Compatible numbers to ten
These exercises and activities are for students to use independently of the teacher to practice number properties. Some of these activities would be suitable for homework. Others require follow-up during teaching sessions.
Number Framework domain and stage:
Basic facts, stage 4, advanced counting
Level 1 number
Numeracy Project book reference:
These activities can be used to follow the teaching episodes based on “tens frames again” and “patterns to 10”, Book 4, page 34 and are for those students who are able to use the associated number properties.
Prior knowledge. Students should be able to:
- Count on.
- Use their “10 and” facts
During these activities, students will meet:
- The equals sign used as a balance (Exs 1, 5)
- Inequality symbols (Ex 2)
- Use of a box or other geometric shape to represent an unknown number (Exs 3, 4)
- Use of several geometric shapes to represent a range of unknown numbers (Ex 5)
- “Ten and” facts (Ex 6)
- Numeracy problems embedded in language-rich contexts (Ex 7)
- Identifying patterns and generalising from these (Exs 8, 9)
Knowing the basic facts to 10, including the compatible numbers, is essential knowledge if students are to advance to becoming part-whole thinkers. For older students, one way to provide practice and reinforcement of these skills is for the students to play maths games – with two dice. (One at least of these should have a number rather than dots). Another way to prepare students for the part-whole leap is to develop their understanding around their facts to 10, so while these exercises are based around compatible numbers to ten, they include a number of other issues that are significant to the learning of students at this stage. This can introduce a level of challenge even for students who initially seem to know their compatible numbers to 10. Important learning is outlined below.
Use of the equals sign
There are a lot of misconceptions around the use of the equals sign. Some students seem to think that it means “work out the answer”. Consequently, no equals signs have been provided in exercises where simple additions are required. Students with such an understanding may also think that 4 + 6 = 10 is a correct way to write the sentence, but not 10 = 4 + 6. (Hence number 25 in exercise 1). Such a question may be posed as part of practice, then discussed at a later teaching session, or could be included as part of a lesson.
The second part of exercise 1 is provided to reinforce the use of the equals sign when two things are identical/equal, and that the things on the left and right balance. This meaning of equals should also be raised as part of the teaching around this topic.
It has been noted that in recent years there has not been the same emphasis on developing an understanding of inequalities in primary mathematics. This is not intended, but may mean that some students come through without the understandings they have had in the past. Consequently some teaching may need to be provided before students understand the signs and the concepts required by this exercise.
Start, change, result unknown
Students do really know a fact until they can recognise and use it in all three formats. Exercise 3 not only provides practice in recognising the facts in these other formats, but also introduces students to lower level algebra. Mental computation (or recall of known facts) of such simple equations is most sensible method of solution.
Use of shapes as unknowns
Students should have been working with shapes as unknowns for quite some time before reaching secondary school, so should be conversant with what is expected in using a shape in a sentence/equation. In this example, however, the meaning of the unknown has changed. Firstly there are two different shapes – which traditionally would mean that they represent different numbers (though there is the special case where they are the same.) Students may need to discuss this before attempting the problem). However, the unknowns do not represent a single number in this context. This too may to be introduced – that there could be lots of possibilities for such an equation (though is likely to arise naturally if you ask them all to think of two numbers that add up to ten.
The link between addition and subtraction
A single addition fact should be able to be turned into related subtraction facts and simple subtractions should be able to be solved using knowledge of basic addition facts. However, for many students, subtraction understanding lags behind addition understanding. (One of the research reports indicated that subtraction was the weakest of the 4 operations – Irwin & Britt, 2004 evaluation report). Making the link between addition and subtraction is thus essential teaching at this level.
One issue with providing word problems in an exercise alongside simple number problems is that some students learn not to read the words, and simply to pull out the numbers “and do the same to them”. To address this problem, this exercise includes a variety of formats of problem. In fact, number one requires a subtraction with the numbers 6 and 4 – rather than an addition, while others include change unknown format – so could equally be an addition or a subtraction that relies on their compatible number knowledge. This exercise thus provides a good basis for a teaching session around “what words tell us that we should be adding the numbers…” In this teaching session, students could be encouraged to develop a list of words commonly used to indicate that the operations of addition and subtraction are to be used.
Discovery based on patterning
Students learn a lot of mathematics (things that is not necessarily directly taught – or intended to be taught) by identifying patterns. Often, the better we are at identifying patterns, the better we are at mathematics. These exercises look to harness patterning to help students realise that knowing these facts to 10 mean that they can also answer a whole load of other problems. Both exercises require follow-up discussion – and additional practice built around consolidating these discoveries. For example, students could make a poster showing how to use their facts to 10 to answer other problems. They should also do some practice work in using this new skill – in all 3 formats, start, change and result unknown.
Ones to make you think.
Extension work if you like – the material to really challenge the thinking – for all students, regardless of their level, need to have their thinking stretched..
The main issue here being the repeated use of a box indicates that the same number has been used repeatedly. Stage 4 students are not likely to be ready to move to the 2? format, so leave this until stage 5, when some multiplication has been learned. Likewise, some may not be ready to work with numbers in their hundreds – though the question could lead to a useful group discussion for all around this size of number.