Download Unit 3 Number and Operations in Base Ten

Unit 3 N
umber and Operations in Base Ten:
Multiplication
Introduction
In this unit, students will learn how to perform multiplication using the
standard algorithm. Students will build on their understanding of how to
compute products of one-digit numbers and multiples of 10,100, and 1,000.
Students will display an understanding of the role played by the distributive
property in the multiplication process. They will also connect diagrams of
areas or arrays to numerical work to develop understanding of general base
ten multiplication methods.
The unit begins with multiplication of single-digit numbers and progresses
to multiplication of multi-digit numbers by 2-digit numbers. Methods
include expanded form, base ten materials, and the standard algorithm
for multiplication.
NOTE: The term “product” in this unit is used to refer to both an expression
involving multiplication and to the evaluation of that expression. So, for
example, the product of 3 and 4 is both 3 × 4 and 12.
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The use of grid paper for multiplication using the standard algorithm helps
student organize their work and line up digits in the appropriate columns.
If students do not use grid paper notebooks in general, you will need to
have lots of grid paper on hand throughout the unit.
Number and Operations in Base Ten
D-1
NBT5-13 Introduction to Multiplication
Pages 35–36
Goals
STANDARDS
preparation for 5.NBT.B.5
Students will write a product as repeated addition. Students will
recognize the commutative property of multiplication, and write a
product for a given array.
Vocabulary
addition
array
column
commutative
multiplication
product
row
PRIOR KNOWLEDGE REQUIRED
Can add single-digit numbers
MATERIALS
BLM Multiplication Charts (pp. D-55–57)
BLM Using the Multiplication Chart to Multiply (p. D-58)
calculators
BLM Multiplication Facts—Commutative Property (p. D-59)
NOTE: The Grade 3 Common Core State Standards require that students
know all one-digit multiplication facts. You can use the exercises in
“How to Learn Your Times Tables in a Week” (p. A-47), and in the BLM
Multiplication Charts and BLM Using the Multiplication Chart to
Multiply to help students learn their facts. We recommend you review basic
facts regularly with students who are struggling.
Review the concept of multiplication as repeated addition. On the
board, write:
5 × 4 = ?
Ask students for the answer. Allow time for the class to agree on an answer.
ASK: How can we draw a diagram to show that 5 × 4 = 20?
that has 4 groups with 5 items in each group is correct, try to direct
students toward a diagram with 5 groups of 4 items in each group, such as:
Students should think of 5 × 4 as adding five 4s. The first number is the
number of times we are adding the second number.
ASK: What addition statement could we write for this diagram?
(4 + 4 + 4 + 4 + 4 = 20) Tell students that 5 × 4 is really just a short form
for adding five 4s.
D-2
Teacher’s Guide for AP Book 5.1
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Although a diagram such as:
Draw the following diagram on the board (which students may have already
suggested for the previous question):
ASK: What addition statement could we write for this diagram? ( 5 + 5 + 5
+ 5 = 20) Remind students that this is a way of adding four 5s to get 20.
ASK: What multiplication question can you write for 5 + 5 + 5 + 5 = 20?
Although 5 × 4 = 20 is correct, direct them toward 4 × 5 = 20 instead.
Students should think of 4 × 5 as adding four 5s.
Exercises: Write each multiplication question as an addition question.
a)5 × 3 = 15
b) 4 × 7 = 28
c) 5 × 1 = 10
d) 3 × 0 = 0
Answers: a) 3 + 3 + 3 + 3 + 3 = 15, b) 7 + 7 + 7 + 7 = 28,
c) 1 + 1 + 1 + 1 + 1 = 5, d) 0 + 0 + 0 = 0
Exercises: Write each addition question as a multiplication question.
a)4 + 4 + 4 = 12
b) 8 + 8 = 16
c) 9 + 9 + 9 + 9 = 36
Answers: a) 3 × 4 = 12, b) 2 × 8 = 16, c) 4 × 9 = 36
Writing a product for a given array. Draw the following diagrams on
the board:
Using dots:
Using squares:
ASK: What product is represented by each diagram?
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If students are having difficulty, have them count the number of rows and
number of columns. Remind them that rows are horizontal, while columns
are vertical.
row
row
row
row
row
row
colcolcolcol
colcolcolcol
Tell students that the product is created by writing the number of rows first,
then the number of columns.
Here, the number of rows is 3 and the number of columns is 4, so we write
the product 3 × 4.
Number and Operations in Base Ten 5-13
D-3
Exercises: Write a product for each diagram.
a)b)c)d)
Answers: a) 4 × 6, b) 2 × 5, c) 3 × 2, d) 4 × 1
Recognizing the commutative property of multiplication. The
commutative property of multiplication states that you can reverse the
numbers in a product, and the answers will be equivalent. For example,
5 × 4 = 4 × 5.
Draw on the board:
ASK: What multiplication is represented by each array? (3 × 4, 4 × 3) ASK:
What can we do to the diagram on the left to make it look like the diagram
on the right? (rotate the diagram 90 degrees)
SAY: Notice that when we rotate the diagram on the left, the total number of
dots doesn’t change! So, 3 × 4 = 4 × 3.
Have students draw an array diagram for 4 × 5, then for 5 × 4. ASK: Does
rotating the diagram on the left give you the diagram on the right? (yes)
ASK: What can we say about 4 × 5 and 5 × 4? (they are equivalent)
ASK: What do you notice about the answer to a multiplication equation
when the numbers are reversed? (the answers are the same)
Ask students for other examples that show this is true. (sample answers:
7 × 4 = 28 and 4 × 7 = 28, 9 × 3 = 27 and 3 × 9 = 27)
Ask students to use their calculators to create similar bonus questions with
greater numbers. (sample answers: 234 × 85 = 19,890 and 85 × 234 = ?)
Using the commutative property to review multiplication facts. Have
students complete the multiplication facts in BLM Multiplication Facts—
Commutative Property. Make sure they notice that each fact they know
automatically gives them another fact.
D-4
Teacher’s Guide for AP Book 5.1
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Bonus: If 13 × 15 = 195, what is the answer for 15 × 13? (195)
Review the perfect squares: 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9, and so on up
to 9 × 9 = 81. The perfect squares form a diagonal in the table.
× 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9
2 2 4
3 3
9
4 4
21
16
5 5
25
6 6
36
7 7
21
49
8 8
64
9 9
81
diagonal of perfect squares
SAY: If you know 7 × 3 = 21, fill in the answer 21. Then travel diagonally
through the line of perfect squares to find the spot for 3 × 7 and fill in the
answer 21.
Have students complete the multiplication table using the facts they know,
and using the commutative property to find another fact.
Extensions
(MP.2)
1.Can we reverse the order of the numbers in an addition statement and
still get the same answer? Provide examples.
Answers: yes; 5 + 7 = 7 + 5, 9 + 6 = 6 + 9, 3 + 4 = 4 + 3
(MP.2)
2.Can we reverse the order of the numbers in a subtraction statement
and still get the same answer? If not, provide a counter example.
Answers: no; 7 – 5 ≠ 5 – 7, 9 – 3 ≠ 3 – 9
(MP.1)
3.The rows and columns of the times tables have been mixed up. Fill in
the missing numbers.
a)
× 7 5 6
b)
4 28
25
10
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3
12
×
6 4
7
5 40 30 20
2
10
42
3
12
4
Answers
a)
× 7 5 6 2
4
5
3
2
Number and Operations in Base Ten 5-13
28
35
21
14
20
25
15
10
24
30
18
12
8
10
6
4
b)
× 8 6 4 5 7
5
2
6
3
1
40
16
64
24
8
30
12
36
18
6
20
8
24
12
4
25
10
30
15
5
35
14
42
21
7
D-5
(MP.1)
4. Use the numbers 1 to 5 to fill in the missing numbers:
×
×
0 4
2
Answers: 2 × 5 = 10, 3 × 4 =12
(MP.1)
5. Use all the numbers from 0 to 9 to fill in the missing numbers.
6
×
2
×
4 5
6
×
3
×
5 3
5
×
2
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Answers: 8 × 3 = 24 or 3 × 8 = 24, 9 × 6 = 54, 7 × 5 = 35 or
5 × 7 = 35, 6 × 5 = 30, 6 × 2 = 12
D-6
Teacher’s Guide for AP Book 5.1
NBT5-14 Multiplication by Adding On
Pages 37–38
Goals
STANDARDS
preparation for 5.NBT.B.5
Students will find products by adding onto smaller products.
PRIOR KNOWLEDGE REQUIRED
Vocabulary
array
column
equation
product
row
sum
Can add and multiply
Can represent multiplication in different ways
Finding a product for a given array. Draw the following diagram on the
board. For each diagram, ask for volunteers to count the number of rows
and number of columns. Remind students that rows are horizontal and
columns are vertical
a)
b)
rows
columns
rows
columns
Answers: a) 3 rows, 4 columns; b) 4 rows, 6 columns
Ask students to write a product for each diagram.
Answers: a) 3 × 4, b) 4 × 6
Finding a product by adding to a smaller product using arrays.
Draw the following diagram on the board:
×
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×
+
×
Ask for three volunteers to write a product for each of the circled arrays.
Answers: 5 × 3, 4 × 3, 1 × 3
SAY: But the number of dots in the diagrams is the same. So the product
on the left must be the same as the sum of the product on the right!
So, 5 × 3 = (4 × 3) + (1 × 3).
Number and Operations in Base Ten 5-14
D-7
Draw the following diagram on the board. SAY: We can put both diagrams
into one diagram. Note that 1 × 3 = 3.
