Model Drawing
For Problem Solving
Mary Flahive
Wendi Kral
What is Singapore Math Model Drawing?
•Base-10 math program used in Singapore
•Three part program
Number sense is the overall
understanding of a number.
Mental math helps to develop
this.
Place Value is a student’s
understanding of a digit’s
position in a number.
Model Drawing is a seven-step
visual method of turning a word
problem into a diagram with
unit bars that represent values.
History of Singapore Math
Another nation that took the NCTM Standards and
other research on problem-based approaches seriously
is Singapore. After its independence in 1965,
Singapore realized that without any natural
resources it would have to rely on human capital
for success, so they embarked on an effort to develop
a highly educated citizenry. Various education reforms
were initiated and in 1980 the Curriculum Development
Institute was established, which developed the Primary
Mathematics program. This program was based on the
concrete, pictorial, abstract approach. This
approach, founded on the work of renowned cognitive
American psychologist Jerome Bruner, encourages
mathematical problem solving, thinking and
communication.
From : The Daily Riff
Bill Jackson Math Helping Teacher
Scarsdale Public Schools
Singapore math is not "drill and kill," facts memorization, or
rote learning of procedures.
In Singapore math, students learn their addition, subtraction
and multiplication facts and eventually memorize them. But they
also learn the structure and patterns behind the facts so if
they forget them, they are able to reconstruct them in their
minds. From early grades, children learn how to compose and
decompose numbers and manipulate them in useful ways in
order to calculate mentally.
Singapore math is about students solving problems, thinking
deeply, sharing their ideas, and learning from one another.
Conceptual, procedural and factual understanding is developed
through problem solving and carefully structured practice and
as a result, students learn how to think deeply and appreciate
mathematics.
In 1983, prior to implementing this
curriculum, Singapore ranked 17
out of 26 countries tested in eighth
grade Mathematics.
Just twelve years later, in 1995,
Singapore ranked number one out
of 41 countries tested at that level
and remained number one in both
1999 and 2003.
copyright 2007 Dr. Richard Bisk
PROBLEM SOLVING WITH MODEL
DRAWING
The model drawing approach takes students from the
concrete to the abstract stage via an intermediary pictorial
stage.
Students create bars and break them down into “units.” The
units create a bridge to the concept of an “unknown” quantity
that must be found.
Students can learn to use this strategy in the primary grades
and continue with it through the middle grades.
copyright 2007 Dr. Richard Bisk
Benefits of Model Drawing
Students have one strategy for solving every problem.
Students have a visual to associate with numbers
that can be abstract.
Students learn to translate the English into math and then back
into English.
Students start to see the relationship behind numerical
values.
Model Drawing is NOT the answer
to every problem.
Courtney starts with 12 birdhouses. She makes three new birdhouses each
week. Which pattern shows the number of birdhouses Courtney has at the
end of each week?
Give one way that a cone and a cylinder are alike. NO
One month Tony’s puppy grew 7/8 of an inch. The next month his puppy
grew 5/8 of an inch. How many inches did Tony’s puppy grow in two months? Y
Which spinner has a probability of 0 for landing on a star, ?
What transformation changed shape 1 to shape 2?
Mrs. Thomas gave the store clerk $25.00 for a pair of jeans. She received
$2.88 back in change. What was the price of the jeans?
Seven Steps for Model Drawing
1.Read the problem.
2.Identify the variables (the who and the what)..
3.Draw a unit bar or bars.
4.Chunk information by rereading the problem
one sentence at a time, and adjust the unit bar or
bars to match the information.
5.Decide on your question mark, and draw it in
the appropriate place.
6.Work the computation.
7.Write a complete and grammatically correct
sentence to answer your question mark.
Step 1 - Read The Problem
Mr. Hobart sells 6 pans of brownies every day. He makes $10
per pan. How much money does Mr. Hobart make in a day?
•We can write a problem on the board and ask the whole
class to read it aloud together.
•We can ask each student to copy a problem from the
board and read it to himself or herself silently.
•We can ask each student to read the problem from his or
her own paper in a low voice.
•We can alternate calling on different students to read
problems we put on the board.
•We can read a problem to students.
Step 2 Identify the Variables
Who does or has what?
How does that relate to what the other person or
people do or have?
Mr. Hobart sells 6 pans of brownies every day. He
makes $10 per pan. How much money does Mr.
Hobart make in a day?
