01-NM6TR-C03-Interior_L01-L02.qxd
Chapter 3
2
12/1/08
10:16 PM
Page 18
Identifying Multiples
STUDENT BOOK PAGES 74–76
GOAL
Identify multiples to solve problems.
PREREQUISITE SKILLS/CONCEPTS
• Identify factors of whole numbers.
• Extend a number pattern by multiplying or adding whole
numbers.
SPECIFIC OUTCOME
N3. Demonstrate an understanding of factors and
multiples by
• determining multiples and factors of numbers less
than 100
• identifying prime and composite numbers
• solving problems involving multiples.
[PS, R, V]
Achievement Indicators
• Identify multiples for a given number and explain the
strategy used to identify them.
• Solve a given problem involving factors or multiples.
Preparation and Planning
Pacing
(allow 5 min for
previous homework)
5–10 min Introduction
10–15 min Teaching and Learning
20–30 min Consolidation
Materials
• rulers
• Optional: counters
Masters
• Number Lines, Masters Booklet p. 33
• Optional: Scaffolding for Lesson 2,
Question 3 p. 77
Recommended
Practising Questions
Questions 2, 3, 5, 8, & 9
Key Question
Question 5
Extra Practice
Mid-Chapter Review Questions 3 & 4
Chapter Review Questions 4 & 5
Workbook p. 18
Mathematical
Process Focus
PS (Problem Solving) and V (Visualization)
Vocabulary/Symbols
multiple
Nelson Website
Visit www.nelson.com/mathfocus and follow
the links to Nelson Math Focus 6, Chapter 3.
18
Chapter 3: Number Relationships
Math Background
In previous grades, students have multiplied factors to
calculate a product. In this lesson, students will approach
multiplication from a different perspective as they
calculate multiples of a number using known
multiplication facts and skip counting. Students will
multiply a given number by sequential whole numbers to
build a list of multiples. For example, to build a list of
multiples of 6, students will multiply 6 by 1, 2, 3, 4, … to
get the multiples 6, 12, 18, 24, …. To use skip counting,
students will count in units of the given number. For
example, to build a list of multiples of 5, students will
count by 5s to get the multiples 5, 10, 15, 20, and so on.
Students use a number line to help them visualize the
pattern in the list of multiples. Students will apply these
skills in various problem-solving contexts.
Number Lines, Masters
Booklet p. 33
Optional: Scaffolding for
Lesson 2, Question 3 p. 77
Copyright © 2010 Nelson Education Ltd.
01-NM6TR-C03-Interior_L01-L02.qxd
12/1/08
10:16 PM
Page 19
1
1 Introduction
2 Teaching and Learning
2
3
(Whole Class) ± 5–10 min
Briefly review some mental math strategies that students have
learned for multiplication. On the board, on a transparency,
or on an interactive whiteboard, write the following
multiplication expressions:
3
4⫻8
4
6⫻7
8⫻5
Ask volunteers to share their strategies for calculating each
product. Try to elicit a variety of strategies.
5
Sample Discourse
“How can you calculate the product of 4 and 8?”
• I used doubling. I know 2 ⫻ 8 = 16,
so 4 ⫻ 8 = 16 ⫹ 16, which is 32.
• I used doubling. I know 4 ⫻ 4 = 16,
so 4 ⫻ 8 = 16 ⫹ 16, which is 32.
“How can you calculate the product of 6 and 7?”
• I skip counted up. I know 6 ⫻ 6 = 36,
so 6 ⫻ 7 = 36 ⫹ 6, which is 42.
• I skip counted down. I know 7 ⫻ 7 = 49,
so 6 ⫻ 7 = 49 ⫺ 7, which is 42.
“How can you calculate the product of 8 and 5?”
• I used doubling. I know 2 ⫻ 5 = 10, so 4 ⫻ 5 = 10 ⫹ 10,
which is 20, and 8 ⫻ 5 is 20 ⫹ 20, or 40.
