Answers: 1,106; Bonus: 8,417
Standard algorithm for addition with regrouping. Now demonstrate using the standard
algorithm alongside the place value addition for the first example you did together (452 + 273):
1
+
4
5
2
2
7
3
7
2
5
You can project a transparency of grid paper onto the board to show the outlines.
C-40
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
Point out that after regrouping the tens, you add the 1 hundred that you carried over from the
tens at the same time as the hundreds from the two numbers, so you get 1 + 4 + 2 = 7
hundreds. NOTE: When adding and subtracting multi-digit numbers, students should be
encouraged to use grid paper to line up the place values.
Exercises: Use the standard notation to add.
a) 358 + 217
b) 475 + 340
c) 695 + 258
d) 487 + 999
e) 1,358 + 7,217
f) 4,658 + 8,347
g) 94,358 + 18,647 h) 862,595 + 198,857
Bonus: 427 + 382 + 975 + 211
Answers: a) 575; b) 815; c) 953; d) 1,486; e) 8,575; f) 13,005; g) 113,005; h) 1,061,452;
Bonus: 1,995
Students who are struggling can use place value charts alongside the standard algorithm.
Bonus: Find a short way to do part d) that doesn’t require regrouping.
Answer: the answer is 1 less than 487 + 1,000 (1,487), so 487 + 999 = 1,486
SAY: You have to make sure the place values are lined up, the ones with the ones, tens with
tens. This can be tricky when the numbers have a different number of digits, but you just have to
make sure the ones digits are aligned and the commas are aligned. Demonstrate the alignment
in a) below:
32,405
+ 9,736
Exercises:
a) 32,405 + 9,736
b) 789,104 + 43,896
c) 999,678 + 1,322
d) 94,358 + 8,647
e) 652,722 + 798
f) 5,973 + 297,588
Bonus: 17,432 + 946 + 3,814 + 568,117
Answers: a) 42,141; b) 833,000; c) 1,001,000; d) 103,005; e) 653,520; f) 303,561; Bonus:
590,309
Word problems practice.
Exercises:
a) Jayden ran 1,294 km one year and 1,856 km the next. How many kilometers did he run
altogether?
b) In an election with three candidates, the candidate who won received 1,052,817 votes. The
other two received 972,435 votes and 71,095 votes. Did the candidate who won get more than
the other two combined?
Answers: a) 3,150 km; b) yes; the other two combined received only 1,043,530 votes
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-41
Using the standard algorithm to subtract 3-digit numbers with regrouping. Write on the
board:
–
3
6
7
1
9
2
−
300
+
60
+
7
100
+
90
+
2
5
SAY: 7 − 2 is easy to subtract, but we can’t subtract 60 − 90 since we want a positive digit in the
answer. (Students will see in Extension 3 how they can use 60 − 90 = −30 to solve this
question.) So let’s take away 100 from 300 and add it to the 60.
200
–
160
300
+
60
+
7
100
+
90
+
2
100
+
70
+
5
SAY: Now it is easy to subtract each place value.
Exercises: Subtract, then check by adding.
a) 358 − 129
b) 346 − 183
c) 862 − 257
Answers: a) 229, b) 163, c) 605, d) 182
d) 309 − 127
Do the following example (852 − 459) together as a class:
Emphasize that you write the second regrouping above the first one, not over the first
regrouping, so that you can see each step easily.
Exercises: Subtract. Then check by adding.
a) 563 − 175
b) 541 − 273
c) 422 − 358
d) 542 − 289
Bonus: Make up your own subtraction question that requires regrouping twice. Ask a partner to
solve your question.
Selected answers: a) 388, b) 268, c) 64, d) 253
C-42
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
Borrowing from zero. Write on the board:
–
5 0 3
1 8 4
ASK: Do I have enough ones to subtract? (no) What do I need to do? (regroup 1 ten as 10
ones) There are no tens to take from; what can I do? (regroup 1 hundred as 10 tens) Have a
volunteer do so (or do it yourself if no one volunteered the strategy):
4
–
10
5 0 3
1 8 4
SAY: Now we can regroup a ten as 10 ones. Have a volunteer do so, then have another
volunteer show the subtraction:
9
4 10 13
–
5 0 3
1 8 4
3 1 9
Exercises: Subtract using the standard algorithm.
a) 402 − 169
b) 501 − 223
c) 402 − 36
d) 62,187 − 41,354
e) 54,137 − 28,052
f) 9,319 − 6,450
g) 4,037 − 2,152
h) 90,319 − 6,405
i) 145,207 − 1,128
j) 3,695 − 1,697
k) 1,000 − 854
l) 10,000 − 4,356
Answers: a) 233; b) 278; c) 366; d) 20,833; e) 26,085; f) 2,869; g) 1,885; h) 83,914; i) 144,079;
j)1,998; k) 146; l) 5,644
Word problems practice.
Exercises:
1. Construction of the Statue of Liberty began in France in 1881. When it was completed, the
statue was shipped to the United States and rebuilt there in 1886. How long ago was it built in
France? How long ago was it rebuilt in the United States?
Answers: The answer depends on current year; e.g., 2014 − 1881 = 133 years ago, and
2014 − 1886 = 128 years ago
2. In 1810, the population of New York City was 96,373. In 2010, the population of New York
City was 8,175,133. How much did the population grow in those 200 years?
Answers: 8,078,760
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-43
Adding multi-digit integers with the same sign. Tell students that they can add integers with
the same sign by adding their absolute values. Write on the board:
+ 5 + 3 = +8
− 5 − 3 = −8
SAY: Adding two gains gets a greater gain and adding two losses gets a greater loss. Tell
students that they can use grid paper to do their rough work.
Exercises: Add the integers.
a) + 7 + 4
b) − 5 − 8
c) + 836 + 749
d) − 25 − 788
Bonus: (−376) + (−145) + (−309) + (−822)
Answers: a) +11; b) −13; c) +1,585; d) −813; Bonus: −1,652
Remind students that they can subtract an integer by adding its opposite.
Exercises: Subtract.
a) −3 − (+17)
b) +31 − (−482)
c) −142 − (+7,483)
d) +8,160 − (−752)
Answers: a) −20; b) +513; c) −7,625; d) +8,912
Word problems practice.
(MP.4) Exercises: Write an addition equation to show the answer to the word problem.
a) The Boomerang Nebula, with a temperature of −458°F, is the coldest place in the universe
known to scientists. The coldest possible temperature is only 2 degrees colder. What is the
coldest possible temperature?
b) The average temperature on Jupiter is −244°F. The average temperature on Neptune is
122°F colder. What is the average temperature on Neptune?
Answers: a) −458°F − 2°F = −460°F, b) −244°F − 122°F = −366°F
Adding multi-digit integers with different signs. Tell students that they can add integers with
different signs by subtracting their absolute values. Write on the board:
+ 5 − 3 = +2
− 5 + 3 = −2
SAY: Adding a gain and a loss reduces the gain or the loss. If the gain is bigger, you get an
overall gain and if the loss is bigger, you get an overall loss. Write on the board:
+ 3,482 − 21,674
ASK: What’s bigger, the gain or the loss? (the loss) SAY: Take away the smaller absolute value
from the bigger absolute value.
C-44
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
Write the subtraction in grid form on the board and ask a volunteer to do the multi-digit
subtraction:
1
11
5
17
2
1
6
7
4
2
1
6
7
4
−
3
4
8
2
−
3
4
8
2
1
8
1
9
2
Write on the board:
So, + 3,482 − 21,674 = −18,192
SAY: The difference tells you the number part and the number with the bigger absolute value
tells you the sign.
Exercises: Add the integers.
a) + 742 − 846
b) + 917 − 156
c) − 18,431 + 17,563
d) − 9,476 + 18,512
Answers: a) −104; b) +761; c) −868; d) +9,036
SAY: Now you will need to decide whether to add or subtract the absolute values to add the
integers.
Exercises: Add the integers.
a) − 543 − 712
b) + 543 − 712
c) + 81,416 + 3,517
d) − 21,416 + 712,183
Answers: a) −1,255; b) −169; c) +84,933; d) +690,767
(MP.4) Exercises: Write an addition equation to show the answer to these word problems.
a) The temperature on Mercury can get as high as 869°F. It can also get 1,167°F colder. What
temperature is that?
b) The average temperature on Neptune is about −340°F and on Venus, it is about 1,210°F
warmer than that. What is the average temperature on Venus?
Answers: a) + 869°F − 1,167°F = −298°F; b) − 340°F + 1,210°F = +870°F
Exercises: Subtract by adding the opposite number.
a) −174 − (−311)
b) +854 − (+5,142)
c) −853 − (−215)
d) −514 − (+312)
e) +704 − (+908)
f) +360 − (−412)
Answers: a) +137; b) −4,288; c) −638; d) −826; e) −204; f) +772
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-45
Extensions
(MP.4) 1. On BLM Balance Model, students will use a pan balance model to understand that
an integer can be subtracted by adding its opposite.
Answers: 1. b) down, c) down, d) up; 2. b) −1, c) −1, d) +1; 3. a) a negative integer, b) a
positive integer, c) a negative integer; 4. a) negative, b) positive
(MP.4) 2. Show students a paper person who starts at sea level (0 inches), goes down 1 inch
for every brick added and goes up 1 inch for every helium balloon added.
a) Where is the person if they are holding …
i) 2 bricks?
ii) 3 helium balloons?
iii) 8 helium balloons?
iv) 5 bricks?
v) 3 bricks and 2 helium balloons?
vi) 4 bricks and 1 helium balloon?
vii) 2 bricks and 5 helium balloons?
b) What happens if you take away 3 bricks?
c) What happens if you take away 4 helium balloons?
d) How is taking away a helium balloon like adding a brick?
Answers: a) i) −2, ii) +3, iii) +8, iv) −5, v) −1, vi) −3, vii) +3, b) moves up 3 inches, c) moves
down 4 inches, d) they both make the person move down an inch
(MP.1, MP.7) 3. Show students a way to subtract positive numbers using what they know about
integers instead of regrouping. Example:
734 − 568 = (700 − 500) + (30 − 60) + (4 − 8)
= 200 − 30 − 4
= 170 − 4
= 166
Students can try this technique with some of the questions they have already done and verify
that it gets the same answer.
C-46
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
NS7-10
Factors
Pages 48–50
Standards: preparation for 7.NS.A.1
Goals:
Students will determine the factors of numbers up to 100 and find the greatest common factor of
pairs of numbers up to 100.
Students will write fractions in lowest terms.
Prior Knowledge Required:
Understands that two fractions are equivalent if one can be obtained from the other by
multiplying the numerator and denominator by the same number
Vocabulary: factor, factor pair, greatest common factor (GCF), lowest terms
Materials:
BLM Puzzles (p. C-90, see Extension 1)
Introduce factors. Remind students that whole numbers are 0, 1, 2, 3, 4, and so on. Decimal
numbers and fractions are not whole numbers. Have students brainstorm ways of multiplying
two whole numbers to get 12. (1 × 12 or 12 × 1, 2 × 6 or 6 × 2, 3 × 4 or 4 × 3) Tell students that
the numbers that appear in products that make 12—1, 2, 3, 4, 6, and 12—are called factors of
12. Then ask students how they could show that 2 is a factor of 6. (use 2 × 3 = 6) ASK: What
other number does this show is a factor of 6? (3) Point out that students can also use skip
counting—2, 4, 6—to check that 2 is a factor of 6.
Exercises: Write a multiplication equation that shows that …
a) 5 is a factor of 10
b) 8 is a factor of 80
c) 5 is a factor of 40
d) 1 is a factor of 2
e) 1 is a factor of 15
f) 1 is a factor of 43
Answers: a) 5 × 2 = 10, b) 8 × 10 = 80, c) 5 × 8 = 40, d) 1 × 2 = 2, e) 1 × 15 = 15, f) 1 × 43 = 43
Students who are struggling can use skip counting to find the missing factor.
1 is a factor of every whole number. ASK: What numbers is 1 a factor of? (all of them; every
