1. Building the Foundation to Algebra
Rational Numbers

2. 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.

3. Hexagon Fractions

4. 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.
Partner activity
Solutions: 7 (if other pieces); 8 if allow hexagon to cover a hexagon
2 trapezoids: 1 = ½ + ½
3 blue rhombus: 1 = 1/3 + 1/3 + 1/3
6 triangles: 1 = 1/6 + 1/6 + 1/6 + 1/6 +1/6 + 1/6
Trapezoid and 3 triangles: 1 = ½ + 1/6 + 1/6 + 1/6
Trapezoid, rhombus, and triangle: 1 + ½ + 1/3 + 1/6
2 blue rhombuses and 2 triangles: 1 = 1/3 + 1/3 + 1/6 + 1/6
Blue rhombus and 4 triangles: 1 = 1/3 + 1/6 + 1/6 + 1/6 + 1/6
May need to prompt participants to create combinations of different shapes. People typically cover the hexagon using only one shape first.
Be sure to take time to discuss how participants knew they found all the ways.

5. Hexagon Fractions 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
Hopefully participants will make different shapes with their three hexagons.
Ask what shapes were used as wholes.
Point out the different shapes of wholes.
If there are no differences, ask whether it is possible to make a different shape with three hexagons?
Do the hexagons have to be touching to form a single region?
Were the fraction names of the different pattern block pieces the same or different?

6. The Large Hexagon Use the large hexagon shape as the whole. (The ONE)
Determine what fractional part each pattern block shape represents:
Hexagon
Trapezoid
Rhombus
Triangle
One of the challenges of this larger shape is that it cannot be covered completely by hexagons. Participants will have to determine the value of the hexagon block based on the values of the other pattern blocks. Be sure to ask how participants determined the fraction for the hexagon.
Let participants realize this added constraint and grapple with the mathematics.
Another important idea is that the sum of the all pieces is one. One way to get at this is to ask:
What could you do to check that you labeled all your pieces correctly?

7. How is this Possible?
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?
The key idea is that the wholes are different.
This is an example of where the same shape represents different fractional parts because the wholes are different.
Names of fractional parts depend on the relationship between the parts and the whole.
It is important to emphasize that fractions can be meaningless unless one thinks of them in reference to the whole. Half a minute is different from half an hour, etc.
It is extremely important that students have experiences in which the whole varies.
Explicitly ask about the relationship between the fraction and the size of the whole, e.g.,
In the first activity, the trapezoid was ½. What was it in triple hexagons? What was it in the large hexagon? Can you predict what it would be in a whole that was equivalent to 6 hexagons? What is the pattern? Why?
The goal is to get participants to articulate the relationship between the scale factor relating the wholes and the fractional part, e.g., When the size of the whole is increased by a factor of 3, the fraction represented by the same sized piece is 1/3 the original fraction (e.g., the piece that was ½ becomes 1/6.)

8. 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?
Again, the key idea is that the wholes are different and names of fractional parts depend on the relationship between the parts and the whole.
This is an example of how students can develop misconceptions if they do not pay attention to the whole in naming and comparing fractions.

9. Making Connections
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?

10. 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?

11. From Parts to Wholes

12. Flight Problem
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?
ANSWER I_________________________________I________________________________I ↑ AWAKE (1/2) ↑ SLEPT (1/3) ↑ ↑ PGH FELL AWOKE(1/6) SF ASLEEP 1/2 + 1/3 + 1/6 = 1 The mathematical challenge lies in the changing wholes: First the “whole” is the trip from Pittsburgh to San Francisco and “I fell asleep after half the trip”. Then the “whole” changes when the statement says, “When I awoke, I still had to travel half the distance that I traveled while sleeping”. Now the “whole” is the distance I was sleeping. In other words within the final ½ of the trip I slept and woke up. The final distance I was awake must be half of the distance I was asleep. The whole then reverts back to the whole trip to determine the fractional part for each of the three segments-awake-asleep-awake.

