Unit 1 Expressions and Equations:
Operations on Rational Numbers
Introduction
This unit is an opportunity for teachers to ensure that students have a fundamental knowledge
of the basic skills needed to succeed in later units in Grade 8 and in further studies in high
school. Students will review multiplication and division, operations with fractions, order of
operations, operations with decimals, and integers.
The amount of time students need for review will vary considerably. Evaluate carefully to ensure
that students have these fundamental skills before they move on to later units. Many challenging
extensions are provided for students who have shown they have mastered the basic skills.
Signaling. In these lesson plans, we often suggest that all students signal their answers
simultaneously (e.g., by flashing thumbs up and thumbs down). For a complete description of
signaling, see Introduction, p. A-20.
Materials. We recommend that students work on grid paper to help organize their work,
especially for the lessons on the standard multiplication and division algorithms. If students
do not have grid paper, you will need to have lots of grid paper available (e.g., from BLM 1 cm
Grid Paper on p. I-1).
Fraction notation. We show fractions in two ways in our lesson plans:
Stacked:
1
2
Not stacked: 1/2
If you show your students the non-stacked form, remember to introduce it as new notation.
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-1
EE8-1
Multiplication Review
Pages 1–4
Standards: preparation for 8.EE.A.4, 8.EE.C.7, 8.F.A.3
Goals:
Students will review the meaning of multiplication and the multiplication algorithm.
Prior Knowledge Required:
Can add two single-digit numbers
Can multiply two single-digit numbers
Vocabulary: algorithm, associative property, column, commutative property, distributive property,
expanded form, multiple, power of 10, product, regrouping, row
Materials:
BLM Multiplication Standard Algorithm (pp. B-102–103), optional
(MP.7) Review the meaning of multiplication. Write on the board:
5+5+5+5
ASK: What operation can be used as a short form for repeated addition? (multiplication) How
many 5s are we adding? (4) How do we write 5 + 5 + 5 + 5 using multiplication? (4 × 5) In the
expression 4 × 5, what does the 4 tell us? (how many times to add) What does the 5 tell us?
(the number to be added) Write on the board:
4×5=5+5+5+5
add four 5s
Exercises:
1. Write using multiplication.
a) 8 + 8 + 8
b) 7 + 7 + 7 + 7 + 7
c) 9 + 9 + 9 + 9
d) 1 + 1 + 1 + 1 + 1 + 1
Answers: a) 3 × 8, b) 5 × 7, c) 4 × 9, d) 6 × 1
2. Write using addition.
a) 4 × 3
b) 5 × 2
c) 3 × 7
d) 5 × 1
Answers: a) 3 + 3 + 3 + 3, b) 2 + 2 + 2 + 2 + 2, c) 7 + 7 + 7, d) 1 + 1 + 1 + 1 + 1
(MP.7) Review the commutative property of multiplication. Draw on the board:
B-2
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
ASK: How many rows of dots are there? (4) How many dots are in each row? (5) Circle the rows
of dots on the board, as shown below:
ASK: How many dots are there? (20) What addition equation can we write to show there are
20 dots? (5 + 5 + 5 + 5 = 20) What multiplication equation can we write as a short form for this
addition? (4 × 5 = 20) Draw the diagram again on the board, but this time circle the columns of
dots, as shown below:
ASK: Has the number of dots changed? (no) What addition equation can we write to show there
are 20 dots? (4 + 4 + 4 + 4 + 4 = 20) What multiplication equation can we write as a short form
for this addition? (5 × 4 = 20) SAY: So 4 × 5 and 5 × 4 both equal 20. Write on the board:
4×5=5×4
commutative property
SAY: Remember, for multiplication, we can change the order of the numbers being multiplied
and the answer will stay the same. This is known as the commutative property.
Exercises:
1. Use the commutative property to rewrite the product in a different order.
a) 5 × 7
b) 9 × 6
c) 3 × 8
d) 4 × 6
Bonus: 1,234 × 576
Answers: a) 7 × 5; b) 6 × 9; c) 8 × 3; d) 6 × 4; Bonus: 576 × 1,234
2. What other operation shows the commutative property? Give an example.
Answer: addition; example: 7 + 5 = 5 + 7
(MP.7) Review regrouping. Draw on the board:
one
ten
hundred
thousand
SAY: Remember, there are 10 ones in a tens block and 10 tens in a hundreds block. Sometimes
when we are representing calculations using blocks, we get a number that uses more than ten
of any of these blocks. Draw on the board:
23 ones
14 tens
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-3
For each example, have a volunteer replace ten of the given blocks with a larger block. Then
have volunteers write the number and name of each block. (see pictures below)
2 tens and 3 ones
1 hundred and 4 tens
SAY: This is called regrouping.
Exercises: Draw base ten blocks to represent the number. Regroup to use the fewest blocks.
a) i) 27 ones
ii) 19 tens
Bonus: 42 hundreds
b) i) 5 tens, 13 ones
ii) 2 hundreds, 12 tens
Bonus: 15 hundreds, 3 tens
c) i) 2 hundreds, 17 tens, 14 ones ii) 3 hundreds, 19 tens, 12 ones
Bonus: 17 hundreds, 13 tens, 18 ones
Answers:
a) i) 2 tens, 7 ones; ii) 1 ten, 9 ones; Bonus: 4 thousands, 2 hundreds
b) i) 6 tens, 3 ones; ii) 3 hundreds, 2 tens; Bonus: 1 thousand, 5 hundreds, 3 tens
c) i) 3 hundreds, 8 tens, 4 ones; ii) 5 hundreds, 0 tens, 2 ones; Bonus: 1 thousand, 8 hundreds,
4 tens, 8 ones
(MP.5, MP.7) Review the multiplication algorithm. Write on the board:
324 =
hundreds +
tens +
ones
Ask a volunteer to fill in the blanks to finish writing 324 using place values. (3, 2, 4) SAY: Suppose
we wanted to multiply 324 by 2. Continue writing on the board:
324
=
3
hundreds +
2
tens +
× 2
4
ones
× 2
=
hundreds +
tens +
ones
ASK: What is 3 hundreds times 2? (6 hundreds) What is 2 tens times 2? (4 tens) What is 4 ones
times 2? (8 ones) Fill in the blanks on the right side of the equation. ASK: What number can we
write for 6 hundreds, 4 tens, and 8 ones? (648) Fill in the blank on the left side of the equation.
SAY: Instead of writing the place values, we can calculate the multiplication using a grid. Write
on the board:
3
2
×
6
4
2
4
8
SAY: Multiply the 2 by each digit in the original number.
B-4
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
Exercises: Multiply using a grid.
a) 132 × 3
b) 413 × 2
c) 121 × 4
Answers: a) 396; b) 826; c) 484; Bonus: 264,804
Bonus: 132,402 × 2
(MP.7) Multiplication with one regrouping. Write on the board:
324 =
3
hundreds +
2 tens +
4
× 3
ones
× 3
=
hundreds +
tens +
ones
Ask a volunteer to fill in the blanks after multiplying the expanded form of 324 by 3. (9, 6, 12)
ASK: How can we regroup 12 ones? (1 ten + 2 ones) SAY: We had 6 tens before. ASK: How
many tens do we have now? (7) Continue writing on the board:
=
9
hundreds +
6
tens +
12
ones
=
9
hundreds +
6
tens +
1
ten +
=
9
hundreds +
7
tens +
2
ones
2
ones
SAY: Using a grid, we show the 1 ten that was regrouped in the line above the multiplication.
Write on the board:
10 ones were regrouped as 1 ten
1
3
2
×
4
3
9
7
2
(3 × 2 tens + 1 ten) = 7 tens
Exercises: Multiply using a grid.
a) 248 × 2
b) 261 × 3
c) 132 × 4
Answers: a) 496; b) 783; c) 528; Bonus: 1,204,852
Bonus: 301,213 × 4
(MP.7) Multiplication with more than one regrouping. Write on the board:
137 =
1
hundred +
3
tens +
× 5
7
ones
× 5
=
hundreds +
tens +
ones
Ask a volunteer to fill in the blanks after multiplying the expanded form of 137 by 5. (5, 15, 35)
ASK: How can we regroup 35 ones? (3 tens + 5 ones) SAY: We had 15 tens before. ASK: How
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-5
many tens do we have now? (18) Continue writing on the board and point out where 35 ones
was regrouped and how we got 18 tens:
=
5
hundreds +
15
tens +
35
ones
=
5
hundreds +
15
tens +
3
tens +
=
5
hundreds +
18
tens +
5
ones
5
ones
ASK: How can we regroup 18 tens? (1 hundred + 8 tens) SAY: We had 5 hundreds before.
ASK: How many hundreds do we have now? (6) Continue writing on the board and point out
where 18 tens was regrouped and how we got 6 hundreds:
=
5
hundreds +
18
tens +
5
=
5
hundreds +
1
hundred +
8
=
6
hundreds +
8
tens +
ones
5
ones
tens +
5
ones
SAY: In a grid, we show the 3 tens and the 1 hundred that were regrouped in the top row. Write
on the board:
10 tens were regrouped
as 1 hundred
30 ones were regrouped as 3 tens
1
3
1
3
×
5
6
(1 × 5 + 1) hundreds
=
6
hundreds
7
8
5
(3 × 5 + 3) tens = 18 tens = 1 hundred + 8 tens
Exercises: Multiply using a grid. You may need to regroup more than once.
a) 485 × 2
b) 219 × 3
c) 736 × 4
Bonus: 324,673 × 4
Answers: a) 970; b) 657; c) 2,944; Bonus: 1,298,692
(MP.5, MP.7) Review multiplying by powers of 10. Write on the board:
324 =
3 hundreds +
2
tens +
× 10
4
ones
× 10
=
hundreds +
tens +
ones
Ask a volunteer to fill in the blanks by multiplying by 10 using place values. (30, 20, 40)
B-6
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
ASK: How can we regroup 30 hundreds? 20 tens? 40 ones? (3 thousands, 2 hundreds, 4 tens)
How many ones do we have after regrouping? (0) Continue writing on the board:
=
30
hundreds +
20
tens +
40
=
3
thousands +
2
hundreds +
ones
4
tens +
0
ones
(MP.8) Ask a volunteer to write the digits for this number. (3,240) ASK: What shortcut could we
have used to multiply 324 by 10? (write a zero at the end of the number) How many zeros
should we write at the end of the number if we are multiplying by 100? (2)
Exercises: Multiply by writing the appropriate number of zeros after the number.
a) 375 × 10
b) 243 × 100
c) 4,526 × 1,000
Bonus: 31,456 × 1,000,000
Answers: a) 3,750; b) 24,300; c) 4,526,000; Bonus: 31,456,000,000
(MP.7, MP.8) Using the associative property to multiply by multiples of 10. Write on
the board:
7 × 10
5 × 100
3 × 1,000
ASK: What is the shortcut to multiply a whole number by a power of 10? (write as many zeros
as the power of 10 at the end of the whole numbers) Ask for volunteers to find the products on
the board. (70, 500, 3,000)
Write on the board:
45 × 30
SAY: Here we are multiplying by a multiple of a power of 10. The number 30 can be written as
3 × 10. Write on the board:
= 45 × (3 × 10)
SAY: The associative property of multiplication means that we can multiply this in a different
order. We can multiply the 45 × 3 first. Write on the board:
= (45 × 3) × 10
ASK: What is 45 × 3? (135) What is a quick way to multiply 135 by 10? (write a zero at the end
of the number) Write on the board:
= 135 × 10
= 1,350
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-7
Exercises: Use the associative property to multiply mentally.
a) 41 × 20
b) 63 × 200
c) 112 × 3,000
Answers: a) 820; b) 12,600; c) 336,000
(MP.4) Review the distributive property. Draw on the board:
8
4
ASK: What calculation can we do to find the total number of squares? (4 × 8) How many
squares are there? (32) Draw on the board to the right of the previous picture:
4
5
3
A
B
SAY: All the shaded squares make a rectangle, which we will call Rectangle A. ASK: What
calculation can we do to find the total number of squares in Rectangle A? (4 × 5) What is 4 × 5?
(20) What calculation can we do to find the total number of squares in Rectangle B? (4 × 3)
What is 4 × 3? (12) What calculation can we do with these products to find the total number of
squares? (add them) ASK: Has the number of squares changed in the two diagrams? (no)
SAY: So the total for each diagram is the same. Write on the board:
4 × 8 = 4 × (5 + 3)
= (4 × 5) + (4 × 3)
SAY: Notice that the multiplication by 4 is applied—or distributed—to both parts of the 8, the 5
and the 3. We can think of this as the 4 being shared with the 5 and the 3. We now have two
products and we are adding the products. This is called the distributive property. Write on
the board:
5 × (6 + 2)
ASK: What operation is in front of the bracket? (multiplication) What operation is inside the
bracket? (addition) How do we rewrite the expression using the distributive property?
(5 × 6 + 5 × 2) Continue writing on the board:
= (5 × 6) + (5 × 2)
B-8
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
Exercises: Use the distributive property to rewrite the expression.
a) 7 × (8 + 3)
b) 9 × (4 + 6)
c) 23 × (10 + 4)
Bonus: 159 × (90 + 4)
Answers: a) (7 × 8) + (7 × 3), b) (9 × 4) + (9 × 6), c) (23 × 10) + (23 × 4),
Bonus: (159 × 90) + (159 × 4)
(MP.7, MP.8) Using the distributive property to multiply multi-digit numbers. Write on
the board:
24 × 43
SAY: If we write 43 as 40 + 3, we get this. Write and draw on the board:
40
3
= 24 × (40 + 3)
24
ASK: Why might this be easier than looking at the multiplication as 24 × 43? (we can break
the question into two easier steps) How can we rewrite this using the distributive property?
(see answer below)
= (24 × 40) + (24 × 3)
SAY: We want to find the answer without using a calculator. Have students calculate the answer
using pencil and paper. ASK: What is 24 × 4? (96) What is a shortcut for calculating 24 × 40?
(write a zero at the end of the answer for 24 × 4) What is 24 × 40? (960) What is 24 × 3? (72)
Continue writing on the board:
= 960 + 72
ASK: What is 960 + 72? (1,032) So what is the answer to 24 × 43? (1,032) Write the answer on
the board. Keep this calculation on the board for use in the next section.
Exercises: Use the distributive property to calculate.
a) 13 × 24
b) 42 × 21
c) 52 × 93
Bonus: 123 × 24
Answers: a) (13 × 20) + (13 × 4) = 312; b) (42 × 20) + (42 × 1) = 882;
c) (52 × 90) + (52 × 3) = 4,784; Bonus: (123 × 20) + (123 × 4) = 2,952
(MP.7) Using the multiplication algorithm to multiply multi-digit numbers. SAY: We just
used the distributive property to calculate 24 × 43. Refer students back to the calculation on the
board from the previous section. SAY: Now I want to calculate 24 × 43 using a grid. ASK: First,
how can we calculate 24 × 40 from 24 × 4? (multiply 24 × 4 and then write a zero at the end)
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-9
Write the following on the board, but do not include the arrow or the answers, which are shown
in italics below:
1
1
2
4
4
3
7
2
9
6
0
0
3
2
×
1
1
×
9
2
4
4
0
6
0
Ask a volunteer to multiply 24 × 3 on the grid to the left. As the volunteer works on the grid,
describe to the class what is happening: 4 × 3 gives 12 ones, which is 1 ten and 2 ones. We
place the 1 ten at the top of the tens column. 2 × 3 gives 6 tens, plus 1 ten carried over, which
gives a total of 7 tens and 2 ones. So the first row shows 72.
SAY: Now, on a new grid, we can multiply 24 × 40. Ask a second volunteer to multiply 24 × 4 tens
on a grid to the right. When multiplying 4 × 40, the answer is 160, which is 1 hundred, 6 tens,
and 0 ones. We place the 1 hundred at top of the hundreds column and then multiply 2 × 4,
which is 8, and add 1, to get 9 hundreds and 6 tens.
Point out that the answers the volunteers have written, 72 and 960, correspond to the answers
from the distributive property. ASK: What was the next step when we used the distributive
property? (add the two numbers) What should we do now? (add the two numbers) Ask a
volunteer to copy the 960 from the grid on the right into the grid on the left, and then add the
two numbers to get the final answer. (1,032)
Exercise: Use two grids to multiply.
a) 47 × 23
b) 54 × 28
Answers: a) 1,081; b) 1,512
(MP.7) Multiplying using the standard algorithm. SAY: We can save some writing by
combining the work into one grid. This procedure for doing the multiplication is known as
the standard algorithm. Write on the board:
regrouping from 23 × 50
1
regrouping from 23 × 4
1
×
B-10
2
3
5
4
23 × 54
= (23 × 4) + (23 × 50)
9
2
23 × 4
1
1
5
0
23 × 50
1
2
4
2
(23 × 4) + (23 × 50)
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
Explain that the two lines of calculation correspond to the two parts of the multiplication using
the distributive property.
Do another example with the class on the board. Write 23 × 75 in a grid on the board. One at
a time, ask one volunteer to multiply 23 × 5 and another volunteer to multiply 23 × 70, and then
have them combine the results, as shown in the grid below:
regrouping from 23 × 70
2
1
regrouping from 23 × 5
23 × 75
= (23 × 5) + (23 × 70)
2
3
7
5
1
1
5
23 × 5
1
6
1
0
23 × 70
1
7
2
5
(23 × 5) + (23 × 70)
×
Exercises: Multiply using the standard algorithm.
a) 47 × 26
b) 38 × 54
c) 73 × 98
Answers: a) 1,222; b) 2,052; c) 7,154; Bonus: 16,192
Bonus: 352 × 46
NOTE: Students who would benefit from more practice can complete BLM Multiplication
Standard Algorithm. (a) 516; b) 736; c) 896; d) 945; e) 1,242; f) 948; g) 2,268; h) 4,410;
i) 2,772; j) 7,521; k) 11,826; l) 18,972) You can use the second page of the BLM to create
additional practice questions.
Extensions
(MP.1) 1. The third number in the sequence 1, 2, 6, 24, … can be written as 1 × 2 × 3. Find the
next number in the sequence.
Answer: The pattern is: 1, 1 × 2, 1 × 2 × 3, 1 × 2 × 3 × 4. The next number is
1 × 2 × 3 × 4 × 5 = 120.
(MP.1) 2. The third number in the sequence 2, 4, 8, 16, … can be written as 2 × 2 × 2. Find the
next number in the sequence.
Answer: The pattern is: 2, 2 × 2, 2 × 2 × 2, 2 × 2 × 2 × 2. The next number is
2 × 2 × 2 × 2 × 2 = 32.
(MP.1) 3. The distributive property also applies when there is multiplication on the outside of the
brackets and subtraction inside the brackets. For example:
24 × (30 − 4) = (24 × 30) − (24 × 4)
Here the multiplication by 24 is applied, or distributed, to both parts of 34: the 30 and the 4.
We then have two products and must subtract.
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-11
Use the distributive property to rewrite the expression.
a) 32 × (20 − 2)
b) 73 × (20 − 6)
c) 41 × (30 − 6)
Answers: a) (32 × 20) − (32 × 2), b) (73 × 20) − (73 × 6), c) (41 × 30) − (41 × 6)
(MP.1) 4. Use the standard order of operations to evaluate each expression in Extension 3.
Answers: a) 640 − 64 = 576; b) 1,460 − 438 = 1,022; c) 1,230 − 246 = 984
(MP.1) 5. Check your answers to Extension 3 by multiplying using a grid.
a) 32 × 18
b) 73 × 14
c) 41 × 24
Answers: a) 576; b) 1,022; c) 984
(MP.1) 6. SAY: You can use the distributive property when there is subtraction inside the
brackets to perform mental math multiplication. For example, instead of multiplying 32 × 38 in
a grid, we can rewrite 38 as (40 − 2). Write on the board:
32 × 38
= 32 × (40 − 2)
= (32 × 40) − (32 × 2)
= 1,280 − 64
= 1,216
Have students perform the following multiplications using mental math.
a) 71 × 68
b) 82 × 29
c) 53 × 57
Hints: a) 71 × (70 − 2), b) 82 × (30 − 1), c) 53 × (60 − 3)
Answers: a) 4,970 − 142 = 4,828; b) 2,460 − 82 = 2,378; c) 3,180 − 159 = 3,021
(MP.1) 7. Find the missing digit(s). Each ? represents a different digit.
a)
b)
3
4
×
c)
?
2
2
?
×
?
?
4
8
×
6 9 2
8 7 2
1
Answers: a) 346 × 2 = 692; b) 218 × 4 = 872; c) 280 × 5 = 1,400
?
?
4
0
0
8. Find the missing digits.
a)
b)
4
×
3
5
?
?
3
4
8
0
?
?
?
?
c)
?
?
?
3
2
7
7
4
1
6
?
?
?
×
?
?
?
?
?
1
?
0
1
2
?
?
?
×
7 8 3 0
1 2 3 8 4
3 5 4 2
Answers: a) 435 × 18 = 3,480 + 4,350 = 7,830; b) 387 × 32 = 774 + 11,610 = 12,384;
c) 253 × 14 =1,012 + 2,530 = 3,542
B-12
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
EE8-2
Division Review
Pages 5–6
Standards: preparation for 8.EE.A.4, 8.EE.C.7, 8.F.A.3
Goals:
Students review the meaning of division and the division algorithm.
