Proportional Reasoning

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 10 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 1
R 2 2
L 3 4
P 4 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 rods 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?
• 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 http//ejad.best.vwh.net/java/pattern
  s/patterns_j.shtml and http//nlvm.usu.edu/en/nav/
  topic_t_1.html