Proportional Reasoning

1
Proportional Reasoning
Two different candles, P and Q, lighted at the
same time were burning at different, but
constant, rates. When candle P had burned 16 mm,
candle Q had burned 10 mm. When candle Q had
burned 35 mm, how many mm would candle P have
burned?

• Strategy 1 Setting up a proportion

• Strategy 2 Coordinating quantities

0 mm
16 mm
32 mm
48 mm
56 mm
0 mm
10 mm
20 mm
30 mm
35 mm
Which strategy demonstrates a conceptual
understanding of proportional relationship?
2
Objective Making Connections

• Proportion

What is a proportion?
3
Objective Making Connections

• Proportion

Beginning Algebra
y mx b
How can we link these two ideas?
4
Key Ideas

• Focus on Co-variation and Invariance

• Identifying quantities

that change (i.e. variables)
Two different candles, P and Q, lighted at the
same time were burning at different, but
constant, rates. When candle P had burned 16 mm,
candle Q had burned 10 mm. When candle Q had
burned 35 mm, how many mm would candle P have
burned?
(a) Identify the quantities in this problem.
These are values of quantities, not quantities!
These are numbers, not quantities!
The length burned for candle P at the first
moment. (given)
The length burned for candle P at the second
moment. (unknown)
The length burned for candle Q at the first
moment. (given)
The length burned for candle Q at the second
moment. (given)
5
Key Ideas

• Focus on Co-variation and Invariance

• Identifying quantities

that change (i.e. variables)
and how those quantities are related
Two different candles, P and Q, lighted at the
same time were burning at different, but
constant, rates. When candle P had burned 16 mm,
candle Q had burned 10 mm. When candle Q had
burned 35 mm, how many mm would candle P have
burned?
(a) Identify the quantities in this problem.
(b) Let p represent the number of mm that candle
P had burned when candle Q had burned q mm. Write
an equation to relate p and q.
Mentally act out the problem situation.
Draw diagrams to represent the problem situation.
6
Two different candles, P and Q, lighted at the
same time were burning at different, but
constant, rates. When candle P had burned 16 mm,
candle Q had burned 10 mm. When candle Q had
burned 35 mm, how many mm would candle P have
burned?

• Which candle is skinner?
• Candle P
• Candle Q
• The same

7
Two different candles, P and Q, lighted at the
same time were burning at different, but
constant, rates. When candle P had burned 16 mm,
candle Q had burned 10 mm. When candle Q had
burned 35 mm, how many mm would candle P have
burned?

10 mm
16 mm

• Which candle is skinner?
• Candle P
• Candle Q
• The same

1st moment
8
Two different candles, P and Q, lighted at the
same time were burning at different, but
constant, rates. When candle P had burned 16 mm,
candle Q had burned 10 mm. When candle Q had
burned 35 mm, how many mm would candle P have
burned?
10 mm
16 mm
2nd moment
1st moment
9
Two different candles, P and Q, lighted at the
same time were burning at different, but
constant, rates. When candle P had burned 16 mm,
candle Q had burned 10 mm. When candle Q had
burned 35 mm, how many mm would candle P have
burned?
What is invariant in this problem?
The burning rate of each candle.
10 mm
16 mm
2nd moment
1st moment
10
Two different candles, P and Q, lighted at the
same time were burning at different, but
constant, rates. When candle P had burned 16 mm,
candle Q had burned 10 mm. When candle Q had
burned 35 mm, how many mm would candle P have
burned?
What is invariant in this problem?
The burning rate of each candle.
10 mm
16 mm
2nd moment
Initially
1st moment
11
Two different candles, P and Q, lighted at the
same time were burning at different, but
constant, rates. When candle P had burned 16 mm,
candle Q had burned 10 mm. When candle Q had
burned 35 mm, how many mm would candle P have
burned?
What is invariant in this problem?
The burning rate of each candle.
Initially
2nd moment
1st moment
12
Two different candles, P and Q, lighted at the
same time were burning at different, but
constant, rates. When candle P had burned 16 mm,
candle Q had burned 10 mm. When candle Q had
burned 35 mm, how many mm would candle P have
burned?
What is invariant in this problem?
The burning rate of each candle.
Initially
13
Two different candles, P and Q, lighted at the
same time were burning at different, but
constant, rates. When candle P had burned 16 mm,
candle Q had burned 10 mm. When candle Q had
burned 35 mm, how many mm would candle P have
burned?
What is invariant in this problem?
The burning rate of each candle.
2nd moment
Initially
14
Two different candles, P and Q, lighted at the
same time were burning at different, but
constant, rates. When candle P had burned 16 mm,
candle Q had burned 10 mm. When candle Q had
burned 35 mm, how many mm would candle P have
burned?
What else is invariant in this problem?
The ratio of 16/10 is invariant.
What does the ratio 16/10, or the value 1.6,
represent?
2nd moment
Initially
15
Two different candles, P and Q, lighted at the
same time were burning at different, but
constant, rates. When candle P had burned 16 mm,
candle Q had burned 10 mm. When candle Q had
burned 35 mm, how many mm would candle P have
burned?
What else is invariant in this problem?
The ratio of 16/10 is invariant.
What does the ratio 16/10, or the value 1.6,
represent?
Length (mm) Burned by Candle P 0 16 x
Length (mm) Burned by Candle Q 0 10 35
16
Two different candles, P and Q, lighted at the
same time were burning at different, but
constant, rates. When candle P had burned 16 mm,
candle Q had burned 10 mm. When candle Q had
burned 35 mm, how many mm would candle P have
burned?
What else is invariant in this problem?
The ratio of 16/10 is invariant.
What does the ratio 16/10, or the value 1.6,
represent?
Candle P burned 1.6 mm for every 1mm burned by
Candle Q.
Length (mm) Burned by Candle P 0 1.6 16 x
Length (mm) Burned by Candle Q 0 1 10 35
17
Two different candles, P and Q, lighted at the
same time were burning at different, but
constant, rates. When candle P had burned 16 mm,
candle Q had burned 10 mm. When candle Q had
burned 35 mm, how many mm would candle P have
burned?
What else is invariant in this problem?
The ratio of 16/10 is invariant.
What does the ratio 16/10, or the value 1.6,
represent?
Candle P burned 1.6 mm for every 1mm burned by
Candle Q.
Length (mm) Burned by Candle P 0 1.6 16 x
Length (mm) Burned by Candle Q 0 1 10 35
Length (mm) Burned by Candle P 0 1.6 3.2 4.8 6.4 8 16 32 48 x
Length (mm) Burned by Candle Q 0 1 2 3 4 5 10 20 30 35
p
q
(b) Let p represent the number of mm that candle
P had burned when candle Q had burned q mm. Write
an equation to relate p and q.
q
p

