Algebra I/Unit 5/Lesson Seed / Interpreting Exponential Data

MSDE Mathematics Lesson Seed

Domain: Interpreting Data
Cluster Statement:Summarize, represent and interpret data on two categorical and
quantitative variables.
Standard:
S.ID.6 Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.
  1. Fit a function to the data; use functions fitted to data to solve problems in the
context of the data. (supporting)
Purpose/Big Idea:
Students will construct an exponential model using data.
Materials:
  • Towers of Hanoi – modeling exponential growth

  • Towers of Hanoi – game sets (if available or can be purchased)
  • Life size model using weights or textbooks or stacking rings

Description of how to use the activity:
  1. Warm-up: Share the legend of the about the monks and the 64 rings that is provided in the Warm Up.
Have students “notice and wonder” about how long it would take / how many moves it would take to move the 64 golden rings to the new tower (see legend). Use as an opportunity to capture student interest and make world geography connection.

  1. Provide students the opportunity to try moving the rings as described. Possible ways of providing this opportunity include:
  • have students work with an applet such as the one found below to experiment with the process of moving the rings.

  • Provide each student a set of staking rings or a variation of them.
  1. Challenge students to compete with a partner to complete the process in as few moves as possible using 1 ring, 2 rings, 3 rings etc.
  2. After students have had the opportunity to play for a while, begin to pull the students back together and collect data on the minimum number of moves used to move the rings as described.
  3. Record the data in a table and display the table for the class.
  4. Instruct students to create a scatter plot of the data.
  5. Students can brainstorm what their model looks like compared to previously studied function. Have students “play” with different equations and share ideas.
  6. Instruct students to create regression models using a graphing calculator
  7. Discussions about the accuracy of the model can be facilitated using students understanding of the r-value when using regression with Ti-calculator.
  8. Close the activity by discussing real world applications of exponential growth. Have students brainstorm other places where this model may be appropriate (i.e. Under Armour profits, Facebook membership, Moore’s Law

Guiding Questions:
  1. How does the linear model compare to the exponential model?
  2. What similarities/differences occur in the linear model vs. the exponential model?
  3. From a T.O.V. what conclusions can be drawn to indicate that the data fits an exponential model compared to a linear model?
Extensions
  1. Have students solve the Tower with 64 disks using their model. Have students estimate the amount of time it would take to complete this task.

Warm Up/ Motivation:

Legend has it that a group of Eastern monks are the keepers of three towers on which sit 64 golden rings. Originally all 64 rings were stacked on one tower with each ring smaller than the one beneath. The monks are to move the rings from this first tower to the third tower one at a time but never moving a larger ring on top of a smaller one. Once the 64 rings have all been moved, the world will come to an end.

How long do you think it would take to complete this task?

The Towers of Hanoi

  1. Play the Towers of Hanoi game using the process described below.

Begin playing with one disk and continue playing, increasing the number of disks that you use by one disk each time you play. Record the number of moves required to move the disks, as described in the legend,in the grid below.

Number of disks used / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8
Moves
  1. Create a scatter plot of the data using the capabilities of a graphing calculator. Sketch the scatter plot on the grid below.

  1. What type of function does the shape of the scatter plot suggest might be used to model this data? Justify your answer.
  1. Use the regression features of a graphing calculator to determine the equation of a model for the data. Record the equation below.
  1. Using your model estimate the number of moves that would be required to complete the

64-rings. How long do you think this would take to complete?

Solutions/Reference:

Number of pieces / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8
Moves / 1 / 3 / 7 / 15 / 31 / 63 / 127 / 255

From this formula you can see that even if it only takes the monks one second to make each move, it will be 2^64 - 1 seconds before the world will end. This is 590,000,000,000 years (that's 590 billion years) - far, far longer than some scientists estimate the solar system will last. That's a really long time!

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