NBT4-28 Multiplying by Adding On
Pages 90–91
STANDARDS
3.OA.B.5, preparation for
4.NBT.B.5
Goals
Students will find products by adding on to smaller products.
PRIOR KNOWLEDGE REQUIRED
Vocabulary
adding on
brackets
Can add and multiply
Can represent multiplication different ways
Brackets tell us what to do first. Tell students that expressions can
include more than one operation. Write on the board:
3×5+2
Point out that in this expression we have two operations, multiplication and
addition, so we have to pick one to do first. Tell students that brackets tell
us what to do first. Write on the board:
(3 × 5) + 2
3 × (5 + 2)
Calculate each expression, describing aloud what you do:
(3 × 5) + 2 = 15 + 2
3 × (5 + 2) = 3 × 7
= 17
= 21
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Point out that we get different answers depending on which operation
we do first.
NOTE: Students who don’t
have their multiplication
facts memorized and who
need help multiplying can
refer to the multiplication
chart on BLM Using the
Multiplication Chart to
Multiply (p. B-45) or BLM
9 × 9 Multiplication
Chart (p. I-3).
Exercises: Have students do the operation in brackets first and then
calculate each expression.
a) (2 + 3) × 4 and 2 + (3 × 4)
b) 7 − (3 × 2) and (7 − 3) × 2
Answers
a) 5 × 4 = 20 and 2 + 12 = 14, b) 7 − 6 = 1 and 4 × 2 = 8
Point out the connection between both expressions in part a) and between
both expressions in part b): The numbers and operations are all the same,
but the answers are different because the brackets, which are in different
positions, tell you which operation to do first.
Review different representations for multiplication. Draw the following
three pictures on the board:
012345678910
Number and Operations in Base Ten 4-28
E-1
Have students write a multiplication equation for each picture. Then have
them draw their own pictures and invite partners to write multiplication
equations for each other’s pictures. Illustrate the various ways of representing
multiplication equations by having volunteers share their pictures.
Adding to get a larger product. Draw 2 rows of 4 dots on the board:
ASK: What multiplication equation do you see? (2 × 4 = 8) (Prompt as
needed: How many rows are there? How many in each row? How many
altogether?) What happens when I add a row? Which numbers change?
Which number stays the same? (The multiplication sentence becomes
3 × 4 = 12; we added a row, so there are 3 rows.) How many dots did we
add? Invite a volunteer to write an equation that shows how 2 × 4 becomes
3 × 4 when you add 4. Answer: (2 × 4) + 4 = 3 × 4.
Look back at the number line you drew above:
012345678910
ASK: What multiplication equation do you see? What addition equation do
you see? (Prompt as needed: Which number is repeated? How many times
is it repeated?) Ensure that all students see 5 × 2 = 10 in this picture. Then
extend the line and add another arrow:
012345678910
11
12
ASK: Now what multiplication equation do you see? What are we adding to
5 × 2 to get 6 × 2? Have a volunteer show how to write this as an equation.
Answer: (5 × 2) + 2 = 6 × 2
Find a product by adding on to a smaller product using arrays. Have
students use arrays to practice representing products as smaller products
and sums. Begin by providing an array with blanks, and have volunteers
come up and fill in the blanks, as shown here.
3×5
4×5
+
5
Have students draw an array (or use counters) to show that:
a)3 × 6 = (2 × 6) + 6 b) 5 × 3 = (4 × 3) + 3 c) 3 × 8 = (2 × 8) + 8
E-2
Teacher’s Guide for AP Book 4.1
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Repeat using the 4 sets of 3 hearts and adding another set of 3 hearts.
Find a product by adding on to a smaller product without using arrays.
Have students do the following questions without using arrays:
a)
c)
e)
g)
If 10 × 2 = 20, what is 11 × 2?
If 11 × 5 = 55, what is 12 × 5?
If 6 × 3 = 18, what is 7 × 3?
If 2 × 7 = 14, what is 3 × 7?
b) If 5 × 4 = 20, what is 6 × 4?
d) If 8 × 4 = 32, what is 9 × 4?
f) If 8 × 2 = 16, what is 9 × 2?
Finally, have students turn products into a smaller product and a sum
without using arrays. Begin by giving students statements with some of
the blanks filled in, then move to statements where students must fill in
all the blanks themselves.
a)5 × 8 = 4 × 8 +
c)7 × 4 =
×
+
b)9 × 4 =
×4+
Extensions
1.Tell students that when two operations in an equation are both
subtraction, doing them in different orders can produce different
answers. (When the two operations are both addition or multiplication,
the answers are the same.) Write on the board:
(8 − 3) − 2
8 − (3 − 2)
Demonstrate getting different answers:
(8 − 3) − 2 = 5 − 2
8 − (3 − 2) = 8 − 1
= 3
=7
Exercises: Have students calculate these expressions.
a)(9 − 5) − 3 and 9 − (5 − 3)
b)10 − (7 − 2) and (10 − 7) − 2
Answers: a) 4 − 3 = 1 and 9 − 2 = 7, b) 10 − 5 = 5 and 3 − 2 = 1
2. Circle the correct statement.
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(2 × 5) + 5 = 2 × 6
or
(2 × 5) + 5 = 3 × 5
Answer: (2 × 5) + 5 = 3 × 5
3. a)For which expressions do you get the same answer whether
you add first or multiply first?
1 × 3 + 4
3 × 1 + 4
4 + 1 × 3
4+3×1
Answers
1 × 3 + 4 = 3 + 4 or 1 × 7, so they give the same answer.
3 × 1 + 4 = 3 + 4 or 3 × 5, so they give different answers.
4 + 1 × 3 = 4 + 3 or 5 × 3, so they give different answers.
4 + 3 × 1 = 4 + 3 or 7 × 1, so they give the same answer.
Number and Operations in Base Ten 4-28
E-3
b)Write another expression that produces the same answer whether
you add first or multiply first. Check your work.
Sample answers
1 × 4 + 9
8 + 3 × 1
1 × 7 + 6
8+9×1
NOTE: Although students do not need to articulate this, in order for
adding first or multiplying first to give the same answer, the 1 needs
to be on the left or on the right, not in the middle, and it needs to be
involved in multiplication only, not addition.
4.Sarah counts the dots below in different ways. Show Sarah’s groupings
and explain her thinking to fill in the blanks.
a)4 × 5 =
dots
b)(2 × 6) + (2 × 4) = 12 + 8 =
c)(6 × 6) − (4 × 4) = 36 − 16 =
dots
dots
Answers
a)b)c)
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Explanation for part c): There are 6 rows of 6 dots, and 4 rows of 4
white dots. Take away the white dots to get the number of black dots.
E-4
Teacher’s Guide for AP Book 4.1
NBT4-29 Multiplying by Tens, Hundreds, and
Pages 92–94
Thousands
STANDARDS
3.NBT.A.3, 4.NBT.A.1
Vocabulary
multiple
place value
Goals
Students will multiply by multiples of 10.
PRIOR KNOWLEDGE REQUIRED
Can represent numbers using base ten materials
Can multiply
MATERIALS
base ten blocks
Multiplying by 10. Give students 9 ones blocks, 9 tens blocks, and
9 hundreds blocks. Have them make the number 6 with their blocks.
Then ASK: What block is ten times as much as a ones block? (a tens
block) Explain that to make 10 × 6, they can use 6 tens blocks. Have
students use this method to find:
a)4 × 10
b) 3 × 10
c) 5 × 10
Have students show you a tens block. ASK: What block is ten times as
much as a tens block? (a hundreds block) What is ten times as much as
3 tens blocks? (3 hundreds blocks) Write on the board:
10 × =
So 10 × 30 = 300
Have students make each of the following numbers using base ten
blocks, and then make 10 times the number by replacing each block
with 10 times its value.
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a)
30b)
4 c)
34d)
20e)
27f)
81
Write on the board:
35 × 10 = 350
Pointing to 35, SAY: This is 3 tens and 5 ones. Pointing to 350, SAY: When
you multiply by 10, you get 3 hundreds and 5 tens; there are 0 ones.
Exercises: Have students use this method to find:
a)7 × 10
f)68 × 10
b) 8 × 10
g) 17 × 10
c) 4 × 10
h) 37 × 10
d) 9 × 10
i) 81 × 10
e) 12 × 10
j) 32 × 10
Bonus: 6,217 × 10
Using base ten blocks to represent products. ASK: What product does
each picture represent?
Number and Operations in Base Ten 4-29
E-5
(3 × 2)
(3 × 20)
Have a volunteer draw a picture for 3 × 200. Tell students that they can
just draw a square for a hundreds block; demonstrate by showing an
example on the board:
not
Ask students if they see a pattern in the multiplication equations that
correspond to these pictures:
(MP.7)
3×2=6
3 × 20 = 60
3 × 200 = 600
Point out that the answer is always obtained by finding 3 × 2 and then
adding the number of zeros in the second number.