whole number) Point out that counting by ones shows that 1 is a factor of every number. When
you count by ones, you can reach every number.
Finding factors. Tell students that you want to find all the factors of 10. Draw on the board:
0×
= 10
1×
= 10
2×
= 10
3×
= 10
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-47
Ask in turn: Is 0 a factor of 10? Is 1 a factor of 10? Is 2? Is 3? Then demonstrate writing the
number that fits in the box; if no number fits, write “x.” (0 × x = 10, 1 × 10 = 10, 2 × 5 = 10,
3 × x = 10) Continue in this way, even going past 10, until students point out that you can stop,
or if no one points it out, until you reach 13. (1, 2, 5, and 10 are the factors of 10) ASK: Can any
number greater than 10 be a factor of 10? (no) Why not? (because even multiplying by 1 makes
it greater than 10) ASK: Did we find all the factors of 10? (yes) Emphasize that by checking the
numbers in order, instead of randomly, you can be sure that you didn’t miss any. Summarize by
saying that you can find all the factors of 10 by checking the numbers from 1 to 10 in order.
Exercises: Find all the factors of …
a) 12
b) 14
c) 15
Answers: a) 1, 2, 3, 4, 6, 12; b) 1, 2, 7, 14; c) 1, 3, 5, 15
Factors come in pairs. Point out that the factors of a number come in pairs. For example,
3 × 4 = 12, so 3 and 4 are a factor pair for 12. ASK: What is another factor pair for 12? (2 and 6,
or 1 and 12) List on the board all the factor pairs for 12. Point out that once you know that
3 × 4 = 12, you automatically know that 4 × 3 = 12. When we say that 3 and 4 are a factor pair,
we don’t have to say that 4 and 3 are, too—they are the same pair of numbers, so they are the
same factor pair.
You can stop when you find a number that already appears in a factor pair. Show students
the chart below. Point out that once students reach a number that appears in an earlier pair,
they don’t need to fill in any more of the chart. For example, 7 is paired up with 4 because 4 is
paired up with 7. On the other hand, any number not paired up so far won’t be paired up at all.
For example, because 7 × 4 = 28, then 8 × 4 is already more than 28, so 8 would have to pair
up with 1, 2, or 3. But we already checked those, so 8 is not paired up at all.
C-48
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
Exercises: Complete similar charts for 40 and 36.
Tricks for eliminating factors.
• If the number is odd, then no even number can be a factor.
• If the ones digit of a number is anything other than 0 or 5, then 5 is not a factor.
• If a number is not a factor, no multiple of that number can be a factor either. Example: If
3 is not a factor, then 6 is also not a factor.
A trick for dividing easily in a special case. SAY: When every digit in a number is divisible by
the same number, it is easy to divide by that number—just divide the digits one at a time. Write
on the board:
93 ÷ 3 = 31
46 ÷ 2 = 23
84 ÷ 2 =
36 ÷ 3 =
Have volunteers write the answers. (42, 12)
Exercises: Divide.
a) 86 ÷ 2
b) 69 ÷ 3
Answers: a) 43, b) 23, c) 320, d) 201
c) 960 ÷ 3
d) 804 ÷ 4
Students who are struggling can use base ten blocks for a few examples to see why this is true.
A shortcut for making charts. SAY: You don’t need to list all the possible first factors in a
chart; you can just list the ones that work. In the exercise below, do part a) together as a class.
Demonstrate that if students know, for example, that 4 is not a factor of 42 (because it is
between 40 and 44), then they don’t need to list it in the chart at all. Also, 5 is not a factor
because the ones digit of 42 is neither 0 nor 5.
Exercises: Find all pairs of numbers that multiply to …
a) 42
b) 30
c) 63
Students may need a calculator to do these:
e) 64
f) 72
g) 60
Answers:
a) 1, 42;
b) 1, 30
c) 1, 63
d) 1, 39
2, 21;
2, 15
3, 21
3, 13
3, 14;
3, 10
7, 9
6, 7
5, 6
e) 1, 64
f) 1, 72
g) 1, 60
Bonus: 1, 180
2, 32
2, 36
2, 30
2, 90
4, 16
3, 24
3, 20
3, 60
8, 8
4, 18
4, 15
4, 45
6, 12
5, 12
5, 36
8, 9
6, 10
6, 30
9, 20
10, 18
12, 15
d) 39
Bonus: 180
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-49
Common factors. Write on the board:
42: 1, 2, 3, 6, 7, 14, 21, 42
30:
SAY: From the chart of factor pairs, it is easy to list the factors in order. Demonstrate going
down the left side and up the right side of the chart. Have a volunteer do the same for 30. (1, 2,
3, 5, 6, 10, 15, 30) Then ASK: What factors do 30 and 42 have in common? (1, 2, 3, and 6) Tell
students that these are called the common factors of 30 and 42. ASK: What is the lowest
common factor? (1) What is the greatest common factor? (6) SAY: The lowest common factor is
always 1, so it’s not very interesting. What is interesting is the greatest common factor because
it tells you what all the other common factors are, too. If 6 is a factor of both numbers, then so
are all the factors of 6.
Exercises: List the factors of both numbers in order from smallest to greatest. Then find their
greatest common factor.
a) 63 and 39
b) 30 and 72
c) 64 and 72
d) 30 and 60
Answers: a) 3, b) 6, c) 8, d) 30
Reducing fractions to lowest terms. Tell students that finding common factors is sometimes
useful for fractions. If the numerator and denominator have a factor in common, other than 1,
then you can divide both by that factor and make an equivalent fraction that might be easier to
work with. Write on the board:
4
6
ASK: Do 4 and 6 have a common factor other than 1? (yes, 2) Write on the board:
4÷2
2
=
6÷2
3
SAY: Now the numbers are smaller, so it might be easier to work with.
Exercises: Make an equivalent fraction by dividing the numerator and denominator by the same
number.
15 ÷ 5
12 ÷ 4
936 ÷ 3
16 ÷ 2
b)
c)
d)
a)
20 ÷ 5
12 ÷ 2
603 ÷ 3
20 ÷ 4
Answers: a) 3/4, b) 3/5, c) 8/6, d) 312/201
SAY: If you divide by the greatest common factor, then you will have made the numerator and
denominator as small as you can.
C-50
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
Exercises: Divide the numerator and denominator by their greatest common factor to make an
equivalent fraction.
8
7
6
30
30
36
b)
c)
d)
e)
f)
a)
10
21
15
72
60
45
Answers: a) 4/5, b) 1/3, c) 2/5, d) 5/12, e) 1/2, f) 4/5
Tell students that fractions are said to be in lowest terms if both numbers are as small as
possible. ASK: Are the fractions you made in lowest terms? (yes) PROMPT: Do their numerator
and denominator have a common factor other than 1? (no) Point out that if they did have a
common factor other than 1, then the original number you divided by wasn’t the greatest
common factor. You could have picked a greater one. SAY: Fractions are in lowest terms when
the greatest common factor of their numerator and denominator is 1.
Exercises: Find the greatest common factor of the numerator and denominator. Is the fraction
in lowest terms?
9
6
12
5
8
9
b)
c)
d)
e)
f)
a)
9
8
10
6
15
15
Answers: a) 2, no; b) 3, no; c) 1, yes; d) 1, yes; e) 3, no; f) 1, yes
Write on the board:
Greatest Common Factor = GCF
Tell students that GCF is often written for the greatest common factor.
(MP.1) Exercises: Divide the numerator and denominator by their GCF. Make sure the fractions
you make are in lowest terms.
10
22
50
44
18
b)
c)
d)
e)
a)
60
12
56
33
15
Answers: a) 6/5, b) 5/6, c) 2/3, d) 5/6, e) 11/14
Extensions
(MP.1, MP.7) 1. This extension provides four puzzles for students to solve. If students have time
to do only one puzzle now, encourage them to continue with more puzzles later.
Ask students to copy the shapes in each puzzle from BLM Puzzles on 1 cm grid paper.
Students should cut the pieces out and try to assemble them into a single large rectangle. After
students have completed Puzzle 1, have them tape or glue the pieces together and set the
rectangle aside (they will need it later).
Let students struggle for a few minutes with Puzzle 2, then point out that it can take a long time
to solve larger puzzles using trial and error. It would be easier if students at least knew the size
of the rectangle they were trying to build. ASK: Is there a way to find the area of the larger
rectangle from the pieces you are trying to build it with? (yes, add the areas of the smaller
pieces) Have students verify that this works for Puzzle 1. (indeed, 2 + 2 + 3 + 4 + 9 = 20, which
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-51
is the area of the 4 × 5 rectangle they created) Demonstrate using the steps below to solve
Puzzle 1 and have students use them to solve Puzzles 2, 3, and 4 (Bonus).
Step 1: Find the area of the rectangle you are trying to build.
Answers: Puzzle 1: 20, Puzzle 2: 56, Puzzle 3: 60, Puzzle 4: 135
Step 2: Use factor pairs to find the possible lengths and widths of rectangles with that area.
Answers: Puzzle 1: 1 × 20, 2 × 10, 4 × 5; Puzzle 2: 7 × 8, 14 × 4, 28 × 2, 56 × 1; Puzzle 3:
1 × 60, 2 × 30, 3 × 20, 4 × 15, 5 × 12, 6 × 10; Puzzle 4: 1 × 135, 3 × 45, 5 × 27, 9 × 15
Step 3: Use logic to eliminate possibilities. For example, in Puzzle 2, the 7 × 8 rectangle won’t
fit the 12 × 1 piece. In Puzzle 3, the 5 × 12 rectangle won’t fit both the 5 × 6 and the 1 × 7 at the
same time.
Answers: Students should be left with the following dimensions in each case: Puzzle 2: 14 × 4,
Puzzle 3: 6 × 10, Puzzle 4: 9 × 15
Step 4: Use the fact that you know the dimensions of the rectangle to solve the puzzle.
(MP.1) 2. Find all number words, from “one” to “million,” whose letters are in alphabetical order.
Hints:
● It’s not a good idea to try and check them all!
● If the letters in the words “one” to “nine” are not in alphabetical order, can the letters in any
word containing “one” to “nine” (e.g., fifty-one) be in alphabetical order?
Answer: forty
Solution: First, check all number words up to “twelve” individually. Second, eliminate any
number words in the “teens” in one go since the letters in “teen” are not in alphabetical order.
Third, eliminate any number word that has a 1-digit number word in it. For example, the letters
in “twenty-seven” cannot in alphabetical order since the letters in “seven” are not. Finally,
eliminate all number words that involve the word “hundred” or “thousand” since neither of these
has letters in alphabetical order either. The only numbers left are multiples of 10 up to 90. If you
try each of these in turn—ten, twenty, thirty, … ninety—you find that only “forty” works.
C-52
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
NS7-11
Adding and Subtracting Fractions with
the Same Denominator
Pages 51–52
Standards: 7.NS.A.1b, 7.NS.A.1c, 7.NS.A.1d
Goals:
Students will add and subtract positive and negative fractions that have the same denominator.
Prior Knowledge Required:
Can add and subtract integers using a number line
Can add and subtract integers without a number line
Can subtract an integer by adding its opposite
Can name fractions from a picture
Vocabulary: integer, negative, opposite number, positive, unit fraction
Materials:
arrows used in Lesson NS7-5
BLM Number Lines from −1 to 1 (Fifths) (p. C-91)
Preparation. You will need the arrows you made for Lesson NS7-5. Change the labels on the
arrows (by taping a cutout over the current labels) to be the fractions …

