Transcript CPM Lesson 1.2.4

Lesson 1.2.4
Concept: Products, Factors, and
Factor Pairs
Vocabulary:
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• Factors – numbers that create new numbers when they are
multiplied. ( 3 and 4 are factors of 12)
• Product – a number resulting from a multiplication or two or
more numbers. ( 25 is a product of 5 x 5)
• Prime Factor – a factor that is also a prime number.
(2 x 3 = 6, 2 and 3 are prime factors of 6)
In mathematics, factors are numbers that create new numbers when
they are multiplied.
• A number resulting from multiplication is called a product.
• In other words, since 2(3) = 6, 2 and 3 are factors of 6, while 6 is
the product of 2 and 3.
• Also, 1(6) = 6, so 1 and 6 are two more factors of 6. T
The number 6 has four factors; 1, 2, 3, and 6. In this lesson, you will
use an extended multiplication table to discover some interesting
patterns of numbers and their factors.
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73. Have you ever noticed how many patterns exist in a simple multiplication
table? Such a table is a great tool for exploring products and their factors.
• Get a Multiplication Chart from your
teacher, or use the 1-73 Student
eTool (CPM).
• Fill in the missing products to complete
the table.
• With your team, describe at least three
ways you could use the table to figure out
the missing numbers (besides simply
multiplying the row and column numbers)
.
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74. Gloria was looking at the multiplication table and noticed an interesting
pattern.
“Look,” she said to her team. “All of the prime numbers show up only two
times as products in the table, and they are always on the edges.”
Discuss Gloria’s observation with your team. Then choose one color to
mark all of the prime numbers on your multiplication chart.
a. Explain why the placement of the prime numbers makes sense?
Multiplication Chart link
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75. See how many patterns you can discover in the multiplication table.
a. Gloria’s observation in problem concept problem #74 related to prime
numbers. What other kinds of numbers do you know about? In your group
write a list of the kinds of numbers you have discussed. (You may want to
look back at Lesson 1.2.3 to refresh your memory.)
Composite numbers, even numbers, odd numbers,
any single number.
Show where these different kinds of numbers appear on the table using
different colors. Make a key/legend.
b. What patterns can you find in the locations of the numbers of each
type? Explain your observations.
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76. Consider the number 36
a. Choose a color or design (circle or x the number) and mark every 36
that appears in the table.
b. Imagine that more rows and columns are added to the multiplication
table until it is as big as your classroom floor. Would 36 appear more
times in this larger table? If so, how many more times and where? If
not, how can you be sure?
c. List all of the factor pairs of 36. (A factor pair is a pair of numbers
that multiply to give a particular product. For example, 2 and 10 make
up a factor pair of 20, because 2 · 10 = 20.)
• How do the factor pairs of 36 relate to where it is found in the
table?
• What does each factor pair tell you about the possible
rectangular arrays for 36?
• How many factors does 36 have?
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77. Frequency is the number of times an item appears in a set of
data. What does the frequency of a number in the multiplication table tell
you about the rectangular arrays that are possible for that number?
a. Gloria noticed that the number 12 appears, as a product, 6 times in the
table. She wonders, “Shouldn’t there be 6 different rectangular arrays
for 12?” What do you think? Work with your team to draw all of the
different rectangular arrays for 12 and explain how they relate to the
table.
b. How many different rectangular arrays can be drawn to represent the
number 48? How many times would 48 appear as a product in a table
as big as the classroom? Is there a relationship between these
answers?
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78. PRIME FACTORIZATION
a. What are all the factors of 200?
1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, and 200
b. A prime factor is a factor that is also a prime
number. What are the prime factors of 200?
• Sometimes it is useful to represent a
number as the product of prime
factors. How could you write 200 as a
product using only prime factors? Writing a
number as a product of only prime numbers
is called prime factorization
•
200
4 · 50
2·2
10 · 5
5·2
How could you write 200 as a product using
only prime factors? Writing a number as a
product of only prime numbers is
called prime factorization
c. What if you used 10 · 20 instead of 4 · 50?
Finish this prime factorization using a prime factor
tree.
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200 = 2 x 2 x 2 x 5 x 5
or
200 = 23 x 52
Learning Log #2
Title this entry “Characteristics of Numbers and Prime Factorization” and
label it with today’s date.
Reflect about the number characteristics and categories that you have
investigated. You may look at your lesson notes in your Concept Notebook
and refer to your toolkit notebook.
In this learning log entry…
1. Summarize the 7 characteristics of numbers you explored in Lesson
1.2.3 (such as prime and composite).
2. Describe how you use these properties to write the prime factorization of
a number.
#93 Checkpoint Quiz
Round the decimals to the specified place in parts (a) through (c). Place the
correct inequality sign (< or >) in parts (d) through (f).
a.
b.
c.
d.
e.
f.
17.1936 (hundredths)
0.2302 (thousandths)
8.256 (tenths)
47.2_____47.197
1.0032 _____1.00032
0.0089 _____0.03
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Tonight’s homework is…
1.2.4 Review & Preview, problems #85 to
#92, #94.
• Label your assignment with your name
and Lesson number in the upper right hand
corner of a piece of notebook paper.
(Lesson 1.2.4)
•Show all work and justify your answers for
full credit.
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