Products of 2-Digit
Numbers, Part 2
Objective To guide children as they extend the partial-products
algorithm to products of any two 2-digit numbers.
a
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Algorithms
Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Apply place-value concepts to find
partial products. [Number and Numeration Goal 1]
• Use addition to add partial products. [Operations and Computation Goal 2]
• Use multiplication facts to calculate
partial products. [Operations and Computation Goal 3]
• Use base-10 blocks and arrays to
model multiplication. [Operations and Computation Goal 6]
Key Activities
Children use the arrays they created in
Lesson 9 10 to extend the partial-products
algorithm to any two 2-digit numbers.
Ongoing Assessment:
Informing Instruction See pages 780
and 781.
Family
Letters
1 2
4 3
Playing Beat the Calculator
(Multiplication)
Interactive
Teacher’s
Lesson Guide
Differentiation Options
ENRICHMENT
Exploring Egyptian Multiplication
Math Masters, pp. 308 and 309
Children explore the Egyptian
multiplication algorithm.
Ongoing Assessment:
Recognizing Student Achievement
Using the Lattice Method
Use an Exit Slip (Math Masters,
page 398). [Operations and Computation Goal 3]
Math Boxes 9 12
Math Journal 2, p. 236
Children practice and maintain skills
through Math Box problems.
Home Link 9 12
Math Masters, p. 307
Children practice and maintain skills
through Home Link activities.
Advance Preparation
Teacher’s Reference Manual, Grades 1– 3 pp. 108, 109
Multiplication and Division
Curriculum
Focal Points
Math Journal 2, p. 282
Student Reference Book, p. 279
per group: calculator
Children practice multiplication facts.
Math Journal 2, pp. 230 and 235
Home Link 9 11
Math Masters, p. 306
transparency of Math Masters, p. 306
(optional) half-sheet of paper red, green,
and blue erasable markers (optional) base -10 blocks (optional) overhead
base -10 blocks (optional)
Unit 9
Common
Core State
Standards
Ongoing Learning & Practice
Materials
778
Assessment
Management
EXTRA PRACTICE
Math Journal 2, p. 235
Math Masters, pp. 310 and 311
Children practice the lattice method
with 2-digit numbers.
ELL SUPPORT
Using a Graphic Organizer
Differentiation Handbook, p. 34
Children use a graphic organizer
from the Differentiation Handbook.
Mathematical Practices
SMP1, SMP2, SMP3, SMP4, SMP5, SMP6, SMP7
Content Standards
Getting Started
3.OA.3, 3.OA.4, 3.OA.5, 3.OA.7, 3.NBT.2, 3.NBT.3,
3.MD.5b, 3.MD.7a, 3.MD.7b, 3.MD.7c, 3.MD.7d
Mental Math and Reflexes
Math Message
Write these problems on a
half sheet. Solve and show your work.
Write the following problems on the board. Have children use mental math
to solve.
5 [70s] 350
30 [6s] 180
40 [2s] 80
80 [50s] 4,000
60 [90s] 5,400
70 [80s] 5,600
70 [400s] 28,000
3,000 [80s] 240,000
90 [7,000s] 630,000
20 × 34 =
680
70 × 48 =
3,360
Home Link 9 11
Follow-Up
Partners share methods for
solving Problems 4 and 5.
Briefly review answers.
Algorithm Project The focus of this
lesson is the partial-products algorithm.
To teach U.S. traditional multiplication,
see Algorithm Project 3 on page A10.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
Children share solution strategies. The partial-products
algorithms are written below.
34
× 20
20 [30s] → 600
20 [4s] → + 80
600 + 80 → 680
48
× 70
70 [40s] → 2,800
70 [8s] → + 560
2,800 + 560 → 3,360
Extending the Partial-Products
WHOLE-CLASS
DISCUSSION
Algorithm
NOTE Math Masters, page 306 is a
duplicate of Math Journal 2, page 230.
Student Page
Date
Time
LESSON
9 10
(Math Journal 2, p. 230; Math Masters, p. 306)
1.
Array Multiplication 3
How many squares are in a 17-by-34 array?
Total squares =
17 × 34 =
578
578
20
Use a transparency or a copy of Math Masters, page 306 to model
the partial-products algorithm. Ask children to turn to the first
array on journal page 230. Show them how to find the total number
of squares in the array using the partial-products algorithm.
