Download Connecting Bar Models to Algebra

Connecting Bar Models to
Algebra
Richard Bisk
Professor of Mathematics
[email protected]
www.worcester.edu/smip
sites.google.com/site/singmathproject/
To find presentation slides on the web
Search for: singmathproject
Sometimes “Bisk math” works too.
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Common Core Mathematical Practices
• MP1 Make sense of problems and
persevere in solving them.
• MP2 Reason abstractly and
quantitatively.
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C→P→A
Concrete:
Pictorial:
Abstract:
?
││││││││
8
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Abstraction
• Gives mathematics its power.
• But abstraction without
understanding??
• Leads to confusion.
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Is this a hard problem?
Than spent 3/5 of his money in the first
week and 1/3 of the remainder in the
second week. He spent $110 altogether.
How much did he have left?
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MODEL DRAWING
and
C→P→A
The model drawing approach takes students from
the concrete to the abstract stage via an
intermediary pictorial stage.
Many complex problems are easier to solve with
algebra….
if you understand the algebra!
Outline of Presentation
• Examples starting at grade 1
• Connections to algebra
• Problems for you to try
•Handout has many of the examples.
Addition
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Connects set and measurement models
0
5
10
15
3
?
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Subtraction
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Operations and Algebra
• CCSS.Math.Content.1.OA.B.4
Understand subtraction as an
unknown-addend problem. For
example, subtract 10 - 8 by finding
the number that makes 10 when
added to 8.
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13 + x = 7
13+x=17
x + 11= 14
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Multiplication
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Division
Suppose Trish scored x goals.
Then Shawn scores 3x goals
x + 3x = 36
4x = 36
x=9
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Example (grade 4): Josh spent 2/5 of his
money on a present for his Dad. The present
cost $12. How much money did he have at first?
Josh’s
Money
?
12
2 units = 12
1 unit = 6
5 units = 30
Josh started with $30.
Josh’s
Money
?
12
2 units = 12
2
x  12
5
1 unit = 6
1
x6
5
5 units = 30.
x  30
Suppose Josh started with x dollars.
Example (grade 5): Bob and Jill share $80 in
the ratio 3:2. How much money did Jill get?
Bob
$80
Jill
?
5 units = 80
2 units =32
1 unit = 16
Jill gets $32.
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Bob and Jill share $80 in the ratio 3:2. How
much money did Jill get?
Suppose Bob gets 3x dollars and
Jill gets 2x dollars.
Then
3x + 2x = 80
5x = 80
x = 16
Jill’s share is 3x = 48
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Three boys, Juan, Seth and Jared shared a number of
stamps in the ratio of 3:5:7. If Seth received 45 stamps,
how many more stamps did Jared receive than Juan?
?
Juan
Seth
45
Jared
Can you finish this?
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''Of the people in attendance at a recent baseball game, one-third had grandstand tickets,
one-fourth had bleacher tickets, and the remaining 11,250 people in attendance had
other tickets. What was the total number of people in attendance at the game?"
Seats
G G
G
G
B
B
B
?
11,250
5 units = 11,250
1 unit =
2,250
12 units = 27,000
The total attendance was 27,000.
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''Of the people in attendance at a recent baseball game, one-third had grandstand tickets,
one-fourth had bleacher tickets, and the remaining 11,250 people in attendance had
other tickets. What was the total number of people in attendance at the game?"
Suppose x people attended the game.
Then
x
1
1
x
x  11, 250
3
4
7
x
x  11, 250
12
5
x 11, 250
12
1
x  2250
12
x  27, 000
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Than spent 3/5 of his money in the
first week and 1/3 of the remainder in
the second week. He spent $110
altogether. How much did he have
left?
Than spent 3/5 of his money in the first week and 1/3 of the remainder
in the second week. He spent $110 altogether. How much did he have
left?
Money
Remainder
110
?
Money
11 units = $110
4 units = $40
1 unit = $10
He has $40 left.
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Than spent 3/5 of his money in the first week and 1/3 of the remainder in
the second week. He spent $110 altogether. How much did he have left?
Shall we do the algebraic solution?
Suppose Than started with x dollars.
The first week, he spent (3/5)x and had (2/5)x remaining.
The second week he spent (1/3) of the remaining money or (1/3)(2/5)x.
Altogether he spent (3/5)x + (1/3)(2/5)x.
If we simplify this we get (11/15)x was spent altogether.
Now we need to solve (11/15)x = 110.
We get (1/15)x = 10.
So x = 150.
He started with $150 and spent $110. So he had $40 left.
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Nick spent $39 on dinner. This was 26% of
his paycheck. How much was his paycheck?
100%
Nick’s pay
?
26% (39)
26% → 39
Nick’s paycheck was
$150
1% → 39/26
100% → 100 x (39/26)
= $150
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Nick spent $39 on dinner. This was 26% of
his paycheck. How much was his paycheck?
100%
Nick’s pay
?
26% (39)
26% of x = 39
26% → 39
0.26x =39
1% → 39/26
100% → 100 x (39/26)
= $150
x = 39/.26 = 150
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Problems to Try
1. Ann has 7 more books than Kim. Together
they have 33 books. How many books does
Kim have?
2. Ribbon A is twice as long as Ribbon B.
Ribbon C is twice as long as Ribbon A. The
total length of the three ribbons is 224 cm.
How long is Ribbon C.
(Adapted from “Primary Math 3 – Challenging
Word Problems”)
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3. On a recent math exam, 1/5 of the grades
were A’s and 1/3 were B’s. There were 6
more B’s than A’s. How many students
took the exam?
4. After spending 2/5 of his allowance on a
shirt, Oscar still has $27 left. How much
was his allowance?
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After spending 2/5 of his allowance on a shirt, Oscar still has
$27 left. How much was his allowance?
Oscar’s
Money
?
27
3 units = 27
1 unit = 9
5 units = 45
Oscar started with $45.
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5. Two kiwis and a pineapple cost $4. Two
kiwis and three pineapples cost $9. What
does a pineapple cost?
6. John and Paul spent $45 together, John and
Henry spent $65 together, and Henry spent 3
times as much as Paul. How much did John
spend? (From Primary Math 5A)
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7. A fish tank weighs 11.5 lbs. when it is 1/8 full
of water and 34 lbs. when it is 3/4 full. How
much does the empty tank weigh? (From
Parker/Baldridge: Elementary Mathematics
for Teachers)
8. A school raised $18,000 in a charity event.
The money raised was shared among three
charities, A, B, and C, in the ratio 1:2:3. How
much money did each organization receive?
(From Math In Focus 6A)
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Common Core Mathematical Practices
• MP1 Make sense of problems and
persevere in solving them.
• MP2 Reason abstractly and
quantitatively.
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