Transcript Model Drawing

Guest Presenters:
Susan Gebhart and Julie Dill
Model drawing is a strategy first used in Singapore to
solve word problems. Bar models or tape diagrams, as
they are sometimes called, are visual models or graphic
organizers of the information that is in the word problem.
Model drawing involves “translating” the words in the
problem to a visual model.
Model drawing is a good match for many of the Math
Practices described in CCSS (Common Core State
Standards) and can be used by students from grade 1
through algebra.
Model drawing makes a strong connection to algebra.
In model drawing the rectangular bars represent the
relationship between quantities in the problem. The
question mark indicates the unknown. In algebra these
same unknowns would be represented with a variable
like “x”. Students are engaged in algebraic thinking
even before they are ready to handle formal algebra.
What Is Model Drawing?
Jim has 2 Lego sets and Lin has
5 Lego sets. How many sets do
they have altogether?
Jim’s Lego sets
Lin’s Lego sets
2
5
?
2+5=7
Jim and Lin have 7 Lego sets altogether.
Give It a Try
Betty, Jill, and Chris shared $78.75. Betty
had 1/2 as much money as Jill, and Chris
had 4 times as much money as Betty.
How much money did Jill have?
An Example of Model Drawing
Here’s the model…
what’s the problem?
Progression of Solving Word Problems
Concrete Objects
3
Picture
7
4
There were two trees.
One had 3 apples and the
other had 4 apples.
Number Bond
Story
(oral or written)
Number Statement
Part-Whole
Shanice got 5 cards on
Monday and 3 cards on
Tuesday. How many cards
does Shanice have now?
Monday’s
Cards
Tuesday’s
Cards
5
?
3
Shanice has _____ cards now.
?
5
?
5
3
Monday’s
Cards
3
Tuesday’s
Cards
80% of word
problems for
grades 1-6
can be solved
using bar
models!
You still need to
teach your students
other heuristics so
after they read a
problem they are
equipped with the
skills to choose
which strategy
would be most
efficient for solving
that particular word
problem.
Step-by-Step Model Drawing Method
1.Read the problem.
2.Rewrite the question in the problem as a
complete sentence leaving space for the answer.
3.Identify the “who” and the “what.”
4.Draw a unit bar to model each variable.
5.Chunk the problem and adjust your unit bars to
match your information. Fill in your question
mark.
6.Work your computation.
7.Write the answer in your sentence.
Give students “Attack Time”.
They need to read the problem,
think to themselves,
and talk or think about where they will begin.
Part – Whole
James got 158 cards on Monday
and 331 cards on Tuesday. How
many cards does James have now?
1
James got 158 cards on Monday and 331 cards on Tuesday. How
many cards does James have now?
James has _______ cards now.
Monday’s
cards
Tuesday’s
cards
158
331
?
James got 158 cards on Monday and 331 cards on Tuesday. How
many cards does James have now?
489 cards now.
James has _______
Monday’s cards
Tuesday’s cards
158
331
?
158
?
331
Part – Whole
Giles read 24 books in the first part
of the year. If he read 59 books in
the entire school year, how many
books did he read in the second
part of the year?
2
Giles read 24 books in the first part of
the year. If he read 59 books in the
entire school year, how many books did
he read in the second part of the year?
first
part
second
part
24
A. 59 – 24 = 35
P. 35 + 24 = 59
59
?
24
59
Giles read 35 books in the second
part of the year.
?
Part – Whole
Calvin had $575 in his savings
account. He bought a bow for $125
and a package of arrows for $90.
How much money did he have left in
his savings account?
3
Calvin had $575 in his savings account.
He bought a bow for $125 and a
package of arrows for $90. How much
money did he have left in his savings
account?
bow
arrows
125
90
A. $125 + $90 = $215
$575 – $215 = $360
P. $125 + $90 + $360 = $575
?
125
575
savings
90
575
Calvin had $360 left in his savings
account.
?
Part – Whole
Deidra was in a 72-mile bike race
over the course of 3 days. How
many miles would she need to ride
the third day to finish the 72-mile
bike race if she rode 23 miles the
first day and 29 miles the second
day?