Prior Knowledge Required:
Can multiply using the standard algorithm
Can divide numbers where the product is less than 100
Vocabulary: array, dividend, divisor, long division, quotient, remainder
Materials:
34 counters
(MP.7) Review the two meanings of division. Write on the board:
12 ÷ 3 = 4
Ask for two volunteers to count the dots in each array. (12) For the first diagram, ask the
volunteer to circle groups of 4 dots in rows. For the second diagram, ask the volunteer to circle
groups of 3 dots in columns. The diagrams should look like this:
ASK: In the first diagram, how many groups are there? (3) How many items are in each group?
(4) Write underneath the first diagram:
12 ÷ 3 = 4
number of items in each group
number of groups
SAY: In the second diagram, how many groups are there? (4) How many items are in each
group? (3) Write underneath the second diagram:
12 ÷ 3 = 4
number of groups
number of items in each group
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-13
SAY: So when we write 12 ÷ 3 = 4, the 3 can either mean the number of items in each group or
the total number of groups. Write on the board:
12 ÷ 4 = 3
ASK: What could the 4 mean here? (either the number of groups or the number of items in each
group) SAY: Notice that this equation also describes the two diagrams we looked at earlier. In
the first diagram, there are 4 items in each group. In the second diagram, there are 4 groups.
SAY: The diagrams can be represented by two division equations: 12 ÷ 3 = 4 or 12 ÷ 4 = 3.
Exercises: Write two division equations for the diagram or description.
a)
b) 20 apples are divided into 5 baskets.
c) 24 people are divided equally into 4 vans.
Answers: a) 15 ÷ 3 = 5, 15 ÷ 5 = 3; b) 20 ÷ 5 = 4, 20 ÷ 4 = 5; c) 24 ÷ 4 = 6, 24 ÷ 6 = 4
(MP.7) Long division with a one-digit divisor and with remainders. SAY: Suppose you have
$34 to share among 6 people. Invite 6 people to the front of the class. Invite another person to
distribute 34 counters to represent dollar bills. Ask the distributor to try to give $6 to each
person. ASK: Was there enough money to give $6 to each person? (no) Have the people return
the counters to the distributor. Ask the distributor to try to give $4 to each person. ASK: Was
there enough money this time? (yes) How much does the distributor have left? ($10) What is
wrong with that? (the student distributing money can give each person at least $1 more) Have
the distributor give each person another dollar. ASK: How much money does the distributor
have left? ($4) Collect all the counters and have students return to their seats.
Draw on the board:
ASK: How many groups are there? (6) How many dollars are in each group? (5) How many
dollars were shared? (30) What calculation did you do to find this? (6 × 5) How many dollars
were not shared? (4) SAY: We can describe this division with a division equation.
Write on the board:
division equation
34
÷
6
dividend divisor
=
5
R4
quotient remainder
Remind students of the following:
• The dividend is the number being divided. It is the total number of items.
• The divisor is the number we are dividing by. In this case, it is the number of groups.
• The quotient is the number of items in each group.
• The remainder is the number of items left over that didn’t fit in a group.
B-14
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
SAY: We can perform this division using long division. Write on the board:
divisor
5
6 34
quotient
dividend
- 30
how many dollars were actually shared
4
remainder
Review and explain each term in this context: the dividend is how many dollars there were at
the beginning, the divisor is the number of people, the quotient is how many dollars each person
got, 30 is the number of dollars actually shared, and the remainder is how many dollars that
were not shared.
ASK: How do we calculate how many dollars were actually shared? (6 × 5) How do we calculate
how many dollars were not shared? (34 − 30) Why didn’t we give each person get 6 dollars
instead of 5? (6 × 6 = 36, but we had only $34 at the beginning) Why didn’t we give each person
4 dollars instead of 5? (4 × 6 = 24; we could have shared more dollars with each person) Write
on the board:
5 43
-
ASK: What is the divisor? (5) What is the dividend? (43) If we were sharing $43 among
5 people, why would a guess of $9 for each person be wrong? (9 × 5 = 45, which is more
money than we started with) Why would a guess of $7 for each person be wrong? (7 × 5 = 35,
so we could give each person more money) If 9 is too high a guess and 7 is too low a guess,
what is a good guess? (8) If we give $8 to each person, how many dollars will be shared? (40)
How did you calculate that? (8 × 5 = 40) How many dollars are left over? (3) How did you
calculate that? (43 − 40) Continue writing on the board:
8
5 43
- 40
8 × 5 = 40
3
43 − 40 = 3
The division equation is: 43 ÷ 5 = 8 R 3
(MP.2, MP.7) Determining if the estimate for the quotient is too high, too low, or just right.
SAY: When you multiply your estimate by the divisor, it might be too high, too low, or just right.
Write on the board:
9
7
8
a) 5 43
b) 4 34
c) 6 49
- 45
- 28
6
- 48
1
too high
too low
just right
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-15
ASK: How can you tell that the estimate in a) is too high? (45 is greater than the dividend, 43)
How can you tell that the estimate in b) is too low? (the remainder is greater than the divisor,
6 > 4) How can you tell that the estimate in c) is just right? (1 < 6)
SAY: If the estimate times the divisor is greater than the dividend, the estimate is too high. If the
remainder is greater than or equal to the divisor, the estimate is too low. If the remainder is less
than the divisor, the estimate is just right.
Exercises:
1. Indicate whether the estimate is too high, too low, or just right.
7
5
38
a)
6
9
47
b)
7
3
25
c)
6
6
42
d)
- 35
- 54
- 21
- 36
4
6
3
Answers: a) just right, b) too high, c) too low, d) too low
2. Use long division to divide. Then write a division equation.
a) 73 ÷ 9
b) 48 ÷ 7
c) 53 ÷ 8
Answers: a) 73 ÷ 9 = 8 R 1, b) 48 ÷ 7 = 6 R 6, c) 53 ÷ 8 = 6 R 5
(MP.7) Long division where the quotient has more than one digit. SAY: We want to share
$93 among 4 people. We can think of $93 as 9 ten-dollar bills and 3 one-dollar bills. Write on
the board:
4 93
SAY: When we try to guess the dividend here, we can see that it will need more than one digit
because the highest one-digit number is not enough: 4 × 9 = 36 is much smaller than 93. So we
know that the quotient will have two digits. This will require regrouping. Circle the first digit in the
dividend that is greater than the divisor. Here it is 9. Circle the digit 9 in the dividend. ASK: If we
divide 9 ten-dollar bills among 4 people, how many ten-dollar bills will go in each group? (2)
Continue writing on the board:
2
4 93
SAY: We write a 2 above the 9 in the quotient to show that 2 ten-dollar bills have been given to
each person. ASK: How many ten-dollar bills have been shared? (8) How did you calculate
that? (2 × 4 = 8) How many ten-dollar bills still have to be shared? (1) How did you calculate
that? (9 − 8 = 1) Continue writing on the board:
2
4 93
-8
1
B-16
2 × 4 = 8 ten dollar bills shared
9 − 8 = 1 ten dollar bill still to be shared
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
ASK: How much money has been shared so far? ($80) How did you calculate that? (8 × 10 = 80)
How much money still has to be shared? ($13) How did you calculate that? (93 − 80) SAY: We
show that in the long division with the 1 ten-dollar bill still to be shared and “bringing down” the
3 ones from the dividend. You can think of this as exchanging the single ten-dollar bill for 10
one-dollar bills and adding 3 more one-dollar bills. So now we have 13 ones to share. Continue
writing on the board.
2
4 93
-8
13
ASK: What is a good estimate for sharing $13 among the 4 people? (3) Why is 4 wrong?
(4 × 4 = 16 and we don’t have $16 to share; we only have $13 to share) SAY: We write the
digit 3 in the quotient above the 3 to show each person will get another $3. ASK: How many
one-dollar bills have been shared? (12) How did you calculate that? (3 × 4 = 12) How many
one-dollar bills are left over? (1) How did you calculate that? (13 − 12 = 1) Continue writing
on the board:
2
4 93
-8
13
- 12
1
ASK: What is the division equation? (93 ÷ 4 = 23 R 1) How much money did each person get?
($23) How much money was left over? ($1)
Repeat with 137 ÷ 4. Note that this time, 13 will be circled at the very first step, as shown in the
answer below:
34
4 137
- 12
17
137 ÷ 4 = 34 R 1
- 16
1
Exercises: Divide using long division. Then write the division equation.
a) 236 ÷ 5
b) 347 ÷ 8
c) 473 ÷ 9
Bonus: 1,359 ÷ 4
Answers: a) 47 R 1; b) 43 R 3; c) 52 R 5; Bonus: 339 R 3
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-17
Extensions
(MP.1) 1. Find the the missing numbers.
?9
8?
a) 2 1??
b) 4 ???
- 12
- 32
??
c) 6 ???
- 48
19
25
23
- 1?
- ??
- ??
?
1
5
Answers:
69
86
83
a) 2 139 , b) 4 345 , c) 6 503
- 12
- 32
- 48
19
25
23
- 18
- 24
- 18
1
1
5
(MP.1) 2. John is giving almonds to his friends. He wants to give the same number of almonds
to each friend. If he has 2 friends, 1 almond is left over. If he has 5 friends, 3 almonds are left
over. John has more than 15 almonds, but fewer than 30 almonds. What is the exact number
of almonds?
Answer: 23
(MP.1, MP.8) 3. When we say a number is “divisible” by another number, we mean that there is
no remainder, or the remainder is zero. For example, 32 is divisible by 4 because, when you
divide 32 by 4, the remainder is zero, as shown below:
8
4 32
- 32
0
Here is a shortcut to find out whether a number is divisible by 3:
Step 1: Add the digits in the number.
Step 2: If the total is divisible by 3, then the number is divisible by 3.
Example: 234 is divisible by 3 because 2 + 3 + 4 = 9 and 9 is divisible by 3. You can check
using long division:
78
3 234
- 21
24
- 24
0
B-18
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
Here is an explanation for why the shortcut works:
234 = (2 × 100) + (3 × 10) + 4
= (2 × 99 + 2) + (3 × 9 + 3) + 4
= (2 × 99) + (3 × 9) + (2 + 3 + 4)
Since 99 is divisible by 3, and 9 is divisible by 3, (2 × 99) + (3 × 9) is also divisible by 3. Whether
the whole expression is divisible by 3 will then depend on whether (2 + 3 + 4) is divisible by 3.
Since 2 + 3 + 4 = 9 and 9 is divisible by 3, then the original number, 234, is also divisible by 3.
Use the shortcut to find whether the number is divisible by 3. Then check your answer using
long division.
a) 423
b) 5,162
c) 7,143
Bonus: 1,843,452
Answers: a) 4 + 2 + 3 = 9, so 423 is divisible by 3; b) 5 + 1 + 6 + 2 = 14, so 4,162 is
not divisible by 3; c) 7 + 1 + 4 + 3 = 15, so 7,143 is divisible by 3; Bonus:
1 + 8 + 4 + 3 + 4 + 5 + 2 = 27, so 1,843,452 is divisible by 3
(MP.1, MP.8) 4. A leap year has 366 days instead of 365 days. A year cannot be a leap year if it
is not divisible by 4. Remember: We say a number is divisible by another number if the
remainder is 0. Which of the following cannot be leap years?
A. 1975
B. 1992
C. 2004
D. 2008
Answer: A, because 1975 cannot be a leap year (1975 ÷ 4 = 493 R 3)
(MP.1, MP.8) 5. Karen’s mother was born on February 29. Could she have been born in
1977? Explain.
Answer: February 29 only occurs during leap years. 1977 ÷ 4 = 494 R 1, so 1977 is not a
leap year. Karen’s mother could not have been born in 1977.
(MP.1) 6. For a number to be divisible by 4, the last two digits of the number must be divisible
by 4. Example: since 28 is divisible by 4, the number 1,928 is also divisible by 4.
Here is an explanation for why the shortcut works:
1,928 = 19 × 100 + 28
Since 100 is divisible by 4, 19 × 100 is divisible by 4. Whether a number is divisible by
4 will then depend on whether the last two digits, 28, are divisible by 4. 28 is 4 × 7,
so 1,928 is divisible by 4.
Use the shortcut to check whether the number is divisible by 4. Then check your answer using
long division.
a) 716
b) 2,734
c) 23,188
Bonus: Will the year 2049 be a leap year? Explain using the shortcut.
Answers: a)16 is divisible by 4, so 716 is divisible by 4; b) 34 ÷ 4 = 8 R 2, so 2,734 is not
divisible by 4; c) 88 is divisible by 4, so 23,188 is divisible by 4; Bonus: 49 ÷ 4 = 12 R 1, so 2049
is not divisible by 4, and 2049 is not a leap year.
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-19
EE8-3
2-Digit Division Review
Pages 7–9
Standards: preparation for 8.EE.A.4, 8.EE.C.7, 8.F.A.3
Goals:
Students review the meaning of division and the division algorithm.
Prior Knowledge Required:
Can divide by a one-digit divisor using the standard algorithm
Can write the division equation after dividing using the standard algorithm
Vocabulary: approximately equal to sign (≈), dividend, divisor, long division, quotient,
remainder, round down, round up, rounding
Materials:
BLM Long Division (pp. B-104–105), optional
(MP.7) Rounding to the nearest ten. SAY: Until now we have been dividing with one-digit
divisors. It is harder to estimate for a two-digit divisor than for a one-digit divisor. We will use
rounding to the nearest ten to help us. Write on the board:
28
tens
ones
SAY: To round to the nearest tens, look at the ones digit. If that digit is 5 or greater, round up to
the next highest multiple of 10. If the digit is below 5, round down to the next lowest multiple of
10. ASK: What is the ones digit in 28? (8) Do we round up or down? (up) What do we round up
to? (30) SAY: Another way to look at rounding to the nearest ten is to ASK: Is 28 closer to 20 or
to 30? (closer to 30)
Exercises: Round to the nearest multiple of 10.
a) 37
b) 24
c) 57
d) 18
Answers: a) 40, b) 20, c) 60, d) 20
(MP.7) Steps 1–3 of the long division with a two-digit divisor. SAY: To help us with our first
estimate for the quotient, we will be rounding the divisor to the nearest multiple of ten and
circling the first group of digits in the dividend larger than the divisor. Write on the board:
23 782
B-20
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
ASK: What is the divisor? (23) What is the dividend? (782) What is the divisor rounded to the
nearest multiple of 10? (20) Reading from left to right, what is the first group of digits in the
dividend larger than the divisor? (78) Write on the board:
23 782
23 ≈ 20
SAY: Remember that we can use the sign (≈) that means “approximately equal to” here.
ASK: What is the approximate answer to 78 ÷ 20? (3) Why is 4 not correct? (4 × 20 = 80, which
is too high) Why is 2 not correct? (2 × 20 = 40, which is much smaller than 78) SAY: So now we
will estimate 3 for the first digit in the quotient. Ask for a volunteer to come to the board and
multiply 23 × 3, as shown below:
23 782
- 69
23
´ 3
69
SAY: So, we have three steps of long division so far: Step 1, round the divisor; Step 2, estimate
the quotient; and Step 3, multiply by the divisor.
Exercises: Perform Steps 1–3 of long division.
a) 667 ÷ 29
b) 769 ÷ 24
Answers:
c) 1,538 ÷ 32
2
29
24
32
3
4
a) 29 667 ´ 2 , b) 24 769 ´ 3 , c) 32 1538 ´ 4
72
128
- 58
58
72
128
(MP.7) Steps 4–5 of long division with a two-digit divisor. SAY: The next steps in long
division are to subtract, check, and then bring down the next digit from the dividend. Write on
the board:
3
23 782
- 69
92
Step 4: Subtract and check
Step 5: Bring down the next digit of the dividend
SAY: Suppose my guess for the first digit of the quotient had been 4. ASK: How would I know
that estimate is too high? (23 × 4 = 92, which is greater than 78, so my estimate is too high)
SAY: An estimate for a digit of the quotient may be too high or too low. If that is the case, we
repeat Steps 2 and 3; in other words, keep revising your estimate for the quotient until it is
just right.
Exercises: Perform the next two steps of long division using your answers to the previous
exercises.
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-21
Answers:
2
3
b) 24 769
a) 29 667
- 58
4
c) 32 1538
- 72
87
- 128
49
258
(MP.7) Steps 6–8 of long division with a two-digit divisor. SAY: The next steps in long
division start the process over again as if you had a new question: Step 6, estimate the next
digit in the quotient; Step 7, multiply by the rounded divisor; and Step 8, subtract and check the
quotient—and go back to the earlier steps if you need to revise your estimate—and find any
remainder. Write on the board:
3
23 782
- 69
92
SAY: Now it’s as if we had a new division: 92 ÷ 23. So in this case, we estimate the next digit
in the quotient will be 4, multiply 23 × 4 = 92, and then subtract and check. Continue writing on
the board:
34
23 782
23
- 69
92
´ 4
- 92
92
0
SAY: We stop dividing when there are no more digits to bring down from the dividend. Then we
can write the division equation. ASK: What is the division equation? (782 ÷ 23 = 34 R 0)
Exercises: Finish the long division from the previous exercises.
Answers:
2
32
48
a) 29 667 , b) 24 769 , c) 32 1538
- 58
- 72
- 128
87
49
258
- 87
- 48
- 256
0
1
2
If you think students would benefit from extra practice, assign BLM Long Division.
(1. a) 47 R 1, b) 39 R 2, c) 45 R 3, d) 62 R 4, e) 173 R 0, f) 281 R 1, g) 253 R 0, h) 232 R 0,
i) 320 R 4, j) 534 R 5, k) 932 R 6, l) 520 R 1; 2. a) 21 R 2, b) 36 R 3, c) 23 R 10, d) 19 R 37,
e) 248 R 22, f) 256 R 0, g) 124 R 23, h) 415 R 4, i) 64 R 19, j) 75 R 32, k) 70 R 22, l) 57 R 28)
B-22
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
Extensions
(MP.1) 1. Find the missing numbers.
25
4?
a) 2? ???
2??
b) ?7 ??? 3
- 46
c) ?? 78 ??
12?
- 148
6?
- ???
-??
6
26
- ??
6?
- ??
???
- 252
22
Answers:
25
41
217
a) 23 581 , b) 37 1543 , c) 36 7834
- 46
- 148
- 72
121
63
63
- 115
- 37
- 36
6
26
274
- 252
22
(MP.1) 2. A prime number is a number greater than 1 that is divisible by only two numbers:
itself and 1. The first ten prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
Use division to find two prime numbers that have a product of …
a) 91
b) 187
c) 391
Answers: a) 13 × 7, b) 17 × 11, c) 23 × 17
(MP.1) 3. The numbers in each puzzle follow a pattern. Find the missing number in the
last puzzle.
542
15

31
557
17
30

17
968
31
2

179
23
?
42

18
9
Answers: 542 ÷ 17 = 31 R 15, 557 ÷ 31 = 17 R 30, 968 ÷ 23 = 42 R 2, and 179 ÷ 18 = 9 R 17.
So the missing number in the last rectangle is 17.
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-23
EE8-4
Fractions Review
Pages 10–13
Standards: preparation for 8.EE.A.2, 8.EE.B.5, 8.EE.C.7
Goals:
Students will review the meaning of a fraction, comparing fractions, adding fractions with the
same denominator, and adding fractions with different denominators.
Prior Knowledge Required:
Can add and multiply single-digit numbers
Vocabulary: denominator, equivalent fraction, lowest common denominator (LCD),
lowest common multiple (LCM), multiple, numerator
Materials:
BLM Finding Prime Numbers (p. B-106, see Extension 2)
BLM Finding the LCM Using Prime Number Factors (p. B-107, see Extension 3)
(MP.7) Review the meaning of a fraction. Draw on the board:
7
10
numerator: number of shaded parts
denominator: total number of parts in the whole
SAY: Remember that the top number of a fraction is called the numerator. It tells how many
parts of the whole are shaded. The bottom of a fraction is called the denominator. It tells the
total number of parts in the whole.
Exercises: Write a fraction for the shaded region.
a)
b)
c)
d)
Answers: a) 5/8, b) 3/4, c) 7/12, d) 17/30
(MP.7) Finding equivalent fractions. SAY: Some fractions are equal to others even though
they have different numerators and denominators. Draw on the board:
B-24
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
Ask a volunteer to shade in parts to show the fraction 3/4. After the volunteer has shaded the
parts, double the number of parts by drawing a dashed horizontal line through the middle, as
shown below:
ASK: How many parts are there now? (8) How many shaded parts are there now? (6) What
fraction is shown by the diagram? (6/8) Has the size of the shaded region changed? (no)
SAY: So the fractions 3/4 and 6/8 represent the same amount even though they have different
numerators and different denominators. We say that 3/4 and 6/8 are equivalent fractions and
write 3/4 = 6/8. ASK: What happened to the number of shaded parts? (doubled) What happened
to the total number of parts? (doubled) Write on the board:
3 3 ´2 6
=
=
4 4 ´2 8
SAY: We can find equivalent fractions by multiplying the numerator and denominator by the
same number.