• I skip counted down. I know 10 ⫻ 5 = 50, so 9 ⫻ 5 =
50 ⫺ 5, which is 45, and 8 ⫻ 5 = 45 ⫺ 5, which is 40.
6
7
8
Copyright © 2010 Nelson Education Ltd.
(Whole Class/Small Groups)
± 10–15 min
Before reading, remind students that a comet is a small body
that orbits the Sun, and it is only visible from Earth at
certain points in its orbit. Comets that appear regularly are
referred to as periodic comets. Together, read about the
comets and then read the central question on Student Book
page 74. Have students set up Oleh’s List and retrace his
steps to show the first multiples of 7. Then direct them to
Léa’s Number Line. Tell students to use their rulers to draw
an open number line with two arrows. Ask them to point out
which number Léa starts with on the number line and how
she gets to the next number. When students have become
comfortable with Léa’s method, have them work through
Prompts A to C in small groups. You may want to discuss the
two methods as a group and have volunteers explain which
method they prefer.
4
5
6
7
8
Sample Discourse
“Which math operations did Oleh use in his method? How is
Oleh’s method different from Léa’s method?”
• Oleh used multiplication to determine the multiples of 7 and
addition to calculate the years the comet would be seen from
Earth. Léa only used addition to figure out the years after
2000 the comet would be seen.
Lesson 2: Identifying Multiples
19
01-NM6TR-C03-Interior_L01-L02.qxd
12/1/08
10:16 PM
Page 20
Answers to Reflecting Questions
D. For example, you create a multiple of 7 by multiplying
7 by a counting number. So any multiple is 7 times a
counting number and 7 must be a factor.
E. For example, any factor of 9 has to be 9 or less, so there
are only 9 possible numbers. But multiples of 9 are
created by continually adding 9s and you can add 9s
forever.
1
2
3 Consolidation ± 20–30 min
Checking (Pairs)
4
Draw students’ attention to the Communication Tip. Ensure
that they are comfortable with the notation “…,” which is
called an ellipsis. If students require additional guidance, refer
them to Oleh’s and Léa’s methods in the example. You may
want to distribute number lines to students; however, students
do not need to use scaled number lines; rather, they can
sketch empty number lines.
5
6
Practising (Individual)
7
These questions give students opportunities to practise
calculating multiples. Students will also explain connections
between factors and multiples. Encourage students to use
mental math strategies in their calculations. Encourage
students to use number lines as visualization tools.
2. Ensure students understand that the “first five multiples”
can be calculated by multiplying by the first five
counting numbers, 1, 2, 3, 4, and 5, or by repeatedly
adding the number to itself until five multiples
are listed.
3. If extra support is required, guide these students and
provide copies of Scaffolding for Lesson 2,
Question 3 p. 77.
7. Students create lists of multiples of two numbers and
then identify the numbers that appear in both lists. In
later grades, students will formalize this understanding
as they learn about common multiples.
8
“Which method is easier for you to use? Explain.”
• Oleh’s method is easier because multiplying to determine the
multiples is faster than adding, and I only have to replace the
last digits of 2000 with the multiples of 7 to get the years.
• Léa’s method is easier because I like adding better than
multiplying.
Answers to Prompts
A. For example, I multiplied 7 by 3 to get 21.
B. For example, I added 7 to 2014 to get the year 2021.
C. For example, I listed the multiples of 7 until I got to 70.
I stopped at 70 because I know 2000 ⫹ 70 = 2070 is
past 2067.
7, 14, 21, 28, 35, 42, 49, 56, 63, 70, …
I added these multiples to 2000 to get these years in
which the comet will likely be seen from Earth: 2007,
2014, 2021, 2028, 2035, 2042, 2049, 2056, and 2063.
Reflecting (Whole Class)
Here students reflect on the relationship between factors and
multiples. Students should recognize that a multiple is the
product of a factor and a counting number.