4
Deidra was in a 72-mile bike race over
the course of 3 days. How many miles
would she need to ride the third day to
finish the 72-mile bike race if she rode
23 miles the first day and 29 miles the
second day?
miles
first day
second
day
23
29
?
A. 23 + 29 = 52
72 – 52 = 20
P. 23 + 29 + 20 = 72
29
23
29
72
72
Deidra will need to ride 20 miles
on the third day.
?
Part – Whole
Jim had 901 baseball cards. He
gave 283 to Sue and then bought
155 new cards. How many cards
did Jim end up with?
5
Jim had 901 baseball cards. He gave
283 to Sue and then bought 155 new
cards. How many cards did Jim end up
with?
baseball
cards
?
to Sue
new
cards
283
29
155
901
Jim ended up with 773 cards.
A. 901 + 155 = 1,056
1,056 – 283 = 773
P. 901 + 155 = 1,056
1,056 – 283 = 773
Part – Whole
Oliver decorated 70 cupcakes for
dessert. 45 cupcakes were eaten.
How many cupcakes were left?
7
Oliver decorated 70 cupcakes for
dessert. 45 cupcakes were eaten. How
many cupcakes were left?
A. 70 – 45 = 25
P. 25 + 45 = 70
eaten
cupcakes
45
?
45
70
70
There were 25 cupcakes left.
?
Differentiating
Carl has ____ candies. He ate ____ candies.
How many candies does he have left?
(10, 8)
2
(25, 17)
(6, 3 )
#
#
?
Additive Comparison
Sharon had 5 stuffed animals. Lola
had 3 more stuffed animals than
Sharon. The girls put their animals
in one basket. How many animals
are in the basket?
8
Sharon had 5 stuffed animals. Lola had
3 more stuffed animals than Sharon.
The girls put their animals in one
basket. How many animals are in the
basket?
Sharon’s
animals
5
Lola’s
animals
5
3
A. 5 + 5 + 3 = 13
P. 13 – 10 = 3
?
8
?
There are 13 animals in the basket.
5
Additive Comparison
Josh and Aaron went to collect
golf balls. Together they found 11
golf balls. Josh found 3 more balls
than Aaron. How many golf balls
did Aaron find?
9
Josh and Aaron went to collect golf
balls. Together they found 11 golf balls.
Josh found 3 more balls than Aaron.
How many golf balls did Aaron find?
Aaron’s
golf balls
?
Josh’s
golf balls
?
3
A. 11- 3 = 8
8÷2=4
P. 4 + 4 + 3 = 11
11
Aaron found 4 golf balls.
11
?+3
?
Additive Comparison
Justin has 18 fewer swimming
ribbons than Nathan. Altogether,
they have 110 swimming ribbons.
How many swimming ribbons does
Justin have?
10
Justin has 18 fewer swimming ribbons
than Nathan. Altogether, they have 110
swimming ribbons. How many
swimming ribbons does Justin have?
Justin’s
ribbons
Nathan’s
ribbons
A. 110 – 18 = 92
92 ÷ 2 = 46
P. 46 + 18 = 64
64 + 46 = 110
46 ?
110
18
Justin has 46 swimming ribbons.
110
18 + ?
?
Additive Comparison
The third grade class raised $350
for their field trip. The girls raised
$30 more than the boys. How
much did the girls raise for the
field trip?
11
The third grade class raised $350 for
their field trip. The girls raised $30
more than the boys. How much did the
girls raise for the field trip?
?
Girl’s $
raised
Boy’s $
raised
46
A. $350 - $30 = $320
$320 ÷ 2 = $160
$160 + $30 = $190
P. $160 + $160 + $30 = $350
30
$350
350
30 + ?
The girls raised $190 for the field trip.
?
Additive Comparison
Ryan and Sharie had the same
amount of money. Ryan lost $10
while his friend Sharie collected
another $25. How much more
money does Sharie have than
Ryan?
