Dividing Fractions.notebook
Dividing Fractions.notebook
February 25, 2016
Dividing Fractions Learning Goals We will: solve problems involving division of simple fractions represent the division of fractions using a variety of tools and strategies Success Criteria I can: divide a whole number by a fraction divide a fraction by a fraction divide a fraction by a whole number use a model to represent division of fractions explain why our strategies for dividing fractions work
Dividing Fractions.notebook
February 25, 2016
Getting Started a) How many times will 1/2 (the fraction represented by the yellow rods) go into whatever whole number is represented by the amount of orange rods you have at your table? Example: if you have 3 orange rods, your question is how many times will 1/2 go into 3 7 orange rods: how many times will 1/2 go into 7? b) How many times will 1/5 go into your whole number?
Dividing Fractions.notebook
February 25, 2016
Let`s review what we already know about division: *The first number in a division question is called the dividend this is the number that gets divided up *The second number in a division question is called the divisor this is the number that determines how many groups the first number gets split into 6 ÷ 2 What does this mean? What are we actually doing here? *We are seeing how many times 2 will go into 6 6 ÷ 2 = 3 *The quotient is the answer to a division question. It tells us how many times the divisor will go into the dividend
Dividing Fractions.notebook
February 25, 2016
More Facts About Division *In when dividing ORDER MATTERS. We cannot flip the dividend and the divisor and expect to get the same answer (like when we are multiplying and adding) 10 ÷ 5 = 2 BUT 5 ÷ 10 = 0.5 *Division is the inverse operation (opposite) of multiplication. We can work backwards and use multiplication to help us solve division problems. if 10 ÷ 5 = 2, then 2 x 5 = 10 So if you were given a more challenging question (example 56 ÷ 7, you could ask yourself "7 times what number gives me 56?) *When the divisor does not "fit perfectly" into the dividend, we will have a remainder 7 ÷ 2 = 3 R1 (2 will go into 7 3 times with one left over) *as intermediate students, we would express our remainders as decimals (or as fractions) *What would our remainder look like as a decimal? *As a fraction? Why?
Dividing Fractions.notebook
February 25, 2016
7 ÷ 2 =
Arrange the stars into two equal groups *How many stars are left over when you can no longer make the groups equal? *How many stars are necessary to make one equal group of stars? (a star in each circle) *So what fraction represents your remainder?
Dividing Fractions.notebook
February 25, 2016
6 ÷ 2
What we are asking is: 'How many groups of two can I get from 6?' **What could you use (other than division) to help you find out the answer to the question above?
Dividing Fractions.notebook
6 ÷ 2 = 3 3 x 2 = 6 6 x 0.5 = 3
What fraction is equivalent to 0.5?
February 25, 2016
Dividing Fractions.notebook
6 ÷ 2 = 3 6 x 0.5 = 3 6 x 1/2 = 3
So, dividing by 2/1 is the same as multiplying by 1/2
How many of you understand why this is true at this point? :)
February 25, 2016
Dividing Fractions.notebook
February 25, 2016
you see 6 x 2/3. It means 6 Recall how we multiply. Getting When Started groups of 2/3. 6 x 2/3 = 4
Recall, the rule for multiplying fractions. 6 x 2/3 is 6 groups of 2/3
6 x 2/3 = 12/3 = 4
Dividing Fractions.notebook
February 25, 2016
With the area model we see that we are multiplying the numerators and multiplying the denominators.
x
1/5
2/3
We are finding the area of the coloured portion with is 1 x 2 in comparison to the area of the whole 5 x 3
Dividing Fractions.notebook
February 25, 2016
Let's start looking at dividing whole numbers by fractions:
a) 4 ÷ 1/2 (This means how many groups of 1/2 can we get out of 4) *I know that I can get 2 groups of 1/2 out of 1 whole, so I can get 8 groups of 1/2 out of 4 wholes (2 x 4) So..... 4 ÷ 1/2 = 4 x 2/1 = 8
Dividing Fractions.notebook
How many groups of 1/2 will fit into 4?
February 25, 2016
Dividing Fractions.notebook
b) 5 ÷ 1/4 (This means how many groups of 1/4 can we get out of 5) **I know:
February 25, 2016
Dividing Fractions.notebook
c) 5 ÷ 1/10 (This means: * I know:
February 25, 2016
Dividing Fractions.notebook
February 25, 2016
What happens when the fractions get more difficult/don't work as well together?
a) How many groups of 5/6 will go into two wholes? b) How many sixths make up those full groups? c) How many sixths are left over? **d) What fraction does that represent?
Dividing Fractions.notebook
February 25, 2016
What happens when the fractions get more difficult/don't work as well together?
a) How many groups of 5/6 will go into two wholes? 2 b) How many sixths make up those full groups? 5/6 + 5/6 = 10/6 c) How many sixths are left over? 12/6 10/6 = 2/6 **d) What fraction does that represent? How many think it is 2/6? 2/5? Why? The correct answer is that 2 ÷ 5/6 = 2 2/5. *This is true because you have two "sticks" left over. You need 5 "sticks" to make a full group of 5/6, so you have 2 of the necessary 5 "sticks" to make another full group
Dividing Fractions.notebook
Fraction ÷ Fraction
1/2 ÷ 1/4 (How many groups of 1/4 will fit into 1/2?)
