multiplication and division of fractions. in reality, no one can teach mathematics. effective...
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In reality, no one can teach mathematics. Effective teachers are those who can
stimulate students to learn mathematics. Educational research offers compelling
evidence that students learn mathematics well only when they construct their own
mathematical understanding
Everybody Counts National Research Council, 1989
Operating With Fractions
Meaning of the denominator (number of equal-sized pieces into which the whole has been cut);
Meaning of the numerator (how many pieces are being considered);
The more pieces a whole is divided into, the smaller the size of the pieces;
Fractions aren’t just between zero and one, they live between all the numbers on the number line;
Understand the meanings for operations for whole numbers.
A Context for Fraction Multiplication
Nadine is baking brownies. In her family, some people like their brownies frosted without walnuts, others like them frosted with walnuts, and some just like them plain.
So Nadine frosts 3/4 of her batch of brownies and puts walnuts on 2/3 of the frosted part.
How much of her batch of brownies has both frosting and walnuts?
Multiplication of FractionsConsider:
How do you think a child might solve each of these?
Do both representations mean exactly the same thing to
children?
What kinds of reasoning and/or models might they use
to make sense of each of these problems?
Which one best represents Nadine’s brownie problem?
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3
3
4
Models for Reasoning About Multiplication
Fraction of a fraction Linear/measurement Area/measurement models Cross Shading
We will think of multiplying fractions as finding a fraction of another fraction.
34
We use a fraction square to represent the fraction .3
4Then, we shade of
We can see that it is the same as .
23
34
612
2
3
3
4
1
2
How much is of ?
2
3
3
4
The Linear Model with multiplication utilizes the number line and partitions the fractions
4
3
3
2of
4
3
3
1of 4
3
3
3of
How much is of ?
2
3
3
4
4
4
1
4
2
4
3
4
1
2
0
2
3
3
4
1
2
3
4
We can also use the linear model with shapes and partition accordingly
How much is of ?
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3
3
4
Identify ¾ of the circleBreak into 3 pieces
Take 2 pieces
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3
3
4
1
2
Answer is ½
In the third method, we will think of multiplying fractions as multiplying a length times a length to get an area.
23
34
This area is X = 612
Width is
Length is
Area
Number of square units
Is 6 out of 12
3 4
2 3
2
3
3
4
1
2
How much is of ?
2
3
3
4
Modeling multiplication of fractions using the length times length equals area approach requires that the children understand how to find the area of a rectangle.
A great advantage to this approach is that the area model is consistently used for multiplication of whole numbers and decimals. Its use for fractions, then is merely an extension of previous experience.
In the fourth method, we will represent both fractions on the same square.
34is
23is
How much is of ?
2
3
3
423
34
2
3
3
4
1
2
Modeling multiplication of fractions using the cross shading approach does produce correct answers. However, many elementary students may not grasp the
“because it is shaded in both directions”
overlapping concept. This may require some additional explanations
Classroom Problem
Eric and his mom are making cupcakes. Each cupcake gets 1/4 of a cup of frosting. They are making 20 cupcakes. How much frosting do they need?
Another student strategy
1/4 of a cupSo, 5, 6, 7, 8 -- that’s 2 cups.
…so 5 cups altogether.
9, 10, 11, 12 -- that’s 3 cups.
17, 18, 19, 20 -- that’s 5 cups.
13, 14, 15, 16 -- that’s 4 cups.
4 of these is 1 cup…
Another student strategy
1/4 + 1/4 + 1/4 + 1/4 = 1
1/4 + 1/4 + 1/4 + 1/4 = 1
1/4 + 1/4 + 1/4 + 1/4 = 1
1/4 + 1/4 + 1/4 + 1/4 = 1
1/4 + 1/4 + 1/4 + 1/4 = 1
5 cups
Q: What’s a number sentence for this problem?
A: 20 x 1/4 = 5 (there are others)
Other Contexts for Multiplication of Fractions
Finding part of a part (a reason why multiplication doesn’t always make things “bigger”)
Pizza (pepperoni on ⅓ of ½ pizza)
Recipes ( 1¾ cups of sugar is used but we want to make ½ a batch)
Ribbon (you have ⅜ yd , ⅓ of the ribbon is
used to make a bow)
Division with Fractions
Sharing meaning for division:
1• One shared by one-third of a group?
• How many in the whole group?
• How does this work?
13
Division With Fractions
Repeated subtraction / measurement meaning
1• How many times can one-third be subtracted
from one?• How many one-thirds are contained in one?• How does this work?• How might you deal with anything that’s left?
13
Division of Fractions examples
How many quarters are in a dollar? Ground beef cost 2.80 for ½ pound. What is the
price per pound? Maggie can walk the 2 ½ miles to school in 3/4 of
an hour. How long would it take to walk 4 miles? Barb had ¾ of a pizza left over from her party. She
wants to store it in plastic containers. Each container holds ⅓ of a pizza. How many containers will she use? How many will be completely full? How full will the last container be?
Division of Fractions examples
You have 1 cups of sugar. It takes
cup to make 1 batch of cookies. How many batches of cookies can you make?
How many cups of sugar are left?
How many batches of cookies could be made
with the sugar that’s left?
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2
1
3
“How many one eighths are in three fourths?”
Our pizza is cut into 8 pieces. If three fourths of a pizza is left, how many slices remain?
?8
1
4
3
Recall: a slice represents one eighth of the pizza
How many one eighths are in three fourths?
To find this we must first find 3/4
of the pizza.We then cut each fourth
into halves to make eighths.
We can see there are 6 eighths in three
fourths.
?8
1
4
3
Pizza
68
1
4
3
Now only half of the pizza is left. How many slices remain?
How many one eighths are in one half?
Using a fraction manipulative, we show one half of a
circle.To find how many one eighths are in one half, we cover the one half with eighths
and count how many we use.
Pizza
We find there are 4. There are four one eighths in one half.
48
1
2
1
?8
1
2
1
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