proportional reasoning
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Proportional Reasoning
• Strategy 1: Setting up a proportion
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 2: Coordinating quantities0 mm
0 mm
16 mm
10 mm
16 mm
10 mm
16 mm
10 mm
32 mm
20 mm
16 mm
10 mm
48 mm
30 mm
8 mm
5 mm
56 mm
35 mm
Which strategy demonstrates a conceptual understanding of proportional relationship?
Proportion
Objective: Making Connections
a xb d=
What is a proportion?
Proportion
Objective: Making Connections
Beginning Algebra
a xb d= y = mx + b
How can we link these two ideas?
y = mx
Key Ideas• Focus on Co-variation and Invariance
(a) Identify the quantities in this problem.
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?
These are values of quantities,
not quantities!
16 35 10
?The length burned for candle P at the first moment. (given)
The length burned for candle Q at the first moment. (given) The length burned for candle Q at the second moment. (given)
The length burned for candle P at the second moment. (unknown)
Identifying quantities that change (i.e. variables)
mm mm mm These are numbers,
not quantities!
Key Ideas• Focus on Co-variation and Invariance
Identifying quantities
(a) Identify the quantities in this problem.
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?
that change (i.e. variables)and how those quantities are related
(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.
Which candle is skinner?a. Candle P b. Candle Qc. The same
P Q
16 mm 10 mm
1st moment
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?
P Q
16 mm 10 mm
Which candle is skinner?a. Candle P b. Candle Qc. The same
1st moment
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?
P Q
10 mm16 mm
PQ
35 mm?
1st moment 2nd moment
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?
P Q
10 mm16 mm
PQ
35 mm?
1st moment 2nd moment
What is invariant in this problem?
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?
The burning rate of each candle.
P Q P Q
10 mm16 mm
PQ
35 mm?
Initially 1st moment 2nd moment
What is invariant in this problem?
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?
The burning rate of each candle.
Initially2nd moment
What is invariant in this problem?
1st moment
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?
The burning rate of each candle.
P Q
Initially
What is invariant in this problem?
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?
The burning rate of each candle.
Initially 2nd moment
What is invariant in this problem?
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?
The burning rate of each candle.
P Q
Initially
PQ
2nd moment
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?
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?
Length (mm) Burned by Candle P
0 16 x
Length (mm) Burned by Candle Q
0 10 35
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?
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 1.6 16 x
Length (mm) Burned by Candle Q
0 1 10 35
Candle P burned 1.6 mm for every 1mm burned by Candle Q.
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 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
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?
Candle P burned 1.6 mm for every 1mm burned by Candle Q.
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?
p
q
q p
(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.
= 1.6
Key Ideas
• Focus on Co-variation and Invariance Focusing on quantities and relationships Making connections among various representations
(c) How else can we show the relationship between the variables?
Time
Length burned (mm)
35
10
16
x
1st Moment 2nd Moment
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?
Candle P
Candle Q
Key Ideas
• Focus on Co-variation and Invariance Focusing on quantities and relationships Making connections among various representations Interpreting slope meaningfully
Time
Length burned (mm)
35
10
16
Candle P
Candle Q
x
1st Moment 2nd Moment
What does the slope represent of each line represent? The burning rate for each candle (value is unknown). How are the slopes related?
1016
T1
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.
16 T1
10 T1
&
16 1.610 1=
16 T1
10 T1
= mP = 1.6mQ
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
16 x10 35= 16 p
10 q=
p = 1.6q
16 x10 35
=
1016
Time
Length burned (mm)
10
16
35
Candle P
Candle Q
x
1st Moment 2nd Moment
x 35= p
q=
pq
16 1.610 1=
x
35pq
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
16 x10 35= 35 x
10 16=
We can solve this problem by setting up a proportion like .35 x10 16
=
Time 1st Moment 2nd Moment
Length burned (mm)
10
16
35
Candle P
Candle Q
?
The ratio 35/10 is equal to 3.5. What is the significance of 3.5 in terms of the burning candles?
20
30
We can solve this problem by setting up a proportion like .
Time 1st Moment 2nd Moment
Length burned (mm)
10
16
35
Candle P
Candle Q
?
The ratio 35/10 is equal to 3.5. What is the significance of 3.5 in terms of the burning candles?
32
48
35 x10 16
=
Two different candles, P and Q, lighted at the same time were burning at different but constant rates. At 8:00pm candle P had burned 16 mm and candle Q had burned 10 mm. At 8:50pm candle Q had burned 35 mm.
Time 8:00pm 8:50pm
Length burned (mm)
7:40pm
1016
35
a. At what time were the two candles lighted?
b. What is the burning rate for candle Q?
?
200
c. Suppose the original length of candles P and Q are 200 mm. Which candle is skinnier?
Two different candles, P and Q, lighted at the same time were burning at different but constant rates. At 8:00pm candle P had burned 16 mm and candle Q had burned 10 mm. At 8:50pm candle Q had burned 35 mm.
20 700
1016
35
t (min)
200
(mm)
bQ vs t graph
bP vs t graph
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.
Two different candles, P and Q, lighted at the same time were burning at different but constant rates. At 8:00pm candle P had burned 16 mm and candle Q had burned 10 mm. At 8:50pm 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.
t (min)
(mm)
bQ vs t graph
hP vs t graph
bP vs t graph200
0
Two different candles, P and Q, lighted at the same time were burning at different but constant rates. At 8:00pm candle P had burned 16 mm and candle Q had burned 10 mm. At 8:50pm candle Q had burned 35 mm.
t (min)
(mm)
hQ vs t graph
bQ vs t graph
bP vs t graph
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.
hP vs t graph0
200
Two different candles, P and Q, lighted at the same time were burning at different but constant rates. At 8:00pm candle P had burned 16 mm and candle Q had burned 10 mm. At 8:50pm candle Q had burned 35 mm.
t (min)
(mm)
hQ vs t graph
bQ vs t graph
bP vs t graph
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
hP vs t graph0
200
Mathematics Teaching in the Middle SchoolVol. 14, No. 8, April 2009
All the contextualized problems in this presentation are from this article in
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