(MP.1)
Multiplying numbers that end in one or more zeros. Write on the board:
30 × 200 =
10 × 200 =
ASK: Which of these problems is exactly like a problem you already know
how to do? (the second one) Have a volunteer solve 10 × 200. ASK: How
can you use the answer to the second problem to find the answer to the
first one? PROMPT: What would you multiply your answer by? (3) Draw the
following picture on the board to illustrate this:
200
10
30
10
Point out that each block is 10 × 200 and the 3 blocks together are 30 × 200.
Write on the board:
30 × 200 = 3 × (10 × 200)
= 3 × 2,000
ASK: Is this problem one you already know how to solve? (yes) Have a
volunteer solve it. SAY: Changing a problem you haven’t seen before into
a problem like many you have seen before is an important strategy for
solving math problems.
E-6
Teacher’s Guide for AP Book 4.1
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10
Exercises
a)40 × 800 = 4 × (10 × 800)
b) 30 × 400 = 3 × (10 × 400)
=4×
=3×
=
=
c)60 × 800 = 6 × (10 × 800)
=6×
=
Answers: a) 4 × 8,000 = 32,000, b) 3 × 4,000 = 12,000,
c) 6 × 8,000 = 48,000
Patterns when multiplying powers of 10. Write on the board:
1,000 × 10
(MP.7, MP.8)
ASK: How many zeros do you add to 1,000 when multiplying by 10? (one)
Write on the board:
1,000 × 10 = 10,000
SAY: There are 3 zeros in 1,000 and we add 1 zero when we multiply by 10,
so the product has 3 zeros + 1 zero = 4 zeros. Now write on the board:
1,000 × 100 = 1,000 × 10 × 10
Point out that since 100 is 10 × 10, students can look at multiplying by
100 as multiplying by 10 twice. ASK: We add 1 zero when multiplying by
10 once, so how many zeros would we add when multiplying by 10 twice?
(two) Point out that there are already 3 zeros in 1,000, and we added two
more, so there are 5 zeros in 1,000 × 100. Have students predict how many
zeros there are in 10,000 × 1,000. (4 + 3 = 7) Point out that there are already
4 zeros in 10,000 and we add a zero every time we multiply by 10. Show
this on the board as
10,000 × 10 × 10 × 10
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SAY: We add a zero three times, so there are 3 more zeros than in 10,000.
Write on the board:
10,000 × 1,000 = 10,000,000
4 zeros+3 zeros = 7 zeros
Exercises: Multiply.
a)100 × 1,000
d)100,000 × 1,000
b)1,000 × 10,000
c)10,000 × 100
Answers: a) 100,000, b) 10,000,000, c) 1,000,000, d) 100,000,000
Patterns when multiplying tens and hundreds. Write on the board:
20 × 700 = 2 × 10 × 7 × 100
Number and Operations in Base Ten 4-29
E-7
(MP.7)
Remind students that multiplying the same numbers in any order will get
the same answer, then rewrite the equation as follows and have volunteers
fill in the blanks:
= 2 × 7 × 10 × 100
=
=
×
ASK: How can you get 20 × 700 from knowing 2 × 7? (add 3 zeros to the
answer) How can you get 30 × 200 from knowing 3 × 2? (add 3 zeros to
the answer) Write on the board:
30 × 200 = 6,000
ASK: How can you get 400 × 7,000 from knowing 4 × 7? (add 5 zeros to
the answer) Demonstrate writing 28, adding 5 zeros, and then putting the
commas in the correct places:
400 × 7,000 = 2, 8 0 0, 0 0 0
Exercises: Multiply.
a)400 × 600
b) 300 × 7,000 c) 40 × 20,000 d) 3,000 × 2,000
Answers: a) 240,000, b) 2,100,000, c) 800,000, d) 6,000,000
Tell students that they can use this shortcut to multiply really large numbers.
Bonus: 600,000,000 × 4,000,000,000
Answer: 2,400,000,000,000,000,000
A special case. Write on the board:
ASK: What is 5 × 8? (40) How many zeros are in the question? (2) So how
many zeros do we add to 40 to get the answer? (2) Write on the board:
50 × 80 = 4,000. Point out that the answer has 3 zeros. It looks like there is
1 extra zero, but that’s just because 40 already has a zero, so when we add
2 more, we get 3 altogether. Tell students that in the next set of questions,
they will have to be careful when counting zeros. The product of the two
1-digit numbers might itself have a zero, but students still have to add all
the zeros from the question to that product.
Exercises: Multiply by adding the correct number of zeros to the product
of the 1-digit numbers.
a)3 × 8 =
b)4 × 9 =
c)5 × 6 =
d)5 × 4 =
e)2 × 7 =
f)6 × 4 =
g)2 × 5 =
E-8
so 300 × 800 =
so 40 × 9,000 =
so 50 × 600 =
so 500 × 4,000 =
so 200 × 70,000 =
so 600 × 40 =
so 200,000 × 500 =
Teacher’s Guide for AP Book 4.1
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50 × 80
Answers: a) 24 and 240,000, b) 36 and 360,000, c) 30 and 30,000,
d) 20 and 2,000,000, e) 14 and 14,000,000, f) 24 and 24,000,
g) 10 and 100,000,000
Estimating to check the reasonableness of answers on a calculator.
Write on the board:
324 × 7,136
Have students calculate the answer on a calculator. (2,312,064) If students
get different answers, write them all on the board. Tell students that people
sometimes punch numbers into their calculators incorrectly, and we
want a way to know if an answer is reasonable. Write a second problem
on the board:
300 × 7,000
ASK: Which problem is easier to do without a calculator? (the second
one) Will the answers be close? (yes) How do you know? (the numbers
are close to each other—324 is close to 300 and 7,136 is close to 7,000)
ASK: Which answer will be larger? (the first one) How do you know? (both
numbers are larger—324 is larger than 300 and 7,136 is larger than 7,000)
SAY: Calculating 300 × 7,000 is a good way to check if your answer to
324 × 7,136 is reasonable. It is easier to calculate by hand, the answer will
be close, and you even know which answer should be larger. Have students
calculate 300 × 7,000 by hand. They should get 2,100,000. ASK: Does the
answer you got on the calculator make sense? (yes, 2,312,064 is close to,
but larger than, 2,100,000)
Exercises: Have students use a calculator to answer these questions
and multiply the same numbers rounded to the leading place value to
check their answers.
a)2,985 × 68
b) 4,012 × 312
c) 297 × 8,888
d) 532 × 8,102
Extensions
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1. Fill in the missing number.
a)
× 300 = 90,000
c)500 ×
= 15,000
b)
× 20 = 60,000
Answers: a) 300, b) 3,000, c) 30
2. Find as many answers as you can to
×
= 80,000.
Sample answers: 80,000 × 1, 8,000 × 10, 800 × 100, 80 × 1,000,
8 × 10,000, 4 × 20,000, 40 × 2,000, 400 × 200, 4,000 × 20,
and 40,000 × 2
Number and Operations in Base Ten 4-29
E-9
NBT4-30 Mental Math
Pages 95–96
STANDARDS
4.NBT.B.5
Vocabulary
array
Goals
Students will use arrays to understand the distributive property. They
will multiply large numbers by breaking them into smaller numbers.
PRIOR KNOWLEDGE REQUIRED
Can apply the distributive property
Can represent multiplication problems using arrays
Can represent numbers using base ten materials
MATERIALS
two colors of base ten blocks (different colors for tens and ones blocks)
Review writing multiplication expressions from arrays. Draw the
rectangles below on the board. Ask students what multiplication expression
(number of rows × number in each row) they see in each picture:
(2 × 5)
(3 × 3)
(3 × 5)
(3 × 10)
(5 × 3)
Now ask students to identify the multiplication expression for the whole
diagram (3 × 13 since there are 3 rows and 13 in each row). ASK: How can
we get 3 × 13 from 3 × 10 and 3 × 3? What operation do we have to use?
How do you know? (You have to add because you want the total number
of squares.) Then write on the board:
(3 × 10) + (3 × 3) = 3 × 13
Repeat with several examples, allowing students to write the final equation
that combines multiplication and addition. Include examples where one part
of the array is 20 squares long instead of 10. When the array is 20 squares
long, demonstrate how to count the squares across by marking every fifth
square. Since 20 objects are harder to recognize as being 20 than 5 objects
are to recognize as being 5, we are turning a harder problem into an easier
one by marking every fifth square.
E-10
Teacher’s Guide for AP Book 4.1
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Writing multiplication equations for arrays divided into two parts. Then
do the same thing for each part of this diagram:
(3 × 10)
(3 × 3)
(MP.2)
Modeling arrays and products using base ten blocks. Students can
make models using base ten blocks to show how to break a product into
the sum of two smaller products. AP Book 4.1 p. 108 Question 2. a) shows
that 3 × 24 = (3 × 20) + (3 × 4). Ask students to make a similar model to
show that 4 × 25 = (4 × 20) + (4 × 5).
Step 1: M
ake a model of the number 25 using colored base ten blocks
(one color for the tens blocks and one color for the ones):
2 tens
5 ones
Step 2: Extend the model to show 4 × 25:
4 × 25
Step 3: B
reak the array into two separate arrays to show that
4 × 25 = (4 × 20) + (4 × 5).
