Use a range of mental strategies including counting on from, counting back from, counting on and back, doubles, near doubles, bridging to ten; use a range of informal recording methods for addition and subtraction including jump and split strategies; record number sentences using drawings, numerals symbols and words; use the language of addition and subtraction.

## Strategy

Students can:

• use a range of mental strategies including counting on from, counting back from, counting on and back, doubles, near doubles, bridging to ten
• use a range of informal recording methods for addition and subtraction including jump and split strategies
• record number sentences using drawings, numerals, symbols and words
• use the language of addition and subtraction.

### Activities to support the strategy

Students need to recognise and use the terms ‘add’, ‘plus’, ‘is equal to’, ‘take away’, ‘minus’ and ‘the difference between’ to describe addition and subtraction. They also need to record number sentences using drawings, numerals, symbols and words. Students need opportunities to both pose and solve problems in addition and subtraction.

In Stage 1 the syllabus focuses on developing a range of mental strategies and informal recording methods for addition and subtraction.

The following activities assist students in making connections between counting and addition and subtraction strategies and provide students with opportunities to demonstrate their understanding of place value and how numbers can be combined and partitioned.

It is important to present number sentence problems to students horizontally e.g. 34 + 49 =, this assists students in reading the numbers from left to right and will develop their understanding of place value and of how to read numbers. This will also provide students with opportunities to look at the whole number, not just the digits, assisting students with estimating the solution.

### Activity 1 – language of mathematics

The Herrington Think Board is one way to organise and solve problems with students that also focuses on the language of mathematics.

• A process for understanding mathematics by Sue Gunningham (PDF 422.58KB) .
• For example, the teacher can supply the story “Mary had five oranges, Tom took two away, how many oranges does Mary have left?”
• Students can then draw a picture of the story, use objects such as counters or play dough to create and work out the problem, then record a number sentence that matches the story.
• To focus on the language, provide students with the number sentence and ask them to write a story to match.

View/print – think board template (PDF 37.99KB)

### Activity 2 – warm up activities

Short, focused, frequent activities are great ways to start or conclude a mathematics lesson. They are an opportunity to repeat skills that need to be practised.

### Roll two dice and add

This is a whole class activity where students sit in a circle and two six-sided dice are thrown. Students share strategies for adding the numbers together.

This activity can be played using subtraction and can be extended by changing the dice to eight- or ten-sided dice or by adding in a third dice.

The third dice will provide opportunities for students to look for doubles, or friends of ten or to use known facts.

Students find today's date on the calendar, count how many days until the end of the month and work out the date 10 days later. Students explain strategies for their addition.

Write a number on the board (less than 20). In pairs students make the number and record their combination. Ask how else they can make it.

### Hands up

Students are asked to come out the front and make a two-digit number e.g. 24 with their hands. Students soon realise they will need two others to help them. Do the same for another two-digit number e.g. 27. Students then discuss how to add the numbers e.g.

grouping all the tens together then adding the ones, they will be able to make an extra ten from the 4 and 7, so the student with seven fingers up changes to ten (and joins the other tens) and the student with four changes to one.

This activity assists students in understanding the place value of tens and ones and how to re-unitise numbers. You can record students’ strategies as they are explaining what they are doing.

You can then move on to representing the numbers using unifix cubes in sticks as tens and loose blocks as ones. These strategies assist students in visualising the numbers.

### Activity 3 – make 100

The teacher removes the picture cards (kings, queens, jacks) from a standard pack of playing cards. The Ace is used to represent one. In small groups, each student is dealt six cards. The aim of the activity is to add all six card numbers together to make the closest total to 100 (but no greater than 100). Each student can nominate one of their cards to be a 'tens' card.

For example, if the student was dealt

they could nominate the 7 card to have the value 70 and add the remaining cards for a total of 93. They should be encouraged to record their calculations and share their strategies.

### Activity 4 – finding the difference

Students are given a number sentence with a missing element (using numbers less than 20) e.g.

