Doubling fives
This activity focuses on doubling multiples of five.
Problem
This challenge is about doubling multiples of 5.
Pick a multiple of 5. Double it.
Pick a different multiple of 5. Double it.
Pick another multiple of 5. Double it.
What do you notice? What is the same about your answers? What is different?
What do you think will happen if you double another, different, multiple of 5? Try it!
Will your noticings always be true?
Test out your ideas by trying some more examples.
Mathematicians aren't usually satisfied with a few examples to convince themselves that something is always true.
Can you create an argument that would convince mathematicians?
Getting Started
You could record your results in a table like this one:
| Multiple of 5 | Double the multiple of 5 | How many tens? |
| 5 | 10 | 1 |
Student Solutions
Thank you to everybody who sent in their ideas about this activity. Lots of children noticed that the answer would always end in a zero, but it turned out to be more difficult to think of an argument that would satisfy mathematicians!
We received a lot of solutions from the children at Clifton Hill Primary School in Australia. Ava used a table to work systematically through the multiples of 5:
First I used a "multiples of 5" chart. This helped me a lot. I did one multiple of 5, double the number then how many tens are in the number. And saw that the tens is just like saying the ones times table. (1,2,3,4,5,6,7,8,9,10.) It's never ending. This probably won't persuade mathematicians but it would persuade me!
Good ideas, Ava. Sometimes it's helpful to think of an explanation that would persuade you, first, before trying to find one that would persuade a mathematician!
Marley worked systematically to find out how many 10s were in each double. You can click on Marley's picture to make it bigger:
Firstly I drew up a grid with 3 categories, multiples, doubles, and how many 10s. After that, it was much easier to work with because I could easily notice stuff and see the patterns.
Rosie from the UK draw a similar table:
Every time you take a multiple of 5 and double it, it will always become a multiple of 10. Then, if you see how many tens the answer has got, it has just been divided by ten again. So, if you had 25 and you doubled it, it would equal 50. Then, if you see how many tens 50 has got, it has 5 and that would happen with all the other ones. This will always happen with every single multiple of 5.
Well done, both of you. It certainly looks like the doubles will always be multiples of 10. I wonder how we can be sure?
Ru thought about the tens and ones digits of a number separately:
The answer to this is that it will make a multiple of 10 as 5 + 5 = 10 and 0 + 0 = 0 and then just add the 10s place together so it will just make something that ends with 0 and will make it a multiple of 10 and 5.
This is a really convincing explanation, Ru. We know that 5 + 5 = 10, and so when we add two multiples of 5 together we will just end up doing 5 + 5 lots of times, which is like adding lots of 10s together.
We also received similar solutions from: Ferdia, Benny, Kayhan and Lucas from Clifton Hill Primary School in Australia; Fred from Hardford in the USA; Harry from St Michael's in England; Lucas from Warcop Primary C of E in England; Georgia, Maddy and Chloe from Australia; and Xander from Canada. Thank you all for sharing your ideas with us!
Teachers' Resources
Why do this activity?
This activity provides a context for doubling multiples of 5. The children will also be noticing and describing patterns.
Possible approach
It would be good to start the lesson by counting in 5s and reminding the class that multiples of 5 end in a 5 or 0.
A possible starting point for the activity is looking at doubling single-digit numbers. The children could try numbers of their choice and record them on whiteboards. As you collate the ideas, you could record the responses systematically using a table like this one:
| Single digit number | Double the single digit number | Odd or even? |
| 1 | 2 | even |
| 2 | 4 | even |
What do children notice about all of the answers? Can they spot any patterns? Can they come up with a rule? "Whenever we double a single digit number ..." They then might conjecture that this will also be true for two-digit numbers and might test this.
Explain to the class that they will be looking at doubling multiples of 5. Do they think the answers are going to be odd or even? This time, they will be exploring the number of 10s that are created when multiples of 5 are doubled. You might encourage systematic recording in a table like this:
| Multiple of 5 | Double the multiple of 5 | How many tens? |
| 5 | 10 | 1 |
Some children start at or move onto the extension tasks (below).
Key questions
What is double this multiple of 5?
How many 10s is this? Is it an odd or even number of 10s?
Which numbers end up with an odd/even number of 10s?
Can you record your work systematically?
What is the same about these numbers? Can you spot a pattern?
Why does this happen?
Possible extensions
Extension 1: How about this time you start with a multiple of 10 and then halve it? Can you come up for a rule for when the answer will end in a 0 and when it will end in a 5?
Extension 2: Have a go at the original activity or the extension for other numbers. You could try starting with multiples of 3 and see what happens when you double them. What happens to multiples of 6 when you halve them?
Possible support
Base 10 apparatus or similar may help children to reason about the halves and doubles.