4 × 20
4× 5
Deciding how to split a product into smaller products. ASK: How can we
use an array to show 3 × 12? Have a volunteer draw it on the board. Then
ask if anyone sees a way to split the array into two smaller rectangular arrays.
Which number should be split, the 3 or the 12? SAY: Let’s split 12 because 3
is already small enough. Write on the board:
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12 =
+
ASK: What’s a nice round number that is close to 12 and is easy to multiply
by 3? (10) Fill in the blanks (12 = 10 + 2), then have a volunteer split the
array. ASK: What is 3 × 10? What is 3 × 2? What is 3 × 12? How did you
get 3 × 12 from 3 × 10 and 3 × 2? Have a volunteer write the equation that
shows this on the board: 3 × 12 = (3 × 10) + (3 × 2).
Splitting a product into smaller products using arrays. ASK: Did we
need to draw the arrays to know how to split the number 12? Ask students
to split the following products in their notebooks without drawing arrays.
Students should split the 2-digit numbers in each product into tens and
ones (for example, to multiply 6 × 21, split 21 into 20 and 1).
a)2 × 24 b) 3 × 13 c) 4 × 12 d) 6 × 21 e) 9 × 31 f) 4 × 22
When students are comfortable splitting products into the sum of two
smaller products, have them solve each problem.
Number and Operations in Base Ten 4-30
E-11
Bonus
These problems require regrouping:
a)2 × 27
b) 3 × 14
c) 7 × 15
(MP.3)
d) 6 × 33
e) 8 × 16
Multiplying 3-digit numbers by 1-digit numbers mentally. ASK: To find
2 × 324, how can we split 324 into smaller numbers that are easy to multiply
by 2? How would we split 24 if we wanted 2 × 24? We split 24 into 2 tens
and 4 ones. What should we split 324 into? (324 = 300 + 20 + 4) So we
can double each part separately.
2 × 324
=(2 × 300) + (2 × 20) + (2 × 4)
Have students multiply more 3-digit numbers by 1-digit numbers by
expanding the larger number and applying the distributive property.
(Exercises: 4 × 221,3 × 123, 3 × 301) Students should record their
answers and the corresponding base ten models in their notebooks.
Then have students solve additional problems without drawing models.
Finally, have students solve problems in their heads. Only include
problems that do not require regrouping.
Bonus
a)3 × 412
b)2 × 31,421 c)3 × 311,213 d)3 × 221,312
1.Using the 10 times table to multiply. Have students use the 10 times
table to multiply by 11 and 12.
For example:
12 × 7 = (10 × 7) + (2 × 7)
= 70 + 14
= 84
Students can also use the 10 times table to multiply by 9. For example:
10 sevens
− 1 seven
9 sevens
Since 9 × 7 is one less 7 than 10 × 7, we can find the former by
subtracting 7 from the latter: 9 × 7 = (10 × 7) − 7 = 70 − 7 = 63. Have
students complete BLM Using the 10 Times Table to Multiply (p. E-45).
E-12
Teacher’s Guide for AP Book 4.1
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Extensions
2.Have students combine what they learned in this lesson with what they
learned about multiplying by multiples of 10 in Lesson NBT4-29. Ask
them to multiply 3-digit numbers by multiples of 1,000 or 10,000.
a)342 × 2,000 b) 320 × 6,000 c) 324 × 2,000 d) 623 × 20,000
3.Have students do Questions 1 and 2 on BLM Using Area to Find
Equal Products (p. E-46). Students will discover that multiplying one
factor in a product by 2 and dividing the other factor by 2 results in the
same answer. They do this by cutting rectangles in half and gluing them
together again a different way:
10
10
10
3
10
3
3
3
(MP.7, MP.2)
So 6 × 10 = 3 × 20
6 ÷ 2 10 × 2
Point out that this can be useful for finding products because
sometimes doubling one factor and halving the other makes the
product easier to find. Have students use this method to multiply
even numbers by 5.
a)16 × 5 =
× 10 =
b) 24 × 5 =
× 10 =
Answers: a) 80, b) 120
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Students can also do BLM Moving Rectangles Around (pp. E-47–E-48)
for further practice. This BLM requires students to imagine cutting and
moving rectangles instead of actually cutting and moving them.
As a further extension, teach students that multiplying one factor in a
product by 3 and dividing the other factor by 3 also results in the same
answer. They can think of this as cutting rectangles into 3 equal parts
instead of 2 equal parts and putting them together another way (side
by side instead of one above the other, for example). If they want to
turn a rectangle into a new rectangle 3 times as wide, they have to cut
the rectangle into 3 parts, all the same height. So they are dividing the
height by 3. Then have students find as many products as they can that
are equal to 6 × 12, using this method of multiplying one factor and
dividing the other factor by the same number. (3 × 24, 2 × 36, 1 × 72,
12 × 6, 18 × 4, 36 × 2, 72 ×1)
Number and Operations in Base Ten 4-30
E-13
NBT4-31 Using Doubles to Multiply
Pages 97–98
STANDARDS
3.OA.B.5, preparation for
4.NBT.B.5
Goals
Students will use doubles and doubling to multiply mentally.
PRIOR KNOWLEDGE REQUIRED
Vocabulary
double
Can skip count by 2s
Can double 1-digit numbers
Understands the relationship between skip counting and multiplying
Review skip counting by 2s to double. Remind students that to double a
number means to add it to itself. Point out that students can skip count by
2s to double. Draw on the board:
7
+7
2 4681012
14
14
Have students find the double of 9 by skip counting by 2s.
Review doubling 2-digit numbers with ones digit less than 5. Remind
students that they can double 2-digit numbers by doubling the digits
separately. Write on the board:
13
+ 13
26
Point out that the 2 tens in 26 are double the 1 ten in 13 and the 6 ones in
26 are double the 3 ones in 13. Exercises: Double these numbers.
a)23b)
14c)31d)
24 e)52f)34g)
63h)
54
Using doubles to multiply. SAY: We know that 2 × 7 is 14. What is 4 × 7?
(28) Point out that 4 is double 2, so 4 sevens is double 2 sevens. Write on
the board:
3 × 7 is
So 6 × 7 is
12 × 7 is
24 × 7 is
Have volunteers answer each question successively (21,42, 84, 168).
Point out that the first number doubles each time, so the product does
as well. Whenever either number in the multiplication expression doubles,
so does the product.
E-14
Teacher’s Guide for AP Book 4.1
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Answers: a) 46, b) 28, c) 62, d) 48, e) 104, f) 68, g) 126, h) 108
Exercises
a)2 × 6 is
b)3 × 8 is
c)4 × 8 is
d)8 × 8 is
e)9 × 8 is
so 4 × 6 is
so 6 × 8 is
so 8 × 8 is
so 16 × 8 is
so 9 × 16 is
Bonus: 12 × 12 is 144 so 12 × 24 is
Review doubling 2-digit numbers with ones digit 5 or more. When the
ones digit is 5 or more, regrouping is required. Write on the board:
36 =
So the double of 36 is
30
+
+
6
=
To fill in the blanks, ASK: What is double 30? (60) What is double 6? (12)
So what is double 36? (60 + 12 = 72)
Continue using doubles to multiply. Write on the board:
3 × 9 is
So 6 × 9 is
12 × 9 is
24 × 9 is
Have volunteers answer each question successively (27, 54, 108, 216).
Exercises: Use doubling to solve the problems.
a)12 × 3 is
so 12 × 24 is
so 12 × 6 is
so 12 × 12 is
b)4 × 9 is
so 16 × 18 is
so 8 × 9 is
so 16 × 9 is
c)4 × 7 is
so 16 × 14 is
so 8 × 7 is
so 8 × 14 is
so 16 × 28 is
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Answers: a) 36, 72, 144, 288, b) 36, 72, 144, 288, c) 28, 56, 112, 224, 448
Bonus: Find ...
a)8 × 13 using 2 × 13
c)8 × 12 using 2 × 12
b) 8 × 25 using 2 × 25
Answers: a) 2 × 13 = 26 so 4 × 13 = 52 and 8 × 13 = 104, b) 50, 100,
200, c) 24, 48, 96
Extensions
1.Teach students to use triples to multiply. For example, 3 × 7 is 21 and
9 sevens is triple 3 sevens, so 9 sevens is 63. Students can complete
BLM Using Triples to Multiply (p. E-49).
Number and Operations in Base Ten 4-31
E-15
2.Remind students that 5 times a number is half of 10 times the
number. Have students multiply 5 by 2-digit numbers, using the
following progression.
• Use 2-digit numbers where both digits are even.
Example: 5 × 48 is half of 480, so 5 × 48 = 240
• Use 2-digit numbers where the ones digit is odd and
the tens digit is even.
Example: 5 × 87 is half of 870, so 5 × 87 = 435
• Use 2-digit numbers where both digits are odd.
Example: 5 × 39 is half of 390, and 390 = 300 + 90,
so half is 150 + 45 = 195
3.Tell students that, to make sure their answers to AP Book 4.1 p. 98
Question 7 are correct, their answer to the top row (parts a), b), and c))
should be 160 less than their answers to the bottom row (parts d), e),
and f)) directly underneath. For example, the answer to part a) plus 160
should equal the answer to part d). Challenge students to figure out why!