February 25, 2016
Dividing Fractions.notebook
February 25, 2016
Working On It
Dividing Fractions.notebook
February 25, 2016
The "Easy" Way
When we are dividing with fractions, the rule is you keep the first fraction the same, and you invert (or flip) the second fraction, then multiply
Example: 1/2 ÷ 1/4 = 1/2 x 4/1 = 4/2 = 2 *Our goal is to UNDERSTAND WHY this works. Our models help us show this. There are other ways that might make a bit more sense to some (stay tuned.....)
Dividing Fractions.notebook
February 25, 2016
Working On It
Thinking about multiplying, what would make sense that we do?
Dividing Fractions.notebook
February 25, 2016
3/4 ÷ 5/8
The "other way" to prove this works
= 3/4 (x 4) ÷ 5/8 (x 4)
multiply both denominators by 4 to cancel out the ÷4 in the dividend (first number in a division problem)
= 3 ÷ 40/8 = 3 (x 8) ÷ 40/8 (x 8) =24/40 = 3/5 (simplified)
multiply the dividend and divisor by 8 to cancel out the ÷8 in the divisor (second number in the division problem)
Dividing Fractions.notebook
February 25, 2016
Homework Questions *Please remember dividing fractions is only a Grade 8 expectation. Any Grade 7s who want to try this should ensure that they have mastered multiplying whole numbers by fractions and that they have no test corrections to do
1. Use create a model to show to answer to 6 ÷ 1/3
2. Find the quotient. **Level 4 opportunity: prove why your answer is correct a) 5 ÷ 2/3
b) 4/7 ÷ 5
c) 7/9 ÷ 3/8
d) 3/8 ÷ 7/9
3. Place the digits 2, 3, 5, and 7 in the boxes to produce the greatest quotient. / ÷ / =
4. Every day, the Randolph Pet Store uses 2 bags of dog food to feed the dogs. For how many days will 1/2 of a bag of dog food last? 5. The elephants at the zoo eat 2 4/5 buckets of bananas each day. The zookeeper bought 5 3/5 buckets of bananas. For how many days will the bananas last? Simplify your answer and write it as a proper fraction or as a whole or mixed number.
Dividing Fractions.notebook
February 25, 2016
February 25, 2016
Dividing Fractions Learning Goals We will: solve problems involving division of simple fractions represent the division of fractions using a variety of tools and strategies Success Criteria I can: divide a whole number by a fraction divide a fraction by a fraction divide a fraction by a whole number use a model to represent division of fractions explain why our strategies for dividing fractions work
Dividing Fractions.notebook
February 25, 2016
Getting Started a) How many times will 1/2 (the fraction represented by the yellow rods) go into whatever whole number is represented by the amount of orange rods you have at your table? Example: if you have 3 orange rods, your question is how many times will 1/2 go into 3 7 orange rods: how many times will 1/2 go into 7? b) How many times will 1/5 go into your whole number?
Dividing Fractions.notebook
February 25, 2016
Let`s review what we already know about division: *The first number in a division question is called the dividend this is the number that gets divided up *The second number in a division question is called the divisor this is the number that determines how many groups the first number gets split into 6 ÷ 2 What does this mean? What are we actually doing here? *We are seeing how many times 2 will go into 6 6 ÷ 2 = 3 *The quotient is the answer to a division question. It tells us how many times the divisor will go into the dividend
Dividing Fractions.notebook
February 25, 2016
More Facts About Division *In when dividing ORDER MATTERS. We cannot flip the dividend and the divisor and expect to get the same answer (like when we are multiplying and adding) 10 ÷ 5 = 2 BUT 5 ÷ 10 = 0.5 *Division is the inverse operation (opposite) of multiplication. We can work backwards and use multiplication to help us solve division problems. if 10 ÷ 5 = 2, then 2 x 5 = 10 So if you were given a more challenging question (example 56 ÷ 7, you could ask yourself "7 times what number gives me 56?) *When the divisor does not "fit perfectly" into the dividend, we will have a remainder 7 ÷ 2 = 3 R1 (2 will go into 7 3 times with one left over) *as intermediate students, we would express our remainders as decimals (or as fractions) *What would our remainder look like as a decimal? *As a fraction? Why?
Dividing Fractions.notebook
February 25, 2016
7 ÷ 2 =
Arrange the stars into two equal groups *How many stars are left over when you can no longer make the groups equal? *How many stars are necessary to make one equal group of stars? (a star in each circle) *So what fraction represents your remainder?
Dividing Fractions.notebook
February 25, 2016
6 ÷ 2
What we are asking is: 'How many groups of two can I get from 6?' **What could you use (other than division) to help you find out the answer to the question above?
