Exponentials Lesson 3 Episode 3 (Teachers)

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Josh and Arobindo do not seem to struggle much with creating an equation to represent the height of the beanstalk on Day 100, though they do not use a symbol to indicate the height of the beanstalk and choose instead to write out the words “height of the plant on Day 100.” Some students may struggle to link the various quantities in this context (starting height, growth rate, and number of days) using appropriate mathematical operations. For example, they might not know whether to add 1 (the starting height) to the expression 3¹⁰⁰ or to multiply it. Since the starting height is 1, they might not even include it, which can be problematic when the starting height is not the multiplicative identity element. 

For use in a classroom, pause the video and ask these questions:

1. [Pause the video at 0:14] Make a prediction for how tall the beanstalk will be on Day 100. Discuss with a partner how you could represent this number. Which quantities from the beanstalk context are important for your representation?
2. [Pause the video at 3:03] Ask your students the following questions, providing time for them to discuss their ideas with a partner for each question before you pose the next one: 
   • What does the 1 represent in the expression 1 × 3¹⁰⁰?  
   • What does the 3 represent in the expression 1 × 3¹⁰⁰?  
   • What does the 100 represent in the expression 1 × 3¹⁰⁰?  
   • What does the 3¹⁰⁰ represent in the expression 1 × 3¹⁰⁰? 
3. [Pause the video at 4:33] Consider Arobindo’s description of the little bumps representing 100 days and his utterance, “times 3, 100 times.” Can you rephrase Arobindo’s explanation and describe how that explanation relates to the expression 3¹⁰⁰?

Josh and Arobindo provide two different explanations for what the 3¹⁰⁰ represents in their diagram. First, they say it represents the height of the beanstalk on Day 100, and then they say it also represents the growth factor from Day 0 to Day 100. Ask your students to discuss these two different meanings for the expression 3¹⁰⁰. You might consider introducing a new timeline that begins on a day that is not zero, say Day 5, and extends past Day 105. Ask them if they can see any place in the diagram that would also represent the expression 3¹⁰⁰.