Download Chapter Audio Summary for McDougal Littell

Chapter Audio Summary for McDougal Littell
Middle School Math, Course 3
Chapter 4 Factors, Fractions, and Exponents
In Chapter 4 you saw how to use prime factorization to find the greatest common factor,
or GCF, of two numbers and to use the GCF to write a fraction in simplest form. You
saw how to find the least common multiple or LCM of two numbers and how to compare
and order fractions. You saw how to use the rules of exponents to simplify expressions
with positive and negative exponents. You also saw how to write a number in scientific
notation.
Turn to the lesson-by-lesson Notebook Review that starts on p. 190 of the textbook.
Review the Check Your Definitions section and the Use Your Vocabulary section, and
then look at the review examples that begin with the lesson numbers.
Lesson 4.1 Can you write a prime factorization?
Important words and terms to know are: prime number, composite number, prime
factorization, factor tree, and monomial.
The goal of lesson 4.1 is to write the prime factorization of a composite number.
Read the review example.
“Write the prime factorization of 504.”
The first step is to write 504 as 2 times 252. The next step is to factor 252, and continue
the process until you have only the prime factors of 504. Remember that prime numbers
have factors of itself and one. 252 divided by 2 is 126, which divided by 2 is 63. 63 can
be written as 3 times 21, and 21 can be written as 3 times 7. You now have the prime
factorization, 504 = 2⋅ 2⋅ 2⋅ 3⋅ 3⋅ 7 . Remember that an exponent shows how many times
a base is multiplied times itself. The prime factorization can also be written as 2 3 ⋅ 32 ⋅ 7 .
Now try Exercises 2, 3, 4, and 5. If you need help, go back to the worked-out examples
on pages 168-170.
Lesson 4.2 Can you find the GCF of two numbers?
Important words and terms to know are: common factor, greatest common factor, and
relatively prime.
The goal of lesson 4.2 is to find the greatest common factor, or GCF, of two numbers.
Read the example.
McDougal Littell: Audio Summary
Factors, Fractions, and Exponents
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Chapter Audio Summary for McDougal Littell
Middle School Math, Course 3
“Find the GCF of 36 and 60.”
Remember that to find the GCF of two numbers, start by finding the prime factorization
of each number, 36 = 22 ⋅ 32 and 60 = 22 ⋅ 3⋅ 5 . Next, find which numbers are common
factors. Both factorizations have a 2-squared term and each has at least one factor of 3,
so the GCF is 2 2 ⋅ 3, or 12. Notice that only one factor of 3 was included in the GCF,
because there is only one 3 in the prime factorization of 60.
Now try Exercises 6, 7, 8, and 9. If you need help, go back to the worked-out examples
on pages 173-175.
Lesson 4.3 Can you write a fraction in simplest form?
Important terms to know are: simplest form and equivalent fractions.
The goal of lesson 4.3 is to write a fraction in simplest form.
Read the example.
“Write 48/72 in simplest form.”
To reduce the fraction to simplest form, you must first find the GCF of 48 and 72.
Because 48 = 24 ⋅ 3, and 72 = 23 ⋅ 32 , the GCF is 2 3 ⋅ 3, or 24. The next step is to divide
the numerator and denominator by the GCF, 24, so the simplest form of the fraction is
2/3.
Now try Exercises 10, 11, 12, and 13. If you need help, go back to the worked-out
examples on pages 179-181.
Lesson 4.4 Can you find the LCM of two numbers?
Important words and terms to know are: multiple, common multiple, and least common
multiple.
The goal of lesson 4.4 is to find the least common multiple of two numbers.
Read the example.
“Find the LCM of 20 and 48.”
Remember that a multiple of a number is that number multiplied by a positive integer.
For example, the first three multiples of 20 are 20, 40, and 60. To find the least common
multiple of 20 and 48, first find the prime factorization of each number:
McDougal Littell: Audio Summary
Factors, Fractions, and Exponents
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Chapter Audio Summary for McDougal Littell
Middle School Math, Course 3
20 = 2 ⋅5 and 48 = 2 ⋅ 3 . Unlike GCF, where you found the smallest number of times a
common factor appeared in the prime factorization, in an LCM you need to identify the
highest number of times a factor appears in either factorization. Choose 24, 3, and 5 as
the highest number of times a factor appears for 20 and 48, so that the LCM is 2 4 ⋅ 3⋅ 5 or
240.