Exercises:
1. Use multiplication to find the equivalent fraction.
a)
3 ´2
=
5 ´2
b)
5 ´3
=
6 ´3
c)
7 ´4
=
8 ´4
Answers: a) 6/10, b) 15/18, c) 28/32
2. What number are the numerator and denominator multiplied by to get the equivalent fraction?
a)
8´
9´
=
24
27
b)
4´
7´
=
24
42
c)
7´
8´
=
35
40
Answers: a) 3, b) 6, c) 5
(MP.7) Finding the lowest common multiple (LCM). Write on the board:
The multiples of 3 are 3, 6, 9, 12, …
ASK: What calculation can I do to find the multiples of 3? (3 × 1, 3 × 2, 3 × 3, 3 × 4, …) Write on
the board:
Multiples of 6 ___, ___, ___, ___, ___, ___
Multiples of 8 ___, ___, ___, ___, ___, ___
Ask volunteers to fill in the blanks. (6, 12, 18, 24, 30; 8, 16, 24, 32, 40)
ASK: What is the first number that appears in both lists? (24) SAY: We call 24 the lowest
common multiple, or LCM, of 6 and 8. In other words, it is the lowest number, or the first number
in the sequence, and it is common to both lists, or shared by both.
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-25
Exercises: Find the LCM of the numbers.
a) 4, 6
b) 6, 9
Answers: a) 12, b) 18, c) 30
c) 10, 15
(MP.7) Comparing fractions with different denominators. Draw on the board:
ASK: What is a fraction for each diagram? (5/8, 3/8) Which fraction is bigger? (5/8) Write on
the board:
5
8
3
8
ASK: What is the same about these fractions? (denominator) What is different? (numerator)
How can we tell just from the numerator which fraction is greater? (5 > 3 so 5/8 > 3/8)
Write the > sign between the two fractions on the board.
SAY: If the denominators are different, we need to do more work before we can find which
fraction is bigger. Draw on the board:
SAY: It is hard to tell just by looking which fraction is bigger. Ask for a volunteer to write the
fraction for each diagram on the board, as shown below:
3
4
5
8
ASK: Are the denominators the same? (no) What is the LCM of 4 and 8? (8) What number can
we multiply the numerator and denominator of 3/4 by to get the same denominator as 5/8? (2)
What will the equivalent fraction be for 3/4? (6/8) Write on the board:
3 ´2 6
=
4 ´2 8
SAY: Notice the denominators are now the same for the two fractions. ASK: Which fraction is
bigger, 6/8 or 5/8? (6/8) How can you tell? (6 > 5) So, which of the original fractions is bigger,
3/4 or 5/8? (3/4)
Write on the board:
Which is bigger,
B-26
3
5
or ?
4
6
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
Ask for a volunteer to write the firsts few multiples of 4 on the board. Ask for a different
volunteer to write the first few multiples of 6 on the board. Finally, ask for a third volunteer to
circle the first multiple that shows up in both lists, as shown below:
4, 8, 12, 16…
6, 12, 18, 24, …
ASK: So, what is the lowest common multiple of 4 and 6? (12) SAY: We call this the lowest
common denominator, or LCD, of the fractions 3/4 and 5/6. SAY: We want to find equivalent
fractions with 12 as the new denominator. Point to 3/4 and ASK: For the first fraction, what
should we multiply the numerator and denominator by to get a denominator of 12? (3) Point to
5/6, and ASK: for the second fraction, what should we multiply the numerator and denominator
by to get a denominator of 12? (2) Ask for volunteers to find the equivalent fractions, as
shown below:
3 ´3
9
=
4 ´3 12
5 ´2 10
=
6 ´2 12
ASK: Which fraction is bigger, 9/12 or 10/12? (10/12) So which of the original fractions is bigger,
3/4 or 5/6? (5/6) Write on the board:
5 3
>
6 4
SAY: We could also write that 3/4 is the smaller fraction. Write on the board:
3 5
<
4 6
Exercises: Find the bigger fraction by first finding equivalent fractions with the same denominator.
a)
2
5
or
3
6
b)
3
7
or
4
12
c)
2
4
or
3
5
d)
5
7
or
8
12
Answers: a) 5/6, b) 3/4, c) 4/5, d) 5/8
(MP.7) Adding or subtracting fractions with the same denominator. Draw on the board:
Ask for a volunteer to write the fraction for the shaded group of parts on the left. (2/8) Ask a
different volunteer to write a fraction for the shaded group of parts on the right. (3/8) ASK: How
many different parts are shaded? (5) How many parts are there in total? (8) What addition
equation can we write for the fractions in the diagram? (2/8 + 3/8 = 5/8) When adding fractions
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-27
with the same denominator, what happens to the denominator? (it stays the same) What
happens to the numerators? (we add them) Write on the board:
3
4
+ =?
11 11
ASK: Are the denominators the same? (yes) What should do with the numerators? (add them)
What is the final answer? (7/11) Write on the board:
5 2
- =?
7 7
ASK: Are the denominators the same? (yes) What should you do with the numerators? (subtract
them) What is the final answer? (3/7)
Exercises: Add or subtract the fractions.
a)
5
6
+
13 13
b)
10 7
17 17
c)
11 3
15 15
d)
17
4
+
23 23
Answers: a) 11/13, b) 3/17, c) 8/15, d) 21/23
(MP.7) Adding or subtracting fractions with different denominators. Draw on the board:
1
8
1 1
+
4 8
1
4
ASK: Why can’t we add these fractions? (they don’t have the same denominator; the pieces
are not the same size) Ask for a volunteer to draw lines on the board to break up the pie into
8 pieces, as shown below:
ASK: How many pieces are in the pie? (8) What fraction is each piece? (1/8) What is an
equivalent fraction for the 1/4 pie we shaded before? (2/8) Why can we add the fractions now?
(all the pieces are the same size; the fractions have the same denominator) What is the total of
the fractions? (3/8) Write on the board:
1 1
+
4 8
2 1
= +
8 8
3
=
8
B-28
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
SAY: If the fractions do not have the same denominator, we have to find equivalent fractions
using the lowest common denominator. Then we add the fractions by keeping the denominator
and adding the numerators. Write on the board:
3 1
+
4 6
ASK: What is the LCD of the fractions? (12) Write on the board:
3´
4´
=
1´
6´
12
=
12
Ask for a volunteer to find equivalent fractions for 3/4 and 1/6 with denominator 12. ASK: What
do we multiply 4 by to get 12? (3) What do we multiply 6 by to get 12? (2) Write on the board:
3 1
+
4 6
3 ´3 1 ´2
=
+
4 ´3 6 ´2
=
9
2
+
12 12
ASK: Can we add the fractions now? (yes, because they have the same denominator) Have a
volunteer add the fractions by keeping the denominator and adding the numerators. (11/12)
Exercises: Add or subtract the fractions.
a)
5
1
+
8 12
b)
7 2
10 5
c)
7
3
24 16
d)
4
1
+
15 20
Bonus:
3 5 2
+ 8 6 3
Answers: a) 15/24 + 2/24 = 17/24, b) 7/10 − 4/10 = 3/10, c) 14/48 − 9/48 = 5/48,
d) 16/60 + 3/10 = 19/60, Bonus: 9/24 + 20/24 − 16/24 = 13/24
(MP.7) Fractions as division. Draw on the board:
SAY: Suppose we have three pizzas. Abigail, Ben, Chelsea, and David want to share the
pizzas. Ask for a volunteer to divide each pizza into quarters. ASK: How many quarters are
there? (12) How many pieces should each person get? (3) Label the pieces as shown below
and say each person’s name as you label them:
A
A
B
B
C
D
A
B
C
C
D
D
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-29
ASK: How do we write 12 quarters as a fraction? (12/4) SAY: We can describe this by writing
12/4 = 3. Notice that 12 ÷ 4 = 3. So writing 12 quarters is the same as writing 12 ÷ 4. Write on
the board:
12
= 12 ¸ 4
4
Exercises: Write the fraction as a division statement.
a)
8
4
b)
15
3
c)
24
6
Answers: a) 8 ÷ 4, b) 15 ÷ 3, c) 24 ÷ 6
Extensions
(MP.1) 1. Find a fraction equivalent to
2
so that …
3
a) the denominator is 3 more than the numerator.
b) the denominator is 5 more than the numerator.
c) the numerator is a multiple of 3.
d) the denominator is a multiple of 5.
Answers: a) 6/9, b) 10/15, c) 6/9, d) 10/15
(MP.1) 2. Distribute BLM Finding Prime Numbers. Have students follow the instructions to use
the Sieve of Eratosthenes to find prime numbers to 100.
Answers: 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
(MP.1) 3. Distribute BLM Finding the LCM Using Prime Number Factors. Have students
follow the instructions to learn how to find the lowest common multiple (LCM) using prime factors.
Answers:
1. a) 2 × 5, b) 2 × 7, c) 3 × 5, d) 3 × 7, e) 2 × 11, f) 3 × 11, g) 5 ×7
2. a) 2 × 3 × 3, b) 2 × 2 × 2 × 2 × 3, c) 2 × 3 × 5, d) 2 × 2 × 3 × 3, e) 2 × 2 × 5, f) 2 × 2 × 2 × 5,
g) 3 × 5 × 5
3. a) 30, b) 24 ,c) 70, d) 72, e) 56, f) 75
(MP.1) 4. The magicicada is a type of cicada in the eastern United States that spends most of
its time feeding underground. After a certain number of years, these insects emerge from the
ground for their mating season. Some cicadas emerge after 13 years, while the others emerge
after 17 years.
a) Is 13 composite or prime? Is 17 composite or prime? Explain.
b) How often will the 13-year cicadas and the 17-year cicadas emerge at the same time?
c) Scientists believe this behavior occurs to avoid having all the cicadas emerge at the same
time and risk being wiped out by predators. How do the numbers 13 and 17 help with this?
Answers: a) both 13 and 17 are prime; b) every 221 years; c) if 13 and 17 had not been prime,
the LCM would have been less than 13 × 17
B-30
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
EE8-5
Reducing Fractions
Pages 14–15
Standards: preparation for 8.EE.A.2, 8.EE.B.5, 8.EE.C.7
Goals:
Students will reduce fractions by finding an equivalent fraction with fewer parts in the whole.
Students will learn how to find the greatest common factor (GCF) of a pair of numbers.
Students will use the GCF to reduce fractions to lowest terms.
Prior Knowledge Required:
Can find equivalent fractions by multiplying the numerator and denominator by the same number
Vocabulary: equivalent fraction, factor, greatest common factor (GCF), lowest terms,
reducing fractions
Materials:
BLM Finding the GCF Using Prime Number Factors (p. B-108, see Extension 2)
BLM Finding the GCF Using Euclid’s Algorithm (p. B-109, see Extension 3)
(MP.7) Reducing fractions by finding an equivalent fraction with fewer parts in the whole.
Draw on the board:
B.
A.
ASK: In Figure A, how many parts are in the whole? (15) How many parts are shaded? (10)
What fraction is represented by Figure A? (10/15) In Figure B, how many parts are in the
whole? (3) How many parts are shaded? (2) What fraction is represented by Figure B? (2/3)
SAY: Look carefully at the two figures. ASK: If you ignore the dashed lines in Figure A, what can
you say about the two figures? (they are the same) SAY: So the two fractions must be equal.
Write on the board:
10 ¸
15 ¸
=
2
3
ASK: For the fraction 10/15, what number can we divide both the numerator and the
denominator by to get 2/3? (5)
SAY: Similar to when we multiplied both the numerator and denominator by the same number to
get equivalent fractions, we can divide both the numerator and denominator by the same
number to get equivalent fractions. This is called reducing fractions.
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-31
Exercises: Divide the numerator and denominator to reduce the fraction.
a)
8 ¸2
=
12 ¸2
b)
12 ¸6
=
18 ¸6
c)
20 ¸4
=
32 ¸4
Bonus:
120 ¸6
=
300 ¸6
Answers: a) 4/6, b) 2/3, c) 5/8, Bonus: 20/50
(MP.7) Finding the greatest common factor (GCF). SAY: To reduce a fraction as much as
possible, we need to divide by the greatest common factor. The greatest common factor, or
GCF, of two numbers is the largest factor that will divide into both numbers. Write on the board:
8 ¸2
=
12 ¸2
SAY: Remember, 2 is a factor of 12 because 12 = 6 × 2. ASK: What are other factors of 6? (1, 3,
4, 6, 12) Why is 4 a factor of 12? (12 = 4 × 3) Why are 1 and 12 factors of 12? (12 × 1 = 12)
Write on the board:
Find the GCF of 12 and 18.
Factors of 12:
Factors of 18:
Ask a volunteer to write all the factors of 12. SAY: If you can think of two numbers that multiply
to give 12, each number is a factor. Ask for a different volunteer to write all the factors of 18.
Make sure the factors of each number are written in order from smallest to largest, as
shown below:
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Ask four different volunteers to circle a number that shows up in both lists. (1, 2, 3, 6) SAY: So,
these are the factors that are common to both lists. ASK: Which is the largest, or greatest,
number that was circled? (6) ASK: What is the GCF of 12 and 18? (6)
Exercises: Write all the factors of each number and then find the GCF.
a) 18, 24
b) 18, 27
c) 20, 24
d) 28, 32
Answers: a) 6, b) 9, c) 4, d) 4, Bonus: 16
Bonus: 48, 80
(MP.7) Finding a fraction in lowest terms. Write on the board:
24 ¸2
=
36 ¸2
24 ¸3
=
36 ¸3
24 ¸4
=
36 ¸4
24 ¸6
=
36 ¸6
24 ¸12
=
36 ¸12
Ask volunteers to divide the numerator and denominator by the given number for each fraction
on the board. (12/18, 8/12, 6/9, 4/6, 2/3) ASK: How do we know each new fraction is equivalent
B-32
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
to the original fraction 24/36? (for each fraction, we divided the numerator and denominator by
the same number)
Ask volunteers to find the GCF of the numerator and denominator for the fractions on the board.
(12/18, GCF = 6; 8/12, GCF= 4; 6/9, GCF=3; 4/6, GCF 2; 2/3, GCF = 1) ASK: Which fraction
has a GCF of 1? (2/3) SAY: A fraction for which the GCF of the numerator and denominator is 1
is in lowest terms.
Exercises:
1. Use the GCF to find the fraction that is in lowest terms.
a)
8
10
5
7
b)
4
9
3
9
c)
7
10
10
15
Answers: a) 5/7, b) 4/9, c) 7/10
2. A fraction has a numerator that is even and a denominator that is even. Explain why the
fraction is not in lowest terms.
Answer: Since they are both even, a common factor is 2.
(MP.7) Reducing fractions to lowest terms by dividing by the GCF. Write on the board:
12
20
SAY: We want to reduce this fraction to lowest terms. Ask a volunteer to list all the factors of 12
and 20. Ask another volunteer to find the GCF from the lists of factors. (see answers below)
12
20
Factors: 1, 2, 3, 4, 6, 12
Factors: 1, 2, 4, 5, 10, 20
GCF = 4
SAY: To find an equivalent fraction in lowest terms, divide the numerator and denominator by
the GCF. Write on the board:
12 ¸4
=
20 ¸4
Ask a volunteer to divide the numerator and denominator to find the fraction in lowest terms. (3/5)
Exercises: Reduce the fraction to lowest terms by finding the GCF, then dividing the numerator
and denominator by the GCF.
a)
15
20
b)
10
14
c)
18
45
d)
21
35
Bonus:
50
75
Answers: a) GCF = 5, 3/4; b) GCF = 2, 5/7; c) GCF = 9, 2/5; d) GCF = 7, 3/5;
Bonus: GCF = 25, 2/3
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-33
(MP.7) Adding or subtracting fractions, then reducing to lowest terms. SAY: When adding
or subtracting fractions, we want our answers to be in lowest terms. We may have to divide
the numerator and denominator by the GCF to reduce the fraction to lowest terms. Write on
the board:
1 3
+
6 10
ASK: Why can’t we add these fractions yet? (the denominators are different) How can we
make the denominators the same? (find the lowest common denominator, LCD) What is the
LCD of 6 and 10? (30) What do we need to multiply the numerator and the denominator of the
first fraction by to make the denominator 30? (5) What do we need to multiply the numerator
and denominator of the second fraction by to make the denominator 30? (3) Write this on the
board and ask for a volunteer to come and find the equivalent fractions, as shown below:
1 ´5 3 ´3
+
6 ´5 10 ´3
5
9
=
+
30 30
ASK: Why are we able to add the fractions now? (the denominators are the same) What is the
sum of these fractions? (14/30) How can we quickly see that this fraction is not in lowest terms?
(both the numerator and denominator are even numbers) What is the GCF of 14 and 30? (2)
What is the reduced fraction? Continue writing on the board to show the sum and then the sum
in lowest terms, as shown below:
14 ¸2
30 ¸2
7
=
15
=
Exercises: Add or subtract the fractions. Reduce your answer to lowest terms.
1 1
6 15
7
1
c)
+
12 15
a)
3
1
+
10 14
2
1
d)
15 21
b)
Answers: a) 5/30 − 2/30 = 3/30 = 1/10, b) 21/70 + 5/70 = 26/70 = 13/35,
c) 35/60 + 4/60 = 39/60 = 13/20, d) 14/105 − 5/105 = 9/105 = 3/35
Extensions
1 2 3
4
, ,
,
,…
2 6 12 20
1
2
3
4
Answers: The fractions can be written as
,
,
,
, ….
1´ 2 2 ´ 3 3 ´ 4 4 ´ 5
5
5
6
6
So the next two fractions are
, and
.
=
=
5 ´ 6 30
6 ´ 7 42
(MP.1) 1. Find the next two fractions in the pattern:
B-34
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
(MP.1) 2. Have students complete BLM Finding the GCF Using Prime Number Factors.
Answers:
1. a) 2 × 5, b) 2 × 7, c) 3 × 5, d) 3 × 7, e) 2 × 11, f) 3 × 11, g) 7 × 5
2. a) 2 × 3 × 3, b) 2 × 2 × 2 × 2× 3, c) 2 × 3 × 5, d) 2 × 2 × 3 × 3, e) 2 × 2 × 5, f) 2 × 2 × 2 × 5,
g) 3 × 5 × 5
3. a) 10 = 2 × 5, 15 = 3 × 5, GCF = 5; b) 8 = 2 × 2 × 2, 12 = 2 × 2 × 3, GCF = 2 × 2 = 4;
c) 10 = 2 × 5, 14 = 2 × 7, GCF = 2; d) 24 = 2 × 2 × 2 × 3, 36 = 2 × 2 × 3 × 3, GCF = 2 × 2 × 3 = 12;
e) 45 = 3 × 3 × 5, 60 = 2 × 2 × 3 × 5, GCF = 3 × 5 = 15; Bonus: 108 = 2 × 2 × 3 × 3 × 3,
162 = 2 × 3 × 3 × 3 × 3, GCF = 2 × 3 × 3 × 3 = 54
(MP.1) 3. Have students complete BLM Finding the GCF Using Euclid’s Algorithm.
Answers: a) 6, b) 8, c) 21, d) 4, Bonus: 12
(MP.1) 4. Find a fraction equivalent to the fraction in the first column and where the sum of the
numerator and denominator is as shown in the middle column.
Sum of Numerator
Equivalent to
Answer
and Denominator
3
4
2
3
5
6
10
16
a)
b)
c)
d)
14
25
44
130
Answers: a) 6/8, b) 10/15, c) 20/24, d) 50/80
(MP.1) 5. a) Add.
1 1 1
1
+ + +
2 4 8 16
1 1 1
1
1
b) Use your answers to a) to predict + + + +
.
2 4 8 16 32
i)
1 1
+
2 4
ii)
1 1 1
+ +
2 4 8
iii)
Answers: a) i) 3/4, ii) 7/8, iii) 31/32, b) 63/64
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-35
EE8-6
Multiplying Fractions
Pages 16–18
Standards: preparation for 8.EE.A.2, 8.EE.B.5, 8.EE.C.7
Goals:
Students will learn that to multiply fractions you multiply the numerators and multiply
the denominators.
Students will learn how to reduce fractions before multiplying.
Prior Knowledge Required:
Can multiply whole numbers
Can write a fraction for a diagram with shaded parts
Vocabulary: denominator, factor, greatest common factor (GCF), improper fractions,
lowest terms, numerator, reducing fractions, unit fraction
(MP.7) Why we multiply the denominators when we are multiplying fractions. Draw on
the board:
ASK: How many parts are in the whole? (4) How many parts are shaded in? (1) What is a
fraction for the shaded region? (1/4) SAY: Suppose I want 1/2 of 1/4. Have a student draw a
dashed line to divide the shaded part into two equal parts. Ask another student to extend the
dashed line through the rest of the rectangle to make 8 equal parts. (see answer below)
SAY: I want only 1/2 of 1/4, so I am going to shade in one of the smaller parts, as shown below:
ASK: How many of the smaller parts are there? (8) How many are shaded in hash marks? (1)
What fraction is shown by the region with harsh marks? (1/8) Write on the board:
1
1 1
of
=
2
4 8
B-36
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
ASK: How many parts were in the original rectangle? (4) After I divided each part into 2 parts,
how many parts are there? (8) Point to the denominators and ASK: What operation could we
have performed on 2 and 4 to get 8? (multiplication)
SAY: If we had 2 groups of 3, we would multiply 2 × 3 to get the total number of objects. In the
same way, if we want to find 1/2 of 1/4, we multiply the numerators, then the denominators.
