Compatible numbers to 20
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 5; early additive part-whole
Level 2 number
Numeracy Project book reference:
These activities can be used to follow teaching episodes that build on tens frames again and patterns to 10 (on compatible numbers 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:
- Recall the compatible numbers to 10
- Solve problems with small numbers using a part-whole strategy
- Recall “10 and facts”
During these activities, students will meet:
- The equals sign used as a balance (Ex 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)
- Linking repeated addition to multiplication, using statements with boxes in them
- Linking addition to subtraction (Ex 6)
- Numeracy problems embedded in language-rich contexts (Ex 7)
- Use of several letters to represent a range of unknown numbers (Ex 9)
Many of these activities parallel those in “compatible numbers to 10” (stage 4). They are designed for students who already have their facts to 10, but need to extend these to 20. However, these students may or may not have had exposure to the algebra aspects of those activities, so they have largely been repeated here. The teaching notes for “compatible numbers to 10” contain information about some significant teaching points embedded in both sets of exercises, so will not be repeated here, but should be accessed and read prior to using these activities.
This exercise is split into 3 parts. The final part has sentences “written in reverse”. Students often think the equals sign means “work out the answer”, so this part is designed to help students realise that the equals sign means that the statements on either side are identical or balance. This understanding is essential for the development of algebra.
When reviewing this exercise with students it can be useful to ask students “why” some of the problems are false, or do not equal 20. For example, students may come up with reasons like “as both of the numbers are less than 10, then the total must also be less than 20”.
This exercise is in two parts, the second part again reverses the sense of the problems so the sum is on the right of the symbol. This is more challenging and will need to be discussed as part of relevant teaching before setting these problems. In this second set of problems students are doing the reverse of comparing a sum to 20. Rather, they are comparing 20 to the results of their calculation. This reverse sense can be problematic as the students are using inequalities, so may want to use the incorrect sign. Reading the sentence in reverse can alleviate this problem.
This exercise looks to develop students’ understanding of algebraic notation, and of multiplication as repeated addition. Most of the questions can be seen to be simple doubles, though students are not initially asked to “work out the missing number”.
Linking addition to subtraction
Exercise 6 aims to encourage students to look at the links between addition and subtraction in a formal manner. In compatible numbers to 10, the answers for the word problems (in exercise 7) indicate that students could write either a subtraction or an addition, which they could then use to solve the problem. At stage 4, this informal linking of addition and subtraction was not more formally explored. Before getting students to work on this exercise, it may pay to introduce a word problem or two, and explore students’ solution strategies, as this is likely to elicit both subtractions and additions. The formal use of notation to link these sentences can then be introduced and explored. A follow-up to this exercise is to pose the following question.
This question is for a group discussion:
Can any subtraction be changed into an addition, and can any addition be changed into a subtraction? Can you prove this? Your teacher will need to be convinced by any explanation or proof that you come up with.
Word problems and more word problem
These exercises look to build the list of ways we can describe additions and subtractions started in “compatible numbers to 10”. The second exercise also pushes out the form of number from “compatible numbers to 20” to higher compatible number pairs.
Letters to stand for a range of unknowns
This exercise builds on the work in “compatible numbers to 10” in exercise 4, as well as exercise 4 question 1 in this set. It should be seen as a logical next step, and not as something that is particularly difficult for students to access.