Mathematics Developmental Continuum P-10

Flexiable Addition and Subtraction: 2.0

Supporting materials

 

Indicator of Progress

At this level students are able to mentally compute a variety of addition and subtraction calculations using a range of methods.

Success depends on a variety of knowledge about the properties of numbers and operations, such as students being aware that the total of three or more numbers can be determined by combining the numbers in any order. They use this knowledge to enable the calculations to be made mentally. Students will split numbers into ones, tens and hundreds for convenience in calculation. They will use a growing facility with number combinations to move from counting in ones to counting and calculating with tens and hundreds.

See: More about the relevant properties of operations

Illustration 1: Thinking about the order of adding

Students know that they can add numbers in any order and still get the same answer. For mental computation, they select an easy order.

Examples:

  • When given 20 + 60, students know that they will get the same answer as 60 + 20 and then count 60, 70, 80.
  • When given a set of numbers to add (e.g. 4, 8, 7, 6, 3) students make choices about which pairs to combine first (e.g. 4 + 6 = 10, 7 + 3 = 10); so the total is 28.

Successful students at this level also know that they have to be MUCH more careful about changing the order of subtraction, even if they are not yet confident about exactly what they can do with subtraction. Whereas addition is commutative, subtraction is not. 60 + 20 = 20 + 60, but 60 – 20 ≠ 20 – 60.

Illustration 2: Splitting numbers based on place value

Students use place value knowledge when given addition or subtraction calculations involving 2 digit numbers.

Examples:

  • Given 23 + 41, students can split the numbers into tens and ones (e.g. 20 + 3 + 40 + 1) and then combine in a convenient order to find the total (20 + 40 + 3 + 1 =  60 + 4 = 64)
  • Given 83 – 60, students think of 83 as 8 tens + 3 ones and 60 as 6 tens, so they know the answer is 2 tens and 3 ones (23)
  • Given 300 + 416, students think 300 + 400 = 700 and 700 +16 is 716
  • Given 1000 - 250, students think 1000 - 200 = 800 and 800 – 50 = 750.

Illustration 3: Adjusting the result of an easier calculation (near doubles)

  • When given 5 + 6, students think of 5 + 5 = 10, then say “I need one more, so that’s 11”
  • When given 15 + 16, students think of 15 + 15 = 30, then say “I need one more, so that’s 31”.

Illustration 4: Adjusting the result of an easier calculation (known facts)

  • When given 25 – 6, students think of 25 – 5 = 20, then say “I need one less, so that’s 19”
  • When given 27 – 9, students think of 27 – 10 = 17, then say “I need one more, so that’s 18”.

Illustration 5: Links to the Mathematics Online Interview

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

  • Question 18 - I have 9 teddies and you have 4 teddies. How many teddies do we have altogether?
  • Question 19 - Work out the number of biscuits left when the start amount was 8 and 3 were eaten.
  • Question 20 - Work out the number of strawberries when the start amount was 12 and 9 were eaten using counting back, counting down or counting up from strategies.
  • Question 21 - Basic addition and subtraction – using counting on, counting back, counting down to or counting up from strategies.

 

Teaching Strategies

The general teaching strategy is to ensure that students form an understanding of the important mathematical ideas using materials before written recording is used.

Activity 1: Choosing which order to add involves providing a series of bags containing counters and students choose an easy order to add the numbers of counters in the bags.

Activity 2: Strengthening visual images for mental computation suggests that number lines and hundreds charts should be used for students to illustrate their mental methods. With ongoing practice students will internalise the visual images and use them for mental calculation.

Activity 3: Games for number fluency suggests some fun ways of improving students’ immediate recall of number combinations

Activity 4: Mixing addition and subtraction is an activity where students write number sentences involving addition and subtraction and discuss legal moves for combining the steps. Illustration on a vertical number line is recommended.

Activity 1: Choosing which order to add

In this activity, counters are provided in clear plastic bags with labels to indicate the number of counters within. For example, five labelled bags could contain 4 red, 8 blue, 7 yellow, 6 green, 3 white counters. Students are asked to find the total number of counters without taking them out of the bags. While some students will have inefficient strategies, highlight those strategies involving physically grouping the bags (4 and 6, 7 and 3) before counting “ten and ten is twenty and eight more is twenty eight”.

Students work in groups finding the number of counters in their bags mentally and discussing strategies. In the subsequent discussion with the teacher and others, students can demonstrate their suggestions by removing the counters from the bags, and putting them into the groups of ten that make calculation easy.

As students explain their calculation methods, teachers can watch for:

  • Children who are counting by ones instead of using number combinations
  • Children who have not yet achieved fluency in basic facts (e.g. knowing complements to ten). The indicator of progress ‘Complements to Ten – 1.5’ has  further information of this
  • Children who do not yet understand the importance of ten for grouping to make calculation easy.

After some examples using bags of counters, the activity can continue without the counters, just using the labels (e.g. 27 + 9 + 3 + 1 can be thought of as “27 + 3 = 30, and 9 + 1 = 10, so then the total is 40”).

Combining to make tens for easy calculation

 

4 + 6 = 10

7 + 3 = 10

8

Total  28

Activity 2: Strengthen visual images for mental calculation

Visual images are a great help with mental calculation. They are also an invaluable aid to classroom discussion. Teachers should use concrete materials as well as internet applets with a data projector or interactive whiteboard. Illustrate mental computation with:

Activity 3: Games for number fluency

Good mental computation depends on fluency with number facts. Time at home and at school should be spent developing fluency, ensuring it is built on understanding.

  • At home, students should engage in mental computation during activities such as shopping. They should also be encouraged to play games where they have to score. Card games, dice games, commercial board games and some computer programs provide good practice.
  • At school, teachers can use the same activities, with the addition of special educational software and equipment such as flash cards.

Since the 1920s research has shown that practice to improve fluency in number combinations should be:

  • based on understanding
  • with immediate feedback for students on right/wrong
  • frequent but for short times
  • partially concentrating on new skills, and
  • include mixed practice of a wide range of skills on each occasion.
Back to Top

Activity 4: Mixing addition and subtraction

Some of the same efficient grouping practices can also be used with subtraction, and with mixed addition and subtraction. Teachers and students can make up stories about travelling up and down a long ladder, or set of stairs.

Example: I climbed up 27 steps, went down 15, climbed back up 16, went down 4, then went down 6 more, then went down 2.  On what step did I finish?

Students write this as a number sentence, and discuss good ways to calculate the answer.

27 – 15 + 16 – 4 – 6 – 2

One good mental strategy would go like this. “Start at 27, then going down 15 and up 16 is the same as going up 1, so you are at step 28. Going down 4 and then 6 is the same as going down 10, so you are at step 18. Then go down 2 to finish on step 16.” The key idea is to keep the story in mind, to help give meaning to the symbolic expression.

Other contexts for similar story problems are turning pages back and forwards in a book, and travelling north and south. Illustrations with a number line especially a vertical number line, assist in visualisation. Number lines can be helpful even if they are ‘empty’ without number labels (see picture). They can operate as a mental guide. Teachers should also include story problems where most of the calculation is about multiples of ten or a hundred.

Further resources

The following resource contains sections that may be useful when designing learning experiences:

McIntosh, A. (2005). Mental computation: a strategies approach. Hobart: Department of Education, Tasmania.

Digilearn object *
Counting beetles: level 3 (L8283) - Students solve addition and subtraction problems using a range of counting strategies.
(https://www.eduweb.vic.gov.au/dlr/_layouts/dlr/Details.aspx?ID=7438)

* Note that Digilearn is a secure site; SMART::tests login required.

Back to Top