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Section A Number Theory

4-1 Divisibility

4-2 Factors and Prime Factorization

4-3 Greatest Common Factor

Section A Quiz

Section B Understanding Fractions

4-4 Decimals and Fractions

4-5 Equivalent Fractions

4-6 Mixed Numbers and Improper Fractions

Section B Quiz

Section C Introduction to Fraction Operations

4-7 Comparing and Ordering Fractions

4-8 Adding and Subtracting Fractions with Like Denominators

4-9 Estimating Fractions Sums and Differences

Section C Quiz

Number Theory & Fraction Unit Test

4-1 Divisibility

Vocabulary

Divisible ______________________________________________________________

______________________________________________________________

Composite Number _____________________________________________________

______________________________________________________________

Prime Number: ________________________________________________________

______________________________________________________________

Divisibility Rules

|A number is divisible by |Divisible |Not Divisible |

| 2 if the last digit is even (0, 2, 4, 6, 8). | | |

| 3 if the sum of the digits is divisible by 3. | | |

| 4 if the last two digits form a number divisible by 4. | | |

| 5 if the last digit is a 0 or 5. | | |

| 6 if the number is divisible by both 2 and 3. | | |

| 9 if the sum of the digits is divisible by 9. | | |

| 10 if the last digit is 0. | | |

Example 1 Checking Divisibility

A) Tell whether 610 is divisible by 2, 3, 4, and 5.

| |Divisible or Not Divisible |Explain how you know |

|2 | | |

|3 | | |

|4 | | |

|5 | | |

So 610 is divisible by ________________.

B) Tell whether 387 is divisible by 6, 9, 10.

| |Divisible or Not Divisible |Explain how you know |

|6 | | |

|9 | | |

|10 | | |

So 387 is divisible by _________________.

Example 2 Identifying Prime and Composite Numbers

Tell whether each number is prime or composite.

A) 45

Divisible by: 1, 3, 5, 9, 15, 45 Prime or Composite

B) 13

Divisible by: ____________________ Prime or Composite

C) 19

Divisible by: ____________________ Prime or Composite

D) 49

Divisible by: ____________________ Prime or Composite

Lightly shade in all the prime numbers.

|1 |2 |3 |4 |

| | | | |

| | | | |

| | | | |

| | | | |

| | | | |

_________ = __________ = _________

The same area is shaded when the rectangle is divided into 8 parts, 12 parts, and 4 parts.

Example 2 Multiplying and Dividing to Find Equivalent Fractions

Find the missing number that makes the fractions equivalent.

A) [pic] B) [pic]

C) [pic] D) [pic]

Example 3 Writing Fractions in Simplest Form

Write each fraction in simples form.

A) [pic]

B) [pic]

C) [pic]

D) [pic]

Think and Discuss

1. Explain whether a fraction is equivalent to itself.

2. Tell which of the following fractions are in simplest form: [pic], [pic], [pic]. Explain.

3. Explain how you know that [pic] is in simplest form.

4-6 Mixed Numbers and Improper Fractions

Vocabulary

Improper Fraction ______________________________________________________

________________________________________________________________

Proper Fraction ________________________________________________________

________________________________________________________________

|Improper and Proper Fractions |

|Improper Fractions |[pic] = 1 [pic] = 1 |

|Numerator equals denominators ( fraction is equal to 1 | |

| |[pic] > 1 [pic] > 1 |

|Numerator greater than denominator ( fraction is greater than 1 | |

|Proper Fractions | |

|Numerator less than denominator ( fraction is less than 1 |[pic] < 1 [pic] < 1 |

Example 1 Astronomy Application

The longest total solar eclipse in the next 200 years will take place in 2186. It will last about [pic] minutes. Write [pic] as a mixed number.

METHOD 1: Use a model.

Draw squares divided into half sections. Shade 15 of the half sections.

|[pic] |[pic] |

|[pic] |[pic] |

|[pic] |[pic] |

|[pic] |[pic] |

|[pic] |[pic] |

|[pic] |[pic] |

|[pic] |[pic] |

|[pic] |[pic] |

1 2 3 4 5 6 7 [pic]

There are ____ whole squares and ___ half square, or [pic] squares, shaded.

METHOD 2: Use division

[pic] = [pic]=

Example 2 Writing Mixed Numbers as Improper Fractions

A) Write [pic] as an improper fraction. Use multiplication and addition.

B) Write [pic] as an improper fraction. Use multiplication and addition.

Think and Discuss

1. Read each improper fraction [pic], [pic], [pic].

2. Tell whether each fraction is less than 1, equal to 1, or greater than 1:

[pic], [pic], [pic], [pic].