Using unifix cubes, students work in pairs to make towers to represent the 13 and 5. They then compare the towers to work out the difference between the two towers. As a follow on activity, you can have students make the number differentiating between tens and ones using two colours. This will link to bridging to ten strategies.

### Activity 5 – using a number line for difference

Use a 1 to 100 number line (commercially made, drawn on the board, created on the IWB or simply use a tape measure/one metre ruler). Mark the place of two numbers, for example 32 and 41 Have students come out and work out the difference between the two numbers.

Be aware that although we generally relate difference to subtraction, some students will use a ‘count on’ not ‘count back’ strategy to solve the problem and therefore it can be related to addition as well. Have students write number sentence to match the working out.

Students can also write sentences using words to describe the working out. e.g. 'The difference between 32 and 42 is ten, so the difference between 32 and 41 must be nine.'

The concept of difference can also be explored when learning about length. Students can compare lengths and discuss the difference between the objects- either informally (for example, using paper clips) or formally (for example using centimetres).

“The ruler is longer than the pen. The rule is 12 cm and the pen is 9 cm. There is 3 cm difference between the objects.

### Australian curriculum

ACMNA029: Explore the connection between addition and subtraction, ACMNA030: Solve simple addition and subtraction problems using a range of efficient mental and written strategies, ACMNA036: Solve problems by using number sentences for addition and subtraction.

### NSW syllabus

MA1-5NA (+ & -) Uses a range of strategies and informal recording methods for addition and subtraction involving one – and two – digit numbers. MA1-8NA: P & A – Creates, represents and continues a variety of patterns with numbers and objects

### NSW numeracy continuum

Aspect 2: EAS: Facile, Aspect 3: Pattern and Number Structure: Part whole to 10 and Part whole to 20, Aspect 4: Place Value: Ten as a unit; Tens and ones.

### NSW literacy continuum

VOCC8M4: Vocabulary knowledge, Cluster 8, Marker 4: Recognises that different words can be used to describe similar concepts, e.g. everyday or technical language, synonyms.VOCC8M5: Vocabulary knowledge, Cluster 8, Marker 5: Shows evidence of capacity to improve vocabulary choices in response to purpose and audience when reviewing and editing writing.

### Other literacy continuum markers

WRIC8M4 Aspects of writing, Cluster 8, Marker 4: Writes for a wider range of purposes, including to explain and to express an opinion.

### Numeracy apps

Number Line Math: Practice addition and subtraction facts 1–10 using a number line as a tool for solving problems. Adjust the options to include any combination of Result, Change, or Start as the unknown quantity. Select the facts to practice.

• Mathoku junior
• Tens frame
• 100's board
• Math kid
• Friends of ten

## Making Subtraction Efficient – Just Think Addition! (Game Included)

Students who are efficient with the count-on, use-doubles, and make-ten strategies for addition have all they need to be proficient with subtraction number facts!

Subtraction is the inverse, or opposite action, of addition. Both operations involve a part-part-total structure. In the example below, 1 and 3 are two parts that make up the number 4.

### Parts of Subtraction

With addition, the parts are known, but not the total; with subtraction, the total and one of the parts are known, but not the other part. Because of this relationship between the two operations, using addition is the most effective thinking strategy for helping students learn the basic subtraction facts. Watch this ORIGO One video to learn more!

Teaching the think-addition strategy for subtraction from ORIGO Education on Vimeo.

### Fact Families

Addition facts and subtraction facts that involve the same parts and total form fact families. Clusters of subtraction facts are named according to the strategy used for the related addition facts.

For example, 8 – 3 = 5 is part of the count-on subtraction fact cluster because its related addition fact 5 + 3 = 8 involves counting on three.

Students need to understand the connections between the number facts within a family and the related strategy.

### Subtraction Games: Teach Subtraction Using Addition

“Think addition to subtract” is one of the most effective strategies for subtracting mentally. This game reinforces the connection between addition and subtraction. The students are encouraged to use their knowledge of addition to make a true subtraction number sentence.

