Addition and Subtraction

Objectives

To investigate statements about number.

To explore whether statements are always, sometimes or never true.

Resources

Always, sometimes or never? (NRICH) | 📄 Always, sometimes or never? sheet | Three sheets of plain A4 paper

Vocabulary

investigate | prove

Today's Lesson

What to Get Ready

Familiarise yourself with the investigation by accessing the NRICH website.

ACCESS - Always, sometimes or never? (NRICH)

The background information to this problem may also be useful to read beforehand.

READ - Always, sometimes or never? Note (NRICH)

This learning will encourage your student to consider some simple statements about number. The idea is to test whether each statement is always, sometimes or never true.

Access the solutions

READ - Always, sometimes or never? Solutions (NRICH)

Print and cut out the statements from 📄 Always, sometimes or never?

Write on top of the three sheets of A4 paper:

Sheet 1 - Always
Sheet 2 - Sometimes
Sheet 3 - Never

Introduction

Tell your student that we will be looking at some more statements, similarly to the Conjecture that we investigated in the previous lesson (Lesson 3).

Tell them that good mathematicians like us need to think about what is true or not true in Maths and number.

To do this, we should read what other people have said about numbers and try to prove them right or wrong. This helps us to be great thinkers about number, and therefore to get better and better at solving problems.

Begin by reading out the first statement on the sheet:

When you add two numbers you can change the order and the answer will be the same.

If your student is very certain that they know whether this is always, sometimes or never true, ask them to begin to start proving it either way.

Say: ‘We need to test it and prove it. To do this, we need to start with a number and do what it says in the statement or conjecture. So, we could start with 12 and add 7 to it. Can you do that?’

Establish that 12 + 7 = 19.

Say: ‘Now we need to change the order of the addition, as that is what the statement is telling us to test. So, can you change the order?’

Establish that the order should now be 7 + 12.

Ask: ‘Does it still total 19?’

Let your student work this out, and then agree that it does still total 19.

Say: ‘Now, this is very important. We can’t say that this statement is always true yet. Why not?’

Encourage your student to realise that we have only tested one example. We need to test more. Suggest that 3 to 5 examples per statement would be enough.

Once they have done enough testing, ask them to place the statement on the ‘Always’ sheet of paper.

This video outlines how to explore and test a mathematical statement to decide whether it is always, sometimes or never true.

Main Activity

Repeat the above task but by testing the rest of the statements.

Encourage your student to test between 3 and 5 times for each statement, placing the statement on the correct sheet of

paper, depending on what they have found.

Whilst testing, encourage your student to talk to you about what they are thinking and how they can test each statement, but allo

w them to work as independently as they can, only helping them when really needed.

Encourage their thinking by asking questions such as:

How do you know that…?
What makes you say that…?
Have you thought about trying…?
Why do you think this is true / untrue / sometimes true?

Citations

[1] nrich.maths.org [2] nrich.maths.org [3] nrich.maths.org