Lattice Multiplication
Objectives To review and provide practice with the lattice
O
method
for multiplication.
m
www.everydaymathonline.com
ePresentations
eToolkit
Algorithms
Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Add single-digit numbers. [Operations and Computation Goal 2]
• Solve basic multiplication facts. [Operations and Computation Goal 3]
• Use the lattice method to solve
multiplication problems with 1- and
2-digit multipliers. [Operations and Computation Goal 4]
Key Activities
Students review the lattice method for
multiplication with 1- and 2-digit multipliers.
They practice using this multiplication
algorithm.
Key Vocabulary
lattice lattice method (for multiplication)
Materials
Math Journal 1, p. 124
Study Link 56
Math Masters, p. 434
transparency of Math Masters, p. 434
(optional) slate index cards
(optional) dictionary (optional)
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
1 2
4 3
Playing Multiplication Top-It
Student Reference Book, p. 264
Math Masters, p. 506
per partnership: 4 each of number
cards 1–10 (from the Everything
Math Deck, if available)
Students practice multiplication facts.
Ongoing Assessment:
Recognizing Student Achievement
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Exploring Fact Lattice Patterns
Math Masters, pp. 161 and 435
Multiplication/Division Facts Table colored pencils
Students explore the use of the lattice grid
for multiplication.
ENRICHMENT
Use Math Masters, page 506. Investigating Napier’s Rods
[Operations and Computation Goal 3]
Math Masters, pp. 158–160
scissors
Students investigate Napier’s Rods, a
seventeenth-century multiplication method.
Math Boxes 5 7
Math Journal 1, p. 125
Students practice and maintain skills
through Math Box problems.
ELL SUPPORT
Study Link 5 7
Creating Visuals for Multiplication
Algorithms
Math Masters, p. 157
Students practice and maintain skills
through Study Link activities.
chart paper markers colored pencils
Students make posters to display
multiplication algorithms.
Advance Preparation
Teacher’s Reference Manual, Grades 4–6 pp. 126–132
Lesson 5 7
349
Mathematical Practices
SMP2, SMP3, SMP5, SMP6, SMP7, SMP8
Content Standards
Getting Started
4.NBT.5, 4.MD.2
Mental Math and Reflexes
Pose multiplication facts and extended facts. Suggestions:
5 ∗ 6 = 30
6 ∗ 3 = 18
6 ∗ 4 = 24
7 ∗ 3 = 21
7 ∗ 4 = 28
9 ∗ 8 = 72
8 ∗ 7 = 56
7 ∗ 7 = 49
6 ∗ 60 = 360
8 ∗ 50 = 400
70 ∗ 90 = 6,300
80 ∗ 30 = 2,400
Math Message
Study Link 5 6 Follow-Up
What do you think the missing digits are?
Ask volunteers to write solutions on the board.
Students indicate thumbs-up if they agree with the
solution. Challenge students to find a way to solve 340 ∗ 50
mentally. Sample answer: Multiply 340 by 100. The answer is
half of this product, or 17,000.
6
?
7
?
2
8
3
1
2
6
4
Algorithm Project The focus of this
lesson is lattice multiplication. To teach U.S.
traditional multiplication, see Algorithm
Project 5 on page A21.
?
1
0
? 5
5
1 Teaching the Lesson
Math Message Follow-Up
Links to the Future
Fluently multiplying whole numbers using
the standard algorithm is expected in
Grade 5.
WHOLE-CLASS
DISCUSSION
Students share answers. The pair of digits in each cell names the
product of two digits outside the lattice—one above the cell of the
lattice and the other to the right of the cell of the lattice. Thus, the
missing digits in the first lattice form the product of 6 and 4 (24).
The missing digits in the second lattice form the product of 6 and
5 (30).
6
2
4
7
2
8
3
1
2
6
4
3
1
0
0 5
5
Tell students that in this lesson they will review the lattice
method for multiplication. Introduced in Third Grade Everyday
Mathematics, this algorithm relies almost entirely on the recall
of basic multiplication facts. If students do not yet have a favorite
multiplication algorithm, the lattice method is a good one to suggest.
350
Unit 5 Big Numbers, Estimation, and Computation
Demonstrating the Lattice
WHOLE-CLASS
ACTIVITY
NOTE The lattice method is a very efficient
Method for 1-Digit Multipliers
algorithm, no matter how many digits are in
the factors. For problems with 1- and 2-digit
multipliers, the lattice method takes about the
same amount of time as the partial-products
algorithm or the traditional multiplication
algorithm. For problems with three or more
digits in the factors, the lattice method is much
faster and much more likely to yield a correct
answer.
