Download Operations With Fractions

Operations With Fractions
Chapter 4
Math 7
Estimating With Fractions and
Mixed Numbers
Lesson 4-1
Using Benchmarks With Fractions
• A benchmark is a convenient number used to
replace fractions that are less than 1.
Estimating Sums and Differences Using
Benchmarks
Estimate as O
when the
numerator is
very small
compared to the
denominator
Estimate as ½
when the
numerator is
about half the
denominator
Estimate as 1
when the
numerator and
the denominator
are nearly equal
Estimate to add
• Estimate ⅞ + ⅗ using benchmarks
• ⅞
• ⅗
1
½
• 1 + ½ = 1 ½ Use benchmarks to estimate each
fraction and then add
Estimating with Mixed Numbers
• When a sum or difference involves mixed
numbers, you can make a reasonable estimate
by rounding to the nearest whole number
• 8⅕-4¾
• 8 – 5 = 3 Round each mixed number and then
subtract
Estimating Products
To estimate a product of mixed numbers, round
each mixed number to the nearest whole
number, then multiply
2 ⅖ x 6 ⅟₉
2 x 6 = 12 round each number then multiply
Estimating with Compatible Numbers
To estimate a quotient of mixed numbers, you
can use compatible numbers
43 ⅟₄ ÷ 5 ⁷⁄₈
42 ÷ 6 = 7
practice
• Work with a partner to complete odds or
evens workbook p. 62
Adding and Subtracting Fractions
Lesson 4-2
Adding Fractions
• If the denominators are the same, add the
numerators and keep the denominator
⅖+⅕=⅗
Adding Fractions
• If the denominators are different, find LCD,
make equivalent fractions, add the
numerators and keep the denominator
⅘+⅔
Find LCD and make equivalent fractions
¹²⁄₁₅ + ¹⁰⁄₁₅ = ²²⁄₁₅ = 1 ⁷⁄₁₅
Subtracting Fractions
• If the denominators are the same, subtract
the numerators and keep the denominator
⅖-⅕=⅕
Subtracting Fractions
• If the denominators are different, find LCD,
make equivalent fractions, subtract the
numerators and keep the denominator
⅘-⅔
Find LCD and make equivalent fractions
¹²⁄₁₅ - ¹⁰⁄₁₅ = ²⁄₁₅
practice
• Work with a partner to complete odds or
evens workbook p. 64
Adding and Subtracting Mixed
Numbers
Lesson 4-3
Adding Mixed Numbers Mentally
• Mental Math
• Find 10¹⁄₅ + 6⅖
Add the whole numbers 10 + 6 = 16
Add the fractions
¹⁄₅ + ⅖ = ⅗
Combine the two parts 16 + ⅗ = 16 ⅗
Adding Mixed Numbers With Unlike
Denominators
• Find 8 ⅓ + 6 ½
• Estimate 8 + 7 = 15
Find the LCD: for ⅓ and ½ is 6
Add
8 ²⁄₆ + 6 ³⁄₆ = 14 ⁵⁄₆
Simplify 14 ⁵⁄₆ is in simplest form
Adding Mixed Numbers with
Renaming
• Find 15⁵⁄₆ + 3 ½
• Stack:
15⁵⁄₆ = 15⁵⁄₆
• Find LCD
+ 3 ½ = 3³⁄₆
18⁸⁄₆
• Rename ⁸⁄₆ as 1²⁄₆
• Add
18 + 1²⁄₆ = 19²⁄₆
• Simplify 19⅓
Practice
• Add mixed numbers
Subtracting Mixed Numbers With
Unlike Denominators
• Find 8 ½ - 6 ⅓
• Estimate 9 - 6 = 3
Find the LCD: for ⅓ and ½ is 6
Add
8 ³⁄₆ - 6 ²⁄₆ = 2 ⅟₆
Simplify
2 ⅟₆ is in simplest form
Subtracting Mixed Numbers with
Renaming
• Find 7 – 2 ⁵⁄₈
Write 7 as a mixed number
Stack the problem
Subtract
Simplify if needed
7 ⁸⁄₈
– 2 ⁵⁄₈
5 ³⁄₈
Another Example
• 11 ⅟₆ - 5 ⅔
Find LCD and stack it up
Rename
Simplify
5 ³⁄₆ = 5 ½
11 ⅟₆ = 10 ⁷⁄₆
- 5 ⁴⁄₆ = 5 ⁴⁄₆
5 ³⁄₆
Practice Time
• Subtract, simplify
Multiplying Fractions and Mixed
Numbers
Lesson 4-4
Three Easy Steps
• Find ⁵⁄₆ x ⅔
• Step 1: multiply numerators: 5 x 2 = 10
• Step 2: multiply denominators: 6 x 3 = 18
• Step 3: simplify: ¹⁰⁄₁₈ = ⁵⁄₉
Multiply x Whole Number
What is ²⁄₇ of 28?
