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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.
a/b * c/d = a * c/b * d
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.
10/12 * 2/3 Now compare each fraction with 12. These fractions do not have the same denominator. The denominators of the fractions are 12, 3, and 2. The least common denominator of these numbers is 12. Each fraction can be rewritten as an equivalent fraction with the common denominator to make it easier to compare them.
Rewrite | Compare with 12, or 612 | |
---|---|---|
10/12 | 10/12 | 10/12 > 6/12 |
2/3 | 2*4/3*4 = 8/12 | 8/12 > 6/12 |
Both fractions are greater than 12. An estimate for the product can then be 1 because the fractions can be rounded to 1. rccc Product: & 10/12 & * & 2/3 [0.5em] & ↓ & & ↓ Estimate: & 1 & * & 1 Since 1 is the identity element of multiplication, this product is also equal to 1. However, that does not say much about the original product, and 1 is not one of the possible answer options. Now think of rounding only one of the fractions to 1 one at a time.
Estimate for 1012* 23 | |
---|---|
Round 1012 to 1 | 1 * 2/3 = 2/3 |
Round 23 to 1 | 10/12 * 1 = 10/12 |
The fraction 1012 can be simplified to 56. The answer is either 23 or 56. The fractions in the options are in simplest form and 23 is among the options. Therefore, the answer is 23.
In addition to a number line, the product can also be represented by an area model. In this model, the denominators are used to divide a rectangular diagram into smaller parts. Consider modeling the following product. 10/12 * 2/3 The denominators are 12 and 3. The rectangle is then divided into 12 columns and 3 rows.
The numerators determine which parts will be shaded. Since 10 is the numerator of 1012, 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 2036= 59.
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 9/1
a/b=.a /9./.b /9.
Simplify quotient
Rewrite 19 as 15+4
Write as a sum of fractions
Calculate quotient
Rewrite 1+4/15 as 1 415
Paulina's PE class lasts for 1 512 hours. The table shows what fraction of the class time is allocated for various activities.
Activity | Fraction |
---|---|
Warm-up | 1/5 |
Instruction | 1/2 |
Playing a game | 1/10 |
Cool-down | 1/5 |
Part of 1 512-hour Class | ||
---|---|---|
Activity | Fraction | |
Warm-up | 1/5 | |
Instruction | 1/2 | |
Playing game | 1/10 | |
Cool-down | 1/5 |
a bc=a* c+b/c
a * 1=a
Add terms
Rewrite 60 as 60/1
Multiply fractions
Multiply
Calculate quotient
Part of 1 512-hour Class | ||
---|---|---|
Activity | Fraction | |
Warm-up | 1/5 | |
Instruction | 1/2 | |
Playing game | 1/10 | |
Cool-down | 1/5 |
a*b/c= a* b/c
Multiply
Calculate quotient
Paulina loves a photo of her playing volleyball and prints it. The diagram shows the dimensions of the photograph.
The length of the photo is 8 34 inches and the width of the photo is 6 23 inches. Area = 8 34 * 6 23 Since the fractional parts of the mixed numbers are greater than 12, the mixed numbers can be rounded up. Area = & 8 34 * 6 23 & ↓ ↓ & 9 * 7 Therefore, the area of the photo is about 9* 7, or 63 square inches.
a bc=a* c+b/c
Multiply
Add terms
Multiply fractions
Multiply
a/b=.a /4./.b /4.
Calculate quotient
Rewrite 175 as 174+1
Write as a sum of fractions
Calculate quotient
Rewrite 58+1/3 as 58 13
a bc=a* c+b/c
Multiply
Add terms
a/b=a * 5/b * 5
a/b=a * 4/b * 4
a/b=a * 5/b * 5
a/b=a * 3/b * 3
Multiply
Add fractions
Add terms
Multiply fractions
Multiply
a/b=.a /6./.b /6.
Calculate quotient
Rewrite 3233 as 3200+33
Write as a sum of fractions
Calculate quotient
Rewrite 64+33/50 as 64 3350
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.
The important point in multiplying fractions is to ensure that the fractions are either proper fractions or improper fractions. 1 26 * 2 = 8/6 * 2/1 The final step usually involves simplifying the resulting fraction. However, to make calculations easier, first check if the two fractions are already in their lowest forms. If not, the fractions can be simplified first before multiplying them. 4 & 8/6 &* 2/1 = 4/3 * 2/1 = 8/3 3 & Consider the challenge presented at the beginning of the lesson. Paulina devotes two-sixths of her free time to exercise.