4×3
5×3
+
3
5 × 3 = (4 × 3) + 3
Exercises: Fill in the missing products and number. Then write an equation.
a)
b)
+
+
Answers
a)5 × 4 = (4 × 4) + 4
b) 3 × 5 = (2 × 5) + 5
Finding a product by adding on to a smaller product without using
arrays. Write the following on the board:
3 × 6 = ( 2 × 6) + 6
5 × 7 = ( 4 × 7) + 7
ASK: In each equation, what is the pattern in the underlined numbers? (the
second underlined number is one less than the first underlined number)
Exercises: Turn each product into a lesser product and a sum.
a)7 × 4 = ( × 4 ) + 4
b) 9 × 5 = ( × 5) + 5
Answers: a) 7 × 4 = (6 × 4) + 4; b) 9 × 5 = (8 × 5) + 5
Exercises: Find the product by using a lesser product and sum.
a)8 × 6 =
b)9 × 4 =
Answers
a)8 × 6 = (7 × 6) + 6
b) 9 × 4= (8 × 4) + 4
= 42 + 6
= 32 + 4
= 48
= 36
Extensions
D-8
(MP.2)
1.Have students use BLM Finding Easier Ways to Multiply (pp. D-60–62)
to learn to turn products into sums of easier products.
(MP.2)
2.For extra practice with mental math, have students follow BLM Using
Doubles to Multiply (pp. D-63–64) and BLM Using Triples to Multiply
(p. D-65).
Teacher’s Guide for AP Book 5.1
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(MP.2)
NBT5-15 Multiplying by Tens, Hundreds,
Pages 39–41
and Thousands
STANDARDS
5.NBT.A.2, 5.NBT.B.5
Vocabulary
multiple
power
Goals
Students will multiply by multiples of 10 and 100.
PRIOR KNOWLEDGE REQUIRED
Can use base ten materials to represent numbers
Can multiply single-digit numbers
MATERIALS
base ten materials
Multiplying a single digit number by multiples of 10. Give each student
9 ones blocks, 9 tens blocks, and 9 hundreds blocks. ASK: Which block is
equal in value to 10 ones blocks? (1 tens block)
Remind students that = 1 = 10
= 100
Ask students to take 4 ones blocks. Write on the board:
10 × 4 = 10 ×
SAY: Since each ones block gets multiplied by 10, replace each ones block
by a tens block. Write on the board:
10 × 4 = 10 ×
=
= 40
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Exercises: Use base ten blocks to model multiplication.
a)10 × 5
b) 7 × 10
c) 10 × 3
Answers: a) 5 tens blocks, b) 7 tens blocks, c) 3 tens blocks
For question b), you may have to remind students that 7 × 10 = 10 × 7.
ASK: What is a shortcut to multiplying by 10? (add a zero to the end of the
number being multiplied by 10) If students are struggling for the answer,
write on the board:
10 × 5 = 50
10 × 7 = 70
10 × 3 = 30
Number and Operations in Base Ten 5-15
D-9
Write on the board:
4 × 50
ASK: How many tens are in 50? (5)
Draw on the board:
4 × 50 = 4 ×
=4×
5
tens
ASK: How many tens are there after we multiply by 4? (20) What number is
the same as 20 tens? (200)
Exercises: Multiply.
a)5 × 30
b) 6 × 40
c) 7 × 30
Answers: a) 150, b) 240, c) 210
ASK: What is a shortcut to multiplying a one-digit number by multiples of
10? (multiply the non-zero digits, then add a zero) If students are struggling
for the answer, write on the board:
5 × 30 = 150
6 × 40 = 240
7 × 30 = 210
Multiplying a multi-digit number by multiples of 10. Write on the board:
10 × 30
ASK: How many tens are in 30? (3) Which block is the same as ten tens
blocks? (a hundreds block) Write on the board:
(MP.5)
= = 300
Exercises: Model using base ten blocks as in the example above.
a)10 × 50
b) 40 × 10
c) 10 × 20
Answers: a) 5 hundreds blocks, b) 4 hundreds blocks,
c) 2 hundreds blocks
For question b), you may have to remind students that 40 × 10 = 10 × 40.
ASK: What is a shortcut to multiplying a multiple of 10 by 10? (multiply
the non-zero digits, then add two zeros) If students are struggling for the
answer, write on the board:
10 × 50 = 500
40 × 10 = 400
10 × 20 = 200
D-10
Teacher’s Guide for AP Book 5.1
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10 × 30 = 10 × Multiplying a multi-digit number by multiples of 100. Write on the board:
10 × 200
ASK: How many hundreds are in 200? (2) What block is the same as
10 hundreds blocks? (a thousands block)
Draw on the board:
10 × 200 = 10 × =
(MP.5)
= 2,000
Exercises: Solve using base ten materials.
a)10 × 300
b) 10 × 400
c) 10 × 600
Answers: a) 3,000, b) 4,000, c) 6,000
Multiplying a multi-digit number by multiples of 100 without blocks.
Write on the board:
30 × 40
ASK: How many tens are in 30? (3) How many tens are in 40? (4)
Write on the board:
30 × 40
= (3 tens) × (4 tens)
= (3 × 10) × (4 × 10)
SAY: With multiplication, we can change the order of the numbers. Write
only the first line of the following problem on the board:
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= (3 × 4) × (10 × 10)
= 12 × 100
= 1,200
Point to the first part of the problem and ASK: How much is 3 × 4? (12)
Write 12 on the board. Point to the second part and ASK: How much is
10 × 10? (100) Finish writing the second line of the problem. ASK: How
much is 12 × 100? (1,200) Write the answer.
Exercises: Multiply.
a)20 × 70
b) 30 × 60
c) 50 × 400
Answers: a) 1,400, b) 1,800, c) 20,000
(MP.2)
ASK: What is a shortcut for this multiplication? (multiply the non-zero digits,
and write all the zeros after) If students need a hint, write on the board:
20 × 70 = 1,400
30 × 60 = 1,800
50 × 400 = 20,000
Number and Operations in Base Ten 5-15
D-11
Identifying patterns when multiplying powers of 10.
Write on the board:
10 × 10 = 100
10 × 100 = 1,000
10 × 1,000 = 10,000
ASK: What do you notice about the total number of zeros in the answer
of each equation compared to the total number of zeros in the question?
(they are the same)
Write on the board:
100 × 1,000
ASK: How many zeros are in 100? (2) How many zeros are in 1,000? (3)
How many zeros are there in total? (5) What number starting with 1 has
5 zeros at the end? (100,000) SAY: So, 100 × 1,000 = 100,000.
Exercises: Multiply.
a)100 × 100
d)100 × 100,000
b) 1,000 × 100
e) 1,000 × 1,000
c) 10,000 × 100
f) 1,000,000 × 100,000
Answers: a) 10,000, b) 100,000, c) 1,000,000, d) 10,000,000, e) 1,000,000,
f) 100,000,000,000
Extensions
1. Find the missing number:
a)
c)
(MP.7)
×
×
= 40,000
= 120,000
3. How many dimes are in these dollar amounts?
a)$5
D-12
b)
× 200 = 80,000
d) 400 ×
= 1,200,000
2.Find as many answers as you can using multiples of 10 for
each question.
a)
b)
(MP.7)
× 300 = 60,000
× 1,000 = 500,000
b)$1,000
c)$100,000
(MP.1)
4.The total weight of all the termites on the planet is approximately
10 times as great as the total weight of humans. The average human
weighs 50 kg. The world population is approximately 7,000,000,000
people. What is the approximate total weight of all the termites in
the world?
(MP.1)
5.Insects first appeared on Earth 350,000,000 years ago. Humans
appeared approximately 130,000 years ago. About how many times
as many years have insects been on Earth? Explain your answer.
Teacher’s Guide for AP Book 5.1
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(MP.2)
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Answers: 1. a) 200, b) 400, c) 500, d) 3,000, 2. a) 1 × 40,000, 5 × 8,000,
8 × 5,000, 4 × 10,000, 10 × 4,000, 50 × 800, 80 × 500, 100 × 400,
200 × 200; b) sample answers: 1 × 120,000, 2 × 60,000, 3 × 40,000,
4 × 30,000, 5 × 24,000, 6 × 20,000, 8 × 15,000, 10 × 12,000,
12 × 10,000, 15 × 8,000, 20 × 6,000, 24 × 5,000, 25 × 4,800, 30 × 4,000,
40 × 3,000, 48 × 2,500, 50 × 2,400, 60 × 2,000, 80 × 1,500, 100 × 1,200,
120 × 1,000, 150 × 800, 200 × 600, 240 × 500, 250 × 480, 300 × 400;
3. a) 50, b) 10,000, c) 1,000,000; 4. 3,500,000,000,000 kg = 3,500,000,000
tonnes; 5. about 300 times as many years
Number and Operations in Base Ten 5-15
D-13
NBT5-16 Patterns in the 6, 7, 8, and 9 Times Tables
Pages 42–43
STANDARDS
preparation for 5.OA.B.3,
5.NBT.B.5
Goals
Students will be fluent with multiplication by 6, 7, 8, or 9.
PRIOR KNOWLEDGE REQUIRED
Can skip count by 6, 7, 8, and 9
Can identify multiples of 6, 7, 8, and 9
Vocabulary
multiple
pattern
times table
Recognizing patterns in the ones digits of the 6 times table.
(MP.7)
Write on the board:
6×1=
6×5=
6×9=
6 × 2 =
6 × 6 =
6 × 3 =
6 × 7 =
6 × 4 =
6 × 8 =
Ask students for answers and fill in.