1
5

2
5

3
5

4
5

1
5

2
5

3
5

4
5
Make sure that the positive fractions are on arrows that point right and the negative fractions are
on arrows that point left.
What it means to add fractions. Draw on the board:
SAY: When I add one piece of size one fourth together with two pieces of size one fourth, I get
three pieces of size one fourth. When adding fractions with the same denominator, the
denominator tells you the size of the pieces you are adding, so that doesn’t change. You add
the numerators to see how many pieces there are altogether.
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-53
Exercises: Add the fractions.
1 4
2 1
5
4
4
6
b) 
c)
d)
a) 


5 5
7 7
11 11
11 11
3
4
6
Bonus:


800 800 800
Answers: a) 3/5, b) 5/7, c) 9/11, d) 10/11, Bonus: 13/800
Using unit fractions to subtract fractions. Write on the board:
1 1 1 1 1 1 1
     
8 8 8 8 8 8 8
ASK: What fraction is this? (seven eighths) Tell students you want to take away two of the
eighths and write on the board:
1 1 1 1 1 1 1
     
8 8 8 8 8 8 8
7 2
 
8 8
Have a volunteer write the answer. (5/8) Tell students that fractions with numerator 1 are used a
lot and have a special name—they are called unit fractions. Point out that subtracting fractions
is just like adding fractions; the denominator tells us the unit fractions we are subtracting, so it
doesn’t change. Then you subtract the numerators.
Exercises: Subtract.
Bonus: These problems involve both addition and subtraction.
Answers: a) 6/8 or 3/4, b) 2/6 or 1/3, c) 1/5, d) 2/8 or 1/4, e) 3/100, Bonus: f) 1/18, g) 2/17
Using arrows to represent positive and negative fractions on number lines. Draw on the
board a number line with the numbers 6 inches apart and show students how to represent 3/5
by starting the “+3/5” arrow at 0.
+ 3
5
4
3
2
1
1
2
3
4
0
1



5
5
5
5
5
5
5
5
Have volunteers show how to represent other fractions on the number line: −2/5,
4/5, −4/5, −3/5.
−1
C-54

Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
Using a number line to add positive and negative fractions. Write on the board:
2 3
 
5 5
Demonstrate how to show the addition on the number line:
–3
5
+ 2
5
−1

4
5

3
5

2
5

1
5
0
1
5
2
5
3
5
4
5
1
Tell students that to show a number, we start at 0, and to add a number to a number that’s
already there, we start where the first number ended. Have volunteers show other additions:
4 2
 
5 5
3 4
 
5 5
1 2
 
5 5
1 3
 
5 5
2 4 3
  
5 5 5
(−2/5, −1/5, −3/5, +2/5, −1/5)
Provide students with BLM Number Lines from −1 to 1 (Fifths) to do the exercises below.
Exercises: Use a number line to add.
1 3
3 2
2 4
3 3
1 3
b)  
c)  
d)  
e)  
a)  
5 5
5 5
5 5
5 5
5 5
3 1 2
Bonus:   
5 5 5
Answers: a) −4/5, b) −1, c) +2/5, d) 0, e) −2/5, f) −2/5, Bonus: −2/5
3 1
f)  
5 5
Tell students that in parts e) and f), they are adding the same two numbers, so they should get
the same answers.
Bonus: Use other pairs of numbers to check that adding the same numbers gets the same
answer, no matter which order you add them in.
Adding positive and negative fractions without using a number line. Write on the board:
+ 3 − 5 = −2
3 5
2
  
