Math 8

Unit 3: Operations with Fractions

Unit 3: Operations with Fractions

What you’ll learn• Estimate the products and quotients of

fractions and mixed numbers• Multiply a fraction by a whole number and by

a fraction• Divide a fraction by a whole number and by a

fraction• Multiply and divide mixed numbers• Use the order of operations with fractions

Unit 3: Operations with Fractions

Why it’s important:• Used with cooking for measuring• Used with shopping (sales ½ off)• Used with construction for measuring

ReviewFractions are used to show parts of a

whole.

Fraction Review

41

Three parts out of four are coloured in.4

3

One part out of four is white.

Fraction Review

Review• The top is called the numerator.• The bottom is called the denominator.• The line is called the fraction bar.

41 numerator

denominator

fraction bar

Fraction ReviewReviewImproper fractions have a numerator larger than the denominator.Mixed fractions are a combination of whole number and fraction.

45 Improper

fraction411 Mixed

fraction

Fraction Review

ReviewEquivalent fractions – have the same value, they are multiples of each other.

41

82

Fraction Review

ReviewFind the smallest equivalent fraction.A. B.

C. D.

42

93

62

1812

• Equivalent fraction animation

Fraction ReviewReviewTo add or subtract fractions they must have the same denominator. Use equivalent fractions to get the same denominator.

41

82

Fraction Review

Make equivalent fractions by doubling the quarter.

The denominator must be the same for both fractions.

Reduce the fraction to the smallest possible equivalent fraction.

2184

82282

82

41

82

Fraction Review

Make equivalent fractions by halving the two eighths.

The denominator must be the same for both fractions.

Reduce the fraction to the smallest possible equivalent fraction.

2142

41141

41

41

82

3.1 USING MODELS TO MULTIPLY FRACTIONS AND WHOLE NUMBERS

Unit 3: Operations with Fractions

3.1 Using Models To Multiply Fractions and Whole Numbers

• Focus: Use models to multiply a fraction by a whole number

3.1 Using Models To Multiply Fractions and Whole Numbers

• To add fraction you must have the same denominator.

38

14+ =

3.1 Using Models To Multiply Fractions and Whole Numbers

• We change the second fraction into eighths.

38

28+ =

3.1 Using Models To Multiply Fractions and Whole Numbers

• We add the numerators.

38

28+ = 3+2

8 = 58

3.1 Using Models To Multiply Fractions and Whole Numbers

• Try:

31

31

21

61

41

31

3.1 Using Models To Multiply Fractions and Whole Numbers

Remember• Multiplication is accelerated addition• 2 + 2 + 2 = 6

• This can also be done as three groups of two

23

2 x 3 = 6

3.1 Using Models To Multiply Fractions and Whole Numbers

• Match the addition and multiplication

a. 3 + 3 + 3 + 3b. 5 + 5c. 2 + 2 + 2 + 2 + 2d. 1 + 1 + 1

•4 x 3•1 x 3•2 x 5•5 x 2

• Remember

155

3.1 Using Models To Multiply Fractions and Whole Numbers

• What is 3 times two thirds?

32

32

32

323

3.1 Using Models To Multiply Fractions and Whole Numbers

• What is 3 times two thirds?

236

32

32

32

236

32

13

323

3.1 Using Models To Multiply Fractions and Whole Numbers

• Write out the following multiplications as additions and draw a diagram.

322 c.

433 b.

214 a.

3.1 Using Models To Multiply Fractions and Whole Numbers

• Homework

3.2 USING MODELS TO MULTIPLY FRACTIONS

Unit 3: Operations with Fractions

3.2 Using Models to Multiply Fractions

• Focus: Use models to multiply fractions

3.3 MULTIPLYING FRACTIONSUnit 3: Operations with Fractions

3.3 Multiplying Fractions

• Focus: Develop an algorithm to multiply fractions

3.3 Multiplying Fractions

• We have seen how to multiply fractions by whole numbers.

1865

3.3 Multiplying Fractions

• Multiply the 5 by 18

1865

6185

690

15

• Reduce by dividing the numerator and denominator by 6

Method 1 - reduce at end

3.3 Multiplying Fractions

• Reduce by dividing 18 and 6 by 6. You must do the same to a numerator and denominator.

1865

118

65

15

• Multiply the numerators, then the denominators.

Method 2 - reduce at start

3

1

3.3 Multiplying Fractions

• Example

21

32

3.3 Multiplying Fractions

• Example

21

32

21

32

1

1

31

Split the coloured sections into two parts.

Keep one of the two parts.

