Download Divisibility - Everyday Math

Divisibility
Objectives To introduce divisibility rules for division by 2, 3,
5,
5 6, 9, and 10; and how to use a calculator to test for divisibility
by a whole number.
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ePresentations
eToolkit
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Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Use divisibility rules to solve problems. [Number and Numeration Goal 3]
• Explore the relationship between the
operations of multiplication and division. [Operations and Computation Goal 2]
Key Activities
Students use a calculator to test for
divisibility by a whole number. They learn
and practice divisibility rules.
Key Vocabulary
factor rainbow divisible by quotient divisibility rule
Materials
Math Journal 1, pp. 13 and 14
Study Link 1 4
calculator overhead calculator (optional)
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
1 2
4 3
Playing Factor Captor
Student Reference Book, p. 306
Math Masters, pp. 453 and 454
counters or centimeter cubes calculator
Students practice finding factors of
a number.
Math Boxes 1 5
Math Journal 1, p. 15
Students practice and maintain skills
through Math Box problems.
Ongoing Assessment:
Recognizing Student Achievement
Use Math Boxes, Problem 4. [Number and Numeration Goal 1]
Study Link 1 5
Math Masters, p. 15
Students practice and maintain skills
through Study Link activities.
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Practicing Divisibility with Counters
per partnership: 66 counters, 3 dice
Students use dice and counters to predict
divisibility relationships between 2 numbers.
EXTRA PRACTICE
Practicing Multiplication Facts
Math Journal 1, p. 9
Math Masters, p. 11
Students use a multiplication facts routine.
ENRICHMENT
Exploring a Test for Divisibility by 4
Math Masters, p. 16
Students use place-value concepts to
investigate a test for divisibility by 4.
ELL SUPPORT
Building a Math Word Bank
Differentiation Handbook, p. 142
Students add the terms divisor, dividend,
quotient, and remainder to their Math
Word Banks.
Advance Preparation
Teacher’s Reference Manual, Grades 4–6 pp. 79–83, 267–269
Lesson 1 5
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Getting Started
Mental Math and
Reflexes
Math Message
Pose basic and extended multiplication/division facts. Have students
write the answers for each set of problems. At the end of each set, ask
students to describe the patterns. Suggestions:
5 ∗ 5 25
5 ∗ 50 250
5 ∗ 500 2,500
5 ∗ 5,000 25,000
6 ∗ 3 18
60 ∗ 3 180
600 ∗ 3 1,800
6,000 ∗ 3 18,000
8 ∗ 4 32
80 ∗ 40 3,200
800 ∗ 400 320,000
8,000 ∗ 4,000 32,000,000
Solve Problems 1 and 2 at the top of
journal page 13.
Study Link 1 4 Follow-Up
Have partners compare answers. Ask the
class how they know that all possible factors
have been listed. Have volunteers model using a factor
rainbow to pair factors for 25, 28, 42, and 100. If there is
an odd number of factors, the middle factor is paired with
itself. Explain that this only happens with square numbers.
1
NOTE Some students may benefit from
doing the Readiness activity before you begin
Part 1 of each lesson. See the Readiness
activity in Part 3 for details.
2
3
4
6
8 12 16 24 48
1 Teaching the Lesson
▶ Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
(Math Journal 1, p. 13)
NOTE If possible, use an overhead calculator
to model the keystrokes and calculator
displays for lesson examples.
Interactive whiteboard-ready
ePresentations are available at
www.everydaymathonline.com to
help you teach the lesson.
Students share solution strategies. Use students’ responses to
emphasize to the class that even numbers are numbers that are
divisible by 2.
▶ Using a Calculator to Test
INDEPENDENT
ACTIVITY
for Divisibility by a Whole Number
(Math Journal 1, p. 13)
NOTE Factor rainbows are introduced in
the Study Link Follow-Up. This tool helps
students identify all of the factors for a
given number. The rainbow is a visual
representation of the factor pairs and
provides a way to check if the factor list is
complete. Share the factor rainbow in the
Study Link Follow-Up with the class. Factor
rainbows will be used again in Lesson 1-6.