8 8
8
these cancel
1 1 1 1 1 1 1 1
       
8 8 8 8 8 8 8 8
these cancel
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-55
SAY: If you can add positive and negative integers, then you can add positive and negative
fractions. You just need to add and subtract the numerators and keep the denominator the
same.
Exercises: Add the fractions by adding and subtracting the numerators.
1 3
3 2
2 4
3
3
2 7
7
2
b)  
c)  
d)  
e)  
f)  
a)  
7 7
10 10
9 9
8 8
11 11
11 11
2 4 1 3 5 6
Bonus:      
7 7 7 7 7 7
Answers: a) −4/7, b) −5/9, c) +2/8 or +1/4, d) 0, e) −5/11, f) −5/11, Bonus: −3/7
ASK: In parts e) and f), why should you get the same answers? (because you are adding the
same two numbers, just in a different order)
Using opposite numbers to add positive and negative fractions.
Exercises: Add and subtract the numerators to add each pair of fractions.
7
4
7
4
5 1
2 3
2 3
5 1
b)  
and  
c)   and  
a)   and  
10 10
10 10
9 9
9 9
8 8
8 8
Answers: a) +5/8 and −5/8, b) +3/10 and −3/10, c) +4/9 and −4/9
(MP.3, MP.6) ASK: In each pair, how do the answers compare? (they are opposite numbers;
they have the same number part but the opposite sign) If some students say that they are
opposite integers, point out that they are opposite numbers but they are fractions, not integers.
Then ASK: How could you have predicted that the answers would be opposite numbers? (the
numbers that are being added are opposite) Use the arrows to demonstrate how −1/5 + 4/5
and +1/5 − 4/5 have opposite sums because each number being added is opposite.
Bracket notation. Remind students that when adding positive and negative numbers, people
often write the two numbers being added in brackets. Write on the board:
 3   5 
     or  3  5
 8   8 
8 8
(+3) + (−5) or + 3 − 5
SAY: You can change the notation back using these rules:
+(+) = +
+(−) = −
Exercises: Write the addition without brackets, then evaluate.
 4  5
 3  4
 3  7
 1  6 
b)     
c)     
d)     
a)     
 10   10 
 10   10 
 8   8 
 9   9 
Answers: a) +4/10 − 5/10 = −1/10, b) −3/10 − 4/10 = −7/10, c) −3/8 + 7/8 = +4/8 or +1/2,
d) −1/9 + 6/9 = 5/9
C-56
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
Subtracting numbers by adding their opposite number. Remind students that you can
subtract numbers by adding their opposite. Write on the board:
+3 − (−5) = +3 + (+5) = + 3 + 5
and
+3 − (+5) = + 3 + (−5) = +3 − 5
SAY: You can change subtraction to addition using these rules:
−(+) = −
−(−) = +
Exercises: Subtract.
 4   1
5  1
2  5
b)     
c)    
a)    
 7   7 
9  9 
8  8 
Answers: a) −4/9, b) +5/7, c) −3/8, d) −7/7 = −1
 2  5
d)     
 7   7 
Extension
(MP.4) If students did Extension 2 from Lesson NS7-6, they can extend the concept to fractional
time zones, again using a number line to help them. Display the time zones shown below:
Time zones
Kathmandu, Nepal
Kabul, Afghanistan
Helsinki, Finland
Rome, Italy
St. John’s, Canada
New York City, USA
Chicago, USA
3
4
1
4
2
+2
+1
1
3
2
−5
−6
5
Give students real-world problems to solve based on these cities. Sample problems:
a) A sporting event is being held in New York City at 2 p.m. Ava lives in Kabul. At what time
should she turn on her television to watch the event live?
b) Amit lives in Kathmandu. His friend Lily lives in St. John’s. Amit wants to call Lily when he gets
home from school at 3:15 p.m. Lily gets up at 7:00 a.m. Will she get the call after she gets up?
c) An acting awards ceremony is being held in Chicago at 7:30 p.m. Ross lives in St. John’s. At
what time should he turn on his television to watch the event live?
Answers: a) 11:30 p.m.; b) no, she will get the call at 6:00 a.m.; c) 10:00 p.m.
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-57
NS7-12
Mixed Numbers and Improper Fractions
Pages 53–55
Standards: preparation for 7.NS.A.1
Goals:
Students will convert between mixed numbers and improper fractions.
Prior Knowledge Required:
Can name the proper fraction shaded in a picture
Vocabulary: improper fraction, mixed number
Naming mixed numbers and improper fractions. Tell students that you made 3 pies, cut one
of them into 4 pieces, and ate a piece. Draw on the board:
ASK: How much pie is left? (2 3/4 pies) Write the mixed number on the board, and tell students
that it is called a mixed number because it is a mixture of a whole number and a fractional part.
Now tell students that we can also write 2 3/4 as a fraction without a whole number part. But to
do that we need to make all the shaded pieces the same size. Cut the other two pies into
quarters to demonstrate.
ASK: How many parts are in one whole? (4) Write this as the denominator. ASK: How many
parts are shaded? (11) Write this as the numerator:
11
4
SAY: Eleven fourths is an improper fraction because the numerator is greater than the
denominator. The amount that an improper fraction represents is greater than 1.
Exercises: Name the improper fraction and mixed number for these models.
Answers: a) 8/3, 2 2/3; b) 9/5, 1 4/5; c) 13/6, 2 1/6
C-58
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
For students who need extra help writing the mixed number, provide the following structure:
Using pictures to convert between mixed numbers and improper fractions.
Exercises: Draw a model for each improper fraction, then write the equivalent mixed number.
Answers: a) 2 1/2, b) 3 2/3, c) 1 3/5
If students are struggling, have them first practice drawing whole pies divided into halves, thirds,
quarters, and so on.
Exercises: Sketch the pies for each mixed number. Divide all the whole pies into the same
number of parts and write the equivalent improper fraction.
Answers: a) 9/4, b) 29/8, c) 11/8, d) 8/3, e) 23/6
If students are struggling, show them how to cut a pie into eighths by first cutting it into fourths,
or how to cut a pie into sixths by first cutting it into thirds.
Comparing and ordering mixed numbers and improper fractions.
Exercises: Draw a picture to show each number. Then decide which is greater.
Bonus: Write the improper fractions as mixed numbers. Then put all the numbers in order from
greatest to least.
1
27
11
1
36
3
3
7
4
4
4
4
4
4
4
Answers: a) 2 3/4; b) 4 3/5; Bonus: written as mixed numbers, they are 3 1/4, 6 3/4, 2 3/4,
7 1/4, 9, 4 3/4. In order from greatest to least, they are 9 > 7 1/4 > 6 3/4 > 4 3/4 > 3 1/4 > 2 3/4.
Using multiplication to convert mixed numbers to improper fractions. Draw on the board:
3
1