Step 3 Drawing Unit Bars
Unit bars provide the visual.
In early grade students can draw individual unit s, like
fish, for unit bars
We would like students to draw squares or rectangles
of the same size for each unit bar.
Mary’s Money
Wendi’s brownies
Sue’s brownies
Step 4 Reread Each Sentence and
Adjust Unit Bars
Chunk information to make it more manageable.
Adjust the unit bar or bars to match the
information in the problem.
This is where it is easy to make a mistake, work
slowly.
Mr. Hobart sells 6 pans of brownies every day. He
makes $10 per pan. How much money does Mr.
Hobart make in a day?
The first sentence tells us he sold 6 pans of brownies every
day.
Next sentence tells us he makes $10 per pan.
You want to write appropriate quantities either right above or
below the bars in grades 3 and up. You can write inside the bars
in lower grades.
Step 5 Adding the Question Mark
We add our question mark to indicate what we need to find out and
where. Sometimes it surrounds multiple unit bars with a big bracket,
and other times, it just relates to one part of one unit bar with a small
bracket or no bracket at all.
Step 6 Work the Computation
Computations are done horizontally to match how we wrote it out in our
model drawing.
Step 7 Write a Sentence
The sentence has to be complete.
The sentence has to address the who and the
what and the question mark.
The sentence should be written underneath
all the work.
Mr. Hobart makes $60 in a day.
Addition Problems-Golden Rules
•Draw unit bars on the smaller side so you can add to them
•You can also use pattern blocks for younger students to represent
the units.
Sean has 2 plastic cars. He also has 5
wooden cars. How many cars does Sean
have altogether?
Try It!
Does your Model Drawing look something like this?
Carlos read 221 pages of his book over the
weekend. Jasmine read 198 pages. How many
pages did Carlos and Jasmine read altogether?
Carlos’ pages
221
?
Jasmine’s pages
198
Jesse had $3.00 more than Clinton. If Clinton had
$10.00, how much money did they have altogether?
Now we have two variables. What should we do?
Try This One
Penny ate 6 ice cream cones more than Jeff did over
the weekend. If Jeff ate 8 ice cream cones, how many
ice cream cones did they eat altogether?
Three Whos
John had 10 pencils, Andrew had 9 pencils, and Calvin had 5
pencils. They decided they'd put their pencils together and share
them equally. How many pencils did each student get?
Try It
Ashley has 6 kittens, Dara has 3 kittens, and Noel has 3
kittens. They put their kittens in the same room to play. At
the end of the day, they want to split the kittens equally.
How many kittens will each child go home with?
Subtraction Problems-Golden Rules
Most subtraction problems require you to draw a longer
unit bar to begin with.
It's really helpful to identify the segment of the unit you're
subtracting and draw a diagonal slash (sometimes
called a vertical slash even though it's not exactly
vertical) through the value. This is a great visual
reminder.
We usually place numerical values outside the unit bars
with subtraction because we manipulate the inside of the
units with sections and slashes.
Nathan had $27.00 to buy gifts for his family. If he spent
$9.00 on a gift for his brother, how much money did he
have left to spend on the rest of his family?
Nathan’s Money
?
$9.00
$27.00
Diane had 106 stamps in her collection. She gave 12 away to a friend who
was starting a collection. How many stamps did Diane have left?
Ryan and Chris started out with an equal amount of baseball cards. Ryan lost
15 cards, and Chris collected another 45 cards. How many more cards did
Chris have in the end?
Ryan’s baseball cards
15
?
Chris’ baseball cards
45
Pablo and Nadine each had an equal number of chips. Pablo lost 12
chips, and Nadine got another 20 chips. How many more chips did Nadine
have in the end?
Multistep Problems
The pep club made 425 buttons to sell on Friday. The club sold 75 more
buttons in the morning than they did in the afternoon. If all the buttons were
sold, how many buttons did the pep club sell in the morning?
2 units = 350
1 unit = 175
We need morning buttons
175 + 75 = 250
The Pep Club sold 250 buttons in the morning.
Your Turn
Belvedere's Chocolates made 350 truffles for
a wedding. They gave away 46 more truffles
at the sign-in table than they did at the dessert
buffet. If all the truffles were passed out, how
many truffles did Belvedere's give away at the
sign-in table?
Now we will practice problems that
come from the textbook and the
Exeter math sheets. Part II will be
working on higher level problems,
fractions, and percents.