Write on the board:
1 1 1
´ =
2 4 8
SAY: Remember that a unit fraction is a fraction with 1 as the numerator. When we multiply unit
fractions, our answer is also a unit fraction. We find the new denominator by multiplying the
denominators. Write on the board:
1 1
´ =
5 3
Ask for a volunteer to write the product on the board. (1/15)
Exercises: Multiply the fractions.
a)
1 1
´
2 3
b)
1 1
´
3 5
c)
1 1
´
4 7
d)
1 1
´
7 9
Bonus:
1 1
´
25 4
Answers: a) 1/6, b) 1/15, c) 1/28, d) 1/63, Bonus: 1/100
(MP.7, MP.8) Why we multiply the numerators when we are multiplying fractions. Draw on
the board:
3
4
ASK: How many parts are in the whole? (4) How many parts are shaded in? (3) What is a
fraction for the shaded region? (3/4) SAY: Suppose I want 2/3 of 3/4. Ask a volunteer to draw
two horizontal dashed lines to divide each of the shaded regions into 3 equal parts. Ask another
student to extend the dashed lines through the rest of the rectangle so that every part in the
original picture is now divided into 3 equal parts. The picture should look like this:
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-37
SAY: I want only 2/3 of 3/4, so I am going to mark 2 of the 3 smaller parts with hash marks in
each shaded region. Shade in 2 of the smaller parts in each of the 3 shaded quarters, as
shown below:
ASK: In the picture now, how many of the smaller parts are there? (12) How many are
shaded in with hash marks? (6) What fraction is shown by the hash-marked region? (6/12)
Write on the board:
2
3
6
of
=
3
4 12
Point to the shaded regions and SAY: We had 3 of the 4 regions shaded. After we divided each
region into 3 parts, we shaded 2 parts of each with hash marks. So we shaded 2 + 2 + 2 parts
with hash marks. Write on the board:
2+2+2
ASK: What multiplication can we use as a shortcut to adding 2 three times? (3 × 2) SAY: So a
shortcut to counting the hash-marked parts is to multiply 3 × 2, which is the same as 2 × 3.
Write on the board:
2 3
6
´ =
3 4 12
Point to the numerators 2 and 3 and SAY: To multiply fractions, we can multiply the numerators
2 and 3 to get 6. Point to the denominators 3 and 4 and SAY: We can multiply the denominators
3 and 4 to get 12. Write on the board:
3 7
´ =
5 8
Ask for a volunteer to multiply the numerators. (21) Ask for a different volunteer to multiply the
denominators. (40) SAY: So 3/5 × 7/8 = 21/40. To multiply fractions, multiply the numerators
and denominators separately.
Exercises: Multiply the fractions.
a)
3 5
´
4 7
b)
2 4
´
5 9
c)
1 3
´
2 8
d)
1 1
´
4 5
Bonus:
21 3
´
25 4
Answers: a) 15/28, b) 8/45, c) 3/16, d) 1/20, Bonus: 63/100
B-38
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
(MP.7) Multiplying improper fractions. Write on the board:
3
2
SAY: Remember: a fraction with the numerator greater than or equal to the denominator is
called an improper fraction. Suppose I had to find 3/4 of 3/2. Draw on the board:
Ask for a volunteer to shade in 1/2 of each picture. The picture should look like this:
SAY: We have 1/2 + 1/2 + 1/2 shaded. This gives us three halves. Divide the first picture into
4 parts, as shown below:
Ask for a volunteer to do the same to the other pictures. SAY: We want 3/4 of 3/2. Shade in
3 out of every 4 parts of the 3/2 with hash marks, as shown below:
ASK: How many parts or small rectangles does each larger rectangle have? (8) In the entire
diagram, how many of the small parts are shaded in with hash marks? (9) SAY: So we have
9 parts, each 1/8. We have shaded 9/8. Write on the board:
3 3 9
´ =
4 2 8
ASK: How could we have calculated the answer without the pictures? (multiply the numerators
and multiply the denominators separately) SAY: So to multiply improper fractions, we follow the
same steps: multiply the numerators and denominators separately.
Exercises: Multiply the fractions.
a)
5 7
´
3 8
b)
8 2
´
7 3
c)
7 3
´
2 2
Answers: a) 35/24, b) 16/21, c) 21/4
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-39
(MP.7) Multiplying fractions and then reducing the answer. SAY: When dealing with
fractions, we like to have fractions in lowest terms. ASK: Why is it easier to deal with fractions
that are in lowest terms? (the numbers are smaller) Write on the board:
3 2
´
4 3
ASK: How do we multiply fractions? (multiply the numerators and multiply the denominators
separately) Ask for a volunteer to multiply the fractions on the board to give the answer. (6/12)
SAY: This fraction is not in lowest terms. ASK: What is the GCF of 6 and 12? (6) Ask a volunteer
to divide the numerator and denominator by 6 to find the answer, as shown below:
6 ¸6
12 ¸6
1
=
2
=
Exercises: Multiply the fractions and then reduce the answer to lowest terms.
5 3
´
9 10
5 18
c) ´
9 15
a)
3 21
´
14 6
4 5 9
Bonus:
´ ´
27 6 10
b)
Answers: a) 1/6, b) 3/4, c) 2/3, Bonus: 1/9
(MP.7) Multiplying fractions by simplifying the multiplication first. SAY: As you noticed in
the previous exercises, sometimes the numbers get large when multiplying fractions. This can
make the product difficult to reduce later. There is a way to simplify the fractions before
multiplying, and doing so will make our work easier. Write on the board:
12 4
´
15 7
ASK: If we multiply the fractions before reducing, what is our answer? (48/105) ASK: What is
the GCF of this fraction? (3) NOTE: Students may find these calculations difficult, but that is
what the question illustrates: large numbers can be difficult to work with. SAY: It is not easy to
find! Write on the board:
48 ¸3
=
105 ¸3
Ask for volunteers to divide the numerator and denominator by 3. (16/35) SAY: These numbers
are still rather large. To reduce before multiplying, we can choose one number from anywhere in
B-40
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
the numerator and one number anywhere in the denominator. Point to the original multiplication
question and continue writing on the board:
12 4
´
15 7
GCF = ?
ASK: What is the GCF of 12 and 15? (3) SAY: We can divide both 12 and 15 by the GCF.
Continue writing on the board:
12 ¸3 4
´
15 ¸3 7
ASK: What is 12 ÷ 3? (4) What is 15 ÷ 3? (5) Write on the board:
=
4 4
´
5 7
SAY: Notice that the numbers are smaller and so are easier to multiply. ASK: What is the final
answer to the multiplication? (16/35) SAY: Note that this is the same answer as when we
reduced 48/105. When we are multiplying fractions, we can even pick the numerator and
denominator from different fractions to reduce. Write on the board:
3 11
´
14 15
ASK: For this new multiplication, choose a number from the numerator and a number from the
denominator that have a common factor. (3 and 15) What is the GCF of 3 and 15? (3) Write on
the board:
=
3 ¸3 11
´
14
15 ¸3
ASK: After dividing by the GCF, what fractions are left to multiply? (1/14 × 11/5) ASK: What is
the final product? Continue writing on the board:
1 11
´
14 5
11
=
70
=
Exercises: Reduce the fractions first and then multiply.
7 12
´
18 11
2 24
c)
´
15 5
a)
14 4
´
9 21
15 27
Bonus:
´
36 20
b)
Answers: a) 14/33, b) 8/27, c) 16/25, Bonus: 9/16
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-41
Extensions
(MP.1) 1. Find all the pairs of unit fractions less than 1 that multiply to give:
a)
1
12
b)
1
32
c)
1
36
Answers: a) 1/2 × 1/6, 1/3 × 1/4; b) 1/2 × 1/16, 1/4 × 1/8; c) 1/2 × 1/18, 1/3 × 1/12, 1/4 × 1/9,
1/6 × 1/6
(MP.1) 2. Find the missing fraction.
a)
1
´
3
=
2
15
b)
´
5 20
=
6 42
Bonus:
´
5 5
5 10 15
(Hint: =
)
=
=
6 8
8 16 24
Answers: a) 2/5, b) 4/7, Bonus: 3/4
(MP.1) 3. Find the missing numbers.
a) 4 ´
1
=2
b)
´
4
4
=4
5
5
c)
´
3
4
=5
7
7
d) 5 ´
6
=4
1
6
Answers: a) 2, b) 6, c) 13, d) 5
(MP.1) 4. a) Multiply.
æ
1öæ
1ö
i) ççç1- ÷÷÷ççç1- ÷÷÷
è 2 øè 3 ø
æ
1öæ
1 öæ
1ö
ii) ççç1- ÷÷÷ççç1- ÷÷÷ççç1- ÷÷÷
è 2 øè 3 øè 4 ø
æ
1öæ
1 öæ
1 öæ
1ö
iii) ççç1- ÷÷÷ççç1- ÷÷÷ççç1- ÷÷÷ççç1- ÷÷÷
è 2 øè 3 øè 4 øè 5 ø
æ
1öæ
1 öæ
1 öæ
1 öæ
1öæ
1ö
b) Use your answers from a) to predict ççç1- ÷÷÷ççç1- ÷÷÷ççç1- ÷÷÷ççç1- ÷÷÷ççç1- ÷÷÷ççç1- ÷÷÷ .
è
2 øè
3 øè
4 øè
5 øè
6 øè
7ø
Answers: a) i) 1/3, ii) 1/4, iii) 1/5; b) 1/7
B-42
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
EE8-7
Dividing Fractions
Pages 19–21
Standards: preparation for 8.EE.A.2, 8.EE.B.5, 8.EE.C.7
Goals:
Students will learn that to divide fractions, you multiply by the reciprocal of the second fraction.
Prior Knowledge Required:
Can multiply whole numbers
Can divide whole numbers
Can label a number line that is divided into fractional units
Can use skip counting to multiply
Can use skip counting to divide
Vocabulary: divisor, improper fractions, mixed fractions, reciprocal, skip counting, unit fractions
(MP.7) Using skip counting to multiply. Draw on the board:
0
1
2
3
4
5
6
7
8
9
10
10 11
12 13
14
15 16
17 18 19
20
ASK: What addition equation does this diagram represent? (4 + 4 + 4 + 4 + 4 = 20) What
multiplication equation does this diagram represent? (5 × 4 = 20) In the diagram, what tells us to
use the number 5 in the multiplication equation? (the number of jumps or skips or steps) What
tells us to use the number 4? (the size of the jump or skips) What tells us to use the number 20?
(where the skips end)
(MP.7) Using skip counting to divide. SAY: This diagram can also be used to represent the
division equation 20 ÷ 4 = 5. ASK: In the equation, where do we find the size of the skips or
jumps or steps? (the number 4) What does the answer 5 represent? (the number of steps) Write
on the board:
20
÷ 4
size of
step
=
5
number of
steps
(MP.7) Using skip counting to divide unit fractions. Draw on the board:
0
1
2
3
4
5
6
7
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
8
9
10
B-43
Have a volunteer write the missing fractions on the number line. Rewrite any mixed fractions as
improper fractions. Then draw arrows on the board to show each step. The number line should
look like this:
0
1
2
3
2
1
2
5
2
3
7
2
9
2
4
5
11
2
6
13
2
7
15
2
8
17
2
9
19
10
2
ASK: What is the size of the step? (1/2) How many steps are there? (12) What is the division
equation for the diagram? (6 ÷ 1/2 = 12) Draw four more arrows of the same size—the arrows
should finish at the number 8. ASK: What is the size of each step I just added? (1/2) How many
steps are there now, ending at number 8? (16) What is the division equation for the diagram?
(8 ÷ 1/2 = 16)
Draw on the board:
1
3
0
2
3
1
4
3
5
3
2
7
3
8
3
3
10
3
11
3
4
13
3
14
3
16
3
5
17
3
6
19
3
20
3
ASK: Where do the arrows end? (at number 4) What is the size of each step? (1/3) How many
steps or jumps are there? (12) What is the division equation for the diagram? (4 ÷ 1/3 = 12)
Write on the board:
8¸
1
= 16
2
4¸
1
= 12
3
ASK: In each of the examples we have looked at now, how could we have used multiplication to
get the answers instead of a number line? (multiply by the denominator: 8 ´ 2 = 16, 4 ´ 3 = 12)
Exercises: Use multiplication to find the answer.
a) 7 ¸
1
2
b) 8 ¸
1
3
c) 9 ¸
1
4
d) 6 ¸
1
5
Bonus: 3 ¸
1
100
Answers: a) 14, b) 24, c) 36, d) 30, Bonus: 300
(MP.7) Dividing fractions by multiplying by the reciprocal. Draw on the board:
1
B-44
2
3
4
5
6
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
Ask for a volunteer to divide each box in the bottom row into 4 equal parts. ASK: What
fraction can I use to label each of the smaller boxes? (1/4) Label each part in the bottom row
with the fraction:
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
ASK: How many 1/4 fractions did we write? (24) What division equation can we write for this
diagram? (6 ÷ 1/4 = 24) How can we rewrite this division equation as a multiplication equation?
(6  4 = 24) Draw a bracket under the first three 1/4 fractions, as shown below. ASK: What
fraction can I use to represent three of these 1/4 fractions? (3/4) Ask for a volunteer to draw
brackets for as many groups of three of the 1/4 fractions as possible. The final picture should
look as follows:
1
1
4
1
4
3
4
2
1
4
1
4
1
4
1
4
3
1
4
3
4
1
4
1
4
3
4
1
4
4
1
4
1
4
1
4
3
4
1
4
3
4
5
1
4
1
4
1
4
1
4
6
1
4
3
4
1
4
3
4
1
4
1
4
1
4
1
4
3
4
ASK: How many 1/4 fractions are there? (24) How many 3/4 fractions are there? (8) What did
we divide by to change 24 fractions to 8 fractions? (3) What division equation does the diagram
represent? (24 ÷ 3/4 = 8) SAY: To find the number of 1/4 fractions, we multiplied 6 × 4. Then to
find the number of 3/4 fractions, we divided by three. In other words, in the row of 6 boxes, we
multiplied 6 by 4 to make 24 quarter parts. Then, to group the fraction boxes into 3/4 fractions,
we divided the total number of fraction boxes by 3, 24/3 = 8, thus 8 groups. Write on the board:
6¸
3
= 6 ´ 4 ¸ 3 = 24 ¸ 3 = 8
4
Point to the denominator in 3/4 and SAY: To divide by the fraction 3/4, we first multiply by the
denominator, 4. Point to the numerator in 3/4 and SAY: Then we divide that product by the
numerator, 3.
Exercises: Divide by first multiplying by the denominator and then dividing by the numerator.
a) 12 ¸
6
7
b) 15 ¸
3
8
c) 16 ¸
4
5
Bonus: 7 ¸
7
100
Answers: a) 14, b) 40, c) 20, Bonus: 100
(MP.7) Review fractions as divisions. Draw on the board:
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-45
ASK: How many parts are in the whole? (4) How many are shaded? (3) What is a fraction for
the shaded region? (3/4) SAY: The fraction 3/4 can also mean 3 ÷ 4. Draw on the board:
SAY: Remember that the fraction 3/4 can mean either 3/4 of one whole—as shown in the
shaded picture above—or dividing 3 wholes by 4. Write the letters A, B, C, and D three times
each, as shown below:
A
A
B C
C D
A
B
B C
D D
Write on the board:
3
can also mean 3 ÷ 4.
4
Exercises: Write the fraction as a division.
a)
5
8
b)
4
3
c)
6
2
Answers: a) 5 ÷ 8, b) 4 ÷ 3, c) 6 ÷ 2
(MP.7) Review the reciprocal of a fraction. SAY: The reciprocal of a fraction is a fraction with
the numerator and denominator switched. Write on the board:
The reciprocal of
3
4
is .
4
3
Exercises: Find the reciprocal.
a)
5
8
b)
7
4
c)
5
11
Answers: a) 8/5, b) 4/7, c) 11/5
(MP.7) Dividing whole numbers by fractions by multiplying by the reciprocal of the
fraction. Write on the board:
6¸
2
3
SAY: We can divide this fraction by multiplying 6 by 3 and then dividing by 2. Continue writing
on the board:
6¸
B-46
2
= 6´3 ¸ 2
3
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
Point to the 3 ¸ 2 part of the statement above and SAY: But 3 ÷ 2 can be written as the fraction
3/2. Write on the board:
6¸
2
3
= 6´
3
2
ASK: What do we call fractions like 2/3 and 3/2? (reciprocals) SAY: So to divide by a fraction,
we can multiply by the reciprocal of the fraction.
SAY: We can write 6 as the fraction 6/1 and then divide the numerator and denominator
before multiplying. Circle the 6 and 2. ASK: What is the GCF of 6 and 2? (2) Continue writing
on the board:
=
6 ¸2 3
´
1
2 ¸2
ASK: What is 6 ÷ 2? (3) What is 2 ÷ 2? (1) Continue writing on the board:
=
3 3
´
1 1
SAY: Now the numbers are smaller. We can multiply the fractions by multiplying the numerators
and denominators separately. ASK: What is the product? (9/1) What whole number is this equal
to? (9) Continue writing on the board:
9
1
=9
=
Exercises: Rewrite the statement using multiplication. Then reduce and multiply to find
the answer.
a) 12 ¸
4
3
b) 12 ¸
3
4
c) 15 ¸
3
5
d) 15 ¸
5
3
Answers: a) 9, b) 16, c) 25, d) 9
(MP.7) Dividing fractions by multiplying by the reciprocal of the divisor. Write on the board:
3 3
¸
4 2
SAY: To find 3/4 ÷ 3/2, first find 3 ÷ 3/2. ASK: How do we use multiplication to find 3 ÷ 3/2?
(multiply the whole number by the reciprocal of the fraction: 3 × 2/3) SAY: This is how many
fractions of 3/2 fit into 3. We are looking for how many fractions of 3/2 fit into 3/4. We only want
1/4 of 3 × 2/3. Continue writing on the board:
3 3 1
2
¸ = of 3 ´
4 2 4
3
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-47
SAY: But 1/4 of 3 = 3/4. Write on the board:
3 3 3 2
¸ = ´
4 2 4 3
Point to the division sign, and then to the multiplication sign, and SAY: Notice that dividing
by a fraction became multiplying. Point to the fraction 3/2 on the left side of the equation.
SAY: Notice that we used the reciprocal of 3/2, which is 2/3 (point to the 2/3), in the
multiplication. SAY: So to divide a fraction by a fraction, we multiply by the reciprocal of
the divisor or the second fraction.
Exercises: Rewrite the division as a multiplication.
a)
2 4
¸
3 5
3
4
b) ¸
6
5
c)
2 7
¸
5 8
Answers: a) 2/3 × 5/4, b) 3/4 × 5/6, c) 2/5 × 8/7
SAY: Let’s continue the previous example. Write on the board:
3 3 3 2
¸ = ´
4 2 4 3
SAY: Let’s simplify the multiplication before multiplying the fractions. Choose a number from the
numerator and a number from the denominator that have a greatest common factor, or GCF,
other than 1, either 2 and 4, or 3 and 3. ASK: What is the GCF of 2 and 4? (2) What is the GCF
of 3 and 3? (3) SAY: We can divide by these GCFs to reduce the terms before multiplying. Write
on the board:
=
3 ¸3 2 ¸2
´
4 ¸2 3 ¸3
NOTE: You might have to do the simplifying separately if students find this step too difficult. Ask
a volunteer to divide and write the simplified fractions on the board:
=
1 1
´
2 1
Ask for another volunteer to multiply the numerators and denominators to find the answer, as
shown below:
=
1
2
Exercises: Rewrite as a multiplication statement. Then reduce and multiply to find the answer.
a)
5 4
¸
6 3
b)
3 9
¸
8 4
c)
5 15
¸
2
8
Answers: a) 5/8, b) 1/6, c) 4/3
B-48
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
Extensions
5 4
(MP.1) 1. The division ¸ can also be written as a complex fraction:
6 3
5
6
4
3
.
Write the complex fraction as a division of fractions.
a)
2
3
4
9
b)
5
8
3
2
4
9
6
27
c)
Answers: a) 2/3 ÷ 4/9, b) 5/8 ÷ 3/2, c) 4/9 ÷ 6/27
(MP.1) 2. Rewrite each division in Extension 1 as a multiplication. Then reduce and multiply.
Answers: a) 2/3 × 9/4 = 3/2, b) 5/8 × 2/3 = 5/12, c) 4/9 × 27/6 = 2
(MP.1) 3. A continued fraction is a unit fraction where the denominator is written as the sum of a
whole number and another unit fraction. Here is an example:
1
1
1+
2
numerator is 1
1 plus another unit fraction
To find the value of the continued fraction, first rewrite the denominator as an improper fraction.
Then write the division as a multiplication by the reciprocal. Then reduce and multiply.
1
1
2
1
=
2 1
+
2 2
1
=
3
2
2
= 1´
3
2
=
3
1+
Find the value of the denominator first.
