Exponentials Lesson 5 (Teachers)

Introduction to Fractional Exponents

The students begin to explore the height of a magic beanstalk at times between Day 0 and Day 1. Specifically, the investigate the height of a beanstalk on Day ½, day ¼, and Day ⅓. 

Episode 1: Making Sense

Josh and Arobindo review the mathematical relationships on a timeline that represents the growth of a beanstalk that triples in height each day and has a height of 1 cm on Day 0.

Episode 2: Exploring

The students use their timeline to find the height of the beanstalk on Day ½.

Episode 3: Repeating Your Reasoning

Josh and Arobindo use their timeline to find the height of the beanstalk on Day ¼.

Episode 4: Repeating Your Reasoning

Arobindo and Josh use their timeline to find the height of the beanstalk on Day ⅓.

Episode 5: Reflecting

The students reflect on how a beanstalk grows over each half day.

Mathematics in this Lesson

Targeted Understandings

This lesson can help students:

  • Understand fractional exponents quantitatively, where the exponent is a unit fraction.
  • Use the beanstalk timeline to explain the relationship between fractional exponents and roots. Specifically, students should be able to explain that since there are b 1/bth-day time periods in a full day, over each 1/bth-day the beanstalk grows by a factor of the bth root of 3. This is because the bth root of 3 is the number that when multiplied by itself b times is 3 (the growth factor over a full day).

Common Core Math Standards

CCSS.MATH.CONTENT.HSN.RN.A.1Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. 

This lesson features Arobindo and Josh making sense of rational exponents in the context of a growing beanstalk. Throughout the lesson, the students work with a timeline that they partition to show fractional parts of days and growing rates. They reason about the growth rate of the beanstalk over one-half, one-third, and one-fourth days when the beanstalk is known to triple in height over one day. They partition their timeline to show halves, thirds, and fourths, and make sense of numbers that when multiplied by themselves (two times, three times, and four times, respectively), result in the growth rate for one full day. They link this to the idea of square, cube, and fourth roots.

Common Core Math Practices

CCSS.MATH.PRACTICE.MP7Look for and make use of structure.

According to the CCSSM, “Mathematically proficient students look closely to discern a pattern or structure… They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects.” Throughout this unit, Arobindo and Josh have been making use of a timeline to reason about a growing beanstalk. This lesson highlights how such a timeline can be used to make sense of fractional exponents. In Episode 1, the students notice that over any two-day period, the beanstalk will grow nine times taller [2:55]. In Episode 2 they use this structure to reason about how the plant grows over half-day intervals, arguing that it must be the square root of 3, since when you multiply it by itself you get three, and the product of the growth rates over two consecutive half-days must also be 3 [5:50; 8:45]. They repeatedly make use of the structure of the timeline to support their arguments. For example, in Episode 4 [0:45] they draw arcs between successive tick marks on their timeline and use those arcs to support their argument that the growth rate between each set of tick marks representing a 1/3-day interval must be