Answer: Use the distributive property.
For example, (16 × 3) + (16 × 10) = 16 × 13.
4. Fill in the missing number.
4 × 8 × 16 =
× 16 × 16
13 × 1 = 13
13 × 2 =
13 × 4 =
13 × 8 =
13 × 16 =
13 × 32 =
13 × 64 =
Have students double successively to finish all these questions. Then
tell students that you want to know 13 × 3. ASK: Which two answers
can I add to find 13 × 3? (3 thirteens = 2 thirteens + 1 thirteen =
26 + 13 = 39) Which two answers can I add to find 13 × 5? (13 × 1
and 13 × 4) Have students do the addition to find 13 × 5. (13 + 52 =
65) Repeat for 13 × 6, 13 × 10, 13 × 17, and 13 × 20.
Then tell students that you want to multiply 13 × 7. ASK: Which three
answers can you add to find 13 × 7? (4 + 2 + 1 is 7, so add 13 × 4,
13 × 2, and 13 × 1) Have students do the addition. (52 + 26 + 13 = 91)
E-16
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
5.Show students how to multiply using the method of the ancient
Egyptians. Explain that the Egyptians knew how to double numbers
and how to add numbers, and this allowed them to multiply any two
numbers they wanted! Write on the board:
ASK: How can I choose three numbers so that they add to 13 × 11?
(use 8 + 2 + 1 = 11, so add 13 × 8, 13 × 2, and 13 × 1) Have students
calculate 13 × 11 using this method. (104 + 26 + 13 = 143)
Ask students to solve the following products independently. This
time, they will have to decide which numbers to add together, with
no prompts from you.
a)13 × 9
b) 13 × 13
c) 13 × 21
d) 13 × 18
Bonus
e)13 × 35
f) 13 × 71
g) 13 × 96
h) 13 × 41
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Have students check their answers with a calculator.
Number and Operations in Base Ten 4-31
E-17
NBT4-32 Standard Method for Multiplication
Page 99
(No Regrouping)
Goals
STANDARDS
4.NBT.B.5
Students will multiply 2-digit numbers by 1-digit numbers using the
standard algorithm (without regrouping ones).
Vocabulary
standard algorithm
PRIOR KNOWLEDGE REQUIRED
Can break 2-digit numbers into a multiple of 10 and a 1-digit number
Can multiply
Review splitting products apart to multiply. SAY: I want to find 42 × 3?
How can I split the 42 to make this easier? (42 = 40 + 2) Why does this
make it easier? (Because I know 4 × 3 is 12; to find 40 × 3, I just add a 0
because 40 × 3 is 4 tens × 3, which is 12 tens.)
Introduce a modified version of the standard algorithm through arrays.
Put the following base ten model on the board for reference, then show
students this modification of the standard method for multiplying:
40 × 3
+
2×3
42
×3
2 × 3 =6
40 × 3 =120
126
ASK: How are the two ways of writing and solving 42 × 3 the same and
how are they different? Put up the following incorrect notation and ask
students to identify the error. (the 6 resulting from 3 × 2 is aligned with
the tens, not the ones)
42
×3
2×3= 6
40 × 3 =120
186
ASK: Why is it important to line up the ones digit with the ones digit and the
tens digit with the tens digit? Emphasize that the incorrect alignment adds
60 and 120 but 2 × 3 is only 6, not 60.
Point out that if students make this mistake, there is an easy way to tell that
the answer is wrong. ASK: What is 50 × 3? (150) Can 42 × 3 be more than
50 × 3? (no) So 42 × 3 cannot equal 186.
E-18
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
(MP.6, MP.3)
Practice using the modified standard algorithm without regrouping.
Have students practice multiplying more numbers using the modified
standard method. They can use grid paper to help them line the numbers
up, if necessary. Examples:
12322143
× 4
× 3
× 8
× 2
2×4=
10 × 4 =
A shortcut—the standard algorithm. Have volunteers write their answers
on the board. Then circle the 0 in the second row of each problem, where
the tens are being multiplied. ASK: Will there always be a 0 here? How do
you know? If there is always going to be a 0 here, is there a way we could
save time and space when we multiply? Suggest looking at the ones digit
of the answer. Where else do students see that digit? What about the other
digits? Where else do they see those? Show students how they can skip
the intermediate step and go straight to the last line, as follows:
42
× 3
126
2 × 3 ones = 6 ones
4 × 3 tens = 12 tens
12 tens + 6 ones = 120 + 6 = 126
Have students use this standard method, or algorithm, to multiply more
2-digit numbers by 1-digit numbers in their notebooks. They should use
arrows to label the number of ones and tens, as is done above.
Bonus: Provide problems with more digits, but still without regrouping.
(Examples: 321 × 4; 342 × 2; 4,221 × 3)
Extensions
1. Fill in the missing numbers.
a)b)c)
4 3
2
1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
×
×
6 4
2
0
4
2
×
3
9
6 2
6
Answers: a) 243 × 2 = 486, b) 132 × 3 = 396, c) 20,134 × 2 = 40,268
2. Fill in the missing numbers with the digits 0 to 5.
2
4
×
8
6
Answer: 5,243 × 2 = 10,486
Number and Operations in Base Ten 4-32
E-19
NBT4-33 Multiplication with Regrouping
Page 100
STANDARDS
4.NBT.B.5
Vocabulary
regrouping
standard algorithm
Goals
Students will multiply any 2-digit number by any 1-digit number using
the standard algorithm.
PRIOR KNOWLEDGE REQUIRED
Knows the standard algorithm for multiplication when no regrouping
is required
MATERIALS
BLM Multiplication—The Standard Algorithm (pp. E-50–E-51)
Review multiplication without regrouping. Remind students how
they used the standard algorithm (modified and unmodified) to multiply
numbers in the previous lesson:
Method 1:
Method 2:
42
× 3
2 × 3 =6
40 × 3 =120
126
42
× 3
126
Introduce problems with regrouping. Then have students do the following
questions using Method 1:
Have volunteers write their answers on the board. ASK: How are these
problems different from the problems we did before? (the product involving
the ones digit is more than 10, you can’t just write the answer as the product
involving the tens digit and then the product involving the ones digit)
Show students how to write out the regrouping as follows for the first
problem above:
2 × 7= 1 ten + 4 ones
+ 40 × 7=28 tens
42 × 7=29 tens + 4 ones
So 42 × 7 = 294
Exercises: Have students write out the regrouping for these questions
(that use only the 2, 3, 4, and 5 times tables):
E-20
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
42 35 28
× 7× 9× 4
2 × 7 =
5 × 9 =
8×4=
40 × 7 =
30 × 9 =
20 × 4 =
a) 2 × 7 =
+ 30 × 7 =
ten +
tens
32 × 7 =
tens + 4 ones
ones
So 32 × 7 =
b) 1 × 5 =
+ 40 × 5 =
ten +
tens
ones
41 × 5 =
tens +
ones
c) 7 × 3 =
+ 20 × 3 =
tens +
tens
ones
27 × 3 =
tens +
ones
ten +
tens
ones
tens +
ones
So 41 × 5 =
So 27 × 3 =
d) 8 × 4 =
+ 20 × 4 =
28 × 4 =
So 28 × 4 =
Bonus: Use questions that use the 6, 7, 8, and 9 times tables.
e)39 × 7
f) 46 × 8 g) 27 × 7
h) 46 × 9
ones
tens
Regrouping the ones using the standard algorithm. Point out that when
using regrouping to multiply, students multiplied the tens and ones separately,
and then they combined the results. Explain that when using the standard
algorithm for multiplication, we also multiply the tens and ones separately;
we multiply the ones first, then the tens. There is no need to explain why we
start with the ones—that explanation will come later. Write on the board:
1
2 ones × 7
×
7
= 14 ones
4
= 1 ten + 4 ones
3 2
×
8
×
8 6
5
×
ones
tens
ones
tens
2 7
Number and Operations in Base Ten 4-33
ones
tens
ones
Explain that we write the 1 above the tens column because it shows how
many tens there are from multiplying the ones. Have volunteers regroup
the ones in these problems.
tens
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4 2
3 4
5
×
3
E-21
2 3
×
2 7
2
×
ones
tens
7 8
3
ones
tens
5 4
8
×
ones
tens
ones
tens
Then have students individually copy the following problems onto grid
paper (demonstrate how to line up the digits in the first one, and where
to write the words “tens” and “ones”) and then regroup the ones.
×
3
ones
tens
hundreds
Regrouping the tens. Add a label for the hundreds column to the problem
already on the board:
1
4 2
×
2 ones × 7
7
= 14 ones
4
= 1 ten + 4 ones
SAY: The ones have already been regrouped; now we have to regroup
the tens too. ASK: How many tens are in the product? Write on the board:
4 tens × 7 = 28 tens
SAY: But we also have 1 ten already regrouped from the ones. Finish
writing out 42 × 7 in terms of tens and ones, explaining as you go
where you get each number.
