building the foundation to algebra

BUILDING THE FOUNDATION TO ALGEBRARational Numbers

Goals Develop a conceptual understanding of

fractions as parts of regions and parts of sets.

Determine fractional parts when the whole varies.

Determine the size of the whole from fractional parts.

Connect mixed number and improper fraction representations.

Hexagon Fractions

Hexagon Fractions

Use a hexagon as a base. Cover the hexagon with other pattern block

pieces. Take another hexagon and build a different

design on top of it. Make as many designs as you can that cover

the hexagon. How many different designs can you build?

How do you know you found all of them? Make fraction number sentences to

describe each of your designs, e.g., 1 = ½ + ½.

Make a triple hexagon shape. Use that shape as the whole.

(The ONE) Determine what fractional part

each pattern block shape represents: Hexagon Trapezoid Rhombus Triangle

Hexagon Fractions

Use the large hexagon shape as the whole. (The ONE)

Determine what fractional part each pattern block shape represents: Hexagon Trapezoid Rhombus Triangle

The Large Hexagon

From her work with pattern blocks in third grade, Lynn always thought that the trapezoid was called ½. But when she made her triple hexagon, the trapezoid wasn’t called ½ anymore!

What happened? How is this possible?

How is this Possible?

Lynn was trying to figure out which was larger, 1/3 or 1/2. “My third grade teacher said that in fractions, larger is smaller and smaller is larger, so 1/2 is larger than 1/3.” But then she looked at the three pattern block problems she just did. “The hexagon is 1/3 and the trapezoid is 1/2. The hexagon is bigger than the trapezoid. So, 1/3 IS larger than 1/2. I knew larger couldn’t be smaller!”

What happened? How is this possible?

How is this Possible?

Why do the same pattern blocks have different values for the hexagon, triple hexagon, and large hexagon?

What is the relationship between the size of the whole shapes and the fractional value of the pattern block pieces?

Making Connections

Looking through Teacher Lenses

How would you characterize the level of this task: High or low cognitive demand?

What mathematical ideas are embedded in the task?

What makes this worthwhile mathematics?

From Parts to Wholes

On the flight from Pittsburgh to San Francisco, I fell asleep after traveling half the trip. When I awoke, I still had to travel half the distance that I traveled while sleeping. For what part of the entire trip did I sleep?

Flight Problem

From Parts to Wholes

What is the whole if… the rhombus is

1/2? the rhombus is

1/3? the rhombus is

1/4? the trapezoid is

3/4? the hexagon is 2/3? the hexagon is 3/5? the rhombus is

2/9?

Use your pattern block pieces to build the following shapes. Sketch your shape on the recording paper:

A triangle that is 1/3 green and 2/3 red. A triangle that is 2/3 red, 1/9 green, and

2/9 blue. A parallelogram that is 3/4 blue and

1/4 green. A parallelogram that is 2/3 blue and 1/3

green. A trapezoid that is 1/2 red and 1/2 blue.Build larger versions of your

solutions with the same fractional parts.

Pattern Block Puzzles

What strategies did you use to solve the puzzles?

What happened when you tried to build a larger version of your puzzle? What patterns did you notice?

Pattern Block Puzzles

Disappearing Cookies

Disappearing Cookies

Patty Peterson put out a plate of freshly baked cookies. As her family came home from a hard day at school, they helped themselves: Peggy took 1/5 of the cookies. Paula took 3/8 of the cookies left on the plate. Porter took 1/3 of the remaining cookies. Pansy took 2/5 of the remaining cookies. Polly took 1/2 of the remaining cookies. Payton took 2/3 of what was left. When Penny got there, there was only one cookie left!How many cookies did Patty Peterson bake?

Fractional Parts of Sets

Chocolate Fractions

Make a set of 12 candies 1/4 = candies

2/4 = candies

3/4 = candies

Make a set of 20 candies 1/5 = candies

3/5 = candies

5/5 = candies

How does knowing the number of candies in a unit fraction help you figure out the number of candies in other fractions?

2/3 of 15?

5/7 of 21?

4/9 of 27?

1/2 of 27?   1/3 of 19?  2/3 of 19?

Describe a method for finding the fractional part of any set.

Chocolate Fractions

59 of 36 = 20

Chocolate Fractions

Materials: 1 small pack of candies per player

Number of Players: 2 – 6 Object of the game:

To score the most points by writing fraction sentences to describe your set of candies.

Name It!

Directions:1. Close your eyes and count out 24 candies. The

24 candies are your “Whole”.2. Write as many fraction statements as you can to

describe your Whole.3. Compute your score.

1 point for each statement that contains 24ths.

3 points for each statement that contains unit fractions (fractions with numerator of 1) with denominators less than 24.

5 points for each statement that contains non-unit fractions (i.e., fractions with numerators of 2 or more) with denominators less than 24.

Name It!

How did you figure out the different fractional parts?

What strategies did you use to increase your number of points?

De-Briefing the Game

Vonnie: “5 blue candies are 1/3 of my whole.”

Colleen: “6 brown ones are 1/4 of my whole.”

Marilee: “8 red ones are 2/5 of my whole.”

Judy: “12 yellow candies are 3/4 of my whole.”

Ken: “15 orange candies are 3/5 of my whole.”

How can you find the number of items in the whole given any fractional part?

How much is the whole?