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+2/3 as 6 23
The diagram shows how Zain multiplies two mixed numbers.
Let's examine the steps Zain followed when multiplying 3 14 and 6 25. We want to figure out if their work is correct.
As we can see, Zain directly multiplies the integer and fraction parts of both mixed numbers with each other. They then add these two products. However, this is an incorrect procedure. We calculate the product of two mixed numbers by rewriting the mixed numbers so that they are improper 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 product!
Finally, let's rewrite the answer as a mixed number and see if our friend was correct.
The result of the product is different from what Zain found. Therefore, Zain is not correct. The answer is C.
We can also calculate the product by using the Distributive Property. Remember, a mixed number a bc is the same as a + bc. Let's rewrite the given expression as follows. 3 14 * 6 25 = (3+1/4) * (6 + 2/5 ) If we apply the Distributive Property here, we get a different expression than the one that Zain found.
There are two missing terms in the Zain's work, 65 and 32. This means that Zain is not correct. Let's continue and find the correct answer.
As we can see, we ended with the same result as before. Therefore, Zain is not correct.
Ali spends 312 of the day at an amusement park. He spends 49 of that time driving a go-kart.
How many hours does he spend driving the go-kart? Write the answer as a mixed number.We know that Ali spent 312 of his day at an amusement park and 49 of that time driving a go-kart. The product of these two fractions will give us what fraction of the day Ali spent driving the go-kart. Let's find it! 3/12 * 4/9 Notice that we can simplify the first fraction to 14 because 12 is 4 times 3. 3/3* 4 * 4/9 = 1/4* 4/9 We can also simplify across the two fractions. There is a common factor between the numerator of 49 and the denominator of 14. 1/4* 4/9 = 1/1 * 1/9 ⇒ 1/9 Since 1 is the identity element of multiplication, the product is 19. This means that Ali spent 19 of his day driving the go-kart. We can calculate how many hours this ride took by multiplying 19 by the 24 hours in a day.
Ali spent 2 23 hours driving the go-kart.
We want to find the total cost of the indoor paint in storage. There are 42 gallons of paint in storage. We know that 59 of that paint is for outdoor use and the rest is for indoor use. Since the total amount of paint is 1 when represented as a fraction, we can determine the fraction of the indoor paint by subtracting 59 from 1.
We found that four-ninths of 42 gallons of paint is for indoor use. 42Liters of Paint ↙ 1.5cm ↘ Outdoor Use Indoor Use 5/9 2cm 4/9 Now we can find the number of gallons of paint for indoor use. To do so, we need to multiply 42 by 49. The result will represent 49 of the 42 gallons of paint in storage. 42 * 4/9 To calculate this product, we will move the whole number to the numerator of the fraction as a factor. Let's do it!
We found that there are 563 gallons of paint for indoor use in storage. Finally, we can find the total cost of the indoor paint in storage. There are 563 gallons paint and each one costs $18. To find the total cost, we need to multiply the fraction and the whole number like we did before.
The total cost of the indoor paint in storage is $336.
We want to compare the quantities on each side of the box and complete the statement. 10/12 * 32/20 10/12 Notice that the expression on the left-hand side is 1012 multiplied by a fraction. When we multiply a number by a fraction, the product can be greater than, less than, or equal to the multiplicand, or the original number. If the multiplier fraction is less than 1, the result will be less than the multiplicand. Let's see an example! 1/2 * 1/2 = 1/4 ⇒ 1/4 < 1/2 If the fraction is equal to 1, the result will be equal to the multiplicand. 1/2 * 2/2 = 1/2 ⇒ 1/2 = 1/2 Finally, if the fraction is greater than 1, the result will be greater than the multiplicand. 1/2 * 4/2 = 2 ⇒ 2 > 1/2 Let's focus on the given statement. The fraction 1012 is multiplied by 3220. Since the numerator 32 is greater than the denominator 20, the fraction 3220 is greater than 1. Therefore, the product of 1012 and 3220 is greater than 1012. With this in mind, we can complete the statement. 10/12 * 32/20 > 10/12