Success depends on students being able to use a broader range of strategies to determine the missing number in a wide range of number sentences, including some with two operations.
Before this, students can find missing numbers using known facts (e.g. 4 + □ = 12, □ × 3 = 15, 24 ÷ 4 = □, 50 + □ = 100).
Success on a wider range of number sentences relies on having a repertoire of strategies to select from, as well as the ability to interpret the placeholder in a meaningful way. Meaning can come from interpreting the number sentence as a story about what was done to a number, or by linking the number sentence to a model situation.
i) Students can solve missing number sentences if they recognise a basic number fact. For example, a student sees 50 + □ = 80 and says “30, I know that 50 + 30 = 80”.
This student might be unable to solve 152 + □ = 182, if they do not have other strategies at their disposal.
ii) A student sees 25 − □ = 17 and says “I think of 17 plus something gives 25, that must be 8”, but this student may not be able to find the missing number in 4 × □ + 6 = 22.
iii) A student at this level can use systematic trial and error (also referred to as guess, check and improve) to find the missing number in various number sentences.
iv) Later, students can use logical arithmetic reasoning to deduce the missing number in various number sentences. This is easiest when the student interprets the number sentence with a story or model.
v) Later, students choose a sequence of inverse operations to solve various number sentences. This is a more formalized strategy which focuses on the written symbols rather than the physical situation, and requires explicit use of inverse operations. It is the beginning of formal algebra.
It is essential that students are encouraged to make sense of a number sentence by modelling with materials and stories. Once this is secure, the transition to working only with abstract mathematical symbols is much easier.
Activity 1: Write your own story and number sentence helps students to establish the link between the materials, the story and the number sentence.
Activity 2: Cover it up provides students with an opportunity to discuss strategies for finding the missing number in a number sentence. The general teaching strategy for the guess-check-improve strategy is to encourage students to work systematically and record their working in order to improve upon previous attempts.
Activity 3: Can you do it another way? provides students with opportunities to discuss and use relational thinking (their understanding of the relations between numbers) to solve special types of missing number sentences. This activity emphasizes the ‘balance’ idea of the equals sign (equivalent expressions), rather than the ‘answer’ idea of the other activities.
Activity 1: Write your own story and number sentence
Provide students with a situation such as the following:
Four plates with three counters on each and a spare two counters on the table.
Ask students to write a simple story and a matching number sentence (see samples below). Move around the class, looking for good examples to discuss with everyone, focusing on the different structures of the stories (e.g. different actions which lead to different operations) rather than focusing on the different contexts (e.g. counters, lollies, biscuits, etc).
Ask the students “How is this story/number sentence the same as or different to …?”
This provides students with a meaningful link between the number sentences and the materials and an appreciation of a ‘ number sentence family’ (analogous to a basic fact family). Place samples of students’ work on the wall for future reference to number sentence families.
This activity can be undertaken at many different levels, depending on the situation provided and students’ ability to write number sentences.
Possible Stories:
1) Multiplication and Addition
If I have 4 plates with 3 counters on each plate and 2 spare counters, how many counters do I have altogether? 4 × 3 +2 =14
2) Division* with remainder
If I have 14 counters to be put onto plates, then how many plates will I need if I can put 3 onto each plate? 14 ÷ 3 = 4 remainder 2
3) Division# with remainder
If I have 14 counters to be shared among 4 plates, then how many on each plate? 14 ÷ 4 = 3 remainder 2
4) Subtraction and Division*
If I have 14 counters and gave 2 away, then how many plates will I need if I can put 3 onto each plate? (14 − 2) ÷3 = 4
5) Subtraction and Division#
If I have 14 counters and gave 2 away, and then need to share among 4 plates, how many counters would be on each plate? (14 − 2) ÷4 = 3
Notes:
Division* is Quotition Division (repeated subtraction of the ‘quota’ of 3)
Division# is Partition Division (equal sharing onto 4 plates)
For more information: More about Construction of Number Sentences (Level 2.5)
Write a number sentence (such as 4 × 3 + 2 =14) on the board and hold a blank card over one of the numbers. For example, 4 × □ + 2 = 14 (hiding the 3). Ask the students “What number is behind the card?” and then “How could we find this number if we did not know what it was?”