1
2 ones × 7 = 14 ones
4 2
4 tens × 7 = 28 tens
7
42 × 7 = 28 tens + 1 ten + 4 ones
4
= 29 tens + 4 ones
Then finish writing the answer:
1
4
7
×
2
2
9
4
(7 × 4) + 1
E-22
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
×
= 1 ten + 4 ones
Have volunteers finish these problems:
a)
b)
4
2
5
3
9
×
c)
4
6
7
8
×
5
d)
1
2
8
(9 × 2) + 4
3
5
×
2
4
6
×
0
4
(8 × 3) + 4
Exercises: Have students copy these questions and finish the multiplication.
a)
b)
1
2
5
4
2
9
7
4
3
e)
3
2
8
f)
4
3
5
4
2
5
× 2
× 5
× 4
× 8
× 8
5
4
5
2
0
0
Some students may need to write the intermediate product instead
of holding it in their heads. Demonstrate as shown in the margin.
2
7
×
3
d)
1
× 3
28
+1
c)
1
Using the standard algorithm. Write problems like the following
on the board:
4
a)13 × 4 b) 35 × 3 c) 23 × 8 d) 46 × 3 e) 64 × 3 f) 32 × 5
g)23 × 7 h) 42 × 6 i) 38 × 3 j) 32 × 9 k) 37 × 5 l) 86 × 5
Demonstrate how to line up the numbers using a grid, and have students
do the problems on grid paper. BLM Multiplication—The Standard
Algorithm provides scaffolding for students who need it.
(MP.3)
Put the following problems on the board and ask students to identify which
one is wrong and to explain why.
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Method 1:
Method 2:
32
× 7
2 × 7 =14
30 × 7 = 210
= 224
Method 3:
32
× 7
2114
1
32
× 7
224
2 × 7 = 14
Point out that Method 2 would work if regrouping wasn’t required.
For example,
32
× 4
128
(MP.3)
It’s only because there are more than ten ones in 2 × 7 that Method 2 doesn’t
work. Method 2 makes it look like 3 tens × 7 is 21 hundreds, but in fact it is
only 21 tens. Discuss with students how they can see that 32 × 7 is not 2,114
Number and Operations in Base Ten 4-33
E-23
by using estimation. Explain to students that estimating is a good way to
check if their answer makes sense. ASK: What is a number close to 32 that
is easy to multiply by? (30 or 40) What is 30 × 7? (210) What is 40 × 7? (280)
ASK: Does it make sense that 32 × 7 is more than 40 × 7? (no) Emphasize
that even if they make this type of mistake, students can use estimating to
catch and correct it.
Extensions
1. Fill in the missing numbers.
a)b)c)
4
9 2
×
3
×
7
4
×
4
6
3
0
8
Answers: a) 42 × 9 = 378, b) 92 × 5 = 460, c) 44 × 7 = 308
2. Use the numbers from 0 to 5 to fill in the missing numbers.
a)
8
×
b)
4
×
2
5
2 2
Answers: a) 84 × 3 = 252, b) 42 × 5 = 210
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
3.
Make up a problem similar to the ones in Extension 1 and
ask a friend to solve it.
E-24
Teacher’s Guide for AP Book 4.1
NBT4-34 Multiplying with the 6, 7, 8, and 9
Page 101–102
Times Tables
Goals
STANDARDS
4.NBT.B.5
Students will multiply using the standard algorithm, including situations
that require using the 6, 7, 8, and 9 times tables.
Vocabulary
PRIOR KNOWLEDGE REQUIRED
standard algorithm
Can apply the standard algorithm in situations that require only
the 1, 2, 3, 4, and 5 times tables
Multiplication using the 6 times table. Review the 6 times table. Have
students determine the 6 times table from the 5 times table by adding the
number being multiplied by 6, and then from the 3 times table by doubling.
Explain that students should check that they get the same answer. Then
allow students to use the 6 times table to find the products of several 2-digit
numbers multiplied by 6. Examples: 38 × 6, 97 × 6, 84 × 6. Encourage
students to only peek at the 6 times table when they need to. Observe
students to see when they no longer need to look at the times table.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
(MP.7)
Multiplication using the 8 times table. Review the 8 times table. Have
students determine the 8 times table from the 2 times table by doubling
twice. Students should check their answers by ensuring that each number
is 8 more than the previous number in the 8 times table. Then allow
students to use the 8 times table to find the products of several 2-digit
numbers multiplied by 8. Examples: 37 × 8, 49 × 8, 56 × 8, 78 × 8, 85 × 8,
93 × 8. Encourage students to only peek at the 8 times table when they
need to. Observe students to see when they no longer need to look at the
times table.
Review a pattern in the 9 times table. If you haven’t already done so,
teach students the trick for multiplying any 1-digit number by 9: Get the
tens digit by subtracting 1 from the number being multiplied (Example: the
tens digit of 9 × 8 is 8 – 1 = 7), then subtract the tens digit of the answer
from 9 to get the ones digit. (See How to Learn Your Times Tables in a
Week in the Mental Mather section of this guide.) (Example: 9 – 7= 2,
so 9 × 8 = 72)
Practice using the 9 times table and the standard algorithm. Exercises:
Have students do these problems:
a)84 × 9
e)66 × 9
i)64 × 9 b) 78 × 9
f) 53 × 9
j) 93 × 9 c) 96 × 9
g) 89 × 9
k) 75 × 9
d) 77 × 9
h) 98 × 9
Solve problems using all the times tables learned to date. Exercises:
a)84 × 8
e)67 × 5
Number and Operations in Base Ten 4-34
b) 76 × 9
f) 77 × 8
c) 37 × 6
d) 48 × 9
E-25
(MP.7)
The 7 times table. Point out to students that if they know all the other
times tables, then the 7 times table is easy to work out. SAY: I know that
7 × 9 is 63. What is 9 × 7? (63 too, because you are multiplying the same
numbers) I know that 7 × 8 is 56. What is 8 × 7? (56 too!) I know that
7 × 5 is 35. What is 5 × 7? (35 too) Have students solve these problems:
a)6 × 7
b) 8 × 7
c) 3 × 7
d) 9 × 7
e) 5 × 7
Tell students that there is only one number multiplied by 7 that they haven’t
figured out yet. ASK: What number is that? (7 × 7, because it’s not from
any other times table) PROMPT: What product is only in the 7 times table?
Tell students that as long as they know all the other times table facts,
7 × 7 = 49 is the only one they have to learn to know all the 7 times table
facts too. Students can add the 5 times table to the 2 times table to check
their answers to the 7 times table. Demonstrate on the board:
7 × 8 = (5 × 8) + (2 × 8)
= 40 + 16
= 56
Have students use the 7 times table to solve these problems. Exercises:
b) 76 × 7
f) 78 × 7
c) 95 × 7
d) 67 × 7
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a)84 × 7
e)47 × 7
E-26
Teacher’s Guide for AP Book 4.1
NBT4-35 Multiplying a Multi-Digit Number by
Pages 103–105
a 1-Digit Number
Goals
STANDARDS
4.NBT.B.5
Students will multiply any 3-digit number by a 1-digit number.
PRIOR KNOWLEDGE REQUIRED
Vocabulary
Can apply the standard algorithm for multiplication
Can multiply a 2-digit number by a 1-digit number using
expanded form and base ten materials
Can regroup
standard algorithm
MATERIALS
base ten materials
BLM Practice with Times Tables (p. E-52)
Review using expanded form, base ten materials, and the standard
algorithm to multiply. Review with students 3 ways of multiplying 34 × 2:
1. Using the expanded form:
3 tens + 4 ones
× 2 6 tens + 8 ones = 68
2. With base ten materials:
3. Using the standard algorithm:34
× 2
68
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(MP.2)
Introduce multiplying a 3-digit number by a 1-digit number. Tell
students that you would like to multiply 213 × 2. Have a volunteer write
213 in expanded form on the board:
hundreds +
tens +
ones
Then add “× 2”:
2
hundreds +
1
ten +
3
ones
× 2 hundreds +
tens +
ones
Have a volunteer multiply each place value by 2 and then write the answer.
Tell students that there is a simpler way to deal with place value than by
writing “hundreds,” “tens,” and “ones.” Show students how the same
problem can be done using expanded form without words.
Number and Operations in Base Ten 4-35
E-27
200 + 10 + 3
× 2 400 + 20 + 6 = 426
Invite another volunteer to solve the problem using base ten materials. Do a
few more problems that do not require any regrouping (Examples: 324 × 2,
133 × 3, 431 × 2). Have volunteers use whichever method they prefer.
The modified standard algorithm for multiplying 3-digit numbers
by 1-digit numbers. Tell students that you want to multiply 213 × 4.
Write on the board:
200 +10 +3
× 4 +
+
=
Have a volunteer fill in the blanks. Then demonstrate the modified standard
algorithm for the same problem:
213
× 4 3 × 4:
12
10 × 4:
40
200 × 4: 800 852
(MP.3)
Emphasize that it is important to line up the place values because we are
adding 12 + 40 + 800, and when adding numbers together, we have to
line up the place values. ASK: How are the two ways of solving the problem
similar and how are they different? (You multiply the ones, tens, and hundreds
separately in both; you line the digits up in the second method.) Then have a
volunteer solve 213 × 4 using base ten materials and have other volunteers
explain how this method relates to the two methods shown above.
