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| 10 Theory slides |
| 10 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
Paulina runs for two-sixths of her free time each day.
If Paulina has 5 hours of free time per day, how many hours does she run in 4 days? Give the exact answer.
The product of two fractions is equal to the product of the numerators divided by the product of the denominators. The resulting fraction is then simplified to its lowest terms, if possible.
ba⋅dc=b⋅da⋅c
Paulina drinks one-third of the water in her bottle before PE class. During class, she drinks ten-twelfths of the remaining water.
Notice that each third is divided into six smaller equal parts. The other third can also be divided into six parts. This will make it easier to find what fraction of the whole the red part represents.
The red part shows what fraction of the water bottle Paulina drinks during class.
Rewrite | Compare with 21, or 126 | |
---|---|---|
1210 | 1210 | 1210>126 |
32 | 3⋅42⋅4=128 | 128>126 |
Estimate for 1210×32 | |
---|---|
Round 1210 to 1 | 1×32=32 |
Round 32 to 1 | 1210×1=1210 |
The fraction 1210 can be simplified to 65. The answer is either 32 or 65. The fractions in the options are in simplest form and 32 is among the options. Therefore, the answer is 32.
The numerators determine which parts will be shaded. Since 10 is the numerator of 1210, 10 of the 12 columns will be shaded. Similarly, 2 of the 3 rows will be shaded.
In this model, the overlapping region represents the product. For this example, the product is 3620=95.
The product of two fractions is equal to the product of the numerators divided by the product of the denominators. Practice finding the product of fractions. Simplify the answer to its lowest terms.
When multiplying fractions by whole numbers or mixed numbers, both factors should be in the form of a proper fraction or an improper fraction.
Rewrite 9 as 19
ba=b/9a/9
Simplify quotient
Rewrite 19 as 15+4
Write as a sum of fractions
Calculate quotient
Rewrite 1+154 as 1154
Paulina's PE class lasts for 1125 hours. The table shows what fraction of the class time is allocated for various activities.
Activity | Fraction |
---|---|
Warm-up | 51 |
Instruction | 21 |
Playing a game | 101 |
Cool-down | 51 |
Part of 1125-hour Class | ||
---|---|---|
Activity | Fraction | |
Warm-up | 51 | |
Instruction | 21 | |
Playing game | 101 | |
Cool-down | 51 |
Rewrite 60 as 160
Multiply fractions
Multiply
Calculate quotient
Part of 1125-hour Class | ||
---|---|---|
Activity | Fraction | |
Warm-up | 51 | |
Instruction | 21 | |
Playing game | 101 | |
Cool-down | 51 |
Paulina loves a photo of her playing volleyball and prints it. The diagram shows the dimensions of the photograph.
acb=ca⋅c+b
Multiply
Add terms
Multiply fractions
Multiply
ba=b/4a/4
Calculate quotient
Rewrite 175 as 174+1
Write as a sum of fractions
Calculate quotient
Rewrite 58+31 as 5831
acb=ca⋅c+b
Multiply
Add terms
ba=b⋅5a⋅5
ba=b⋅4a⋅4
ba=b⋅5a⋅5
ba=b⋅3a⋅3
Multiply
Add fractions
Add terms
Multiply fractions
Multiply
ba=b/6a/6
Calculate quotient
Rewrite 3233 as 3200+33
Write as a sum of fractions
Calculate quotient
Rewrite 64+5033 as 645033
To multiply a fraction by a whole number, the whole number is multiplied by the numerator of the fraction. To multiply mixed numbers, the mixed numbers can be converted into improper fractions before multiplying. Practice finding the product of fractions. Simplify the answer to its lowest terms.
Start by finding the number of hours Paulina spends running in a day.
Rewrite 20 as 18+2
Write as a sum of fractions
Calculate quotient
Rewrite 6+32 as 632
Find each product. Write the result in simplest form.
We want to find the product of 36 and 811. 3/6 * 8/11 When we multiply fractions, we need to remember that the product of two fractions is equal to the product of the numerators divided by the product of the denominators. Let's find the given product!
