Adding and subtracting within 10
Counting on, taking away, part-whole number bonds, and how adding and subtracting fit together
About four to five lessons of 30 to 40 minutes
You already add and take away all day
You had 3 toy cars and a friend gave you 2 more. How many now? You just added. You had 5 grapes and you ate 2. How many left? You just took away. Adding and taking away are things you do every single day, at snack time, at tidy-up time, and every time you count your fingers.
Today we give those everyday moves their proper names and learn to write them down. By the end you will count on to add, count back to take away, and see how adding and taking away are really two sides of the same picture.
- 3 teddies and 2 more teddiescount them all: 3 then 4, 5, that is 5 teddies, 3 + 2 = 5
- Show 4 fingers, then pop up 3 morecount on from 4: 5, 6, 7, so 4 + 3 = 7
- 6 apples in the bowl, take 2 outcount back: 6, then 5, 4, that leaves 4, so 6 - 2 = 4
- 5 cars in the garage, 2 drive away5 take away 2 leaves 3 cars, 5 - 2 = 3
What students will be able to do
Students will add and subtract within 10 by counting on and counting back, model an adding or taking-away story with objects and drawings, use part-whole number bonds to see the two parts inside a whole, and begin to add and subtract within 5 from memory.
- I can act out an adding story and a taking-away story with objects.
- I can count on from the bigger number to add.
- I can count back to take away.
- I can break a number into two parts and name the parts.
- I can add and subtract small numbers up to 5 in my head.
Standards this unit teaches
- K.OA.A.1Common Core (US)Represent addition and subtraction
Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g. claps), acting out situations, verbal explanations, expressions, or equations.
- K.OA.A.2Common Core (US)Add and subtract within 10
Solve addition and subtraction word problems, and add and subtract within 10, e.g. by using objects or drawings to represent the problem.
- K.OA.A.5Common Core (US)Fluently add and subtract within 5
Fluently add and subtract within 5.
- AC9MFN04Australian Curriculum v9 (ACARA)Model adding and taking away (Foundation)
Represent practical situations that involve adding to, taking away from and combining collections, using materials and counting strategies to model everyday adding, taking-from and how-many situations.
- AC9MFN03Australian Curriculum v9 (ACARA)Part-part-whole to 10 (Foundation)
Split and combine collections of up to ten using part-part-whole thinking, naming the two parts that make the whole. This is the number-bond idea this unit rests on.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Add
- put groups together to find how many there are altogether
- Sum (total)
- the answer when you add, how many there are altogether
- Subtract (take away)
- start with an amount and take some off to find how many are left
- Difference
- the answer when you subtract, how many are left or how many more
- Number bond
- two parts that join to make a whole, such as 3 and 4 make 7
- Part and whole
- the whole is the total, the parts are the smaller groups inside it
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Adding is putting groups together
ConcreteStart with real things in your hands. Put 3 counters in one group and 2 counters in another. To add is to push them together and find how many altogether. Slide the two groups into one and count them all: 1, 2, 3, 4, 5. There are 5. We write it as 3 + 2 = 5 and read it as three plus two equals five.
The plus sign means put together. The equals sign means is the same as, so 3 + 2 is the same amount as 5. Say the whole sentence out loud together: three plus two equals five.
Do it again with fingers. Hold up 3 on one hand and 2 on the other, then count every raised finger. Fingers, counters, teddies, it is always the same move: join the groups and count them all.
- If I put 4 blocks with 1 more block, how many altogether?
- What does the plus sign tell us to do?
- Which number is the total, the answer or the parts we started with?
2. Counting on to add
PictorialCounting all works, but there is a faster way: counting on. To add 4 + 3, start at 4 and take 3 hops forward along the number line, saying 5, 6, 7 as you go. You do not go back to 1, you start from the number you already have. You land on 7, so 4 + 3 = 7.
The trick is to start counting at the next number, not at the number you are on. Standing on 4, the first hop lands on 5, not on 4. Count the hops, three of them, and read off where you stop.
Always start from the bigger number so you take fewer hops. For 2 + 5, start at 5 and hop 2, which is quicker than starting at 2 and hopping 5. You still get 7 either way.
Use counting on to work out 6 + 3.
- Start on the bigger number, 6.
- Take 3 hops forward, saying 7, 8, 9.