10 [30s]
10 [4s]
7 [30s]
7 [4s]
300 + 40 + 210 + 28
→
→
→
→
→
34
× 17
300
40
210
+ 28
578
10
0
2.
10
20
How many squares are in a 22-by-28 array?
30
Total squares =
22 × 28 =
616
616
20
10
NOTE Encourage children to add the partial products mentally. They may think
300 plus 200 is 500. 500 plus 40 is 540. 540 plus 10 is 550. 550 plus 20 is 570.
570 plus 8 is 578.
0
10
20
30
Math Journal 2, p. 230
204-239_EMCS_S_MJ2_G3_U09_576418.indd 230
4/11/11 3:26 PM
Lesson 9 12
779
34
⫻ 17
30 ⫻ 10
30 ⫻ 7
7⫻4
10 ⫻ 4
Adjusting the Activity
ELL
Have children draw lines connecting each pair of digits as they multiply.
Explain that the lines will form a bowtie shape if they have found all the partial
products. Note that the partial products may be found in any order. (See margin.)
A U D I T O R Y
K I N E S T H E T I C
T A C T I L E
V I S U A L
Ongoing Assessment: Informing Instruction
Watch for children who need support with partial products of multiples of 10 and
2-digit numbers. Have them write one number in expanded notation before
multiplying. For example, in 70 × 48, 48 would be written as 4 tens 8 ones, or
40 + 8. The multiplication becomes 70 × 40 and 70 × 8.
NOTE When using the partial-products
algorithm to multiply 2-digit numbers, children
are applying the Distributive Property of
Multiplication over Addition two times.
For example:
Children refer to the array diagram in their journals and match
each part of the diagram with a partial product as you do the
same at the overhead.
17 × 34 = 17 × (30 + 4)
= (17 × 30) + (17 × 4)
= [(10 + 7) × 30] + [(10 + 7) × 4]
= (10 × 30) + (7 × 30) + (10 × 4) + (7 × 4)
= 300 + 210 + 40 + 28
= 578
10
The diagram to the right provides a visual
representation of the partial products in
34 × 17.
0
10
20
30
●
There are 30 green squares in each of 10 rows, so there are
300 green squares (10 [30s]) in all.
●
There are 4 red squares in each of the 10 rows to the right of
the green squares, or 40 red squares (10 [4s]) in all.
●
There are 30 red squares in each of the 7 rows above the green
squares, or 210 red squares (7 [30s]) in all.
●
There are 4 blue squares in each of 7 rows, or 28 blue squares
(7 [4s]) in all.
●
There are 578 squares (300 + 40 + 210 + 28) total in
the array.
With children’s help, use the algorithm to solve the second problem
on the journal page.
20 [20s]
20 [8s]
2 [20s]
2 [8s]
400 + 160 + 40 + 16
780
Unit 9 Multiplication and Division
→
→
→
→
→
28
× 22
400
160
40
+ 16
616
Student Page
Date
Again, children match each part of the diagram with the
completed problem on the board as you do the same at the
overhead projector.
Time
LESSON
2-Digit Multiplication
9 12
Multiply. Compare your answers with a partner. If you disagree,
discuss your strategies with each other. Try the problem again.
1.
2.
24
× 16
4.
10
5.
59
× 79
10
20
6.
42
× 53
2,000
100
120
+ 6
2,226
300
60
120
+ 24
504
Sample answer:
First, I multiplied 10 × 20 to get 200. Then I multiplied
10 × 4 to get 40. Next, I multiplied 6 × 20 to get 120.
After that, I multiplied 6 × 4 and got 24. Finally, I added
the partial products to solve the problem.
200 + 40 + 120 + 24 = 384
7. Describe in words how you solved Problem 1.
Give children additional problems of this type as needed, such as
23 × 46 1,058; 13 × 28 364; and 31 × 22 682.
Finding Products of
800
160
70
+ 14
1,044
36
× 14
3,500
630
450
+ 81
4,661
0
12
× 87
1,200
60
40
+ 2
1,302
200
40
120
+ 24
384
20
3.
42
× 31
PARTNER
ACTIVITY
Math Journal 2, p. 235
204-239_EMCS_S_MJ2_G3_U09_576418.indd 235
2/18/11 1:54 PM
2-Digit Numbers
(Math Journal 2, p. 235)
Have children work with partners to solve the problems on journal
page 235. Remind them to write the number model for each
partial product. For example, 10 [20s] → 200 or 10 × 20 → 200.
