NBT4-34 Multiplication Charts
Page 97-99
STANDARDS
3.OA.7, 3.OA.9,
preparation for 4.NBT.5
Vocabulary
column
multiplication chart
row
Goals
Students will complete multiplication charts and look for
simple patterns in them.
PRIOR KNOWLEDGE REQUIRED
Can multiply using arrays
MATERIALS
BLM 2-cm Grid Paper (p. I-2)
BLM 9 x 9 Multiplication Chart (p. I-3)
BLM Using the Multiplication Chart to Multiply (p. E-62)
BLM The Blank 10 x 10 Multiplication Chart (p. E-63)
BLM The Blank 12 x 12 Multiplication Chart (p. E-64)
Review rows and columns. Tell students that you will be using the words
“row” and “column” a lot in this lesson, and then remind them what these
words mean with a visual aid.
c
r
o
w
l
u
m
n
Keep this visual aid on display throughout the lesson.
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A rectangle as an array of squares. Review multiplying using arrays.
Explain that students can think of a rectangle as an array of squares. Draw
on the board:
3 rows of 4 dots is 12 dots
3 × 4 = 12
3 rows of 4 squares is 12 squares
3 × 4 = 12
Explain to students that they can tell what multiplication equation a
rectangle represents by counting the squares down and across.
1234
1
2
3
Number and Operations in Base Ten 4-34
number of squares down
× number of squares across
total number of squares
3
× 4
12
E-1
Draw several rectangles on the board. Exercises: Have students
write the number of squares down and across. Then have students
write the corresponding multiplication equation by counting the total
number of squares.
a)
b)
Answers: a) 2 × 5 = 10, b) 3 × 7 = 21
Now have students do more such questions, but this time have students
write the total number of squares in the bottom right square of the
rectangle. Example:
15
2
3
4
1
2
3
4
5
12
5
Introduce multiplication charts. Draw a blank 5 × 5 multiplication chart
on BLM 2-cm Grid Paper as shown in the margin. Project it onto the board.
Demonstrate how to find 3 × 4. Make a rectangle 3 squares down and 4
squares across, and write how many squares are in the rectangle in its bottom
right square. Have students complete Student Book p. 97 Question 1.
Using completed charts to multiply. Copy BLM 9 × 9 Multiplication
Chart onto a transparency and project it onto the board. Demonstrate
how to draw a rectangle to find products. For example, to determine 3 × 4,
draw a rectangle with 3 rows of 4 squares each, starting at the top left corner
of the chart. The answer is the number in the bottom right corner of the
rectangle: 3 × 4 = 12. Erase the rectangle, and have volunteers solve
more such problems: 6 × 2, 4 × 7, 8 × 3. Tell students how they can use
the chart to multiply without drawing rectangles. Students can use their
fingers or two sheets of paper to define the rectangles that correspond to
different products. For example, to see the rectangle for 6 × 8, they can
place one finger or sheet of paper under row 6 (to cover all rows below
6) and another to the right of column 8 (to cover all columns to the right
of 8). The number in the bottom right corner is the answer to 6 × 8. Have
volunteers use this method to multiply 5 × 7, 4 × 9, and 9 × 8. Then have
students complete BLM Using the Multiplication Chart to Multiply.
Completing a multiplication chart. Ask students to draw a new, blank
5 × 5 multiplication chart on grid paper. Tell students that they will now fill
in the entire chart. Emphasize the fact that each square in the chart is the
bottom right square of a rectangle that corresponds to a product. Students
should start by filling in the correct square for these products:
a)3 × 5
b) 2 × 4
c) 5 × 2
d) 1 × 3
e) 4 × 1
Be sure students fill in the correct squares accurately for these products
before they continue with the remaining products. Students who need to
can use their fingers or two sheets of paper to define the rectangles that
correspond to different products.
E-2
Teacher’s Guide for AP Book 4.1
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× 1
Bonus
Students can complete a 3 × 12 chart after they complete the 5 × 5 chart.
Patterns in the rows. Ask students to describe any patterns they see in
the rows of their completed 5 × 5 charts and to predict the pattern for
the sixth row.
Using the rows to complete the columns. ASK: How do the patterns in the
columns compare to the patterns in the rows? (they are the same) The first
column is the same as the first row, the second column is the same as the
second row, and so on. Why is that the case? Encourage students to write
out not just the numbers in the second row, but the products they represent:
2 × 1, 2 × 2, 2 × 3, 2 × 4, 2 × 5. Repeat for the numbers in the second
column: 1 × 2, 2 × 2, 3 × 2, 4 × 2, 5 × 2. ASK: Why are the numbers in
the second row the same as the numbers in the second column? (because
they are the result of the same numbers being multiplied, just in a different
order) Explain to students that because the order doesn’t matter when
multiplying, if they have a row completed, they can also complete the
corresponding column.
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Learning the times tables. Explain to students that each row in a
multiplication chart is a times table for a different number—it tells what
that number is when multiplied by many numbers in turn (1, 2, 3, ...).
Give students time to complete a 10 × 10 or 12 × 12 multiplication
chart. (Students can use BLM The Blank 10 × 10 Multiplication Chart
or BLM The Blank 12 × 12 Multiplication Chart.) Students can fill in
each row by skip counting. For example, to complete the “× 6” row, skip
count by 6 starting at 6. They can use the relationship between the rows
and columns to make less work for themselves.
Emphasize that you will not be evaluating students’ work in any way. It
doesn’t matter how much of the chart they complete or how quickly they
work. One purpose of this exercise is to help students become more
familiar and comfortable with multiplication charts and times tables; they
will find math much easier if they know their multiplication facts. Another
purpose is to demonstrate the value and effect of practice in learning.
Let your students know that remembering their tables will get easier with
practice. It’s okay to struggle with a concept or skill, particularly when you
are first learning it. Eventually, with practice, it will no longer be a struggle!
