Two Ways

Years 5 - 8

This page would be more informative
with a photo from your classroom.
doug@blackdouglas.com.au

Summary

This activity arose in a discussion with a group of secondary teachers on the potential for using Threaded Activities in their classrooms. Six Plus was used as an example to help the group see that a rich mathematical activity experienced for frequent, small amounts of time in a familiar context but with a fresh challenge, can help students construct their own learning. Two Ways developed from this by asking if Six Plus could be adapted to help children learn to add and subtract integers. 		 

Materials

  • 1 calculator for each person

Background

The success of the activity depends on:
  1. the teacher saying as little as possible, and
  2. using a calculator that can enter and display negative numbers.
Usually simple four function calculators have a change sign button (+/-) for entering negative numbers and usually the number is entered first and then given its positive or negative 'polarity' or 'state'. (This is the reverse of the way negative numbers are written by hand.)

Annoyingly too, some (most?) calculators write the negative using their subtraction sign. Subtraction is an operation involving two numbers; negative is a characteristic of a particular number. This distinction is easier to learn if, for example, six subtract negative 4 is written as 6 - -4, with the negative as a superscript. Similarly 6 - +4.

Also note in this last sentence the convention that if no sign is used, the number is positive.

 

Content

  • addition facts beyond 10
  • addition facts to 10
  • mathematical conversation
  • negative numbers
  • operations - integers
  • operations - whole number
  • properties of number
  • properties of zero
  • recording - calculator
  • recording - written
  • subtraction
  • writing numerals

The teacher is asked to say as little as possible after the introduction because the challenge is for children to discover the calculator's rules for adding and subtracting positive and negative numbers. Part of the expectation is that they will find the need to use negative numbers, so be careful about hinting.

Procedure

The teacher asks for a student volunteer, who comes to the front with their calculator. The teacher also has a calculator.
I'm going to set you a challenge today that is called Two Ways. It will probably take a few days to figure this one out. Amy is going to help me demonstrate it.

Amy I am going to press some buttons on my calculator. You don't do anything yet.

As you press the buttons, say what you are doing, but don't show what you are doing. Start with something as straight forward as 6 + 9 = 15.
Okay, first I am pressing 6.
Now I am pressing add or subtract; I'm not telling you which one.
Now I am pressing one number ... now equals.
Show Amy the answer and ask her to tell the class what it is.
Now it's your turn Amy. You have to reconstruct what I did. If you get to my answer in one try you get a point.
How did I start? ... Okay write that on your calculator.
What did I do next? ... And then ...
Amy will undoubtedly reproduce 6 + 9 = 15 and with appropriate joy you assign her one point.
Now I am going to do it again using the same rules.
Make a point of starting with the same number and saying the steps above in exactly the same way. However this time you will be pressing 6 - -9 =15. Again ask Amy to announce the answer.
But it's the same?
Comment that it is indeed exactly the same as last time.
I started with the same number as the first time.
I ended with the same number as the first time.
I even followed the same rules in the middle.
But I did do it a different way ... The challenge is to find my different way.
Thank Amy for helping and open the challenge to everyone.
I am only going to give you a couple of minutes on this today, but if you do figure it out, then please come out here and show me Two Ways.
At this stage you might feel tempted to press on with the activity for the whole lesson. We hope you won't. This is not intended to be a lesson on integer arithmetic. It is simply a moment to introduce a challenge. By returning to the challenge for a few minutes each lesson for the next few lessons:
  • students will realise that you really do expect them to be the discoverers
  • you will be providing time for students to construct their own learning through experiment and peer group discussion
  • you will demonstrate that you value what will become known as adding and subtracting positive and negative numbers (integers).
So, that's the challenge. If I give you any start number and any finish number, you have to show me you can get from one to the other in two ways using our rules.

I will give you a few minutes in each of the next few lessons to work on it together. But I won't be telling you. You have to tell me.

Your roles through the threaded activity are to:
  • begin the discussion with a Start Number / End Number pair, for example a pair that involves starting negative and finishing more negative, or a pair that involves moving across zero, or a pair that includes zero as one of the numbers.
  • invite children to show you their knowledge to date
  • encourage students to lead discussions with the class
  • support the development of an agreed class explanation of how the calculator does this arithmetic.

Enrichment

The expected outcome of the Two Ways is that students are effective and efficient at integer arithmetic. However, even though they have discovered them for themselves, they are only working from rules. Further meaning can be added by exploring how these rules are applied in one or more of the various models that have been developed to support teaching integer arithmetic. Maths300 offers three:
  • Lesson 32, Walk The Plank
  • Lesson 76, Protons & Anti-Protons
  • Lesson 158, Goods & Bads


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