powers 5t8

							                             Powers of Two
Subject Area: General Sciences, Mathematics

Grade Level: 5-8


                                       Overview
This activity introduces students to the concept of exponents by working with
powers of two. Students will examine two simple scenarios, one that uses linear
growth and the other that uses exponential growth. From this comparison they
will observe that although exponential growth starts slowly, it eventually leaves
linear growth far behind. Students will calculate the growth of a variable (money)
in a data table using simple math and then will make a bar graph of the results.

                                       Objectives

      Students will tabulate the growth in a quantity that occurs from multiplication (2*2,
       2*3, 2*4)
      Students will tabulate the growth in a quantity that occurs exponentially (22, 23,
       24)
      Students will compare the two growth rates through the use of data tables and
       through the use of graphs
      Students will explain why exponential growth leads to far bigger quantities than
       linear growth


                            Connections to Standards

      Math: Computation, data and representations, generalizations, patterns and
       sequences, mathematical ideas in other disciplines, exponent relations
      Science: Use evidence to develop explanations and predictions, collect and
       display data
                                          Materials

This is a paper and pencil activity, although there would be many ways to include manipulatives
if the teacher wished to do so (in which case you will want to use fewer repetitions in the data
table). The student handout (below), a pencil and scratch paper are the only supplies that are
necessary.
                                         Procedure
The students can go through the procedure in a step-by-step fashion once they have received
the handout for the activity. They should 1) Read the first part of the worksheet and answer
question1 2) Fill out the data table 3) Answer the remainder of the worksheet questions 4) Make
the bar graph. An introduction that prepares the class for the activity would be helpful. The text
for a possible introduction is given below.

Multiplication can serve as a shortcut for addition. We can see
this by looking at a set of boxes.

Count how many little boxes make up the big square (16). Is
                   there a shortcut for finding the number of
                   little boxes? Sure – there are four in each
                   row, so just add the number per row four
                   times: 4 + 4 + 4 + 4 = 16. Is there a
                   shorter shortcut? Sure – multiply the number
                   of squares in one row by the number of rows:
                   4 x 4 = 16. If there were five rows, we
would take 4 x 5 = 20. The first case, 4 x 4 = 16, is somewhat
special – we are multiplying a number by itself. Another way to
write this expression is 42 = 16. The little two is called an
exponent, and it tells us how many times the four is multiplied
by itself. 72 is the same as 7 x 7, 562 is the same as 56 x 56,
and so on. If we use an exponent of two, we say we are squaring
the number (can you see why we would use the term ‘squaring’?).

                   Now let’s look at a three-dimensional figure.
                   How many small boxes are in the big box? We
                   could add all the boxes one by one, or we
                   could add 4 + 4 + 4 … a total of sixteen
                   times. Multiplication gives us a good
                   shortcut. There are 4 boxes in a row, there
                   are 4 rows in a layer, and there are 4 layers
                   in the figure. So we can take 4 x 4 x 4 to
                   get the total number of boxes, which is 64.
                   Since we are multiplying the four by itself
                   three times, we can write 43 = 64. The
                   exponent is a three, which means we are
                   cubing the four (can you see why we would use
the term ‘cubing’ in this instance?). Exponents, also called
powers, are useful when we multiply numbers by themselves. In
this activity you will compare the results of regular
multiplication (2 x 3, 2 x 4) to the results of using powers (23,
24).
                                Handouts for Students
                                Using Powers of Two
A father decided that he would help his two children, Carmen and Zach, learn about
powers of two. He told Carmen "I will pay you every day for making your bed, and if you
do a good job I will double your salary every day. I'll start with two pennies the first day."
But to Zach the father said "If you do a good job making your bed every day, I will give
you 10,000 pennies the first day. Then I will multiply 10,000 by the day number for each
following day (10,000 x 2, 10,000 x 3, etc.)". Zach was very excited. He thought that he
got a much better deal than Carmen.

Question 1: Carmen and Zach both felt that Zach would make the most money. Why
do you think they felt this way? Do you agree with them?




Now let’s find out how much money they each would earn by making their beds for
twenty consecutive days. You can keep track of their earning by filling out the table on
the next page. Remember: Zach’s earnings are 10,000 multiplied by the day number.
Carmen’ earnings get doubled each day. Fill out the table now.

When you finish filling out the table, please answer the questions below.

   2. Who ended up making the most money after twenty days? Were their earnings
      on day 20 close to each other, or was one quite different from the other?




   3. On what day were their daily earnings the closest to each other?




   4. If you wanted to take the multiplication shortcut to find Zach’s daily earnings, you
      could take (10,000 pennies x the number of the day). If you wanted to calculate
      Carmen’s daily earnings, you could take (previous day’s earnings)2. What does
      this activity teach us about expressions that have exponents? Do the values of
      these expressions grow at a constant rate?



Now you will make a graph of the children’s’ earnings on the graph paper you
have been given.
                          Getting Rich By Making Your Bed
                                     Data Table

Fill in each empty box in the table below. Zach’s earnings are 10,000 times the number
of days he has made his bed. Carmen’s earnings are just doubled each day.

 Day         Zach’s Earnings for the Day          Carmen’ Earnings for the Day
Number               (pennies)                             (pennies)

    1                    10,000                                  2
    2                    20,000                                  4
    3
    4
    5
    6
    7
    8
    9
   10
   11
   12
   13
   14
   15
   16
   17
   18
   19
   20

Warning: The story of Zach and Carmen is not true! In real life kids do not earn this much
for making their beds. Don't try Carmen's strategy on your parents! (hee, hee)
Now make a graph of the money that the children made each day. Make a bar graph for which height of the bars will represent
the amount of money they earned. Use one color of bar for Carmen and a different color for Zach, and put the bars side-by-
side for each day. You will need to estimate the heights of the bars when you color them. Be sure to include a legend.

                                                                       Getting Rich by Making Your Bed


                                     1000000


                                      900000


                                      800000
   Earnings for the Day (Pennie s)




                                      700000


                                      600000


                                      500000


                                      400000


                                      300000


                                      200000


                                      100000


                                           0
                                               1   2   3   4   5   6   7     8    9    10   11      12   13   14   15   16   17   18   19   20
                                                                                       Day Number