Project Impact CURR 231 Curriculum 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 Think first and estimate your answer before attempting 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 this mentally.

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/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Presentation 8a Finding and Using Equivalent Fractions Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All 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. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Begin with a unit square. Each side of the unit square is The area of the unit square is also 1. 1 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

It can be subdivided vertically. The area of the unit square can be subdivided several ways into equal parts. Thirds Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

It can be subdivided horizontally. Fourths Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

It can be subdivided both vertically and horizontally. Twelfths Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

If parts of an equally subdivided unit square 2323 are shaded a different color, image of a fraction is presented. a clear visual Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

3434 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

7 12 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Fraction squares can also provide clear visual images for equivalent fractions If we begin with a fraction using vertical subdivisions, we can visualize another name for that fraction if we subdivide the parts horizontally = Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

If we begin with a fraction using horizontal subdivisions, we can visualize another name for that fraction if we subdivide the parts vertically = 3434 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

When enough examples have been accumulated, children can readily recognize a pattern that suggests how to find equivalent fractions. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Suppose we begin with another fraction. If we cut the parts using one horizontal line, 4646 = 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 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

We can find another name for the same fractional amount if we multiply both the numerator and denominator by the same number = X Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

5 10 = X Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

12 18 = X Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

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. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

For example, it is difficult to tell which of these fractions is greater Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

If we rename the fractions using the same fractional units, = = 9 15 the comparison is easy. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

= = > Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

A similar process can be used to compare these fractions Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

We rename the fractions using the same fractional units = = Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

< = = Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

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 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

We can do this to compare any two fractions. 15 Multiply the denominator by 3. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Multiply the numerator by 3. 9 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Multiply the denominator by Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Multiply the numerator by Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Both fractions have the same denominator. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

So the numerator tells which fraction is greater. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

If we know the denominators will be the same Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

we only need to compare the numerators. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

We will use the same procedure to compare two other fractions. We know that 35 will be the denominator of both fractions. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

We will use the same procedure to compare two other fractions. So all we need to compute are the two numerators. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

We will use the same procedure to compare two other fractions. The numerators will tell us which fraction is greater. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

This numerator is greater. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

So this is the greater of the two original fractions. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

This is the greater of the two original fractions. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

The notion of equivalent fractions is also used when we add unlike fractions. Remember that we always add like units Suppose we want to add fractions with unlike fractional units. We need to rename those fractions so the units will be the same. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

= = = = Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Suppose we want to add these fractions: Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

= = = = Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

So it turns out that when we have unlike fractions to add, product of the two denominators as the common denominator. we can always use the We can use 24 as the common denominator. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

We multiply this numerator and denominator by 8. = Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

We multiply this numerator and denominator by = = Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Chapter 8: Fractions: Working with Units Smaller Than One Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Presentation 8b Modeling Fraction Multiplication Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

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. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

We will begin by thinking of fraction multiplication as finding a fraction of a fraction. We will think of 2323 X 3434 as meaning the same as 2323 of Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

In order to find 2323 of, 3434 we will start with 3434 and find of it Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

A Fraction of a Fraction of X = 6 12 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

of X = X Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

3 4 of X = X 2 5 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

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 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Let’s examine how this approach works with fraction multiplication. 3 4 This length is Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

2 3 This length is 3 4 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

This area is X Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

This area is X It is also X = 6 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

1 2 This length is X This area is X Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

1 2 This length is X This area is X X 4 10 = It is also equal to 4 10 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

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. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Chapter 9: Decimals and Percents: Working with Base-Ten Units Smaller Than One and Using Hundredths as a Common Denominator Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Presentation 9 Fraction Comparison and the Meaning of Percent Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

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. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

For example, suppose we have these three fractions: We can rename the fractions using 30 ( that is, 2 X 3 X 5) as the denominator. = = = Now it is easy to compare any two of the fractions or arrange the fractions in order. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

These same three fractions can be renamed using any number as the denominator To accomplish this, we begin by multiplying the numerator and the denominator by 17. = = = For example, we could use 17 as the denominator. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

These same three fractions can be renamed using any number as the denominator Then we divide the numerator and the denominator by the original denominator. = = = For example, we could use 17 as the denominator. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

These same three fractions can be renamed using any number as the denominator = = = For example, we could use 17 as the denominator. Divide this numerator and denominator by 2. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

These same three fractions can be renamed using any number as the denominator = 8.50 = = For example, we could use 17 as the denominator. 17 Divide this numerator and denominator by 2. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

These same three fractions can be renamed using any number as the denominator = 8.50 = = For example, we could use 17 as the denominator. 17 Divide this numerator and denominator by 3. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

These same three fractions can be renamed using any number as the denominator = 8.50 = For example, we could use 17 as the denominator. 17 = Divide this numerator and denominator by 3. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

These same three fractions can be renamed using any number as the denominator = 8.50 = For example, we could use 17 as the denominator. 17 = Divide this numerator and denominator by 5. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

These same three fractions can be renamed using any number as the denominator = 8.50 = For example, we could use 17 as the denominator. 17 = Divide this numerator and denominator by 5. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

These same three fractions can be renamed using any number as the denominator = 8.50 = For example, we could use 17 as the denominator. 17 = Now it is easy to compare any two of the fractions or arrange the fractions in order. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Or, we could use 100 as the denominator To accomplish this, we begin by multiplying the numerator and the denominator by 100. = = = Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Then we divide the numerator and the denominator by the original denominator. = = = Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

= = = Divide this numerator and denominator by 2. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

= = = Divide this numerator and denominator by 2. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

= = = Divide this numerator and denominator by 3. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

= = = Divide this numerator and denominator by 3. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

= = = Divide this numerator and denominator by 5. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

= = = Divide this numerator and denominator by 5. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

= = = Now it is easy to compare any two of the fractions or arrange the fractions in order. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

When the denominator is 100, the numerator is called the percent. The symbol for percent is % = = = = 50% (One half equals 50 percent.) = 66.67% (Two thirds equals percent.) = 40% (Two fifths equals 40 percent.) Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

The term percent literally means “per hundred” 1212 = = 50% or “out of one hundred.” 50 out of one hundred Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

The term percent literally means “per hundred” 1212 = = 50% or “out of one hundred.” 50 per hundred 50 percent Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Video – Math Talks  Catherine Pieck 6 th 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?