developing proportional reasoning through...
TRANSCRIPT
developing proportional reasoning
through manipulatives
Kevin Dykema
Mattawan (MI) Middle School
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
Email me to join
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
