Title: Building Understanding for Fractions using Fraction Bars and Fraction Bar Charts
1Building Understanding for Fractionsusing
Fraction Bars and Fraction Bar Charts
- Jim Rahn
- www.jamesrahn.com
- James.rahn_at_verizon.net
2An Activity to Build Estimation Skills and Number
Sense
- Four Digit Fun with Fractions
3- You will be given four digits.
- You must create problems which meet the following
specifications - Use all four digits to create a fraction as close
to zero as possible. - Use all four digits to create a fraction that is
about one half. - Use all four digits to create two fractions that
sum as close to 1 as possible.
4- Use all four digits to create a fraction as close
to zero as possible. - Use all four digits to create a fraction that is
about one half. - Use all four digits to create two fractions that
sum as close to 1 as possible.
3, 5, 7, 9
5- Use all four digits to create a fraction as close
to zero as possible. - Use all four digits to create a fraction that is
about one half. - Use all four digits to create two fractions that
sum as close to 1 as possible.
2, 4, 1, 9
6Only Sixteenths Game
- Recognizing and Creating Equivalent Fractions
7- Sort through a standard deck of playing cards,
putting the 4s, 8s and kings in one stack and the
aces, 2s and 3s in the other stack. The kings
will represent sixteenths. - The 4s, 8s and kings stand for the denominator,
or bottom number of the fraction. - The aces, 2s and 3s will stand for the numerator,
or top number of a fraction.
8- Each player places the magnified inch template in
a Communicator. - The numerator and denominator cards are placed
face down in two separate piles. - Players alternately choose a numerator and
denominator card from the top of the deck. - If the fraction is not in sixteenths, they change
the fraction to an equivalent form using
sixteenths. - They then mark that distance of sixteenths on the
giant inch, each time starting where they left
off the last time. - The first player to have their mark extend beyond
an inch wins the round. - The first player to win 12 rounds (a foot) wins
the game. - When the cards are all used, each stack is
reshuffled and the numerator and denominator
piles are once again created.
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10Multiplying Fractions
- Using the Area Model to Understand Multiplication
of Fractions
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12- Draw in the vertical and horizontal lines to
complete the square.
- If the square was 1 unit by 1 unit, what is the
area of this square?
- Draw in the other vertical and horizontal lines
to complete the smaller rectangles.
- What is the area of each of the small rectangles?
13- What is the area of each of the rectangle whose
dimensions are
14- Draw two sides of the square and show fifths and
fourths.
- Draw in the vertical and horizontal lines.
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16Multiplying with Mixed Numbers
Label the dimensions
Find the area of each piece of the rectangle
What is the total area?
17Using Fraction Bars to Visualize Multiplication
Find a piece that will allow you to divide each
of those pieces into sixths.
18Using Fraction Bars to Visualize Multiplication
Find a piece that will allow you to divide each
of those pieces in half
19Using Fraction Bars to Visualize Multiplication
20Division of Fractions
- Building Conceptual Understanding for Division
21What are these statements asking?
22What is this question asking? How many one
quarters are in 1?
23USE THE FRACTION BARS
24Picture this division problem
What do we want to know?
We need to look at eighths and quarters.
25First locate one quarter.
Now find how many eighths cover the same space.
Count the number of eighths below one quarter.
26What does this problem mean?
What is this question asking?
Estimate the quotient.
Visualize this problem by finding the quarters
and twelfths.
27First locate three quarters.
Now find how many five twelfths cover the same
space.
We see one full set of five twelfths fits and
four out of five pieces of another set are
needed.
28Which is the dividend?
Which is the divisor?
29Picture these problems on your communicator or
with the fraction bars
Find the answers to each problem.
30Each problem can be written in another form
31This still means how many one eighths are in
three quarters.
But this time there is a fraction in both the
numerator and the denominator . We call this a
complex fraction.
32We dont see many complex fractions because there
are ways to simplify them.
33Rewriting the denominator to be equal to 1.
34Using fraction bars we already know the answer is
6.
Now we see the answer is 6 another way.
35Write the problem as a complex fraction.
There is a fraction in the denominator.
We want to change the fraction so the denominator
is 1.
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39Write the problem as a complex fraction.
There is a fraction in the denominator.
We want to change the fraction so the denominator
is 1.
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43In the past
- Theres no need to wonder why. Just invert and
multiply. - Unfortunately, that approach has resulted in the
fact that many adults dont know what happens
when one divides a fraction, and they cannot
remember what to do. - The phrase has led to frustration and no
understanding.
44Instead
- Take time to go through the steps we have just
used - to visually solve the problem with fractions bars
to develop visual understanding for division - to change to complex fractions because the steps
are based upon number properties students already
know
45Using Mental Math
- Place the Using Mental Math to Divide Fractions
in your communicator.
46Study all the fractions on the page.
What do you notice about the denominators in each
problem?
47 2 2 4 3
2
Write all the answers to the problems in Row 1.
What does division mean?
What appears to be an efficient way to answer
these problems?
When will this work?
48 5 8 16 8
32
Write the answers to Row 2.
How are these problems different from Row 1?
What appears to be an efficient way to answer
these problems?
49Write the answers to Row 3.
2 6 6 3
2
How are these problems different from Row 1?
Could you multiply by the reciprocal?
Could you do them like Row 1?
50Write the answers to Row 4.
What do you notice about each of these problems?
How are they different from Row 1?
Can you picture the fraction bars?
51Write the answers to Row 5.
Why are all these quotients fractions rather than
whole numbers?
Can you picture the fraction bars?
52Write the answers to Row 6.
How can these problems be completed mentally
without using the reciprocal?
Can you picture the fraction bars?
53Write the answers to Row 7.
Change the fractions so they have the same
denominator.
What do you do next?
54- With whole numbers, it is frequently easier to
complete problems using mental math. - Other times it is best to write a few
intermediate steps to determine a final answer. - Other times it is best to estimate an answer and
then use the calculator to find the exact answer.
55- We have looked at all four operations with
fractions. - You will see 8 fraction problems.
- You will try to decide which method to use to
find the correct answer mental math, paper and
pencil, or estimation and calculator. - To prepare for this we will do several mental
math problems.
56- Place Transparency 39 in your communicator.
- Circle the correct answer by using problem
solving, estimation and number sense.
57How did I solve them so quickly?
How did I think about each problem?
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59What is wrong with choices A and B?
60Is the answer bigger or smaller than 1?
What is another way to think about this problem?
Is the answer smaller than either fraction?
61Between what two integers is this answer? Why?
Is there more than one choice for the answer?
62Now answer the question.
63What would double this number be?
64Now can you answer the subtraction?
65Now can you estimate the sum?
66Things to think about
- What is the name of an answer in an addition
problem? - Will the sum of two fractions of the kind we have
been working with always be greater than either
of the two addends? - What is the name of the answer to a subtraction
problem? - Will the difference of two fractions of the kind
we have been working with always be smaller than
the minuend?
67- What is the name of the answer to a
multiplication problem? - Will the product of two proper fractions always
be greater than either proper fraction? - What is the name of the answer to a division
problem? - Will the quotient of two proper fractions always
be larger than either divisor or dividend?
68Building Understanding for Fractionsusing
Fraction Bars and Fraction Bar Charts
- Jim Rahn
- www.jamesrahn.com
- James.rahn_at_verizon.net