Dividing Fractions.notebook
6 ÷ 2 = 3 3 x 2 = 6 6 x 0.5 = 3
What fraction is equivalent to 0.5?
February 25, 2016
Dividing Fractions.notebook
6 ÷ 2 = 3 6 x 0.5 = 3 6 x 1/2 = 3
So, dividing by 2/1 is the same as multiplying by 1/2
How many of you understand why this is true at this point? :)
February 25, 2016
Dividing Fractions.notebook
February 25, 2016
you see 6 x 2/3. It means 6 Recall how we multiply. Getting When Started groups of 2/3. 6 x 2/3 = 4
Recall, the rule for multiplying fractions. 6 x 2/3 is 6 groups of 2/3
6 x 2/3 = 12/3 = 4
Dividing Fractions.notebook
February 25, 2016
With the area model we see that we are multiplying the numerators and multiplying the denominators.
x
1/5
2/3
We are finding the area of the coloured portion with is 1 x 2 in comparison to the area of the whole 5 x 3
Dividing Fractions.notebook
February 25, 2016
Let's start looking at dividing whole numbers by fractions:
a) 4 ÷ 1/2 (This means how many groups of 1/2 can we get out of 4) *I know that I can get 2 groups of 1/2 out of 1 whole, so I can get 8 groups of 1/2 out of 4 wholes (2 x 4) So..... 4 ÷ 1/2 = 4 x 2/1 = 8
Dividing Fractions.notebook
How many groups of 1/2 will fit into 4?
February 25, 2016
Dividing Fractions.notebook
b) 5 ÷ 1/4 (This means how many groups of 1/4 can we get out of 5) **I know:
February 25, 2016
Dividing Fractions.notebook
c) 5 ÷ 1/10 (This means: * I know:
February 25, 2016
Dividing Fractions.notebook
February 25, 2016
What happens when the fractions get more difficult/don't work as well together?
a) How many groups of 5/6 will go into two wholes? b) How many sixths make up those full groups? c) How many sixths are left over? **d) What fraction does that represent?
Dividing Fractions.notebook
February 25, 2016
What happens when the fractions get more difficult/don't work as well together?
a) How many groups of 5/6 will go into two wholes? 2 b) How many sixths make up those full groups? 5/6 + 5/6 = 10/6 c) How many sixths are left over? 12/6 10/6 = 2/6 **d) What fraction does that represent? How many think it is 2/6? 2/5? Why? The correct answer is that 2 ÷ 5/6 = 2 2/5. *This is true because you have two "sticks" left over. You need 5 "sticks" to make a full group of 5/6, so you have 2 of the necessary 5 "sticks" to make another full group
Dividing Fractions.notebook
Fraction ÷ Fraction
1/2 ÷ 1/4 (How many groups of 1/4 will fit into 1/2?)
February 25, 2016
Dividing Fractions.notebook
February 25, 2016
Working On It
Dividing Fractions.notebook
February 25, 2016
The "Easy" Way
When we are dividing with fractions, the rule is you keep the first fraction the same, and you invert (or flip) the second fraction, then multiply
Example: 1/2 ÷ 1/4 = 1/2 x 4/1 = 4/2 = 2 *Our goal is to UNDERSTAND WHY this works. Our models help us show this. There are other ways that might make a bit more sense to some (stay tuned.....)
Dividing Fractions.notebook
February 25, 2016
Working On It
Thinking about multiplying, what would make sense that we do?
Dividing Fractions.notebook
February 25, 2016
3/4 ÷ 5/8
The "other way" to prove this works
= 3/4 (x 4) ÷ 5/8 (x 4)
multiply both denominators by 4 to cancel out the ÷4 in the dividend (first number in a division problem)
= 3 ÷ 40/8 = 3 (x 8) ÷ 40/8 (x 8) =24/40 = 3/5 (simplified)
multiply the dividend and divisor by 8 to cancel out the ÷8 in the divisor (second number in the division problem)
Dividing Fractions.notebook
February 25, 2016
Homework Questions *Please remember dividing fractions is only a Grade 8 expectation. Any Grade 7s who want to try this should ensure that they have mastered multiplying whole numbers by fractions and that they have no test corrections to do
1. Use create a model to show to answer to 6 ÷ 1/3
2. Find the quotient. **Level 4 opportunity: prove why your answer is correct a) 5 ÷ 2/3
b) 4/7 ÷ 5
c) 7/9 ÷ 3/8
d) 3/8 ÷ 7/9
3. Place the digits 2, 3, 5, and 7 in the boxes to produce the greatest quotient. / ÷ / =
4. Every day, the Randolph Pet Store uses 2 bags of dog food to feed the dogs. For how many days will 1/2 of a bag of dog food last? 5. The elephants at the zoo eat 2 4/5 buckets of bananas each day. The zookeeper bought 5 3/5 buckets of bananas. For how many days will the bananas last? Simplify your answer and write it as a proper fraction or as a whole or mixed number.
Dividing Fractions.notebook
February 25, 2016