Extensions
1. The rows and columns of the times tables have been mixed up.
Fill in the missing numbers.
a)
× 7
5
b)
×
6 4
6
4 28
7
5 40 30 20
25
10
2
10
3
42
12
3
12
4
Number and Operations in Base Ten 4-34
E-3
Answers: a) 7, 5, 6, 2 across by 4, 5, 3, 2 down, b) 8, 6, 4, 5, 7 across
by 5, 2, 6, 3, 1 down
2. Use the numbers 1 to 5 to fill in the missing numbers.
×
×
0
4
2
Answers: 4 × 5 = 20 (or 5 × 4 = 20), 3 × 4 = 12
(MP.1)
3. Use the numbers 0 to 9 to fill in the missing numbers.
6
×
×
2
4
5
6
×
3
×
5
5
×
3
2
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Answers: 8 × 3 = 24 (or 3 × 8 = 24), 9 × 6 = 54, 5 × 7 = 35
(or 7 × 5 = 35), 6 × 5 = 30, 6 × 2 = 12
E-4
Teacher’s Guide for AP Book 4.1
NBT4-35 Multiplying by Adding On
Page 100-101
STANDARDS
3.OA.5, preparation for
4.NBT.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
Point out that we get different answers depending on which operation we
do first.
Exercises: Have students do the operation in brackets first and then
calculate each expression. Allow students to use their multiplication charts
from the previous lesson.
a)(2 + 3) × 4 and 2 + (3 × 4)
b) 7 − (3 × 2) and (7 − 3) × 2
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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 4-35
E-5
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 (as in Student Book
p. 100 Question 2) and have volunteers come up and fill in the blanks,
as shown here.
4×5
3×5
+
5
Find a product by adding on to a smaller product without using arrays.
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-6
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.
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.
(2 × 5) + 5 = 2 × 6
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or
(2 × 5) + 5 = 3 × 5
Answer: (2 × 5) + 5 = 3 × 5
3. a)For which of these expressions would multiplying first and adding
first produce the same answer?
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.
b)Write another expression that produces the same answer whether
you add first or multiply first. Check your work.
Number and Operations in Base Ten 4-35
E-7
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 and 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 same dots in different ways. See margin.
Show Sarah’s groupings and explain her thinking.
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 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-8
Teacher’s Guide for AP Book 4.1
NBT4-36 Multiplying by Tens, Hundreds, and
Page 102-104
Thousands
STANDARDS
3.NBT.3, 4.NBT.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 4-36
E-9
(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
(MP.7)
Ask students if they see a pattern in the multiplication equations that
correspond to these pictures:
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.
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
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
E-10
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
(MP.1)
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.
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
(MP.7, MP.8)
Patterns when multiplying powers of 10. Write on the board:
1,000 × 10
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
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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
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
Number and Operations in Base Ten 4-36
E-11
(MP.7)
Patterns when multiplying tens and hundreds. Write on the board:
20 × 700 = 2 × 10 × 7 × 100
Remind students that multiplying the same numbers in any order will get
the same answer, then re-write 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
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 =
E-12
so 300 × 800 =
so 40 × 9,000 =
so 50 × 600 =
so 500 × 4,000 =
so 200 × 70,000 =
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
50 × 80
f)6 × 4 =
g)2 × 5 =
so 600 × 40 =
so 200,000 × 500 =
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.
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a)2,985 × 68
b) 4,012 × 312
c) 297 × 8,888
d) 532 × 8,102
Extensions
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.
Possible 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 4-36
E-13
NBT4-37 Finding Easier Ways to Multiply
Page 105-107
STANDARDS
3.MD.7, 3.OA.5, 3.OA.7,
preparation for 4.NBT.5
Goals
Students will understand that the area of a rectangle can be obtained
from the product of its side lengths. Students will discover and apply the
distributive property (without using that phrase) in order to remember the
times tables up to 9 × 9.
Vocabulary
area
column
product
rectangle
row
sum
PRIOR KNOWLEDGE REQUIRED
Can multiply using arrays
Can multiply using repeated addition
Can comfortably add 2-digit numbers
Knows that area is additive
MATERIALS
Review the word “area.” Hold up a sheet of paper and an index card. ASK:
Which one is bigger? (the paper) How do you know? (the index card fits
onto the paper and there is a lot of space left over) Explain to students that
we use the word “area” to describe how much space a shape takes up.
Now photocopy 2-cm grid paper (e.g., from BLM 2-cm Grid Paper) onto a
transparency, and draw a rectangle 2 squares high by 5 squares wide and
another rectangle 3 squares high by 3 squares wide. ASK: Which rectangle
is taller? Which rectangle is wider? Which one is bigger? How can you tell?
(the 2 × 5 rectangle has 10 squares and the 3 × 3 rectangle only has 9
squares, so the 2 × 5 rectangle is bigger) If students are not sure, tell them
to imagine the rectangles as two pieces of birthday cake. Which one would
they want, the one with 9 squares or the one with 10 squares?
Relating area of rectangles to multiplication. Remind students that they
can think of a rectangle as an array of squares instead of an array of dots.
Draw on the board the following picture to emphasize this connection:
3 rows of 5 dots is
3 × 5 = 15 dots
E-14
3 rows of 5 squares is
3 × 5 = 15 squares
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
index card
sheet of paper
BLM 2-cm Grid Paper (p. I-2)
grid paper
scissors
BLM Putting Rectangles Together (p. E-65)
Exercises: Have students express the area of various rectangles in terms
of multiplication.
b)
a)
(4 × 6 = 24)
(6 × 7 = 42)
c)
(5 × 5 = 25)
Making bigger rectangles from smaller rectangles. Give students grid
paper and have them cut out two rectangles: one rectangle consists of
2 rows of 3 squares and the other consists of 4 rows of 3 squares. Challenge
students to find a way to put the two rectangles together to make a new
rectangle. The area of this larger rectangle will be the sum of the areas of
the original rectangles.
ASK: What size rectangle did you get? (6 rows of 3 or 3 rows of 6, depending
on how they orient the rectangles) Explain that if you have 2 rows of 3 and
4 rows of 3, you can put one above the other to make 6 rows of 3.
Repeat the exercise with 2 rows of 5 and 4 rows of 5. When students are
finished, draw on the board the picture in the margin. Explain that if you
have one rectangle that is 2 rows of 5 and another rectangle that is 4 rows of
5, you can make a bigger rectangle that is 6 rows of 5. Now draw these two
rectangles on the board:
Have students draw in their notebooks the rectangle they think they would
get by putting one rectangle above the other (students should draw 6 rows
of 7). When students are done, emphasize that the rows in a rectangle are
just objects, and we can count them (and add them) the same way we do
other objects. Write on the board:
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2 apples + 4 apples = 6 apples
2 rows of 7 + 4 rows of 7 = 6 rows of 7
Write the areas as products to see the distributive property. Remind
students that a rectangle with 2 rows of 7 squares has 2 × 7 squares. ASK:
How many squares does a rectangle with 4 rows of 7 squares have? (4 × 7)
How many squares does a rectangle with 6 rows of 7 squares have? (6 × 7)
Then write on the board:
2 apples + 4 apples = 6 apples
2 rows of 7 + 4 rows of 7 = 6 rows of 7
(2 × 7) + (4 × 7) = 6 × 7
The last equation illustrates the distributive property of multiplication over
addition. Students do not need to know what the property is called but
should recognize and be able to apply it. Write on the board:
Number and Operations in Base Ten 4-37
E-15
(3 × 7) + (5 × 7) =
×7
ASK: 3 rows of 7 plus 5 rows of 7 is how many rows of 7? Write 8 in
the blank.