Recall for students the class discussion on the review of divisibility
in Lesson 1-4. Remind students that a whole number (the dividend)
is divisible by a whole number (the divisor) if the remainder in the
division is zero. The result or quotient, must be a whole number. If
the remainder is not zero, then the number being divided is not
divisible by the second number.
If your students use calculators that display answers to division
problems as a quotient and a whole number remainder, you might
want to demonstrate the procedure. With the TI-15 calculator, this
is done by pressing the Int÷ key instead of the ÷ key. With the
Casio fx-55, use the
key. For example, If you press 27 Int÷ 5
, or 27
5
, the display will show a quotient of 5 with a
remainder of 2.
38
Unit 1
Number Theory
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Student Page
Date
Adjusting the Activity
Time
LESSON
䉬
Math Message
1.
Circle the numbers that are divisible by 2.
2.
What do the numbers that you circled have in common?
28
dividend
divisor
quotient
remainder
K I N E S T H E T I C
T A C T I L E
57
33
112
123,456
211
Allow 5 to 10 minutes for students to complete Problems 3–10 on
the journal page 13.
705
Example 2: Is 122 divisible by 5?
To find out, divide 122 by 5.
135 / 5 ⫽ 27
122 / 5 ⫽ 24.4
The answer, 27, is a whole
number. So 135 is divisible by 5.
The answer, 24.4, has a decimal
part. So 122 is not divisible by 5.
Use your calculator to help you answer these questions.
3.
Is 267 divisible by 9?
No
5.
Is 809 divisible by 7?
7.
Is 4,735 divisible by 5?
No
Yes
9.
Is 5,268 divisible by 22?
No
Yes
4.
Is 552 divisible by 6?
6.
Is 7,002 divisible by 3?
8.
Is 21,733 divisible by 4?
Yes
No
10.
Is 2,072 divisible by 37?
Yes
Math Journal 1, p. 13
WHOLE-CLASS
ACTIVITY
PROBLEM
PRO
PR
P
RO
R
OB
BLE
BL
L
LE
LEM
EM
SOLVING
SO
S
OL
O
LV
VING
VIN
IIN
NG
Ask: How can you know that a number is divisible by 2 without
actually doing the division? Numbers that end in 0, 2, 4, 6, or 8
are divisible by 2. Can you tell whether a number is divisible by 10
without dividing? Yes; numbers that end in 0 are divisible by 10.
Can you tell whether a number is divisible by 3 without dividing?
Allow students to explore this question before continuing. There
are rules that let us test for divisibility without dividing or using a
calculator.
1. Go over the divisibility-by-3 rule on journal page 14: A number
is divisible by 3 if the sum of its digits is divisible by 3.
2. Illustrate by using the rule to test several examples.
●
399
V I S U A L
When testing for divisibility with a calculator that does not display
remainders, the first number is not divisible by the second number
if the quotient has a decimal part. Ask students to use their
calculators to test whether 27 is divisible by 9. 27 is divisible by 9
because the result is 3—a whole number. Test whether 27 is
divisible by 5. 27 is not divisible by 5 because the result is 5.4—
not a whole number.
(Math Journal 1, p. 14)
900
Suppose you divide a whole number by a second whole number. The answer may be a
whole number, or it may be a number that has a decimal part. If the answer is a whole
number, we say that the first number is divisible by the second number. If the answer
has a decimal part, the first number is not divisible by the second number.
Example 1: Is 135 divisible by 5?
To find out, divide 135 by 5.
▶ Introducing Divisibility Rules
5,374
They are all even numbers.
135 ÷ 5 = 27 R0
A U D I T O R Y
Divisibility
15
Write the number model from the first example on journal page 13 on
the board with each number appropriately labeled, including a remainder of zero.
Is 237 divisible by 3? Yes. 2 + 3 + 7 = 12, and 12 is
divisible by 3.
