1 lesson 2.3.3 multiplying fractions. 2 lesson 2.3.3 multiplying fractions california standard:...
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1
Lesson 2.3.3Lesson 2.3.3
Multiplying FractionsMultiplying Fractions
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Lesson
2.3.3Multiplying FractionsMultiplying Fractions
California Standard:Number Sense 1.2Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers.
What it means for you:You’ll practice multiplying fractions, and you’ll extend this to multiplying fractions by integers and mixed numbers.
Key words:• area model• mixed numbers
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Multiplying FractionsMultiplying FractionsLesson
2.3.3
You’ve multiplied fractions in other grades — but it’s still not an easy topic.
In this Lesson you’ll get plenty more practice at it.
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Multiplying FractionsMultiplying Fractions
Area Models Show Fraction Multiplication
Lesson
2.3.3
Multiplying a fraction by another fraction means working out parts of a part.
You can show this graphically using an area model.
1
5
2
3For example, × means “one-fifth of two-thirds.”
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Multiplying FractionsMultiplying FractionsLesson
2.3.3
The part showing × is the part that represents one-fifth
of two-thirds — this is the part that’s shaded in both colors.
1
5
2
3
To use an area model, you start by drawing a rectangle.
Shade in of the rectangle in one direction:1
5
Then shade in of it in the other direction,
using a different color:
2
3
2
3
1
5
There are 2 squares shaded out of a total of 15, so × = .2
15
1
5
2
3
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Multiplying FractionsMultiplying Fractions
Example 1
Solution follows…
Lesson
2.3.3
Solution
Calculate × using the area model method.3
4
1
3
You need to work out three-quarters of one-third — so shade in three-quarters of the rectangle in one direction, and one-third in the other direction: 1
3
3
4
There are 3 out of 12 squares shaded
in both colors, so × = .3
4
1
3
3
12
7
Calculate these fraction multiplications by drawing area models:
1. ×
2. ×
3. ×
1
3
2
5
3
4
1
5
3
4
1
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Multiplying FractionsMultiplying Fractions
Guided Practice
Solution follows…
Lesson
2.3.3
2
15
3
20
3
24
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Multiplying FractionsMultiplying Fractions
You Can Multiply Fractions Without Drawing Diagrams
Lesson
2.3.3
When you draw an area model, the total number of squares is always the same as the product of the denominators of the fractions you’re multiplying.
You’ve already seen the area model for × :1
5
2
3
Multiply the denominators:The total number of squares is 5 × 3 = 15.
2
3
1
5
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Multiplying FractionsMultiplying FractionsLesson
2.3.3
That means you can work the product out without drawing the area model.
Also, the number of squares shaded in both colors is always the same as the product of the numerators.
Multiply the numerators:The total number of squares shaded in both colors is 1 × 2 = 2.
2
3
1
5
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Multiplying FractionsMultiplying Fractions
Example 2
Solution follows…
Lesson
2.3.3
Solution
Calculate × without drawing a diagram.3
4
1
3
Multiply the numerators:
So × = .3
4
1
3
3
12
Multiply the denominators:
3 × 1 = 3
4 × 3 = 12
Now write this as a fraction: 3
12
numerator
denominator
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Multiplying FractionsMultiplying FractionsLesson
2.3.3
The solution to Example 2 could be simplified a bit more.
That means you could write × = .3
4
1
3
1
4
3
12
1
4
÷3
÷3The numerator and denominator can both be divided by 3…
…so represents
the same amount, but simplified
1
4
Simplifying just means writing the solution using smaller numbers, but so that the fraction still means the same thing.
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Multiplying FractionsMultiplying Fractions
Guided Practice
Solution follows…
Lesson
2.3.3
Calculate these fraction multiplications without drawing area models. Simplify your answer where possible.
4. × 5. × 6. ×
7. × 8. × 9. ×
5
6
1
2
2
3
1
3
1
3
6
7
3
8
1
4
5
7
3
5
11
12
6
7
5
12
2
9
3
7
2
7
3
32
11
14
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Multiplying FractionsMultiplying Fractions
First Convert Whole or Mixed Numbers to Fractions
Lesson
2.3.3
To multiply fractions by mixed numbers, you can just write out the mixed numbers as a single fraction and carry on multiplying as normal.
The same is true if you need to multiply a fraction by an integer — you can write the integer as a fraction and use the multiplication method from before.
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Multiplying FractionsMultiplying Fractions
Example 3
Solution follows…
Lesson
2.3.3
Calculate: (i) 3 × , (ii) × 8
Solution
1
2
1
4
4
5
Then just multiply out the fractions as normal:
(i) Convert 3 to a fraction:1
23 = =
1
2
7
2
(3 × 2) + 1
2
3 × = × =1
2
1
4
7
2
1
4
7
8
Solution continues…
15
× 8 = × =4
54
5
8
1
32
5
(ii) The integer 8 can be written as .8
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Multiplying FractionsMultiplying Fractions
Example 3
Lesson
2.3.3
Calculate: (i) 3 × , (ii) × 8
Solution (continued)
1
2
1
4
4
5
So you can multiply as normal:
16
Calculate the following.Simplify your solutions where possible.
10. 1 × 11. × 2
12. 1 × 13. 3 ×
14. 1 × 15. 1 × 1
1
3
2
3
1
5
1
3
1
3
2
3
2
7
1
4
1
5
5
7
1
2
Multiplying FractionsMultiplying Fractions
Guided Practice
Solution follows…
Lesson
2.3.3
4
15
7
9
6
7
1
4
10
9
1
9or 1
18
7
4
7or 2
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Multiplying FractionsMultiplying Fractions
Independent Practice
Solution follows…
Lesson
2.3.3
Find the product and simplify each calculation in Exercises 1–3.
1. ×
2. – ×
3. – × 2
2
3
7
10
4
9
1
3
7
15
3
15
5
12
4
45–
35
36–
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Multiplying FractionsMultiplying Fractions
Independent Practice
Solution follows…
Lesson
2.3.3
4. A positive whole number is multiplied by a positive fraction smaller than one. Explain how the size of the product compares to the original whole number.
5. A rectangular patio measures 8 feet wide and 12 feet long. What is the area of the patio?
6. A recipe for 12 muffins calls for 3 cups of flour. How many cups of flour are needed to make 42 muffins?
The product will always be smaller than the original whole number.
103 square feet1
8
11 cups3
8
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Multiplying FractionsMultiplying Fractions
Round UpRound Up
Lesson
2.3.3
Multiplying fractions is OK because you don’t need to put each fraction over the same denominator.
If you need to multiply by integers or mixed numbers, just turn them into fractions too.