Transcript Slide 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
16
32
48
56
0 mm
10
20
30
35
0 mm
16 mm
10 mm
16 mm
10 mm
10 mm
16 mm
8 mm
5 mm
Which strategy demonstrates a conceptual understanding of
proportional relationship?
Objective: Making Connections
Proportion
a
b
=
x
d
What is a proportion?
Objective: Making Connections
Proportion
a
b
=
Beginning Algebra
x
d
y = mx + b
y = mx
How can we link these two ideas?
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,
These
are
numbers,
16 mm
10 mm
quantities!
notnot
quantities!
?
35 mm
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)
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.
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?
a. Candle P
b. Candle Q
c. The same
P
Q
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?
10 mm
16 mm
Which candle is skinner?
a. Candle P
b. Candle Q
c. The same
P
Q
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?
10 mm
16 mm
35 mm
?
P
Q
Q
P
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?
What is invariant in this problem?
The burning rate of each candle.
10 mm
16 mm
35 mm
?
P
Q
Q
P
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?
What is invariant in this problem?
The burning rate of each candle.
10 mm
16 mm
35 mm
?
P
Q
Initially
P
Q
Q
P
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?
What is invariant in this problem?
21ndstInitially
moment
The burning rate of each candle.
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?
P
Q
Initially
The burning rate of each candle.
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?
Initially
The burning rate of each candle.
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?
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
Initially
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?
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
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
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
3.2
4.8
6.4
8
16
32
48
px
Length (mm)
Burned by
Candle Q
0
1
2
3
4
5
10
20
30
35
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 = 1.6 p
Key Ideas
• Focus on Co-variation and Invariance
 Focusing on quantities and relationships
 Making connections among various representations
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?
Length burned (mm)
x
Candle P
35
Candle Q
16
10
Time
1st Moment
2nd Moment
Key Ideas
• Focus on Co-variation and Invariance
 Focusing on quantities and relationships
 Making connections among various representations
 Interpreting slope meaningfully
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
Length burned (mm)
16
T1
16
T1
10
T1
10
T1
&
=
16
10
x
Candle P
35
Candle Q
16
10
16
10
Time
T1
1st Moment
2nd Moment
=
1.6
1
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
10
=
x
35
16
10
=
p
q
16
10
=
x
35
16
10
=
1.6
1
=
x
35
=
p
q
p = 1.6q
Length burned (mm)
Candle P
x
Candle Q
35
x
p
16
35
p q
10
16
q
10
Time
1st Moment
2nd Moment
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
10
=
x
35
35
10
=
x
16
We can solve this problem by setting up a proportion like 35 = x .
10
16
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
We can solve this problem by setting up a proportion like 35 = x .
10
16
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
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.
a. 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
7:40pm
?
8:00pm
8:50pm
Time
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.
(mm)
bP vs t graph
200
bQ vs t graph
35
16
10
0
20
70
t (min)
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.
(mm)
bP vs t graph
200
bQ vs t graph
hP vs t graph
0
t (min)
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.
Let hQ be the height in mm of candle Q at time t.
(mm)
bP vs t graph
200
bQ vs t graph
hP vs t graph
0
hQ vs t graph
t (min)
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.
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
hP vs t graph
0
hQ vs t graph
t (min)
All the contextualized problems in this
presentation are from this article in
Mathematics Teaching in the Middle School
Vol. 14, No. 8, April 2009