Project Impact CURR 231 Curriculum and Instruction in Math Session 6 Chapters 8 & 9

Project Impact CURR 231Curriculum and Instruction in Math

Session 6

Chapters 8 & 9

Outcomes Number Talk – Middle School Example Share Website Reflections Text – Teaching Math 8 & 9 Video – Math Talks Make and Take – Fraction Tiles and/or Circle

Pizza Game time – student led

Number Talk

9.8 + 8.7Think first and estimate your answer beforeattempting to solve the problem.

Mentally solve the problem.

Share with a partner how you solved this. I will Listen and post some of your strategies for solving thismentally.

Share Website Reflections Each participant will share the highlights of

their favorite math related website.

Chapter 8: Fractions:Working with Units Smaller Than One

Tucker/Singleton/WeaverTeaching Mathematics to ALL Children, Second Edition

Copyright ©2006 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

All rights reserved.

Presentation 8a Finding and Using Equivalent Fractions


Copyright ©2006 by Pearson Education, Inc.

Upper Saddle River, New Jersey 07458All rights reserved.

One of the most effective models for fractions is a pictorial model rather than a physical model.

The fraction square is an excellent tool for establishing mental imagery for a wide variety of fraction concepts.




Begin with a unit square.

Each side of the unit square is 1.

1

1

The area of the unit square is also 1.

1




It can be subdivided vertically.

The area of the unit square can be subdivided several ways into equal parts.

Thirds




It can be subdivided horizontally.

Fourths




It can be subdivided both vertically and horizontally.

Twelfths




If parts of an equally subdivided unit square

2

3

are shaded a different color, image of a fraction is presented.

a clear visual




3

4




7

12




Fraction squares can also provide clear visual images for equivalent fractions.

2

3

If we begin with a fraction using vertical subdivisions, we can visualize another name

for that fraction if we subdivide the parts horizontally.

4

6=




If we begin with a fraction using horizontal subdivisions, we can visualize another name

for that fraction if we subdivide the parts vertically.

9

12=

3

4




When enough examples have been accumulated, children can readily recognizea pattern that suggests how to find equivalent fractions.




Suppose we begin with another fraction. If we cut the parts using one horizontal line,

4

6=

every part is cut into two pieces. We have 2 times as many parts.Every shaded part is also cut into two pieces. We have 2 times as many shaded parts.

X 2

X 22

3




We can find another name for the same fractional amount if we multiply both the numerator and denominator by the same number.

12

16=

X 4

X 43

4




5

10=

X 5

X 51

2




12

18=

X 6

X 62

3




There is one big idea that determines what we do procedurally when making comparisons—

People who compare unlike things are said to be “comparing apples and oranges.” Comparison of fractions is much easier when the fractional units are the same.

we compare like units.




For example, it is difficult to tell which of these fractions is greater.

2

3

3

5




2

3

3

5

If we rename the fractions using the same fractional units,

10

15= =

9

15

the comparison is easy.




2

3

3

5

10

15= =

9

15

2

3

3

5>




A similar process can be used to compare these fractions.

1

4

2

7




1

4

2

7

We rename the fractions using the same fractional units.

7

28= =

8

28

1

4

2

7




1

4

2

7<

1

4

2

7

7

28= =

8

28

1

4

2

7




1

4

2

7

Note that the process we have been using results in our renaming the fractions using a common denominator which is the product of the two original denominators.

1 X 7

4 X 7= =

4 X 2

4 X 7=

7

28=

8

28




3

5

1

3

We can do this to compare any two fractions.

15

Multiply the denominator by 3.




3

5

1

3

15

Multiply the numerator by 3.

9




Multiply the denominator by 5.

3

5

1

3

15

9

15




Multiply the numerator by 5.

3

5

1

3

15

9

15

5




3

5

1

3

15

9

15

5

Both fractions have the same denominator.




3

5

1

3

15

9

15

5

So the numeratortells which

fraction is greater.




3

5

1

3

15

9

15

5

If we know the denominators will be the same




3

5

1

3

15

9

15

5

we only need tocompare thenumerators.




5

7

3

5

We will use the same procedure to compare two other fractions.

We know that 35 will be the denominator of both fractions.




5

7

3

5


So all we need to compute are the two numerators.




5

7

3

5


The numerators will tell us which fraction is greater.




5

7

3

5

25 21

This numerator is greater.




5

7

3

5

25 21

So this is the greater of the two original fractions.




1615 3

8

2

5

This is the greater of the two original fractions.




The notion of equivalent fractions is also used when we add unlike fractions.

Remember that we always add like units.

2

3

3

5+

Suppose we want to add fractions with unlike fractional units.

We need to rename those fractions so the units will be the same.




2

3

3

5

10

15= =

9

15

2

3

3

5+

10

15= +

9

15

19

15=




Suppose we want to add these fractions:

2

7

1

4+




1

4

2

7

7

28= =

8

28

1

4

2

7

2

7

1

4+

7

28= +

8

28=

15

28




So it turns out that when we have unlike fractions to add,

1

8

2

3+

product of the two denominators as the common denominator.

we can always use the

We can use 24 as the common denominator.