- You stop on 9, so that is the total.
Answer: 6 + 3 = 9.
- To add 5 + 2, which number do you start on and how many hops?
- Standing on 4, does your first hop land on 4 or on 5?
3. Number bonds: the two parts inside a whole
PictorialEvery whole is made of parts. Draw a bar for the whole, 7, and split it into two parts, 4 and 3. This is a number bond: 4 and 3 are the two parts that join to make the whole 7. The parts always add back to the whole, so 4 + 3 = 7.
One whole can split in many ways. 7 can be 4 and 3, or 5 and 2, or 6 and 1. Each is a number bond for 7. Learning the bonds for numbers up to 10 by heart is the single most useful thing you can do in early maths, because it turns adding and taking away into something you just know.
The bar shows the parts sitting inside the whole. When you know two of the three numbers, you can always find the third: two parts make the whole, and the whole take one part leaves the other.
- The whole is 7 and one part is 4. What is the other part?
- Show me two different number bonds that both make 6.
4. Counting back to take away
PictorialTo subtract is to take some away and find how many are left. On the number line we count back. For 7 - 3, start on 7 and take 3 hops backward, saying 6, 5, 4. You land on 4, so 7 - 3 = 4. The minus sign means take away.
Counting back is counting on in reverse. Start on the number you have and hop the other way. Count the hops, three of them, and read off where you stop.
Taking away makes the amount smaller, so on the number line you always move toward 0. If you find yourself moving the bigger way, you are adding, not taking away.
Use counting back to work out 9 - 2.
- Start on 9.
- Take 2 hops back, saying 8, 7.
- You stop on 7, so that is how many are left.
Answer: 9 - 2 = 7.
- For 8 - 3, which number do you start on and which way do you hop?
- Does taking away make the number bigger or smaller?
5. Adding and taking away go together
AbstractHere is the big idea that ties the unit together. One number bond gives you four facts at once. Look at the whole 7 with parts 4 and 3. The two parts make the whole two ways, and the whole take one part leaves the other two ways. So the same picture tells you 4 + 3 = 7, 3 + 4 = 7, 7 - 3 = 4, and 7 - 4 = 3.
This is why bonds are so powerful. If you know 4 + 3 = 7, you already know 7 - 3 = 4 without any counting, because taking away 3 from the whole just leaves the other part. Adding and taking away undo each other.
For the smallest numbers, up to 5, aim to just know these facts without hopping at all. 2 + 3 = 5 and 5 - 2 = 3 should become as automatic as knowing your own name.
The whole is 6, and one part is 2. Write all four facts.
- The other part must be 4, because 2 and 4 make 6.
- The two adding facts: 2 + 4 = 6 and 4 + 2 = 6.
- The two taking-away facts: 6 - 2 = 4 and 6 - 4 = 2.
Answer: 2 + 4 = 6, 4 + 2 = 6, 6 - 2 = 4, 6 - 4 = 2.
- If 5 + 1 = 6, what is 6 - 1 straight away?
- Why does knowing one number bond give you four different facts?
Common misconceptions and how to address them
MisconceptionWhen counting on to add, the child starts the count on the number instead of the next one, so 4 + 3 comes out as 6.
Why it happens: It feels natural to say four while pointing at the 4, which uses up a hop before any have been taken.
How to address it: Put a finger on 4 and say the number is where we stand, then say the first hop out loud as five. Count only the hops, not the standing spot. Trace three hops on the number line together.
MisconceptionThe child counts an object twice or skips one, so the total is wrong even when the method is right.
Why it happens: In a scattered group it is hard to keep track of which objects have already been counted.
How to address it: Move each object to a new pile as it is counted, or line them up in a row and touch each once. One touch, one number.
MisconceptionAdding and taking away are treated as the same because both just involve counting.
Why it happens: Young children see counting in both and do not yet attach a direction to the two operations.
How to address it: Tie plus to hopping forward and getting more, and minus to hopping back and getting fewer. Act out both with objects: adding brings more in, taking away sends some out.
MisconceptionThe equals sign is read as here comes the answer, so 7 = 3 + 4 looks wrong to the child.
Why it happens: Equals is almost always met at the end of a sum, so it gets heard as now write the answer rather than is the same as.