3. Explain why any mixed number written as a fraction will be improper.

4-7 Comparing and Ordering Fractions

Vocabulary

Like Fractions _________________________________________________________

________________________________________________________________

Unlike Fractions _______________________________________________________

________________________________________________________________

Common Denominator __________________________________________________

________________________________________________________________

Example 1 Comparing Fractions

Compare. Write , =.

A) [pic] [pic]

B) [pic] [pic]

Example 2 Cooking Application

A) Ray has [pic] cup of nuts. He needs [pic] cup to make cookies. Does he have enough nuts for the recipe?

Compare [pic] and [pic].

B) Rachel and Hannah have [pic] cups of cabbage. They need [pic] cups to make potstickers. Do they have enough for the recipe?

Compare [pic] and [pic].

Example 3 Ordering Fractions

A) Order [pic], [pic], and [pic] from least to greatest.

[pic] = ___ [pic] = ___ [pic] = ___ Rename with like denominators

The fractions in order from least to greatest are _____________________________.

B) Order [pic], [pic], and [pic] from least to greatest.

[pic] = ___ [pic] = ___ [pic] = ___ Rename with like denominators

The fractions in order from least to greatest are _____________________________.

Think and Discuss

1. Tell whether the values of the fractions change when you rename two fractions so that they have common denominators.

2. Explain how to compare [pic] and [pic].

4-8 Adding and Subtracting with Like Denominators

Example 1 Life Science Application

A) Sophie plants a young oak tree in her backyard. The distance around the trunk grows at a rate of [pic] inch per month. Use pictures to model how much this distance will increase in two months, then write your answer in simplest form.

| | | | |

| | | | |

| | | | |

| | | | |

| | | | |

| | | | |

( =

[pic] + [pic]

[pic] + [pic] = [pic] Add the numerators. Keep the same denominators

= [pic] Write your answer in simplest form.

The distance around the trunk will increase by _____ inch.

B) Snow was falling at a rate of [pic] inch per hour. How much snow fell after two hours? Write your answer in simplest form.

After two hours _______ inch of snow fell.

Example 2 Subtracting Like Fractions and Mixed Numbers

Subtract. Write each answer in simplest form.

A) 1 - [pic]

___ - ___ = ___

Check:

B) 1 - [pic]

___ - ___ = ___

Subtract. Write each answer in simplest form.

C) [pic] - [pic]

[pic] - [pic] Subtract the fractions. Then subtract the whole numbers.

[pic]

[pic] Write your answer in lowest terms.

Check:

D) [pic] - [pic]

Example 3 Evaluation Expressions with Fractions

Evaluate each expression for x = [pic]. Write each answer in simplest form.

A) [pic] - x

[pic] - x Write the expression

[pic] - [pic] = [pic] Substitute [pic] for x and subtract the numerators. Keep the same denominator.

= [pic] Write your answer in simplest form.

B) x + [pic]

C) x + [pic]

Evaluate each expression for x = [pic]. Write each answer in simplest form.

A) [pic] - x

B) x + [pic]

Think and Discuss

1. Explain how to add or subtract like fractions.

2. Tell why the sum of [pic] and [pic] is not [pic]. Give the correct sum.

3. Describe how you would add [pic] and [pic]. How would you subtract [pic] from [pic]?

4-9 Estimating Fraction Sums and Differences

Example 1 Estimating Fractions

Estimate each sum or difference by rounding to 0, [pic], or 1.

A) [pic] + [pic]

[pic] + [pic] Think: [pic] rounds to 1 and [pic] rounds to 0.

1 + 0 = 1

[pic] + [pic] is about 1.

B) [pic] - [pic]

[pic] - [pic] Think: [pic] rounds to ____ and [pic] rounds to ____.

__ - __ = ___

[pic] - [pic] is about ____.

C) [pic] + [pic]

[pic] + [pic] Think: [pic] rounds to ____ and [pic] rounds to ____.

__ + __ = ___

[pic] + [pic] is about _____.

D) [pic] - [pic]

[pic] - [pic] Think: [pic] rounds to ____ and [pic] rounds to ____.

__ - ___ = ____

[pic] - [pic] is about _____.

Example 2 Sports Application

|Nature Club’s |

|Biking Distances |

|Day |Distances (mi) |

|Monday |[pic] |

|Tuesday |[pic] |

|Wednesday |[pic] |

|Thursday |[pic] |

A) About how far did the Nature Club ride

on Monday and Tuesday?

B) About how much farther did the Nature Club ride on Wednesday than Thursday?

C) Estimate the total distance that the Nature Club rode on Monday, Tuesday, and Wednesday.

Think and Discuss

1. Tell whether each fraction rounds to 0, [pic], or 1: [pic], [pic], [pic].

2. Explain how to round mixed numbers to the nearest whole number.

3. Determine whether the Nature Club met their goal to ride at least 35 total miles.

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