12
Ryan and Sharie had the same amount
of money. Ryan lost $10 while his friend
Sharie collected another $25. How
much more money does Sharie have
than Ryan.
A. $10 + $25 = $35
Ryan’s $
46
10
25
Sharie’s $
?
Sharie has $35 more than Ryan.
Additive Comparison
Blaine is 8 years older than Jade
and 2 years younger than Ramel.
The total of their ages is 63 years.
How old is Ramel?
13
Blaine is 8 years older than Jade and 2
years younger than Ramel. The total of
their ages is 63 years. How old is
Ramel?
A. 8 + 8 + 2 = 18
63 – 18 = 45
3u = 45
u = 15
15 + 8 + 2 = 25
P. $125 + $90 + $360 = $575
Blaine’s
age
8
Jade’s
age
Ramel’s
age
?+8
63
8
2
?
?
63
?+8+2
Ramel is 25 years old.
Multiplication
Ling put 3 photos on each page of
her album. If there were 6 pages,
how many photos did Ling put in
her album?
14
Ling put 3 photos on each page of her
album. If there were 6 pages, how
many photos did Ling put in her album?
A. 6 x 3 = 18
P. 3 + 3 + 3 + 3 + 3 + 3 = 18
Ling’s
photos
3
3
3
3
3
3
?
Ling put 18 photos in her album.
Multiplication
Luis ran 2 miles each day during
the month of September. How
many total miles did Luis run in
September?
15
Luis ran 2 miles each day during the
month of September. How many total
miles did Luis run in September?
miles run
2
2
X 30
Luis ran 60 miles in September.
?
A. 30 x 2 = 60
P. 30 + 30 = 60
Do you really want
your students to write
“2” thirty times?
Is that working
efficiently????
Division
Devi saved $40 in 5 weeks. She
saved the same amount each
week. How much did Devi save
each week?
16
Devi saved $40 in 5 weeks. She saved
the same amount each week. How
much did Devi save each week?
A. $40 ÷ 5 = $8
P. $8 x 5 = $40
week 1
money
saved
each
week
week 2
week 3
week 4
week 5
?
40
Devi saved $8 each week.
Multiplicative Comparisons
Deana had 3 times as many books as
Hilary. If they had 16 books
altogether, how many books did
Deana have?
17
Deana had 3 times as many books as
Hilary. If they had 16 books altogether,
how many books did Deana have?
A. 16 ÷ 4 = 4
3 x 4 = 12
16
Hillary’s
books
Deanna’s
P. 4 + 4 + 4 = 12
Deanna’s
Deanna’s
46
16
?
Deanna had 12 books.
?
Multiplicative Comparisons
Wyatt had 4 times as many stamps in
his collection as Shannon. If they had
45 stamps altogether, how many
more stamps did Wyatt have than
Shannon?
18
Wyatt had 4 times as many stamps in
his collection as Shannon did. If they
had 45 stamps altogether, how many
more stamps did Wyatt have than
Shannon?
A. 45 ÷ 5 = 9
9 x 3 = 27
Wyatt’s
stamps
Shannon’s
stamps
46
45
?
Wyatt had 27 more stamps than Shannon.
P. 5 x 9 = 45
3 x 9 = 27
Choose one of the models and write
a problem to go with it.
Scoring Model Drawing
 Reads the problem.
______ (1)
 Rewrites the question in the problem as a complete
sentence, leaving space for the answer.
______ (1)
 Identifies the “Who” and the “What.”
______ (1)
 Draws a unit bar for each variable.
______ (1)
 Chunks the problem and adjusts the unit bars
to match the information. Fills in the question mark. ______ (3)
 Computes and solves problem correctly.
______ (2)
 Writes the answer in the sentence.
______ (1)
Total
______ (10)
Multiplication Bar Modeling
Solve Length Word Problems Using a Bar Model
Solve Comparison Problems Using Bar Models
Interpret Division as an Unknown Factor Problem Using a Bar Model
Online
and now
also an
app!
Singapore Math Bar Models App from Houghton
Mifflin Harcourt
Evaluations
Enjoy your lunch!