## Power play

The purpose of this activity is to engage students in evaluating the powers of whole numbers and generalising the repeating sequence in the ones digit of those powers.

## About this resource

This collection of learning activities is designed to provide engaging contexts in which to explore the achievement objectives of The 2007 New Zealand Curriculum. The activities themselves do not represent a complete coverage of learning at this level. Rather, they provide an opportunity to apply the mathematics learned in previous lessons. The activities are intended to reflect the range of approaches that make up a differentiated classroom.

This activity assumes the students have experience in the following areas:

- Multiplication and division of whole numbers and decimals, including division with remainders.
- Evaluating powers of whole numbers with whole number exponents.
- Continuing sequential patterns and finding rules for the next term.

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.

# Power play

## Achievement objectives

NA5-2: Use prime numbers, common factors and multiples, and powers (including square roots).

NA5-9: Relate tables, graphs, and equations to linear and simple quadratic relationships found in number and spatial patterns.

## Description of mathematics

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.

## Required materials

See **Materials that come with this resource** to download:

*Power play activity*(.pdf)

## Activity

The last (ones) digit of 3^{2} is 9 since 3 x 3 = **9**

The last (ones) digit of 3^{3} is 7 since 3 x 3 x 3 = **27**

The last (ones) digit of 3^{4} is 1 since 3 x 3 x 3 x 3 = **81**

- What is the ones digit of 3
^{2019}? - Do similar patterns exist in the powers of other whole numbers? For example, what is the ones digit of each of the following?
- 5
^{2019} - 2
^{2019} - 4
^{2019} - 8
^{2019} - 7
^{2019}

- 5

The following prompts illustrate how this activity can be structured around the phases of the Mathematics Investigation Cycle.

### Make sense

Introduce the problem. Allow students time to read it and discuss it in pairs or small groups.

- What are the important words and symbols? (Students need to understand the meaning of powers expressed in exponent notation, e.g., 34 = 3 x 3 x 3 x 3.)
- What would be good examples to try? That involves replacing x with different whole numbers. (The task suggests using 3, 5, 4, 8, and 7 as bases.)
- What do I already know that will be needed to solve this problem? (Students may recognise that the ones digit can only be predicted if there is a repeating pattern in the powers.)
- Can I imagine or visualise what the numbers look like?
- What part of the problem could I get started with? Working on one base to start with simplifies the task and enables pattern spotting. (Once a pattern for one base is found, the strategy can be applied to powers of different bases.)

### Plan approach

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.

- What are the maths skills I need to work this out? Multiplication and division are required to make sense of the patterns generated by powers. (Looking for repetition in the powers is also required.)
- What could the solution look like? (The problem demands a rule to predict the ones digit of any power for a given base.)
- What strategies can I use to get started? (Listing many examples in sequence will help with pattern spotting.)
- Can I notice a pattern to write down and explore?
- What simple numbers, for the base, could I try first?
- What tools (digital or physical) could help my investigation? (Digital calculation will save time and ease the load on working memory.)

### Take action

Allow students time to work through their strategy and find a solution to the problem.

- Have I created enough examples to find a common property among the differences?
- Have I recorded my ideas in a way that helps me see patterns?
- Are there any patterns?
- How might I describe and represent the pattern?
- What is the best way to express the patterns I see? (Algebra is important for this task.) (Letters can be used as generalised numbers, e.g., x used to represent any base.)
- Can I express my rule for finding the ones digit as an algorithm (set of steps)?

### Convince yourself and others

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.

- What is the solution?
- Show and explain how you worked out your solution.
- Is the solution (algorithm) the same or different for the powers of each base? Why are the algorithms different?
- How does my solution answer the question? (Do my algorithms work reliably, or do they need refinement?)
- What connections can I see to other situations? Why would this be? (Could students extend their reasoning to deal with ones digits for powers of other bases, such as 11, 13, and 27?)
- Do similar patterns exist in the tens, hundreds, thousands, … digits? Why or why not?
- Which ideas would convince others that my findings answer the investigation question?

## Examples of work

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 be freely available. For example, students can be asked to programme 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.

Row 23 |
21 |
10460353203 |
---|---|---|

Row 24 |
22 |
31381059609 |

Row 25 |
23 |
94143178827 |

Row 26 |
24 |
2.8243E + 11 |

Row 27 |
25 |
8.47289E + 11 |

Row 28 |
26 |
2.54187E + 12 |

The limitation is good in that students will need to predict the pattern. Students should identify that the ones digits follow a repeating sequence of 1, 3, 9, 7, 1, 3, 9, 7, etc.

Students can use their scientific calculator, though the same restrictions apply to the length of the number displayed. For example, 3^{20} will display as 3486784401. However, 3^{21} will not display correctly as it requires 11 digits. So even with the support of digital technology, students still need to apply a conceptual approach to find the ones digit of 3^{2019}.

The student uses multiplicative thinking, particularly division with remainders, to predict the ones digit of powers.

I noticed that if "x" is a multiple of 4, then the ones digit is 1.

- For example: 3
^{12 }= 53144**1**

If it (x) is not a multiple of 4, I still divide by 4 to get a remainder.

- For example: 3
^{19 }19 ÷ 4 = 4 r 13

So, the ones digit should be the thrid number in the pattern: 1, 3, 9, **7**.

Test it!

3^{19 }= 116226146**7**

Hang on!

3^{20} will have 1 as the ones digit. So, the remainder of 3 means go to the last number in the pattern: 1, 3, 9, **7**.

Therefore, 3^{2019} will have 7 as the ones digit because 3^{2020} will have 1 as the ones digit since 2020 is divisible by four (remainder zero).

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