Student Page
Date
Time
LESSON
15
䉬
Divisibility Rules
For many numbers, even large ones, it is possible to test for divisibility without
actually dividing.
Here are the most useful divisibility rules:
䉬 All numbers are divisible by 1.
䉬 All even numbers (ending in 0, 2, 4, 6, or 8) are divisible by 2.
䉬 A number is divisible by 3 if the sum of its digits is divisible by 3.
Example: 246 is divisible by 3 because 2 + 4 + 6 = 12, and 12 is divisible by 3.
䉬 A number is divisible by 6 if it is divisible by both 2 and 3.
Example: 246 is divisible by 6 because it is divisible by 2 and by 3.
䉬 A number is divisible by 9 if the sum of its digits is divisible by 9.
Example: 51,372 is divisible by 9 because 5 + 1 + 3 + 7 + 2 = 18, and
18 is divisible by 9.
䉬 A number is divisible by 5 if it ends in 0 or 5.
䉬 A number is divisible by 10 if it ends in 0.
●
Is 415 divisible by 3? No. 4 + 1 + 5 = 10, and 10 is not
divisible by 3.
1.
Divisible. . .
Number
Students complete Problems 1–3 independently. Have them check
each other’s work.
by 2 ?
7,960
384
by 3 ?
by 6 ?
by 9 ?
✓
75
3. Ask students to provide examples of a number that is divisible
by 3 and a number that is not. Encourage them to apply the
divisibility-by-3 test first. Then have them check that it works
by carrying out the division on their calculators.
Assign small groups to present examples for the remaining
divisibility rules (5, 6, or 9).
Test each number below for divisibility. Then check on your calculator.
✓
✓
✓
✓
✓
✓
✓
3,725
90
36,297
✓
✓
✓
2.
Find a 3-digit number that is divisible by both 3 and 5.
3.
Find a 4-digit number that is divisible by both 6 and 9.
by 5 ?
by 10 ?
✓
✓
✓
✓
✓
✓
Sample answers: 735; 540
Sample answers: 1,800; 5,454
Math Journal 1, p. 14
Lesson 1 5
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Student Page
Date
Time
LESSON
15
1.
Circle the numbers that are divisible by 3.
221
381
474
922
2.
726
Round 3,045,832 to the nearest…
a.
million.
b.
thousand.
c.
ten-thousand.
3,000,000
3,046,000
3,050,000
11
3.
4 249
Complete the table.
Fraction
4.
Decimal
Percent
0.60
0.25
60%
0.50
50%
70%
3
ᎏᎏ
5
ᎏ1ᎏ
4
ᎏ1ᎏ
2
7
ᎏᎏ
10
25%
0.70
0.85
85
ᎏᎏ
100
Write an 8-digit numeral with
5 in the hundredths place,
8 in the tens place,
3 in the ones place,
8 in the thousands place,
4 in the hundreds place,
and 6 in all other places.
6 6 8, 4 8 3 . 6 5
85%
Complete.
a.
b.
c.
d.
e.
PARTNER
ACTIVITY
▶ Playing Factor Captor
(Student Reference Book, p. 306;
Math Masters, pp. 453–454)
Students practice finding factors of a number by playing Factor
Captor. Students have the option of playing any of the two
Factor Captor grids. If students are using Grid 2 for the first time,
suggest that they omit the last two rows of the gameboard.
4 30 31
80 90
5.
2 Ongoing Learning & Practice
Math Boxes
6.
56,000
400 5,000 ⫽ 2,000,000
70
6,300 ⫽
90
300
21,000 ⫽ 70 900
720,000 ⫽ 800 70 800 ⫽
Pencils are packed 18 to a box. How many
pencils are in 9 boxes?