Rewrite as:
²⁄₇ x ²⁸⁄₁
Simplify before multiplying: ²⁄₁ x ⁴⁄₁
Multiply numerators: 2 x 4 = 8
Multiply denominators: 1 x 1 = 1
Simplify: ⁸⁄₁ = 8
Mixed Number x Mixed Number
• Find 2 ³⁄₅ x 4 ½
Step 1: rewrite as improper fraction: ¹³⁄₅ x ⁹⁄₂
Step 2: multiply numerators: 13 x 9 = 117
Step 3: multiply denominators: 5 x 2 = 10
Step 4: simplify and write as a mixed number:
¹¹⁷⁄₁₀ = 11 ⁷⁄₁₀
Partner Practice
• multiply
Dividing Fractions and Mixed
Numbers
Lesson 4-5
Definition
Facts/Characteristics
Two numbers that when
multiplied their product is 1
to find the reciprocal of a fraction
interchange, or “flip” the
numerator and denominator
Vocabulary
Word
⅔
and
³⁄₂
Reciprocals
⅔ = ⁴⁄₆
are reciprocals of each other
Examples
Non-Examples
Fraction ÷ Fraction
• Find ⅔ ÷ ⁵⁄₆
⅔ ÷ ⁵⁄₆ = ⅔ ●⁶⁄₅ Step 1: rewrite and multiply by
the reciprocal of the divisor
= 2 ● 6 = 12 Step 2: multiply
3 ● 5 15
=4
Step 3: simplify
5
* did you cancel?
Dividing Mixed Numbers
• Rewrite the mixed numbers as improper
fractions and then follow the steps
9½ ÷ 2³⁄₄ = ¹⁹⁄₂ ÷ ¹¹⁄₄ rewrite as improper fractions
= ¹⁹⁄₂ x ⁴⁄₁₁
multiply by reciprocal of
divisor
= ⁷⁶⁄₂₂
= ³⁸⁄₁₁ = 3 ⁵⁄₁₁ simplify
Partner Practice
• Divide Mixed Numbers
Divide Fractions and Whole Numbers
• Remember all whole numbers can be written
with a denominator of 1
Find : ³⁄₄ ÷ 5
Find: 5 ÷ ³⁄₄
= ³⁄₄ ÷ ⁵⁄₁
= ⁵⁄₁ ÷ ³⁄₄
= ³⁄₄ x ¹⁄₅
= ⁵⁄₁ x ⁴⁄₃
= ³⁄₂₀
= ²⁰⁄₃
= 6⅔
Practice
• Divide
Solving Equations With Fractions
Lesson 4-6
Solving equations with fractions
• Use inverse operations to get the variable
alone on one side of the equation
Ex: x- ⅓ = ⁵⁄₆
+ ⅓ = ⁵⁄₆ + ⅓
x = ⁵⁄₆ + ⅓
x = ⁵⁄₆ + ²⁄₆
x = ⁷⁄₆ = 1¹⁄₆
add ⅓ to each side
find common denominator
simplify
Partner Practice
• Solve these equations