2008 may 29fractions: grades 3-5: slide 1 welcome welcome to content professional development...

2008 May 29 Fractions: Grades 3-5: slide 1

Welcome

Welcome to content professional development sessions for Grades 3-5. The focus is Fractions.

Fractions in Grades 3-5 lays critical foundation for proportional reasoning in Grades 6-8, which in turn lays critical foundation for high school algebra.

The goal is to help you understand this mathematics better to support your implementation of the Mathematics Standards.


Introduction of Facilitators

INSERTthe names and affiliations

of the facilitators


Introduction of Participants

In a minute or two:1. Introduce yourself.2. Describe an important moment in your life that contributed to your becoming a mathematics educator.3. Describe a moment in which you hit a “mathematical wall” and had to struggle with learning.


Overview

Some of the problems may be appropriate for students to complete, but other problems are intended ONLY for you as teachers.

As you work the problems, think about how you might adapt them for the students you teach.

Also, think about what Performance Expectations these problems might exemplify.


Problem Set 1

The focus of Problem Set 1 is representing a single fraction.

You may work alone or with colleagues to solve these problems.

When you are done, share your solutions with others.


Problem Set 1

Think carefully about each situation and make a representation (e.g., picture, symbols) to represent the meaning of 3/4 conveyed in that situation.


Problem 1.1

John told his mother that he would be home in 45 minutes.


Problem 1.2

Melissa had three large circular cookies, all the same size – one chocolate chip, one coconut, one molasses.

She cut each cookie into four equal parts and she ate one part of each cookie.


Problem 1.3

Mr. Albert has 3 boys to 4 girls in his history class.


Problem 1.4

Four little girls were arguing about how to share a package of cupcakes.

The problem was that cupcakes come three to a package.

Their kindergarten teacher took a knife and cut the entire package into four equal parts.


Problem 1.5

Baluka Bubble Gum comes four pieces to a package.

Three children each chewed a piece from one package.


Problem 1.6

There were 12 men and 3/4 as many women at the meeting.


Problem 1.7

Mary asked Jack how much money he had.

Jack reached into his pocket and pulled out three quarters.


Problem 1.8

Each fraction can be matched with a point on the number line.

3/4 must correspond to a point on the number line.


Problem 1.9

Jaw buster candies come four to a package and Nathan has 3 packages, each of a different color.

He ate one from each package.


Problem 1.10

Martin’s Men Store had a big sale – 75% off.


Problem 1.11

Mary noticed that every time Jenny put 4 quarters into the exchange machine, three tokens came out.

When Mary had her turn, she put in twelve quarters.


Problem 1.12

Tad has 12 blue socks and 4 black socks in his drawer.

He wondered what were his chances of reaching in and pulling out a sock to match the blue one he had on his left foot.


Reflection

Even a “simple” fraction, like 3/4, has different representations, depending on the situation.

How do you decide which representation to use for a fraction?

How can we help students learn how to choose a representation that fits a given situation?


Problem Set 2

The focus of Problem Set 2 is representing different fractions.




Problem 2.1

Represent each of the following:

a. I have 4 acres of land. 5/6 of my land is planted in corn.

b. I have 4 cakes and 2/3 of them were eaten

c. I have 2 cupcakes, but Jack as 7/4 as many as I do.


Problem 2.2

The large rectangle represents one whole that has been divided into pieces.

Identify what fraction each piece is in relation to the whole rectangle. Be ready to explain how you know the fraction name for each piece. A ___ B ___ C ___ D ___ E ___ F ___ G ___ H ___

A

B C D

E

F

G H


Problem 2.3

What is the sum of your eight fractions? What should the sum be? Why?


Problem 2.4

Mom baked a rectangular birthday cake. Abby took 1/6.

Ben took 1/5 of what was left. Charlie cut 1/4 of what remained. Julie ate 1/3 of the remaining cake. Marvin and Sam split the rest.

Was this fair? How does the shape of the cake influence your answer?


Problem 2.5

If the number of cats is 7/8 the number of dogs in the local pound, are there more cats or dogs?

What is the unit for this problem?


Problem 2.6

Ralph is out walking his dog. He walks 2/3 of the way around this circular fountain.

Where does he stop?

start


Problem 2.7

Ralph is out walking his dog. He walks 2/3 of the way around this square fountain.

Where does he stop?

START --------->


Reflection

Why is it important for students to connect their understanding of fractions with the ways they represent fractions?

How do you keep track of the unit (that is, the value of 1) for a fraction?

How can you help students learn these things?


Problem Set 3

The focus of Problem Set 3 is unitizing.




Describing Unitizing

Unitizing is thinking about different numbers of objects as the unit of measure.

For example, a dozen eggs can be thought of as:

12 groups of 1, 6 groups of 2,

4 groups of 3, 3 groups of 4,

2 groups of 6, 1 group of 12


Applying Unitizing

4 eggs is 1/3 of a dozen since it is 1 of the 3 groups of 4

4 eggs = 1 (group of 4)12 eggs = 3 (group of 4)

so

4 eggs / 12 eggs = 1 (group of 4) / 3 (group of 4)

= 1/3


Thinking about the Unit

4 eggs can be thought of as a unit which measures thirds of a dozen.

2/3 of a dozen = 2 groups of 4 eggs = 8 eggs

5/3 of a dozen = 5 groups of 4 eggs = 20 eggs


Usefulness of Unitizing

Skill at unitizing (that is, thinking about different units for a single set of objects) helps develop flexible thinking about “the unit” for representing fractions.

Flexible thinking is a critical skill in understanding fractions deeply and in developing a base for proportional reasoning.


Problem 3.1

Can you see ninths? How many cookies will you eat if you eat 4/9 of the cookies?

