The purpose of this activity is to engage students in evaluating powers of whole numbers and generalise the repeating sequence in the ones digit of those powers.
This activity assumes the students have experience in the following areas:
The problem is sufficiently open ended to allow the students freedom of choice in their approach. It may be scaffolded with guidance that leads to a solution, and/or the students might be given the opportunity to solve the problem independently.
The example responses at the end of the resource give an indication of the kind of response to expect from students who approach the problem in particular ways.
The last (ones) digit of 32 is 9 since 3 x 3 = 9
The last (ones) digit of 33 is 7 since 3 x 3 x 3 = 27
The last (ones) digit of 34 is 1 since 3 x 3 x 3 x 3 = 81.
What is the ones digit of 32019?
Do similar patterns exist in the powers of other whole numbers? For example, what is the ones digit of each of the following?
The following prompts illustrate how this activity can be structured around the phases of the Mathematics Investigation Cycle.
Introduce the problem. Allow students time to read it and discuss in pairs or small groups.
Discuss ideas about how to solve the problem. Emphasise that, in the planning phase, you want students to say how they would solve the problem, not to actually solve it.
Allow students time to work through their strategy and find a solution to the problem.
Allow students time to check their answers and then either have them pair share with other groups or ask for volunteers to share their solution with the class.
The student uses calculations, with the support of digital technology, to identify elements of repeat in the ones digit of powers.
Digital technology allows for powers to be calculated easily. Since the purpose of the task is for students to identify and apply sequential patterns, rather than perform mental or written calculations, it is appropriate that digital technology is freely available. For example, students can be asked to program a spreadsheet to calculate the powers of three. They might begin with a layout like this. Note that a recursive formula as shown will take the cell content above and multiply that number by three.
A limitation is that most spreadsheets limit the number of digits available in a cell to 15 or 16. That will mean that the display below occurs:
The limitation is good in that students will need to predict the pattern. Students should identify that the ones digits follow a repeating sequence 1, 3, 9, 7, 1, 3, 9, 7, …
Students can use their scientific calculator to though the same restrictions apply to the length of the number displayed. For example, 320 will display as 3486784401. However, 321 will not display correctly as it requires 11 digits. So even with support of digital technology students need to apply a conceptual approach to find the ones digit of 32019.
The student uses multiplicative thinking, particularly division with remainders, to predict the ones digit of powers.
Click on the image to enlarge it. Click again to close.
Therefore, 32019 will have 7 as the ones digit because 32020 will have 1 as the ones digit, since 2020 is divisible by four (remainder zero).
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/power-play at 8:53pm on the 26th February 2024