1

8

2

3+

We multiply this numerator and denominator by 8.

=16

24




We multiply this numerator and denominator by 3.

1

8

2

3+ =

16

24+

3

24=

19

24




Chapter 8: Fractions:Working with Units Smaller Than One




Presentation 8b ModelingFraction Multiplication




In the literature, you can find two different approaches for modeling multiplication of fractions that are supported by research:

A Fraction of a Fraction

Length X Length = Area

We will examine each of these two methods.




We will begin by thinking of fraction multiplication as finding a fraction of a fraction.

We will think of2

3X

3

4

as meaning the same as2

3of .

3

4




In order to find2

3of ,

3

4

we will start with 3

4

and find of it. 2

3




A Fraction of a Fraction

34

23

of

23

34X = 6

12




23

12

of

12

23X = 2

6

12

23X




34

of

34

25X = 6

20

34

25X

25




Now, we will think of fraction multiplication as multiplying lengths of the sides of a rectangle to find its area.

to get this area.We multiply this length times this length




Let’s examine how this approach works with fraction multiplication.

34This length is




23This length is

34




23

34

This area is X34

23




23

34

This area is X34

23

It is also612

34

23X =

612




12This length is

45This length is

12

45X

This area is12

45X




12This length is

45This length is

12

45X

This area is12

45X

12

45X 4

10=

It is also equal to4

10




Either of these approaches to the modeling of fraction multiplication works well with children.

Both methods do an effective job of building mental imagery for the process.

Both methods do a good job of convincing children that the answers make sense—that they must be correct.

And, consequently, both methods produce results that can be used as the basis for generalizing the fraction multiplication algorithm.




Chapter 9: Decimals and Percents:Working with Base-Ten Units Smaller Than One and Using Hundredths as a Common Denominator




Presentation 9 Fraction Comparison and theMeaning of Percent




Recall that there is one big idea for comparison—compare like units.

We have already applied this big idea in the comparison of two fractions. We renamed the fractions with the same denominator (the fractional unit) and then the comparison was easy.

If we want to compare more than two fractions, we need to rename them so that all of them have the same denominator.




For example, suppose we have these three fractions:

1 2

23

25

We can rename the fractions using 30 ( that is, 2 X 3 X 5) as the denominator.

= 1530

= 2030

= 1230

Now it is easy to compare any two of the fractions or arrange the fractions in order.




These same three fractions can be renamed using any number as the denominator.

1 2

23

25

To accomplish this, we begin by multiplying the numerator and the denominator by 17.

= 1734

= 3451

= 3485

For example, we could use 17 as the denominator.





1 2

23

25

Then we divide the numerator and the denominator by the original denominator.

= 1734

= 3451

= 3485






1 2

23

25

= 1734

= 3451

= 3485


Divide this numerator Divide this numerator and denominator by 2.and denominator by 2.





1 2

23

25

= 8.50 = 3451

= 3485


17






1 2

23

25

= 8.50 = 3451

= 3485


17






1 2

23

25

= 8.50 = 3485


17= 11.33

17






1 2

23

25

= 8.50 = 3485


17= 11.33

17






1 2

23

25

= 8.50 =


17= 11.33

176.8017






1 2

23

25

= 8.50 =


17= 11.33

176.8017





Or, we could use 100 as the denominator.

1 2

23

25

To accomplish this, we begin by multiplying the numerator and the denominator by 100.

= 100200

= 200300

= 200500




1 2

23

25

Then we divide the numerator and the denominator by the original denominator.

= 100200

= 200300

= 200500




1 2

23

25

= 100200

= 200300

= 200500





1 2

23

25

= 50 100

= 200300

= 200500





1 2

23

25

= 50 100

= 200300

= 200500





1 2

23

25

= 50 100

= 66.67 = 200500100





1 2

23

25

= 50 100

= 66.67 = 200500100





1 2

23

25

= 50 100

= 66.67 = 40 100100





1 2

23

25

= 50 100

= 66.67 = 40 100100





When the denominator is 100, the numerator is called the percent. The symbol for percent is %.

23

25

1 2

= 50 100

= 66.67

= 40 100

100

= 50% (One half equals 50 percent.)

= 66.67% (Two thirds equals 66.67 percent.)

= 40% (Two fifths equals 40 percent.)




The term percent literally means “per hundred”

1 2

= 50 100

= 50%

or “out of one hundred.”

50 out 50 out of one of one hundredhundred




The term percent literally means “per hundred”

1 2

= 50 100

= 50%

or “out of one hundred.”

50 per 50 per hundredhundred

50 percent50 percent




Video – Math Talks Catherine Pieck 6th Number Talk

Make and Take Activity Fraction Tiles

Activity

GAME TIME!!!

Each week, students will take turns leading the class in a math game.

Closing Final thoughts, comments? Making connections – Anything to add to

your reflection?

Documents

Project Impact CURR 231 Curriculum and Instruction in Math Session 6 Chapters 8 & 9