162 pencils
(unit)
INDEPENDENT
ACTIVITY
▶ Math Boxes 1 5
(Math Journal 1, p. 15)
18
19 20
Math Journal 1, p. 15
NOTE As students continue to develop their
strategies for Factor Captor, they will find that
as more numbers are used, the scoring rules
increasingly reward a player for planning
ahead and anticipating an opponent’s moves.
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lessons 1-7 and 1-9. The skills in
Problems 5 and 6 preview Unit 2 content.
Writing/Reasoning Have students write a response to the
following: Explain how you solved Problem 6. Sample
answer: Because there are 18 pencils per box and 9 boxes
total, I multiplied 18 ∗ 9: 10 ∗ 9 is 90 and 8 ∗ 9 is 72;
90 + 72 = 162. There are 162 pencils in all.
Ongoing Assessment:
Recognizing Student Achievement
Date
Use Math Boxes, Problem 4 to assess students’ understanding of place
value. Students are making adequate progress if they are able to correctly
position and identify digits and their values in whole numbers through the
hundred-thousands and decimals through the hundredths.
Study Link Master
Name
Math Boxes
Problem 4
[Number and Numeration Goal 1]
Time
Divisibility Rules
STUDY LINK
15
䉬
䉬 All even numbers are divisible by 2.
11
䉬 A number is divisible by 3 if the sum of its digits is divisible by 3.
▶ Study Link 1 5
䉬 A number is divisible by 6 if it is divisible by both 2 and 3.
䉬 A number is divisible by 9 if the sum of its digits is divisible by 9.
INDEPENDENT
ACTIVITY
(Math Masters, p. 15)
䉬 A number is divisible by 5 if it ends in 0 or 5.
䉬 A number is divisible by 10 if it ends in 0.
1.
Use divisibility rules to test whether each number is divisible by 2, 3, 5, 6, 9, or 10.
Number
夹
998,876
夹
36,540
5,890
Divisible…
by 2?
by 3?
by 6?
by 9?
by 5?
by 10?
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
33,015
1,098
Home Connection Students use divisibility rules to test
whether numbers are divisible by 2, 3, 5, 6, 9, or 10. They
learn the divisibility rule for 4 and recheck the numbers
for those that are also divisible by 4.
A number is divisible by 4 if the tens and ones digits form a number that is
divisible by 4.
Example: 47,836 is divisible by 4 because 36 is divisible by 4.
It isn’t always easy to tell whether the last two digits form a number that is divisible by 4. A
quick way to check is to divide the number by 2 and then divide the result by 2. It’s the same
as dividing by 4, but is easier to do mentally.
Example: 5,384 is divisible by 4 because 84 / 2 ⫽ 42 and 42 / 2 ⫽ 21.
2.
Place a star next to any number in the table that is divisible by 4.
3.
250 º 7 ⫽
Practice
5.
1,750
(20 ⫹ 30) º 5 ⫽
4.
250
6.
1,931 ⫹ 4,763 ⫹ 2,059 ⫽
78 ⫼ 6 ⫽
8,753
13
Math Masters, p. 15
40
Unit 1
Number Theory
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Teaching Master
Name
3 Differentiation Options
READINESS
▶ Practicing Divisibility
LESSON
15
䉬
PARTNER
ACTIVITY
15–30 Min
with Counters
To explore the concept of divisibility using a concrete model, have
students use counters to determine whether a number is divisible
by the numbers 1–6.
Partners take turns rolling three dice. Make a two-digit number
with two of the dice, and count out that number of counters. They
predict whether the number of counters is divisible by the number
on the third die. Then partners check the prediction by dividing
the counters into the number of groups indicated on the third die.
EXTRA PRACTICE
▶ Practicing Multiplication Facts
SMALL-GROUP
ACTIVITY
Date
2.
10 cubes
100 cubes
1,000 cubes
1.
Time
Divisibility by 4
What number is shown by the base-10 blocks?
1 cube
1,111
Which of the base-10 blocks could be divided evenly into 4 groups of cubes?
The groups of 1,000 cubes and
100 cubes
3.
Is the number shown by the base-10 blocks divisible by 4?