(Math Masters, p. 434)
Demonstrate the lattice method using the following examples.
Students should show their work using the computation grids on
Math Masters, page 434.
Example: Use the lattice method to multiply 3 ∗ 45.
4
5
3
4
4
1
5
1
5
Multiply 3 ∗ 5. Write the answer as shown.
2 5
Adjusting
the Activity
3
5
1
A lattice usually consists of two or more cells
with diagonals. Write 4 and 5 above the cells of
the lattice. Write 3 on the right side.
Have students look up the term lattice in
the dictionary and think of places where
they might see lattices, such as a gate, a
window, or a patio. Have students discuss
why this word is used to describe the
multiplication algorithm.
Multiply 3 ∗ 4. Write the answer as shown.
3
AUDITORY
4
1
1
5
1
2 5
3 5
3
ELL
KINESTHETIC
TACTILE
VISUAL
Starting at the right, add the numbers inside
the lattice along each diagonal. Read the answer
starting on the left side of the lattice and
continuing across the bottom. 3 ∗ 45 = 135.
Student Page
The sum of the numbers along a diagonal may be a 2-digit number.
When this happens, write the ones digit in the answer space and
the tens digit at the top of the next diagonal.
Date
䉬
51 6
6 6 3
2 3
7
The sum of the numbers in the middle diagonal
is 12. Write 2 in the answer space and 1 at the
top of the next diagonal. Then add the numbers
in that diagonal.
Read the answer: 7 ∗ 89 = 623.
Practicing the Lattice Method
3 ⴱ 56 5
Example: Use the lattice method to multiply 7 ∗ 89.
9
Lattice Multiplication
5 7
Use the lattice method to find the products.
1.
8
Time
LESSON
Students complete Problems 1–5 independently. Partners check
each other’s work.
3
2 1
2 4 2
4 2
9 2 1
6 1
PARTNER
ACTIVITY
(Math Journal 1, p. 124)
6
1 1 3
1 5 8
6 8
1,890
4. 6 ⴱ 315 3 1 5
1 0 3
1 8 6 06
8 9 0
2,961
6. 47 ⴱ 63 6
for 1-Digit Multipliers
168
8.
4
2.
518
2
6
7 4
41 2
11 4 8
2 6 8
5 9 87
0 8
1 8
2,556
5. 9 ⴱ 284 2 8 4
11 7 3 9
2 8 2 6
5 5 6
676
7. 26 ⴱ 26 2 6
0 1 2
4 2
8 ⴱ 26 208
7,568
4 7 3
0 01 0 1
4 7 3
2 4 1 6
7 4 2 8
5 6 8
19
7 ⴱ 74 1 3 6
6 2 6
7 6
7
16 ⴱ 473 3.
9.
37,030
8 0 5
3 01 2
3 2 0 04
4 0 3 6
7 8 0 0
0 3 0
46 ⴱ 805 Math Journal 1, p. 124
Lesson 5 7
351
Demonstrating the Lattice
WHOLE-CLASS
ACTIVITY
Method for 2-Digit Multipliers
(Math Masters, p. 434)
Demonstrate the lattice method for 2-digit multipliers using the
example below. Use a transparency of Math Masters, page 434 or
copy the lattices onto the board. Students should show their work
on the lattice grids.
Adjusting the Activity
Consider these suggestions
for students who need help with the
lattice method.
0 1
6 8
0 2
8 4
KINESTHETIC
2
Multiply 4 ∗ 6. Then multiply 4 ∗ 2.
0
Write the answers as shown.
4
6
6
4
2
VISUAL
11
6 8
0 2
8 8 4
8 4
tho
nd
(Math Journal 1, p. 124)
Students complete Problems 6–9 independently. Partners check
each other’s work.
Unit 5 Big Numbers, Estimation, and Computation
3
4
3
4
PARTNER
ACTIVITY
with 2-Digit Multipliers
352
4
6
0
hu
Practicing the Lattice Method
3
6
11
6 8
0 2
8 8 4
8 4
Read the answer: 34 ∗ 26 = 884.
The lattice method for multiplication
may seem like a trick. It is not, of course,
and place value is evident within the lattice.
8
0
Starting at the right, add the numbers inside
the lattice along each diagonal. The sum of
the numbers along the second diagonal is
8 + 2 + 8 = 18. Write 8 in the answer space
and 1 at the top of the next diagonal.