1.6
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Key Ideas

• Focus on Co-variation and Invariance

• Focusing on quantities and relationships

• Making connections among various representations

19
Two different candles, P and Q, lighted at the
same time were burning at different, but
constant, rates. When candle P had burned 16 mm,
candle Q had burned 10 mm. When candle Q had
burned 35 mm, how many mm would candle P have
burned?
(c) How else can we show the relationship
between the variables?
Candle P
Candle Q
20
Key Ideas

• Focus on Co-variation and Invariance

• Focusing on quantities and relationships
• Making connections among various representations

• Interpreting slope meaningfully

21
What does the slope represent of each line
represent?
The burning rate for each candle (value is
unknown).
How are the slopes related?
Candle P burned 1.6 times as fast as Candle Q.
Slope of line for Candle P is 1.6 times that of
Candle Q.
mP 1.6mQ
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Key Ideas

• Focus on Co-variation and Invariance

• Focusing on quantities and relationships
• Making connections among various representations
• Interpreting slope meaningfully

• Recognizing that ratio is invariant

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Key Ideas

• Focus on Co-variation and Invariance

• Focusing on quantities and relationships
• Making connections among various representations
• Interpreting slope meaningfully

• Recognizing that ratio is invariant

• Relating the meaning of ratio to the context of
  the problem

25
We can solve this problem by setting up a
proportion like .
The ratio 35/10 is equal to 3.5. What is the
significance of 3.5 in terms of the burning
candles?
Length burned (mm)
?
Candle P
35
30
Candle Q
20
16
10
Time
1st Moment
2nd Moment
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We can solve this problem by setting up a
proportion like .
The ratio 35/10 is equal to 3.5. What is the
significance of 3.5 in terms of the burning
candles?
Length burned (mm)
?
Candle P
48
35
32
Candle Q
16
10
Time
1st Moment
2nd Moment
27
Two different candles, P and Q, lighted at the
same time were burning at different but constant
rates. At 800pm candle P had burned 16 mm and
candle Q had burned 10 mm. At 850pm candle Q had
burned 35 mm.

1. At what time were the two candles lighted?

b. What is the burning rate for candle Q?
c. Suppose the original length of candles P and Q
are 200 mm. Which candle is skinnier?
Length burned (mm)
200
35
16
10
Time
800pm
850pm
740pm
?
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Two different candles, P and Q, lighted at the
same time were burning at different but constant
rates. At 800pm candle P had burned 16 mm and
candle Q had burned 10 mm. At 850pm candle Q had
burned 35 mm.
Let t be the of minutes since the lighting of
the candles. Let bP be the length of candle P
that has burned at time t. Let bQ be the length
of candle Q that has burned at time t.
(mm)
bP vs t graph
200
bQ vs t graph
35
16
10
t (min)
20
70
0
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Two different candles, P and Q, lighted at the
same time were burning at different but constant
rates. At 800pm candle P had burned 16 mm and
candle Q had burned 10 mm. At 850pm candle Q had
burned 35 mm.
Let t be the of minutes since the lighting of
the candles. Let bP be the length of candle P
that has burned at time t. Let bQ be the length
of candle Q that has burned at time t. Let hP be
the height in mm of candle P at time t.
(mm)
bP vs t graph
200
bQ vs t graph
hP vs t graph
t (min)
0
30
Two different candles, P and Q, lighted at the
same time were burning at different but constant
rates. At 800pm candle P had burned 16 mm and
candle Q had burned 10 mm. At 850pm candle Q had
burned 35 mm.
Let t be the of minutes since the lighting of
the candles. Let bP be the length of candle P
that has burned at time t. Let bQ be the length
of candle Q that has burned at time t. Let hP be
the height in mm of candle P at time t. Let hQ
be the height in mm of candle Q at time t.
(mm)
bP vs t graph
200
bQ vs t graph
hQ vs t graph
hP vs t graph
t (min)
0
31
Two different candles, P and Q, lighted at the
same time were burning at different but constant
rates. At 800pm candle P had burned 16 mm and
candle Q had burned 10 mm. At 850pm candle Q had
burned 35 mm.
Write an equation to relate the variables in each
pair and briefly describe the meaning of the
slope and y-intercept of your equation. (i) bP
and t (ii) bQ and t (iii) hP and t(iv) hQ
and t (v) hP and bP (vi) bP and bQ
(mm)
bP vs t graph
200
bQ vs t graph
hQ vs t graph
hP vs t graph
t (min)
0
32
All the contextualized problems in this
presentation are from this article in
Mathematics Teaching in the Middle School Vol.
14, No. 8, April 2009