Exercises: Fill in the blanks for these questions:
a)(4 × 3) + (6 × 3) =
b)(5 × 2) + (4 × 2) =
c)(6 × 5) + (4 × 5) =
d)(3 × 6) + (2 × 6) =
×3
×2
×
×
Then have students replace each description below with a product.
a)
b)
c)
d)
3 rows of 2 + 3 rows of 5 = 3 rows of 7
4 rows of 3 + 4 rows of 5 = 4 rows of 8
2 rows of 4 + 2 rows of 5 =
rows of
3 rows of 4 + 3 rows of 5 =
rows of
Answers
a) (3 × 2) + (3 × 5) = 3 × 7, b) (4 × 3) + (4 × 5) = 4 × 8,
c) 2 rows of 9, (2 × 4) + (2 × 5) = 2 × 9,
d) 3 rows of 9, (3 × 4) + (3 × 5) = 3 × 9
If kids need extra practice, have students do BLM Putting Rectangles
Together.
Review the model below for addition. Remind students that addition can
be represented by drawing objects with a line separating the quantities
being added. Draw the picture in the margin. SAY: This picture shows
4 + 3 = 7. Have students write individually what the following pictures show.
a)
b)
Answers: a) 2 + 5 = 7, b) 3 + 5 = 8
SAY: This picture shows 3 × 4 = 12 because there are 3 groups of 4 dots,
and if we count them, we find that there are 12 dots altogether. Have
students write individually what the following pictures show.
a)
b)
E-16
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
Review the model below for multiplication. Remind students that
multiplication can be modeled by drawing groups of the same size:
Answers: a) 5 × 4 = 20, b) 6 × 3 = 18
Combining the models for addition and multiplication to illustrate the
distributive property. Draw and write on the board:
4 dots + 2 dots = 6 dots
4 groups of three + 2 groups of three = 6 groups of three
Explain that it doesn’t matter what the objects are: 4 of them plus 2 more of
them is 6 of them, and we can show that by drawing a line between the first
four and the next two. Point to each part of the last sentence on the board and
ASK: What multiplication does this show? Write the products that correspond
to each part of the sentence to create an equation below it, like this:
4 groups of three + 2 groups of three = 6 groups of three
(4 × 3)
+(2 × 3)
=(6 × 3)
Now draw this on the board:
ASK: What multiplication does this show? (7 × 4 = 28) How would you
draw a line to show that 2 groups of four + 5 groups of four = 7 groups of
four? Have a volunteer demonstrate.
Applying the distributive property without using pictures. Explain to
students that 8 × 7 means adding 8 sevens together, so we can split 8
sevens into 5 sevens + 3 sevens:
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7+7+7+7+7+ 7+7+7
Again, remind students that sevens are just objects, like circles or apples or
squares. If you have 5 of them and 3 more of them, you will have 8 of them.
So 8 × 7 = (5 × 7) + (3 × 7). Write on the board:
6+6+6+6+6+ 6+6
6+6+6+6+ 6+6+6
7 × 6 = (5 × 6) + (2 × 6)
7 × 6 = (4 × 6) + (3 × 6)
=
+
=
+
=
=
Using the 5 times table to multiply. SAY: If you have memorized the times
tables from 1 to 5, you can always use 5 times a number to multiply it by a
larger number. Write the following example on the board:
9 × 7 = (5 × 7) + (4 × 7)
Number and Operations in Base Ten 4-37
E-17
SAY: 5 × 7 and 4 × 7 are both in the times tables from 1 to 5, so this is
a good way to split the product into smaller products. Exercises: Have
students split these products into 5 times a number plus a smaller product:
a)8 × 9
b) 9 × 8
c) 7 × 6 d) 6 × 7
e) 8 × 6 f) 6 × 8
Students should ensure their answers to each pair—a) and b), c) and d), e)
and f)—are the same. Write on the board:
7×9=5×9+
×
Then write each product in words:
7 nines = 5 nines +
ASK: What do we have to add to 5 nines to get 7 nines? (2 nines) Write that
in the blank: 7 nines = 5 nines + 2 nines, so 7 × 9 = 5 × 9 + 2 × 9.
Have students copy and finish these equations individually:
a)6 × 7 = 5 × 7 +
6 sevens = 5 sevens +
b)8 × 8 = 5 × 8 +
×
8 eights = 5 eights +
c)7 × 8 = 5 × 8 +
×
×
7 eights = 5 eights +
Then have students use the smaller products to calculate each answer.
Extensions
1. Split the 2-digit number into tens and ones to find the product.
b) 3 × 28
c) 4 × 17
d) 6 × 34
e)
8 × 29
Answers
a) 7 × 16 = (7 × 10) + (7 × 6) = 70 + 42 = 112,
b) 3 × 28 = (3 × 20) + (3 × 8) = 60 + 24 = 84,
c) 4 × 17 = (4 × 10) + (4 × 7) = 40 + 28 = 68,
d) 6 × 34 = (6 × 30) + (6 × 4) = 180 + 24 = 204,
e) 8 × 29 = (8 × 20) + (8 × 9) = 160 + 72 = 232
2. Fill in the missing numbers.
4 + 8 + 12 = 4 ×
40 + 80 + 120 = 4 ×
E-18
Teacher’s Guide for AP Book 4.1
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a)7 × 16 NBT4-38 Advanced Arrays
Page 108
Goals
STANDARDS
4.NBT.5
Students will use arrays to understand the distributive law.
PRIOR KNOWLEDGE REQUIRED
Vocabulary
Can apply the distributive property
Can represent multiplication problems using arrays
array
MATERIALS
two colors of base ten blocks (different colors for tens and ones blocks)
Review writing multiplication expression 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)
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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)
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.