The standard algorithm for multiplying 3-digit numbers by
1-digit numbers. Show students how to use the standard algorithm
to multiply 213 × 4:
2
1
×
8
5
3
3 ones × 4 = 12 ones = 1 ten + 2 ones
4
1 ten × 4 = 4 tens
2
2 hundreds × 4 = 8 hundreds
Altogether, 8 hundreds + 4 tens + 1 ten + 2 ones
= 8 hundreds + 5 tens + 2 ones
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)
E-28
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
1
Exercises: Have students solve additional problems in their notebooks.
Students should solve each problem using base ten materials, expanded
form, and the standard algorithm.
a)112 × 6
b) 321 × 6
c) 215 × 5
d) 312 ×7
Tell students to be sure that they get the same answer all three ways.
If they do not, they should look for a mistake.
Exercises: Have students solve these problems using the standard algorithm.
a)213 × 5
(MP.6)
b) 213 × 6
c) 213 × 7
Tell students you are not going to check that their answers are correct.
They can check their answers themselves by adding 213 to 213 × 4
(obtained previously as 852). Do they get 213 × 5? ASK: If not, why
should you look for a mistake? Students can check their answers to
213 × 6 and 213 × 7 in the same way.
The standard algorithm using the 6, 7, 8, and 9 times tables. Some
students may need to review the methods for determining the 6, 7, 8,
and 9 times tables. Ask students to multiply 596 by 6, 7, 8, and 9, using
the standard algorithm, and then check their work by adding 596 to each
previous answer. For example, make sure that (596 × 6) + 596 = 596 × 7.
Point out that an easy way to add 596 is to add 600 and subtract 4.
The special case where the 3-digit number has a 0-digit.
Write on the board:
5
306
×
9
2 7 5 4
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Describe each step of the process, pointing at each number or
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: Have students do several problems with 0 in the 3-digit number.
a)406 × 9
b) 460 × 9
c) 807 × 9
d) 870 × 9 e) 708 × 9
Bonus: 12,009 × 6
Students who need more practice with the times tables can complete
BLM Practice with Times Tables.
Number and Operations in Base Ten 4-35
E-29
Extensions
(MP.3)
1.Explain why a 3-digit number multiplied by a 1-digit number must
have at most 4 digits. Have students investigate, by using their
calculators, the maximum number of digits that the answer can
have when multiplying:
a) 2-digit numbers by 2-digit numbers
b) 1-digit numbers by 4-digit numbers
c) 2-digit numbers by 3-digit numbers
d) 3-digit numbers by 4-digit numbers
e) 3-digit numbers by 5-digit numbers
Answers: a) 4, b) 5, c) 5, d) 7, e) 8
Challenge students to predict the maximum number of digits when
multiplying a 37-digit number by an 8-digit number. Answer: 37 + 8 = 45
(MP.8)
2.Challenge students to find a shortcut for multiplying a 3-digit number by
a 1-digit number when there are 0 tens in the 3-digit number. Answer:
Multiply the hundreds and ones separately, and write the two answers
next to each other. For example, 709 × 8 is 5,672 because 700 × 8 =
5,600 and 9 × 8 is 72. Then 5,600 + 72 = 5,672.
3. Name the ones digit of the following:
a)(2 × 6) + (20 × 6) + (100 × 6)
b) 99 × 91 × 19 × 11
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) The only term that affects the ones digit is 2 × 6 = 12,
so the ones digit is 2, b) The ones digit will be the ones digit of
9 × 1 × 9 × 1 = 81, so the ones digit is 1.
E-30
Teacher’s Guide for AP Book 4.1
NBT4-36 Word Problems with Multiplying
Page 106
STANDARDS
4.NBT.B.5, 4.OA.A.2
Goals
Students will gain a greater conceptual understanding of multiplication.
PRIOR KNOWLEDGE REQUIRED
Vocabulary
Can multiply different ways
pair
product
sum
Word problems in point form. Review word problems with students.
Write the following problems on the board and have students write both
addition and multiplication equations to model each problem:
a) 4 bowls
b) 4 weeks
3 apples each 7 days in each week
How many apples? How many days?
c) 5 minutes
d) 3 cm
60 seconds in each minute
10 mm in each cm
How many seconds?
How many millimeters?
One-step word problems in full sentences. Have students solve the
following problems after first rewriting them in point form:
a)There are 2 apples in each bowl. There are 5 bowls. How many apples
are there?
b) There are 6 horses. Each horse has 4 legs. How many legs are there?
c)There are 5 glasses. Each glass holds 120 mL. How much do all the
glasses hold together?
(MP.4)
Well-known information is sometimes left out. In each question below,
discuss which piece of information could be left out and why.
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d)There are 5 weeks before Katie’s birthday. There are 7 days in
each week. How many days are there before Katie’s birthday?
e)Tina’s pencil is 4 cm long. Each centimeter has 10 mm.
How many millimeters long is the pencil?
f)Natalia just turned 3 years old. There are 12 months in each year.
How many months old is Natalia?
Explain that when the problem involves information that is well known, such
as how many days are in a week or how many months are in a year, the
information is often not stated: I don’t need to tell you that there are 7 days
in a week because I assume you know that. Have students write the piece
of information that was left out of these word problems:
a) Julie is exactly 19 days old. How many hours old is she?
b) Shawn is exactly 4 years old. How many months old is he?
Number and Operations in Base Ten 4-36
E-31
c)Mary’s heart beats 94 times a minute. How many times would it beat
in an hour?
Students can make up similar questions (in which well-known information
is not stated) and have a partner solve them.
Extensions
1.Give students tiles or counters and ask them to solve these riddles
by making models.
a) I am less than 10. You can show me with
• 2 equal rows of tiles, or
• 3 equal rows of tiles.
Solution: The number is 6. To see this, students could find all the
numbers they can make with 3 equal rows and decide which of those
numbers they can also make with 2 equal rows. They can also start
with 2 equal rows instead of 3, but this will be more work.
(MP.1)
Encourage students to solve the problem both ways and compare the
solutions. Discuss why it takes more work to start with 2 equal rows.
(There are more possibilities with 2 equal rows, and so more numbers
to try to put into 3 equal rows.) ASK: Why are there more possibilities
with 2 equal rows than with 3 equal rows? (every second number can
be made with 2 equal rows, whereas only every third number can be
made with 3 equal rows)
b) I am between 15 and 25. You can show me with
• 2 equal rows of tiles, or
• 5 equal rows of tiles.
Discuss with students whether it is better to start by finding all the
numbers with 2 equal rows or with 5 equal rows. (start with 5 equal
rows because there are fewer such numbers to check)
a)
children each have
do the children have?
b) Roger ran
km each day for
pencils. How many pencils
days. How far did he run?
c)Make up your own word problem involving multiplication
and solve it.
E-32
Teacher’s Guide for AP Book 4.1
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2. Add missing information to the word problems and solve them.
NBT4-37 Multiplying 2-Digit Numbers by
Pages 107–108
Multiples of 10
STANDARDS
4.NBT.B.5
Vocabulary
array
double
multiple
standard algorithm
Goals
Students will multiply 2-digit numbers by 2-digit multiples of 10 (10, 20,
30, …, 90) by using the standard algorithm.
PRIOR KNOWLEDGE REQUIRED
Can multiply using arrays
Can apply the distributive property
Can multiply 2-digit numbers by 1-digit numbers using
the standard algorithm
Can multiply 2-digit numbers by 2-digit multiples of 10
without regrouping using mental math
MATERIALS
BLM Cutting an Array into Ten Strips (p. E-53)
Review multiplying 1-digit numbers by multi-digit numbers mentally.
Use numbers that require either no regrouping at all (Examples: 2 × 324,
3 × 132, 4 × 201) or regrouping at the largest place value only (Examples:
4 × 612, 2 × 804, 3 × 430).
Using arrays to multiply by multiples of 10. Draw this picture on the board:
20
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32
Tell students that the picture represents an array that is 20 squares high and
32 squares long. (If students have trouble visualizing the individual squares,
give them BLM Cutting an Array into Ten Strips.) Ask students to imagine
dividing the array into smaller strips that are 2 squares high and 32 squares
long. Draw this picture to show the strips:
20
32
Each strip is 2 squares high.
ASK: What multiplication statement gives the number of squares in each
strip? (2 × 32, because each strip is 32 squares long and 2 squares high)
Number and Operations in Base Ten 4-37
E-33
20
There are 2 × 32 squares in a strip.
32
ASK: How many strips are in the array? (10, because 10 strips of height 2
make an array that is 20 squares high) There are 10 strips, each containing
2 × 32 squares, so 20 × 32 can be rewritten as 10 × (2 × 32).
Write a new height and length on your diagram, but do not erase the
ten 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 how high each strip is (given that the array has been cut into
ten strips). Exercises: Use the following dimensions with the same array:
a)
b)
c)
40
30
57
50
77
42
Solution to a) Each strip is 3 squares high (since 3 × 10 = 30).