ASK: Do you see a pattern in the ones digits of the answers? If students
have difficulty seeing the pattern, write on the board:
06 12 18 24 30 36 42 48 54
9876543210987654321098765432109876543210
Ask a student to come to the board and circle the ones digits from the
answers on the row that counts down repeatedly from 9.
9876543210987654321098765432109876543210
ASK: What is the pattern? (ones digits are all four numbers apart)
SAY: If you have trouble remembering the numbers, picture the pattern and
count four spots each time.
a)6 × 5
b) 6 × 7
c) 6 × 9
Bonus
d)6 × 80
e) 60 × 400
Answers: a) 30, b) 42, c) 54, Bonus: d) 480, e) 24,000
Recognizing patterns in the ones digits of the 7 times table.
Write on the board:
7×1=
7×5=
7×9=
7 × 2 =
7 × 6 =
7 × 3 =
7 × 7 =
7 × 4 =
7 × 8 =
Ask students for answers and fill in.
D-14
Teacher’s Guide for AP Book 5.1
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Exercises: Multiply the numbers.
ASK: Do you see a pattern in the ones digits of the answers? If students
have difficulty seeing the pattern, write on the board:
07 14 21 28 35 42 49 56 63
987654321098765432109876543210
Ask a student to come to the board and circle the ones digits from the
answers on the row that counts down repeatedly from 9.
9 8 7 6 5 4 3 2 1 0 9 8 7 6 5 4 3 2 1 0 9 8 7 6 5 4 3 2 1 0
ASK: What is the pattern? (ones digits are all three numbers apart) SAY: If
you have trouble remembering the numbers, picture the pattern and count
three spots each time.
Exercises: Multiply.
a)7 × 4
b) 7 × 6
c) 7 × 9
Bonus
d)7 × 80
e) 70 × 300
Answers: a) 28, b) 42, c) 63, Bonus: d) 560, e) 21,000
Recognizing patterns in the ones digits of the 8 times table.
Write on the board:
8×1=
8×5=
8×9=
8 × 2 =
8 × 6 =
8 × 3 =
8 × 7 =
8 × 4 =
8 × 8 =
ASK: Do you see a pattern in the ones digits of the answers? If students
have difficulty seeing the pattern, write on the board:
08 16
24 32 40 48 56 64 72 9 8 7 6 5 4 3 2 1 0 9 8 7 6 5 4 3 2 1 0
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Ask a student to come to the board and circle the ones digits from the
answers on the row that counts down repeatedly from 9.
9 8 7 6 5 4 3 2 1 0 9 8 7 6 5 4 3 2 1 0
ASK: What is the pattern? (ones digits are all two numbers apart) SAY: If
you have trouble remembering the numbers, picture the pattern and count
two spots each time.
Exercises: Multiply.
a)8 × 5
b) 8 × 8
c) 8 × 6
d) 8 × 90
e) 80 × 300
Answers: a) 40, b) 64, c) 48, d) 720, e) 24,000
Number and Operations in Base Ten 5-16
D-15
Recognizing patterns in the ones digits of the 9 times table.
Write on the board:
9×1=
9×5=
9×9=
9 × 2 =
9 × 6 =
9 × 3 =
9 × 7 =
9 × 4 =
9 × 8 =
ASK: Do you see a pattern in the ones digits of the answers? If students
have difficulty seeing the pattern, write on the board:
09
18
27 36 45 54 63 72 81
9 8 7 6 5 4 3 2 1 0 9 8 7 6 5 4 3 2 1 0
Ask a student to come to the board and circle the ones digits from the
answers on the row that counts down repeatedly from 9.
9
8
7
6
5
4
3
2
1
0
ASK: What is the pattern? (ones digits are all one number apart) SAY: If you
have trouble remembering the numbers, picture the pattern and count one
spot each time.
Exercises: Multiply.
a)9 × 4
b) 9 × 7
c) 9 × 5
d) 9 × 80
Bonus: 90 × 600
Answers: a) 36, b) 63, c) 45, d) 720, Bonus: 5,400
NOTE: For other patterns in the times tables see “How to Learn Your
Times Tables in a Week” in the Mental Math section of this guide.
Extensions
(MP.2, MP.8)
1. Have students complete BLM Circle Charts (p. D-66).
D-16
multiples of 4
multiples of 4
Teacher’s Guide for AP Book 5.1
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Answers
2. Have students complete BLM Divisibility Rules (pp. D-67–68).
(MP.2)
multiples of 8
multiples of 3
Answers
1. A number is a multiple of 2 if it ends in 2, 4, 6, 8, or 0.
2. A number is a multiple of 5 if it ends in 5 or 0.
3. A number is a multiple of 3 if the sum of its digits is a multiple of 3.
4. A number is a multiple of 9 if the sum of its digits is a multiple of 9.
5. 42 divisible by 2, 3; 36 divisible by 2, 3, 9; 60 divisible by 2, 3, 5; 135
divisible by 3, 5 ,9; 525 divisible by 3, 5; 143,172 divisible by 2, 3, 9
3.Ask students to describe any patterns they see in the following products:
2 × 6 = 12
4 × 6 = 24
6 × 6 = 36
8 × 6 = 48
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Answer: Students should notice that when you multiply 6 by an even
number, the ones digit of the answer is the same as the number you are
multiplying by. The tens digit is half the number you are multiplying by.
Number and Operations in Base Ten 5-16
D-17
NBT5-17 Multiplying by Powers of 10
Pages 44–45
Goals
STANDARDS
5.NBT.A.1, 5.NBT.A.2,
5.NBT.B.5
Students will write a power as a product and a product as a power.
PRIOR KNOWLEDGE REQUIRED
Can multiply multi-digit numbers by 2-digit numbers
Knows the commutative property of multiplication
Knows multiplication is a short form for repeated addition
Vocabulary
base
exponent
power
product
Introduce the parts of a power. ASK: How can we check that 5 × 4 = 20?
(5 × 4 = 4 + 4 + 4 + 4 + 4 = 20) What does the 5 tell us? (how many of
the second number we should add) SAY: There is a similar short form for
multiplying. Write on the board:
45 = 4 × 4 × 4 × 4 × 4
SAY: Instead of writing 4 × 4 × 4 × 4 × 4, we can write 45. ASK: What does
the 5 tell us this time? (how many times to multiply the number by itself)
SAY: It is important to learn the correct vocabulary for this short form. Write
on the board:
base
45
exponent
power
SAY: 4 is the base, 5 is the exponent, and 45 is a power of 4.
Exercises: State the base, exponent, and power.
a)63
b)84
c)210
Answers: a) base = 6, exponent = 3, power = 63; b) base = 8,
exponent = 4, power = 84; c) base = 2, exponent = 10, power = 210
45 = 4 × 4 × 4 × 4 × 4
multiply the base 5 times
Exercises: Write each power as a product.
a)24
b)63
c)72
d)81
Answers: a) 24 = 2 × 2 × 2 × 2, b) 63 = 6 × 6 × 6, c) 72 = 7 × 7, d) 81 = 8
Evaluating a power. Write on the board:
34 = 3
D-18
×
3
×
3
×
3
Teacher’s Guide for AP Book 5.1
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Writing a power as a product. SAY: The exponent tells us how many times
to multiply the base. Write on the board:
SAY: To evaluate the power, we need to start multiplying and keeping track
of the product as we go along. What is 3 × 3? (9) Write the answer in the
first box. What is 9 × 3? (27) Write the answer in the second box. What is
27 × 3? (81) Write the final answer. Finish the equation on the board:
34 = 3
×
9 3 ×
27 3 ×
81
3
= 81
Exercises: Evaluate.
a)24
b)53
c)82
d)103
Answers: a) 16, b) 125, c) 64, d) 1,000
(MP.7)
Evaluating products involving a power of 10. Write on the board:
2 × 102 = 2 × (10 × 10) = 2 × 100 = 200
2 × 103 = 2 × (
)=2×
2 × 104 = 2 × (
)=2×
=
=
(2,000, 20,000)
ASK: What pattern do you see in the power and the answer? (the number
of zeros added to the digit is the same as the exponent) If a hint is needed,
write on the board:
2 × 102 = 200
2 × 103 = 2,000
2 × 104 = 20,000
Exercises: Evaluate without writing a product.
a)3 × 103
b)7 × 105
c)9 × 106
d)32 × 105
Answers: a) 3,000, b) 700,000, c) 9,000,000, d) 3,200,000
Extensions
1. Write as a power of 10.
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a)100
b)10,000
c)1,000,000,000
2. Write as a power of 2.
a)
4b)
8c)
32
3.Evaluate.
a)102 × 103
b)104 × 102
c) 106 × 103
(MP.2)
4. How can you predict the number of zeros when evaluating 107 × 108?
(MP.1, MP.8)
5. Have students complete BLM The Power of Doubling (pp. D-69–70).
Answers: 1. a) 102, b) 104, c) 109; 2. a) 22, b) 23, c) 25; 3. a) 100,000,
b) 1,000,000, c) 1,000,000,000; 4. add the exponents; 5. the total after 30
days is $2,147,483,647
Number and Operations in Base Ten 5-17
D-19
NBT5-18 Arrays and Multiplication
Pages 46–48
Goals
STANDARDS
5.OA.B.3, 5.NBT.B.5
Students will write a product for an array using the distributive property.
Students will write a product using the distributive property, without
using an array.