Write as a multiplication by the reciprocal.
Find the value of the continued fraction.
a)
1
1
1+
3
b)
1
1
2+
4
c)
1
3+
1
5
Answers: a) 3/4, b) 4/9, c) 5/16
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-49
EE8-8
Mixed Numbers and Improper Fractions
Pages 22–24
Standards: preparation for 8.EE.A.2, 8.EE.B.5, 8.EE.C.7
Goals:
Students will write a fraction as an improper fraction and as a mixed number, and convert
between the two formats.
Students will add and subtract fractions written as mixed numbers.
Students will multiply and divide fractions written as mixed numbers by first converting to
improper fractions.
Prior Knowledge Required:
Can add, subtract, multiply, and divide fractions written as improper fractions
Can find the remainder mentally when dividing whole numbers less than 100
Can divide and write the division equation, including the remainder
Vocabulary: improper fraction, lowest common denominator (LCD), mixed number,
proper fraction, whole number
Materials:
15 quarter circles
tape
(MP.7) Recognizing proper and improper fractions. SAY: An improper fraction has a
numerator larger than or equal to its denominator. Write on the board:
3
4
5
2
7
4
9
10
7
6
1
2
4
3
8
8
Ask for a student to circle the improper fractions (5/2, 7/4, 7/6, 4/3, 8/8) ASK: What is true about
the numerators and denominators in the remaining fractions? (the numerator is smaller than its
denominator) SAY: These are called proper fractions.
(MP.7) Recognizing mixed numbers. SAY: Mixed numbers are numbers written with a whole
number and a proper fraction. Write on the board:
2
whole
number
3
4
proper fraction
ASK: Why can we say that the fraction part of the mixed number here is a proper fraction? (the
numerator is smaller than its denominator)
B-50
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
(MP.7) Converting a mixed number to an improper fraction. Tape 11 quarter circles to the
board in a random fashion. Ask three students to come to the board to each make a circle from
the quarter circles. The results should look like this:
ASK: How many whole circles did we get? (2) Why couldn’t we get another full circle? (we
needed 4 quarters and we only had 3 quarters) How many quarters are in the last circle? (3)
SAY: We have 2 wholes and 3 quarters. ASK: How many quarters did we start with? (11)
Write on the board:
2
3
4
=
11
4
ASK: How many quarters are in the first 2 whole circles? (8) How can you get this number by
multiplying? (2 × 4) How many quarters are in the incomplete circle? (3) Write underneath the
quarter circles on the board:
2×4
+
3
= 11 quarters
Point to the mixed number in the fraction equation and SAY: We can calculate the number of
quarters by multiplying the whole number (2) by the denominator (4) and adding the numerator
(3). Draw arrows and operations to show this, as shown below:
3 +
=
× 24
11
4
Exercises: Convert the mixed number to an improper fraction by multiplying the whole number
by the denominator and then adding the numerator.
a) 3
1
2
b) 2
5
6
c) 4
2
3
Bonus: 10
99
100
Answers: a) 7/2; b) 17/6; c) 14/3; Bonus: 1,099/100
(MP.7) Converting an improper fraction into a mixed number. Write on the board:
11
can mean 11 ÷ 4
4
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-51
Ask a student to divide 4 into 11 including a remainder using the division algorithm. The result
should look like the following:
2
4 11
8
3
ASK: What is the divisor? (4) What is the quotient? (2) What is the remainder? (3) What is
the division statement? (11 ÷ 4 = 2 R 3) SAY: From before, we saw that 11/4 = 2 3/4. Write
on the board:
11
3
=2
4
4
11 ÷ 4 = 2 R 3
ASK: Where does the quotient appear in the mixed number? (the whole number) Where does
the divisor show up? (the denominator) Where does the remainder appear? (the numerator)
Exercises: Change the improper fraction into a mixed number by dividing.
a)
14
3
b)
13
5
c)
31
6
Bonus:
233
30
Answers: a) 4 2/3, b) 2 3/5, c) 5 1/6, Bonus: 7 23/30
(MP.5, MP.7) Regrouping a mixed number having an improper fraction as a mixed
number with a proper fraction. SAY: When adding mixed numbers, we sometimes have an
answer with mixed numbers that still have improper fractions. Write on the board:
3
9
4
Point to the parts of the fraction and SAY: 3 is the whole number. 9/4 is the fraction, but it is
an improper fraction because its numerator, 9, is greater than or equal to its denominator, 4.
We would like to write this as a mixed number but with a proper fraction.
SAY: Let’s look only at the improper fraction part of this mixed number. Write on the board:
9
4
ASK: If we divide 9 by 4, what is the quotient and remainder? (2 R 1) How do we write the
fraction as a mixed number? (2 1/4) SAY: So we can write 3 and 9/4 as 3 and 2 1/4. Write
on the board:
3
B-52
9
1
= 3+2
4
4
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
Point to the whole numbers 3 and 2, and SAY: We can add the whole numbers to get 5 1/4.
Continue writing on the board:
3
9
1
1
= 3+2 = 5
4
4
4
Exercises: Write as a mixed number with a proper fraction.
a) 3
11
4
b) 2
8
5
c) 5
14
3
Bonus: 5
301
100
Answers: a) 5 3/4, b) 3 3/5, c) 9 2/3, Bonus: 8 1/100
(MP.5, MP.7) Adding mixed numbers by leaving fractions in mixed number form. Write on
the board:
Two Grade 8 classes had pizza parties. Class A ate 3
5
7
pizzas. Class B ate 2 pizzas.
8
8
What was the total number of pizzas eaten?
ASK: What operation do we need to find the answer? (addition) Write on the board:
3
5
7
+2
8
8
SAY: We can add the fractions by adding the whole numbers and fractions separately.
ASK: What is 3 + 2? (5) What is 5/8 + 7/8? (12/8) Continue writing on the board:
5
7
+2+
8
8
12
= 5+
8
= 3+
SAY: We don’t want an improper fraction in the answer so we will have to change that.
ASK: What is 12/8 written as a mixed fraction? (1 4/8) Continue writing on the board:
= 5 +1
4
8
SAY: We’d like to write the answer as a mixed number with a proper fraction. ASK: What is
5 + 1? (6) Continue writing on the board:
=6
4
8
ASK: Is the answer in reduced form? (no) How can you tell? (the GCF of 4 and 8 is 2) What is
4/8 in reduced form? (1/2) Continue writing on the board:
=6
1
2
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-53
SAY: So the two classes ate 6 1/2 pizzas in total.
Exercises: Add. Then write the answer as a mixed number with a proper fraction. Reduce
fractions where possible.
5
6
a) 4 + 1
4
6
3
8
b) 2 + 5
7
8
2
3
c) 5 + 4
2
3
Bonus: 99
79
23
+3
100
100
Answers: a) 6 1/2, b) 8 1/4, c) 10 1/3, Bonus: 103 1/50
(MP.5, MP.7) Adding mixed numbers with different denominators. Write on the board:
2
3
5
+1
4
6
ASK: What is different about this question compared with our previous ones? (these fractions
have different denominators, 4 and 6) To add the numbers, what do we need to find first? (the
lowest common denominator, LCD) What is the LCD of 4 and 6? (12) What is an equivalent
fraction for 3/4 and with the denominator 12? (9/12) What is an equivalent fraction for 5/6
with the denominator 12? (10/12) SAY: So we can rewrite the question. Continue writing on
the board:
=2
9
10
+1
12
12
Ask a volunteer to add the whole numbers and add the fractions to write the answer, as
shown below:
=3
19
12
ASK: So should we stop with the answer in this form? Help students to see that the answer
combines a whole number and improper fraction, so they need to change the answer to a mixed
number with a proper fraction. ASK: How do we rewrite 19/12 as a mixed number? (1 7/12)
Continue writing on the board:
7
= 3 +1
12
ASK: What is the final answer as a mixed number with a proper fraction? (4 7/12) Write the final
answer on the board:
=4
7
12
Exercises: Add. Write your answer as a mixed number with a proper fraction.
2
3
a) 2 + 1
3
4
2
5
b) 3 + 4
5
6
Bonus: 4
43
37
+2
50
100
Answers: a) 4 5/12, b) 8 7/30, Bonus: 7 23/100
B-54
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
(MP.5, MP.7) Subtracting mixed numbers that have the same denominator without
regrouping. Write on the board:
5
1
3 -1
8
8
SAY: We can subtract mixed numbers by subtracting the whole numbers and fractions
separately, and then adding the results. ASK: What is 3 − 1? (2) What is 5/8 − 1/8? (4/8)
Continue writing on the board:
æ 5 1ö
= (3 - 1) + çç - ÷÷÷
çè 8 8 ø
2 +
=
4
8
ASK: Can the fraction 4/8 be reduced? (yes) How can you tell? (the GCF of 4 and 8 is 4) What
is the reduced fraction equivalent to 4/8? (1/2) What is the final answer? (2 1/2)
Exercises: Subtract. Reduce to lowest terms if possible.
3
4
a) 4 - 1
1
4
5
8
b) 7 - 3
3
8
Bonus: 50
37
33
- 20
40
40
Answers: a) 3 1/2, b) 4 1/4, Bonus: 30 1/10
(MP.5, MP.7) Subtracting mixed numbers with the same denominators by regrouping.
SAY: Sometimes it is difficult to subtract mixed numbers because the fraction part of the first
number is smaller than the fraction part of the second number. Here is an example. Write on
the board:
4
1
3
-1
4
4
ASK: What makes this subtraction trickier? (3/4 is bigger than 1/4 so we can’t subtract)
SAY: We can regroup the first number, 4 1/4, to make this easier. Draw on the board:
ASK: What mixed number can we write for this picture? (4 1/4) SAY: I am going to break up the
last full circle into 4 quarters. Draw on the board:
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-55
ASK: How many whole circles do I have? (3) How many quarters are left? (5) SAY: I can write
this as 3 5/4. In other words, I regrouped 1 whole as 4 quarters. This will help us when we can’t
subtract the fractions. Write on the board:
1
3
-1
4
4
5
3
= 3 -1
4
4
4
ASK: How has the regrouping helped us? (the first fraction is now larger than the second
fraction) Ask for a volunteer to subtract the whole numbers and subtract the fractions on the
board. (2 2/4) Ask for another volunteer to reduce the fraction to lowest terms. (2 1/2)
Exercises: Subtract by regrouping the first mixed number.
3
8
a) 4 - 2
5
8
1
6
b) 7 - 3
4
6
Bonus: 3
7
9
-1
50
50
Answers: a) 1 3/4, b) 3 1/2, Bonus: 1 24/25
(MP.5, MP.7) Regrouping mixed numbers with different denominators. SAY: Sometimes
when subtracting, you have to use all the methods we discussed: changing fractions so they
have the same denominators, regrouping, and reducing to lowest terms. Write on the board:
1
2
6 -2
4
3
ASK: Are the denominators the same? (no) What is the LCD of these fractions? (12) What is the
equivalent fraction for 1/4 with the denominator 12? (3/12) What is the equivalent fraction for 2/3
with the denominator 12? (8/12) Continue writing on the board:
=6
3
8
-2
12
12
ASK: Can we subtract the fractions? (no) Why not? (the first fraction is smaller than the second
one) How can we regroup 6 3/12 to help? (change it to 5 15/12) Continue writing on the board:
=5
15
8
-2
12
12
ASK: Can we subtract now? (yes) Why? (the fraction part of the first number is larger than the
fraction part of the second one) Ask for a volunteer to subtract the whole numbers and subtract
the fractions separately. (3 7/12)
Exercises: Subtract the mixed numbers.
1
4
a) 3 - 1
5
6
1
8
b) 6 - 3
3
4
c) 5
3
7
-1
10
15
Answers: a) 1 5/12, b) 2 3/8, c) 3 5/6
B-56
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
(MP.7) Multiplying mixed numbers. SAY: It is convenient to leave numbers in the form of
mixed numbers when adding or subtracting, but when multiplying or dividing, it is much easier
if we first convert the mixed numbers to improper fractions. Write on the board:
3
3
2 ´1
4
5
Ask for two volunteers to each change one of the mixed numbers into an improper fraction, as
shown below:
=
11 8
´
4 5
ASK: What can we do to make the numbers smaller before multiplying? (reduce) What number
from the numerator and number from the denominator have a greatest common factor other
than 1? (8, 4) What can we divide both numbers by? (4) Ask for a volunteer to reduce the
fractions on the board, as shown below:
=
11
8 ¸4
´
4 ¸4 5
=
11 2
´
1 5
ASK: Now that we have reduced the numbers, how do we multiply the fractions? (multiply the
numerators and denominators separately) What is the final answer? (22/5) Write the answer
on the board.
Exercises: Multiply the mixed numbers. Remember to reduce to lowest terms.
2
3
a) 4 ´ 3
1
7
4
5
b) 2 ´1
4
7
3
8
c) 3 ´ 3
5
9
Answers: a) 14 2/3, b) 4 2/5, c)12
(MP.7) Dividing mixed numbers. SAY: Just as we did when multiplying mixed numbers, when
dividing mixed numbers we need to change them to improper fractions first. Write on the board:
3
1
1
¸2
4
2
Have two volunteers each change a mixed number into an improper fraction, as shown below:
=
13 5
¸
4
2
ASK: What should we do to divide these fractions? (multiply the first fraction by the reciprocal of
the second fraction) Continue writing on the board:
=
13 2
´
4 5
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-57
ASK: Before we multiply, what should we do first? (reduce the fractions) Which numerator from
either fraction and which denominator from either fraction have a GCF other than 1? (2 and 4)
What can we divide both numbers by? (2) Ask for a volunteer to reduce the fractions on the
board, as shown below:
13
2 ¸2
´
4 ¸2 5
13 1
=
´
2 5
=
ASK: Now that we have finished reducing, how do we multiply the fractions? (multiply the
numerators and denominators separately) What is the final answer? (13/10) Write the answer
on the board.
Exercises: Divide the fractions. Reduce the answer to lowest terms.
1
4
a) 2 ¸ 1
4
5
3
8
b) 3 ¸ 2
1
4
4
7
c) 3 ¸ 2
1
2
Answers: a) 1 1/4, b) 3/2, c) 1 3/7
Extensions
(MP.1) 1. The sum of the digits in a mixed number is 8. If the mixed number is less than 2, what
are the possible fractions?
Answers: 1 1/6, 1 2/5, 1 3/4
3
4
(MP.1) 2. Find the missing mixed number: 2 ¸
3
=1
11
Answer: 121/56
(MP.1) 3. Fill in the blanks using only the digits 2, 3, 4, 5, 6, and 8 with no repetition, to find the
largest possible answer to …
a)
+
b)
-
c)
´
d)
¸
Answers: a) 8 4/5 + 6 2/3 = 15 7/15, b) 8 5/6 − 2 3/4 = 6 1/12, c) 8 4/5 × 6 2/3 = 58 2/3,
d) 8 5/6 ÷ 2 3/4 = 3 7/33
B-58
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
EE8-9
Order of Operations
Pages 25–26
Standards: preparation for 8.EE.A.2, 8.EE.B.5, 8.EE.C.7
Goals:
Students will apply the standard order of operations to calculations involving fractions.
Prior Knowledge Required:
Can add, subtract, multiply, and divide fractions
Vocabulary: operation, order of operations, priority, scientific calculator
Materials:
scientific calculators
a pair of dice for each pair of students
BLM Random Operations with Fractions (pp. B-110–111)
scissors and tape
(MP.5, MP.7) Review the order of operations. Write on the board:
2+5×3
ASK: What operations appear in this question? (addition, multiplication) SAY: There seems to
be a choice here: doing 2 + 5 first or doing 5 × 3 first. Have half of the class calculate 2 + 5 first,
and then multiply by 3. Have the other half of the class calculate 5 × 3 first, and then add 2.
ASK: What are the answers? (21, 17) SAY: Mathematicians decided they wanted only one
answer for this calculation. So they created a standard order of operations, which is a rule to
help everyone do the questions in the same order and get the same answer every time.
SAY: Here are the operations we will use. They tell us to perform the operations addition,
subtraction, multiplication, division, or brackets. Write on the board:
+
−
×
÷
()
ASK: Which of these operations must be done first? (brackets) Which two operations come
next? (×, ÷) SAY: Multiplication and division come next and are equally important. Of these two
operations, whichever comes first as you read a statement from left to right should be done first.
ASK: Which two operations come last? (+ , −) SAY: Addition and subtraction come next and are
equally important. Of these two operations, whichever comes first, from left to right, should be
done first. Write on the board:
5 × (6 + 2)
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-59
Ask for a volunteer to circle the operation that should be done first. (6 + 2) ASK: What is 6 + 2?
(8) What is 5 × 8? (40) Write on the board:
5 × (6 + 2)
=5×8
= 40
Write on the board:
16 ÷ 2 × 4
ASK: What are the two operations here? (division, multiplication) SAY: These operations are
equally important. ASK: Which do we do first? (division) Why? (because it comes first in the
question) What is 16 ÷ 2? (8) What is 8 × 4? (32) Continue writing on the board:
=8×4
= 32
ASK: If we wanted to perform the question 2 × 4 first, how could we change the question? (add
brackets around 2 × 4) Write on the board:
16 ÷ (2 × 4)
ASK: What is 2 × 4? (8) What is 16 ÷ 8? (2) Continue writing on the board:
= 16 ÷ 8
=2
SAY: Notice that the answers are different when we do the multiplication in brackets before
the division.
(MP.5) Using a scientific calculator to check order of operations. Tell students that a
scientific calculator has been programmed to follow the same order we just discussed. SAY: On
your calculator, input 2 + 5 × 3 =. The answer should be 17. If your calculator gave you a different
answer, it has not been programmed for the order of operations that mathematicians have
agreed on.
Exercises: Use the standard order of operations to calculate. Then check your answer using a
scientific calculator.
a) 3 + 12 ÷ 6
b) 9 × (7 − 3)
c) 7 + 3 × 4
Answers: a) 5, b) 36, c) 19
B-60
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
(MP.7) Order of operations with multiple steps. SAY: If there are several operations to be
done, do them one at a time. As noted earlier, do the most important operation first using the
rule that mathematicians determined—in other words, any brackets, then multiplication or
division, then addition or subtraction—then rewrite the rest of the question and proceed to the
next operation. Write on the board:
5 + 3 × (7 + 2)
ASK: Which operation should be done first? (brackets) What is 7 + 2? (9) Continue writing on
the board:
=5+3×9
SAY: There are two operations left: addition and multiplication. ASK: Which is more important?
(multiplication) ASK: What is 3 × 9? (27) What is 5 + 27? Continue writing on the board:
= 5 + 27
= 32
(MP.7) Order of operations within fractions. SAY: Sometimes the numerator and
denominator of a fraction have calculations that have to be done first. Write on the board:
6 + 5´2
12 - 8 ¸ 2
SAY: We treat the numerator and denominator separately as if each has brackets around it.
Write on the board:
=
(6 + 5 ´ 2)
(12 - 8 ¸ 2)
SAY: Remember that the fraction line actually means division. So we can think of it as this.
Continue writing on the board:
= (6 + 5 × 2) ÷ (12 − 8 ÷ 2)
SAY: Brackets have to be done first. Do the brackets separately and then divide the
answers from the two brackets. ASK: In the numerator, which operation should be done first?
(multiplication) What is 5 × 2? (10) What is 10 + 6? (16) In the denominator, what operation
should be done first? (division) What is 8 ÷ 2? (4) What is 12 − 4? (8) Continue writing on
the board:
(6 + 10)
(12 - 4)
16
=
8
=
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-61
SAY: So the numerator is 16 and the denominator is 8. Remember that the fraction line can be
interpreted as division. What is 16 ÷ 8? (2) Write the answer on the board.
Exercises: Evaluate the expression.
a)
2 + 3´4
9 - 1´ 2
b)
15 - 35 ¸ 7
12 - 3 ´ 2
c)
2 + 3´5 - 3
6 - 12 ¸ 4 + 1
Answers: a) 2 , b) 5/3, c) 7/2
(MP.7) Order of operations involving fractions. The order of operations applies to operations
with fractions. Write on the board:
3
5
3
+
¸
4 12 8
ASK: What are the operations between the fractions? (addition, division) Which operation must
be done first according to the rule? Which has higher priority? In other words, which operation is
more important and should be done first? (division) What is the method for dividing fractions?
