Number lines
Problem
Leah and Tom both have number lines and a counter. They always start with their counters at zero.
Leah's number line goes along from left to right like this:
First Leah made a jump of three along her number line and then a jump of four. Where did she land?
Next Leah made a secret jump along her number line. Then she made a jump of five and landed on $9$.
How long was her secret jump?
Then Leah made a jump of six and another secret jump. She landed on $10$.
How long was her second secret jump?
Tom's number line goes up and down like this:
First Tom made a jump of three up his number line and then a jump of two. Where did he land?
Next Tom made a secret jump up his number line. Then he made a jump of six and landed on $8$.
How long was his secret jump?
Then Tom made a jump of four and another secret jump. He landed on $7$.
How long was his second secret jump?
There are some printable NRICH Roadshow resources for the game that this task is based on: Instructions and Number Line.
Getting Started
It might help to have a number line of your own to work with.
You could print off this sheet of number lines.
If Leah did a jump of five to get to $9$, where did she start?
How could you find out?
So how big was the jump she had to do to get there from zero?
Student Solutions
It was interesting to read the solutions that came in. While doing this I found that the question had been interpreted in two ways. It was all about whether each new step meant a new start at zero or not. I think that it is right to accept both interpretations. This sometimes happens in the classroom with challenges or investigations.
Firstly, here's a solution from Badger Class at Fawkham C of E School:
First Leah made a jump of three along her number line and then a jump of four.
She landed on seven. We worked it out by adding four and three.
Next Leah made a secret jump along her number line.
Then she made a jump of five and landed on nine. We worked out by counting back three from seven and then add on five.
Then Leah made a jump of six and another secret and she landed on ten. So the secret number is four. We worked it out by the sum 6 + 4 = 10
TOM
First Tom made a jump of three up his number line and then a jump of two. So the number is five. We worked it out by adding three and two so that makes five.
Next Tom made a secret jump up his number line. Then he made a jump of six and landed on eight. He went back three jumps and forward six jumps.
Then Tom made a jump of four and another secret jump. He landed on seven. His second secret jump was three.
Thank you to Daniel from Englefield Green Infant School and Daina and Emma from Kelso State School for their solutions to the Number Lines problem. Daniel said:
Leah's first jump landed at 7, 3+4=7
Leah's first secret jump was 4, 4+5=9
Leah's second secret jump was 4, 6+4=10
Tom's first jump landed at 5, 3+2=5
Tom's first secret jump was 2, 2+6=8
Tom's second secret jump was 3, 4+3=7
Lastly, this was sent in by Alexander who goes to Freemantle Primary School in Australia:
Leah's first secret jump was 4, and her second secret jump was 4.
Tom's first secret jump was 2, and his second secret jump was 3.
If you come to this challenge after November 2013 please send in any further solutions and ways in which you worked them out.
Teachers' Resources
Number Lines
Leah and Tom both have number lines and a counter. They always start with their counters at zero.
Leah's number line goes along from left to right like this:
First Leah made a jump of three along her number line and then a jump of four. Where did she land?
Next Leah made a secret jump along her number line. Then she made a jump of five and landed on $9$.
How long was her secret jump?
Then Leah made a jump of six and another secret jump. She landed on $10$.
How long was her second secret jump?
Tom's number line goes up and down like this:
First Tom made a jump of three up his number line and then a jump of two. Where did he land?
Next Tom made a secret jump up his number line. Then he made a jump of six and landed on $8$.
How long was his secret jump?
Then Tom made a jump of four and another secret jump. He landed on $7$.
How long was his second secret jump?
There are some printable NRICH Roadshow resources for the game that this task is based on: Instructions and Number Line.
Why do this problem?
This activity is a good one to try once children are familiar with number lines, and they have begun to use them for addition. It could be a good way to talk about subtraction.
The problem requires higher-order thinking compared with simply adding or subtracting. It encourages children to be confident in "undoing" operations, and therefore introduces them to the idea of subtraction being the inverse of addition and vice versa. However, you don't need to use this vocabulary at first.
The idea of having an unknown number (the secret jump) in a calculation (further jumps along the number line) is the essence of algebra, and this problem introduces that important concept but in an appropriate context.
Possible approach
Start with a number line with which the children are familiar and then introduce a vertical number line if they have not met them before.
Involve the children in suggesting numbers to add using both number lines before introducing the idea of the 'secret jump'. Next, it might be appropriate to try some of your own examples with the group, asking them how they are going to find the secret number of jumps. Encourage pupils to explain how they will do it in their own words and discuss whether or not this will work.
Key questions
Possible extension
Use a number line which goes to up to $20$ or $100$.
Possible support
Make sure that children are counting the jumps they make, and not the starting point. If there are difficulties with a vertical number line they could focus on using a horizontal number line instead.