Proportional Reasoning

Equivalents

Integer Rods

WhiteWhite 1 cm x 1 cm WW 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

Equivalent Rods

Equivalents How many combinations can be made of each

type of rod? Is there a pattern?

How many different ways are there to make W?

How many different ways are there to make R?

How many different ways are there to make L?

How many different ways are there to make P?

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 ?

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?

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

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

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?

R

W

P

R

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?

L

W

G

R

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?

P

W

N

R

What rational number does this represent?

What rational number does this represent?

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/patterns/patterns_j.shtml and http://nlvm.usu.edu/en/nav/topic_t_1.html