Skip to main content
### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Odds, Evens and More Evens

### Why do this problem?

### Possible approach

### Possible support

### Possible extension

## You may also like

### Days and Dates

### Summing Consecutive Numbers

### Paving Paths

Links to the University of Cambridge website
Links to the NRICH website Home page

Nurturing young mathematicians: teacher webinars

30 April (Primary), 1 May (Secondary)

30 April (Primary), 1 May (Secondary)

Or search by topic

Age 11 to 14

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

This problem offers a really straightforward starting point for discussion of sequences that can lead on to generalisations, and perhaps for some students thinking about orders of infinity!

Begin by displaying the sequences below, or give students the top section of this worksheet, and ask:

"What do you notice?"

$A_0 = 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...$

$A_1 = 2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42...$

$A_2 = 4, 12, 20, 28, 36, 44, 52, 60...$

$A_3 = 8, 24, 40, 56, 72, 88, 104...$

$A_4 = 16, 48, 80, 112, 144...$

$A_5 = 32, 96, 160...$

$A_6 = 64...$

$A_7 = ...$

.

.

.

Allow students some time to consider on their own or in pairs, noting down their thoughts before sharing them with the class.

Then pose the following question and allow students to continue working on their own or in pairs:

**"Which sequences will contain the number 1000?**"

After a few minutes, hand out this scaffolding sheet

"When you've finished or can't make any further progress, look at the worksheet showing three approaches used by students working on this task."

"What might each student do next? Can you take each of their starting ideas and develop it into a solution?"

Here are some prompts that could be offered to students working on each approach if they get stuck:

For Alison's approach:

"What happens to the numbers as you go down the rows?"

"So what happens as you go up the rows?"

For Bernard's approach:

"Which numbers end in a 0 in row $A_2$?"

"Which numbers end in a 0 in row $A_3$?"

"Which of these sequences will hit 1000?"

For Charlie's approach:

"Can you find a similar method to Charlie's to describe the other rows?"

"Which descriptions include 1000?"

Select a few students to report back on how each approach was developed, and invite students to share their own alternative approaches.

**In a follow-up lesson,** return to the very first question "What do you notice?".

Invite students to phrase their noticings as questions and conjectures.

Here are some **key questions** that students might suggest, or which could be offered if students' ideas are not forthcoming (these appear at the bottom of the worksheet):

- How many of the numbers from 1 to 63 appear in the first sequence? The second sequence? ...

- Do all positive whole numbers appear in a sequence?
- Do any numbers appear more than once?
- Which sequence will be the longest?

- Given any number, how can you work out in which sequence it belongs?
- How can you describe the $n^{th}$ term in the sequence $A_0$? $A_1$? $A_2$? ... $A_m$?

Shifting Times Tables provides some preliminary work on sequences that may prepare students for tackling this task.

Ask students to provide convincing proofs of their answers to two of the questions above:

- Do all positive whole numbers appear in a sequence?
- Do any numbers appear more than once?

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

How many different ways can I lay 10 paving slabs, each 2 foot by 1 foot, to make a path 2 foot wide and 10 foot long from my back door into my garden, without cutting any of the paving slabs?