2
2
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-59
ASK: How many halves are in 3 1/2? (7) Write “7” as the numerator. Then show how to get 7 by
using multiplication and addition:
3×2
+
1
SAY: 3 wholes × 2 halves in each whole = 6 halves altogether. Then add the 1 extra half.
Exercises:
1. How many halves are in …
a) 4
b) 7
c) 2
d) 2
1
2
e) 5
1
2
Answers: a) 8, b) 14, c) 4, d) 5, e) 11, f) 23, Bonus: 2,001
1
2
f) 11
1
2
Bonus: 1,000
2. Write the mixed number as an improper fraction.
1
1
3
5
5
b) 5
c) 4
d) 7
e) 6
a) 3
6
9
8
6
7
Selected solution: a) 3 × 7 + 1 = 22, so 3 1/7 = 22/7
Answers: b) 31/6, c) 39/9, d) 61/8, e) 41/6, Bonus: 42,011/60
Bonus: 700
11
60
Using division to convert improper fractions to whole numbers. Write on the board:
3=
6
2
because 3 × 2 = 6
SAY: Three equals six halves, because there are two halves in each whole, so six halves in
three wholes. SAY: You can also go the other way. If you have 6 halves and there are 2 halves
in each whole, then there are 3 wholes. Write on the board:
6÷2=3
Exercises: Write each improper fraction as a whole number.
Answers: a) 9, b) 7, c) 6, d) 4, Bonus: 9
Using division with remainders to convert improper fractions to mixed numbers. Draw on
the board:
C-60
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
Point to the 2 in 7/2 and SAY: I know there are 2 parts in each whole, so the denominator in the
mixed number is 2 (point to the 2 on the other side of the equation). SAY: We want to know how
many wholes—how many groups of 2 halves—are in 7 halves, and how many halves are left
over. ASK: What operation should I use to find out? (division) Draw on the board:
So
7÷2=
7
1
3
2
2
Exercises: For each improper fraction, use division with remainders to write the mixed number.
Draw a picture to show your answer.
Answers: a) 14 ÷ 3 = 4 R 2, so 14/3 = 4 2/3; b) 11 ÷ 8 = 1 R 3, so 11/8 = 1 3/8; c) 11 ÷ 4 = 2 R
3, so 11/4 = 2 3/4; d) 10 ÷ 3 = 3 R 1, so 10/3 = 3 1/3
(MP.1) The fraction part of a mixed number is less than 1. Point to the division on the board
(7 ÷ 2 = 3 R 1) and SAY: The remainder is less than the number being divided by. Point to the
mixed number on the board (3 1/2) and SAY: The numerator in the mixed number is less than
the denominator. Point out the connection between the two statements. SAY: The remainder in
the division statement is the numerator in the mixed number, and the number being divided by
is the denominator.
If fractions are equivalent, so are their opposites. Remind students that if two positive
numbers are equivalent, then their negative opposites are equivalent, too. Write on the board:
3
6
6
so 3  
2
2
Exercises:
1. Write the mixed number for the improper fraction or the improper fraction for the mixed
number.
7
1
9
3
1
b) 2
c) 
d) 6
e) 5
a) 
4
5
5
5
2
Answers: a) −1 3/4, b) −11/5, c) −1 4/5, d) −33/5, e) −11/2
2. Use the location of the positive fractions and mixed numbers to place their opposites. Hint:
Use 0 as a mirror line.
1
15 1
1
1
4
5
4
4 3
−5
−4
−3
−2
−1
0
1
2
3
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
4
5
C-61
Answers:
1
15
−
−4
3
4
−5
−1
−4
3. Show 3
−5
−2
−
1
5
−1
0
1
2
3
4
5
1
2
3
4
5
3
4
5
1
1
and − 3 on the number line.
4
4
−4
Answers:
−3
−5
−3
1
4
−3
−2
−1
0
1
4
−4
3
−3
−2
−1
0
1
2
1
4
If students struggle with Question 2 or 3, emphasize that to find −3 1/4, for example, they should
move left from −3, going 1/4 of the way from −3 to −4.
Extensions
1. The fraction 3/5 represents the fraction of Stick A that Stick B is. Turn the fraction upside
down to make it 5/3. What does this fraction represent?
Answer: The fraction of Stick B that Stick A is.
2. Jane needs 3 2/3 cups of flour. She has two measuring cups. Cup A holds 1/3 cup and Cup B
holds 1/6 cup.
a) If Jane uses Cup A, how many scoops would she need?
b) If Jane uses Cup B, how many scoops would she need?
Answers: a) 11, b) 22
3. If twice “tar” is “tartar,” then 4/3 of “tar” is “tart.” Using the same logic, what is …
6
3
4
of “tin”?
b)
of “table”?
c)
of “eras”?
a)
5
2
3
Answers: a) tint, b) tablet, c) eraser
C-62
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
NS7-13
Adding and Subtracting Mixed Numbers
Pages 56−58
Standards: 7.NS.A.1d
Goals:
Students will add and subtract positive and negative mixed numbers.
Prior Knowledge Required:
Can add and subtract positive and negative fractions
Can convert between mixed numbers and improper fractions
Vocabulary: negative, opposite, positive
Materials:
BLM Adding and Subtracting Mixed Numbers—Advanced (pp. C-92–93, see Extension 4)
Adding mixed numbers with the same denominator, no regrouping required. Draw on the
board:
+
1
1
6
+
=
2
1
6
=
ASK: How many whole pies are shaded altogether? (3) Shade the three whole circles in the
picture. ASK: How many sixths are shaded? (2) Shade two sixths in the picture. Have a volunteer
write the sum. (3 2/6) Point out that when you add two mixed numbers, you can add the wholes
and the parts separately. SAY: 1 plus 2 is 3 and 1/6 plus 1/6 is 2/6. So, 1 1/6 plus 2 1/6 is 3 2/6.
Exercises: Add.
3
1
1
2
Bonus: 1  2  4  3
8
8
8
8
Answers: a) 7 3/5, b) 5 5/8, c) 3 6/7, d) 8 7/10, Bonus: 10 7/8
Adding mixed numbers by converting to improper fractions. Remind students that they can
change any mixed number to an improper fraction, then have them change 1 1/6 and 2 1/6 to
improper fractions (7/6 and 13/6). Write on the board:
1
1 7 13
1 2  
6
6 6
6
20

6
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-63
SAY: When the question is given in terms of mixed numbers, the answer has to be in terms of
mixed numbers, too. Write on the board:
3
2
1
3
6
3
ASK: Is this the same answer we got before? (yes)
Exercises: Do the same additions from the last exercise but, this time, change the mixed
numbers to improper fractions. Make sure you get the same answer as you did before.
3
1
1
2
Bonus: 1  2  4  3
8
8
8
8
Answers: a) 17/5 + 21/5 = 38/5 = 7 3/5, b) 11/8 + 34/8 = 45/8 = 5 5/8, c) 10/7 + 17/7 = 27/7 =
3 6/7, d) 53/10 + 34/10 = 87/10 = 8 7/10, Bonus: 11/8 + 17/8 + 33/8 + 26/8 = 87/8 = 10 7/8
Choosing between ways of adding mixed numbers: changing to improper fractions or
adding the parts and wholes separately. Write on the board:
3
4
2 1
5
5
ASK: What do the whole numbers add to? (3) What do the parts add to? (7/5) ASK: Is the
answer three and seven fifths? (no) Why not? (3 7/5 is not a mixed number) SAY: You can still
add the fractions this way, but now there is an extra step. Write on the board:
3
7
2
 3 1
5
5
Have a volunteer write the answer on the board. (4 2/5) Have students add the same fractions
by changing them to improper fractions. If they don’t get the same answer (4 2/5), they should
check with someone who did to find the mistake.
Remind students that, when changing to improper fractions, they need to remember to change
the answer back to a mixed number.
(MP.5) Exercises: Add both ways. Circle the easier method.
4
9
3
2
3
4
2
7
b) 1  1
c) 1  1
d) 3  2
a) 4  7
8
8
5
5
10
10
10
10
1
1
1
1
1
Bonus: 4  7  3  8  6
2
2
2
2
2
Sample Solution: b) 8/5 + 9/5 = 17/5 = 3 2/5 or 2 + 7/5 = 2 + 1 2/5 = 3 2/5
Answers: a) 11 5/8; c) 2 9/10; d) 6 3/10, Bonus: 30 1/2
C-64
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
(MP.5) ASK: Why is it harder to use improper fractions to add? (you have to remember to
change your answer back to a mixed number; you are adding bigger numbers, so more likely to
make a mistake) PROMPT: What makes you more likely to make a mistake?
Then ASK: Why is it harder to add the parts and wholes? (when the parts add to more than 1
whole, you have to regroup) SAY: Some of you will like to change the fractions to improper
fractions all the time and, if you do it right, it will always get the right answer. But some of you
will like to add the other way because it will often save you work, and reduce the chance of
making a mistake. Both ways are okay.
Subtracting mixed numbers without regrouping. Draw on the board:
3
2
4 1
5
5
ASK: How many wholes are we taking away? (1) Cross out one whole in the picture. How many
fifths are we taking away? (2) Cross out two parts in the picture. ASK: How many wholes are
left? (3) How many parts are left? (1) Point out that you subtracted the wholes (4 − 1 = 3) and
the parts (3/5 − 2/5 = 1/5) separately:
3
2
1
4 1  3
5
5
5
Exercises: Subtract the wholes and the parts separately.
3
1
Bonus: 290  286
5
5
Answers: a) 1 2/7, b) 5 2/8 or 5 1/4, c) 3, Bonus: 4 2/5
Subtracting mixed numbers by converting to improper fractions.
Exercises: Do the same subtractions from the last exercise but, this time, change the mixed
numbers to improper fractions. Make sure you get the same answer as you did before.
3
1
Bonus: 290  286
5
5
Answers: a) 1 2/7, b) 5 2/8 or 5 1/4, c) 3, Bonus: 4 2/5
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-65
Choosing between ways of subtracting mixed numbers: changing to improper fractions
or subtracting the parts and wholes separately. Write on the board:
3
4
2 1
5
5
SAY: I want to subtract one whole and four fifths. Show a first attempt:
SAY: I have only three fifths to take away, but I need to take away another fifth. ASK: What can
I do to make another fifth to take away? (split the extra whole into fifths) Demonstrate doing so,
then take away the other fifth:
ASK: What is left? (4/5) Write on the board:
3
4
8
4 4
2 1  1 1 
5
5
5
5 5
SAY: We changed the picture from showing 2 3/5 to showing 1 8/5. We could then take away
1 4/5.
Exercises:
1. Take away 1 from the whole number part and add 1 whole to the fraction part.
Hint: It might help to draw a picture.
3
5
1
2
b) 5
c) 6
d) 3
a) 8
4
5
8
2
Answers: a) 7 7/4, b) 4 3/2, c) 5 13/8, d) 2 7/5
(MP.1) 2. Subtract by subtracting the parts and wholes separately. Then do the same questions
by changing both numbers to improper fractions. Make sure you get the same answer both
times.
5
2
1
3
4
9
9
7
b) 4  2
c) 7  2
d) 3  1
a) 7  3
10
10
10
10
8
8
5
5
Answers: a) 4 3/8, b) 1 3/5, c) 4 5/10 or 4 1/2, d) 2 2/10 or 2 1/5
(MP.3) 3. Sara said that 4 2/9 − 1 7/9 = 3 5/9. What is the mistake?
Answer: The fraction parts were subtracted in the wrong order, 7/9 − 2/9, but 2/9 is the fraction
part of the bigger number. The correct answer is 2 4/9.
C-66
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
1
3
(MP.5) 4. How would you subtract 9  4 ? ______
5
5
A. Separate the wholes and the parts.
B. Change the mixed numbers to improper fractions.
Explain your choice.
Sample answer: “I would use improper fractions because subtracting the parts requires
regrouping.”
(MP.5) Adding positive and negative fractions.
Exercises: Add the positive and negative numbers by first changing them to improper fractions.
Remember to change your answer back to a mixed number if you need to.
1
1
2
1
2
2
1
b)  7  1
c)  3  4
d)  5  4
e)  3  6
a)  2  1
2
3
3
3
2
3
3
Answers: a) −1/2, b) −5 1/2, c) 1 2/3, d) −2/3, e) −10
Subtracting positive and negative fractions. SAY: You can subtract a number by adding the
opposite number. Write on the board:
−(+) = −
−(−) = +
Exercises: Subtract.
1 
1
1 
1
1  3
4  2
b) 2  5 
c) 3  2 
d) 2  7 
a) 3  4 