(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
Number and Operations in Base Ten 4-38
E-19
the sum of two smaller products. Student Book p. 108 Question 2a, 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: Make a model of the number 25 using different 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: Break the array into two separate arrays to show that
4 × 25 = (4 × 20) + (4 × 5).
4 × 20
4× 5
Extension
Using the 10 times table to multiply. (NOTE: Students in Massachusetts
can do this extension to satisfy MA.5a. In other states, this extension can be
used as a Bonus for students who work quickly.)
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-66).
E-20
Teacher’s Guide for AP Book 4.1
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Have students use the 10 times table to multiply by 11 and 12. For example:
NBT4-39 Mental Math
Page 109
Goals
STANDARDS
4.NBT.5
Students will multiply large numbers by breaking them into
smaller numbers.
PRIOR KNOWLEDGE REQUIRED
Can represent multiplication problems using arrays
Can apply the distributive property
Understands that multiplication is commutative, a × b = b × a
Can represent numbers using base ten materials
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, as in the last lesson. 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:
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
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When students are comfortable splitting products into the sum of two
smaller products, have them solve each problem.
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 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 (see the margin on the next page). 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 them
Number and Operations in Base Ten 4-39
E-21
2 × 324
solve additional problems without drawing models. Finally, have them
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
Extensions
1.Have students combine what they learned in this lesson with what they
learned about multiplying by multiples of 10 in Lesson NBT4-36. Ask
them to multiply 3-digit numbers by multiples of 1,000 or 10,000.
= (2 × 300) + (2 × 20) + (2 × 4)
a)342 × 2,000 b) 320 × 6,000 c) 324 × 2,000 d) 623 × 20,000
2.Have students do Questions 1 and 2 on BLM Using Area to Find
Equal Products (p. E-67). 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 =
Students can also do BLM Moving Rectangles Around (p. E-68–E-69)
for further practice. This BLM requires students to imagine cutting and
moving rectangles instead of actually cutting and moving them.
As 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 the other 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)
E-22
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) 80, b) 120
NBT4-40 Using Doubles to Multiply
Page 110-111
STANDARDS
3.OA.5, preparation for
4.NBT.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
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Answers: a) 46, b) 28, c) 62, d) 48, e) 104, f) 68, g) 126, h) 108
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. 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.
Number and Operations in Base Ten 4-40
E-23
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 at least 5. 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
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-70).
E-24
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) 36, 72, 144, 288, b) 36, 72, 144, 288, c) 28, 56, 112, 224, 448
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 Student Book p. 111
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 questions
(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
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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:
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)
Number and Operations in Base Ten 4-40
E-25
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.
E-26
Teacher’s Guide for AP Book 4.1
NBT4-41 Multiplying in Different Ways
Page 112
Goals
STANDARDS
4.NBT.5
Students will multiply the same numbers in different ways in order to
verify their own answers.
Vocabulary
double
half
how many more
how many times as many
product
PRIOR KNOWLEDGE REQUIRED
Can use doubles to multiply
Can multiply by splitting a product into two smaller and easier products
MATERIALS
BLM The Blank 12 × 12 Multiplication Chart (p. E-64)
BLM The 10 × 10 Times Table Chart (p. E-71)
BLM The 12 × 12 Times Table Chart (p. E-72)
(MP.7)
Finding the same product many ways. Together as a class, find 8 × 5
four ways.
a)
b)
c)
d)
Double 4 × 5. (double 20 is 40)
Add 5 × 5 and 3 × 5. (25 + 15 = 40)
Subtract 2 × 5 from 10 × 5. (50 – 10 = 40)
Find half of 8 × 10. (half of 80 is 40)
Emphasize that you got the same answer all four ways.
Exercises: Find 8 × 6 two ways.
a)8 × 6 is
b)8 × 6 is
more than 8 × 5 = 40, so 8 × 6 =
more than 7 × 6 = 42, so 8 × 6 =
.
.
Answers: a) 8, 40 + 8 = 48, b) 6, 42 + 6 = 48
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Point out that if students do not get the same answer both ways, they
should look for a mistake.
Bonus
Use times as many to multiply 8 × 6 three more ways.
c)8 × 6 is
times as many as 2 × 6 = 12, so 8 × 6 =
d)8 × 6 is
times as many as 8 × 2 = 16, so 8 × 6 =
e)8 × 6 is
times as many as 4 × 6 = 24, so 8 × 6 =
.
.
.
Answers: c) 4, 12 × 4 = 48, d) 3, 16 × 3 = 48, e) 2, 24 × 2 = 48.
Double Bonus
1.Fill in the blanks to find 6 × 7 four ways. Be sure to get the same
answer all four ways.
a)6 is
times as many as 3, so 6 × 7 is
3 × 7. So 6 × 7 is
×
=
Number and Operations in Base Ten 4-41
times as many as
E-27
b) 6 is
times as many as 2, so 6 × 7 is
2 × 7. So 6 × 7 is
×
=
c) Use 6 × 7 = (5 × 7) + 7 to find 6 × 7.
d) Use 6 × 7 = (6 × 5) + (6 × 2) to find 6 × 7.
times as many as
Answers: a) 2, 2, 2 × 21 = 42, b) 3, 3, 3 × 14 = 30 + 12 = 42,
c) 35 + 7 = 42, d) 30 + 12 = 42
(MP.8)
Another way to compute the 9 times table. Write on the board:
1×9=
2×9=
3×9=
4×9=
5×9=
1
2
3
4
9
8
7
6
5
6×9=
7×9=
8×9=
9×9=
10 × 9 =
Discuss and explain any patterns that students notice in the first part of
the 9 times table. (Answers: The tens digits increase by 1, the ones digits
decrease by 1. This occurs because adding 9 is the same as adding 10 and
then subtracting 1, so to add 9 to a number from 9 to 90, add 1 to the tens
digit and then take 1 from the ones digit.) Have students predict the rest of
the times table by using the patterns in the tens and ones digits. Discuss
any other patterns students may notice. Example: The two digits add to 9.
