developing proportional reasoning through...

developing proportional reasoning

through manipulatives

Kevin Dykema

Mattawan (MI) Middle School

kdykema@mattawanschools.org

Math for Real

Show how math is

used to solve a real-

world problem

Preferably from a

profession

300 words or less,

including several

problems

ONline book club

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kdykema@

mattawanschools.org

An ancient saying

I hear and I forget

I see and I remember

I do and I understand

I can’t Remember the formula

The definition of insanity

Doing the same thing over and over again and

expecting different results

a great quote

• “Without manipulatives, children are too often lost in a world of abstract symbols for which they have no concrete connection or comprehension.” Marilyn Burns

another great quote

• “If students cannot learn the way we teach them, then we must teach them the way they learn.”

• Dr. Kenneth Dunn from Queens college, author of several books on learning styles

implications for instruction

Concrete

Pictorial

Abstract

“Proportional reasoning is both the capstone of

elementary mathematics and the cornerstone of

higher mathematics.”

Lesh, Post, Behr, 1988

proportions and equivalent

fractions

Equivalent Fractions

Same Whole

Different parts of whole

Proportions

Different Whole

Same part of whole

NCTM Research Brief

Proportional Reasoning involves understanding

that

Equivalent ratios can be created by iterating

and/or partitioning into a composed unit

If one quantity in a ratio is multiplied or divided

by a factor, then the other quantity must be

multipled or divided by the same factor to

maintain the proportional relationship

Not just “Cross Multiply and Divide”!

Difficult for students who don’t understand what

is meant by a particular proportional situation or

why a given solution strategy works

Students often use more sophisticated reasoning

when not doing “cross multiply and divide”

Proportions

A company makes charms for bracelets. For

every 3 hearts, it makes 2 diamonds. If the

company makes 15 hearts, how many diamonds

does it make?

Modeling this Problem

• Use Cuisenaire rods to find the rods that have a

3:2 ratio.

• Build a train of 3 red rods and build a one-color

train using only 2 rods underneath the red train.

• Next, build a train of 9 red rods and build the

green train with the same length.

• Finally, build a train of 15 red rods and build the

corresponding green train.

sketch a model

3 = ?

2 6

Abstract Stage

Solve without sketching

Writing: What question do you ask yourself to

solve the prior problem?

Sequencing Problems

Step 1) Ability to do unit rate without “messiness”

I buy 3 books for $12. How much will 15 books

cost?

3 = 15

12 x

Step 2): Ability to do unit rate, but with

“messiness”

I buy 4 books for $9. How much will 20 books

cost?

4 = 20

9 x

Step 3)

I buy 4 books for $9. How many can I buy with

$63?

4 = x

9 63

Step 4)

I buy 6 books for $9. How many will 15 books

cost?

6 = 15

9 x

a new problem

The ratio of benches to trees in a park is 2:4. If

there are 12 trees, how many benches are there

in the park?

Let’s use Color Tiles to model this problem

grade 6 ccss critical areas

1. Connecting ratio and rate to whole number

multiplication and division and using concepts of

ratio and rate to solve problems.

grade 7 ccss critical areas

1. Developing understanding of and applying

proportional relationships

3. Solving problems involving scale drawings

Strawberry Picking

• Each spring, Paul and his family go to

Grandpa's farm to pick strawberries. Paul eats

2 strawberries for every 9 strawberries he puts

in his basket. If Paul ate 8 strawberries, how

many strawberries did he put in his basket?

Modeling this Problem

• Use one color of Base Ten units to model the

strawberries that Paul eats and a different color

of units to model the strawberries that Paul

picks.

• We could make a table to help organize our

work.

The table

x y

2 9

4 18

6 27

8 36

Why Model this Problem?

• It gives a visual approach to solving the

problem.

• It helps make an abstract problem more

concrete.

• In the future, students can visualize using the

blocks to solve similar problems.

Research shows that the systematic use of visual

representations and manipulatives may lead to statistically

significant or substantively important positive gains in math

achievement. (Pages 30-31)

The evidence indicates, in short, that manipulatives can provide

valuable support for student learning when teachers interact

over time with the students to help them build links between the

object, the symbol, and the mathematical idea both represent.

