Download Chapter Three Notes

Name:______________________Chapter 3 Test Date:______
Chapter 3: Number Theory and Fraction Concepts
Lesson 3.1 Factors, Multiples, and Divisibility
Objective: Use divisibility rules to determine if one number is divisible by
another.
Warm-Up:
1. 75 ÷ 5 =____ 2. 48 ÷ 4 = _____3. 52 ÷ 5 =_____4. 104 ÷ 6=_____
A multiple of a given number is the product of that number and a whole
number.
A whole number is divisible by another whole number when the quotient is a
whole number and the remainder is 0.
When a number is divided by any of its factors, or divisors, the remainder is
0.
Divisibility Rules
A whole number is divisible by
2 - if the number is even
3 – if the sum of the digits is divisible by 3
4 – if the last two digits are divisible by 4
5 – if the last digit is 0 or 5
6- if the number is divisible by 2 and 3
9 - if the sum of the digits is divisible by 9
1
Examples
Lesson 3.1 Practice Problems:
1. Tell whether 135 is divisible by 2, 3, 4, 5, 6, 9 or 10.
2: Is the ones digit even? _____
3: Is the sum of the digits divisible by 3? _____
1 + 3 + 5 = ____ ÷ 3 = ____
4: Are the last two digits divisible by 4? _____
5: Is the ones digit a 0 or a 5? _____
6: Is the number divisible by both 2 and 3? ______
9: Is the sum of the digits divisible by 9? _____
1 + 3 + 5 = ___ ÷ 9 = ___
10: Is the ones digits a 0? _____
Something to think about:
1. Based on what you know, how would you explain the Divisibility Rules to
your parents?_______________________________________________
_________________________________________________________
_________________________________________________________
2. How would you adapt the Divisibility Rules to create a strategy or
mnemonic device to remember the rules?__________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
Lesson 3.2 Prime and Composite Numbers
Objective: Identify numbers as prime or composite, and give the prime
factorization for numbers.
Warm-Up: Write a multiplication sentence for each product.
1. 12____________
2. 7 ____________
3. 72____________
4. 33____________
** Every whole number greater than 1 is either a prime or composite number.
Two is the only even prime number.
A prime number is a whole number greater than 1 and has exactly two
factors, 1 and itself.
2
A composite number is a whole number greater than 2 and has more than two
factors.
Prime factorization is the set of primes whose product is a given composite.
Steps to find the Prime Factorization of any Composite Number:
1. Begin with any two factors of the number.
2. Continue to write factors until all the factors are prime.
3. Circle the prime factors.
4. Write the prime factors in order from least to greatest.
5. Write the factors in exponential form.
**Use a factor tree.
Practice Problems:
Write the prime factorization of each number.
1.
48
2. 66
Something to think about:
1. Based on what you have learned, how would you explain prime
factorization?______________________________________________
_________________________________________________________
_________________________________________________________
2. Compare and Contrast prime and composite numbers.
3
Lesson 3.3 Greatest Common Factor
Warm-Up: List the factors of each number.
1. 12
2. 31
3. 54
4. 100
Common factor is a factor that is the same for two or more numbers.
The greatest common factor (GCF) is the largest number that is a factor of
two or more numbers.
***Hint: The GCF is ALWAYS
≤ (the smallest # in a given set).
Two ways to find the GCF are:
1. List the factors of each number.
(T Charts/Number Buddies)
Find the GCF of 36 and 42 ____________.
_________________________________________________________
2. Use prime factorization.
1. List the prime factors of each number.
36:
42:
2. Circle the factors that the numbers have in common.
36:
42:
3. Multiply the common prime factors.
36:
42:
Something to think about:
1. Based on what you have learned, how would you explain GCF to your
parents.___________________________________________________
_________________________________________________________
2. Identify two ways to find the GCF of a set of numbers.
_________________________________________________________
_________________________________________________________
4
Lesson 3.4 Least Common Multiple
Warm-Up: List the first five multiples of each number.
1. 8:
2. 11:
3. 19:
4. 21:
5. 30:
6. 15:
A common multiple is a multiple that is the same for two or more numbers.
The least common multiple (LCM) is the smallest number, other than zero,
that is a multiple of two or more numbers.
***Hint: The LCM is ALWAYS
≥ (the largest # in a given set).
Two ways to find the LCM are:
1. List the multiples of each number.
1. List multiples of the numbers.
4: 4, 8, 12, 16, 20, 24
5: 5, 10, 15, 20, 25
2. Find the lowest number they have in common.
LCM of 4 and 5 is 20.