To prompt students to notice that the two digits add to 9, SAY: The tens digit
always increases by 1 and the ones digit always decreases by 1. How does
the sum change? (it doesn’t—it stays the same) Emphasize that when one
number goes up and the other number goes down by the same amount,
their sum stays the same. ASK: What do the two digits always add to? (9)
SAY: I wonder if we can use the fact that the digits always add to 9 to make
knowing the 9 times table easier? Tell students you will give them the tens
digit of various multiples of 9, as hints, and they just have to tell you the
ones digit. Examples:
b) 8 × 9 = 7
e) 7 × 9 = 6
c) 5 × 9 = 4
SAY: If only we had an easy way of knowing what the tens digit is. Tell
students to look for a pattern. ASK: How can we get the tens digit from the
number we are multiplying 9 by? (subtract 1) Prompt students to see this by
circling the relevant numbers in the previous examples:
a)6 × 9 = 5
d)9 × 9 = 8
b) 8 × 9 = 7
e) 7 × 9 = 6
c) 5 × 9 = 4
Now, have students fill in only the tens digit for these products:
a)8 × 9 =
c)4 × 9 =
e)9 × 9 =
b)6 × 9 =
d)7 × 9 =
f)5 × 9 =
Then have students fill in both digits for the same problems as above.
(Subtract 1 from the number that 9 is multiplied by to find the tens digit,
then subtract the tens digit from 9 to get the ones digit.)
E-28
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
a)6 × 9 = 5
d)9 × 9 = 8
ACTIVITIES 1–2
1.Find 9 × 8 in as many different ways as possible. Do this all
together as a class or challenge students to find more answers
than you, the teacher. (If you choose to play “against” the class,
ensure that the class wins. Give hints and/or give up after three or
four turns.) Here are 10 ways to find 9 × 8:
• Skip count by 9s until holding up 8 fingers
• Skip count by 8s until holding up 9 fingers
• Double 9 × 4 = 36 to get 9 × 8 = 72
• Multiply 3 × 8 = 24 by 3 to get 9 × 8 = 72
• Multiply 9 × 2 = 18 by 4 to get 9 × 8 = 72
• Use 10 eights – 1 eight = 9 eights
• Add 5 × 8 = 40 and 4 × 8 = 32 to get 9 × 8 = 72
• Add 9 × 5 = 45 and 9 × 3 = 27 to get 9 × 8 = 72
• Add 8 × 8 = 64 and 1 × 8 = 8 to get 9 × 8 = 72
• Subtract 1 from the number you are multiplying 9 by, to get the
tens digit. (8 – 1 = 7) Subtract the tens digit from 9 to get the
ones digit. (9 – 7 = 2) So 9 × 8 = 72.
If the class reaches 10 ways quickly, challenge them to see how
many more they can find altogether.
2.Students can play the following game in pairs: Player 1 picks
two numbers (up to 12) and multiplies them. Player 1 should
choose numbers that will produce a 2-digit product. For example,
5 × 7 = 35. Player 2 multiplies the digits in the product. In this
case, Player 2 multiplies 3 × 5 = 15. Players continue multiplying
the digits in each other’s products until they reach a product with
only 1 digit, in which case play stops. The goal is to have at least 4
turns in each sequence (2 per player). Examples:
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
Player 1: 9 × 8 = 72
Player 2: 7 × 2 = 14
Player 1: 1 × 4 = 4; play ends, goal not reached.
Player 2: 7 × 7 = 49
Player 1: 4 × 9 = 36
Player 2: 3 × 6 = 18
Player 1: 1 × 8 = 8; play ends, goal reached.
Player 2: 8 × 11 = 88
Player 1: 8 × 8 = 64
Player 2: 6 × 4 = 24
Player 1: 2 × 4 = 8; play ends, goal reached.
Notice that the player whose turn ends a sequence is not the one
to start the next one. Before pairs play independently, create a
few sequences together as a class. Start with numbers that will
produce sequences that are too short (2 or 3 turns). Examples:
Number and Operations in Base Ten 4-41
E-29
5 × 4 (2 turns), 3 × 7 (2 turns), 12 × 6 (3 turns). Point out that
none of these examples produce a sequence with 4 turns, which is
what you want students to find.
Pairs should try to produce as many different 4-turn sequences
as possible. Students should keep track of all the numbers they
start with and the sequences these produce, so that they don’t
repeat a sequence. (Students could use BLM The Blank 12 × 12
Multiplication Chart to keep track of all the pairs of numbers they
start with, put an X in the boxes that don’t work, and a checkmark in
the boxes that do.) After pairs have played for some time, give them
an opportunity to share their successful sequences with the class.
What have students learned so far? Distribute copies of BLM The 10 × 10
Times Table Chart or BLM The 12 × 12 Times Table Chart. Ask students
to try to complete all the times tables in either multiplication chart. As with
the first time you asked students to do this, at the beginning of the unit
(Lesson NBT4-34), emphasize that you will not be evaluating the students’
work in any way. When the time allotted for this exercise has elapsed, tell
students to compare their work on the chart now with their work on the
chart at the beginning of the unit. Did they complete more of the chart? Did
they do it more quickly? Was it easier to work on the chart? How did the
tricks and shortcuts they’ve learned in the unit so far help them to complete
the chart this time? Emphasize that practice and learning make us able to
do things better and more quickly. If students are struggling with a new
concept or skill, encourage them to keep at it. Eventually, they will “get it”
and it will no longer be a struggle.
Extension
Teach students to use the diagonal entries of a multiplication chart (the
perfect squares) to find the entries directly above and below the diagonal
entries. First show an array like the following:
Remind students that the area of the big rectangle is the sum of the areas
of the square and the gray rectangle. Write on the board:
E-30
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
Students should check their times tables by verifying that various
combinations of rows (or columns) add to the correct larger-numbered row
(or column). For example, the row for 3 and the row for 5 should add to
the row for 8, the row for 3 and the row for 7 should add to the row for 10.
Using the row for 10 is particularly convenient if students are familiar with
pairs adding to 10.
Area of square + Area of gray strip
×
+
=
=
Total area
×
Count the number of squares across and down and label the rows
and columns as shown:
123456
1
2
3
4
5
Have students write the equation for the sum of the areas individually.
Answer: (5 × 5) + 5 = 5 × 6
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Distribute copies of BLM Using Diagonals to Multiply (p. E-73–E-74). After
students finish Question 1, point out that they can use 5 × 5 to find 5 × 6 by
adding 5: 25 + 5 = 30, so 5 × 6 = 30. But 5 × 6 + 6 = 6 × 6 (demonstrate
this by adding another row to the array) so students can check their answer
to 5 × 6 by adding 6. Do they get 6 × 6? Yes, because 30 + 6 = 36 = 6 × 6.
Have students do Questions 2–4. Then introduce the process on the
second page of the BLM. Explain to students that if they can remember the
diagonal entries in the multiplication chart, then they can find the entries
above and below the diagonal entries. Plus, they can check their answers.