(multiply by the reciprocal of the second fraction) What is the reciprocal of 3/8? (8/3) Continue
writing on the board:
=
3
5 8
+ ´
4 12 3
SAY: So, we must complete the division by multiplying by the reciprocal. However, when we do
that, we need to reduce the numbers in the numerator and denominator first as much as we
can. ASK: How can we reduce? What pair of numbers, one from the numerator and one from
the denominator have a GCF other than 1? (8 and 12) What is the GCF of 8 and 12? (4)
SAY: Let’s divide each of these numbers by 4. Continue writing on the board:
=
3
5
8 ¸4
+
´
4 12 ¸4 3
ASK: What is 8 ÷ 4? (2) What is 12 ÷ 4? (3) Continue writing on the board:
=
3 5 2
+ ´
4 3 3
SAY: Now we can multiply. ASK: What is 5/3 × 2/3? (10/9) Continue writing on the board:
=
3 10
+
4
9
SAY: Now the only operation we have left is the addition of fractions. ASK: What is the LCD of
4 and 9? (36) What are the equivalent fractions using the denominator 36? (27/36 and 40/36)
B-62
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
Continue writing on the board:
=
27 40
+
36 36
ASK: Now that the denominators are the same, how do we add the fractions? (add the
numerators) What is 27 + 40? (67) Continue writing on the board:
=
67
36
SAY: Since this is an improper fraction, we must change the answer to a mixed number. ASK: If
we wanted to write this as a mixed number, what would it be? (1 31/36)
Exercises: Evaluate the expression.
a)
3 5 1
+ ¸
4 6 2
b)
2 1 4 1
+ ´ 3 2 3 6
c)
3 3 2 1
¸ - ´
4 2 3 8
Answers: a) 2 5/12, b) 1 1/6, c) 5/12
Activity
Students work in pairs. Give each pair of students a pair of dice, one half of BLM Random
Operations with Fractions (1), and a copy of BLM Random Operations with Fractions (2).
Have students cut out the die from the first page of the BLM and tape together the cutout to
create a die.
Have each pair create questions using the dice and the second page of the BLM. For each
question, partners take turns rolling the paper die and writing the resulting operations in the
circles. Then they take turns rolling the numbered dice and writing the resulting numbers in
the boxes.
operation
operation
Might become:
æ5
æ9
3ö
7ö
6 ´çç + 2 ÷÷÷ or 12 ¸ çç - 1 ÷÷÷
çè12
ç
è4
4ø
8ø
Example:
number
fraction
mixed number
NOTE: For questions involving subtraction, students should place the larger number first. For
mixed numbers, they should choose the larger number for the denominator so that the fraction
part will be a proper fraction.
Partners find the answer individually, and then check each other’s work before creating the
next question.
(end of activity)
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-63
Extension
(MP.1) Place brackets in the correct places to make the equation true.
a) 3 + 5 × 4 = 32
b) 7 + 5 × 3 + 2 = 38
c) 7 + 5 × 3 + 2 = 60
d)
2 + 5´3
=3
17 - 5 ´ 2
e)
2 + 5´3 -1
=2
1+ 3 ´ 2
g)
5 3 35 1
¸ ´
=
8 7 12 2
h)
5 2 1 1 5
¸ ¸ + =
7 3 3 4 8
f)
1
2 7
+ 3´ =
2
3 3
Answers: a) (3 + 5) × 4 = 32, b) (7 + 5) × 3 + 2 = 38, c) (7 + 5) × (3 + 2) = 60,
æ1
ö 2 7
(2 + 5)´(3 - 1)
2 + (5 ´ 3) - 1
(2 + 5)´ 3
5 æ 3 35 ö 1
= 2 or
= 2 , f) çç + 3÷÷÷´ = , g) ¸ çç ´ ÷÷÷ = ,
= 3 , e)
ç
è2
ø 3 3
8 çè 7 12 ø 2
1+ 3 ´ 2
1+ 3 ´ 2
17 - 5 ´ 2
5 æ 2 æ 1 1 öö 5
h) ¸ ççç ¸ ççç + ÷÷÷÷÷÷ =
7 è 3 è 3 4 ø÷ø 8
d)
B-64
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
EE8-10
Decimal Review
Pages 27–29
Standards: preparation for 8.EE.A.4
Goals:
Students will review how decimals are created from decimal fractions.
Students will review how to mentally multiply and divide decimals by powers of 10.
Prior Knowledge Required:
Can add, subtract, multiply, and divide fractions
Can find the value of a digit depending on its place value
Vocabulary: decimal, decimal fraction, decimal point, expanded form, place value, powers of 10
Materials:
BLM Do You Believe in Magic? (p. B-112, see Extension 3)
(MP.5, MP.7) Review powers of 10. Write on the board:
10, 100, 1,000, 10,000
Remind students that these numbers are called powers of 10. ASK: What operation can you
perform to get from a power of 10 to the next power of 10? (multiply by 10) Point to the numbers
on the board and ASK: What are the next two powers of 10? (100,000; 1,000,000) How can we
get to the next power of 10 without multiplying? (write another zero at the end of the number)
Write on the board:
7
23
536
459
,
,
,
10 100 1,000 10,000
ASK: What is true about the denominators of these fractions? (they are all powers of 10) SAY: A
fraction with a power of 10 as the denominator is called a decimal fraction. Write on the board:
7
=
10 100
SAY: I want to find an equivalent decimal fraction for 7/10 that has 100 in its denominator
instead of 10. ASK: What can I multiply the numerator and denominator of 7/10 by to make the
new denominator 100? (10) What is 7 × 10? (70) Continue writing on the board:
7 10
70
´ =
10 10 100
SAY: So the fraction 70/100 is a decimal fraction equivalent to 7/10.
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-65
Exercises: Find the missing number to make the decimal fractions equivalent.
a)
5
=
10 100
b)
3
=
10 1,000
c)
23
=
100 1,000
Bonus:
10
=
70
700
=
100
Answers: a) 50; b) 300; c) 230; Bonus: 7, 1,000
(MP.7) Writing decimals in expanded form. Write on the board:
332
SAY: The digit 3 in the number 332, has a different value depending on where it appears in the
number. ASK: How much is the first 3 worth? (300) How much is the next 3 worth? (30) What
determines the value of the digit? (the place value) Reading from left to right, what are the place
values in this number? (hundreds, tens, ones) When reading from left to right, what operation
can we perform to find the next place value? (divide by 10) Write on the board:
Place Values
100
10
1
3
3
2
Digits
Point to the place values and have students confirm that to get from 100 to 10, and from 10 to 1,
you divide by 10 each time. SAY: To find the value of a digit, multiply the digit by its place value.
Point to the number 332 and ASK: What is the first 3 worth? (300) How did you calculate that?
(3 × 100) What is the next 3 worth? (30) How did you calculate that? (3 × 10) What is the 2 worth?
(2) What can we multiply the 2 by to find its value? (2 × 1) How do we find the total value? (add
300, 30, and 2)
SAY: Remember that, with decimals, there are also place values to the right of the ones column.
We again divide by 10 to find the next place to the right. Point to the place values and SAY: We
divide 1 by 10 to get the place value 1/10, we divide 1/10 by 10 to get the place value 1/100,
and we divide 1/100 by 10 to get the place value 1/1,000. Draw on the board:
Place
Values
Digits
1,000
100
10
1
1
10
1
100
1
1,000
0
2
3
6
SAY: The place values to the left of the decimal place, in words, are thousands, hundreds, tens,
and ones. Write these place values in words in the blank top row. ASK: What are the place
values after the decimal place in words—in other words, to the right of the decimal place?
(tenths, hundredths, thousandths) Point out the difference in the spelling; that is, the added “th”
sound. Write the remaining place values in words in the top row, as shown below:
Place
Values
Digits
B-66
Thousands
Hundreds
Tens
Ones
1,000
100
10
1
0
Tenths
1
10
2
Hundredths Thousandths
1
1
1,000
100
3
6
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
SAY: The decimal 0.236 is made from 2 tenths, 3 hundredths, and 6 thousandths. If we multiply
each digit by its place value, we get the following. Write on the board:
0.236 = 2 ´
1
1
1
+ 3´
+ 6´
10
100
1,000
SAY: We can also write it this way:
0.236 =
2
3
6
+
+
10 100 1,000
SAY: This is called the expanded form.
Exercises: Write the decimal in expanded form.
a) 0.157
b) 0.83
c) 0.4628
Bonus: 0.0034
Answers: a) 1/10 + 5/100 + 7/1,000; b) 8/10 + 3/100; c) 4/10 + 6/100 + 2/1,000 + 8/10,000;
Bonus: 0/10 + 0/100 + 3/1,000 + 4/10,000 or 3/1,000 + 4/10,000
(MP.7) Write a number in expanded form as a decimal. SAY: If a number is in expanded
form, we can write it as a decimal by writing the numerators in the correct columns of the place
value chart. Write on the board:
2+
1
3
7
+
+
10 100 1,000
Thousands
Place
Values
Digits
1,000
Hundreds Tens
100
10
Ones
1
Tenths
1
10
Hundredths Thousandths
1
1
1,000
100
Ask a volunteer to copy each numerator into the correct column (1 in the tenths column, 3 in the
hundredths, 7 in the thousandths) SAY: The 2 just goes in the ones column. ASK: So, what is
the decimal form of the number? (2.137)
Now erase 2.137 from the place value chart and write on the board:
2+
1
7
+
10 1,000
Ask a volunteer to write the digits for this new number in the correct columns of the place value
chart. Point out that there is no fraction with 100 in the denominator. The chart should have a 2
in the ones column, a 1 in the tenths column, and a 7 in the thousandths column. There should
be no digit in the hundredths column. ASK: What digit should we place in the hundredths
column? (0) What is the decimal form of the number? (2.107)
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-67
Exercises: Write in expanded form.
a) 3.14
b) 7.05
c) 8.3004
Answers: a) 3 + 1/10 + 4/100; b) 7 + 5/100; c) 8 + 3/10 + 4/10,000
(MP.7) Writing a decimal as a decimal fraction. Write on the board:
0.34
ASK: How do we write this in expanded form? (3/10 + 4/100) SAY: Let’s add the fractions to
find a single fraction. Write on the board:
3
4
+
10 100
ASK: What is the LCD of 10 and 100? (100) What should we multiply the numerator and
denominator in the first fraction by to make the denominator 100? (10) Continue writing on
the board:
=
3 ´10
4
+
10 ´10 100
ASK: What is 3 × 10? (30) What is the equivalent fraction for the first fraction now? (30/100)
SAY: Now the denominators are the same for both fractions, so we can add the numerators.
ASK: What is 30 + 4? (34) Continue writing on the board:
=
34
100
SAY: So 0.34 written as a decimal fraction is 34/100. ASK: How many decimal digits are
there to the right of the decimal point? (2) How many zeros are in the power of 10 in the
denominator? (2)
Exercises: Write a decimal fraction for the decimal.
a) 0.27
b) 0.4
c) 0.361
Answers: a) 27/100; b) 4/10; c) 361/1,000; Bonus: 27/1,000
Bonus: 0.027
(MP.7) Writing a decimal fraction as a decimal. Write on the board:
37
100
ASK: How many zeros are in the power of 10 in the denominator? (2) SAY: So we need
2 decimal digits to the right of the decimal point. Continue writing on the board:
= 0.37
B-68
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
Count the digits to the right of the decimal point and show that it is the same as the number of
zeros in the denominator. SAY: Let’s try another example. Write on the board:
9
100
ASK: How many zeros are in the power of 10 in the denominator? (2) How many decimal digits
should there be to the right of the decimal point? (2) Continue writing on the board:
= 0.9
ASK: Is this correct? (no) Why? (there is only 1 decimal digit to the right of the decimal point)
How can we have 2 decimal digits and still have the decimal equal the fraction 9/100? (put a
zero before the 9, so that it is 0.09) Revise the number 0.9 on the board by inserting a zero to
the left of the digit 9, as shown below:
= 0.09
Ensure that students notice the change. ASK: There are 2 decimal digits to the right of the
decimal point— why is 0.90 not correct? (0.90 has 2 decimal points but 0.90 = 90/100, which is
not the same as 9/100 or 09/100 or 0.09.)
Exercises: Write a decimal for the decimal fraction.
a)
7
10
b)
81
100
c)
3
100
Bonus:
8
10,000
Answers: a) 0.7, b) 0.81, c) 0.03, Bonus: 0.0008
(MP.7, MP.8) Multiplying decimals by powers of 10. SAY: Suppose we want to multiply a
decimal by a power of 10. Write on the board:
0.28 × 10
ASK: How can we write 0.28 as a decimal fraction? (28/100) How can we write 10 as a fraction?
(10/1) SAY: Let’s multiply in fraction form. Continue writing on the board:
=
28 10
´
100 1
SAY: Instead of reducing first, let’s multiply the numerators and denominators right away.
ASK: What is 28 × 10? (280) What is 100 × 1? (100) Continue writing on the board:
=
280
100
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-69
ASK: How can we write this as a mixed number? (2 8/100) SAY: We can write this mixed
number in expanded form as 2 + 80/100. Continue writing on the board:
= 2+
80
100
ASK: How can we write this as a decimal? (2.80) Write on the board:
0.28 × 10 = 2.8
ASK: What is the shortcut for multiplying by 10? (move the decimal point to the right by 1 position)
Write on the board:
0.28 × 10 = 0 2 8
ASK: What is the shortcut for multiplying by 100? (move the decimal point to the right by
2 positions) What is the shortcut for multiplying by 1,000? (move the decimal point to the right
by 3 positions) SAY: It helps to imagine zeros in empty columns when we are either multiplying
or dividing decimals by powers of 10. These imaginary zeros serve as placeholders—they hold
the place so we don’t lose the place value for a column. Write on the board:
3.2 × 1,000
ASK: How many places to the right should we move the decimal point when multiplying 3.2 by
1,000? (3 places) Draw on the board:
3
2
ASK: What is the new number? (3,200.000) SAY: We can leave out the zeros at the end
because when we multiply by their place values, the answer does not change.
Exercises: Multiply by the power of 10 by moving the decimal point to the right the correct
number of places.
a) 2.35 × 10
b) 0.37 × 1000
c) 1.034 × 100
Bonus: 0.49 × 100,000
Answers: a) 23.5, b) 370, c) 103.4, Bonus: 49,000
(MP.7, MP.8) Dividing decimals by powers of 10. SAY: Suppose we want to divide a decimal
by a power of 10. Write on the board:
0.28 ¸ 10
B-70
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
ASK: How can we write 0.28 as a decimal fraction? (28/100) How can we write 10 as a fraction?
(10/1) SAY: Let’s divide in fraction form. Continue writing on the board:
=
28 10
¸
100
1
ASK: What do we have to do to divide by a fraction? (multiply by the reciprocal) What is the
reciprocal of 10/1? (1/10) Continue writing on the board:
28
1
´
100 10
28
=
1,000
=
ASK: How can we write this as a decimal? (0.028) NOTE: Some students may answer 0.28—in
this case, point out that 0.28 = 28/100, not 28/1,000. Write on the board:
0.28 ÷ 10 = 0.028
ASK: What is the shortcut for dividing by 10? (move the decimal point to the left by 1 position)
Draw an arrow to show how the decimal point moved, as shown below:
0.28 ÷ 10 = 0.028
ASK: What is the shortcut for dividing by 100? (move the decimal point to the left by 2 positions)
For dividing by 1,000? (move the decimal point to the left by 3 positions) SAY: Remember, we
can imagine zeros in empty columns like placeholders. Write on the board:
3.2 ÷ 1,000
ASK: How many places to the left should we move the decimal point when dividing by 1,000?
(3) Draw on the board:
3
2
ASK: What is the new number? (0.0032) SAY: We can leave out the zeros at the end because
when we multiply by their place values, the answer does not change. We cannot leave out the
zeros immediately after the decimal point because the place values for 3 and 2 would change.
Exercises: Divide by the power of 10 by moving the decimal point to the left the correct number
of places.
a) 23.5 ÷ 10
b) 283.7 ÷ 100
Bonus: 0.49 ÷ 100,000
c) 10.34 ÷ 1,000
Answers: a) 2.35, b) 2.837, c) 0.01034, Bonus: 0.0000049
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-71
Extensions
(MP.1) 1. The number system we use is called decimal. It uses 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8,
and 9. The decimal number place values are powers of ten: 1, 10, 100, 1,000, etc.
Computers use a number system called binary. This is a binary number:
10111
Binary numbers use only the digits 0 and 1. The binary number place values are powers of two:
1, 2, 4, 8, 16, etc. Here are the place values of the binary number 10111:
Place Values
Digits
16
1
8
0
4
1
2
1
1
1
To find the decimal value of a binary number, multiply each digit by its place value, and then
add the results. The decimal value of the binary number 10111 is:
(1 × 16) + (0 × 8) + (1 × 4) + (1 × 2) + (1 × 1)
= 16 + 0 + 4 + 2 + 1
= 23
Find the decimal value of the binary number.
a) 11011
b) 10101
c) 11101
Answers: a) 27, b) 21, c) 29
(MP.1) 2. In the decimal number system, the decimal place values continue:
1
1
1
,
,
,
10 100 1,000
1
, etc. In contrast, in the binary number system used with computers, the binary place
10,000
1 1 1 1
1
, , ,
,
, etc.
values have denominators that are powers of two:
2 4 8 16 32
1
1
1
1
Place Values
2
4
8
16
0
Digits
1
0
0
1
As in Extension 1, to find the decimal value of a binary number, multiply each digit by its place
value and then add the results. The decimal value of the binary decimal 0.1001 is:
æ 1ö÷ æ
ö æ
ö æ
ö
ç1´ + ç0 ´ 1 ÷÷ + çç0 ´ 1÷÷ + çç1´ 1 ÷÷
÷
÷
÷
ççè 2 ÷÷ø èçç
ç
ç
4ø è
8 ø è 16 ø
1
1
+0+0+
2
16
8
1
=
+
16 16
9
=
16
=
B-72
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
Find the decimal value of the binary number.
a) 0.0101
b) 0.1100
c) 0.1011
Answers: a) 1/4 + 1/16 = 5/16, b) 1/2 + 1/4 = 3/4, c) 1/2 + 1/8 + 1/16 = 11/16
(MP.1) 3. Students work in pairs. Distribute BLM Do You Believe in Magic?. The BLM has
5 tables of numbers and each table is labeled with a letter, M, A, G, I, or C. Each student
secretly chooses a number from 1 to 31. Student A tells their partner all the tables the number
is found in, and then Student B tries to guess Student A’s secret number. The partners then
switch roles.
SAY: The tables M, A, G, I, and C represent the binary place values. The place value for each
table is found at the top left corner of each table. So the M table represents 16, A represents 8,
G represents 4, I represents 2, and C represents 1. To find the secret number, you add the
place values for the tables where the number is found.
Example: A secret number appears in tables M, G, and I. These tables stand for the place
values, 16, 4, and 2.
M
A
G
I
C
1
Place Values
16
8
4
2
0
Digits
1
0
1
1
Thus, the secret number is: 16 + 4 + 2 = 22.
Have students repeat the game in pairs, but instead of guessing, students should add place
values to find their partner’s secret number.
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-73
EE8-11
Operations with Decimals
Pages 30–32
Standards: preparation for 8.EE.A.4
Goals:
Students will review how to add, subtract, multiply, and divide multi-digit decimals.
Prior Knowledge Required:
Can add, subtract, multiply, and divide fractions
Can add, subtract, multiply, and divide multi-digit whole numbers
Vocabulary: decimal digits, decimal fractions, decimals, whole numbers
Materials:
grid paper or BLM 1 cm Grid Paper (p. I-1)
BLM Order of Operations Challenge (pp. B-113–114, see Extension)
(MP.7) Adding and subtracting decimals as decimal fractions. Write on the board:
0.24 =
0.43 =
Ask two students to come to the board and write each decimal as a decimal fraction. (24/100,
43/100) ASK: Why should there be 2 zeros in the denominator? (because there are two digits to
the right of the decimal point) SAY: Let’s add the decimal fractions. ASK: Since the denominators
are the same (both are 100) how do we add the fractions? (add the numerators) What is
24 + 43? (67) Continue writing on the board:
24
100
43
+ 0.43 =
100
67
=
100
0.24 =
ASK: How can we write 67/100 as a decimal? (0.67) Write the answer on the board and
SAY: Look only at the left side of the equations. ASK: How could we have added the decimals
without changing the decimals to decimal fractions? (add the decimal digits) Where do we place
the decimal points? (line up the decimal points) Point out in the example on the board how the
decimals points in the question and the answer line up; emphasize by drawing a line to show
aligning the decimal points. Write on the board:
0.63 + 0.245
B-74
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
SAY: To add or subtract decimals, we have to rewrite the decimals underneath each other so
that the decimal points are lined up. We should also add zeros if necessary to make the two
numbers the same length. Write on the board and point to the additional zero in the first line:
0.630
+0.245
SAY: Now we just add the decimals as if they were whole numbers. When we are done, we put
the decimal point in the answer underneath the other decimal points. Have a volunteer add the
decimals. (0.875) Point out that the decimal points line up.
Write on the board:
0.4 − 0.137
Ask a volunteer to write the two decimal numbers one underneath the other so the decimal
points line up. Ask another volunteer to add zeros to the decimals if necessary and then
subtraction. Remind them that regrouping may be necessary. The answer should look like this:
3 9 10
0. 4 0 0
-0.1 3 7
0.2 6 3
Exercises: Add or subtract the decimals.
a) 0.23 + 0.478
b) 0.34 − 0.156
Answers: a) 0.708, b) 0.184, Bonus: 0.493
Bonus: 0.453 + 0.23 − 0.19
(MP.7) Multiplying a decimal by a whole number. Write on the board:
1.23 × 3
ASK: How can we write 1.23 as a mixed number? (1 23/100) How can we write it as an
improper fraction? (123/100) How can we write 3 as an improper fraction? (3/1) Continue writing
on the board:
=
123 3
´
100 1
ASK: How do we multiply fractions? (multiply the numerators and denominators separately)
What is 123 × 3? (369) What is 100 × 1? (100) Continue writing on the board:
=
369
100
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-75
SAY: Notice that to multiply 1.23 × 3, we ended up multiplying 123 × 3 for the numerator.
