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Title: Building Understanding for Fractions using Fraction Bars and Fraction Bar Charts


1
Building Understanding for Fractionsusing
Fraction Bars and Fraction Bar Charts
  • Jim Rahn
  • www.jamesrahn.com
  • James.rahn_at_verizon.net

2
An 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
6
Only 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.

9
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10
Multiplying Fractions
  • Using the Area Model to Understand Multiplication
    of Fractions

11
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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.

15
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16
Multiplying with Mixed Numbers
Label the dimensions
Find the area of each piece of the rectangle
What is the total area?
17
Using Fraction Bars to Visualize Multiplication
Find a piece that will allow you to divide each
of those pieces into sixths.
18
Using Fraction Bars to Visualize Multiplication
Find a piece that will allow you to divide each
of those pieces in half
19
Using Fraction Bars to Visualize Multiplication
20
Division of Fractions
  • Building Conceptual Understanding for Division

21
What are these statements asking?
22
What is this question asking? How many one
quarters are in 1?
23
USE THE FRACTION BARS
24
Picture this division problem
What do we want to know?
We need to look at eighths and quarters.
25
First locate one quarter.
Now find how many eighths cover the same space.
Count the number of eighths below one quarter.
26
What does this problem mean?
What is this question asking?
Estimate the quotient.
Visualize this problem by finding the quarters
and twelfths.
27
First 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.
28
Which is the dividend?
Which is the divisor?
29
Picture these problems on your communicator or
with the fraction bars
Find the answers to each problem.
30
Each problem can be written in another form
31
This 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.
32
We dont see many complex fractions because there
are ways to simplify them.
33
Rewriting the denominator to be equal to 1.
34
Using fraction bars we already know the answer is
6.
Now we see the answer is 6 another way.
35
Write the problem as a complex fraction.
There is a fraction in the denominator.
We want to change the fraction so the denominator
is 1.
36
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37
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38
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39
Write the problem as a complex fraction.
There is a fraction in the denominator.
We want to change the fraction so the denominator
is 1.
40
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41
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42
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43
In 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.

44
Instead
  • 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

45
Using Mental Math
  • Place the Using Mental Math to Divide Fractions
    in your communicator.

46
Study 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?
49
Write 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?
50
Write 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?
51
Write the answers to Row 5.
Why are all these quotients fractions rather than
whole numbers?
Can you picture the fraction bars?
52
Write the answers to Row 6.
How can these problems be completed mentally
without using the reciprocal?
Can you picture the fraction bars?
53
Write 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.

57
How did I solve them so quickly?
How did I think about each problem?
58
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59
What is wrong with choices A and B?
60
Is the answer bigger or smaller than 1?
What is another way to think about this problem?
Is the answer smaller than either fraction?
61
Between what two integers is this answer? Why?
Is there more than one choice for the answer?
62
Now answer the question.
63
What would double this number be?
64
Now can you answer the subtraction?
65
Now can you estimate the sum?
66
Things 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?

68
Building Understanding for Fractionsusing
Fraction Bars and Fraction Bar Charts
  • Jim Rahn
  • www.jamesrahn.com
  • James.rahn_at_verizon.net
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