LCM can be used as the common denominator.
_____________________________________________
2. Use prime factors to find the LCM.
1. List the prime factors of each one
4: 2x2
5: 5
2. Circle the greatest number of times each different
number appears.
4: 2x2
5: 5
3. Find the product of the numbers you circled.
2 x 2 x 5 = 20
The LCM of 4 and 5 is 20.
5
Something to think about:
1. Suppose everyone but Mr. Cappotelli could tell you what LCM means.
Explain what you should do to teach him this mathematical concept.
_________________________________________________________
_________________________________________________________
_________________________________________________________
2. Recommend a strategy to make LCM’s work for you.________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
3. Compare and Contrast GCF and LCM using a Venn Diagram.
6
Lesson 3.5 Make a Table
1. Set up the table with the correct labels.
2. Enter known data.
3. Look for a pattern. Extend the table as needed.
4. Find the answer in the table and label the answer.
Sara’s mother runs every day. For every 6 miles she runs, Sara runs 1 mile.
Last week Sara ran 5 miles. How far did her mother run?
Complete practice problems in workbook on page 35 (front) 1 and 2.
Something to think about:
1. What can you point out about the problem solving strategy
Make A Table?_____________________________________________
_________________________________________________________
_________________________________________________________
2. Explain the steps necessary when using the Make A Table Strategy.
_________________________________________________________
_________________________________________________________
_________________________________________________________
Lesson 3.6 Understanding Fractions
Warm-Up: Write the number for the word given.
1. one half _____
2. five eighths ______
3. two thirds _____
4. six sevenths _____
A fraction is a number that can be used to describe a part of a set or a part
of a whole.
7
The denominator gives the total number of objects in the set or the number
of equal parts in the whole. (The bottom number in a fraction.)
The numerator gives the number of objects or equal parts being considered.
(The top number in a fraction.)
3
8
Go over examples in workbook page 37 front.
Something to think about:
1. What can you point out about fractions?_________________________
_________________________________________________________
_________________________________________________________
2. Name the two parts of a fraction._____________________________
_________________________________________________________
Lesson 3.7 Equivalent Fractions
Warm-Up: Find the common factors for each pair of numbers.
1. 6, 9 __________
2. 10, 15 __________
3. 4, 2 __________
4. 24, 36 __________
Equivalent fractions are fractions that are equal.
Two ways to write equivalent fractions:
1. Multiply (x) the numerator and denominator by the same number.
2
=
3
Non-example=
2
=
3
5
6

2
=
3
2
=
3
10
24
**Hint: Whatever you multiply the denominator by you will
multiply the numerator by.
8
2. Divide (÷) the numerator and denominator by the same number.
150
30
150
3
150
75
=
=
=
40
4
100
200
200
200
**Hint: Whatever you divide the denominator by you will divide
the numerator by.
The least common denominator (LCD) is the least common multiple of the
denominators of two or more fractions.
Simplest form means a fraction is in lowest terms.
Practice Problems on workbook page 38 front.
Something to think about:
1. Describe how you can find equivalent fractions.___________________
_________________________________________________________
_________________________________________________________
2. Explain how you can put fractions in simplest form. ________________
_________________________________________________________
_________________________________________________________
Lesson 3.8 Improper Fractions and Mixed Numbers
Warm-Up: Solve
1. 9 ÷ 2 = _____
2. 14 ÷ 3= _____ 3. 28 ÷ 5 = _____ 4. 20 ÷ 8 = _____
A proper fraction is number less than one. Examples:_________________
An improper fraction has a numerator that is greater than or equal to its
denominator. Examples:_______________________________________
A mixed number combines a whole number and a fraction, which is greater
than one. Examples:__________________________________________
9
1. Converting an improper fraction to a mixed number
12
Write
as a mixed number.
5
Step 1: Divide the numerator by the denominator.
Step 2: The quotient is the whole number in the mixed number.
Step 3: The remainder is the numerator.
Step 4: The denominator stays the same.
2. Converting a mixed number to an improper fraction.
2
Write 3 as an improper fraction.
5
Step 1: Multiply the denominator by the whole number.
Step 2: Then add the numerator.
Step 3: Write the answer to steps 1 and 2 as the numerator.
Step 4: Use the original denominator.
Practice Problems on workbook page 39 front.
Something to think about:
1. Identify the steps when converting an improper fraction to a mixed
number.___________________________________________________
_________________________________________________________
_________________________________________________________
2. Identify the steps when converting a mixed number to an improper
fraction. _________________________________________________
_________________________________________________________
_________________________________________________________
Lesson 3.9 Estimating Fractional Amounts
To estimate a fractional amount, use a benchmark fraction that is close to
the actual amount.