4.
Circle the numbers that you think are divisible by 4.
324
5,821
7,430
No
35,782,916
Use a calculator to check your answers.
5.
Use what you know about base-10 blocks to explain why you only need to
look at the last two digits of a number to decide whether it is divisible by 4.
Sample answer: Because 1,000 and 100
are divisible by 4, the numbers that the
thousands place and the hundreds place
represent are always divisible by 4. So you
have to look at only the number formed by
the tens and ones digits.
Math Masters, p. 16
5–15 Min
(Math Journal 1, p. 9; Math Masters, p. 11)
To provide additional practice with basic multiplication facts,
have students use the facts routine introduced in Lesson 1-3. See
Teacher’s Lesson Guide, pages 28 and 29 to review the procedure.
ENRICHMENT
▶ Exploring a Test
PARTNER
ACTIVITY
5–15 Min
for Divisibility by 4
(Math Masters, p. 16)
To further explore divisibility, have students use
place-value concepts to investigate why only the last
2 digits in a number determine whether the number is
divisible by 4.
ELL SUPPORT
▶ Building a Math Word Bank
SMALL-GROUP
ACTIVITY
5–15 Min
(Differentiation Handbook, p. 142)
To provide language support for division, have students use
the Word Bank Template found on Differentiation Handbook,
page 142. Ask students to write the terms divisor, dividend,
quotient, and remainder; draw a picture representing each term;
and write other related words. See the Differentiation Handbook
for more information.
Lesson 1 5
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Name
LESSON
13
Date
Multiplication Facts
Copyright © Wright Group/McGraw-Hill
A List
3 ∗ 6 = 18
6 ∗ 3 = 18
3 ∗ 7 = 21
7 ∗ 3 = 21
3 ∗ 8 = 24
8 ∗ 3 = 24
3 ∗ 9 = 27
9 ∗ 3 = 27
4 ∗ 6 = 24
6 ∗ 4 = 24
4 ∗ 7 = 28
7 ∗ 4 = 28
4 ∗ 8 = 32
8 ∗ 4 = 32
4 ∗ 9 = 36
9 ∗ 4 = 36
5 ∗ 7 = 35
7 ∗ 5 = 35
5 ∗ 9 = 45
9 ∗ 5 = 45
6 ∗ 6 = 36
6 ∗ 7 = 42
7 ∗ 6 = 42
6 ∗ 8 = 48
8 ∗ 6 = 48
6 ∗ 9 = 54
9 ∗ 6 = 54
7 ∗ 7 = 49
7 ∗ 8 = 56
8 ∗ 7 = 56
7 ∗ 9 = 63
9 ∗ 7 = 63
8 ∗ 8 = 64
8 ∗ 9 = 72
9 ∗ 8 = 72
9 ∗ 9 = 81
Time
B List
3 ∗ 3= 9
3 ∗ 4 = 12
4 ∗ 3 = 12
3 ∗ 5 = 15
5 ∗ 3 = 15
4 ∗ 4 = 16
4 ∗ 5 = 20
5 ∗ 4 = 20
5 ∗ 5 = 25
5 ∗ 6 = 30
6 ∗ 5 = 30
5 ∗ 8 = 40
8 ∗ 5 = 40
6 ∗ 10 = 60
10 ∗ 6 = 60
7 ∗ 10 = 70
10 ∗ 7 = 70
8 ∗ 10 = 80
10 ∗ 8 = 80
9 ∗ 10 = 90
10 ∗ 9 = 90
10 ∗ 10 = 100
Bonus Problems
11 ∗ 11 = 121
11 ∗ 12 = 132
5 ∗ 12 = 60
12 ∗ 6 = 72
7 ∗ 12 = 84
12 ∗ 8 = 80
9 ∗ 12 = 108
10 ∗ 12 = 120
5 ∗ 13 = 65
15 ∗ 7 = 105
12 ∗ 12 = 144
6 ∗ 14 = 84
11
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