Demonstrate using the example in the gray box on the BLM. Then have
students complete the BLM.
Number and Operations in Base Ten 4-41
E-31
NBT4-42 The Standard Multiplication Method
Page 113
(No Regrouping)
Goals
STANDARDS
4.NBT.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 multiple of 10 and 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-32
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
(MP.6, MP.3)
Practice 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 © 2012 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 0-5.
a)
2
4
×
8
6
Answers: 5,243 × 2 = 10,486
Number and Operations in Base Ten 4-42
E-33
NBT4-43 Multiplication with Regrouping
Page 114
STANDARDS
4.NBT.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 (p. E-75–E-76)
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-34
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2012 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 =
32 × 7 =
ten +
tens
ones
tens + 4 ones
So 32 × 7 =
b) 1 × 5 =
+ 40 × 5 =
41 × 5 =
ten +
tens
ones
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 4-43
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-35
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
1
4
2
Then finish writing the answer as shown in the margin.
2
7
×
9
4
Have volunteers finish these problems:
a)
(7 × 4) + 1
b)
4
2
×
5
9
5
(9 × 2) + 4
E-36
= 29 tens + 4 ones
3
×
c)
4
6
8
8
7
×
d)
1
2
5
0
2
3
×
4
6
4
(8 × 3) + 4
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
×
= 1 ten + 4 ones
Exercises: Have students copy these questions and finish the multiplication.
a)
b)
1
2
+1
2
9
7
4
3
e)
3
2
8
f)
4
3
5
4
2
5
× 3
× 2
× 5
× 4
× 8
× 8
5
4
5
2
0
0
Using standard algorithm. Write problems like the following on the board:
2
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
7
×
3
d)
1
Some students may need to write the intermediate product instead of
holding it in their heads. Demonstrate as shown in the margin.
28
4
5
c)
1
4
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.
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 B would work if regrouping wasn’t required.
For example,
32
× 4
128
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
(MP.3)
It’s only because there are more than ten ones in 2 × 7 that Method B
doesn’t work. Method B 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 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.
Number and Operations in Base Ten 4-43
E-37
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.
8
4
×
×
2
5
2
2
Answers: 84 × 3 = 252, 42 × 5 = 210
COPYRIGHT © 2012 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-38
Teacher’s Guide for AP Book 4.1
NBT4-44 Multiplying with the 6, 7, 8, and 9
Page 115-116
Times Table
Goals
STANDARDS
4.NBT.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.
(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.
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Review a pattern in the 9 times table. Review 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.
(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
(MP.7)
b) 76 × 9
f) 77 × 8
c) 37 × 6
d) 48 × 9
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
Number and Operations in Base Ten 4-44
E-39
numbers) I know that 7 × 8 is 56. What is 8 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
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
a)84 × 7
e)47 × 7
E-40
Teacher’s Guide for AP Book 4.1
NBT4-45 Multiplying a Multi-Digit Number by a
Page 117-119
1-Digit Number
Goals
STANDARDS
4.NBT.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-77)
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
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
(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 4-45
E-41
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
213
× 4
3 × 4:
12
10 × 4:
40
200 × 4: 800
852
×4
+
+
=
Have a volunteer fill in the blanks. Then demonstrate the modified standard
algorithm for the same problem as shown in the margin.
(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:
1
×
4
8 5 2
3 ones × 4 = 12 ones = 1 ten + 2 ones
1 ten × 4 = 4 tens
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)
Have students do additional problems in their notebooks. Students should
solve each problem using base ten materials, expanded form, and the
E-42
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
2 1 3
standard algorithm. Exercises:
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
Describe each step of the process as you demonstrate it, 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.
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
Have students do several problems with 0 in the 3-digit number. Exercises:
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.
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:
Number and Operations in Base Ten 4-45
E-43
a)
b)
c)
d)
e)
2-digit numbers by 2-digit numbers
1-digit numbers by 4-digit numbers
2-digit numbers by 3-digit numbers
3-digit numbers by 4-digit numbers
3-digit numbers by 5-digit numbers
Answers: a) 4, b) 5, c) 5, d) 7, e) 8
Challenge your 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 © 2012 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-44
Teacher’s Guide for AP Book 4.1
NBT4-46 Word Problems with Multiplying
Page 120
Goals
STANDARDS
4.NBT.5, 4.OA.2
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 your 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.
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
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 down 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 4-46
E-45
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 it.
Extensions
1.Give your 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 story problem involving multiplication and
solve it.
E-46
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
2. Add the missing information to the story problems and solve them.
NBT4-47 More Mental Math
Page 121
Goals
STANDARDS
4.NBT.5
Students will multiply 2-digit numbers mentally when one of the numbers is a multiple of 10.
Vocabulary
double
multiple
PRIOR KNOWLEDGE REQUIRED
Can double multi-digit numbers by using the distributive property
Can multiply round numbers by round numbers
MATERIALS
BLM 20 × 32 (p. E-78)
Review multiplying 1-digit numbers by multi-digit numbers mentally.
Only include numbers that require either no regrouping at all (Examples:
2 × 324, 3 × 132, 4 × 201), or only regrouping at the largest place value
(Examples: 4 × 612, 2 × 804, 3 × 430).
Using arrays to mutiply by multiples of 10. Show the following array on
the board (e.g. from BLM 20 × 32):
32
10
20
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
10
Write on the board:
20 × 32 = 2 × (10 × 32)
= 2 × 320
= 640
(MP.3, MP.2)
Ask a volunteer to explain how the picture shows why this works (a 20 × 32
array is the same as two 10 × 32 arrays put together.)
ASK: How many 10 × 32 arrays are in a 30 × 32 array? Have students write
a multiplication statement to show this: 30 × 32 = 3 × (10 × 32). ASK:
What is 10 × 32? So what is 30 × 32? (3 × 320 = 460)
Number and Operations in Base Ten 4-47
E-47
Exercises: Solve these problems by splitting the arrays into arrays with one
side having length 10.
a)20 × 14
b) 30 × 23
14
23
20
42
30
30
Answers
14
a)
b)
10
10
c) 30 × 42
c)
23
10
10
10
10
10
10
42
2 × 140 = 280
3 × 230 = 690 3 × 420 = 1,260
Mentally multiplying by multiples of 10 without arrays. Have students
solve these problems by splitting the products into products of 3 numbers,
one of which is 10. Demonstrate the first solution yourself. Exercises:
a)20 × 13
b) 20 × 24
c) 40 × 21
d) 30 × 81
Answers: a) 260, b) 480, c) 840, d) 2,430
Sample Solution: 20 × 13 = 2 × (10 × 13) = 2 × 130 = 260
Challenge students to find several products mentally by following these
steps. Example: 30 × 21
Step 1: Multiply the second number by 10. Example: 21 × 10 = 210
Step 2: Multiply the result by the tens digit of the first number.