Vocabulary
array
distributive property
expanded
place value
product
PRIOR KNOWLEDGE REQUIRED
Can write a product for a given array
MATERIALS
grid paper
Use the distributive property to write a product for an array. Draw the
following diagram on the board:
ASK: What is a product that represents this array of squares? (3 × 14) On
the board, shade in the last 4 columns of the array. Draw on the board:
3 × 14
Ask students to come to the board to write a product for the shaded
squares and the unshaded squares. (3 × 10 and 3 × 4) SAY: But the total
number of squares hasn’t changed just because we shaded in some of the
rectangles! So what can we say about the expressions 3 × 14 and (3 × 10)
+ (3 × 4)? (they are equal)
Exercises: Write products for the entire diagram, the unshaded squares,
and the shaded squares. Then write an equation for the diagram.
a)
b)
D-20
Teacher’s Guide for AP Book 5.1
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Answers: a) 4 × 13, 4 × 10, 4 × 3, 4 × 13 = (4 × 10) + (4 × 3); b) 5 × 26,
5 × 20, 5 × 6, 5 × 26 = (5 × 20) + (5 × 6)
SAY: Instead of drawing all the squares, we can draw a diagram and write
down how many squares there would have been. Draw on the board:
Instead of
We draw
4
(MP.7)
10
3
4 × 10
4×3
Exercises: Write products for the entire diagram, the unshaded squares,
and the shaded squares. Then write an equation for the diagram.
10
a)
2
6
20
b)
3
5
Answers: a) 6 × 12, 6 × 10, 6 × 2, 6 × 12 = (6 × 10) + (6 × 2);
b) 5 × 23, 5 × 20, 5 × 3, 5 × 23 = (5 × 20) + (5 × 3)
(MP.8)
Using the distributive property to write a product for an array without
using an array. NOTE: Students don’t need to know the term “distributive
property,” but we recommend you introduce them to the term.
Distributive Property: A property that allows multiplication of a sum by
multiplying each term, then adding the products.
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Example:
3 × (2 + 4) = (3 × 2) + (3 × 4)
3 × 6
=3 × 2
+3 × 4
SAY: Let’s look at the answers from our last two questions. Write on
the board:
6 × 12 = (6 × 10) + (6 × 2)
5 × 23 = (5 × 20) + (5 × 3)
Number and Operations in Base Ten 5-18
D-21
What pattern do you see in how the right side of the equation was created?
(the first number is shared in each bracket; the second number is broken
into a multiple of 10 and the remainder) If students have difficulty seeing the
pattern, write on the board:
6 × 12 = (6 × 10 ) + (6 × 2 )
5 × 23 = (5 × 20 ) + (5 × 3 )
(MP.8)
Exercises: Rewrite each product in expanded form.
a)5 × 63
b) 7 × 84
c) 9 × 48
Answers: a) 5 × 63 = (5 × 60) + (5 × 3); b) 7 × 84 = (7 × 80) + (7 × 4);
c) 9 × 48 = (9 × 40) + (9 × 8)
Bonus: Write each product as a sum of three or more products.
a)7 × 125
b) 5 × 348
c) 4 × 1,375 d) 9 × 2,486 e) 3 × 23,756
Answers: a) 7 × 125 = (7 × 100) + (7 × 20) + (7 × 5); b) 5 × 348 =
(5 × 300) + (5 × 40) + (5 × 8); c) 4 × 1,375 = (4 × 1,000) + (4 × 300) +
(4 × 70) + (4 × 5); d) 9 × 2,486 = (9 × 2,000) + (9 × 400) + (9 × 80) +
(9 × 6); e) 3 × 23,756 = (3 × 20,000) + (3 × 3,000) + (3 × 700) +
(3 × 50) + (3 × 6)
Performing the standard algorithm for multiplication in two steps.
Write on the board:
5 × 63 = (5 × 60) + (5 × 3)
SAY: We can use the right side of the equation to multiply in two steps:
5 × 3 and 5 × 60
Write on the board, without the answers:
×
3
5
6 0
5
×
Have students copy the equations into their notebooks, on grid paper. ASK:
What is 5 × 3? (15) SAY: Write the answer in the grid on the left underneath
the first line. ASK: What is 6 × 5? (30) Then what is 60 × 5? (300) Remind
students that to multiply by multiples of 10, we can just add the zero at the
end. SAY: Write the answer in the grid on the right.
Write on the board:
5 × 63 = (5 × 60) + (5 × 3)
D-22
Teacher’s Guide for AP Book 5.1
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+
SAY: We’ve done each of the multiplications, but we haven’t done the
addition. Write on the board:
3
5
×
1 5
+ 3 0 0
3 1 5
6 0
5
×
3 0 0
SAY: Copy the answer from 60 × 5 underneath the answer from 3 × 5,
being careful to line up the numbers under the correct place value. Now we
can finish the final step by adding! Have students finish the question. Ask a
volunteer to come up and write the answer on the board.
Exercises: Use two steps to calculate each product.
a)74 × 2
b) 83 × 2
c) 46 × 3
d) 57 × 4
Answers: a) 148, b) 166, c) 138, d) 228
Encourage students to do simple calculations mentally.
Exercises: Multiply using mental math.
a)14 × 2
b) 23 × 3
c) 43 × 2
d) 72 × 3
e) 61 × 4
Answers: a) 28, b) 46, c) 86, d) 216, e) 244
Extension
Fill in the missing numbers.
(MP.2)
a)
0 b)
0 c)
0
4
7
9
×
×
×
2 8 0
5
0
7 0
Answers
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a)
7 0 b)
3 0
8 0 c)
4
9
7
×
×
×
2 8 0
2 7 0
5 6 0
Number and Operations in Base Ten 5-18
D-23
NBT5-19 Standard Method for Multiplication
Pages 49–50
Goals
STANDARDS
5.NBT.B.5
Students will multiply a 2-digit number by a 1-digit number using the
standard algorithm.
Vocabulary
PRIOR KNOWLEDGE REQUIRED
algorithm
area
distributive property
multiple
regrouping
Can multiply a 2-digit number by a 1-digit number using the
distributive property
MATERIALS
grid paper
Use 2 separate grids to multiply a 2-digit number by a 1-digit number.
Draw on the board:
3
45
ASK: What multiplication statement represents the area of the rectangle?
(45 × 3) How can we use a multiple of 10 to break up 45? (40 + 5) Add a
line to the diagram so it looks like this:
3
40
5
ASK: What multiplication statements give the area of each rectangle?
(40 × 3 and 5 × 3) SAY: But the area is the same! So:
45 × 3 =
(40 + 5) × 3
= (40 × 3) + (5 × 3)
We can perform these multiplications using two grids and adding
the results:
3
5
1 5
+ 1 2 0
1 3 5
×
D-24
4 0
3
×
1 2 0
Teacher’s Guide for AP Book 5.1
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Exercises
(MP.7)
1.Use a multiple of 10 and the distributive property to break up the
first factor.
a)36 × 4
b) 54 × 7
c) 73 × 5
d) 87 × 3
Answers: a) 36 × 4 = (30 + 6) × 4 = (30 × 4) + (6 × 4);
b) 54 × 7 = (50 + 4) × 7 = (50 × 7) + (4 × 7);
c) 87 × 3 = (80 + 7) × 3 = (80 × 3) + (7 × 3)
2.Now use two separate grids to find the products for the questions above.
Selected solution
a)
6
4
×
2 4
+ 1 2 0
1 4 4
3 0
4
×
1 2 0
Answers: b) 378, c) 365, d) 261
Using the standard algorithm to multiply a 2-digit number by a 1-digit
number with no regrouping. SAY: We can combine our work into one grid.
Write on the board:
42 × 3
Step 1: M
ultiply 2 × 3 = 6. Place the 6 in the ones column in the bottom
row of the grid.
4 2
×
3
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4 2
×
3
6
2×3
Step 2: M
ultiply 4 tens by 3 = 12 tens. Place the ones digit of 12 in the tens
column in the bottom row of the grid. Place the tens digit of 12 in
the hundreds column in the bottom row of the grid.
SAY: It’s important to place the digits in the correct columns in the bottom
row, since we are really multiplying 40 × 3, not 4 × 3.
4 2
×
3
1 2 6
4 × 3 = 12
4 tens × 3 = 12 tens
Exercises: Use the standard algorithm to multiply.
a)72 × 3
b) 54 × 2
c) 32 × 4
Answers: a) 216, b) 108, c) 128
Number and Operations in Base Ten 5-19
D-25
Using the standard algorithm to multiply a 2-digit number by a 1-digit
number with regrouping. SAY: Sometimes the first step may involve a
product greater than 9. This will require regrouping.
Write on the board:
4 5
×
3
Step 1: SAY: Notice that the first step will give us 5 ones × 3 = 15 ones.
ASK: How many tens are in 15 ones? (1) How many ones are left? (5)
We write the 5 remaining ones in the ones column in the bottom row.
We write the tens digit of 15 in the tens column in the row above the grid.
1
tens digit from 5 × 3 = 15
4 5
×
3
5
ones digit from 5 × 3 = 15
Step 2: S
AY: Multiply 4 tens by 3 = 12 tens. Add the 1 ten from the
regrouping. ASK: How many tens do we have? (13) How many
hundreds are in 13 tens? (1) How many tens are left? (3)
SAY: Write the 13 tens so that the 1 is in the hundreds column in the bottom
row, and the 3 is in the tens column in the bottom row.