13. From Parts to Wholes What is the whole if… the rhombus is 1/2?
the trapezoid is 3/4?
the hexagon is 2/3?
the hexagon is 3/5?
the rhombus is 2/9?
Traditionally, we have only shown students the whole and asked them what fractional part a certain piece is. This fractional reasoning or “fraction sense” is more challenging because we don’t often ask students to reverse their thinking. The first three questions use unit fractions so their solutions should be straightforward—simply iterating the unit fraction.
The last 4 should be more challenging. Probe for participants’ strategies.
End by asking for a general solution, I.e., “How could you find the whole, given any fractional part?”

14. Pattern Block Puzzles
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.
Sample solutions:
For #1, a very
common incorrect
solution is a green
triangle on a
red trapezoid.

15. 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?
Have participants share their solutions-are everyone’s the same? If they are different-How are they different?
Have participants share their strategies for solving the puzzles in the larger group. Make sure everyone understands the thinking of others. Asking participants to build larger models with the same fractional parts of the original puzzle is also challenging.
Insist on similar shapes, not any triangle, parallelogram or trapezoid will do.The larger versions must be mathematically similar--same shape, proportional dimensions--different size. Mathematical similarity may need some definition/clarification; e.g., that all rectangles are not similar shapes may be a new idea for some participants.
Investigate the larger shapes and how they relate to the original. Are they twice as large? The sides should be twice the length but the actual shape will be 4 times as large in area. For example, a shape that is 3 times the side length would be 9 times as large in area.

16. Disappearing Cookies

17. 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?
Answer: 30 cookies
One of the goals of this problem is to introduce/remind participants about working backwards as a problem solving strategy.
Working back:
Penny’s one cookie is 1/3 of the cookies on the plate when Payton took his 2/3. So there were 3 cookies on the plate when Payton took 2/3.
The 3 cookies are ½ the cookies on the plate when Polly took his ½. So there were 6 cookies when Polly took ½.
The 6 cookies are what was left after Pansy took his 2/5. So 6 is 3/5 of what was there. If 6 is 3/5, then there were 10 cookies when Pansy took 2/5.
The 10 cookies are what was left after Porter took his 1/3. So 10 is 2/3 of the cookies. There were 15 cookies when Porter took 1/3.
The 15 cookies are what was left after Paula took 3/8; so 15 is 5/8 of the cookies. So there were 24 cookies when Paula took his 3/8.
The 24 cookies are what was left after Peggy took 1/5; so 24 is 4/5 of the total amount. So, Patty Peterson baked 30 cookies.

18. Fractional Parts of Sets

19. 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
A central mathematical idea is that non-unit fractions are multiples of unit fractions. I.e., If I know 1/3 of a number is 6, then I can find 2/3 of that number by multiplying 2 x 6, because 2/3 = 2 x 1/3.
How does knowing the number of candies in a unit fraction help you figure out the number of candies in other fractions?

20. Chocolate 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?
Answers:
2/3 of 15 = 10
5/7 of 21 = 15
4/9 of 27 = 12
One strategy is to find the unit fraction, then multiply that by the numerator of the fraction.
Be sure to probe them about why this works.
The second set of problems is to get participants to realize that fractional parts of sets can be fractions—they don’t have to be whole numbers.
13½
61/3
122/3
Same strategy applies.
Describe a method for finding the fractional part of any set.

21. 5 of 36 = 20 9 Chocolate Fractions
Discussion should make the mathematics explicit:
9 “guzzinta” 36 produces the # of objects in the unit fraction.
The multiplication is simply multiplying the # of objects in the unit fraction by the number of unit fractions you have.

22. Name It! 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.

23. Name It!
Directions:
Close your eyes and count out 24 candies. The 24 candies are your “Whole”.
Write as many fraction statements as you can to describe your Whole.
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.

24. De-Briefing the Game
How did you figure out the different fractional parts?
What strategies did you use to increase your number of points?

25. How much is the whole? 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?
The wholes are:
Vonnie: 15 M&Ms
Colleen: 24 M&Ms
Marilee: 20 M&Ms
Judy: 16 M&Ms (Look for participants who do 12 ÷ 4 instead of 12 ÷ 3. ) Be sure to ask why you divided by 3 instead of 4. She has 4ths doesn’t she?
Ken: 25 M&Ms (Again, look for participants who do 15 ÷ 5 instead of 15 ÷ 3.) Again, ask why you divided by 3 instead of 5?