{syllabusref}

Item

Year 3 Q 13, 32
Year 5 Q 17
Year 7NC Q 2, 5
Year 9NC Q 8

Item Descriptor

Solves a single-step problem involving addition
Solves a problem involving reasoning, addition and subtraction
Uses compensation strategy for the addition of two four-digit numbers
Solves a problem involving subtraction of 3-digit numbers

Statements of Learning for Mathematics pp. 19, 25

Students have the opportunity to:

  • use known strategies and results to assist mental computation involving the addition and subtraction of one- and two-digit numbers
  • read and interpret practical problems, identify appropriate operations to use, express them mathematically and solve them
  • form reasonable mental estimates to computations involving a single operation.

Quality Teaching Framework

Intellectual quality - Deep knowledge
Quality learning environment - Engagement


STAGE 2

Number - Addition and Subtraction

Strategies

Students can

  • use a range of mental strategies for addition and subtraction, including the split strategy and compensation strategy

Activities to support the strategies

The Mathematics K–6 Syllabus 2002 requires students to use mental strategies such as jump, split and compensation strategies to add and subtract up to 6-digit numbers. Before teaching jump, split and compensation strategies, teachers need to establish how students would typically work out the answer.

Example: Students may use a variety of strategies to solve the following addition.

A child who says '38, 39, 40 ....' is counting by ones. Such children need to be taught what a representation of 38 and 23 looks like using concrete materials to represent tens and ones.

Children then need to be taught how to manipulate the concrete representation using different strategies, including:

  • Counting by tens and then ones
  • Count 38, 48, 58 and then 59, 60, 61

  • Split Strategy where the tens are grouped and the ones are grouped.
  • 30 + 20 = 50 and 8 + 3 = 11

    50 + 11 is the same as 60 + 1

  • Compensation Strategy
  • 38 + 23 is seen as 40 - 2 + 23 = 63 – 2 = 61

When children understand how these strategies work using the concrete representations they are able to mentally manipulate the groups. This underpins using mental strategies.

Some activities for developing mental computation in addition and subtraction are detailed below.

Activity 1 – Hundreds Chart

A variety of warm-up activities can use the hundreds chart. Each class should have a large hundreds chart for whole class activities as well as a class set of individual charts for student use. Using a transparency of a hundreds chart is one way to focus students on the task at hand.

  • Find 1 more than, 1 less than, 10 more than, 10 less than on the hundreds chart.
  • Count by tens off the decade, forwards and backwards.
  • Arrows - Start at a number, say 24, flash a large arrow.

Ask: Where are we now?

What did we do to get to this number?

Do this activity as subtraction
Add 23 to 54, Subtract 35 from 78.

Link this to the jump method of addition and subtraction.

  • Expand the hundreds chart into the number system so that students can use it to add and subtract larger numbers, such as 235 + 78.
  • Remove the hundreds chart and ask the students to visualise it. Then ask students to add and subtract. Link this to the jump method of addition.
  • Ask students to mentally work out some addition and subtraction number sentences, such as 249 + 321.
  • Provide opportunities for students to explain the strategies they used. The teacher records the various strategies on the board and allows the students to discuss the most efficient methods.

Activity 2 – Fancy Dice

This is a great game to move the students from using a hundreds chart as a game board to scoring on an empty number line. This game is suitable for two players as there is a need to wait for a turn.

The first player throws five dice (1 - 6 dot dice).

The second player is allowed to remove any dice that show 2 or 5.

The first player then adds the remaining dice and moves his or her playing piece forward the corresponding number of places on their hundreds chart, starting at zero.

The first player continues their turn by rolling the remaining dice and adding those dice that are left after the second player takes away any 2s and 5s.

The game continues until the second player has all five dice.

Now, the second player rolls the five dice and the first player removes the 2s and 5s.

Deadly play No. 1: If a player rolls two 6's during their turn, they immediately turn over all the dice to the other player, losing their turn.

Deadly play No. 2: If a player rolls three 6's during their turn, they not only lose their turn, but they go back to the start.

