The Big One

Years 2 - 8

Summary

A calculator game for two players that helps develop an understanding of the inverse nature of multiplication and division and a sense of the effect of the operation of division. Suitable for threading.

Materials

  • One calculator per pair

Note: This investigation has been included in Maths At Home. In this form it has fresh context and purpose and, in some cases, additional resources. Maths At Home activity plans encourage independent investigation through guided 'homework', or, for the teacher, can be an outline of a class investigation.
  • Visit the Home Page for more Background.
  • For this specific activity click the Learners link and on that page use Ctrl F (Cmd F on Mac) to search the task name.

Procedure

  • Player A is the 'hider'. Player B is the 'finder'.
  • Player A thinks of a mystery number and 'hides' it in the calculator by entering ÷ [mystery number] = .
    Note: You may want to begin the game by limiting the choice to, say, 2-digit numbers.
  • If clear is not pressed, the calculator is now set up to divide by the mystery number whenever = is pressed.
  • Player B has to find the mystery number by entering [guess number] = .
    Note: Some calculators show a ÷ sign when [guess number] is pressed to remind about the operation which has been entered.
  • If the guess is correct, a special number (the Big 1) will appear on the screen. If the guess is not correct, it will provide information for the next guess.
  • Without pressing clear Player B continues to enter [guess number] = until the Big One appears.
  • Keep a record of each guess. It is also useful to comment on whether the guess was too big or small.
  • Score one point for each trial.
  • Players swap roles. Winner is the player with the lower total score after, say, six rounds.
  

Content

  • decimal interpretation
  • division
  • equations: creating/solving
  • estimating number
  • multiplication
  • operations - whole number
  • pattern generalisation
  • pattern interpretation
  • pattern recognition
  • problem solving
  • recording - calculator
  • recording - written
  • times tables
Consider playing the game as a threaded activity - a few minutes a day, 3 or 4 times a week for a number of weeks - and building in class discussion about the clues which guide students to find The Big One.

Variations

The game may not be the appropriate starting point with your class, but the concept behind it still has much to offer.
Sometimes you use a familiar activity and it develops in a new direction. During my preparation for working with Year 5/6 at St. Francis Xavier school in Lake Cargelligo, I was thinking through The Big One and I realised that beyond the game, there was potential to make stronger connections between division and multiplication and to develop a deeper link between division and the patterns in the decimals that result. Basically, the game is about making the number 1 appear on the screen when the calculator has been 'taught' to divide by (say) 3. But it begs the questions:
  • What happens if you want other whole numbers to appear?
  • What does it mean if non-whole numbers appear?
The lesson turned out to be quite stimulating - for teacher and children I think - so I recorded the details and, with the help of others since, the two lesson variations below have developed.
Doug. Williams, Consultant
Discovering Times Tables
Today we start by teaching your calculator something. I want you to turn your calculator on and press these buttons...
Record on the board as you say:
÷ ... 3 ... =
divide (pause) three (pause) equals
Emphasise that these are buttons by drawing each one inside a small square.
Now we have taught your calculator to divide by 3. I'll prove it to you. Who knows what you get if you start with 6 and divide it by 3? ... Correct, 2! ... Now just press 6, then =. What answer does your calculator give?
On the board, using something like the whiteboard image above, draw and explain to show that 6 was entered and pressing = makes the calculator do what we taught it, so the answer 2 appears on the screen. Emphasise that the calculator will only continue to do this if Clear, or other buttons, aren't pressed.
  • Ask children to give you a couple of other numbers to test, such as 18 or 12 and record them the same way.
  • Also ask for a number that won't give a whole number answer. Make sure the children can actually identify the decimal point in the screen display and mention that these numbers are called decimals.
Okay, let's find as many numbers as we can that can be divided exactly by 3. Try any number you like and when you find one, come and write it in this box on the board.
A bit of excitement tends to develop at this stage as children take ownership of the searching and recording. When sufficient data has been collected, at least one number from each child, stop the search and encourage a deeper look at the data. 		The Big One in High School

Lilydale High School was working hard to build their curriculum around Working Like A Mathematician. They added many resources to offer teachers alternatives to text-based learning. One of those was Calculating Changes membership and the six coaches (a big school!) began their exploration of our site with The Big One. They were able to fill two poster paper sheets with features of the activity likely to encourage learning, one of which was the concept of threading, then trialed it in their classrooms.

Threading, that is using this same activity 2 or 3 times a week for 10 minutes each time over 2 or 3 weeks, led to welcome changes in the classrooms. Students did learn mathematics related to division and multiplication, but they also became more confident, interested and involved in their learning.

It is very likely there are some incorrect entries and the following process helps children to find them. Equally, it illustrates that there is a link between multiplication and division.

Children record in their journals as a class record is built on the board guided by:

What's the smallest number that worked? ... Okay, write zero at the top. ... We are going to organise our data into a list of numbers that worked. ... What's the next smallest? ... Write it. ... and the next? ...
Continue this organised recording until someone suggests that there is something special about these numbers. They are the three times table!
Hey, how about that. We started out by teaching our calculator to divide by three and we discovered the three times table. We got multiplication by starting with division.
  • Ask the children to make a brief journal entry describing how the activity has developed.
  • On another day teach the calculator to divide by another number. In other words, thread this variation of The Big One to lead into using the game version above.
Discovering Decimal Patterns
Decimals appeared as part of the times tables discovery above and they offer a further opportunity to explore.
The other day we taught our calculator to divide by 3. Can you remind me how to do that?
Record the button presses on the board so they will become the 'heading' for the central of three columns.
Thea, tell me any whole number and let's see what happens when we divide it by three.
In the first column write Thea's number. In the third, write the calculator's answer. Between them use an arrow pointing up to the column heading, or dots (...), to indicate the calculation above that the calculator did.

Collect a few more numbers and answers from the children.

Okay mathematicians, your investigation today is to keep exploring any numbers you like divided by 3. In a while I will be asking you what you discover.
Try to avoid organising the data for children. Learning to work like a mathematician involves making decisions about how to organise data in the hope of finding patterns or connections on which to build hypotheses. Eventually though you would hope to find records such as:

0
1
2
3
4
5
6
7
etc. 		...
...
...
...
...
...
...
...
...
0
0.3333333
0.6666666
1
1.3333333
1.6666666
2
2.3333333
etc.

Discuss and record children's discoveries and encourage them to make their own record of the investigation in their journal.


Return to Calculating Changes Activities

Calculating Changes ... is a division of ... Mathematics Centre