LF.0 Linear Function RefresherTEACHER: Burning Candles

Practice 3 Construct viable arguments and critique the reasoning of others.

Practice 4. Model with mathematics.

Practice 8. Look for and express regularity in repeated reasoning.

A-CED.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 
This is a review of 8th grade CCSS.
Use functions to model relationships between quantities.
  • 8.F.4-Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
  • 8.F.5– Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

The launch uses students experience to state the assumptions students need for this task
  • Taller candles of the same thickness burn slower.
  • Thicker candles of the same height burn slower.
/
Ask them to write a sentence to activate their intuition about the task ahead. /
Part 1:
Show them just 1a, by putting page 1 of the student task under the document camera and cover up parts b and c
OR
Cut up parts a, b and c to give them to you students upon completion of the prior task.
a. All candles burn at the same rate, so they are the same thickness, so this is a graph of Group3.
b. Z; y = -1/2x +3;
Y; y = -1/2x +4;
X; y = -1/2x +7
The candles burn 1 cm in 2 minutes.
The candles are 3, 4 and 7 cm tall.
c. The equation must have the same slope. /

Give students part 2 when they have complete part 1.
a. All 3 candles are 8 cm before they are burned, so they are all the same height which is Group 1.
b. P: y = x + 8
Q: y = 2/3 x + 8
R: y = 1/2x + 8
All lines have the same y-intercept, and different slopes. /

This is an extension for differentiation.
Give students part 3 when they have complete part 2. /

Task Launch

In each pair of candles, determine which candle burns out faster?

Write a sentence that compares the burning of candles.

Part 1:

Examine the graph below and the groups of candles in the image. Each line in the graph represents a different candle.

  1. Which group of candles could be represented by the graph? Which candle goes with which line? Why?
  2. Write an equation to represent each candle’s burning. What does the slope tell you about the candle? What does the y-intercept (starting point) tell you about the candle?
  3. Write an equation for a fourth candle, candle N, whose graph is parallel to the graphs of candles X, Y, and Z. Sketch the graph of your equation on the same grid with the graphs for candles X, Y, and Z. Explain how you know this equation will produce a graph that is parallel to the graphs for candles X, Y, and Z.

Part 2:

Examine a different graph below and a different grouping of candles. Each line in the graph represents a different candle.

  1. Which group of candles could be represented by the graph? Which candle goes with which line? Why?
  2. Write an equation to represent each candle’s burning. How are the three equations alike and different? How does this relate to the image of the group of candles you selected?
  3. Write an equation for a fourth candle, candle S, whose graph will intersect the point (0, 8) like the other three lines. Sketch the graph of your equation on the same grid with the graphs for candles P, Q, and R. Explain how you know this equation will produce a graph that intersects (0, 8).

Part 3:

  1. Study the group of candles above, create a graph of the candles’ burning that would represent the amount of candle burned across time
  2. How is the graph similar and different from the graphs above? Why?
  3. How did you think differently in order to create this graph?

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From Reconceptualizing Mathematics by Sowder, Sowder, and Nickerson (2010).