20
Chapter 3: Number Relationships
Answers to Key Question
5. a) 8, 16, 24, 32, 40, 48, 56, 64, 72, 80 plates
b) For example, 10 packages; Pauline needs to buy plates
for 80 people, and 80 plates are in 10 packages.
c) 12, 24, 36, 48, 60, 72, 84, 96, 108, 120
d) For example, 7 packages; Pauline needs to buy at least
80 glasses, and 6 packages have 72 glasses, which is
too little, but 7 packages have 84 glasses, which is
enough.
Copyright © 2010 Nelson Education Ltd.
01-NM6TR-C03-Interior_L01-L02.qxd
12/1/08
10:16 PM
Page 21
Closing (Whole Class)
Question 9 allows students to reflect on and consolidate their
learning for this lesson as they think about multiples of a
number.
Follow-Up and Preparation for Next Class
Have students list the multiples of 2 from 2 to 48. Challenge
students to explain which of the numbers they listed has the
most factors.
Answer to Closing Question
9. Disagree; for example, numbers like 1, 10, 19, and
28 are 9 apart but none are multiples of 9. The list
would have to start at 0, 9, or a multiple of 9 for the
numbers to be all multiples of 9.
Opportunities for Feedback: Assessment for Learning
What you will see students doing
When students understand
If students misunderstand
• Students multiply by counting numbers and/or use skip counting to calculate
multiples.
• Students may have difficulty using mental math to calculate multiples. (See
Extra Support 1.)
Key Question 5 (Problem Solving, Visualization)
• Students use mental math to calculate multiples of 8 up to 80 and multiples
of 12 up to 120 and use their calculations to solve a problem.
• Students may have difficulty choosing the correct number of plates and cups.
(See Extra Support 2.)
• Students may have difficulty using mental math to calculate multiples. (See
Extra Support 1.)
Differentiating Instruction: How you can respond
EXTRA SUPPORT
1. Discuss mental math strategies for multiplying by counting numbers:
Doubling: Students can multiply a known factor by 2. For example, since
2 ⫻ 6 = 12, then doubling the counting number will result in 4 ⫻ 6 = 24.
Doubling can be repeated. For example, 8 ⫻ 6 = 48.
2. Guide students to skip count by 8s using a 100 chart until they reach a
number between 70 and 80, circling each multiple of 8. Repeat with 12s,
circling each multiple of 12 with a different colour.
Skip counting: Students can skip count from a known factor. For example,
since 5 ⫻ 6 = 30, then 6 ⫻ 6 = 30 ⫹ 6, which is 36. Students can also
skip count down. For example, since 5 ⫻ 6 = 30, then 4 ⫻ 6 = 30 ⫺ 6,
which is 24.
EXTRA CHALLENGE
• Have students research and write a problem about an event that occurs every
number of years, for example, the Olympics or leap years. Then have students
exchange their problems with a partner and solve the problems.
b) Andrea’s 21st birthday is in the year 2016, and she wants to know if the
same year will have an Olympic Games. Which Olympic Games, if any, is
occurring that year?
Example:
a) The summer and winter Olympics both occur every four years. Calculate
the years for the next five Olympic Summer Games, starting with 2008.
Then calculate the years for the next five Olympic Winter Games, starting
with 2006.
SUPPORTING DEVELOPMENTAL DIFFERENCES
• Some students may be able to determine the multiples but have difficulty
adding them to a first number, like to the year 2007. The addition component
might be eliminated for these students.
• Other students might have difficulty calculating multiples without concrete
support. Provide counters to help students create equal groups to determine
multiples.
SUPPORTING LEARNING STYLE DIFFERENCES
• Some students may benefit from comparing visual representations of different
sets of multiples. For example, on a 100 chart, they can colour the multiples
of 6, 8, and 9 in different colours to see how the multiples of 9 are more
spread out than the multiples of 6 or 8.
Copyright © 2010 Nelson Education Ltd.
Lesson 2: Identifying Multiples
21