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Making Math Magic

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Fractions, Decimals, and Percents

Are they related???

Making Math Magic

M3 MAKING MATH MAGIC

We believe……

- that all students can and should learn math!
- that if children like math and feel successful at math - they will learn math!
- there are 3 basic stages that children need to go
- through when learning math:
- –Stage 1: Using Manipulatives
- –Stage 2: Developing a Mental Image
- –Stage 3: Using Symbols

Making Math Magic

Students can be more successful when they have plenty of opportunities to:

- Build it! Concrete
- Draw it! Mental Image
- Write it! Symbolic

SAY IT! throughout

Making Math Magic

We believe……

that the HOW in math instruction

is as important

as the WHAT,

if not more so…

Making Math Magic

Review the Standards for Mathematical Practice:

Make sense of problems and

persevere in solving them

Reason abstractly and quantitatively

(contextualize and decontextualize)

Construct viable arguments and

critique the reasoning of others

Standards for Mathematical Practice:

Model with mathematics

Use appropriate tools strategically

Attend to precision

Look for and make use of structure

Look for and express regularity in

repeated reasoning

If we pay particular attention to the learning stages mentioned earlier, which Standards for Math Practice would/could be naturally addressed?

With your talking partner discuss:

- Build it!
- Draw it!
- Write it!

Say It!

Throughout

Making Math Magic

- Understand the relationship between the names (fraction, decimal, percent)
- Move freely between the “part”, the “name”, and the “whole”
- Explore Benchmark numbers to understand size/location, and comparisons.

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Why do middle school and high school students still struggle with fractions, decimals, and per cents?

Our goal is to help students that struggle make sense of rational numbers!

Making Math Magic

Are there some fractions, decimals, per cents that are more prevalent in real life?

How long has it been since you had to deal with 9.0768% in real life????

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Let’s concentrate on those numbers as we revisit to develop understanding.

Halves, fourths, eighths

Thirds, sixths

Tenths

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We are going to start with percents and we are going to start with 50%.

WHY?????

Pre-assessment

Write down (or draw) some things that you know

about 50%.

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Shade 50% of your paper strip.

- Compare your 50% with your talking partner.
- Did you agree?
- How do you know you are correct?
- If I wanted to describe what I shaded with a common fraction, what fraction did I shade?

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Use a number line and mark where 50% would belong.

0% 100%

Let’s compare the strip model and the number line model.

Making Math Magic

0% 50% 100%

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Area model using the paper strip

Linear model using the paper strip

Let’s look at another model.

Use one of the 4” squares to shade 50%

You will also need this Transparency Grid Model

Making Math Magic

If this entire square represents my unit (my 1), what is the name for each of my inside pieces?

What is my shaded part called?

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How do I write that as a fraction?

How do I write that as a decimal?

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If this entire square represents my unit (my 1), what is the name for each piece?

What would be the name for my shaded part?

Making Math Magic

How do I write this as a fraction?

How do I write this as a decimal?

Making Math Magic

Let’s look at all of the ways we have represented our 50%:

50%, , , 0.5, , and 0.50

These are all equivalent and can be used interchangeably.

1

2

5

10

50

100

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Let’s look at other area or linear models..

Fraction Circles

Fraction Rectangles

Cuisenaire Rods

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At your table, each person use a different model to show a 50% (etc.) model..

Fraction Circles

Fraction Rectangles

Cuisenaire Rods

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Introducing the discrete or set model

- Use either a paper circle or a paper square to fold and shade 50%.
- If you have 6 beans, how can you
- place them on your model to be “fair”?
- What is 50% of 6?

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Reverse the process…..

- If 4 represents 50% of a number, what is the number?
- 50% of what number is 4?

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Using the number line to solve problems

Some people prefer one number line

0 ? 24

0% 50% 100%

Some people prefer two number lines

0% 50% 100%

0 ? 24

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Using the double number line ….

Some people prefer the two number lines put together

0 ? 24

0% 50% 100%

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Practicing with the models

- Choose a model to show and solve:
- 50% of what number is 18?
- What is 0.5 of 32?
- What fraction of 30 is 15?
- Twenty-one is 0.50 of what number?

Making Math Magic

Moving on to Fourths……

- Why would we do fourths next?
- Start back with folding the strips.
- Have the students use an 8.5 paper strip and find half. Ask them to leave this folded and fold in half again. Have them predict how many parts they will have.
- How should we label these if I want to use fractions?

one-fourth

one-fourth

one-fourth

one-fourth

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Fourths……

Use one of the 8.5” number lines to label fourths . They can used their folded strip as a reference or they can fold the number line.

one-fourth

one-fourth

one-fourth

one-fourth

0

4

1

4

2

4

3

4

4

4

Discuss the difference in the labels.

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Fourths……

Ask them to use other paper strips and number lines to label using percents.

You may have to lead them to the discovery of “If the whole strip is 100%, then one section is 25%”.

25%

25%

25%

25%

0% 25% 50% 75% 100%

Again, discuss the difference in the labels.

Making Math Magic

Fourths….

Because students may struggle with finding half of 0.5, we are going to use the 4” square and the grid model before we ask them to do the decimal version of the number line.

Making Math Magic

Let’s look at another model.

Use one of the 4” squares to shade

25% or one-fourth.

You will also need this Transparency Grid Model

Making Math Magic

1

4

? s

10

2½

10

=

= 0.2½

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1

4

2½

10

=

= 0.2½

= 0.25

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Let’s do some math…..

Area model

Find an area model for fourths or fold one of the circles or squares into fourths.

What is ¼ of 12?

What is ¾ of 12?

Making Math Magic

Area model

25% of some number is 5, what is the number?

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Area model

75% of some number is 12, what is the number?

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Number line

What is 0.75 of 40?

0 0.25 0.50 0.75 1.00

??

0 40

Making Math Magic

Number line

75% of what number is 15?

0 25% 50% 75% 100%

0 15

??

Making Math Magic

Try these……..

25% of what number is 7?

What is 0.5 x 64?

¾ of what number is 21?

What per cent of 40 is 30?

Making Math Magic

We are debating on doing “eighths” here or moving to thirds and coming back to eighths later.

THOUGHTS?????

Making Math Magic

Thirds……..

Because of the fractions involved in thirds when they are written as a decimal and as a percent, we decided to approach them using the grid model first.

Making Math Magic

1

3

Use the paper grid models to shade

How many tenths?

How many hundredths?

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Tenths

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