The reason for the first question is to highlight that the focus of the second question should be about strategies to find the missing number, not what it is. (Let students know that after this discussion, you will provide other number sentences where they do not know the hidden number and they will need to try various strategies.)
The table below lists possible student responses. The intention of the class discussion is to shift students’ responses to those further down the table. As students work on the problem, move around the group and note the different strategies in use. Call students showing evidence of guess-check-improve to show their method first. Then call some students using logical arithmetic reasoning or other sophisticated method to demonstrate their work. Discuss how to improve each method. Then offer other number sentences for more ‘ cover-up’ teaching sequences.
Possible student responses
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Comment |
“I just knew it was 3”
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May reveal no thought about strategies |
“We could guess some numbers, and see if they work”. “You could try 10, but that doesn’ t work, or you could try 1 but that doesn’ t work and you would keep trying until you got 3”.
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Random guessing |
“We could guess some numbers. Let me try 10, this gives 42, so 10 is too big Let me try 1; this gives 6, so 1 is too small. Let me try 5; this gives 22, so 5 is too big. I need to try a number between 1 and 5, maybe 3. This gives 14 which is right, so 3 is the missing number”. |
Evidence of guess-check-improve. To reduce memory load, it is a useful idea to arrange the numbers in a table (see below). This encourages students to 'improve' on each try rather than guessing randomly. |
“If we had 14 counters altogether and 2 were left on the table, then there would be 12 on the plates. If there were 12 counters placed equally on 4 plates that would be 3 on each plate.” |
This student has used logical arithmetic reasoning, which is a precursor to the more formal technique of using inverse operations. |
“We could write the number sentence in a different way like this: (14 − 2) ÷ 4 = 3” |
This student has noticed the 'number sentence family' from Activity 1: Write your own story. |
The student points to the symbol group (4 × □) and says, “What number, when I add 2 gives 14? It must be 12”. Now the student points to the □ and says, “What number, when I multiply it by 4 gives 12? It must be 3.” We could write it like this: |
This is another sophisticated logical reasoning strategy, and is the basis of algebraic manipulations working towards Level 5. Treating the symbol group as an entity distinguishes this stage. |
Guess-check-improve strategies can be much improved when students use labelled tables to organise their work, and to help them work systematically. Teachers should highlight the importance of the ‘improve’ step – the next try uses information from the earlier try, it is not just another guess.
Missing number |
4 × □ + 2 |
Is it equal to 14? |
|
Try 1 |
10 |
42 |
Too big, need a number smaller than 10 |
Try 2 |
1 |
6 |
Too small, need a number bigger than 1 |
Try 3 |
5 |
22 |
Too big, need a number smaller than 5 |
Try 4 |
3 |
14 |
Right, so 3 is the missing number |
Activity 3: Can you do it another way?
Ask students to find the missing numbers in the following number sentences.
There are two common methods. For the first question, a student might work out 6 + 13 as 19, and then work out 19 – 4. This is a good method, but in this case, there is a better method which might be encouraged because it is closer to the structural thinking that students will need when learning algebra. These number sentences have been carefully chosen to make the next method better.
A student who appreciates that the equals sign indicates equivalence (or balance) between the quantities on either side can find the missing number using relational thinking. In the first number sentence 4 + □ = 6 + 13 this student might say “For the sides to balance, I need a number which is 2 more than 13, so it must be 15.” Note that this student did not need to calculate the total of 19 as an intermediate step. Students may point at the numbers as they describe their strategy; alternatively, drawing arrow diagrams like the following will help to explain the strategy to other students and make a record for later discussion.
As well as preparing students for algebra, relational thinking frequently offers short cuts in mental calculation. For example, it is much easier to determine that 36 is the answer to 96 − 60 instead of 92 − 56. Relational thinking enables a student to replace the calculation of 92 – 56 by the easier calculation.
Likewise, instead of calculating 122−95, a student can replace it by 127 – 100, and 27 is clearly the answer.
4+ □ = 6 + 13 | 47 + 68 = 50 + □ |
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92 − 56= □ − 60 | 122−95 = □ −100 |
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