1
4 5
×
3
1 3 5
4 × 3 + 1 = 13
Exercises: Multiply using the standard algorithm.
b) 54 × 7
c) 93 × 4
Answers: a) 74, b) 378, c) 372
Extensions
1. Find the missing numbers.
a)b)c)
D-26
6
4
×
×
×
3
7
1 2 6 7 1 6 2
Teacher’s Guide for AP Book 5.1
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a)37 × 2
d)e)f)
9
×
×
9
8
8 7
2 2 9 7
×
5
2. Use the digits from 0 to 5 to complete the multiplications.
8
5
×
4
×
2
5
2 9
2
4 6
×
5
2
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Answers: 1. a) 42 × 3 = 126, b) 61 × 7 = 427, c) 54 × 3 = 162,
d) 62 × 9 = 558, e) 94 × 8 = 752, f) 99 × 3 = 297; 2. 84 × 3 = 252,
51 × 4 = 204, 92 × 6 = 552
Number and Operations in Base Ten 5-19
D-27
NBT5-20 Multiplying a Multi-Digit Number by a
Pages 51–52
1-Digit Number
Goals
STANDARDS
5.NBT.B.5
Students will multiply a multi-digit number by a 1-digit number using
the standard algorithm.
Vocabulary
PRIOR KNOWLEDGE REQUIRED
expanded form
regrouping
Can multiply a 2-digit number by a 1-digit number using the
standard algorithm
MATERIALS
base ten materials
Review multiplying a 2-digit number by a 1-digit number in 3 ways:
expanded form, base ten materials, and the standard algorithm.
(MP.5)
a) Using expanded form.
Write on the board:
42 × 3
ASK: How can we write 47 in expanded form? (4 tens + 7 ones)
Write on the board:
4 tens + 2 ones
×3
12 tens + 6 ones
= 1 hundred + 2 tens + 6 ones
= 126
ASK: What is 4 tens × 3? (12 tens) What is 2 ones × 3? (6 ones)
How many hundreds are in 12 tens? (1) How many tens are left? (2)
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b) Using base ten materials.
Write on the board:
42 × 3
Draw:
D-28
Teacher’s Guide for AP Book 5.1
ASK: What do you get when you put all the tens blocks together?
(12 tens blocks)
What can we exchange 10 tens blocks for? (a hundreds block)
Draw:
to get the answer 126
c) Using the standard algorithm.
Write on the board:
4
3
×
1
2
2
6
ASK: What is 2 × 3? (6) What is 4 tens × 3? (12 tens) How many
hundreds are there in 12 tens? (1) How many tens are left? (2)
So 42 × 3 = 126.
Using base ten materials, the standard algorithm, and expanded form,
multiply a 3-digit number by a 2-digit number without regrouping.
a) Using expanded form.
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Tell students you want to multiply 423 × 2. ASK: How can we write 423
in expanded form? (4 hundreds + 2 ten + 3 ones) Write on the board:
4 hundreds + 2 ten + 3 ones
×2
8 hundreds + 4 tens + 6 ones
ASK: What is 4 hundreds × 2? (8 hundreds) What is 2 tens × 2?
(4 tens) What is 3 ones × 2? (6 ones) What is the product? (846)
Number and Operations in Base Ten 5-20
D-29
b) Using base ten materials: Draw on the board:
423 × 2
ASK: How many hundreds are there altogether? (8) How many tens are
there altogether? (4) How many ones are there altogether? (6) What is the
product? (846)
c) Using the standard algorithm.
Write on the board:
4
2
2
×
8
3
4
6
ASK: What is 3 × 2? (6) What is 2 × 2? (4) What is 4 × 2? (8)
So 423 × 2 = 826.
Using expanded form, base ten materials, and the standard algorithm,
multiply a 3-digit number by a 2-digit number with regrouping.
SAY: Sometimes regrouping may be involved. Write on the board:
467 × 2
a) Using expanded form.
Write on the board:
4 hundreds + 6 tens + 7 ones
×2
8 hundreds + 12 tens + 14 ones
ASK: What is 4 hundreds × 2? (8 hundreds) What is 6 tens × 2?
(12 tens) What is 7 ones × 2? (14 ones) What is the product? (846)
Write on the board:
D-30
8 hundreds +
= 8 hundreds + (
12 tens
hundreds +
+
tens) + (
14 ones
tens +
ones)
Teacher’s Guide for AP Book 5.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
ASK: How can we write 456 in expanded form? (4 hundreds + 6 tens +
7 ones)
Ask a volunteer to come up to fill in the blanks. (1, 2, 1, 4) Write on
the board:
hundreds +
=
tens +
ones
Ask another volunteer to gather up the hundreds, tens, and ones to
fill in the blanks. (9, 3, 4) Write the answer:
= 934
b) Using base ten materials.
Draw on the board:
467 × 2
ASK: How many hundreds are there altogether? (8) How many tens are
there altogether? (12) How many ones are there altogether? (14) If we
exchange 10 ones for a tens block, and 10 tens for a hundreds block,
what will the diagram look like?
12 tens
become
1 hundred
+ 2 tens
14 ones
become
1 ten +
4 ones
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ASK: How many hundreds are there altogether? (9) How many tens are
there altogether? (3) How many ones are there altogether? (4) What is
the final product? (934)
c) Using the standard algorithm.
Write on the board, without the top row filled in:
1
4
×
6
7
2
4
ASK: What is 7 × 2? (14) How do we write 14 in expanded form? (1 ten
and 4 ones) SAY: We write the 1 ten in the row above the grid, and we
write the 4 in the ones column in the bottom row of the grid. As you fill
in the numbers, describe to students what each number represents.
Number and Operations in Base Ten 5-20
D-31
1
1
4
6
7
2
×
3
4
ASK: What are 6 tens × 2? (12 tens) SAY: But we have 1 ten from
multiplying 7 × 2, so we really have 13 tens altogether. ASK: How do
we write 13 tens in expanded form? (13 tens = 1 hundred + 3 tens)
SAY: So we write the 3 in the tens column in the bottom row, and we
write the 1 in the hundreds column in the row above the grid.
1
1
4
6
2
×
9
7
3
4
ASK: What is 4 hundreds × 2? (8 hundreds) SAY: But we have 1
hundred from multiplying 6 tens × 2, so really we have 9 hundreds.
Write the 9 in the hundreds column in the bottom row of the grid.
Together, solve:
• problems that require regrouping ones to tens (examples:
219 × 3, 312 × 8, 827 × 2)
• problems that require regrouping tens to hundreds (examples:
391 × 4, 282 × 4, 172 × 3)
• problems that require regrouping both ones and tens (examples:
479 × 2, 164 × 5, 129 × 4)
Have students solve additional problems in their notebooks.
a)112 × 5
b) 321 × 8
c) 215 × 7
d) 312 × 9
Answers: a) 560, b) 2,568, c) 1,505, d) 2,808
Tell students to be sure that they get the same answer all three ways.
If they do not, they should check their work to find the mistake.
Bonus: Find the products.
a)2,456 × 3
b) 5,234,562 × 7
Answers: a) 7,368, b) 36,641,934
D-32
Teacher’s Guide for AP Book 5.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Exercises: Use base ten materials, expanded form, and the standard
algorithm to solve each problem.
Exploring the special case in which the 3-digit number has a 0 digit.
Write on the board:
5
3
0
6
9
×
2
7
5
4
Describe each step of the process, pointing to each digit as you say it:
• 6 ones × 9 is 54 ones, so that’s 5 tens and 4 ones
• 0 tens × 9 is 0 tens, then add the 5 tens
• 3 hundreds × 9 is 27 hundreds, so that’s 2 thousands and
7 hundreds
Exercises: Find the product.
a)406 × 9
b) 460 × 8
c) 807 × 6
d) 870 × 5
e) 708 × 3
Bonus: 12,009 × 7
Answers: a) 3,654, b) 3,680, c) 4,842, d) 4,350, e) 2,124, Bonus: 84,063
Extensions
(MP.1)
1.Using only the digits 2, 3, 4, and 6, find the greatest product that can be
made by multiplying a 3-digit number by a 1-digit number.
Answer: 2,592
(MP.2)
2.Using only the digits 4, 5, 6, and 9, find the least product that can be
made by multiplying a 3-digit number by a 1-digit number.
Answer: 2,276
(MP.2)
3.What is the greatest product possible when multiplying a 3-digit number
by a 1-digit number?
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Answer: 999 × 9 = 8,991
(MP.1)
4. Try the following number trick with a friend.
a)
b)
c)
d)
e)
f)
Pick a number from 1 to 9.
Multiply your number by 100.
Add 3 to your answer.
Multiply your answer by 6.
Subtract 18.
Ask for the answer.
To guess the number, remove the zeros at the end of the number, then
divide by 6. That will be the number your friend started with.
Try it with your friend, then have your friend try it with you. Can you
figure out why it works?
Number and Operations in Base Ten 5-20
D-33
Answer: Multiplying by 100 moves the digits to the left two places.
Adding 3, multiplying by 6, then subtracting 18, gives zeros at the end
of the number. When you remove the zeros, all that is left is the original
number multiplied by 6.
(MP.7)
5.Complete BLM Circle Magic (p. D-71).
Answers: a) 142,857, b) 285,714, c) 428,571, d) 571,428, e) 714,285,
f) 857,142; all the products contain the digits 1 4 2 8 5 7
(MP.7)
6.Use mental math to multiply by 9. To multiply by 10, add a zero. To
multiply by 9, multiply by 10, then subtract. Example:
3 × 9 = (3 × 10) – (3 × 1)
3 × 10
3×9
3×1
3×9
To calculate 457 × 9:
Step 1: Calculate 457 × 10 = 4,570
Step 2: Calculate 457 × 1 =
457
Step 3: Subtract:
4,113
Use mental math to calculate.
a)127 × 9
b) 248 × 9
c) 1,234 × 9
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) 1,143, b) 2,232, c) 11,106
D-34
Teacher’s Guide for AP Book 5.1
NBT5-21 Word Problems I
Page 53
Goals
STANDARDS
5.NBT.B.5
Students will solve word problems involving multiplication that may
require multiplying multi-digit numbers by 1-digit numbers.