Materials needed:

• Take or Tally game board (access this subtraction game board file here)
• 2 blank cubes, marked as follows: (Be sure to underline the 9s and 6s!)
• Write 1, 2, 3, 1, 2, 3 on one cube
• Write 4-9 on the other.

Game Directions: The aim is to complete twelve true number sentences.

• The first player rolls the two number cubes.
• The player then writes the two numbers in one of the number sentences on his or her game board. The completed number sentence must be true.

Example: Sue rolls 4 and 3. She completes the number sentence 7 – 4 = 3.

• If a true number sentence cannot be made, the player makes a tally in the space provided at the bottom of his or her game board.
• The first player to complete twelve number sentences before making a total of ten tallies is the winner.

Strategy Tip: Ask, How did you know to place your numbers in that sentence?

Extending the game:

• Change the rules so two players share the one game board. Each player has his or her own column of number sentences to complete. The remaining rules can stay unchanged.
• Use the Take or Tally Again game board for greater numbers. Make a new number cube by writing numerals 6-11 and replace the cube with 4-9 from the original game.

Encourage children to apply the think-addition strategy for subtraction to numbers beyond the basic facts. For example, the strategy can be extended to solve 106 – 89. This is specifically highlighted in the ORIGO One video referenced earlier in this blog!

• ORIGO Education is dedicated to making learning meaningful, enjoyable and accessible for all students with Pre-K and Elementary print and digital instructional materials, as well as professional learning for mathematics.

Here are lots of “thinking tricks” you can use to make addition easier.

Use the ones that make sense to you!

Hint: start from the larger number.

2 + 6 is Harder: “2 … 3, 4, 5, 6, 7, 8”

6 + 2 is Easier: “6 … 7, 8”

### Jump Strategy

We can also count by 2s or 10s, or make any “jumps” we want to help us solve a calculation.

Think “4 … 14 … 15, 16”

See if any numbers add to 10. They don't have to be next to each other.

• 7+3 is 10,
• 8+2 is another 10, which makes 20,
• Plus 5 is 25

### Do The Tens Last

Break big numbers into Tens and Units, add the Units, then add on the Tens.

### Example: 14+5

1. Break the “14” into Tens and Units: 10 + 4
2. Add the Units: 4 + 5 = 9
3. Now add the Tens: 10 + 9 = 19
4. Think “4 plus 5 is 9, plus 10 is 19”
• Break into Tens and Units: 10 + 4 + 10 + 2
• Add the Units: 4 + 2 = 6
• Now add on the Tens: 6 + 10 + 10 = 26

### Aim for Ten

When a number is close to ten we can “borrow” from the other number so it reaches ten.

 9 is only 1 away from 10 so take 1 from the 7: 9 + 1 + 6 and give it to the 9: 10 + 6 = 16Think “9 plus 1 is 10 … 7 less 1 is 6 … together that is 16”

8+2=10, so lets take 2 from the 5: 8 + 2 + 3 and give it to the 8: 10 + 3 = 13

We can also move backwards to ten, by making the other number bigger as needed:

Reduce 12 by 2:  12 − 2 = 10 Increase 7 by 2:  7 + 2 = 9

12 + 7 = 10 + 9 = 19

### Compensation Method

“Compensation” is where you round up a number (to make adding easier) and then take away the extra after you have added.

It is easier to do 20 + 16 = 36

Then take away the extra 1 (that made 19 into 20) to get: 35

It is easier to do 400 + 126 = 526

Then take away the extra 5 (that made 395 into 400) to get: 521

### Example 5 + 5 = 2 x 5 = 10

5 + 6 = two 5s + 1 = 10 + 1 = 11

7 + 9 = “8 less 1” + “8 add 1” = two 8s = 16

We can also use the Addition Table to help us.