How to address it: Say equals means is the same amount as, and show a balance: 3 + 4 on one side is the same as 7 on the other. Write sums both ways, 3 + 4 = 7 and 7 = 3 + 4, so the sign keeps its real meaning.
MisconceptionIn a part-whole bar, the child thinks the whole is just another part, so they might say 7 and 4 make 3.
Why it happens: The three numbers sit together and their roles are not yet clear.
How to address it: Point to the long bar and call it the whole, the total. The two short bars inside it are the parts. Parts join to make the whole, the whole is never one of the parts.
MisconceptionWhen taking away, the child counts back including the starting number, so 7 - 3 lands on 5.
Why it happens: It is the mirror of the counting-on slip: the standing spot gets counted as the first hop.
How to address it: Start on 7 and say the first hop back is six, not seven. Count the hops away from the start, three of them, and read where you land.
Guided practice (with answers)
1. Work out 3 + 4 by counting on from the bigger number.
Answer: 7. Start on 4 and hop 3: 5, 6, 7.
2. Find the missing part: the whole is 8 and one part is 5.
Answer: 3, because 5 and 3 make 8.
3. Work out 8 - 3 by counting back.
Answer: 5. Start on 8 and hop back 3: 7, 6, 5.
4. Write a number bond that makes 10.
Answer: Any pair that totals 10, such as 6 and 4, or 7 and 3, or 5 and 5.
5. If 6 + 2 = 8, what is 8 - 2?
Answer: 6. Taking the part 2 off the whole 8 leaves the other part, 6.
6. There are 5 birds on a branch and 2 fly away. How many are left?
Answer: 3, because 5 take away 2 is 3.
Independent practice worksheets
Set the matching ChalkBee worksheets for independent work. The answer keys are computed in code, so they are never wrong. Start with number bonds and counting to keep the parts and the whole visible, then move to the adding and subtracting sets.
Differentiation
- Stay concrete for longer: keep real counters and fingers in play before moving to the number line.
- Work within 5 first, both adding and taking away, until those facts are secure, then stretch to 10.
- Give a pre-drawn number line so the child only needs to hop and count, not draw the track.
- Use a part-whole mat with two circles for the parts and one for the whole, and move counters between them.
- Ask for every number bond of a number, such as all the ways to make 8, and record them in order.
- Pose missing-number problems: 4 + ? = 9, or ? - 2 = 5.
- Introduce simple word problems with a change: there were some in the box, 3 more went in, now there are 7, how many at the start?
- Begin adding and subtracting within 20 as a bridge to the Grade 1 unit.
Assessment: exit ticket
A three-question exit ticket for the last few minutes. Children answer on a slip or by showing with fingers. It samples adding, taking away, and the part-whole bond.
1. Work out 5 + 3.
Answer: 8, by counting on from 5: 6, 7, 8.
2. Work out 9 - 4.
Answer: 5, by counting back from 9: 8, 7, 6, 5.
3. The whole is 6 and one part is 2. What is the other part?
Answer: 4, because 2 and 4 make 6.
Teacher notes and timings
- Rough timing across four to five lessons: Lesson 1 adding by putting together (section 1), Lesson 2 counting on (section 2), Lesson 3 number bonds (section 3), Lesson 4 counting back to take away (section 4), Lesson 5 the add and subtract link plus the exit ticket (section 5 and assessment).
- Language to keep saying: count the hops not the spot, start on the bigger number, the whole and its two parts, equals means the same amount as. These four phrases pre-empt most of the misconceptions.
- Keep counters and a part-whole mat on desks throughout. When a child is stuck on the number line, hand them objects and let them build the same fact.
- The number lines run 0 to 10 with whole-number labels only, which is exactly the range this grade knows, so nothing on the diagram is beyond them.
- Push fluency within 5 in short daily bursts (flash a fact, all show the answer on fingers). Within 10 stays a counting-on and counting-back skill at this stage.
- Curriculum note and a US and AU alignment: the US sets adding and subtracting within 10 in Kindergarten (K.OA.A.1, K.OA.A.2) with fluency within 5 (K.OA.A.5). ACARA covers the same ground in Foundation, modelling adding and taking away (AC9MFN04) and part-part-whole to 10 (AC9MFN03), so this unit sits at the very start of both curricula.
- Present mode and print both work: use the Print button for a clean teacher copy or a student handout, and project the page to teach straight from the diagrams.