PRIOR KNOWLEDGE REQUIRED
Can use the standard algorithm to multiply multi-digit numbers by
1-digit numbers
Extensions
(MP.1)
1. Tim earns $9 per hour at his summer job.
a) How much does Tim earn in a 40-hour week?
b) How much will he earn in 8 weeks?
c)How many more weeks will he need to work to buy computer
equipment that costs $3,600?
Answers: a) $360, b) $2,880, c) 80 hours
(MP.1)
2. A T-shirt company sold 1,250 T-shirts at $8 each.
a) How much money did the company earn?
b)For each T-shirt, the company paid $2 for heat, lighting, and
electricity. What are the costs for 1,250 shirts?
c) What profit did the company make on the T-shirts?
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Answers: a) $10,000, b) 2,500, c) $7,500
Number and Operations in Base Ten 5-21
D-35
NBT5-22 Multiplying 2-Digit Numbers by
Pages 54–55
Multiples of 10
Goals
STANDARDS
5.NBT.A.1, 5.NBT.A.2,
5.NBT.B.5
Students will use the standard algorithm to multiply 2-digit numbers
by 2-digit multiples of 10 (10, 20, 30,…, 90).
PRIOR KNOWLEDGE REQUIRED
Vocabulary
array
associative property
commutative property
multiples
product
regrouping
rounding
standard algorithm
Can multiply using arrays
Can apply the distributive property
Can use the standard algorithm to multiply 2-digit numbers by
1-digit numbers
Can use mental math to multiply 2-digit numbers by 2-digit multiples
of 10 without regrouping
MATERIALS
BLM Cutting an Array into Ten Strips (p. D-72)
Review mentally multiplying 1-digit numbers by multi-digit numbers.
Choose examples that require no regrouping (examples: 2 × 324, 3 × 132,
4 × 201) or only regrouping at the greatest place value. (examples:
4 × 612, 2 × 804, 3 × 430)
Using arrays to multiply by multiples of 10.
Draw on the board:
32
20
Tell students that the picture represents an array that is 20 squares high and
32 squares long. If they have trouble visualizing the individual squares, give
them BLM Cutting an Array into Ten Strips.
ASK: What multiplication gives the total number of squares? (20 × 32
because the width is 20 squares and the length is 32 squares)
Ask students to imagine dividing the array into smaller strips that are
2 squares high and 32 squares long.
D-36
Teacher’s Guide for AP Book 5.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
32
each strip is 2 squares high
and 32 squares long
20
ASK: What multiplication gives the number of squares in each strip?
(2 × 32 because each strip is 32 squares long and 2 squares high)
32
2 × 32
20
ASK: How many strips are in the array? (10; each strip is 2 squares high
and the array is 20 squares high)
SAY: There are 10 strips each containing 2 × 32 squares, so you can
rewrite 20 × 32 as:
10 × (2 × 32)
number of strips
number of squares in each strip
Using the same diagram on the board, write a new height and length on
your diagram but do not erase the 10 strips. Make sure the height is a
multiple of ten. Ask students to write a multiplication statement for the new
dimensions. Students must first determine the height of each strip. Remind
them that there are 10 strips.
Exercises: Write a multiplication statement for the array.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
a)
57
30
77
b)
40
42
c)
50
Answers: a) 10 × (3 × 57), b) 10 × (4 × 77), c) 10 × (5 × 42)
Number and Operations in Base Ten 5-22
D-37
Review why using arrays works. Tell students they can prove that
20 × 32 = 10 × (2 × 32) without using a diagram. They can use the
commutative and associative properties of multiplication.
Write on the board:
20 × 32
= (10 × 2) × 32
= 10 × (2 × 32)
(20 = 2 × 10 or 10 × 2, commutative property)
(associative property)
Remind students that the associative property allows us to multiply three
numbers by either multiplying the first and second numbers together, or by
multiplying the second and third numbers together. Example: 2 × 3 × 4 =
(2 × 3) × 4, or 2 × 3 × 4 = 2 × (3 × 4).
SAY: Now rewrite the product to multiply a 2-digit number by a multiple of 10:
20 × 32 = 10 × (2 × 32) = 10 × 64 = 640
Exercises: Calculate the product.
a)30 × 12
b) 40 × 22
c) 50 × 31
Answers: a) 30 × 12 = (10 × 3) × 12 = 10 × (3 × 12) = 10 × 36 = 360;
b) 40 × 22 = (10 × 4) × 22 = 10 × (4 × 22) = 10 × 88 = 880;
c) 50 × 31 = (10 × 5) × 31 = 10 × (5 × 31) = 10 × 155 = 1,550
Multiplying by multiples of 10 to estimate products. Write on the board:
32 × 79
ASK: Is 32 closer to 30 or to 40? (30) Is 79 closer to 70 or to 80? (80)
Write on the board:
32 × 79 ≈
30 × 80
≈ 10 × (3 × 80)
≈ 10 × 240
≈ 2,400
Exercises: Estimate by rounding each number to the nearest ten.
b) 78 × 21
c) 48 × 89
Answers: a) 1,000, b) 1,600, c) 4,500
Use a chart to find products involving multiples of 10. Tell students
they can use a chart to find products such as 40 × 57 = 10 × (4 × 57)
by following these steps:
Step 1: W
hen you multiply a number by 10, you add a 0. So write a
0 in the ones place because you will multiply 10 by (4 × 57).
D-38
Teacher’s Guide for AP Book 5.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
a)46 × 23
×
5
7
4
0
0
rite a 0 here because you
W
will multiply 4 × 57 by 10.
Step 2: Now multiply 57 × 4 using the standard algorithm.
2
×
2
2
Multiply 4 × 7. Write the 2 for the
regrouping in the hundreds column.
5
7
4
Multiply 4 × 5. Add the 2 from the
0 regrouping in the hundreds column.
0
8
Exercises: Multiply.
a)40 × 32
b) 60 × 45
c) 80 × 23
Answers: a) 1,280, b) 2,700, c) 1,840
Extension
(MP.1)
The strongest animal on Earth is the rhinoceros beetle. It weighs 80 grams
and can lift 850 times its own weight. How much can it lift? Use properties
of numbers to break the product into simpler products. Explain how you
found your answer:
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answer: 68,000 g = 68 kg; multiply 85 × 80, then multiply by 10 by
adding zero
Number and Operations in Base Ten 5-22
D-39
NBT5-23 Multiplying 2-Digit Numbers by
Pages 56–58
2-Digit Numbers
STANDARDS
5.NBT.B.5
Vocabulary
algorithm
array
distributive property
double
multiples
regrouping
rounding
standard algorithm
Goals
Students will use the standard algorithm to multiply 2-digit numbers
by 2-digit numbers.
PRIOR KNOWLEDGE REQUIRED
Can multiply using arrays
Can apply the distributive property
Can use the standard algorithm to multiply 2-digit numbers by
1-digit numbers
Can multiply 2-digit numbers by 2-digit multiples of 10
MATERIALS
BLM 1 cm Grid Paper (p. I-1)
Introduce multiplying 2-digit by 2-digit numbers. Write on the board:
45 × 32
ASK: How is this multiplication different from any we have done so far?
(neither number is a multiple of 10—we have only estimated the product in
such cases)
Splitting a problem into easier problems. Tell students that you would like
to find a way to split the problem into two easier problems, both of which
they already know how to do. Have students list all the types of problems
they know how to do that might be helpful:
ASK: How can we break up one of the numbers so that it contains a 2-digit
multiple of 10? (45 = 40 + 5; 32 = 30 + 2) SAY: Let’s pick 32 = 30 + 2
(although it would also work with 45 = 40 + 5).
Using the distribute property to multiply 2-digit numbers by 2-digit
numbers in separate steps.
Draw on the board:
32
45
D-40
Teacher’s Guide for AP Book 5.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
• Multiply a 1-digit number by a 1-digit number.
• Multiply a 2-digit number by a 1-digit number.
• Multiply a 2-digit number by a 2-digit multiple of 10.
ASK: What multiplication statement gives the area? (45 × 32) SAY: Let’s
break up 32 into 30 and 2.
Draw on the board:
30
2
45
32
ASK: What multiplication statement gives the area of the rectangle on the
left? (45 × 30) What multiplication statement gives the area of the rectangle
on the right? (45 × 2) Write on the board and SAY: Since the total area of
the two small rectangles is the same as the large rectangle, we have:
45 × 32 =
45 × (30 + 2)
= (45 × 30) + (45 × 2)
SAY: We know how to use a grid to multiply each of these multiplication
statements. We can add the separate parts in a grid as well.
45 × 2
1
×
4 5
2
9 0
45 × 30
45 × 30 + 45 × 2
1
4 5
3 0
×
1 3 5 0
1
1
4
3
9
1 3 5
1 4 4
×
5
2
0
0
0
Exercises
1.Use BLM 1 cm Grid Paper to multiply using the three separate steps.
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a)37 × 21
b) 48 × 73
c) 53 × 37
Answers: a) 777, b) 3,504, c) 1,961
2.Check the reasonableness of your answers by rounding each factor
to the nearest 10.
Selected solution: a) 37 × 21 ≈ 40 × 20
Using the standard algorithm to multiply a 2-digit number by a
2-digit number.