There are 3 × 57 squares in each strip, so there are 10 × (3 × 57)
squares in the array.
Why using arrays works. Tell students that they can prove that 20 × 32 =
10 × (2 × 32) without using a diagram. They can use the commutative
and associative properties of multiplication.
(2 × 3) × 4 = 2 × (3 × 4)
Point out to students that they can now multiply a 2-digit number
by a multiple of 10 by rewriting the product:
20 × 32 = 10 × (2 × 32) = 10 × 64 = 640
Exercises: Calculate the products.
a)30 × 12
b) 40 × 22
c) 50 × 31
Answers: a) 360, b) 880, c) 1,550
When the multiple of 10 is the second factor. Write 50 × 31 and 31 × 50
on the board. ASK: How are these problems the same? How are they
different? (the same numbers are multiplied in a different order; they have
the same answer) Tell students that they have only seen the multiple of 10
E-34
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
By the commutative property, 20 can be written as either 2 × 10 or 10 × 2.
So 20 × 32 can written as (10 × 2) × 32. But by the associative property,
this statement can be written as 10 × (2 × 32). If students don’t recall the
associative property, remind them that they can multiply three numbers in
either of the orders shown below:
as the first number, but they can still do problems having the multiple of 10
as the second number. Demonstrate the solution to the first exercise below.
Exercises:
a)32 × 40
b) 61 × 30
c) 73 × 20
Answers: a) 32 × 40 = 40 × 32 = 1,280, b) 1,830, c) 1,460
(MP.7)
Teach the same method for multiples of 100 and 1,000. For example, to
find 2,000 × 32, multiply 32 × 2 = 64 and then add 3 zeros to get 64,000.
Here is the longer way to represent this:
2,000 × 32 = 2 × (1,000 × 32) = 2 × 32,000 = 64,000
Rounding to estimate products. Finally, review rounding with students, then
teach them to estimate products of 2-digit numbers by rounding each factor
to the tens digit. Example: 25 × 42 ≈ 30 × 40 = 1,200.
Exercises: Estimate the sum or product.
a)28 + 45
b) 37 + 42
c) 46 × 71
d) 83 × 94
Answers: a) 30 + 50 = 80, b) 40 + 40 = 80, c) 50 × 70 = 3,500,
d) 80 × 90 = 7,200
Bonus
a) Estimate products of three 2-digit numbers.
Example: 27 × 35 × 41 ≈ 30 × 40 × 40 = 48,000.
b)Estimate products of two 3-digit numbers by rounding to the nearest
multiple of a hundred. Example: 271 × 320, becomes 300 × 300 = 900.
Remind students that if the ones digit is 4 or less, they round down, and if
the ones digit is 5 or more, they round up: 34
30; 35
40.
More multiplying. Give students practice with products that require
regrouping. (See Question 5 on page 108 of AP Book 4.1 for examples
of how students can do their rough work using the standard method
for multiplying a 2-digit number by a 1-digit number.)
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Exercises:
a)20 × 35
b) 30 × 45
c) 50 × 52
Answers: a) 700, b) 1,350, c) 2,600
Tell students that they can use a chart to find products like 40 × 57 =
10 × (4 × 57) by following these steps (which are also in AP Book 4.1
on page 108):
Step 1: When you multiply a number by 10, you add a zero. So write
a zero in the ones place because you will multiply (4 × 57) by 10.
×
Number and Operations in Base Ten 4-37
5 7
4 0
0
E-35
Step 2: Now multiply 57 by 4.
2
×
2
5 7
5 7
×
4 0
4 0
8 0 2 2 8 0
Emphasize the similarity between multiplying 37 × 20 and 37 × 2.
Write the following products on the board:
1
1
3 7
3 7
× 2
× 2 0
7 4 7 4 0
Discuss the similarities and differences between the two algorithms. Why
can we multiply 37 × 20 as though it is 37 × 2 and then just add a 0? Why
did we carry the 1 to the tens column in the first problem but to the hundreds
column in the second problem?
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(MP.3)
E-36
Teacher’s Guide for AP Book 4.1
NBT4-38 Multiplying 2-Digit Numbers by
Pages 109–111
2-Digit Numbers
Goals
STANDARDS
4.NBT.B.5
Students will multiply 2-digit numbers by 2-digit numbers.
PRIOR KNOWLEDGE REQUIRED
Vocabulary
Can multiply a 2-digit number by a 1-digit number
Can multiply a 2-digit number by a 2-digit multiple of 10
double
multiple
MATERIALS
grid paper
Introduce the lesson topic. Write on the board 28 × 36. ASK: How is
this multiplication different from any we have done so far? (We have
never multiplied a 2-digit number by a 2-digit number of which neither
is a multiple of 10—we have only estimated the product in such cases.)
(MP.3)
Splitting a problem into easier problems. Tell students that you would
like to think of 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:
• 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.
Allow students time to think of a way to split the problem into two easier
products that they already know how to do. Possibilities include:
28 × 36 = (20 × 36) + (8 × 36)
28
OR
28 × 36 = (28 × 30) + (28 × 6)
Read these out loud as “20 thirty-sixes plus 8 thirty-sixes” or “30 twentyeights plus 6 twenty-eights.”
6
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Write the equations out with blanks if students need a prompt. Example:
30
28 × 36 = 28 ×
+ 28 ×
SAY: 28 is a 2-digit number. How can we split 36 into 2 numbers so that we
know how to do both products? Draw the picture in the margin on the board.
(MP.2)
Remind students that the area of the whole rectangle is the sum of the two
smaller rectangles. Write on the board: 28 × 36 = (28 × 30) + (28 × 6),
and SAY: 36 twenty-eights is 30 twenty-eights plus 6 twenty-eights.
Also remind students that we write brackets to show what operations we do
first. SAY: We first find the areas of the two smaller rectangles (point to the
two products as you say this), and then we add them together to get the
area of the whole rectangle (point to the addition as you say this).
Number and Operations in Base Ten 4-38
E-37
Ask students to rewrite each of the following products as a pair of
singular products:
a)36 × 27
b) 43 × 31
= 36 × (20 + 7)
= 36 × 20 + 36 × 7
c) 52 × 24
After students have completed the exercises above, ask them to calculate
the actual products.
Sample answer:
a)36 × 27
= 36 × (20 + 7)
= 36 × 20 + 36 × 7
= 720 + 252
= 972
Students can do their rough work for these exercises as shown in Question 1
on page 109 of AP Book 4.1.
The standard algorithm. Show students how to multiply 16 × 34 using the
standard algorithm.
Step 1: Multiply 16 × 4.
2
1 6
3 4
6 4 = 4 × 16
regrouping
for 30 × 16
Step 2: Multiply 16 × 30.
1
2
1 6
3 4
6 4
0
×
Write a zero here
because you are
multiplying by 30. Step 3: Add the two products.
1
1
3
6
4 8
6
4
4
0 = 30 × 16
Write the product
3 × 16 here.
2
1
3
×
6
4 8
5 4
×
2
6
4
4
0
4 = 64 + 480
Students can practice the steps of the standard algorithm in Questions 2–5
on pages 110–110 in AP Book 4.1.
Exercises: Use the standard algorithm to do these problems:
a)73 × 46
E-38
b) 54 × 35
c) 46 × 71
d) 84 × 96
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
×
Answers: a) 3,358, b) 1,890, c) 3,266, d) 8,064
Bonus
Add enough zeros to the product of two 2-digit numbers to multiply:
a)780 × 640
b) 3,400 × 250
c) 8,700 × 9,400
Answers: a) 499,200, b) 850,000, c) 81,780,000
Estimating sums and products. Explain to students that they don’t always
need an exact answer. Often they only need to know about how big the
answer is. Do some examples together. For example, 32 + 85 is about
30 + 90 = 120, and 32 × 85 is about 30 × 90 = 2,700 (using rounding).
ACTIVITY
Hot and cold. This is a game for pairs. Player 1 picks two numbers
between 50 and 100 and estimates the product of these numbers.
Player 2 uses a calculator to find the actual product and gives Player 1
an appropriate clue about the estimate, using the words “hot,” “warm,”
“warmer,” “cold,” “colder,” “freezing,” and so on. Player 1 revises his or
her estimate until it is “burning hot,” or within 100 of the correct answer.
Players switch roles and play again.
(MP.6)
Checking the reasonableness of an answer when using a calculator.
SAY: John multiplied 32 × 86 on a calculator and got 1,978. How can he
tell he is wrong? (the answer should be more than 30 × 80 = 2,400, so the
answer is too low) Explain to students that John input 23 × 86 by mistake.
This is a type of mistake anyone can make, even if they know the math,
so it is really important to check that your answer makes sense. Now tell
students that some of the following products were input incorrectly into
a calculator. See if they can tell which ones by estimating. Suggest that
students round both numbers up to get a high estimate and both numbers
down to get a low estimate. Then they can be sure that their answer should
lie in between the two estimates.
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a)27 × 52 = 3,744
b) 38 × 94 = 1,862
c) 12 × 74 = 888
Answers: a) and b) were input into the calculator incorrectly. Students can
signal their answers showing “thumbs up” when the product is reasonable
and “thumbs down” when it is not.