## 7 practical tips for mental math (that ANYONE can use!)

You have most likely heard about mental math — the ability to do calculations in one's head — and how important it is for children to learn it.

But why is it important? Because mental math relates to NUMBER SENSE: the ability to manipulate numbers in one's head in various ways in order to do calculations. And number sense, in return, has been proven to predict a student's success in algebra.

Essentially, what we do with variables in algebra is the same as what students can learn do with numbers in the lower grades.

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People with number sense use numbers flexibly. They are able to take them apart and put them together in various ways in order to do calculations. It is quite similar to being able to “PLAY” with words in order to make interesting sentences, or being able to play with chords and melodies in order to make songs.

But mental math/number sense isn't just for “math whizzes” — quite the contrary! EVERYONE can learn the basics of it, and it will make learning math and algebra so much easier! We expect our children to learn lots of English words and to be able to put those words together in many different ways to form sentences, so why not expect them to do the same with numbers? And they can, as long as they are shown the basics and shown examples of how it happens. So let's get on to the practical part of this writing: mental math strategies for EVERYONE.

1. The “9-trick”.

To add 9 to any number, first add 10, and then subtract 1. In my Math Mammoth books, I give children this storyline where nine really badly wants to be 10… so, it asks this other number for “one”. The other number then becomes one less. For example, we change the addition 9 + 7 to 10 + 6, which is much easier to solve.

But this “trick” expands. Can you think of an easy way to add 76 + 99? Change it to 75 + 100. How about 385 + 999?

How would you add 39 + 28 in your head? Let 39 become 40… which reduces 28 to 27. The addition is now 40 + 27. Yet another way is by thinking of compensation: 39 is one less than 40, and 28 is two less than 30. So, their sum is three less than 70.

2. Doubles + 1.

Encourage children to memorize the doubles from 1 + 1 through 9 + 9. After that, a whole lot of other addition facts are at their fingertips: the ones we can term “doubles plus one more”. For example, 5 + 6 is just one more than 5 + 5, or 9 + 8 is just one more than 8 + 8.

Once you know that 7 + 8 = 15, then you will also be able to do all these additions in your head:

• 70 + 80 is 15 tens, or 150
• 700 + 800 is 15 hundreds, or 1500
• 27 + 8 is 20 and 15, which is 35. Or, think this way: since 7 + 8 is five more than ten, then 27 + 8 is five more than the next ten.

## Think Addition to Solve Subtraction

I have had several mentor teachers give me sage advice over the years and this advice has slowly evolved and melted together and has made me what I am today.  The best advice I was ever given was simple, “they will learn in spite of you.

” So even on a bad day, when a lesson has not gone my way, or misbehavior has thrown us entirely off track, or schedule changes flip my plans upside down, this advice comforts me.  First graders are sponges, and I know that I can pick back up tomorrow and everything will be okay.

Remember this as you teach and don't sweat the little stuff and let it build up to force you into giving up.  There's always tomorrow, and I have had many “do-over” lessons.

This advice is good to keep in mind especially during the holidays when lessons go off the rails all the time!

Need: Cut lots of 1″ strips of green and red paper. And possibly be prepared to cut more in the middle of your lesson. At least 3 index cards per student.

Now back to what we are here for… it is not always easy to incorporate a craft activity into a math lesson, but I plan to do exactly that.  I will have them build a model for an addition problem and then identify the subtraction facts that are related to this model.

First, I will write the problem 2+3=5 on one side of an index card. Then I will be building a model using 2 red strips and 3 green strips to build a model of 5 total. I will build the links and hook them together one color at a time.

Then show them how the 2 red combined with the 3 green equal a chain of 5. Now I will ask them to help me figure out what subtraction problems are related to this equation. I will turn the card over and write what they say, which should be 5-3=2 and 5-2=3.

Watch this model video and look at the completed chains.

Now I will let them pick their beginning addition model and construct their own model. It will be very important to walk the room and support each student in this process. Some will finish fast and I am going to encourage them to build more and to increase in complexity.