Write on the board:
28 × 34
Number and Operations in Base Ten 5-23
D-41
Step 1: Multiply 28 × 4.
regrouping from 8 × 4 = 32
3
2 8
3 4
×
1 1 2
4 × 2 + 3 = 11
28 × 4 = 112
Step 2: Multiply 28 × 30.
regrouping from
8 × 3 = 24
2
×
28 × 3 = 84
3
2
3
1 1
8 4
8
4
2
0
add 0 because we
are multiplying by 30
Step 3: Add the two products.
2
×
3
2
3
1 1
8 4
9 5
8
4
2
0
2
product from 28 × 4
product from 28 × 30
112 + 840
Mentally finding the product of 2-digit numbers multiplied by 2-digit
numbers. Encourage students to mentally find simple products of pairs of
2-digit numbers. Examples:13 × 11, 21 × 12, 22 × 31. (143, 252, 682)
Write the product 12 × 14 on the board. Ask the class to find the
product mentally. After students have had time to do this, ask
volunteers to tell you what strategy they used to find the product.
Write their answers on the board in abbreviated form. For instance, if
a student says “I thought of 12 as 10 plus 2. I multiplied 14 by 10 and
got 140, then I multiplied 14 by 2 and got 28. I added 140 and 28 and
got 168.” You could write on the board:
12 × 14
= (10 + 2) × 14
= 10 × 14 + 2 × 14
= 140 + 28
= 168
D-42
Teacher’s Guide for AP Book 5.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
ACTIVITY
Encourage students to share unusual strategies. For example, “I know
14 is double 7. So 12 × 14 is double 12 × 7. I know 12 × 7 is 84.
And double 84 is 168.” (NOTE: This exercise was inspired by a talk
by Marilyn Burns at the NCTM National Conference in 2013.)
Exercises: Use the standard algorithm to multiply.
a)23 × 47
b) 45 × 62
c) 36 × 58
d) 29 × 36
Answers: a) 1,081, b) 2,790, c) 2,088, d) 1,044
Extensions
(MP.5)
1.Instead of dividing the rectangle for 45 × 32 into two rectangles, we
could divide it into four rectangles. We use the idea that 45 = 40 + 5
and that 32 = 30 + 2.
Write a multiplication statement for each rectangle.
30
2
40
A
B
5
C
D
Answers: A: 40 × 30, B: 40 × 2, C: 5 × 30, D: 5 × 2
To calculate 45 × 32, add up the areas of the individual rectangles:
45 × 32
= (40 × 5) (30 + 2)
= (40 × 30) + (40 × 2) + (5 × 30) + (5 × 2)
= 1,200 +
80 + 150 + 10
= 1,440
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Have students use this technique to find the products.
a)56 × 38
b) 82 × 41
c) 73 × 39
Answers: a) 2,128, b) 3,362, c) 2,847
Have students check the reasonableness of their answers by rounding
each factor to the nearest 10.
Selected solution: a) 56 × 38 ≈ 60 × 40
2.Use the four-rectangle method in a grid to multiply 2-digit numbers by
2-digit numbers.
Number and Operations in Base Ten 5-23
D-43
45
40 +5
× 32 30 + 2
10 80 150 + 1,200 1,440
2×5
2 × 40
30 × 5
30 × 40
Tell the students that the 2 in 32 is multiplied by the 5 and the 40, and
the 30 in 32 is multiplied by the 5 and the 40. To find the answer, add
the four products.
Have students practice this grid method to find the products.
a)62 × 53
b) 28 × 14
c) 37 × 93
Answers: a) 3,286, b) 392, c) 3,441
(MP.7)
3.Distribute BLM Patterns in Multiplication (p. D-73). Have students
discover an easy way to find the product of a 2-digit number ending in
5 that is multiplied by itself (examples: 15 × 15, 25 × 25, 35 × 35).
After students complete the BLM, summarize their answers. ASK: What
are the digits in the tens column and ones column in every case? (25)
ASK: Using the tens digit of the original number, what multiplication
statement will give the remaining digits in the answer? (multiply the
tens digit by 1 more than the tens digit)
Write on the board:
35 × 35 =
1
2
3×4
2
5
5×5
Write on the board:
65 × 65
So 65 × 65 = 4,225
Exercises: Use this technique to find the product.
a)75 × 75
b) 45 × 45
c) 35 × 35
d) 85 × 85
Answers: a) 5,625, b) 2,025, c) 1,225, d) 7,225
Challenge: Find the product.
a)175 × 175 (Hint: Calculate 17 × 18 to find the first 3 digits.)
b)105 × 105
c) 995 × 995
d)1,005 × 1,005
e) 9,995 × 9,995
Answers: a) 30,625, b) 11,025, c) 990,025, d) 1,010,025, e) 99,900,025
D-44
Teacher’s Guide for AP Book 5.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
ASK: What is the tens digit? (6) What is one more than the tens digit?
(7) What is the product of 6 × 7? (42) SAY: These are the first two digits
of the answer. The last two digits will always be 5 × 5 = 25.
NBT5-24 Multiplying 3-Digit Numbers by
Pages 59–61
2-Digit Numbers
Goals
STANDARDS
5.NBT.B.5
Students will use the standard algorithm to multiply 2-digit numbers
by 2-digit numbers.
Vocabulary
PRIOR KNOWLEDGE REQUIRED
area
distributive property
multiples
standard algorithm
Can use the standard algorithm to multiply 2-digit numbers by
1-digit numbers
Can use the standard algorithm to multiply 2-digit numbers by
2-digit numbers
Can multiply 2-digit numbers by 2-digit multiples of 10
MATERIALS
BLM 1 cm Grid Paper (p. I-1)
Introduce multiplying 3-digit numbers by 2-digit numbers. Write on
the board:
456 × 34
ASK: How is this multiplication different from any we have done so far?
(one of the numbers has 3 digits and the other has 2 digits)
Splitting a problem into easier problems. Tell students that you would like
to find a way to split the problem into two easier problems, both of which
they already know how to do. Have students list all the types of problems
they know how to do that might be helpful:
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
• Multiply a 1-digit number by a 1-digit number.
• Multiply a 2-digit number by a 1-digit number.
• Multiply a 3-digit number by a 1-digit number.
• Multiply a 2-digit number by a 2-digit multiple of 10.
ASK: How can we break up the 2-digit number so it involves a multiple
of 10? (34 = 30 + 4)
Using the distributive property to multiply 2-digit numbers by 2-digit
numbers in separate steps.
Draw on the board:
34
456
Number and Operations in Base Ten 5-24
D-45
ASK: What multiplication statement gives the area? (456 × 34) SAY: Let’s
break up 34 into 30 and 4.
Draw on the board:
30
4
456
34
ASK: What multiplication statement gives the area of the rectangle on the
left? (456 × 30) ASK: What multiplication statement gives the area of the
rectangle on the right? (456 × 4) Write on the board and SAY: Since the
total area of the two small rectangles is the same as the large rectangle,
we have:
456 × 34 =
456 × (30 + 4)
= (456 × 30) + (456 × 4)
Multiplying by a multiple of 10. Remind students that multiplying by a
multiple of 10 requires you to add a 0.
Write on the board:
3 × 4 = 12
7 × 5 = 35
9 × 6 = 54
so 3 × 40 =
so 7 × 50 =
so 9 × 60 =
Ask for volunteers to fill in the answers. (120, 350, 540)
Challenge:
71 × 5 = 355
234 × 3 = 702
so 71 × 50 =
so 234 × 30 =
Write on the board:
456 × 34 =
456 × (30 + 4)
= (456 × 30) + (456 × 4)
SAY: We know how to use a grid to multiply each of these.
456 × 4
2
2
4 5 6
4
×
1 8 2 4
D-46
456 × 30
1
1
4 5 6
3 0
×
1 3 6 8 0
add a 0 because we
are multiplying by 10
Teacher’s Guide for AP Book 5.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: 3,550, 7,020
SAY: We’ve calculated 456 × 30 and 456 × 4. All that’s left to do is to
add the two parts:
456 × 4
2
(456 × 30) + (456 × 4)
456 × 30
2
1
4 5 6
4
×
1 8 2 4
1
1
4 5 6
3 0
×
1 3 6 8 0
1
2
2
4 5
3
1 8 2
1 3 6 8
1 5 5 0
×
6
4
4
0
4
Exercises: Multiply using three separate steps.
a)372 × 65
b) 729 × 48
c) 637 × 49
Answers: a) 24,180, b) 34,992, c) 31,213
Multiplying 3-digit numbers by 2-digit numbers using one grid.
SAY: To save paper, combine the grids we’ve used into one grid.
Write on the board: Erase or cover the tens unit
in 34 so that it looks like:
×
4 5 6
3 4
×
4 5 6
4
SAY: Perform the multiplication 456 × 4 first, as though we were only
multiplying a 3-digit number by a 1-digit number.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
2
×
2
4 5 6
4
1 8 2 4
Number and Operations in Base Ten 5-24
D-47
Replace the 3 you previously erased, and temporarily replace the ones digit
in 34 with a 0. SAY: Now we multiply 456 × 30.
2
×
2
4 5 6
3 0
1 8 2 4
0
add a 0 because we
are multiplying by 10
SAY: To avoid confusion, cross out the regrouping we did when we
multiplied 456 × 4. Remember to put the new regrouping starting in the
hundreds column.
1
1
2
1
2
4 5
3
1 8 2
1 3 6 8
×
1
2
6
0
4
0
4 5
3
1 8 2
1 3 6 8
1 5 5 0
×
2
6
4
4
0
4
SAY: All that’s left to do is to add the answers from 456 × 4 and 456 × 30.
Do this in the same grid. Be sure to replace the 4 in 34.