Example: 3 × 210 = 630
Exercises
b) 70 × 21
c) 40 × 72 d) 30 × 53
Answers: a) 1,280, b) 1,470, c) 2,880, d) 1,590
Relating this lesson to the associative property. Have students write each
number as a product of two numbers, where one of the numbers is 10.
a)40 =
b) 30 =
c) 70 =
Point out that 40 × 12 = (4 × 10) × 12, and we found earlier that
40 × 12 = 4 × (10 × 12) using arrays. Write on the board:
(4 × 10) × 12 = 4 × (10 × 12)
Point out that this is a property of any three numbers. You can multiply the
first two first, or you can multiply the last two first. Have students verify this
with several examples.
E-48
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
a)40 × 32
Exercises: Be sure to get the same answer both times.
a) Calculate (3 × 2) × 4 and 3 × (2 × 4)
b) Calculate (2 × 5) × 6 and 2 × (5 × 6)
c) Calculate (2 × 2) × 3 and 2 × (2 × 3)
Answers: a) 6 × 4 = 24 and 3 × 8 = 24, b) 10 × 6 = 60 and 2 × 30 = 60,
c) 4 × 3 = 12 and 2 × 6 = 12
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
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 your students,
then teach your students 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
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
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 your students that if the ones digit is 4 or less, they round down,
30; 35
40.
and if the ones digit is 5 or more they round up: 34
Number and Operations in Base Ten 4-47
E-49
NBT4-48 Multiplying 2-Digit Numbers by
Page 122
Multiples of 10
STANDARDS
4.NBT.5
Vocabulary
standard algorithm
Goals
Students will multiply 2-digit numbers by 2-digit multiples of 10 (10, 20,
30 and so on to 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 20 × 36 (p. E-79)
Review multiplying 2-digit numbers by 1-digit numbers using the
standard algorithm. Exercises:
a)37 × 2
b) 45 × 3
c) 38 × 7
d) 32 × 5
Use arrays to review the distributive property. Write on the board:
36 × 20 = (6 × 20) + (30 × 20)
and show the following picture from BLM 20 × 36:
20
6
+
30
Write on the board:
6 × 20 = 120
+ 30 × 20 = 600
36 × 20 = 720
E-50
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
36
Emphasize the similarity between multiplying 36 × 20 and 36 × 2. Write
on the board:
6 × 2 = 12
+ 30 × 2 = 60
36 × 2 = 72
SAY: We are really doing the same thing to find 36 × 20 as we did to find
36 × 2. We just added an extra zero to both terms in the sum. Have students
solve more problems this way. Exercises:
a)27 × 30
b) 34 × 60
c) 23 × 50
d) 18 × 30
Answers
a)(7 × 30) + (20 × 30) = 210 + 600 = 810
b)(4 × 60) + (30 × 60) = 240 + 1,800 = 2,040
c)(3 × 50) + (20 × 50) = 150 + 1,000 = 1,150
d)(8 × 30) + (10 × 30) = 240 + 300 = 540
(MP.8)
The standard algorithm for multiplying 2-digit numbers by 2-digit
multiples of 10. Ensure students understand that the standard algorithm
follows the same steps as above, but takes short cuts:
Step 1: Multiply 6 × 20.
1
36
× 20
202 × 6 = 12, so 20 × 6 = 120
We can write the tens and ones in the answer because there won’t be
any more tens or ones from multiplying 30 × 20. But we write the 1 in a
temporary hundreds column because there will still be hundreds to add
from 30 × 20.
Step 2: Multiply 30 × 20.
1
36
× 20
7203 × 2 = 6, so 30 × 20 = 600
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
Put your answer in the hundreds column of the answer (don’t forget to add
the 1 hundred from 20 × 6).
Have students practice the first step in each problem:
3
a)
29b)
34c)
29d)
73
× 40
× 70
× 70
× 50
60
Students should be able to explain how this step records, for example, the
fact that 9 × 40 = 360, and why the 3 is in the hundreds column, the 6 is in
the tens column, and the 0 is in the ones column.
Number and Operations in Base Ten 4-48
E-51
When students have mastered this, remind your students that we want to
finish solving 29 × 40 and so far, we have only found 9 × 40. ASK: What
else do we need to do? (We need to find 20 × 40 and then add the result to
9 × 40 to find 29 × 40.)
Show students how to finish solving 29 × 40 using the standard algorithm:
20 × 40 is 800 and we add the regrouped 300 to get 1,100.
9
29
× 4 0
1160
Have students finish b) to d) above.
Some students may need to write the intermediate product instead of
holding it in their heads. Demonstrate as follows:
8
+3
29
×
40
1160
Exercises: Have students copy these questions onto grid paper and
combine both steps of the standard algorithm:
a)35 × 40
b) 62 × 70
c) 84 × 50
d) 53 × 60
Emphasize the similarity between multiplying 37 × 20 and 37 × 2. Write
the following products on the board:
(MP.3)
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?
(MP.7)
When the multiple of 10 is the first factor instead of the second. Then
show students the following problem: 30 × 53. ASK: How is this different
from the other problems we’ve seen so far? (the multiple of 10 is the first
number, not the second) ASK: How can you do this problem? (change it to
53 × 30, since they will have the same answer) Point out that changing to a
problem students already know how to do is a really clever way of solving
a problem. If they know two problems have the same answer, they might as
well do the easier problem. Exercises:
a)20 × 34
b) 30 × 58
c) 60 × 17
Answers: a) 680, b) 1,740, c) 1,020
E-52
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
1
1
3 7
3 7
× 2
× 2 0
7 4 7 4 0
NBT4-49 Multiplying 2 Digits by 2 Digits
Page 123-125
Goals
STANDARDS
4.NBT.5
Students will multiply 2-digit numbers by 2-digit numbers.
PRIOR KNOWLEDGE REQUIRED
Vocabulary
Can multiply a 2-digit 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
Write the equations out with blanks if students need a prompt. Example:
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
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).