Benchmark fractions: (common fractions)
1
1
1
2
3
,
,
,
,
4
3
2
3
4
10
Steps to Estimating Fractional Amounts:
1. Use benchmark fractions:
1
1
1
2
,
,
,
,
4
3
2
3
3
4
Draw a rectangle:
2.
Use rounding:
98
is close to
389
100
400
=
3. Draw a picture:
3
a.
5
c.
5
7
1
4
b. 1
4
9
d. 2
1
6
Something to think about:
1. Recommend at least one way to remember benchmark fractions. _______
_________________________________________________________
_________________________________________________________
297
2. Explain how you could tell which benchmark fraction is close to
?
400
_________________________________________________________
_________________________________________________________
3. What can you conclude about benchmark fractions?
_________________________________________________________
_________________________________________________________
11
Lesson 3.10 Connecting Fractions and Decimals
Warm-Up: Write each word phrase as a decimal.
1. one tenth____________
2. one hundredth______________
3. one and forty-seven hundredths ______________
4. five and sixteen thousandths ________________
**Hint: A fraction and a decimal can both represent the same value.
1. Fractions to Decimals:
4
5
=
100
= .80 (eight tenths or
eighty hundredths)
*Whatever you do to the bottom you do to the top.
2. Memorize these benchmark fractions converted to decimals:
1
= .25
4
1
= .33
3
1
= .50
2
2
= .66
3
3
= .75
4
3. Divide the numerator by the denominator:
9
30
4. Key Concepts:
Terminating decimals are_________________________________
____________________________________________________.
Examples:
12
Repeating Decimals are __________________________________
___________________________________________________.
Examples:
Something to think about:
3
, what is this fraction’s decimal
20
equivalent? ________________________________________________
_________________________________________________________
2. Explain how to convert decimals to fractions and fractions to decimals.
_________________________________________________________
_________________________________________________________
_________________________________________________________
3. Recommend at least one helpful hint when converting fractions and
decimals. __________________________________________________
_________________________________________________________
4. What can you conclude about today’s math class? _________________
_________________________________________________________
_________________________________________________________
1. Suppose you have the fraction
Lesson 3.11 Comparing Fractions and Decimals
Warm-Up: Write each fraction as a decimal.
3
6
7
1.
_______
2.
_______
3.
_______
4
9
8
1. Comparing Fractions:
a. Compare denominators
b. Compare numerators
c. Use the inequality symbols:
>, < or =
13
4.
5
______
12
Examples:
1
4
3
5
9
10
12
30
Strategies: use decimals, find the LCD, or use cross products.
2. Ordering Fractions
a. greatest to least (find common denominators)
1 2 3
,
,
8 5 4
1. Find common denominators (LCD)
2. Whatever you multiply the denominator by you multiply the
numerator by.
3. Order the original numbers on the answer line.
3. Ordering Fractions and Decimals
a. greatest to least (convert to decimals)
1 4
, , .35
4 5
1. Write fractions as decimals.
2. Then compare.
3. Order original numbers on the answer line.
**Hint: Convert fractions to decimals.
4. Use a number line to order fractions and decimals.
________________________________
Something to think about:
1. Suppose you have the 0.7,
25 6
, . How would you order them from
30 10
greatest to least?
_________________________________________________________
_________________________________________________________
14
2. Explain how to compare and order fractions.
_________________________________________________________
_________________________________________________________
3. Recommend one helpful hint when converting decimals to fractions and
fractions to decimals. How is it similar to the water cycle?
_________________________________________________________
_________________________________________________________
Lesson 3.12 Multiple-Step Problems
Key Idea: Identifying hidden questions helps you solve multiple-step
problems. Figure out what you are looking for.
Steps to Solving Multiple-Step Problems
1. Pick out/highlight the important information.
2. Choose the correct operation or operations (+, -, x, ÷).
3. Reread the question several times to check for understanding.
4. Does my answer make sense?
5. Write the answer to the question in a complete sentence.
Complete practice problems on workbook page 43 (front) #1-3.
Something to think about:
1. Suppose you are working on a multiple-step problem, what takes do you
need to use to successfully complete the problem? ___________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
2. Recommend a helpful hint when doing multiple-step problems. ________
_________________________________________________________
_________________________________________________________
3. Create your own multiple-step problem._________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
What should YOU do to prepare for the Chapter 3 Test???
1.
2.
3.
4.
5.
15