2  2
3 
3
4  4
5  5 

 3 


2  3
4  2
4
2 
7
e) 5   2 
f) 1   2 
g) 3   7 
h) 5   3 









 5 
 11  11
9  9
5
8  8
Answers: a) 8, b) −7 2/3, c) −2/4 or −1/2, d) 4 3/5, e) 7 5/8, f) −1 1/5, g) 3 7/9, h) −1 6/11
(MP.4) Word problems practice.
Exercises:
1
a) Mindy’s time zone is 8 hours behind Sam’s time zone. Sam’s time zone is 4 . What is
2
Mindy’s time zone?
2
7
b) A gulper eel is at a depth of −1
miles. A frilled shark is at a depth of −
miles. How
10
10
much lower is the eel than the shark?
Answers: a) −3 1/2, b) 5/10 mile or 1/2 mile
NOTE: If students struggle with Question 12 on AP Book 7,1 p.58, remind them how to locate
3
3
mixed numbers on a number line. For example, 2 is of the way from −2 to −3.
4
4
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-67
Extensions
1. The police overhear a phone conversation where Nina tells Ted that her time zone is 10 3/4
hours ahead of Ted’s. The police know that Ted lives in time zone −5. What does this tell the
police in their search for Nina? Be as specific as you can. Hint: Use the Internet.
Answer: Nina’s time zone is +5 3/4. According to an Internet search, this places her in Nepal.
2. Subtract mentally.
2
3
b) 1 
a) 1 
3
5
3
3
d) 6  5 
e) 5  1 
7
10
Answers: a) 1/3, b) 2/5, c) 3/7, d) 4/7, e) 3 7/10, f) 3 1/2
4

7
1
f) 8  4 
2
c) 1
3. a) Teach students how to subtract fractions by separating the parts and the wholes, but
without the need to regroup even when it looks as though it should be required:
2 5
2
5
4  1  (4  1)  
 9 9 
9
9
 3
 3  
 9 
2
6
9
Have students do more like this:
1
4
1
7
ii) 6  3
i) 5  2
5
5
8
8
4
7
iii) 190  185
9
9
b) Add or subtract.
4 
1
i) 2  4 

5  5
1
2
iii) 4  3
5
5
3
2
ii) 3  4
7
7
c) What is the mistake?
1
2
3
4  3  1
5
5
5
Answers: a) i) 2 2/8 or 2 1/4, ii) 2 2/5, iii) 4 6/9 or 4 2/3; b) i) 1 2/5, ii) 6/7, iii) −4/5; c) The
student added the fraction parts as though both were positive, when in fact the 1/5 in −4 1/5 is
a −1/5. The correct answer is −4/5.
4. Teach students to add and subtract positive and negative fractions another way. See BLM
Adding and Subtracting Mixed Numbers—Advanced.
C-68
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
NS7-14
Adding and Subtracting Fractions with
Different Denominators
Pages 59–60
Standards: 7.NS.A.1d
Goals:
Students will add and subtract positive and negative fractions, including mixed numbers, that
have different denominators
Prior Knowledge Required:
Can add and subtract positive and negative fractions, including mixed numbers, that have the
same denominator
Can find the lowest common multiple of two numbers
Vocabulary: denominator, lowest common denominator (LCD), lowest common multiple
(LCM), numerator
Adding fractions with unlike denominators. Write on the board:
1 2

3 5
1 2

5 5
Discuss how they are different (like or unlike denominators). Then SAY: One fifth plus two fifths
is three fifths, but one third plus two fifths isn’t three of anything, because thirds are not the
same size as fifths. If you want to add one third and two fifths, you have to make them have the
same denominator. Write on the board:
1× 5
=
3× 5
2× 3
=
5× 3
SAY: The easiest way to make two fractions have the same denominator is to use the product
of the denominators. Have volunteers finish the equivalent fractions. (5/15 and 6/15) Then SAY:
Now the fractions are easy to add. Write on the board:
1 2
5
6
11
 


3 5 15 15 15
Exercises: Add the fractions.
1 3
1 2
1 1
3 2
a) 
b) 
c) 
d) 
3 5
4 5
3 2
5 7
Answers: a) 14/15, b) 13/20, c) 5/6, d) 31/35
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-69
Using the LCD of the fractions to add. Write on the board:
3 1
 

4 6 24 24
SAY: I put 24 as the denominator because 24 is 4 × 6. Have volunteers dictate both numerators
(18 and 4), then the sum of the fractions. (22/24 or 11/12) SAY: We ended up with denominator
12, but we could have started with denominator 12. We know 12 would work, too, because 12 is
a multiple of 4 and 6. Write on the board:
3 1
 

4 6 12 12
Have volunteers dictate both numerators (9 and 2), then the sum of the fractions (11/12). ASK:
Did we get the same answers both times? (yes) SAY: You’ll get the same answer no matter
which multiple of both 4 and 6 you use, so you might as well use the lowest common multiple.
Exercises: Use the lowest common multiple of the denominators to add the fractions.
a)
5 1
 


9 6 18 18
b)
4 5




9 12 36 36
c)
3 7




8 12 24 24
d)
5
3




8 10 40 40
Answers: a) 10, 3, 13/18; b) 16, 15, 31/36; c) 9, 14, 23/24; d) 25, 12, 37/40
Remind students that, to find the lowest common multiple of two numbers, they can list the
multiples of the larger number, other than 0, until they find a multiple of the smaller number.
Write on the board:
8 and 10
10, 20, 30, 40
6 and 10
10 and 15
4 and 8
Have volunteers find the LCM for the other three pairs (30, 30, and 8). Point out that if one of
the numbers is a multiple of the other, then that is the LCM of the two numbers.
Exercises: Find the lowest common multiple.
a) 4 and 10
b) 3 and 6
c) 12 and 15
e) 5 and 8
f) 3 and 10
g) 8 and 12
Answers: a) 20, b) 6, c) 60, d) 12, e) 40, f) 30, g) 24, h) 48
d) 4 and 12
h) 12 and 16
Tell students that the lowest common multiple of the denominators of two fractions is called the
lowest common denominator of the fractions and is often written as just LCD.
C-70
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
Exercises: Use the LCD of the fractions to add the fractions.
a)
5 1

6 9
b)
2 1

3 4
c)
1 3
+
6 8
d)
3
2

10 15
e)
3 1

8 4
Answers: a) 17/18, b) 11/12, c) 13/24, d) 13/30, e) 5/8
Adding and subtracting positive and negative fractions. SAY: If you can add and subtract
positive fractions, then you can add and subtract negative fractions, too. Write on the board:
+(+) = +
+(−) = −
−(+) = −
−(−) = +
Exercises: Add or subtract.
7  3
9  3
7  1
3  5
b)    
c)    
d)    
a)    



