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Building Understanding for Fractions using Fraction Bars and Fraction Bar Charts

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## Building Understanding for Fractions using Fraction Bars and Fraction Bar Charts

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### Building Understanding for Fractionsusing Fraction Bars and Fraction Bar Charts

### Building Understanding for Fractionsusing Fraction Bars and Fraction Bar Charts

Jim Rahn

www.jamesrahn.com

James.rahn@verizon.net

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.

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

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

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.

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.

To represent draw a horizontal line segment on the Communicator® and divide it into four equal parts

- To represent draw a unit vertical segment and mark the segment into thirds.

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?

Use the same square and label and

- What is the area of each of the rectangle whose dimensions are

Draw two sides of the square and show fifths and fourths.

- Draw in the vertical and horizontal lines.

- What part of the square is represented by a rectangle with the dimensions of

- What part of the square is represented by a rectangle with the dimensions of

Multiplying with Mixed Numbers

Draw an appropriate rectangle whose dimensions are

Label the dimensions

Find the area of each piece of the rectangle

What is the total area?

Using Fraction Bars to Visualize Multiplication

First find a representation of

Find a piece that will allow you to divide each of those pieces into sixths.

Find

Using Fraction Bars to Visualize Multiplication

First find a representation of

Find a piece that will allow you to divide each of those pieces in half

Find

How many one quarters are in 1?

There are two eighths in one quarter so

First locate one quarter.

Now find how many eighths cover the same space.

Count the number of eighths below one quarter.

What does this problem mean?

What is this question asking?

Estimate the quotient.

Visualize this problem by finding the quarters and twelfths.

So there are 1 and five twelfths in three quarters.

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.

Which is the divisor?

Picture these problems on your communicator or with the fraction bars

Find the answers to each problem.

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.

We don’t see many complex fractions because there are ways to simplify them.

Is there a number we can multiply by to change it to 1?

Rewriting the denominator to be equal to 1.

If we start with

how can we make the denominator equal to 1?

Using fraction bars we already know the answer is 6.

Now we see the answer is 6 another way.

We know the answer is 2 because two fit in .

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.

How can we change to 1 WITHOUT changing the value of ?

To change to 1, we need to multiply by its reciprocal which is

But to do this AND keep the value of theoriginal problem aswe must multiply by a form of 1.

We’ll use

We know the answer is because two and two thirds fit in using fraction bars.

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.

How can we change to 1 WITHOUT changing the value of ?

To change to 1, we need to multiply by its reciprocal which is

But to do this AND keep the value of theoriginal problem aswe must multiply by a form of 1.

We’ll use

In the past…

- “There’s no need to wonder why. Just invert and multiply.”
- Unfortunately, that approach has resulted in the fact that many adults don’t know what happens when one divides a fraction, and they cannot remember what to do.
- The phrase has led to frustration and no understanding.

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

Using Mental Math

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

Study all the fractions on the page.

What do you notice about the denominators in each problem?

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?

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?

How are these problems different from Row 1?

2 6 6 3 2

Could you multiply by the reciprocal?

Could you do them like Row 1?

What do you notice about each of these problems?

How are they different from Row 1?

Can you picture the fraction bars?

Why are all these quotients fractions rather than whole numbers?

Can you picture the fraction bars?

How can these problems be completed mentally without using the reciprocal?

Can you picture the fraction bars?

Change the fractions so they have the same denominator.

What do you do next?

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.

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.

Place Transparency 39 in your communicator.

- Circle the correct answer by using problem solving, estimation and number sense.

How did I solve them so quickly?

How did I think about each problem?

What is wrong with choices A and B?

Is the answer bigger or smaller than 1?

What is another way to think about this problem?

Is the answer smaller than either fraction?

Between what two integers is this answer? Why?

Is there more than one choice for the answer?

Now answer the question.

Round to the nearest whole number.

What would double this number be?

Round to the nearest whole number.

Now can you answer the subtraction?

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?

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?

Jim Rahn

www.jamesrahn.com

James.rahn@verizon.net

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