Exercises
1.Distribute BLM 1 cm Grid Paper. Use one grid to multiply.
a)374 × 21
b) 486 × 73
c) 539 × 37
Answers: a) 7,854, b) 35,478, c) 19,943
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
2.Check the reasonableness of your answers by rounding the first factor
to the nearest hundred, and the second factor to the nearest ten
(example: 374 × 21 ≈ 400 × 20 ≈ 8,000).
D-48
Teacher’s Guide for AP Book 5.1
Extensions
1.Multiply a 3-digit number by a 2-digit number using the rectangle
method. For example:
456
400 + 50 + 6
× 34 30 + 4
24 200 1,600 180 1,500 +
12,000 15,504
4×6
4 × 50
4 × 400
30 × 6
30 × 50
30 × 400
Have students use this method to check their answers from the
previous Exercises multiplying 3-digit numbers by 2-digit numbers.
(MP.1)
2. Super Challenge: Find the missing numbers.
a)
b)
4
2
9 6 8
0
×
4 7
×
2
2
1
1 6
Answers
4
2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
1
Number and Operations in Base Ten 5-24
4 8
5
×
9 6
2 4 2 0
2 5 1 6
1
3
1
5
4
3 2
2
4
×
8
2 2 9
0
1 3 1 2
8 1 5 4 1
8
7
6
0
6
D-49
NBT5-25 Sequences and Multiplication (Advanced)
Pages 62–63
STANDARDS
5.OA.B.3
Vocabulary
pattern rule
sequence
term
T-table
Goals
Students will compare number sequences using T-tables.
PRIOR KNOWLEDGE REQUIRED
Can add, subtract, and multiply whole numbers
Can extend a pattern given a verbal rule
Is familiar with T-tables
Introduce extending sequences by adding, subtracting, or multiplying
the same number. Remind students that when they have a rule for a
sequence, and they know the first number, they can extend the sequence.
For example, if the rule says “Add 5,” and the first number is 1, they get a
sequence by adding 5 to the previous number to get each new number: 1,
6, 11, 16, and so on. Remind students as well that numbers in a sequence
are called terms.
Exercises: Extend the sequence to four terms.
a)
b)
c)
d)
e)
Add 4: 3,
,
Subtract 5: 76,
Multiply by 2: 1,
Multiply by 3: 2,
Multiply by 2: 5,
,
,
,
,
,
,
,
,
,
Answers: a) 7, 11, 15; b) 71, 66, 61; c) 2, 4, 8; d) 6, 18, 54; e) 10, 20, 40
Extending sequences in a T-table. Remind students that they can use
T-tables to compare sequences. Remind them that when they extend
sequences in a T-table, they work with each column separately. Students
who are having trouble focusing on one column can cover the second
column with a sheet of paper.
Exercises: Extend the sequences in the T-table to four terms each.
a)
Add 3
6
Multiply by 2 b)Multiply by 10
3
7
Subtract 10
75
Comparing sequences made by addition in a T-table. Remind students
that they used T-tables with two sequences to find a rule to get the numbers
in the second sequence from the numbers in the first sequence. Show the
table below as an example. ASK: Is there a number you could add to the
D-50
Teacher’s Guide for AP Book 5.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Remind students that two sequences created using the same operation
(example: multiplying by 3) are not necessarily the same sequence. If
the sequences start with different numbers, they will not give the same
sequence of numbers. ASK: Are there any sequences like that in the
exercises you’ve just done? (yes, c) and e))
numbers in the first sequence to get the numbers in the second sequence?
(no) Is there a number that you could subtract? (no) How could you get
from 5 to 10? (add 5 or multiply by 2) Will adding 5 help with the first or the
third row? (no) Will multiplying by 2 help with the third row? (yes) the fourth
row? (yes) Let’s look at the first row. The number in the first sequence is
zero. How much is 0 × 2? (0) Does the rule work? (yes)
(MP.7)
Add 5
Add 10
0
0
5
10
10
20
15
30
Exercises: Extend the sequences. What is the rule for multiplying the
numbers in the first sequence to get the numbers in the second sequence?
a)
Add 2
Add 8
b)
0
0
c)
Add 4
Add 8
0
0
d)
Add 2
Add 10
0
0
Add 3
Add 12
0
0
Answers: a) multiply by 4, b) multiply by 5, c) multiply by 2, d) multiply by 4
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
(MP.7)
Ask students to look at the tables they worked with. Point out that the
sequences were all very special: they all were made by addition, and
they all started at 0. Ask students to use these very special rules for two
sequences to try to figure out a way to tell what number to multiply by.
Students should notice that the numbers they add to make the sequence
(the difference in each sequence) are also connected by the same rule.
Present the tables below one after the other and have students guess the
rule. Then have them extend the tables to three terms and check whether
the rule they predicted works. (Note that it will not work in d).)
a)
Add 2
Add 6
b)
2
6
c)
Add 5
Add 10
d)
1
2
(MP.6)
Add 3
Add 9
3
9
Add 3
Add 6
1
3
ASK: Is there a table where the rule you predicted did not work? (yes, d))
What rule could you predict from the differences? (multiply by 3 for a)
and b), and multiply by 2 for c)) Did the rule work for any of the rows? (no)
Number and Operations in Base Ten 5-25
D-51
Did it work for the first row? (no) Explain that it is not enough to predict the
rule from the difference; it should also work for the first row to work for the
other rows.
Multiply by 2
Multiply by 2
1
3
2
6
4
12
8
24
Comparing pairs of sequences made by multiplying by the same
number. Present the T-table in the margin and have students help you fill in
the numbers.
Ask students to try to find a rule to get from the first sequence to the
second sequence. PROMPT: Is there a number you could add? subtract?
multiply by? (multiply by 3) Repeat with the tables below.
Exercises: What is the rule for how you get the numbers in the second
sequence from the first sequence?
a)
Multiply by 3
Multiply by 3 b)Multiply by 10 Multiply by 10
2
1
2
12
Answers: a) multiply by 2, b) multiply by 6
ASK: Could you tell from the topmost row in the table how the sequences
are connected? (no) Why not? (the instructions are the same) Could you
tell that from the second row in the table? (yes, the number you multiply
the first term in the first sequence by to get the first term in the second
sequence is the number you multiply by in every row)
Ask students to think why the rule to get from one sequence to the other
works. To prompt students to see the connection, have them think how they
get from one row to the next, then to the next:
Multiply by 2 Multiply by 2
×2
×2
Multiply by 2 Multiply by 2
1
5
512
(MP.8, MP.3)
D-52
1
×3
3
2
×?
6
4
×?
12
8
×?
24
×2
×2
×2
To get from 1 to 24, you can either multiply by 3 to get to the first number
in the second sequence, then go down the second sequence, and multiply
by 2 three times. Or you can go down the first sequence, and multiply by
2 three times, then multiply by some number to get 24. What should that
number be? (3) Why? (order does not matter in multiplication, 1 × 3 × 2 ×
2 × 2 = 1 × 2 × 2 × 2 × 3)
Predicting terms from one sequence to the other. Draw the table in the
margin. Tell students you have a riddle for them: you extended one of the
sequences for a few terms, and you want them to figure out what the term
would be in the same row in the second sequence. Ask students to explain
how they found the answer. Students need to figure out the rule to get from
one sequence to the other, then to use the rule to get the number. In the
Teacher’s Guide for AP Book 5.1
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×2
example in the margin, the rule is “Multiply by 5” and the missing number is
512 × 5 = 2,560.
Exercise: The rows between the first and the last are hidden! What is the
missing number in the table?
a)
Multiply by 2
Multiply by 2
1
3
b) Multiply by 10 Multiply by 10
6
1,024
24
6,000,000
Answers: a) 3,072, b) 24,000,000
Point out that it was very hard to check the answers in the Exercises
above, since there are very many terms hidden in between the first and
the last rows.
ACTIVITY
Students play in pairs. Players draw a T-table with a pair of sequences
made by multiplying by the same number. They extend the sequences
to five terms. Then each player makes a riddle for the other player as in
the last Exercise above. The other player has to figure out the rule and
find the missing term. The first player then checks the answer.
Extensions
(MP.3)
1. Correct the mistakes in the tables.
a)Multiply by 5 Multiply by 5 b)
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Multiply by 4
Multiply by 4
1
3
2
40
15,625
48,000
32,768
655,350
Answers: a) change 48,000 to 46,875, b) change 655,350 to 655,360
(MP.7, MP.8)
2.In later grades, students will need to create formulas connecting two
varying quantities. A simple case of this is a formula for sequence: a
rule to get a term in the sequence from the term number. Students can
use BLM Finding Rules for Sequences (pp. D-74–77) to investigate
ways to create a rule for a sequence.
Number and Operations in Base Ten 5-25
D-53
NBT5-26 Word Problems II
Page 64
STANDARDS
5.NBT.B.5
Goals
Students will solve word problems involving multiplication that may
require multiplying a multi-digit number by a 2-digit number.
PRIOR KNOWLEDGE REQUIRED
Can multiply a multi-digit number by a 1-digit number using the
standard algorithm
Extensions
1. Light travels at the speed of approximately 186,000 miles per second.
a) How far does it travel in 1 minute?
b) How far does it travel in 1 hour?
Answers: a) 11,160,000 miles, b) 66,960,000 miles
2.A passenger jet travels from New York to London, England. It travels
620 miles per hour.
a)London is approximately 5,580 miles from New York. Will the plane
make it to London in 9 hours?
b)The Concorde supersonic jet was able to travel 1,395 miles per
hour for the same trip. How much time could be saved by flying
on the Concorde?
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Answers: a) yes, b) the trip took 4 hours, so it saved 5 hours
D-54
Teacher’s Guide for AP Book 5.1