Mathematics Developmental Continuum P-10

Mental Addition and Subtraction: 3.25

Supporting material

Indicator of Progress

At this level students are able to estimate and mentally compute multi-digit addition and subtraction calculations using a variety of strategies, including adjusting the result of an easier calculation, and changing to easier numbers with the same answer. They can carry out these processes confidently and fluently when required in everyday situations and in their mathematical work (e.g. to check reasonableness of results of calculator or pencil and paper calculation). They use empty number lines to assist with informal recording of intermediate steps in these non-algorithmic strategies.

See: More about mental calculations for addition and subtraction.

Mental calculations for addition and subtraction and the illustrations below explain more about a variety of strategies that students can use. Although the calculations here are written, the intention is that these calculations would be carried out by students either entirely mentally or verbally, or supported by informal jottings when the calculations are more difficult. A key goal for mental calculation is that it should be quick – especially quicker than using written calculation or getting a calculator.

Illustration 1: Adding three numbers efficiently

Kilian, Zafar, Max and Michael needed an estimate of the total distance from Longreach to Winton (180 km), to Cloncurry (338 km) to Mount Isa (117 km).

Zafar, demonstrating good mental strategies based on rounding, quickly thought like this:

“180 is approximately 200, 338 is approximately 340 so I add 200 to 340, which is 540. Then 117 is approximately 100, so I add another 100, to get 640. The distance from Longreach to Mt Isa is approximately 640 km.”

Kilian also showed good mental strategies based on rounding, but he also noted whether he was over-estimating or underestimating. In this instance, Kilian has found an ‘upper bound’ for the answer: a number that is definitely too large. This is more sophisticated than Zafar’s approach, even though Zafar, in this case, happens to be closer to the accurate answer (635 km).

“180 is less than 200, and 338 is less than 340. When I add 200 to 340, I get 540, which is an overestimate. The total distance will be less than 540 + 117, which is 640 + 17, which is less than 660. The total distance from Longreach to Mt Isa is near 660km but less than it.”

Kilian could also quickly find a lower bound.

Max also showed good mental strategies and got a very close estimate quickly. He also used knowledge of complements to 100 and the commutative property of addition. Max thought:

“If I start with 180 km, I need 20 to make 200. I can nearly get this 20 from the 117.  180 + 117 is approximately 180 + 20 + 100, which is 300. Then I need to add 338, giving 638. The distance from Longreach to Mt Isa is very close to 638 km. In fact, I added 3 too many by rounding 117 up to 120, so the correct answer is 635 km.”

Michael generally finds mental calculation difficult, but he immediately knew that the distance was more than 500km. He thought:

“The first distance is more than 100km, the second distance is more than 300km and the third is more than 100km, so the total is more than 500km.”

At this level, he will work on making better estimates.

Note how the three good mental calculators tend to work from one number and then add the place value components of subsequent numbers in stages.

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Illustration 2: Subtraction methods

Kilian, Zafar and Max wanted to know how much further it was to drive to Ballarat from Melbourne via Geelong (75km + 87 km), than to drive direct (113km). Their mental calculations split numbers into convenient components, and adjust the results of easier calculations.

Zafar used good estimation skills, including subtraction in stages, when he thought:

“75 + 87 is approximately 70 + 80, which is 150. Then I subtract 100, leaving 50 and subtract the remaining 13 to get approximately 40. It is about 40km further to go through Geelong.”

Kilian first worked out 75 + 87 by thinking

“70 + 80 is 150 and 5 + 7 is 12, so the distance through Geelong is 152 km”. Then subtracted 113 in stages, thinking: “152 – 100 is 52, and 52 – 10 is 42 and 42 – 3 is 42 – 2 – 1, which is 39. It is 39 km further to go through Geelong.”

Max knew it was 152 km through Geelong and needed to subtract 113 km. He noted that the subtraction 153 – 113 is easier, and then adjusted the result of this easier calculation. He thought:

“To calculate 153 – 113, I first subtract the 3, so the calculation becomes 150 – 110. Then I subtract 100, so the calculation becomes 50 – 10, which is 40. But by adding 1 to the 152 at the start, I have made the answer 1 to big. Hence the answer is 39. It is 39km further to go through Geelong.”

Although it is complicated to follow the methods that these boys have used when they are written down, all of these methods can be used spontaneously by students with good mental strategies.