3
1
2
8 4
2
8 4
TACTILE
0
Multiply 3 ∗ 6. Then multiply 3 ∗ 2.
3
s
4
ten
AUDITORY
3
4
6
1
2
8
3
6
1
0 1
6 8
0 2
8 8 4
8 4
Extend the diagonal
lines through the
lattice as shown.
Write 26 above the lattice. Write 34 on the
right side.
es
2
Cross out the factors
of the problem before
adding so those
numbers are not
accidentally added
with the numbers
within the grid.
6
on
1
8
1 6
0 2
8 8 4
8 4
2
us
an
ds
ds
Use an index
card to mark the
place when adding
along each diagonal.
6
11
re
5
Example: Use the lattice method to multiply 34 ∗ 26.
Student Page
2 Ongoing Learning & Practice
Playing Multiplication Top-It
Date
Time
LESSON
Math Boxes
5 7
䉬
1.
This is a circle graph of what Seema does on a typical day.
a.
What does she spend the least amount of time doing?
b.
She spends about the same amount of time
at school and doing homework as she does
Soccer practice
PARTNER
ACTIVITY
sleep
sleeping
(Student Reference Book, p. 264; Math Masters, p. 506)
About what fraction of the day
does she spend doing chores,
eating, and relaxing?
c.
Students play Multiplication Top-It to maintain automaticity with
multiplication facts. See Lesson 3-3 for additional information.
Seema’s 24-Hour Day
2.
1
5
soccer practice
Estimate the sum. Write a number model
to show how you estimated.
3.
Complete.
Rule:
715 1,904 688
a.
Sample answers:
700 1,900 700 3,300
Number model:
NOTE For facts practice through 12 ∗ 12, have students include number cards
11 and 12.
867 2,346 3,596
b.
school and
homework
chores
eating
relaxing
.
ⴱ 80
in
out
50
4,000
70
5,600
90
Number model:
900 2,300 3,600 6,800
7,200
100
8,000
45
3,600
181
Ongoing Assessment:
Recognizing Student Achievement
Math Masters
Page 506
Use Math Masters, page 506 to assess students’ automaticity with basic
multiplication facts. Students are making adequate progress if they can name the
product of the factors generated by the two cards. Some students may play a
variation of the game and demonstrate the ability to mentally solve 2-digit by
1-digit multiplication problems.
4.
162–166
Write each number using digits.
5.
a.
twenty-six million, nineteen
thousand, eighteen
a.
Look at the grid below.
26,019,018
B
b.
three hundred fifty-two million, eight
hundred thousand, two hundred
b.
352,800,200
In which column
is the triangle
located?
A
B
C
1
2
3
In which row is
the triangle
located?
3
4
144
Math Journal 1, p. 125
[Operations and Computation Goal 3]
Math Boxes 5 7
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 125)
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 5-5. The skill in Problem 5
previews Unit 6 content.
Writing/Reasoning Have students write a response to the
following: In Problem 1, about how many minutes does Seema
spend doing chores, eating, and relaxing each day? About 288
minutes Explain how you got your answer. Sample answer: I
multiplied 24 by 60 to find out how many minutes are in a day:
1 of a day on those
24 ∗ 60 = 1,440 minutes. Seema spends about _
5
activities, so 1,440 / 5 = 288 minutes.
Study Link 5 7
INDEPENDENT
ACTIVITY
Study Link Master
Name
57
䉬
Time
Lattice Multiplication
Use the lattice method to find the following products.
1.
5 º 46 230
4
3.
(Math Masters, p. 157)
Home Connection Students use the lattice method to
solve multiplication problems. Students are asked to solve
Problem 7 by using both the lattice method and the
partial-products method.
Date
STUDY LINK
5.
7.
2.
6
2 3 5
2 0 0
3 0
7 º 836 5,852
8 3 6
5 2 4 7
5 6 1 2
8 5 2
775
25 º 31 3 1
0 0
0 6 22
1 0 5
7 5 5
7 5
4.
6.
536
6 7
41 5
5 8 6 8
3 6
4 º 329 1,316
3 2 9
1 01 3 4
1 2 8 6
3 1 6
49 º 52 2,548
5 2
2 01 4
2 0 8
4 1
5 5 89
19
8 º 67 4 8
Use the lattice method and the partial-products
method to find the product.
84 º 78 6,552
Practice
8.
10.