210

56

3.3 Multiplying Fractions

• Example

210

56

2

1

Divide 5 out of 10 in the numerator and out of 5 in the denominator.Divide 2 out of 6 (or 2) in the numerator and out of 2 in the denominator.

1

3

6 16 or

3.4 MULTIPLYING MIXED NUMBERSUnit 3: Operations with Fractions

3.4 Multiplying Mixed Numbers

• Focus: Apply knowledge of multiplying fractions to multiply mixed numbers

3.4 Multiplying Mixed Numbers

To multiply mixed fractions, change the fraction into an improper fraction.

45 Improper

fraction411 Mixed

fraction

Remember:

3.4 Multiplying Mixed Numbers

• To change from mixed to improper:

newnumerator

wholenumber

denominator + numerator= x412

49

4142

3.4 Multiplying Mixed Numbers

• Convert to improper fractions:

321

412

543

412

3.4 Multiplying Mixed Numbers

• To convert to mixed fractions subtract the denominator from the numerator.

25

212

231

3.4 Multiplying Mixed Numbers

25

3.4 Multiplying Mixed Numbers

25

231

3.4 Multiplying Mixed Numbers

25

212

3.5 DIVIDING WHOLE NUMBERS AND FRACTIONS

Unit 3: Operations with Fractions

3.5 Dividing Whole Numbers and Fractions

• Focus: Use models to divide proper fractions and whole numbers

Remember:• Division is determining how many groups you

can make out of a tot

How many groups of 3 can you make out of 15?315

Remember:• Division is determining how many groups you

can make out of a total

How many groups of 3 can you make out of 15?315

5 groups

How many halves are in 1?

211

Try the following:

212

213

214

Try the following:

212

213

214

Is there a pattern? What is it?

Try

311

312

313

Is there a pattern? What is it?

Try

322

324

326

Try

322

324

326

• A “flipped” fraction is called an inverse or a reciprocal.

sreciprocal are 23 and

32

34 of inverse theis

43

Reciprocals are two numbers that multiply to produce 1.

sreciprocal are 23 and

32

1

23

32

23

32

1 1

1 1

• To divide fractions, invert the second fraction and multiply.

23

52

32

52

35

21

53

21

TryA. B. C.

51

54

31

65

51

313

3.6 DIVIDING FRACTIONSUnit 3: Operations with Fractions

3.6 Dividing Fractions

• Focus: Develop algorithms to divide fractions.

3.7 DIVIDING MIXED NUMBERSUnit 3: Operations with Fractions

3.7 Dividing Mixed Numbers

• Focus: apply knowledge of dividing fractions to divide mixed numbers

3.8 SOLVING PROBLEMS WITH FRACTIONS

Unit 3: Operations with Fractions

3.8 Solving Problems with Fractions

• Focus: Identify the operation required to solve a problem involving fractions

3.8 Solving Problems with Fractions

• A pancake recipe makes enough pancakes for 12 people. Sally has to make enough pancakes for 7 people. The recipe needs 10 cups of pancake mix. How much pancake mix should Sally use?

3.8 Solving Problems with Fractions

• The student’s spirit group does a survey. 2/3 of the students want a Christmas dance. If 150 students were surveyed, how many do not want a dance?

3.8 Solving Problems with Fractions

• A sports store examines its receipts. 3/5 of the clothing sold was men’s wear. If the store has 9 000 ft2 of clothing area, how much area should be men’s wear?

3.8 Solving Problems with Fractions

• For a wedding reception 3/5 of the meals are beef, 1/3 of the meals are chicken, while the rest were vegetarian.

a. What fraction of the meals are vegetarian?b. If there are 90 guests, how many people

ordered beef?

3.9 ORDER OF OPERATIONS WITH FRACTIONS

Unit 3: Operations with Fractions

3.9 Order of Operations with Fractions

• Focus: Use the order of operations to evaluate expressions

3.9 Order of Operations with Fractions

Remember• Order of operations:BracketsExponentsDivisionMultiplicationAdditionSubtraction

}} Done at the same time from

left to right

Done at the same time from left to right

3.9 Order of Operations with Fractions

• Try

51

54

12

51

58

59

Multiply first

Add second

3.9 Order of Operations with Fractions

• Try

56

13

34

Divide first, invert second fraction and multiply

65

13

34

Simplify and multiply1

2

25

34

615

68

623

Add

Reduce

3.9 Order of Operations with Fractions• Try

61

31

21 There is a hidden multiply between

the ½ and bracket. Add first by finding a common denominator.

61

62

21

Add in the brackets

21

21

63

21

41

Reduce 3/6 and get rid of brackets

Multiply fractions