6  4
8  6 
8  4
5  4
Answers: a) 13/8 or 1 5/8, b) −21/20 or −1 1/20, c) −17/12 or −1 5/12, d) −11/24
Remind students that an easy way to add and subtract positive and negative mixed numbers is
to change all the mixed numbers to improper fractions. They just have to remember to change
the answer to a mixed number.
Exercises: Add or subtract the mixed numbers by first changing them to improper fractions.
3 
7
3  5
1 
2
5 
7
b) 2  1 
c) 1  2 
d) 2  3 
a) 3  5 
4  10 
4 
3
6  10 
4  8 
Answers: a) −2 5/12, b) −4 9/20, c) 7/8, d) 6 16/30 or 6 8/15
Point out that for part d), even though 30 was the lowest common denominator of the fractions,
we ended up with a smaller denominator—that can happen sometimes.
Extensions
(MP.1) 1. Write each fraction as a sum of exactly three fractions, each with numerator 1.
5 1 1 1
Example:   
6 2 6 6
Hint: How does finding the factors of the denominator help?
7
17
13
13
11
a)
b)
c)
d)
e)
8
18
12
15
15
Answers: a) 7/8 = 1/4 + 1/2 + 1/8; b) 17/18 = 1/2 + 1/3 + 1/9; c) 13/12 = 1/2 + 1/3 + 1/4;
d) 13/15 = 1/3 + 1/3 + 1/5; e) 11/15 = 1/3 + 1/3 + 1/5
2. The LCM of 4, 6, and 9 is 36. Add or subtract the fractions by changing them all to have
denominator 36.
 3 1 1
5  1 1
a)    
b)    
 4 6  9
6  4 9 
Answers: a) 37/36 or 1 1/36, b) 25/36
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-71
(MP.3) 3. Would it be helpful to reduce the fractions to lowest terms before adding 3/12 + 4/12?
Explain.
Answer: No, it’s easier to add fractions with the same denominator, so leave them as is.
4. What fraction of the whole rectangle is shaded? Hint: First determine what fraction of the
whole rectangle each shaded part is. Then add those fractions.
Answer: 1/8 + 1/16 + 1/12 = 13/48
C-72
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
NS7-15
Estimating Sums and Differences of Mixed
Numbers
Pages 61–62
Standards: 7.NS.A.1d
Goals:
Students will estimate sums and differences of positive and negative mixed numbers by
rounding the mixed numbers to the nearest whole number.
Prior Knowledge Required:
Can compare fractions
Can convert improper fractions to mixed numbers
Vocabulary: rounding
Review the many ways to write one half. Draw several pictures of one half on the board:
Have a volunteer write the different names, or equivalent fractions, for one half. SAY: In a
picture showing one half, there are always twice as many parts in the whole as there are in the
shaded part. So, you can double the top number to get the bottom number.
Exercises: Write the missing denominator.
Answers: a) 10, b) 14, c) 24, Bonus: 8,264
SAY: If you know the bottom number of a fraction equivalent to 1/2, you can divide by 2 to get
the top number.
Exercises: Write the missing numerator.
Answers: a) 5, b) 18, c) 25, d) 60
Comparing a fraction to one half. Write on the board:
5
9
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-73
ASK: Is the circle more than half shaded? (yes) SAY: If more pieces are shaded than unshaded,
then the fraction is more than half. ASK: How many pieces are shaded? (5) How can you tell
that from the fraction? (the numerator tells you how many pieces to shade) How many pieces
are unshaded? (4) How can you tell that from the fraction? (the denominator minus the
numerator) Point out that the denominator tells you how many pieces are in one whole and the
numerator tells you how many pieces are shaded, so the number of unshaded pieces is the total
(point to the denominator) minus the shaded pieces (point to the numerator). Write on the board
the table below (but without the answers in brackets):
Fraction Shaded Unshaded
3
(3)
(4)
7
7
(7)
(5)
12
4
(4)
(5)
9
More or Less Than Half
(less)
(more)
(less)
Have students signal the answers in brackets (by raising the number of fingers for the first two
columns, and thumbs up for more and thumbs down for less).
Exercises: Is the fraction more than half or less than half?
Answers: a) more, b) more, c) less, d) less
SAY: Another way to compare a fraction to one half is to double the numerator. If more than half
the pieces are shaded, then double the number of shaded pieces will be more than the total
number of pieces. For example, if 5/8 is shaded, then double the number of shaded pieces is
10. The total number of pieces is 8, and 10 is more than 8, so 5/8 is more than 1/2.
Exercises: Double the numerator. Is the fraction more than half, less than half, or equal to half?
600
5
47
36
b)
c)
d)
a)
1,000
9
100
75
Answers: a) 10, more; b) 94, less; c) 1,200, more; d) 72, less
Rounding mixed numbers to the nearest whole number. Write on the board:
3
2
5
3
4
Tell students you want to round this number to the nearest whole number. ASK: Is this closer to
3 or to 4? (3) How do you know? (because 2/5 is less than half) SAY: Because 2/5 is less than
half, 3 2/5 is less than halfway from 3 to 4, so it is closer to 3. You just have to look at the
fraction part of a mixed number to decide which whole number it is closest to.
C-74
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
Exercises: Round to the nearest whole number.
3
3
1
17
b) 7
c) 6
d) 8
a) 4
5
3
8
35
Answers: a) 4, b) 7, c) 7, d) 8
Rounding improper fractions to the nearest whole number. Write on the board:
2
3
5
13
5
Point to the mixed number and ASK: What is this number, rounded to the nearest whole
number? (3) How do you know? (3/5 is more than half) Point to the improper fraction and repeat
the questions. (also closer to 3, because 13/5 = 2 3/5) Point out that it would be harder to tell
what whole number 13/5 is closest to, or even which two whole numbers it is between, without
changing it to a mixed number, or at least doing the division. Write on the board:
13 ÷ 5 = 2 R 3, so
13
3
2
5
5
Exercises: Change the improper fraction to a mixed number. Then round to the nearest whole
number.
61
37
14
26
39
37
b)
c)
d)
e)
f)
a)
8
3
4
3
5
3
Answers: a) 5, b) 8, c) 7, d) 9, e) 10, f) 12
SAY: Sometimes, the fraction is equally close to both whole numbers. In such cases, you round
up.
Exercises: Round to the nearest whole number.
1
7
10
850
b)
c)
d)
a) 3
2
100
4
2
Answers: a) 4, b) 4, c) 3, d) 9
Rounding negative numbers to the nearest integer. SAY: You can round negative numbers
the same way you round positive numbers. Since we know that 1 3/4 is closer to 2 than to 1, we
also know that −1 3/4 is closer to −2 than to −1. Show this on a number line.
−2
−1
0
1
2
Remind students that they can imagine folding the number line around the zero mark to see
where the opposite number would be. So, if a number is closer to 2 than to 1, then its opposite
is closer to −2 than to −1. SAY: Always round to the opposite of what you would round the
positive number to, even when the number is equally close to two integers.
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-75
Exercises: Round to the nearest integer.
1
3
13
5
b) 8
c) 
d) 
a) 2
5
4
2
4
Answers: a) −2, b) −9, c) −3, d) −3, e) −6, f) −9
e) 
19
3
f) 
26
3
Estimating sums and differences by using rounding. Write on the board:
1
7
3 5
4
9
SAY: I want to estimate the answer, which means that I don’t need the exact answer, just
something close. ASK: What integer is closest to 3 1/4? (3) What integer is closest to 5 7/9? (6)
Write on the board:
1
7
3  5 ≈ 3 − 6 = −3
4
9
Exercises: Estimate the answer by rounding both numbers to the nearest integer.
3 
1
1  2
1  2
a) 8  3 
b) 6  5 
c) 6  2 



4 
6
5  7
4  3
6  11
1  3
8  17 
d) 3  4 
e)    
f)    


5  2
2  5
3  4 
Answers: a) 8 − (−3) = 11, b) 6 − (−6) = 12, c) −7 + 2 = −5, d) −4 + 5 = +1, e) −1 + 6 = 5,
f) +3 − 4 = −1
Students who are struggling may need to be reminded of these rules:
+(+) = +
+(−) = −
−(+) = −
−(−) = +
Bonus: Estimate the answer by rounding all numbers to the nearest integer.
 5   11  15   35 
          
 2   5   4   9 
Answer: − 3 + 2 − 4 − 4 = −9
Deciding whether the estimate is too big or too small. Write on the board:
72
1
≈
4
Ask a volunteer to rewrite the subtraction using the closest whole numbers. (7 − 2) Then ask
another volunteer to write the estimated answer. (5) ASK: Do you think 5 is higher or lower than
the actual answer? (higher) PROMPT: When you take away 2, are you taking away more or
less than you are really supposed to? (less) SAY: When you take away a smaller number than
you need to, you get a bigger number than you need to. Have students verify that the answer is
C-76
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
less than 5 by doing the subtraction. (4 3/4) When students finish, allow volunteers to say how
they did the subtraction. If the solutions below do not come up, bring them up yourself:
1) Start at 2 1/4 and count up to 3, then to 7, so the answer is 4 3/4.
2) Change both numbers to improper fractions and subtract (28/4 − 9/4 = 19/4 = 4 3/4)
Exercises:
a) Predict whether the estimate is too high or too low.
2
2
3
1
ii) 7  4 ≈ 7 − 4
Bonus: 12  4 ≈13 − 4
i) 9 − 5 ≈ 9 − 6
3
5
5
3
b) Check your predictions by doing the calculations.
Answers: a) i) too low, ii) too high, Bonus: too high (because 13 is too big and taking away 4 is
not enough); b) i) 3 1/3 is indeed higher than the estimated 3, ii) 2 3/5 is indeed lower than the
estimated 3, Bonus: 8 4/15 is indeed lower than the estimated 9
Extension
a) Predict whether the estimate is too high or too low.


