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
DATE
NAME
Using the 10 Times Table to Multiply
If you know the 10 times table, you can find the 11 and 12 times tables too!
10 × 7
2×7
12 sevens= 10 sevens + 2 sevens
12 × 7= (10 × 7) + (2 × 7)
= 70 + 14
= 84
1. Write the 10 times table, and then add to write the 11 times table.
+
1
2
3
×10
10
20
30
×11
11
22
33
4
5
6
7
8
9
10
11
12
11
12
2. Write the 10 times table and the 2 times table, and then add to write the 12 times table.
+
1
2
3
×10
10
20
30
×2
2
4
6
×12
12
24
36
4
5
6
7
8
9
10
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3.Cover the top of this page. Use 10 times the number to find 11 or
12 times the number.
a) 12 × 8
b) 11 × 11
c) 12 × 3
d) 12 × 11
e) 12 × 7
f ) 11 × 12
4. Use 10 times the number to find 9 times the number.
a) 10 sevens − 1 seven 9 sevens
d) 9 × 4
= 70
b) 10 eights
=
c) 10 twelves = − 7 − 1 eight
= − 1 twelve = 63
= 9 × 7 = 63
e )9 × 3
f )9 × 5
=
=
=
g) 9 × 11
h)9 × 6
i)9 × 9
Blackline Master — Number and Operations in Base Ten — Teacher’s Guide for AP Book 4.1
E-45
DATE
NAME
Using Area to Find Equal Products
1. a) Cut out the 4 × 6 rectangle and then cut it in half along the thick line.
b) Place the two halves side by side. Tape them together and glue them here:
c) What size rectangle did you make in part b)?
d) Finish the equation:
4 × 6 =
×
2.Cut out another 4 × 6 rectangle, but cut this rectangle in half the other way.
Place the new rectangles one above the other.
So 4 × 6 =
×
3. Find the missing number.
a)4 × 3 = 2 ×
b)6 × 5 = 3 ×
d) 4 × 10 = 8×
e)14 × 5 = 7 ×
BONUS
3 × 604,828,644 = 6 ×
BONUS
8 × 2,113,403 = 4 ×
E-46
c)9 × 12 = 18 ×
f ) 14 × 50 = 7 ×
Blackline Master — Number and Operations in Base Ten — Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2013 JUMP MATH: TO BE COPIED.CC EDITION
DATE
NAME
Moving Rectangles Around (1)
Jane’s rectangle has 4 rows of 3 squares.
She cuts her rectangle into two parts.
She glues the two parts side by side.
Now she has a new rectangle.
The new rectangle has 2 rows of 6 squares.
Jane didn’t change the area of the rectangle by cutting and gluing, so 4 × 3 = 2 × 6.
1.Draw Jane’s new rectangle. Then fill in the blanks to show that the two rectangles
have the same area.
a)
So
×
=
×
×
=
×
×
=
×
b)
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So
c)
So
Blackline Master — Number and Operations in Base Ten — Teacher’s Guide for AP Book 4.1
E-47
DATE
NAME
Moving Rectangles Around (2)
Ahmed’s rectangle has 3 rows of 4 squares.
He cuts the rectangle into two parts.
He glues the two parts one above the other.
Now he has a new rectangle.
It has 6 rows of 2 squares.
Ahmed didn’t change the area of the rectangle by cutting and gluing, so 3 × 4 = 6 × 2.
2. a) Draw Ahmed’s new rectangle.
i)
ii) iii)
b) Fill in the blanks to show that the two rectangles have the same area.
i)
×
=
×
iii)
×
=
×
ii)
×
=
×
3. a) Write the missing number.
i)5 × 8 = 10 ×
BONUS
ii) 7 × 10 = 14 ×
500 × 12 = 1,000 ×
b) Circle the product in each pair that is easiest to find, and then find both products!
E-48
Blackline Master — Number and Operations in Base Ten — Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2013 JUMP MATH: TO BE COPIED.CC EDITION
NBT4-30 Mental Math
1. Write a product for each array.
a)
b)
3 × 20
c)
d)
2.Write a product for the whole array and for each part of the array
(as shown in part a).
3 × 24
a)
b)
3 × 20
3×4
d)
c)
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3. Fill in the blanks (as shown in part a).
2 × 24
a)
2 × 20
b)
2×4
2 × 24 = (2 × 20) + (2 × 4)
c)
4 × 25
d)
Number and Operations in Base Ten 4-30
95
To multiply 3 × 23, Rosa rewrites 23 as a sum:
3 × 23 = 69
23 = 20 + 3
She multiplies 20 by 3 and then she multiplies 3 by 3:
3 × 20 = 60 and 3 × 3 = 9
Finally she adds the results: 60 + 9 = 69
3 × 20 = 60
The picture shows why Rosa’s method works:
3×3=9
3 × 23 = (3 × 20) + (3 × 3) = 60 + 9 = 69
4. Rewrite each multiplication statement as a sum.
a)2 × 24 =
2 × 20
+
c) 3 × 32 =
2×4
+
b)2 × 23 =
+
d) 4 × 12 =
+
5. Multiply using Rosa’s method.
a) 3 × 13 =
3 × 10
+
3×3
=
30 + 9
39
=
b) 3 × 21 =
+
=
=
c)2 × 14 =
+
=
=
d) 3 × 213 =
3 × 200
+
3 × 10
3×3
+
=
600 + 30 + 9
=
e)2 × 231 =
+
+
=
=
f) 2 × 342 =
+
+
=
=
639
6. Multiply in your head by multiplying the digits separately.
b)2 × 31 =
c) 4 × 12 =
e) 4 × 21 =
f)2 × 43 =
g)2 × 32 =
d)5 × 11 =
h) 3 × 33 =
i) 4 × 112 =
j) 2 × 234 =
k) 3 × 233 =
l) 5 × 111 =
m) 3 × 132 =
n)2 × 422 =
o) 4 × 212 =
p) 3 × 333 =
7.Yen planted 223 trees in each of 3 rows. How many trees did she plant altogether?
8.Paul put 240 marbles in each of 2 bags. How many marbles did he put in the bags?
96
Number and Operations in Base Ten 4-30
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a) 3 × 12 =