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Now try Exercises 14 through 18. If you need help, go back to the worked-out
examples on pages 186 and 187.
Turn to the lesson-by-lesson Notebook Review that starts on p. 210 of the textbook.
Review the Check Your Definitions section and the Use Your Vocabulary section, and
then look at the review examples that begin with the lesson numbers.
Lesson 4.5 Can you compare and order fractions?
An important term to know is: least common denominator.
The goal of lesson 4.5 is to compare and order fractions.
Read the example.
“Compare 11/18 and 3/4.”
To compare the fractions 11/18 and 3/4, you must first find the least common
denominator. Remember that the least common denominator is the LCM of the
denominators of the fractions. The LCM of 18 and 4 is 36, so the least common
denominator of the fractions is 36. To compare the fractions, you need to write each
fraction as an equivalent fraction with the common denominator. Determine what you
need to multiply each denominator by to get 36. Because 18 times 2 equals 36, you must
also multiply the numerator, 11, by 2 to get an equivalent fraction. So 11/18 = 22/36. To
find a fraction equivalent to 3/4 with a denominator of 36, multiply both the numerator
and denominator by 9 which gives you 27/36. Now compare the fractions. Because 22 is
less than 27, 22/36 is less than 27/36. Using the original fractions, 11/18 < 3/4.
Now try Exercises 2, 3, 4, and 5. If you need help, go back to the worked-out examples
on pages 192 and 193.
Lesson 4.6 Can you use the rules of exponents?
The goal of lesson 4.6 is to use the rules of exponents to simplify multiplication and
division expressions with exponents.
McDougal Littell: Audio Summary
Factors, Fractions, and Exponents
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Chapter Audio Summary for McDougal Littell
Middle School Math, Course 3
Read the example.
“Multiply or divide. Write your answer as a power.”
Remember that when you multiply two powers with the same bases, add the exponents.
In example a, x 7 ⋅ x 8 = x 7 +8 . Add 7 + 8 to get x 15 .
In example b, when you divide powers with the same base, you can subtract their
exponents. Subtract 6 – 3 to get a3.
Remember that a power expresses the number of times the base is multiplied by itself.
You can always check your answers by expanding the powers and then simplifying.
Now try Exercises 6, 7, 8, and 9. If you need help, go back to the worked-out examples
on pages 196-198.
Lesson 4.7 Can you use negative exponents?
The goal of lesson 4.7 is to rewrite an expression containing negative exponents using
only positive exponents.
Read the example.
“Write x-4 • x-3 using only positive exponents.”
First, use the rules of exponents to combine the terms in the example. –4 + –3 equals –7,
so x-4 • x-3 equals x-7. Remember that a base to a negative exponent is equal to the
reciprocal of that base to the positive exponent, so x-7 = 1/ x7.
Now try Exercises 10, 11, 12, and 13. If you need help, go back to the worked-out
examples on pages 201 and 202.
Lesson 4.8 Can you write a number in scientific notation?
An important term to know is: scientific notation.
The goal of lesson 4.8 is to write numbers in scientific notation.
Read the example.
“Write a number in scientific notation.”
McDougal Littell: Audio Summary
Factors, Fractions, and Exponents
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Chapter Audio Summary for McDougal Littell
Middle School Math, Course 3
In example a, to write 980,000,000 in scientific notation, you must convert the number to
a number between 1 and 10 multiplied by a power of 10. Move the decimal 8 places to
the left to get 9.8. Moving the decimal to the left eight places means that you are
multiplying by 10 eight times. So the answer is 9.8 x 108.
For example b, move the decimal 5 places to the right. Remember that when you move
the decimal to the right you are dividing by 10. Use a negative exponent for the power of
10 to indicate division. So the answer is 1.2 x 10-5.
Now try Exercises 14 through 17. If you need help, go back to the worked-out
examples on pages 205 and 206.
McDougal Littell: Audio Summary
Factors, Fractions, and Exponents
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