1
1
1
ii) − 8 + 4  ≈ − 8 − 4 = −12
i) 7  3   7  3 ≈ 7 + 3 = 10
 3 

4
4
 5   2
1
4
iii) 2  2  2  3  1
iv) 1   3   2  3  1
 6   5 
4
5
b) Check by doing the calculations.
Answers: a) i) too low, ii) too high, iii) too high, iv) too low; b) actual answers are i) 10 1/4,
ii) −12 1/3, iii) 11/20, iv) 47/30 = 1 17/30
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
C-77
NS7-16
Multiplying a Fraction and a Whole Number
Pages 63–65
Standards: 7.NS.A.2
Goals:
Students will multiply positive and negative fractions by whole numbers.
Prior Knowledge Required:
Can add positive and negative fractions with the same denominator
Introduce fractions of a number. Tell students that you can take a fraction of a whole number,
too. ASK: If there are six friends and half of them are girls, how many are girls? If the distance to
a store is six miles, how far away is the halfway point? If I want to finish a race in six hours,
when should I be at the halfway point? Explain that since all of these questions have the same
numeric answer, we can say that the number 3 is half of the number 6.
Using pictures to show unit fractions of numbers. Draw on the board:
1
of 6 = ______
2
1
of 6 = ______
3
SAY: If I have 6 cherries and I want to eat half and give half away, then I have to divide the
cherries into two equal groups. Point to the first picture and ASK: How many are half of 6? (3)
SAY: There are 3 in each group, so 3 is half of 6.
Then tell students that if you want to eat only one third of the cherries, then you need to make 3
equal groups and take one of the three groups. ASK: What is one third of 6? (2)
Exercises: Draw pictures to show the fraction of the number.
1
1
1
of 4
b)
of 15
c)
of 15
a)
2
3
5
Answers: a) 2, b) 5, c) 3
Using division to find unit fractions of numbers. SAY: When you divide 6 objects into 3
equal groups, the size of each group is 6 ÷ 3 = 2. Write on the board:
1
of 6 = 6 ÷ 3 = 2
3
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Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
Exercises: Write a division statement to find the fraction of a number.
Answers: a) 12 ÷ 3 = 4, b) 10 ÷ 5 = 2, c) 16 ÷ 4 = 4, d) 12 ÷ 4 = 3, e) 12 ÷ 2 = 6
Using multiplication and division to find any fraction of a whole number. ASK: If I know
1/3 of 12 is 4, what is 2/3 of 12? (8) Draw a picture to help explain that 2/3 of 12 is twice as
many as 1/3 of 12.
SAY: If 1/3 of 12 is a group of 4, then 2/3 of 12 is 2 groups of 4, or 2 × 4 dots. So, 2/3 of 12 is 8.
Exercises: Use division and multiplication to find the fraction of the number.
Answers: a) 20 ÷ 5 = 4, and 2 × 4 = 8; b) 6, c) 9, d) 20, e) 32
“Of” can mean multiply. Tell students that the word “of” can mean multiply. For example, with
whole numbers, 2 groups of 3 means 2 × 3 objects. SAY: “Of” can mean multiply with fractions,
too: 1/2 of 6 means 1/2 of a group of 6 objects, or 1/2 × 6.
Exercises: Multiply by finding the fraction of the whole number.
Answers: a) 12, b) 8, c) 6, d) 15, e) 28
Multiplication as a short form for addition. Remind students that 3 × 4 is a short form of
writing 4 + 4 + 4, so 3 × 1/4 is a short form of 1/4 + 1/4 + 1/4.
Exercises: Rewrite the product as a sum.
Answers: a) 1/5 + 1/5 + 1/5, b) 2/5 + 2/5 + 2/5, c) 3/7 + 3/7 + 3/7 + 3/7, d) 5/13 + 5/13
Exercises: Rewrite the sum as a product.
Answers: a) 4 × 1/3, b) 5 × 3/11, c) 3 × 4/9
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
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Point out that the addition involves all identical fractions and, therefore, like denominators. So
the addition itself is quite simple.
Exercises: Multiply by adding.
Answers: a) 6/7, b) 9/5, c) 10/7, d) 8/5
The rule for multiplying a whole number by a fraction. Write on the board:
ASK: How did I get the 6? (2 + 2 + 2) SAY: But that’s 3 × 2. Write on the board:
ASK: How would you get the numerator? (multiply 5 times 3) Write 15 as the numerator.
Exercises: Multiply.
Answers: a) 3/12, b) 14/5, c) 18/8, d) 10/16
Review converting improper fractions to mixed numbers. Use division (21 ÷ 8 = 2 R 5, so
21/8 = 2 5/8). Remind students that some improper fractions might represent whole numbers.
For example, when you try to convert 24/3 to a mixed number, 24 ÷ 3 = 8 R 0, so there is no
fractional part.
Exercises: Write the number as a mixed number or whole number.
10
7
15
22
20
b)
c)
d)
e)
a)
4
3
2
5
4
Answers: a) 3 1/2, b) 3 3/4, c) 4 2/5, d) 3 1/3, e) 5
Multiplication of a whole number and a fraction commutes (i.e., order does not matter).
SAY: We know that order does not matter in multiplication; for example, 3 × 5 = 5 × 3.
Exercises: Multiply the same numbers in different orders. Do you get the same answer both
times? If not, find the mistake.
Answers: a) 4, b) 9, c) 10
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Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
Multiplying negative fractions by whole numbers. Write on the board:
3×2=2+2+2=6
3 × (−2) =
ASK: What repeated addition would you use for the second problem (have a volunteer show it):
3 × (−2) = (−2) + (−2) + (−2)
Remind students that they can write addition without brackets:
3 × (−2) = (−2) + (−2) + (−2) = − 2 − 2 − 2
Have a volunteer do the addition (−6). Write on the board:
 1  1  1  1
3            
 2   2   2   2 
Have a volunteer write the addition without brackets:
1 1 1
  
2 2 2
SAY: To multiply 3 times −1/2, you have to add −1/2 three times. We know how to do that
because we know how to add negative fractions. Have a volunteer write the answer. (−3/2)
Exercises:
a) Write the multiplication as repeated addition. Then write the answer.
 2
 1
 4
ii) 3  
iii) 5  
i) 4  
 3 
 5 
 5 
b) Multiply. Compare your answers with part a). What do you notice?
1
4
2
i) 4 
ii) 3 
iii) 5 
5
5
3
Answers: a) i) −4/5, ii) −12/5, iii) −10/3; b) i) 4/5, ii) 12/5, iii) 10/3; they are the opposite answers
Point out that the answer to 4 × (−1/5) is the negative of the answer to 4 × 1/5, because instead
of adding four 1/5s, you are adding four −1/5s. Show on the board:
 1
4  
 5 
−1
−
4
5
−
3
5
−
2
5
 1
4  
 5 
−
1
5
0
1
5
2
5
3
5
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
4
5
1
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Write on the board:
 2
7  
 3 
SAY: You can multiply 7 times −2/3 by first multiplying 7 times 2/3. ASK: What is 7 × 2/3? (14/3)
So what is 7 × (−2/3)? (−14/3) Write the answer on the board:
 2
14
7    
 3 
3
Exercises: Multiply.
 2
 2
 9
 3
b) 8  
c) 5   d) 6  
a) 3  
 5 
 7 
 10 
 5 
Answers: a) −6/5, b) −16/7, c) −45/10, d) −18/5
Extensions
1. Change the mixed number to an improper fraction and multiply.
Answers: a) 16/5 × 10 = 32, b) 8/3 × 9 = 24, c) 13/3 × 6 = 26
2. Point out the connection between the fact that 12 ÷ 2 × 3 = 12 × 3 ÷ 2 and the commutativity
of multiplication, since it is really just the fact that 12 × (1/2) × 3 = 12 × 3 × (1/2).
3. a) Find the fraction of the number.
2
3
8
of 3
of 5
of 9
5
3
9
b) Use the pattern to predict 354/502 of 502.
Answers: a) 2, 3, 8; b) 354
(MP.1) 4. a) Find the fraction of 20.
3
7
3
4
of 20
ii)
of 20 iii)
of 20
iv)
of 20
i)
4
10
5
5
b) Use the answers to write the fractions in order from least to greatest.
c) Use your answers to part a) to add.
3
4
7
7
3
3
i)
+
ii)
+
iii)
+
4
5
10
10
5
4
Answers: a) i) 15, ii) 14, iii) 12, iv) 16; b) ordered from least to greatest: 3/5, 7/10, 3/4, 4/5; c) i)
15 + 16 = 31, so 31/20; ii) 14 + 12 = 26, so 26/20 or 13/10; iii) 15 + 14 = 29, so 29/20
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Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
NS7-17
Problems and Puzzles
Pages 66–67
This lesson is a cumulative review.
Extensions
(MP.4) 1. Teach students that a small fraction of a large number can still be a large number.
Show students a model of 1/60.
a) Is 1/60 a large fraction or a small fraction?
b) If 1/60 of the total number of deaths in the United States in 2009 was due to lack of health
insurance, and there were 2,436,000 deaths, how many were due to a lack of health insurance?
(Challenge students to do this without a calculator.) Is that a large number or a small number?
Answers: a) small, b) 1/6 of 2,400,000 is 400,000 and 1/6 of 36,000 is 6,000, so 1/6 of
2,436,000 is 406,000, so 1/60 of 2,436,000 is 40,600
Explain that people who want to convince you that a problem isn’t very important might quote
the fraction of people instead of the number of people.
2. Complete the magic square. The fractions in each row, column, and diagonal of the magic
square add to −1.
1

2
5

6

Answers:


1
2
3
2
0



1
6
5
6
1
3
1
6



2
3
5
6
7
6
Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System
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3. Name two fractions that have different denominators and a difference between 1 1/2 and 2.
Sample answer: 3/4 and 5/2 work because 5/2 − 3/4 = 7/4, which is between 1 1/2 and 2.
4. Fill in the blanks.
adding −1
undoes
equals
subtracting ______
subtracting ______
equals
undoes
adding _______
Answers: Adding −1 undoes subtracting −1, which equals adding +1. Adding −1 equals
subtracting +1 which undoes adding +1. Both of these give +1 in the bottom right blank.
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Teacher`s Guide for AP Book 7.1 — Unit 2 The Number System