Practice adding the two parts. Write on the board: 23 × 7 = 161 and
23 × 50 = 1,150. ASK: What is 23 × 57? (161 + 1150 = 1,311)
Number and Operations in Base Ten 4-49
E-53
Now write the 23 times table on the board:
232323232323232323
× 1
× 2
× 3
× 4
× 5
× 6
× 7
× 8
×9
23 46 69 92115138161184207
ASK: What is 23 × 6? (138) What is 23 × 60? (1,380) How do you know?
(because 23 × 60 is 23 × 6 × 10) What is 23 × 67? (1,380 + 161 = 1,541)
Have students use these facts to multiply 23 by various 2-digit numbers.
Exercises: 23 × 45; 23 × 78; 23 × 46; 23 × 64.
Answers: 1,035; 1,794; 1,058;1,472.
If students need scaffolding, provide the following structure, either on the
board or to individual students (to find 23 × 45):
23 × 4 =
and 23 × 5 =
Bonus: 23 × 4 =
and 23 × 5 =
so 23 × 40 =
so 23 × 45 =
so 23 × 400 =
so 23 × 405 =
Now draw a 43 × 27 rectangle on the board and have students copy the
rectangle and separate it into two smaller rectangles so that the area of
each is easier to find. Take different answers. (43 × 20 and 43 × 7 OR
40 × 27 and 3 × 27) Have students find the areas of the smaller rectangles
and then add them together to find the area of the larger rectangle. Have
students use this method to find several products of 2-digit numbers.
Exercises: 54 × 45; 36 × 44; 47 × 68, Bonus: 43 × 502.
Answers: 2,430; 1,584; 3,196, Bonus: 21,586.
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:
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
20
7
3
3 × 20
40 × 7
3×7
E-54
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
(MP.7, MP.2)
Then draw this picture.
50
2
30
9
ASK: What product can you determine by finding the area of the big
rectangle? (52 × 39) What products can you determine by finding the areas
of the small rectangles? Give students time to write down the four products.
Have students check their answers with a partner. Then take up the
answers: 50 × 30; 2 × 30; 50 × 9; and 2 × 9. ASK: What makes these four
products so easy to find? (we just have to know how to multiply single-digit
numbers, and multiples of 10) Have students use the four smaller products
to find 52 × 39 then check their answers with a partner. Point out before
they check with a partner that if they do not get the same answer, they
should then check to see if one of the four products is different, and then if
they are all the same, look at the addition.
(MP.8)
Relating the standard algorithm to using the sums of the four easy
products. Now show students the solution to this problem using the
standard algorithm:
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
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)
(MP.3)
Exercise: Have students show how each of the four products that are
added were obtained:
Number and Operations in Base Ten 4-49
E-55
73
× 49
27(
630(
120(
+ 2,800(
3,577
×
×
×
×
)
)
)
)
Then have students rewrite the problem using the standard algorithm
notation. Ensure that students write the regrouping in the correct column:
12
73
× 49
657
+ 2,920
3,577
Notice that students will need to show regrouping twice: once when
multiplying 3 × 9 (carry the 2 to the tens column) and once when
multiplying 3 × 40 (carry the 1 to the hundreds column). This will be easier
for students to do if they use grid paper.
Repeat for other products, first having students write how the four products
were obtained, then having them write the product using the standard
algorithm notation. Exercises: 46 × 58; 34 × 29; 67 × 76.
Answers: 2,668; 986; 5,092.
Exercises: Use the standard algorithm only to do these problems:
a)73 × 46
b) 54 × 35
c) 46 × 71
d) 84 × 96
Answers: a) 3,358, b) 1,890, c) 3,266, d) 8,064
Bonus
Add enought zeros to a product of two 2-digit numbers to multiply:
a)780 × 640
b) 3,400 × 250
c) 8,700 × 9,400
Estimating sums and products. Explain to students that they don’t always
need an exact answer, but often 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.
E-56
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) 499,200, b) 850,000, c) 81,780,000
(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 you 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.
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
1. a)Give students 20 estimation problems to do, including 10 additions
and 10 multiplications (2-digit by 2-digit):
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AdditionMultiplication
32 + 85
32 × 85
41 + 24
41 × 24
36 + 54
36 × 54
72 + 41 72 × 41
72 + 49
72 × 49
89 + 51
89 × 51
39 + 68
39 × 68
45 + 56
45 × 56
32 + 87
32 × 87
33 + 82 33 × 82
Students could check their estimates using an online tool. See
http://teacherlink.org/estimate for one such tool. Students could
record the number of good estimates they produced for the 10
addition problems and the 10 multiplication problems on their first
attempts, or they could count how many attempts they needed to
estimate correctly. Either way, students will get an idea of whether
they are better at estimating sums or products.
(MP.8)
b)Have students construct a test to determine if they are better at
estimating products of two 2-digit numbers or products of a 3-digit
number and a 1-digit number.
2.On BLM Patterns in Multiplication (p. E-80), students discover an
easy way to multiply a 2-digit number having ones digit 5 by itself:
15 × 15, 25 × 25, 35 × 35, and so on.
Number and Operations in Base Ten 4-49
E-57
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
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.
E-58
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
a)105 × 105
d)9,995 × 9,995
NBT4-50 Topics in Multiplication
Page 126
Goals
STANDARDS
4.NBT.5, 4.OA.2
Students will consolidate their understanding of the concepts learned
so far in multiplication.
Vocabulary
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-81)
BLM Define a Number (p. E-82)
This lesson is mostly a review of concepts learned so far.
ACTIVITY
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
(MP.3)
Race against time. 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 which
column the fact should go into. Students may choose to come back to
a phrase that they cannot decide on or that they have already placed.
Students win if all phrases are in the correct column when the time is up.
Variation: To make the game more difficult, you can assign secret
volunteers to purposefully argue for the wrong answer. Be sure to tell the
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 if even 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.
Number and Operations in Base Ten 4-50
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(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 Student Book p. 126
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 of 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.
Answer: 4 × 53 = (4 × 50) + (4 × 3) and 5 × 43 = (5 × 40) + (5 × 3).
Since 4 × 50 = 5 × 40, so 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 more because 8 × 32 is more than 7 × 32, and
8 × 700 = 7 × 800.
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Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
Now challenge students to compare 4 × 53 to 5 × 43 using the same
strategy.
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.
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
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.
Number and Operations in Base Ten 4-50
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