Circulate and assist as necessary.
Ongoing Assessment: Informing Instruction
Watch for children who have difficulty with partial products. Have them continue
to use base-10 blocks to model the problems.
Student Page
Date
Time
LESSON
Math Boxes
9 12
Links to the Future
1.
2.
Find the area of the rectangle.
40
×
length of
short side
80
=
length of
long side
3,200
area
in.2
156
3.
4,368
5 6
31 4 7
4 5 2
4 4 8
3 0 8
6 8
40 in.
Do not expect all children to master the partial-products algorithm for multiplying
2-digit numbers by 2-digit numbers at this time. Using strategies to multiply
2-digit numbers by 2-digit numbers is a Grade 4 Goal.
Practice lattice multiplication.
56 × 78 =
80 in.
70–72
Taylor has 74 inches of string. He
4. Darius took two $5 bills and four
wants to tie equal-sized pieces of
$1 bills to the store. He bought
string to 8 toy cars. How long should
shoelaces for $1.27 and 2 packs
each piece of string be?
of batteries for $3.59 each. What is
the smallest amount of money he
Number model:
can give to the cashier?
74 ÷ 8 = ?
or 8 × ? = 74
9 inches
Each string should be
One $5 bill and four
$1 bills
long.
There are
left over.
2
inches
191
259 260
5. Fill in the oval for the best answer.
6.
The degree measure
of the angle is
less than 90°.
less than 180°.
more than 180°.
Match the tool with its use.
find weight
ruler
measure length
clock
tell time
scale
167 168
236
Math Journal 2, p. 236
204-239_EMCS_S_MJ2_G3_U09_576418.indd 236
3/11/11 1:45 PM
Lesson 9 12
781
Home Link Master
Name
Date
HOME LINK
Time
2 Ongoing Learning & Practice
2 Digits × 2 Digits
9 12
Family
Note
The class continues to practice the partial-products algorithm and the lattice method, now
with any 2-digit numbers. Encourage your child to try these problems both ways and to
compare the answers to be sure they are correct.
Playing Beat the Calculator
68 – 72
Please return this Home Link to school tomorrow.
Use the lattice method and the partial-products algorithm.
735
1. 21 × 35 =
731
2. 17 × 43 =
3. 58 × 62 =
1 7
0 21 4
0 4 8
0 2 3
7 3 1
3 1
5 8
3 4 6
3 0 8
1 1 2
5 0 6
9 6
35
× 21
600
100
30
+ 5
735
43
× 17
400
30
280
+ 21
731
62
× 58
3,000
100
480
+ 16
3,596
2
1
(Multiplication)
3,596
0 0 3
0 6 3
1 0 5
7 0 5
3 5
SMALL-GROUP
ACTIVITY
(Math Journal 2, p. 282; Student Reference Book, p. 279)
Children develop automaticity with multiplication facts by playing
Beat the Calculator. Have children use the Multiplication Fact
Power Table, journal page 282, to record the facts whose products
they provide correctly when playing the role of the Brain. Have the
Caller select facts from the shaded portion of the table. For Fact
Power Table directions, see Lesson 4-5. For game directions, see
page 279 in the Student Reference Book. Remind children to fill in
the product when they have 3 check marks for a fact.
Practice
On the back of this page, use your favorite method to solve these problems.
4. 55 × 49 =
2,695
5. 91 × 33 =
3,003
Ongoing Assessment:
Recognizing Student Achievement
Math Masters, p. 307
267-318_EMCS_B_MM_G3_U09_576957.indd 307
Exit Slip
2/18/11 7:37 PM
Use an Exit Slip (Math Masters, page 398) to assess children’s progress toward
demonstrating automaticity with multiplication facts through 10 × 10. Children
record the facts from the Multiplication Fact Power Table for which they recorded
at least one check mark. Children are making adequate progress if they record
over half of the facts from the shaded portion of the table. Some children may
record most of the facts.
NOTE To provide an additional assessment
of children’s fact recall, consider administering
the Facts Survey (see Lesson 7-2 for
directions) in the next week or so.
[Operations and Computation Goal 3]
Math Boxes 9 12
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 236)
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 9-10. The skill in Problem
6 previews Unit 10 content.