ASK: How we do write the answer as a decimal? (3.69) SAY: So, to multiply a decimal by
a whole number, we multiply the numbers as if there were no decimals and then place the
decimal point in the answer. The number of decimal digits in the answer will be the same as
in the decimal part of the question because the denominator of the fraction remains the same.
1
2
×
3
3
3
6
9
Exercise: Multiply using a grid. You may need to regroup.
a) 2.14 × 4
b) 5.42 × 6
c) 1.034 × 2
Answers: a) 8.56, b) 32.52, c) 2.068
(MP.7) Multiplying a decimal by a decimal. Write on the board:
2.35 × 1.4
ASK: How do we write each decimal as a decimal fraction? (235/100, 14/10) Write on the board:
=
235 14
´
100 10
SAY: Let’s multiply the numerators and denominators without reducing first. ASK: What is
235 × 14? Allow students time to multiply using grid paper or BLM 1 cm Grid Paper. (3,290)
ASK: What is 100 × 10? (1,000) Write on the board:
=
3,290
1,000
SAY: Notice that, once again, for the numerator we multiplied the numbers as if there were
no decimal places, as 235 × 14. ASK: How many zeros are there in the denominator of the
answer? (3) Why are there 3? (2 came from the denominator of the first multiplied number, 100,
and the other came from the second denominator, 10) How can we write the decimal fraction
answer as a decimal? (3.290)
SAY: To multiply a decimal by a decimal, we multiply as if the numbers were whole numbers
and then place the decimal point. The total number of decimal digits in the answer is the sum of
the decimal digits in the multiplication statement.
B-76
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
Write 2.35  1.4 on a grid on the board. Ask for a volunteer to multiply the decimals as if they
were whole numbers. The grid should look like this:
1
2
2
3
5
1
4
9
4
0
2
3
5
0
3
2
9
0
×
ASK: How many decimal digits are in the first decimal in the question? (2) How many decimal
digits are in the second number? (1) What is the total? (3) SAY: So we need 3 decimal digits in
the answer. Count backward 3 spaces from the end of the answer to the multiplication and
place the decimal point in the correct space to make the number 3.290.
Have students do Questions 5–8 on AP Book 8.1 pp. 30–31.
(MP.7, MP.8) Multiplying a decimal by a power of 10. Write on the board:
2
×
5
3
1
0
Ask a volunteer to complete the multiplication on the board. (25.3) Write on the board:
2.53 × 10 = 25.3
ASK: Is there a shortcut for multiplying the decimal by 10? (move the decimal point to the right
1 place) What is the shortcut for multiplying by 100? (move the decimal point to the right 2
places) What is the shortcut for multiplying by any power of 10? (move the decimal point to
the right the same number of places as there are zeros in the power of 10)
Exercises: Multiply by moving the decimal point.
a) 23.65 × 10
b) 8.34 × 100
Bonus: 0.0000025 × 1,000,000
c) 0.0134 × 1,000
Answers: a) 236.5, b) 834, c) 13.4, Bonus: 2.5
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-77
(MP.7, MP.8) Dividing a decimal by a decimal. SAY: It is easier to divide a decimal by a whole
number than by a decimal. Remember, you can make equivalent fractions by multiplying the
numerator and denominator by the same number. Write on the board:
5.31 ÷ 0.3
SAY: This can be written using fraction notation as 5.31/0.3. We can multiply both the top and
bottom of this fraction by 10. Write on the board:
5.31´10
0.3 ´10
ASK: What is 5.31 × 10? (53.1) What is 0.3 × 10? (3) Write the following on the board and
SAY: So the fraction is equivalent to:
=
53.1
3
SAY: This fraction is the same as the division. Write on the board:
3 53.1
SAY: Notice that a shortcut is to move the decimal point one place to the right in both the divisor
and the dividend. Draw arrows to show this, as shown below:
3. 53.1
SAY: To divide a decimal by a whole number, first divide as if all numbers were whole numbers.
Ask for a volunteer to divide on the board. (see answer below)
17 7
3 53.1
-3
23
- 21
21
- 21
0
SAY: As a last step, we have to place the decimal point in the quotient. It should be placed
directly above the decimal in the dividend. Write the decimal point in the dividend so the answer
is 17.7.
Exercises: Divide using long division.
a) 2.548 ÷ 0.7
b) 17.04 ÷ 0.6
Answers: a) 3.64, b) 28.4, Bonus: 657
B-78
Bonus: 26.28 ÷ 0.04
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
(MP.7) Writing a fraction as a decimal. SAY: We can change any fraction into a decimal by
dividing the numerator by the denominator. We can add as many zeros as we need to the right
of the decimal point in the dividend. Write on the board:
5
8
8 5.00000
Ask for a student to do the division on the board. Tell the student to keep dividing until the
remainder is zero, as shown below:
0.625
8 5.00000
-48
20
- 16
40
- 40
0
SAY: So the fraction 5/8 is equivalent to the decimal 0.625.
Exercises: Write as a decimal.
a)
7
8
b)
1
8
c)
3
4
Bonus:
5
16
Answers: a) 0.875, b) 0.125, c) 0.75, Bonus: 0.3125
Extension
(MP.1) Distribute BLM Order of Operations Challenge. This activity can be spread out over
several days or even several weeks.
Students can work in pairs. The object of the challenge is to create an expression using the digits
1, 9, 5, and 7 in that order for as many whole numbers from 1 to 100 as possible. NOTE: The
digits do not have to appear individually. For example: 19 + 57 is a valid expression for the
number 76. Other mathematical symbols can be used as long as students don’t use any digits
other than 1, 9, 5, or 7. You may want to introduce some of the following symbols one at a time
over several weeks:
• absolute value: -5 = 5
• square root: 9 = 3
• round down: êë5.7úû = 5
• exponents: 19 = 1
• round up: éê5.7ùú = 6
Here are some sample answers, both simple and not so simple:
8 = 1 + 9 + 5 – 7, 61 = 1 + 9 + 57 , 88 = 19 ´ 5 - 7 , 25 = 19 + éê5.7ùú , 38 = 19 - 57
At the end of the activity, students can share their answers on chart paper. For each number in
the chart, students can write different ways to produce the same number.
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-79
EE8-12
Rounding Decimals
Pages 33–34
Standards: preparation for 8.EE.A.4
Goals:
Students will review how to round decimals to the nearest tenth or hundredth.
Students will learn how to round money to the nearest penny.
Prior Knowledge Required:
Can add, subtract, multiply, and divide decimals
Vocabulary: approximately equal to symbol (≈), expanded form, hundredths, place value,
rounding, tenths, thousandths
(MP.7) Review expanded form of a number. Write on the board:
Hundreds
Tens
Ones
Tenths
Hundredths
Thousandths
1
3
4
2
0
5
Ask volunteers to write a whole number or fraction to represent the place value for each column.
(100, 10, 1, 1/10, 1/100, 1/1,000) SAY: Remember that in the expanded form, we multiply each
digit by its place value and find the total. Ask a volunteer to write the number in expanded form
on the board. (see answer below)
æ
è
(1´100) + (3 ´10) + (4 ´1) + ççç2 ´
1 ö÷ æç
1 ö÷ æç
1 ö÷
÷
÷÷ + çç0 ´
÷÷ + çç5 ´
10 ø è
100 ø è
1,000 ÷ø
Exercises: Write the number in expanded form.
a) 23.56
b) 9.102
Bonus: 2080.0401
Answers:
a) (2 × 10) + (3 × 1) + (5 × 1/10) + (6 × 1/100)
b) (9 × 1) + (1 × 1/10) + (0 × 1/100) + (2 × 1/1,000)
Bonus: (2 × 1,000) + (0 × 100) + (8 × 10) + (0 × 1) + (0 × 1/10) + (4 × 1/100) + (0 × 1/1,000)
+ (1 × 1/10,000)
(MP.7) Finding the next-smallest place value. SAY: The place values of digits in a number
get smaller as we move from left to right. For example, to the right of the hundreds is the tens
column. To the right of the tenths column is the hundredths column.
Exercise: What is the next-smallest place value?
a) tens
b) hundredths
c) ones
d) hundreds
Answers: a) ones, b) thousandths, c) tenths, d) tens, e) hundredths
B-80
e) tenths
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
(MP.7) Rounding by looking at the next-smallest place value. Write on the board:
2.348
SAY: Suppose we want to round this decimal to the nearest hundredth. One way of rounding is
to look at the next smallest place value. Ask a volunteer to underline the digit in the hundredths
column. (4) ASK: What is the name of the next-smallest place value? (thousandth) Ask a different
volunteer to circle the digit in the thousandths column. (8) The decimal should look like this:
2.348
SAY: To round the decimal, we look at the circled number. We are going to remove this number.
Before we do, we have to decide whether to leave the underlined number as it is or to increase
it by one. If the circled number is less than 5, we leave the underlined number as it is. If the
circled number is 5 or greater, we add 1 to the underlined number. ASK: Is the circled number
5 or greater? (yes) What should we do with the underlined number, 4? (add 1) What should we
do with the circled number? (remove it) Continue writing on the board:
2.348 ≈ 2.35
SAY: Note that we use the approximately equal symbol to show that 2.348 is approximately
equal to 2.35. Now, let’s try rounding the same number to the nearest tenth. Write on the board:
2.348
Ask a volunteer to underline the digit in the tenths column. (3) ASK: What is the next-smallest
place value column? (hundredths) Ask another volunteer to circle the digit in the hundredths
column. (4) ASK: Is the circled digit 5 or greater? (no) SAY: So we don’t add 1 to the underlined
digit, and we just remove the rest of the numbers. Continue writing on the board:
2.348 ≈ 2.3
Exercises:
1. Round to the nearest tenth.
a) 2.574
b) 9.7163
c) 7.355
Bonus: 9.962
Answers: a) 2.6, b) 9.7, c) 7.4, Bonus: 10.0
2. Round to the nearest hundredth.
a) 2.574
b) 9.7163
Bonus: 9.962
c) 7.355
Answers: a) 2.57, b) 9.72, c) 7.36, Bonus: 9.96
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-81
(MP.7) Rounding money to the nearest penny by rounding to the nearest hundredth. Write
on the board:
$82.46
ASK: How can we get this amount of money using only ten-dollar bills, one-dollar bills, dimes,
and pennies? (8 ten-dollar bills, 2 one-dollar bills, 4 dimes, and 6 pennies) Label each digit on
the board, as shown below:
tens
dimes
$82.46
ones
pennies
SAY: When we calculate sales tax, we sometimes get digits to the right of the hundredths
column. For example, if there is a 3 percent sales tax on $82.46, the calculation gives us this
number. Write on the board:
$2.4738
SAY: Let’s show this amount using only one-dollar bills, dimes, and pennies. ASK: How many
dollar bills are there? (2) How many dimes? (4) How many pennies? (7) SAY: The 3 and the 8
refer to fractions of pennies. We need to round the number to the nearest penny. ASK: What is
the name of the place value that refers to pennies? (hundredths) Underline the 7 on the board,
and circle the 3. ASK: Is the circled number 5 or greater? (no) So, how do we round this amount
to the nearest penny? (remove the last two digits) Continue writing on the board:
$2.4738 ≈ $2.47
SAY: If there is a 4 percent sales tax on $82.46 instead of 3 percent, the calculation gives us
this number. Write on the board:
$3.2984
SAY: We want to round the number to the nearest penny. ASK: Which digit do I underline to
show that it refers to pennies? (9) Which digit do I circle to show that it is being rounded? (8)
Underline the 9 and circle the 8. ASK: Is the circled digit 5 or greater? (yes) So, how do we
round this amount to the nearest penny? (add 1 to the number of pennies) What is the sales
tax? ($3.30) NOTE: Some students might think the answer is $3.20—remind them that there
were 29 pennies, and if we add 1 more, we get 30 pennies. Continue writing on the board:
$3.2984 ≈ $3.30
Exercises: Round to the nearest penny.
a) $2.8764
b) $23.7249
Answers: a) $2.88, b) $23.72, c) $30.00
B-82
Bonus: $29.9956
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
Extensions
(MP.1) 1. You can round in a different way using arithmetic. To round 2.348 to the nearest
hundredth, add 5 to the thousandths column.
2.348
You can do this by adding the number 0.005:
2.348
+ 0.005
2.353
Now drop all the digits after the hundredths column.
2.353
So 2.348 ≈ 2.35.
To round the same number to the nearest tenth, add a 5 to the hundredths column.
2.348
You can do this by adding the number 0.05:
2.348
+ 0.05
2.398
Now just drop all the digits after the tenths column.
2.398
So 2.348 ≈ 2.3.
Use this rounding technique to round …
a) 8.46 to the nearest tenth
b) 1.749 to the nearest hundredth
c) 1.53 to the nearest tenth
d) 8.791 to the nearest hundredth
Answers: a) 8.5, b) 1.75, c) 1.5, d) 8.79
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-83
(MP.1) 2. Here are the results of the men’s 100-meter final at the 2012 Summer Olympics:
Runner
Time (seconds)
Usain Bolt
9.63
Yohan Blake
9.75
Justin Gatlin
9.79
Tyson Gay
9.80
Ryan Bailey
9.88
Churandy Martina
9.94
Richard Thompson
9.98
Asafa Powell
11.99
a) Round each time to the nearest tenth of a second.
b) If we rounded to the nearest tenth of a second, how many runners are tied for second place?
c) In one of the earlier races, two runners tied with a time of 10.16 seconds. How could we have
measured the time differently to break the tie?
Answers: a) 9.6, 9.8, 9.8, 9.8, 9.9, 10.00, 12.00; b) 3; c) measure to the nearest thousandth of
a second
(MP.1) 3. Different countries have different currencies. Although Americans and Canadians both
use dollars and the $ symbol, the dollars look different and have different values. Leo is going to
visit Canada in 2015, so he must buy Canadian dollars ($ Cdn) with American dollars ($ US). In
January 2015, $1.00 US had the same value as $1.19863 Cdn, so to convert American dollars
to Canadian dollars, multiply the number of US dollars by 1.19863.
a) Convert $200 US into Canadian dollars, keeping all the decimal places.
b) Round your answer from part a) to the nearest penny.
c) In January 2010, $1.00 US had the same value as $1.03293 Cdn. Convert $200 US into
Canadian dollars at that rate, keeping all the decimal places.
d) Round your answer from part c) to the nearest penny.
e) Why is it better for Leo to visit Canada in 2015 than in 2010?
f) In January 2015, Mary converted $200 US into Canadian dollars by rounding $1.19863 to
the nearest penny first. What amount does she calculate this way?
g) Which method would Mary prefer? Explain.
Answers: a) $239.726; b) $239.73; c) $206.586; d) $206.59; e) In 2015, Leo gets more
Canadian dollars for his American dollars; f) $240; g) Mary would get more money this way
B-84
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
EE8-13
Integers
Pages 35–37
Standards: preparation for 8.EE.A.4, 8.EE.C.7, 8.F.A.3
Goals:
Students will review the use of integers, how to order integers, and how to add and
subtract integers.
Prior Knowledge Required:
Can add, subtract, and order whole numbers
Vocabulary: integers, negative, opposite integers, positive, whole numbers
Materials:
overhead projector
transparent red and black counters, a dozen of each color
BLM The Integer Game (pp. B-115–116)
(MP.7) Review integers and the number line. Draw on the board:
0
Ask a volunteer to label the number line with the counting numbers 1, 2, 3, 4, 5 to the right of
zero. Ask another volunteer to count backward and label the number line with numbers −1, −2,
−3, −4, −5 to the left of zero. The number line should look like this:
−5
−4
−3
−2
−1
0
1
2
3
4
5
SAY: Remember, integers include positive whole numbers, negative whole numbers, and zero.
We do not usually put a + sign in front of positive numbers.
(MP.7) Review ordering integers. ASK: Which number is greater, 2 or 3? (3) Write on the board:
2<3
less than
or
3>2
greater than
SAY: To show this, we can either write “2 is less than 3” or “3 is greater than 2.” On a number
line, greater numbers appear to the right. Smaller numbers appear to the left. Point to the “less
than” symbol and SAY: Notice that the “less than” symbol points to the left, just like the arrow on
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-85
the left side of the number line. Point to the “greater than” symbol and SAY: Notice that the
“greater than” symbol points to the right, just like the arrow on the right side of the number line.
Write on the board:
3
5
7
4
2
6
Ask volunteers to place either the < symbol or > symbol between them so that the statement is
true. (3 < 5, 7 > 4, 2 < 6)
SAY: Just like positive numbers, negative numbers to the right are bigger. Numbers to the left
are smaller. Draw on the board:
−5
−4
−3
−2
−1
0
1
2
3
4
5
ASK: Which number is farther to the right, −1 or −4? (−1) So which number is bigger? (−1)
Write on the board:
−4
−1
−1
−4
Ask volunteers to place either the < symbol or > symbol between each pair of numbers to make
the statement true. (−4 < −1, −1 > −4)
Exercises: Write < or > to make the statement true.
a) −3 5
b) −2
−6
Answer: a) <, b) >, c) <
c) −4
7
(MP.7) Applying integers. SAY: Integers can be used to represent situations where we are
comparing things to zero or when we want to record an increase or a decrease. Things that are
below zero are represented using a negative sign. Things that are above zero are represented
using a positive sign. An increase in something is represented with a positive sign. A decrease
in something is represented with a negative sign. Write the following situations on the board in
turn and ask volunteer to write an integer including the sign for each:
a) An increase in height of 2 inches
b) A loss of $5 from your bank account
c) 20 degrees above zero
d) 15 degrees below zero
e) A golf score that is 4 strokes below par
(a) +2, b) −5, c) +20, d) −15, e) −4)
(MP.4) Using counters to represent integers. Use an overhead projector to display a
transparent black counter and a transparent red counter. Point to the black counter and
SAY: We will use a black counter to represent positive numbers. Point to the red counter and
SAY: We will use a red counter to represent negative numbers. In business, the expression “in
B-86
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
the red” means a company is losing money because of the way that financial records are kept.
The expression “in the black” means a company is making a profit. Ask volunteers to come to
the overhead projector and use the red or black counters to represent +3, −4, −2, and +1.
(3 black counters, 4 red counters, 2 red counters, 1 black counter)
(MP.4) Using
and
to represent integers. SAY: We can represent the counters using
symbols. Draw on the board:
+1
−1
ASK: How can we represent the integer −4? (four
integer +3? (three
symbols)
symbols) How can we represent the
Exercises: Write an integer for the diagram.
b)
a)
c)
d)
Answers: a) +4, b) −2, c) −3, d) +1
(MP.4) Using
and
to represent zero. SAY: In business, if we gain $1 and then we lose
$1, we have no overall gain and no overall loss in money. If we gain one pound in weight, and
then lose one pound in weight, then we have had no overall gain or loss in weight. We can use
the plus symbols and the minus symbols to show this. Draw on the board:
0
no gain, no loss
Exercises: Draw plus or minus symbols to make the diagram show no loss and no gain.
b)
Bonus:
a)
Answers: a) draw 3 minus symbols, b) draw 2 plus symbols, Bonus: draw 1 minus symbol
(MP.4, MP.7) Adding integers using the concept of zero. Write on the board:
+3 + (−5)
SAY: For now, we will include the sign in front of a positive number and put brackets around
negative numbers to make our work more organized. Ask for a volunteer to place black counters
on an overhead projector to represent +3. Ask a different volunteer to place red counters on the
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-87
projector to represent −5. Arrange the counters so each black counter has a red counter
immediately below it. Draw on the board:
On the overhead projector, gather each pair of black and red counters and remove them as you
SAY: A positive one and a negative one make zero. On the board, draw an oval around each
pair of a positive sign and a negative sign and SAY: A positive one and a negative one make
zero. Write a zero below each pair. The final picture should look like this:
0
0
0
Point to the three pairs of positive and negative counters and ASK: If we ignore the counters
that now make up zero, what counters are left? (2 negative ones) Write on the board:
So +3 + (−5) = −2
Repeat with 5 black counters and 2 red counters to demonstrate that +5 + (−2) = +3.
Write on the board:
(−2) + (−3)
ASK: What should I draw to represent the −2? (2 negative signs) What should I draw to
represent the −3? (3 negative signs) Draw them on the board, as shown below:
(−2) + (−3)
ASK: What is different about the question this time? (all the signs are negative signs) Do any of
the counters add to give zero? (no) How many negative signs do we have? (5) What is the
answer to (−2) + (−3)? (−5)
Exercises: Use counters to add.
a) 3 + (−2)
b) 4 + (−7)
Answers: a) 1, b) −3, c) −9
B-88
c) (−5) + (−4)
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
(MP.4, MP.7) Adding integers without counters. Draw on the board:
Answer
Question
+2
+5
+
+
(−4)
(−2)
(−3)
+
+2
(−2)
+
+5
(−3)
+
(−2)
Then ask students to use counters or plus and minus symbols to find the answers. Have
volunteers write the answers on the board in the last column, leaving the middle columns blank.
