1. Proportional Reasoning
Equivalents

2. Integer Rods White 1 cm x 1 cm W Red 2 cm x 1 cm R Lime 3 cm x 1 cm L
Purple 4 cm x 1 cm P
Yellow 5 cm x 1 cm Y
Green 6 cm x 1 cm G
Black 7 cm x 1 cm K
Brown 8 cm x 1 cm N
Blue 9 cm x 1 cm B
Orange cm x 1 cm E

3. Equivalent Rods Equivalents
How many combinations can be made of each type of rod?
Is there a pattern?

4. How many different ways are there to make W?

5. How many different ways are there to make R?

6. How many different ways are there to make L?

7. How many different ways are there to make P?

8. What is the pattern?
How many different ways are there to make Y?
Rods
# of Units in Rod
# of Equivalents W 1 R 2 L 3 4 P 8
Generalization n ?

9. How many different ways are there to make G?
What is the pattern?
Number of Equivalents = 2(n-1), where n is the number of units in a rod
Should you assign your students to find all of the equivalents for K or N?

10. On a test or quiz you will have to give a semi-concrete model of the rods
It is important that your semi-concrete models be as accurate as you can make them
The letter representing the color of the rod should be placed in each rod’s representation once and only once – see class notes

11. Equivalent Fractions How do we represent fractions using integer rods?
Part to whole
Whole changes as necessary to make equivalents
A train is two rods put together
We will ALWAYS use the least number of rods possible to make a representation
Do NOT draw more lines on representations than necessary

12. One half is W over R:
One half is R over P:
One half is ? over ?:
How many half equivalents are there up to an EE train?

13. One third is W over L:
One third is R over G:
One third is ? over ?:
How many third equivalents are there up to an EE train?

14. One fourth is W over P:
One fourth is R over N:
One fourth is ? over ?:
How many fourth equivalents are there up to an EE train?

15. What rational number does this represent?

16. Other Manipulatives
We have just looked at two manipulative that can be used to model rational numbers, there are MANY others
Check out some other electronic manipulative listed under and