Now we will split the numerator and denominator into prime factors. This will enable us to find the greatest common factor (GCF) of the numbers.
Number | Prime Factorization | GCF |
---|---|---|
24 | 2^3 * 3 | 2* 3=6 |
66 | 2* 3 * 11 |
We can simplify the fraction by dividing the numerator and denominator by their GCF.
We will start again by multiplying the numerators, followed by multiplying the denominators. Then we will reduce the fraction if necessary. Let's do it!
Find each product. Express the result in the simplest form.
We want to multiply a fraction and a whole number. To do so, we will first rewrite the whole number as a fraction with a denominator of 1. Then, we will multiply the numerators and denominators with each other. If necessary, we will also simplify the obtained product. Let's do it!
To find the given product, we will rewrite the whole number as a fraction. Then, we will multiply the numerators and denominators with each other like we did in Part A.
We can simplify this fraction because they have a common factor. 14 = 2 * 7 21 = 3 * 7 The greatest common factor (GCF) of the numbers is 7. Let's dividing the numerator and denominator by the GCF.
Evaluate each product. Write the answer as a mixed number.
We want to find the product of a mixed number and a whole number. Let's start by rewriting them as improper fractions. Then, their product will be equal to the product of the numerators divided by the product of the denominators.
Now, we need to write it as a mixed number.
This time we are asked to find the product of two mixed numbers. To do so, we will first rewrite them as improper fractions. Let's do it!
Notice that the numerator of 2110 and the denominator of 1514 have a common factor of 7. Additionally, the numerator of 1514 and the denominator of 2110 have a common factor of 5. We can use this information to simplify the fractions before we multiply them. Let's do it by factoring the numbers! 21/10 * 15/14 = 3 * 7/2 * 5 * 3* 5/2 * 7 The fractions in the product can be simplified to 32. Let's find this product!
We are asked to write the answer as a mixed number. Let's convert it!
Estimate each product.
We want to estimate the product of a mixed number and a whole number. 5 45 * 4 To do so, we will round the mixed number. Recall that if the fractional part of a mixed number is greater than or equal to 12, we round the mixed number up. If not, we round it down. Now let's check the fractional part of our mixed number! 4/5 >1/2 because 8/10 > 5/10 Since the fractional part 45 is greater than 12, we can round the mixed number up. This means that 5 45 is about 6. 5 45 * 4 ↓ 6 * 4 Since 6*4 is equal to 24, the given product is about 24.
We want to estimate the product of two mixed numbers.
1 67 * 6 16
We can round the first mixed number up because 67 is greater than 12. We can round the other mixed number down because its fractional part is less than 12.
6/7 >1/2 & because & 12/14 > 7/14 [0.8em]
1/6 <1/2 &because & 1/6 < 3/6
Let's now multiply the rounded numbers.
1 67 * 6 16
↓ ↓
2 * 6
The product of the mixed numbers is about the product of 2 and 6, or 12.
1 67 * 6 16 ≈ 12
The diagram models the product of two fractions.
In area model, the shaded columns represents one fraction and the shaded rows represents another. If we break down the model into two parts, we can understand what this means. Let's do it!
The figure on the left is divided into 5 parts and 3 of them are shaded. This means that the shaded columns represent 35. In the same way, the figure on the right is divided into 4 parts and 1 of them is shaded, so the shaded rows represent 14. The diagram models the multiplication of these fractions. 3/5 * 1/4
Let's multiply the fractions. Remember that we multiply the numerators with each other and the denominators with each other.
The product of the fractions is 320, which cannot be simplified further.
We can verify that the product is 320 by looking at the diagram. Let's focus on the region where the blue and yellow regions overlap.
This region consists of 3 of the 20 small squares. Notice also that 3 and 20 are the product of the numerators and denominators, respectively. 3/20 The green region represents the product of the multiplication written in Part A and matches the product of the two fractions. 3/5* 1/4 = 3/20 ✓