O O O O O OO O O O O OO O O O O O


Problem 3.2

Can you see twelfths? How many cookies will you eat if you eat 5/12 of the cookies?



Problem 3.3

Can you see sixths? How many cookies will you eat if you eat 5/6 of the cookies?



Problem 3.4

Can you see thirty-sixths? How many cookies will you eat if you eat 14/36 of the cookies?



Problem 3.5

Can you see fourths? How many cookies will you eat if you eat 3/4 of the cookies?



Reflection

Was it easy for you to think about different units for “measuring” the size of a set of objects?

How can we help students think about different units for a set?


Problem Set 4

The focus of Problem Set 4 is more unitizing.




Problem 4.1

16 eggs are how many dozens?

26 eggs are how many dozens?


Problem 4.2

You bought 32 sodas for a class party.

How many 6-packs is that?

How many 12-packs?

How many 24-packs?


Problem 4.3

You have 14 sticks of gum.





Problem 4.4

There are 4 2/3 pies left in the pie case.

The manager decides to sell these with this plan: Buy 1/3 of a pie and get 1/3 at no extra charge.

How many servings are there?


Problem 4.5

There are 5 pies left in the pie case.

The manager decides to sell these with this plan: Buy 1/3 of a pie and get 1/3 at no extra charge.

How many servings are there?


Problem 4.6

Although “unitizing” is a word for adult (and not children), how might work with unitizing help children understand fractions?


Reflection

Would it be easy for students to think about different units for “measuring” the size of a set of objects?

How can we help them learn that?


Problem Set 5

The focus of Problem Set 5 is keeping track of the unit.




Problem 5.1

How do you know that 6/8 = 9/12?

Give as many justifications as you can.


Problem 5.2

Ten children went to a birthday party.

Six children sat at the blue table, and four children sat at the red table.

At each table, there were several cupcakes.

At each table, each child got the same amount of cake; that is they “fair shared.”

At which table did the children get more cake?

How much more?


Problem 5.2

Blue table: 6 children Red table: 4 children

(a) blue table: 12 cupcakes red table: 12 cupcakes

(b) blue table: 12 cupcakes red table: 8 cupcakes


Problem 5.2Blue table: 6 children Red

table: 4 children

(c) blue table: 8 cupcakes red table: 6 cupcakes

(d) blue table: 5 cupcakes red table: 3 cupcakes

(e) blue table: 2 cupcakes red table: 1 cupcake


Problem 5.3

Would you purchase the following poster?

Why or why not?


Reflection

Why is it so important to keep track of the unit for fractions?


Problem Set 6

The focus of Problem Set 6 is in between.




Problem 6.1

Find three fractions equally spaced between 3/5 and 4/5.

Justify your solutions.


Problem 6.2

We know that 3.5 is halfway between 3 and 4, but is 3.5/5 halfway between 3/5 and 4/5?

Explain.


Problem 6.3

Find three fractions equally spaced between 1/4 and 1/3.

Justify your solutions.


Problem 6.4

We know that 3.5 is halfway between 3 and 4, but is 1/3.5 halfway between 1/4 and 1/3?

Explain.


Reflection

How do you know when fractions are equally spaced?

Is it important for students in Grades 3-5 to be able to do determine this?

Where would this idea appear in the K-8 Mathematics Standards?


Problem Set 7

The focus of Problem Set 7 is variations on fraction tasks.




Problem 7.1

What number, when added to 1/2, yields 5/4?

Write at least 5 different answers.


Problem 7.2

Write two fractions whose sum is 5/4.



Problem 7.3

Write two fractions, each with double-digit denominators, whose sum is 5/4.



Problem 7.4

Which of problems 7.1, 7.2, and 7.3 is the most “unusual”?

Why?


Reflection

Do your curriculum materials include “unusual” problems?

Why is it important for students to have experience with “unusual” problems?


Problem Set 8

The focus of Problem Set 8 is modifying fractions.




Problem 8.1

What happens to a fraction if (a) the numerator doubles (b) the denominator doubles (c) both numerator and denominator double(d) both numerator and denominator are halved(e) numerator doubles, denominator is halved (f) numerator is halved, denominator doubles


Problem 8.2

What happens to a fraction if (a) the numerator increases (b) the denominator increases (c) both numerator and denominator increase (d) both numerator and denominator decrease (e) numerator increases, denominator decreases (f) numerator decreases, denominator increases


Problem 8.3

The letters a, b, c, and d each stand for a different number selected from {3, 4, 5, 6}.

Solve these problems and justify each answer.(a) Write the greatest sum: a/b + c/d(b) Write the least sum: a/b + c/d(c) Write the greatest difference: a/b - c/d (d) Write the least difference: a/b - c/d


Reflection

Which of these problems could be presented to students as “mental math” problems?

Which of these problems would students need to explore over a long period of time?


Problem Set 9

The focus of Problem Set 9 is reflection on thinking.




Problem 9.1

Write a division story problem appropriately solved by division so that the quotient has a label different from the labels on the divisor and the dividend.

What does “divisor” mean? What does “dividend” mean?


Problem 9.2

Write a story problem appropriately solved by division that demonstrates that division does not always make smaller.


Problem 9.3

Is a fraction a number?

Explain.


Problem 9.4

Why are fractions called equivalent rather than equal?


Reflection

What knowledge for teachers do these problems address?

Why is this important knowledge for teachers?


Closing Comments

Implementing the K-8 Mathematics Standards will require a deeper focus of mathematics ideas at each grade.

Personal understanding of these ideas will make the implementation process easier.

2008 may 29fractions: grades 3-5: slide 1 welcome welcome to content professional development...

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