Illustration 3: Links to the Mathematics Online Interview

Examples of the types of tasks that would be illustrative of mental addition and subtraction aligned from the Mathematics Online Interview:

  • Question 22 – Derived strategies
  • Question 23 – Multi-digit strategies
  • Question 24 – How many digits?
  • Question 25 – Estimating and calculating addition
  • Question 26 – Estimating and calculating subtraction
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Teaching Strategies

Students need to be able to choose flexibly from various strategies when faced with a given computation. Over time, a wide range of strategies should be discussed with students. Many opportunities arise in the course of ordinary classroom life to focus on these and practise them. Over time, all of the strategies in the illustrations should be discussed. It is important to do both mental computation (exact answers) and estimation, which is a very common requirement for daily life (e.g. checking change when shopping).

A key part of this indicator is that students can carry out mental computation fluently, confidently and when it is appropriate in their everyday lives. Mental calculation that is not quick and carried out confidently is not useful. This is best achieved by encouraging students to use mental computation in all circumstances where it is sensible: in lessons, in the playground, when helping with classroom organisational tasks. Explain to parents the importance of having children assist them with mental calculations at home, and when shopping.

Activity 1: Illustrating mental computation with hundreds charts and number lines provides students with 3 different visual representations that they can use to explain mental computations to others and also which they can use mentally to guide their own computations.

Activity 2: Adjusting numbers to make subtractions easier suggests an appropriate explanation that students need to see to adapt subtraction questions correctly. Students should often be asked not only to explain WHAT they do, but also WHY it works.

Activity 1: Illustrating mental computation with hundreds charts and marked and empty number lines

The aim of this activity is to give students three representations with which they can explain mental strategies and visualise when doing mental calculations. Students are provided with addition and subtraction problems to complete mentally with reference to a hundreds chart, marked number lines and unmarked (empty) number lines.

Note: Providing the answer to the calculation removes this as the focus; rather the focus is that there are many strategies that can be used and the three representations can be used to record intermediate steps. The empty number line is probably the most useful.

Provide students with laminated hundreds charts (PDF - 14Kb), which can be wiped clean, or use a classroom display hundreds board with each group. Use many of the same numerical examples for both addition and subtraction to emphasize that addition and subtraction are inverse operations.

The example below shows 26 + 37 using addition in stages. Choose examples for all of the strategies in the illustrations. See also: Mental calculations for addition and subtraction.

Verbalisation

“To do 26 + 37, I will first add 4 to 26 to get 30. I still have 33 to add on. Next I will add 30 to get 60, and finally add the remaining 3 to get the answer 63.”

Illustration on a hundreds chart

Illustration on a marked number line

Illustration on an empty number line.

The empty number line is only schematic, which makes it easy to draw.

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Activity 2: Adjusting numbers to make subtractions easier

It is reasonably easy for students to understand how to adjust numbers to make additions easier. For example, even very young children understand that 5 + 3 is the same as 4 + 4, because they can easily visualise one of the items making up the collection of 5 going across to the collection of 3 items, making 4 items in each pile.

Mental calculations for addition and subtraction – Adjusting numbers to make subtraction easier is also a powerful strategy, but it is harder to understand. Students can easily make mistakes using it if they do not understand the principles very firmly.

Explain this strategy beginning with an example such as 9 – 4 = 5. Make two lines of counters and discuss that the difference between the two piles does not change if we add one counter to both piles. The key feature here is that subtraction is being explained as ‘compare’ (also called ‘difference between’ rather than being explained as ‘take away’ (see explanation of meanings of operations in More About Construction of Number Sentences).

From the model of 9 – 5, add or subtract equal numbers of counters to both rows, showing that the difference between the top and bottom row always stays the same (namely 5).  Ask students what other things could we do to both the top and bottom rows so that the difference stays the same? This is an opportunity for creativity and for students to indulge their fascination with big numbers. As long as the same numbers are added or subtracted from each row, the difference will stay the same.

The difference between 9 and 4 is 5.

 

The difference between 9 and 4 is 5.

 

Extend to larger numbers. The difference between 27 and 19 is 8 and this stays the same if we increase both lines by 1 (difference between 28 and 20 is still 8)

Ask students: Which is easier to calculate without the counters? 27 – 19 or 28 – 20? Discuss how this property leads to a strategy for mental computation.

To conclude: sometimes we are given a subtraction to calculate and we can choose to change the numbers in certain ways that keep the difference the same.

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Further Resources

Manual on Mental Computation

McIntosh, A. (2005). Mental Computation: A Strategies Approach. (Module 4: Two-digit whole numbers). Department of Education, Hobart.

National Library of Virtual Manipulatives: Electronic number lines

National Library of Virtual Manipulatives (http://nlvm.usu.edu) – These number line applets can be used to for students to illustrate their mental computation, with computer or data projector or interactive whiteboard.

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