1

Do they see the

multiplicative relationship?

Tom Mullen

Math Coach

East Side Community High School

Please do not duplicate or distribute without permission.

2

3

Think of a picture that this number represents.

3

3

Think of a picture that this number represents.

I bet many people thought of a picture like this—3 objects.

4

3

Think of a picture that this number represents.

I bet very few people thought of a picture like this.

If we move away from thinking of a number as representing a singe quantity

and instead think about the fact that numbers often represent the multiplicative

relationship between two quantities, this picture can be represented by the

number 3. There are 3 times as many apples as oranges or for every 1 orange,

there are 3 apples.

5

3

Think of a picture that this number represents.

6

2=

People are so used to thinking of numbers as quantities, that it is difficult to

think of numbers as representing a multiplicative relationship between

quantities. Students who have difficulty thinking this way have difficulty with

many crucial concepts throughout middle school and high school.

In this presentation, I will share some concepts that math teachers at East Side

Community High School have noticed give students difficulty because of their

limited understanding of fractions.

6

2

3

6

10

Which is greater?

.6 .49

49%60%

Which is greater 2/3 or 6/10?

Some students tend to think, 6 and 10 are greater than 2 and 3 so 6/10 is

obviously greater!

Understanding this isn’t true is much more important than the algorithms used

to perform operations with fractions.

Which is greater .6 or .49?

Students tend to look at the decimals as whole numbers: 49 is greater than 6, so

.49 is greater than .6. Students don’t always think in relation to 100 or 10.

Percents? No problem—back to a single quantity.

7

B

A

C

B

A

C

Which angle is greater?

Our 10th grade teacher and I were surprised to realize some students didn’t

really understand what an angle is.

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B

A

C

B

A

C

Which angle is greater?

m!ABC = 30°

m!ABC = 30°

These students tend to think that the longer the legs, the greater the angle.

They are focusing on a quantity, rather than a relationship.

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360°

330°

300°

270°

240°

210°

180°

150°

120°

90°

60°

m!ABC = 30°

B

A

C

30

360

Students who have not been taught to see the multiplicative relationship

between quantities, may not really understand what an angle is. This is

because, these students see 30 as an independent quantity. In fact, 30 is an

insignificant quantity when not considered in relation to 360.

10

MEANING OF SLOPE

How are the lines represented by these equations different?

y = 3

4x + 2

y = 3

5x + 2

Many students looking at these equations would not readily say that the line

represented by the top equation was steeper. Again, without seeing the

relationships between these numerators and denominators, students have

trouble visualizing what the lines represented by these equations look like.

Even though they can identify the slope of a line, they have to graph them to

see which is steeper.

11

!

circumference

diameter ! 3.14

Our students are exposed to the string and circle activity 2-3 times throughout

middle and high school and they can see the relationship between the

circumference and the diameter—the circumference is always a little more

than 3 times the diameter.

But this understanding goes out the window when 3.14 is brought into the mix.

Students grab onto the quantity and leave the ratio behind. They gravitate to

what they are comfortable with. Students who are not comfortable recognizing

the multiplicative relationship between quantities, never really understand !.

12

WHAT IS A LINEAR RELATIONSHIP?

x y

2.0 3.0

4.0 6.0

8.0 12.0

10.0 15.0

14.0 21.0

20

15

10

5

10 20

We have noticed that many students have trouble understanding the meaning

of “linear”.

Just looking at the table, many students would have trouble deciding if the

relationship is linear. If you are not comfortable recognizing the multiplicative

relationship between 2 quantities, you will have trouble seeing when different

pairs of quantities have the same relationship.

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What is special about exponential

growth or decay?

If students don’t truly understand what a linear relationship is, we have noticed

that they don’t grasp what makes exponential growth and decay special.

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It will take more than

shaded diagrams

1

4=

Many teachers teach the meaning of factions strictly using shaded

diagrams. These diagrams alone do not emphasize the multiplicative

relationship between the numerator and denominator. Because of this—to

many students—fractions are commonly seen as two independent quantities.

If this is the extent of student understanding of fractions, their grasp of more

advanced concepts, like the one’s mentioned in this presentation, will be

restricted.