Proportional Reasoning Two different candles P and Q

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 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 y = mx + b x d y = mx How can we link these two ideas?

Key Ideas • Focus on Co-variation and Invariance ü 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 ü 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 1 st 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 1 st 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 1 st moment P Q 2 nd 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 1 st moment P Q 2 nd 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 1 st moment P Q 2 nd 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. 21 ndst. Initially 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. P Q Initially

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 2 nd 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 Initially P Q 2 nd 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 1 mm 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 1 mm 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 ü Focusing on quantities and relationships ü 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 1 st Moment 2 nd Moment

Key Ideas • Focus on Co-variation and Invariance ü Focusing on quantities and relationships ü Making connections among various representations ü 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. m. P = 1. 6 m. Q Length burned (mm) 16 T 1 10 T 1 & = 16 10 x Candle P 35 Candle Q 16 10 10 16 Time T 1 1 st Moment 2 nd Moment = 1. 6 1

Key Ideas • Focus on Co-variation and Invariance ü ü 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. 6 q Length burned (mm) Candle P x Candle Q 35 x p 16 10 10 16 q p q 35 Time 1 st Moment 2 nd Moment

Key Ideas • Focus on Co-variation and Invariance ü ü ü 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 1 st Moment 2 nd 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 1 st Moment 2 nd Moment

Two different candles, P and Q, lighted at the same time were burning at different but constant rates. At 8: 00 pm candle P had burned 16 mm and candle Q had burned 10 mm. At 8: 50 pm 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: 40 pm ? 8: 00 pm 8: 50 pm Time

Two different candles, P and Q, lighted at the same time were burning at different but constant rates. At 8: 00 pm candle P had burned 16 mm and candle Q had burned 10 mm. At 8: 50 pm candle Q had burned 35 mm. Let t be the # of minutes since the lighting of the candles. Let b. P be the length of candle P that has burned at time t. Let b. Q be the length of candle Q that has burned at time t. (mm) b. P vs t graph 200 b. Q 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: 00 pm candle P had burned 16 mm and candle Q had burned 10 mm. At 8: 50 pm candle Q had burned 35 mm. Let t be the # of minutes since the lighting of the candles. Let b. P be the length of candle P that has burned at time t. Let b. Q be the length of candle Q that has burned at time t. Let h. P be the height in mm of candle P at time t. (mm) b. P vs t graph 200 b. Q vs t graph h. P 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: 00 pm candle P had burned 16 mm and candle Q had burned 10 mm. At 8: 50 pm candle Q had burned 35 mm. Let t be the # of minutes since the lighting of the candles. Let b. P be the length of candle P that has burned at time t. Let b. Q be the length of candle Q that has burned at time t. Let h. P be the height in mm of candle P at time t. Let h. Q be the height in mm of candle Q at time t. (mm) b. P vs t graph 200 b. Q vs t graph h. P vs t graph 0 h. Q 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: 00 pm candle P had burned 16 mm and candle Q had burned 10 mm. At 8: 50 pm 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) b. P and t (ii) b. Q and t (iii) h. P and t (iv) h. Q and t (v) h. P and b. P (vi) b. P and b. Q (mm) b. P vs t graph 200 b. Q vs t graph h. P vs t graph 0 h. Q 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