Teaching Master
Name
LESSON
9 12
Date
Time
Egyptian Multiplication
Writing/Reasoning Have children write an answer to the
following: Explain how you decided which bills Darius
could give the cashier in Problem 4. Sample answer: The
total amount Darius needs to give the cashier is $8.45. The
smallest amount of money he can give the cashier is $9.00: one $5
bill and four $1 bills.
With a partner, carefully study the Egyptian multiplication algorithm
below. Then solve a problem using this method.
Example: 13 × 28
Step 1: Write the first factor in the first column (13).
Then write 1 in the first row below the factor.
Double 1 and write 2 in the row below.
Continue to double the number above until you
get a number that is equal to or greater than
the first factor. Cross out that number if it is
greater than the first factor. 16 is crossed out.
Step 2: Write the second factor in the second
column (28). Then write that number again
in the box below. (It should be next to the 1
in the first column.) Double that number in
each new line until the last number lines up
with the last number of the first column
(224 lines up with 8).
Step 3: Starting with the greatest number in column 1
(8), circle the numbers that add up to be the
first factor (13). 8 + 4 + 1 = 13
2nd factor:
1st factor:
2nd factor:
13
1
2
4
8
16
13
1
2
4
8
16
1st factor:
13
1
2
4
8
16
28
28
56
112
224
2nd factor:
28
28
56
112
224
Home Connection Children find the product of two
2-digit numbers using the lattice method and the
partial-products algorithm.
13
1
2
4
8
16
2nd factor:
p
Check the answer by solving the problem using an
algorithm you already know.
1st factor:
28
28
56
112
224
Math Masters, p. 308
267-318_EMCS_B_MM_G3_U09_576957.indd 308
782
Unit 9 Multiplication and Division
INDEPENDENT
ACTIVITY
(Math Masters, p. 307)
g
Step 4: Add the numbers in the second column that
are not crossed out. 28 + 112 + 224 = 364
Answer: 13 × 28 = 364
Home Link 9 12
py g
Cross out the row of numbers that you did not use to
make the first factor (2 and 56).
1st factor:
2/18/11 7:37 PM
Teaching Master
Name
3 Differentiation Options
Date
LESSON
Time
Egyptian Multiplication
9 12
continued
Work with a partner to solve each problem following the steps of the Egyptian
algorithm. Check your answers by solving the problems using an algorithm you
already know.
ENRICHMENT
Exploring Egyptian Multiplication
PARTNER
ACTIVITY
Answer: 24 × 32 =
To further explore multiplication, have children study an ancient
Egyptian algorithm for multiplication on Math Masters, page 308
and use the algorithm to solve another problem. They record their
work on Math Masters, page 309.
Using the Lattice Method
2nd factor:
32
32
64
128
256
512
24
1
2
4
8
16
32
15–30 Min
(Math Masters, pp. 308 and 309)
EXTRA PRACTICE
1st factor:
1. 24 × 32
768
Try This
Do your own.
INDEPENDENT
ACTIVITY
Answers vary.
×
2.
Answer:
×
1st factor:
2nd factor:
=
5–15 Min
(Math Journal 2, p. 235; Math Masters,
pp. 310 and 311)
To offer children more experience with the lattice method, have
them use the lattice method to do the problems on journal page
235. You might want to give them copies of Math Masters, page
311 (blank 2-digit by 2-digit grids). Math Masters, page 310
(blank 2-digit by 3-digit grids) is available for practicing
multiplication problems involving multiplying 3-digit numbers by
2-digit numbers.
ELL SUPPORT
Using a Graphic Organizer
Math Masters, p. 309
267-318_EMCS_B_MM_G3_U09_576957.indd 309
2/18/11 7:37 PM
NOTE The lattice method was developed by
the Egyptians more than 4,000 years ago and
is used today in Russia, Ethiopia, the Arab
world, and the Near East. Point out the
location of Egypt on a world map.
SMALL-GROUP
ACTIVITY
5–15 Min
(Differentiation Handbook, p. 34)
Teaching Master
To provide language support for multiplication, have children
create a graphic organizer for the word multiplication. They
should write words, numbers, symbols, or draw pictures that are
related to the word multiplication. Have them extend the graphic
organizer as appropriate. See the Differentiation Handbook,
page 34 for more details.
Name
Date
Time
LESSON
9 12 Lattice Grids (2-Digit ×2-Digit)
py g
g
p
Math Masters, p. 311
267-318_EMCS_B_MM_G3_U09_576957.indd 311
2/18/11 7:37 PM
Lesson 9 12
783