Binary Thinking
3 out of 3
In the binary number system, we only use the digits 0 and 1. What can you discover about this system?
Problem
Binary Thinking printable sheet
In the binary number system, there are only two digits: 0 and 1. This is a 'base 2' system instead of our usual 'base 10' system, which means that as soon as we get to the number 2, we have to use a new place value column to show this.
The binary place value columns look like this:
| 4s | 2s | 1s |
Which means that the numbers from 0 to 7 are written like this in binary:
| Base 10 | Binary |
| 0 | 0 |
| 1 | 1 |
| 2 | 10 |
| 3 | 11 |
| 4 | 100 |
| 5 | 101 |
| 6 | 110 |
| 7 | 111 |
We wouldn't say '111' as 'one hundred and eleven' here - we would say 'one one one', and we could think about it as 'one 4, one 2 and one 1'.
Have a play around with this idea. Can you work out how we came up with all the binary numbers in our table? What other numbers can you write using the binary number system?
Once you've got the hang of writing numbers using this system, have a go at some calculations. Is there a way of changing your usual methods of adding, subtracting, multiplying and dividing so that they work with binary numbers? Or do you need new methods altogether?
There might be other things you would like to investigate about the binary number system. Let us know what you find out!
Getting Started
We use ten digits in our usual number system: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. When we count past 9, the ones column resets to 0, and the tens column goes up by 1, to make the number 10. The number 10 is written with a one and a zero because it means 'one ten and zero ones'.
How is the binary number system similar to this?
You might find it helpful to draw out a much bigger place value grid, working out what each column would represent. Once you have this, you can write numbers by putting 1s in some of the columns and adding up those column headers. For example, the number 5 in binary is written as 101 because it is 4+1, so there is a 1 in the 4s column and a 1 in the 1s column:
| 4s | 2s | 1s |
| 1 | 0 | 1 |
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
This investigation can be used to introduce multibase work, which can help learners gain a deeper understanding of how our usual base ten number system works. Adapting methods of calculating to work with this new system can give learners more insights into what is happening when they use formal written methods of adding, subtracting, multiplying or dividing.
Possible approach
Introduce the concept of there only being two symbols to represent numbers, rather than our normal ten. Give learners the opportunity to discuss how our usual place value system could be adapted to cope with there only being two digits.
You might like to choose some questions to focus on as a whole class (such as how column addition would work in the binary number system), and then leave the open-ended extension as a 'simmering' task. Leaving a working board up in the classroom will enable learners to add further ideas over the next few days.
Key questions
What would the next place value column be?
Can you see a pattern? Why might the number be doubling each time?
How could you adjust that method to add binary numbers? What do you notice?
Possible support
Some children will need a big place value grid drawn out for them, perhaps with the columns 32, 16, 8, 4, 2 and 1. They might also find it easier to think about a light being 'on' (instead of a 1) or 'off' (instead of a 0) in every column. The ones, or lights, are indicating when the column header is being used in a number. There is more information about this in the Getting Started page.
Possible extension
Provide plenty of opportunities for learners to explore their own ideas, once they are confident with how to write numbers in binary. Finding square roots of binary numbers is a particularly interesting concept for learners to explore.
Once the binary number system has been fully explored, the problem Sometimes We Lose Things could be used to introduce the idea of other bases.