Exponents: It's Gotta Be the Power of 3

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Objective

SWBAT take notes using the Cornell style of note-taking and explore the patterns of the powers of three.

Big Idea

Students use a thorough note-taking strategy and explore patterns of the powers of 3

Do Now

15 minutes

Before class, students line up outside of the room and wait to be greeted one by one. When they go to their seat they will see a vocab index card and sheet of paper (Do Now) that says “Don't Turn Over. Sit silently and wait for instructions. The game will begin in 3 minutes”.

Once everyone is seated, I will ask students to write their names on the current side of the paper. Then, I'll give the directions for the game "Mad Minute Maestros” (English pronunciation) or as I like to say it, “Mad Minute Maestros” (Spanish pronunciation). Spanish speaking students will share the translation of the word in Spanish, "teachers".

The Game

Objective: Students must complete all ten problems, check their work, and then find teammates to help given 2 minutes. I make sure to explain what I mean by "help". Students are NOT to give their teammates answers.

Some possible ways to ask questions rather than give comments:

  • Are you sure you did the correct operation for this one?
  • What number are you multiplying? How many times?
  • Which is the base? Which is the exponent? What does that mean?

How to play: Students must attempt to finish all ten problems before time is up. They should be encouraged to check their work. Once they complete the sheet they must raise it in the air and turn it in to me. Then, they need to find teammates who have not finished and help them finish by asking questions only. No answers may be given. Once this student finishes he/she turns in their paper.Both students can continue to help other students who have not finished and expanding their group. 

Scoring: Students will receive a total score of number of correctly answered problems by the collective group. If the person who initiates any given group incorrectly responds to any questions, these points will be taken off the group score. The group with the largest score wins.

Students may not merge groups. The only way they can expand their group is by finding people to help. Stickers are a useful way to visually keep track of groups by giving the initial students a different sheet.

Once all worksheets have been turned in, I let students know that I will reveal the results the next day. 

Intro to Lesson

10 minutes

“Your task today requires you to explore patterns of the powers of three. For homework you’re going to show me what you know about constructing numerical expressions using exponents to solve word problems. Before you begin your task, we need to review the basics. I’m also going to take this opportunity to introduce you to the “Cornell Style” of note-taking. Let’s review the basics:”

Cornell style

  • the left margin is for topics and concepts (or key words and categorizers); the right margin is for any relevant notes or diagrams.
  • The bottom portion of the paper is saved for a summary

After briefly explaining the structure of “Cornell Notes”, I show students this Prezi:

http://tinyurl.com/mpucraa.

I ask my students to copy the definition and I give students 2-3 sample problems for review:

  1. 5^0 = 1 (Anything to the power of 0 is 1. The "why" behind this may be explored later this year)
  2. 5^1 = 5 (Anything to the power of 1 results in the base. The "why" behind this may be explored later this year)
  3. 5^2 = 5 x 5 = 25
  4. 10^7=10,000,000 ("Can anyone share the rule for powers of 10? How can we figure out the answer without multiplying?")

I will ask students to read and copy the facts on the board. “Power”, “base” and “exponent” should be the topics written on the left margin and the notes that define each term will be written on the right. Included in the Prezi are real world examples of the use of powers. I plan to ask students to pick 1-2 of those examples to write in their notes. Students also complete the included word problem on their paper.  This problem is already included on their paper to avoid copying the problem (time saver!).

 

Task

20 minutes

After notes have been taken, students will read the task situation. 

Mr. Corcoran and I were looking at the powers of 3. He said he could tell me what the last digit would be for three to the eleventh power, three to the thirteenth power, and three to the forty-fifth power! Find the last digit to these exponents and see if you can figure out how Mr. Corcoran might be able to predict the last digit of these and other powers of three.

I have my students write the verbal powers written above as numerical representations on their paper.

My students are instructed to work in groups to discover the pattern. Before students dive into the task we discuss the importance of establishing strategies for solving complex problems like these. I explain to students that the answer to this question is not a simple one word or even a one sentence answer. They will need to first determine how to organize their work so that they can discover a pattern. I ask them to think back to the earlier grades when they were first introduced to patterns. How were these questions written? I ask them to give me an example of a "pattern" question so that we can get to the idea of listing as an organization strategy. 

Students are then sent to work in groups. If after 5 minutes most groups are still struggling with the organization of the work, I list the first 8 powers of three on the board, without giving students the numerical answers:

3^1 = ____

3^2 = ___

3^3 = ___

3^4 = ___

3^5 = ___

3^6 = ___

3^7 = ___

3^8 = ____ 

I ask my students to fill out the answers on their own and try to discover the pattern again. If after 5 more minutes students are still struggling, I add all the answers by cold calling around the room.

3^1 = 3

3^2 = 9

3^3 = 27

3^4 = 81

3^5 = 243

3^6 = 729

3^7 = 2187

3^8 = 6561

At this point some students may begin to see the pattern (MP8). However if other students still cannot see it, the last two clues I offer are to underline each of the last digits and circle the exponents. I ask students to think about the relationship between the last digit and the exponent and to consider multiples. 

We close the activity by discussing homework: students are given the option to complete the task or if they prefer, to complete exponent word problems (attached to the lesson). Students are also expected to reflect on the group work this week and evaluate one person on their team.

The original idea for this activity came from:

http://www.rda.aps.edu/mathtaskbank/pdfs/instruct/6-8/i68Power3.pdf

Closing

10 minutes

Students will stick their group’s chart paper on the board, side by side. We will discuss the results as a class, with the focus being around student-to-student conversations. Some questions that could be asked by the teacher to guide the discussion:

  • Does anyone notice a strategy being used in more than one group?
  • Does anyone notice any similarities between groups?
  • Which explanation of the pattern (answer) do you prefer? Why?
  • Do you notice anything interesting you had not noticed about powers with your group earlier?
  • Which poster do you think best represents the pattern visually?

If time permits, Journal Entry -

Pick your favorite single digit number. Find the first ten powers of this number. Can you find a pattern within your own example? Explain the pattern.

The first person who finishes will draw 5 Popsicle sticks with student names on them. These are the 5 journal entries that get checked.