Encourage students to check their estimates against the actual products
(found using a calculator).
Extensions
(MP.8)
1.On BLM Patterns in Multiplication (p. E-54), students discover an
easy way to multiply a 2-digit number with ones digit 5 by itself:
15 × 15, 25 × 25, 35 × 35, and so on.
After students complete the BLM, summarize their answers. ASK: What
are the tens and ones digits always? (25) How can you get the number
of hundreds in the answer from the tens digit of the number being
Number and Operations in Base Ten 4-38
E-39
multiplied by itself? (multiply the tens digit by 1 more than the tens digit)
Explain to students that they can now multiply some 2-digit numbers in
their heads (and this is a shortcut that not even most mathematicians
know about!). Tell students to not look at the answers they just wrote
down. Write on the board:
35 × 35 =
ASK: What are the last two digits? (25) How do you know? (because
they are always 25—that is easy to remember) What are the first two
digits? (3 × 4 = 12) How do you know? (because that is the pattern we
found; 4 is one more than 3, so multiply 3 × 4) Show this on the board
as follows:
35 × 35 = 1,2 2 5
3×4
Have students do these questions mentally:
a)75 × 75 =
b)65 × 65 =
c)45 × 45 =
d)85 × 85 =
e)95 × 95 =
If students are engaged, you could tell them that this same shortcut
works for multiplying any number with ones digit 5 by itself. Show this
on the board:
175 × 175 = 3 0 ,6 2 5
17 × 18
Challenge students to calculate these products:
b) 995 × 995
c) 1,005 × 1,005
Answers
a) 10 × 11 = 110, so 105 × 105 = 11,025
b) 99 × 100 = 9,900, so 995 × 995 = 990,025
c) 100 × 101 = 10,100, so 1,005 × 1,005 = 1,010,025
d) 999 × 1,000 = 999,000, so 9,995 × 9,995 = 99,900,025
Encourage students to check the reasonableness of their answers by
estimating using rounding. For example, a) should be a little more than
100 × 100 = 10,000, which it is.
(MP.7, MP.2)
E-40
2.Using area to divide a product of 2-digit numbers into four easy
products. Remind students that even 43 × 20 was a combination of
smaller products: 40 × 20 and 3 × 20. ASK: How can we write 43 × 7
as a combination of smaller products? (40 × 7 and 3 × 7) Summarize
by saying that 43 × 27 is actually a sum of four very easy products:
Teacher’s Guide for AP Book 4.1
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a)105 × 105
d)9,995 × 9,995
43 × 27 =
(40 × 20) + (3 × 20) + (40 × 7) + (3 × 7)
= 800 + 60 + 280 + 21
= 1,161
Draw this picture on the board to summarize.
40
3
20
3 × 20
40 × 7
7
3×7
Ask students to draw a picture for the product 39 × 52 and compare the
resulting calculation to the standard algorithm.
Answer:
50
2
30
9
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1
52
becomes 52
× 39
× 39
18(2 × 9)
468
(52 × 9)
450(50 × 9)+ 1,560(52 × 30)
60(2 × 30) 2,028
+ 1,500(50 × 30)
2,028
ASK: Where does the 1 (point to the 1 written above the 5) come from?
(2 × 9 is 18 which is 1 ten and 8 ones, so we write 1 in the tens column
and 8 in the ones column.) How do we use the 1 when multiplying 52 × 39?
(When we multiply 50 × 9 = 450 = 45 tens, we add 1 to the number of
tens, so now we have 46 tens.)
Students can complete BLM Using Area to Multiply (p. E-55) for more
practice using the area model of multiplication.
Number and Operations in Base Ten 4-38
E-41
NBT4-39 Topics in Multiplication
Page 112
STANDARDS
4.NBT.B.5, 4.OA.A.2
Vocabulary
Goals
Students will consolidate their understanding of the concepts learned
so far in multiplication.
PRIOR KNOWLEDGE REQUIRED
double
multiple
Can multiply 2-digit numbers by 2-digit numbers
Can apply the distributive property
Can use inferred information to solve word problems
MATERIALS
BLM Always/Sometimes/Never True (Numbers) (p. E-56)
BLM Define a Number (p. E-57)
This lesson is mostly a review of concepts learned so far.
ACTIVITY
Give each student a copy of BLM Always/Sometimes/Never True
(Numbers). As a class, students sort the statements into three
columns on the board: Always True, Sometimes True, Never True. Each
student who wants to should make their case for where the statement
should go, and then the class can vote on where to put it. Students
may choose to come back to a phrase that they cannot decide on or
that they have already placed.
Variation: To make the game more difficult, you can assign secret
volunteers to purposefully argue for the wrong answer. Be sure to tell
students that you have assigned such volunteers, but don’t reveal who
the volunteers are! The volunteers should argue for the correct answer
often enough that people believe them when they argue for a wrong
answer. To assign such volunteers, you could use any sort of randomizer.
For example, each student rolls a die and those who roll 6 become the
secret volunteers. Or, write on many small pieces of paper a circle or a
square. (Use as many circles as you want volunteers.) Fold the papers
and distribute them to students. Those with the circles become the
volunteers. This is particularly useful because some students will feel
more comfortable saying what they think the answer is even if the teacher
doesn’t know who the volunteers are. If you use this variation, be sure
students understand that you believe they need a challenge, and the
purpose of the volunteers is to make the game more challenging. The
volunteers are not trying to “win” against the rest of the class. They are
only trying to make the game more of a challenge, because activities that
are more challenging are more fun. If the game turns into a competition,
apologize and stop playing.
E-42
Teacher’s Guide for AP Book 4.1
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(MP.3)
(MP.1, MP.3)
(MP.3)
After students play the game above, they should complete BLM Always/
Sometimes/Never True (Numbers) individually. More consolidation of
number sense facts can be found on BLM Define a Number.
After students have had a chance to work through AP Book 4.1 p. 112
Question 7, discuss how students determined the greatest and least
products. In particular, discuss the problem-solving strategy of making an
organized list. Explain that you have to use each of the three digits 3, 4, and
5 once, and you want to list all the ways of putting the 3 digits in the boxes
so that you can find the greatest and least numbers. ASK: How can I make
sure I don’t miss any ways? Suggest trying to put each digit in the box for
the single-digit number, in order. Write on the board:
× 3
× 4
× 5
ASK: If 3 is the 1-digit number, what can the 2-digit number be? Tell
students there are only two possible answers. Have students find all the
possibilities this way.
(MP.7)
Now that you know what all the possible products are, you want to find
out which product is greatest, but without having to compute any of the
products! ASK: What is larger, 3 × 45 or 3 × 54? (3 × 54) How do you
know? (because 54 is larger than 45) Repeat for the products 4 × 35 and
4 × 53, and then 5 × 34 and 5 × 43. Explain that you just reduced the
number of multiplications you have to do by half! The largest product is:
A. 3 × 54, B. 4 × 53, or C. 5 × 43
Challenge students to compare 3 × 54 to 4 × 53 without doing any
multiplications. PROMPT:
3 × 54 = (3 × 50) + (3 ×
)
4 × 53 = (4 × 50) + (4 ×
)
Answer: 3 × 4 = 4 × 3, so we only need to compare 3 × 50 to 4 × 50.
This is easy, so in fact, 4 × 53 is greater than 3 × 54.
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Now challenge students to compare 4 × 53 to 5 × 43 using the same
strategy.
Answer: 4 × 53 = (4 × 50) + (4 × 3) and 5 × 43 = (5 × 40) + (5 × 3).
Since 4 × 50 = 5 × 40, we only need to compare 4 × 3 to 5 × 3.
This is easy, so in fact 5 × 43 is greater than 4 × 53.
Bonus: Without multiplying, explain why 8 × 732 is greater than 7 × 832.
PROMPT:
8 × 732 = 8 × 700 + 8 ×
7 × 832 = 7 × 800 + 7 ×
Answer: 8 × 732 is greater because 8 × 32 is greater than 7 × 32,
and 8 × 700 = 7 × 800.
Number and Operations in Base Ten 4-39
E-43
Extensions
1.Teach multiplication of three numbers using three-dimensional arrays.
Give students blocks and ask them to build a “box” that is 4 blocks
wide, 2 blocks deep, and 3 blocks high.
Have students write the equation that corresponds to the box as
positioned in the margin: 3 × 4 × 2.
Have students pick up their box and turn it around or have them look
at their box from a different perspective: from above or from the side.
Now what equation do they see? Take various answers so that students
see the different possibilities. ASK: When you multiply 3 numbers, does
it matter which number goes first? Would you get the same answer in
every case? (Yes, because the total number of blocks doesn’t change.)
2. What four numbers being multiplied does this model show?
Answer: 3 × 5 × 2 × 4
3. Decide which is larger without multiplying: 63 × 52 or 62 × 53.
Hint: Compare both products to 62 × 52.
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Answer: 63 × 52 is 52 more than 62 × 52, while 62 × 53 is 62 more
than 62 × 52. So 62 × 53 is greater than 63 × 52.
E-44
Teacher’s Guide for AP Book 4.1