39.57
33.67 5.9
71.44 37.67 33.77
9.
68.4 5.82 11.
71.15
74.22
101.06 29.91
Math Masters, p. 157
Lesson 5 7
157
353
Teaching Master
Name
Date
LESSON
Time
57
䉬
1.
Look at a Multiplication/Division Facts Table. Find the shaded diagonal showing
the doubles facts.
2.
Find the doubles facts on the Fact Lattice on Math Masters, page 435. Shade the
doubles facts lightly with a colored pencil.
3.
Compare the two fact tables.
a.
3 Differentiation Options
Fact Lattice Patterns
List 3 things the tables have in common.
Both show multiplication and division facts for the factors
1–9; both are a square shape; both show the square
products along a diagonal; both show the factors along
the top and one side.
b.
Exploring Fact Lattice Patterns
(Math Masters, pp. 161 and 435)
b.
5.
To explore the use of the lattice grid for multiplication, have
students find patterns in the Fact Lattice (Math Masters,
page 435).
Sample answers:
Not counting zero as a factor, in the 9s columns and rows,
the digit of the 10s place goes up by 1 while the digit
for the 1s place goes down by 1.
Describe 2 patterns that you see on the Fact Lattice.
a.
5–15 Min
List 3 things that are different on the Fact Lattice.
Lists factors in reverse; shows factors from 0–9, not 1–10;
shows zeros in 10s place for numbers 10; has diagonal
lines between the 10s and 1s place for products; shows
factors on the right; and does not shade square products.
4.
PARTNER
ACTIVITY
READINESS
Sample answers:
The bottom (1s) digits in the 2s column and rows repeat
a pattern on 0, 2, 4, 6, 8.
Which of your Fact Lattice patterns is also in the Multiplication/Division Facts Table
in your journal?
INDEPENDENT
ACTIVITY
ENRICHMENT
Investigating Napier’s Rods
15–30 Min
(Math Masters, pp. 158–160)
Sample answer: The 9s pattern is in the fact table.
To apply students’ understanding of lattice multiplication, have
them use Napier’s Rods to solve problems.
Math Masters, p. 161
Discuss students’ responses to Problem 5. Students may note that
when solving a problem with 0 as a digit in one of the factors, a
space is left between the rods where the 0 rod would be.
1
1
0
2
1
2
2
0
2
4
1
1
5
4
5
ELL SUPPORT
Creating Visuals for
Teaching Master
Name
Date
LESSON
䉬
Scottish mathematician John Napier (1550–1617) devised a multiplication method using
rods made of bone, wood, or heavy paper. These rods were used to solve multiplication
and division problems.
Example 1:
Example 2:
4 º 67 268
8 º 5,239 41,912
6
1
2
3
4
5
6
7
8
9
7
0
5
0
6
1
7
1
4
2
1
2
1
2
8
2
1
3
8
4
2
4
3
3
0
3
5
5
2
6
4
6
4
4
2
4
9
7
6
8
3
9
5
8
5
6
4
2
2
0
3
0
5
1
0
0
1
5
0
2
0
0
2
5
3
1
0
5
4
4
5
4
6
1
2
6
3
7
4
2
8
2
8
7
1
1
2
4
4
8
1
0
2
6
2
6
7
4
2
8
4
1
9
1
2
Cut out the rods on Math Masters, page 159. Use the rods and the board on
Math Masters, page 160 to solve the following problems and some of your own.
Use another method to check your answers.
1.
3.
5 º 79 1,416
395
6 º 236
2.
4.
644
52,569 9 º 5,841
7 º 92 Try This
5.
Show a friend how you would use Napier’s Rods to solve 3 º 407 or 9 º 5,038.
Math Masters, p. 158
354
15–30 Min
To provide language support for multiplication, have students
make posters to illustrate the multiplication algorithms they have
learned. Display the posters for students to refer to during the rest
of the unit.
For use in Part 1 and the optional Enrichment activity in Part 3
of Lesson 5-8, borrow a ream of copy paper and an empty carton
used to pack 10 reams of paper.
5
5
8
4
1
5
6
4
2
1
0
SMALL-GROUP
ACTIVITY
Planning Ahead
7
3
1
2
1
8
2
1
1
0
3
2
9
2
9
8
1
4
2
1
6
0
6
3
7
0
3
4
0
0
7
9
0
2
2
Multiplication Algorithms
Time
Napier’s Rods
57
0
Unit 5 Big Numbers, Estimation, and Computation