(−2, +3, −1, +3, −5)
Write the following headings for the middle columns: “Number of Positives,” “Number of
Negatives,” “Who had more?,” and “By how many?” Ask different volunteers to write the number
of positives, the number of negatives, who had more, and by how many for each question. The
final table should look like this:
Number of
Negatives
+2
+5
+
+
(−4)
(−2)
Number of
Positives
2
5
(−3)
+
+2
2
3
negatives
1
−1
(−2)
+
+5
5
2
positives
3
+3
(−3)
+
(−2)
0
5
negatives
5
−5
Question
4
2
Who had
more?
negatives
positives
By how
many?
2
3
Answer
−2
+3
For each row, point out that the sign of the final answer depends on who had more: the
positives or the negatives. For example, in the first row, +2 + (−4), there were more negatives
than positives, so the final answer is negative.
SAY: To add without using a chart or symbols, ask yourself two questions: Who had more: the
positives or the negatives? By how many? Write on the board:
(−7)
+
+4
ASK: How many positives are there? (4) How many negatives? (7) Who had more? (the
negatives) By how many? (3) SAY: So the answer is −3.
Exercises: Add the integers.
a) +4 + (−1)
b) +3 + (−8)
c) (−2) + (−4)
Bonus: +997 + (−999)
Answers: a) +3, b) −5, c) −6, d) +7, Bonus: −2
d) +5 + +2
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-89
(MP.4, MP.7) Review modeling of subtraction with whole numbers. Write on the board:
5−2
Place 5 black counters on the overhead projector. Ask for a volunteer to model how to subtract
2 of them—in other words, remove 2 of the 5 counters to leave 3. SAY: We can show this on the
board by drawing counters and then drawing arrows to show some are being taken away. Draw
on the board:
(MP.4, MP.7) Modeling subtraction of integers using counters. SAY: We can use the same
idea with integers. We just have to be careful to take away positives or negatives depending on
the question. Write on the board:
+5
−
+2
SAY: Here I am subtracting 2 positives from 5 positives. ASK: How many positive counters
should I draw? (5) How many positive counters do I need to subtract? (2) How can I show this?
(circle the 2 positives and draw an arrow) Draw on the board:
ASK: What counters are left? (3 positive ones) So what is +5 – (+2)? (+3) Write on the board:
+5
−
(−2)
SAY: This time we need to subtract 2 negatives from 5 positives. Let’s do this on the overhead
projector. ASK: How many black or positive counters do we need? (5) Place 5 black counters
on the overhead projector. SAY: Last time we subtracted 2 black counters because we wanted
to subtract 2 positives. ASK: This time what are we subtracting? (2 negatives) SAY: But the
negative counters are red and we don’t have any red counters to take away. This is a bit of
a problem.
Add a black counter and a red one to the overhead projector—place the black and red counters
underneath each other so students can see you add them as a pair. SAY: Remember that a
pair of 1 positive counter and 1 negative counter make up zero—in other words, I can add a
black counter and a red one, and it doesn’t change the answer. ASK: Why doesn’t the answer
change? (1 black + 1 red = zero, and adding zero doesn’t change the answer) Can I take away
2 red counters yet? (no) What can I do that will allow me to take away 2 red counters? (add
another pair of black and red counters) Place another pair of black and red counters together on
the overhead projector. ASK: How many black counters are there altogether? (7) How many red
ones are there? (2) SAY: We were supposed to subtract 2 negatives—can I do this now? (yes)
How? (remove 2 red counters from the overhead projector) What is left? (7 black counters) So
what is +5 − (−2)? (+7)
B-90
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
(MP.4, MP.7) Modeling subtraction of integers by drawing
and
. Write on the board:
+5 − (−2)
Ask a volunteer to draw positive and negative symbols that will allow us to subtract 2 negatives.
(They should draw 2 positive symbols and 2 negative symbols) Circle each pair of positive and
negative and point out that they are just adding zero with each pair, so they haven’t changed the
answer. The picture should look like this:
Now erase the two circles around the pairs of negative and positive symbols and ASK: How can
we subtract 2 negatives now? (remove the 2 negative symbols that were part of the zeros added)
Circle the 2 negative symbols and draw an arrow to show that they are being removed. The final
picture should look like this:
ASK: What is left? (7 positives) So what is 5 − (−2)? (7) SAY: Notice that when we added the
pairs of positives and negatives, and then took away the negatives, we were left two new
positives. So subtracting −2 was really the same as adding 2 positives.
SAY: Let’s try another example. Write on the board.
(−2) − (+3)
SAY: This question asks us to subtract 3 positives from 2 negatives. But we don’t have any
positives to subtract. ASK: How can we add some positives and negatives that won’t change the
answer? (add 3 pairs of positives and negatives) Continue drawing on the board:
Point out that we added 3 pairs of positive and negative, which are the same as 3 zeros, so the
answer hasn’t changed. ASK: How can we subtract 3 positives now? (take away the 3 positives)
ASK: What is left? (5 negatives) So what is (−2) − +3? (−5)
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-91
(MP.7) Subtracting an integer by adding its opposite. Draw on the board:
−5
−4
−3
−2
−1
0
1
2
3
4
5
SAY: On a number line, when two integers are the same distance away from zero, we call them
opposite integers. In the example here, −4 and +4 are opposite integers. ASK: What is the
opposite of −2? (+2) What is the opposite of +3? (−3) Draw on the board:
5 − (−2)
(−2) − (+3)
SAY: We saw before that subtracting −2 from 5 was the same as adding +2. In a similar way,
subtracting +3 from −2 was the same as adding −3. Continue writing on the board:
= 5 + (+2)
=7
= (−2) + (−3)
= −5
SAY: So subtracting an integer from a number is the same as adding the opposite integer. Write
on the board:
To subtract an integer, add its opposite integer.
5 − (−3)
ASK: What is the opposite of −3? (+3) Instead of subtracting −3, what do we add? (+3)
Continue writing on the board:
= 5 + (+3)
Exercises: Rewrite the subtraction as an addition by adding the opposite of the second integer.
a) 3 − (−4)
b) (−2) − (−1)
c) (−5) – (−4)
d) (−5) – (+4)
Answers: a) 3 + (+4), b) (−2) + (+1), c) (−5) + (+4), d) (−5) + (−4)
Activity
Distribute BLM The Integer Game to each pair of students. Have students play one or more of
the following versions in pairs:
Competitive versions: The object of the game is to get a higher running total at the end of the
game. After students have had an opportunity to play several games, have a class discussion to
share different strategies students used.
Version A—Each player: keep a running total of your score on the scorecard. As you choose
a number, add that integer to your running score.
B-92
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
Version B—Each player: keep a running total of your score on the scorecard. As you choose
a number, subtract that integer from your running score.
Cooperative version: Player 1 adds and Player 2 subtracts numbers to get a common running
total. The goal is to get the highest possible running total after 10 turns per player.
NOTE: Only assign a competitive version of this activity if your class dynamics allow for a
friendly competition.
(end of activity)
Extension
(MP.1) In a magic square, the sum of every row, column, and diagonal is the same. Find the
missing numbers in the magic square.
a)
b)
2
−5
1
−15 −8
−1
−1
−11
−4
−12
0
−10
−6
4
3
2
6
−13
−14
−7
Answers: a) 0, −3, 1, −2, 3; b) 8, 7, −9, −2, −3, −4, 5, −5, 9
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-93
EE8-14
Operations with Integers
Pages 38–41
Standards: preparation for 8.EE.A.4, 8.EE.C.7, 8.F.A.3
Goals:
Students will review the multiplication and division of integers.
Students will use the order of operations to evaluate expressions involving integers.
Students will add and subtract fractions involving integers.
Prior Knowledge Required:
Can add and subtract integers
Can multiply and divide whole numbers
Can add and subtract fractions
Vocabulary: associative property, commutative property, fact family, integers, whole numbers
Materials:
overhead projector
transparency of BLM Multiplication Patterns (p. B-117)
(MP.7) Review multiplication as a short form for addition. Write on the board:
2+2+2+2+2
ASK: How many 2s are we adding? (5) How can we write this addition as a multiplication?
(5 × 2) What is 5 × 2? (10) Write on the board:
5+5
ASK: How many 5s are we adding? (2) How can we write this addition as a multiplication?
(2 × 5) What is 2 × 5? (10) SAY: Notice that 2 × 5 = 5 × 2. Multiplication is commutative, so we
can change the order and the answer will be the same.
(MP.7) Multiplying integers that are opposite in sign. Write on the board:
(−2) + (−2) + (−2) + (−2) + (−2)
ASK: How many −2s are we adding? (5) How can we write this addition as a multiplication?
(5 × (−2)) What is (−2) + (−2) + (−2) + (−2) + (−2)? (−10) Write on the board:
5 × (−2) = −10
B-94
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
Read the multiplication aloud. SAY: When we multiply a positive number by a negative number,
we get a negative answer. Because of the commutative property, we can change the order of
the numbers being multiplied. Write on the board:
(−2) × 5 = −10
Read the multiplication aloud. SAY: When we multiply a negative number by a positive number,
we also get a negative answer.
SAY: Here we had 5 × (−2) = −10 and (−2) × 5 = −10. We can write a general rule from this:
(+) × (−) = (−)
(−) × (+) = (−)
Exercises: Multiply the integers.
a) 6 × (−3)
b) (−4) × 7
c) 9 × (−5)
Answers: a) −18, b) −28, c) −45, Bonus: −143
Bonus: 13 × (−11)
Have students do Question 1 on AP Book 8.1 p. 38.
(MP.7) Multiplication patterns. Project BLM Multiplication Patterns on an overhead
projector. ASK: What is the pattern in the numbers in the first column of the first chart? (they go
down by 1) Have a volunteer continue the pattern by completing the first column of the first chart
on the BLM. SAY: Remember that when you multiply a positive number by a negative number,
the answer will be negative. ASK: What is 2 × (−3)? (−6) What is 1 × (−3)? (−3) SAY: Look at
the numbers in the last column: −12, −9, −6, −3. Draw on the board:
−12 −11 −10 −9 −8 −7 −6 −5
−4 −3 −2 −1
0
1
2
3
4
5
6
7
8
9
ASK: What is happening to these numbers? (increasing by 3) What will be the next numbers in
the pattern? (0, 3, 6, 9, 12) Have a volunteer fill in the last column of the chart. (0, 3, 6, 9, 12)
SAY: Notice that the chart now says that (−1) × (−3) = 3, (−2) × (−3) = 6, and so on. ASK: In
each case, when we multiplied a negative number by a negative number, what was true about
the answer? (the number is positive)
Repeat the process with the second table starting with 4 × (−2) = −8.
SAY: In both tables, when we multiplied a negative number by a negative number, we got a
positive number. We can write this rule as:
(−) × (−) = (+)
Exercises: Multiply the two negative numbers to get a positive answer.
Bonus: (−25) × (−25)
a) (−3) × (−4)
b) (−5) × (−7)
c) (−8) × (−6)
Answers: a) 12, b) 35, c) 48, Bonus: 625
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-95
(MP.7) Fact families using whole numbers. Write on the board:
5 × 4 = 20
4 × 5 = 20
20 ÷ 4 = 5
20 ÷ 5 = 4
SAY: We call these equations a fact family. Notice that each multiplication equation in the fact
family can be read backward as a division equation. Write on the board:
5 × 4 = 20
20 ÷ 4 =5
4 × 5 = 20
20 ÷ 5 = 4
(MP.7) Fact families using integers. Write on the board:
(−3) × 4 = −12
SAY: Read this equation backward to form a division equation. ((−12) ÷ 4 = −3) Write the
division equation on the board. SAY: Notice that we divided a negative number by a positive
number and got a negative answer.
SAY: Since (−3) × 4 = −12, we can use the commutative property to write the following. Write on
the board:
4 × (−3) = −12
SAY: Read this equation backward to form a division equation. ((−12) ÷ (−3) = 4) Write the
division equation on the board. SAY: Notice that we divided a negative number by a negative
number and got a positive answer.
Exercises: Write the remaining members of the fact family.
a) 4 × (−2) = −8
b) (−12) ÷ (−4) = 3
c) (−15) ÷ 5 = −3
Answers:
a) (−2) × 4 = −8, (−8) ÷ (−2) = 4, (−8) ÷ 4 = −2
b) (−12) ÷ 3 = −4, 3 × (−4) = −12, (−4) × 3 = −12
c) (−15) ÷ (−3) = 5, (−3) × 5 = −15, 5 × (−3) = −15
(MP.7) Summary of multiplication and division of integers. Write on the board:
Multiplication: (+) × (+) = (+)
(−) × (+) = (−)
(+) × (−) = (−)
(−) × (−) = (+)
B-96
Division:
(+) ÷ (+) = (+)
(−) ÷ (+) = (−)
(+) ÷ (−) = (−)
(−) ÷ (−) = (+)
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
SAY: This can be a lot to remember. There is an easier way to remember the patterns: For both
multiplication and division, if the signs are the same, the answer is positive. If the signs are
different, the answer is negative. Write on the board:
5 × (−8)
ASK: What is 5 × 8? (40) In this question, are the signs the same or are they different?
(different) SAY: Then the answer is negative. So 5 × (−8) = −40. Write on the board:
(−24) ÷ (−6)
ASK: What is 24 ÷ 6? (4) In this question, are the signs the same or are they different? (same)
SAY: Then the answer is positive. So (−24) ÷ (−6) = +4.
Exercises: Multiply or divide the integers.
a) (−3) × (−4)
b) 24 ÷ (−3)
c) (−16) ÷ (−2)
d) 7 × (−3)
e) 12 ÷ (−6)
f) (−4) × 9
Bonus: (−200) ÷ (−4)
Answers: a) 12, b) −8, c) 8, d) −21, e) −2, f) −36, Bonus: 50
(MP.7) Making expressions easier to read. SAY: To make expressions and equations easier
to read, mathematicians like to write positive numbers without the plus sign or brackets. Write
on the board:
Instead of writing
we write
(−3) + (+7)
(−3) + 7
SAY: Whenever we can, we write subtraction as addition. We often leave brackets around
negative numbers to avoid confusion. Write on the board:
Instead of writing
we write 3 + (−5)
3−5
Exercises: Rewrite as an addition.
a) 5 − 8
b) (−4) − (−5)
c) 2 − 5 − (−3)
Answers: a) 5 + (−8), b) (−4) + 5, c) 2 + (−5) + 3
(MP.7) Adding integers by grouping positives and negatives. Write on the board:
3+5+2+7
SAY: The associative property of addition means that we can add these four numbers in any
order and get the same answer. Let’s try two different orders for this question. Write the addition
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-97
with the numbers in a different order, as shown on the right, below:
3+5+2+7
5+7+2+3
Ask half the class to add the numbers using the order on the left. Ask the other class to add the
numbers using the order on the right. Have volunteers write their answers on the board. They
should be similar to this:
3+5+2+7
=8+2+7
= 10 + 7
= 17
5+7+2+3
= 12 + 2 + 3
= 14 + 3
= 17
SAY: Notice that the answers are the same. Write on the board:
4 − 3 − 5 − (−6)
SAY: If we rewrite this expression using addition, we can add them in any order we like. Change
all the subtraction to addition by adding the opposite. ASK: What will the expression be?
(4 + (−3) + (−5) + 6) Continue writing on the board:
= 4 + (−3) + (−5) + 6
SAY: Because we can change the order, we can add the positives and negatives separately.
Continue writing on the board:
= 4 + 6 + (−3) + (−5)
ASK: How many positives do we have? (10) How many negatives do we have? (8) Continue
writing on the board:
= 10 + (−8)
ASK: What is 10 + (−8)? (+2) SAY: We don’t have to write the + sign in front of the 2. Continue
writing on the board:
=2
Exercises: Evaluate.
a) 5 − 8 + 4 − 7 − (−6)
Answers: a) 0, b) 3, c) −7
B-98
b) (−2) + 7 − 4 − (−2)
c) 2 − 3 − (−6) − 8 + (−4)
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
(MP.7) Order of operations and integers. SAY: The order of operations rules apply to integers.
Write on the board:
5 − (−12) ÷ 4 × (−2)
ASK: What operations are involved here? (subtraction, division, multiplication) Which operation
should come first? (division) Why not multiplication? (division comes first when we read from left
to right) What is (−12) ÷ 4? (−3) Continue writing on the board:
= 5 − (−3) × (−2)
ASK: What operations are left? (subtraction, multiplication) Which operation should come next?
(multiplication) What is (−3) × (−2)? (+6) SAY: We don’t need to write the + sign on the 6.
Continue writing on the board:
=5−6
ASK: How do we subtract integers? (add the opposite of the second integer) What is the
opposite of 6? (−6) What is 5 + (−6)? (−1) Continue writing on the board:
= 5 + (−6)
= −1
Exercises: Evaluate.
a) 2 − (−4) × 3
b) (−12) ÷ (−4) − 4 × (−2)
Answers: a) 14, b) 11, c) −10
c) (−5) − (2 − (−3))
(MP.7) Integers and fractions. SAY: Remember that a fraction can be interpreted as division.
Write on the board:
10
5
SAY: This fraction can be interpreted as 10 ÷ 5, which is 2. This gets a little more interesting
when we use integers in the numerator or denominator. Write on the board:
(-10)
5
10
(-5)
æ10 ö
-çç ÷÷÷
çè 5 ø
Ask for volunteers to write a division expression for each fraction. (see answers below)
= (−10) ÷ 5
= 10 ÷ (−5)
= −(10 ÷ 5)
NOTE: In the last example, you may have to remind students that brackets should be done first.
ASK: In the first two expressions, are the signs the same or different? (different) When we are
dividing integers and the signs are different, is the answer positive or negative? (negative) So in
both the first two expressions, what is the answer? (−2)
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-99
SAY: In the last expression, brackets should be done first. ASK: What is 10 ÷ 5? (2) SAY: Now
we have to put the − sign back in front of the 2. ASK: So what is the answer? (−2) Write the
answer under each expression.
SAY: Notice that the answer to each expression was −2. Write on the board:
æ10 ö
(-10) 10
=
= -çç ÷÷÷
ç
è5ø
5
-5
SAY: This is an important fact to remember: When we have a negative sign in a fraction, the
result is the same whether the negative sign is in the numerator, in the denominator, or on the
outside of the fraction. This fact will help us calculate expressions involving fractions that have
integers in the numerator or denominator. Write on the board:
3
1
+
8 -8
ASK: Before we can add fractions, what must be true about the denominators? (they must be
the same) Are the denominators the same here? (no) How can we rewrite the second fraction
so the denominators are the same? (write the negative sign in the numerator instead of the
denominator) Continue writing on the board:
=
3 -1
+
8
8
ASK: Now that the denominators are the same, what can we do with the numerators? (add them)
What is 3 + (−1)? (+2) Continue writing on the board:
=
2
8
SAY: Note that we don’t have to write the positive sign in front of the 2. ASK: Can we reduce
the fraction? (yes) What is the greatest common factor, or GCF, for 2 and 8? (2) Divide both
the numerator and denominator by 2. ASK: What fraction do we get? (1/4) Continue writing on
the board:
=
1
4
Exercises: Evaluate.
-2
4
3
-3
1
2
1
Answers: a) - , b)
, c)
2
3
12
a)
3
1
+
-4 4
B-100
b)
2
3
c) - -
3
-4
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
Extensions
(MP.1) 1. Mental math with integers. A quick way to calculate an expression such as:
2 − 3 + 5 − 7 − 4 − 6 + 4 is to consider each of the subtractions as adding the opposite.
Then we can add the positive and negative numbers separately.
2 − 3 + 5 −7 − 4 − 6 + 4
= 2 + 5 + 4 + (−3) + (−7) + (−4) + (−6)
=
11
+
(Do this step mentally.)
(−20)
= −9
Use this technique to calculate …
a) 3 − 7 + 4 + 5 − 8 − 9 + 4
Answers: a) −8, b) −6
b) 2 − 5 − 7 + 6 + 3 − 9 + 4
(MP.1) 2. Create expressions that use only the numbers 1, 2, 3, 4 in that order to give the
answers 1, 2, 3, 4, 5, and so on in turn. The numbers can be positive or negative, and when
you evaluate the expression, you must use the standard order of operations. Create expressions
for as many of the answers as you can.
Sample answers: 1 × 2 + 3 − 4 = 1, (1 + 2) × ((−3) + 4) = 3
(MP.1) 3. You can subtract mixed numbers without converting to improper fractions or
1
4
regrouping by using integers. Example: You can subtract 4 - 1
3
by subtracting the whole
4
numbers and improper fractions separately, as shown below:
4
1
3
-1
4
4
= 4 - 1+
1 3
4 4
Subtracting the whole numbers gives 4 − 1 = 3. Subtracting the fractions gives us
1
4
3
4
So 4 - 1 = 3 -
1 3 -2
- =
4 4
4
2
1
1
= 3- = 2 .
4
2
2
Use this technique to subtract the fractions.
1
4
a) 3 - 1
5
6
1
8
b) 6 - 3
3
4
c) 5
3
7
-1
10
15
Answers: a) 1 5/12, b) 2 3/8, c) 3 5/6
Teacher’s Guide for AP Book 8.1 — Unit 1 Expressions and Equations
B-101