# Stage 2 - addition and subtraction

Use a range of mental strategies for addition and subtraction, including the jump, split and compensation strategies; explain and record methods for adding and subtracting

## Strategy

Students can

• use a range of mental strategies for addition and subtraction, including the jump, split and compensation strategies
• explain and record methods for adding and subtracting

## Activities to support the strategy

In the NSW Syllabuses for the Australian Curriculum, Mathematics K-10 requires students to use mental strategies such as jump, split and compensation strategies to add and subtract up to 4-digit numbers. Before introducing a range of strategies, teachers need to establish how students would typically work out the answer using their own mental strategies.

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

A student who says '38, 39, and 40’ is counting by ones. Such students need to be taught what a representation of 38 and 23 looks like using concrete materials to represent tens and ones. This will help build their understanding of place value and how numbers are structured.

Students 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
• Jump strategy where a number line is used to count forwards or backwards first by tens then ones (the ones may be split to bridge to a decade)

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

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 an interactive whiteboard version of a hundreds chart is one way to incorporate basic number skills into your warm up activities.

Here are just a few ideas to use to start your lesson using a hundred chart

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• 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.

See Unit 5: Hundreds Chart from DET Teaching activities, Counting On pp. 69-76

• Expand the hundreds chart into the number system so that students can use it to add and subtract larger numbers, such as 235 + 78. Create different charts that range from 101- 200, 201- 300 and so on…
• 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. Some students may require a blank hundreds chart grid to represent where the numbers are as a visual assistance.
• 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 player 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 player 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

• Create a set of number cards with a selection of numbers 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 with 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.

• See Red dragonfly mathematics challenge as an iPad app at Mathlinks.

### Activity 6 -first to twenty dollars

Students work in groups of three, two players and a ‘banker’. Students take turns to roll a dice marked with amounts of money (5c, 10c, 20c, 50c, \$1, \$2). They place the correct coins on the grid. As they go, they can trade up (e.g. give the banker 2 x 50c and 2 x \$2 to receive a \$5 note) until they reach \$20.00.

• First player to \$20.00 is the winner.
• During this activity students learn to recognise equivalent amounts of money using different denominations. Students need experience with money and changing money and linking addition facts and strategies to 100 with money.
• You can play a variation of this activity and start with the \$20.00 and work down.

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## References

#### Australian curriculum reference

ACMNA054: Recognise and explain the connection between addition and subtraction; ACMNA055: Recall addition facts for single – digit numbers and related subtraction facts to develop increasingly efficient mental strategies for computation. ACMNA083: Use equivalent number sentences involving addition and subtraction to find unknown quantities. ACMNA080: Solve problems involving purchases and the calculation of change to the nearest five cents with and without digital technologies

#### NSW syllabus reference

MA2-5NA: uses mental and written strategies for addition and subtraction involving two, three –, four – and five – digit numbers; MA2-7NA: models compares and represents decimals of up to two decimal places

#### NSW numeracy continuum reference

Aspect 2: EAS – Facile

Aspect 4: Place value: Hundreds, tens and ones; Decimal place value

#### NSW literacy continuum reference

WRIC10M4: Aspects of writing, cluster 10, marker 4: rereads and revises text to check and improve meaning, deleting unnecessary information or adding new information

#### Other literacy continuum markers

REAC9M2: Reading texts, cluster 9, marker 2: uses visual representations, e.g. photographs, tables, charts to enhance meaning when reading factual texts. REAC10M4: reading texts, cluster 10, marker 4: Uses topic knowledge, vocabulary knowledge and context to read unknown words when engaging with subject texts.  COMC9M7: Comprehension, cluster 9, marker 7: Analyses a text by discussing visual, aural and written techniques used in the text.  COMC10M6: comprehension, cluster 10, marker 6: Analyses and evaluates the relative importance of key ideas and information in a text to construct an overview.

## Numeracy app

Greater Than: Greaterthan is a children's app that is meant to help them learn which numbers are bigger than others. It tests their knowledge of greater than and less than for numbers up to 10 in a fun, game-like manner in order to keep the children's interest. The app uses fun sounds in response to the child's selection and encourages them as they go along. The app has a status bar at the bottom which graphically shows the child how well they're doing!