(Page 354)

Research Summary

Classroom Inventory p 31

A

C

R

Concrete

Abstract

Representational

Hands-On Learning Instructional Cycle

Hands-On Learning Instructional Cycle

When students are

exposed to hands-on

learning on a weekly

rather than a monthly

basis, they prove to

be 72% of a grade

level ahead in

mathematics (Page 27)

Impact on Student Performance

Classroom Inventory p 31

What’s your prediction?

Create a bag with 16 Color Tiles, some red, some

blue, and some green.

Have students take turns sampling the tiles in the

bag. Each time, draw 1 tile from the bag without

looking inside. Record the color of the tile, then return

it to the bag. Shake the bag to mix the tiles for the

next draw.

Continue sampling until you are ready to predict the

number of each color of tile in the bag

questions to ask

How did you decide when to stop sampling and

make your predictions?

Did you change your predictions at any time?

Why or why not?

How sure were you about your final predictions?

Why?

Next steps

Combine group results to get class totals.

Allow groups to change predictions.

Have groups check to see what was in each bag

final questions

How close were your original predictions to the

actual number of tiles of each color in the bag?

Did you change your predictions after examining

the class data? If so, why? Were your new

predictions more accurate than your original

predictions?

Do you think there is any connection between the

number of tiles sampled and the accuracy of a

prediction? Explain.

If the World Were a Village

Author: David J. Smith

Money and Possessions

• If all the money in the village were divided

equally, each person would have about $6200

per year. But in the global village, money isn't

divided equally.

• The richest 20 people each have more than

$9000 a year.

• The poorest 20 people each have less than $1 a

day.

• The other 60 people have something in

between.

• Use your Base Ten blocks to model the 20

people who are making more than $9000 a year.

• Construct a table relating the number of people

making more than $9000 and the number of

people in the village.

• Now, create a new village by combining your

original village with your neighbors and record

the data.

• Combine with a third person's original village

and record the data.

• Finally, combine with a fourth person's original

village and record the data.

• What type of relationship do we have here?

• Do you see the equivalent fractions also?

Nationalities

• Of the 100 people in the global village:

• 61 are from Asia

• 13 are from Africa

• 12 are from Europe

• 8 are from South America, Central America (including

Mexico) and the Caribbean

• 5 are from Canada and the United States

• 1 is from Oceania (an area that includes Australia, New

Zealand and the islands of the south, west, and central

Pacific)

Modeling this Problem

• Let's focus on Africa.

• Use a Base Ten flat to represent your village

and use units to represent the 13 from Africa.

• How many would be from Africa if you had two

villages?

• Combine your Base Ten flat with your neighbor's

flat.

is it a proportional

relationship?

A herring swims 3 kilometers in 30 minutes.

Another day, the herring swims 7 kilometers in 70

minutes. Is this a proportional relationship?

visual/ representational

Use centimeter grid paper to determine if the

relationship is proportional:

If 20 people are ahead of you in the lunch line, it

takes 12 minutes to get your lunch. If 30 people

are ahead of you, it takes 18 minutes. Is the

relationship proportional.

abstract stage

Explain how you make a graph to determine if

some data are in a proportional relationship.

constant of proportionality

Liam took his dog for a walk. In 4 minutes, he

had walked 2 blocks. In 10 minutes, he had

walked 5 blocks. If the number of blocks is

proportional to the number of minutes, what is

the constant of proportionality for the

relationship?

Equations of proportional

relationships

Kelly loves to meet new people. When she

moved to a new school, she decided to meet

three new people every two days. How many

people will she have met after 10 days? After 16

days? Write an equation for the number of

people Kelly will have met after x days.

Results from a Study on

Learning styles

In 1996

35-50% were auditory

35% were visual

15-30% were kinesthetic

In 2005

5-20% were auditory

Mathematically Proficient Students’

strands of knowledge

Strategic Competence: ability to problem solve

Adaptive Reasoning: ability to explain and justify

Conceptual Understanding: understand why idea is

important and how it connects

Procedural Fluency: knowing how to solve problems

efficiently

Productive Disposition: seeing math as worthwhile activity

National Research Council, 2001

Resources

Hands-on Standards series of books by

ETAhand2mind

Facebook- ETAhand2mind

Edweb.net- Implementing Common Core

Standards in Math

Kevin Dykemakdykema@mattawanschools.org