Play continues until someone reaches 100 plus.

For the next game, the players start on 100 and subtract their score.

Encourage students to move through the hundreds chart without counting by ones. Ask questions that encourage students to see the partitioning of numbers. That is, if they are on 27 and they have a score of 9, ask them, How many do I need to get to 30? (3) and how many left over? (6), so go to 36.

As students get more confident on jumping through the hundreds chart, introduce the Empty Number Line as a method of scoring.

Activity 3 – Red and Black

Variation 1

Remove the court cards from a pack of cards, shuffle the remaining cards and turn them face down in a pile between two players. Both players decide a starting score between 100 and 500 and record it on their empty number line.

The first player guesses what colour they think the top three cards will be, either black or red. If correct, that is, two out of the three cards selected are of the nominated colour, then the first player adds their cards and adds the total to the starting score.

If incorrect, that is, two out of the three cards are not of their nominated colour, the first player adds their cards together and subtracts the total from the starting score.

The second player now has their guess.

If a player guesses a colour and all three cards are of that colour, the player adds their cards and doubles the total before adding it to their score. Conversely, if all three cards are the wrong colour, the player must add all three, double the total, then subtract the total from their score.

Play continues until all the cards are used. The winner is the player with the highest total.

Variation 2

Remove the court cards from a pack of cards, shuffle the remaining cards and turn them face down in a pile between two players. Both players decide a starting score less than 500 and record it on their empty number line.

The first player guesses what colour they think the top three cards will be, either black or red. If correct, that is, two out of the three cards selected are of the nominated colour, they arrange the two correctly coloured cards into the largest two-digit number possible and adds the number to their score.

If incorrect, that is, two out of the three cards are not of their nominated colour, they arrange the two incorrect cards into the smallest possible two-digit number and subtract the total from their starting score.

The second player now has their guess.

If a player guesses a colour and all three cards are of that colour, the player arranges all three cards into the highest possible number before adding it to their score.

Conversely, if all three cards are the wrong colour, the player arranges all three cards into the lowest possible number then subtracts it from their score.

Play continues until all the cards are used. The winner is the player with the highest total.

Variation 3

This is a good game to reinforce the strategies students use to move around the hundreds chart. Follow similar rules to those described in Red and Black Variation 1 or 2, but each player has a hundreds chart as a game board and they start the game by placing their game piece on 50. (Transparent counters allow players to see the numbers.)

Activity 4 – Secret Number Scramble

Provide a set of number cards up to 10 000. Each student draws a number card.

The teacher calls out the secret number, such as 8000, and the students must find, within 2 minutes, someone with a number that, when added or subtracted to their number, is nearest to this secret number.

Students are encouraged to round their number to the nearest thousand and then to the nearest hundred and then look for a number that they need. Once they find a number, they sit with their partner/s.

The students are then asked to share how they worked out their total.

The pair with the answer nearest to the secret number wins.

Activity 5 – Missing Digits

Pose this problem to the class:

I solved a subtraction task but I can only remember the answer.

It looked like this:

What might the missing numbers be?

Students determine some of the possible numbers and describe how they worked them out.

Links

Curriculum Support

Developing Efficient Numeracy Strategies Stage 2, NSW Department of Education and Training, 2003, p. 38-39, 284-291

Count Me In Too, NSW Department of Education and Training

Recording My Strategies Stage 2, from Mathematics K-6 Assessment and Work Samples, Board of Studies, CD Rom

Demonstrating the split method

This animation shows how the split method can be used to add two digit numbers.

Mathematics K-6 Sample Units of Work (2003) Stage 2 – Addition and Subtraction, pp. 87-90

Interactive Games

http://www.bbc.co.uk/skillswise/numbers/wholenumbers/addsubtract/mental/

http://www.bbc.co.uk/skillswise/numbers/wholenumbers/addsubtract/written/

http://www.bbc.co.uk/schools/ks1bitesize/numeracy/numbers/index.shtml

http://www.woodlands-junior.kent.sch.uk/maths/