Counting to tell how many
The last number you say is the total, and it does not matter which object you start with
About three lessons of 25 to 35 minutes
The last number you say is the answer
Line up 6 toy cars and count them: 1, 2, 3, 4, 5, 6. The last word you said, six, is not just the label on the last car, it is how many cars there are altogether. That is a genuinely clever idea, and it is one you now use without even noticing.
Here is the surprising part: it does not matter which car you start with. Count the same 6 cars starting from the other end, or from the middle one first, and you still land on six. Today you will practise saying the last number as the total, and prove to yourself that the order you count in never changes how many there are.
- 6 toy cars in a rowcount them: 1, 2, 3, 4, 5, 6, the last word, six, is the total
- 9 buttons spilled on the tablescattered objects are trickier, touch each one exactly once
- The same 5 puzzle pieces, counted twicecounted left to right or right to left, it is still 5
- 14 cookies on two trayscounting on past 10 still ends on one last number, the total
What students will be able to do
Students will count up to 20 objects arranged in a line and up to 10 scattered objects, understand that the last number counted tells how many there are in total, and recognise that the total stays the same no matter which object is counted first or the order counted in.
- I can count objects in a row and say the last number as the total.
- I can count up to 20 arranged objects without losing my place.
- I can count up to 10 scattered objects by touching each one exactly once.
- I can explain that counting the same group in a different order still gives the same total.
Standards this unit teaches
- K.CC.B.4Common Core (US)Connect counting to quantity
Understand that the last number said when counting tells how many objects there are, and that the count stays the same regardless of order.
- K.CC.B.5Common Core (US)Count to answer how many
Count up to 20 arranged objects, or up to 10 scattered objects, to answer how many there are.
- AC9MFN02Australian Curriculum v9 (ACARA)Count and compare to 20 (Foundation)
Count and compare collections of up to at least twenty objects, explaining the reasoning behind which group has more or fewer.
- AC9MFN06Australian Curriculum v9 (ACARA)Connect number names to quantities (Foundation)
Connect number names, numerals and quantities, and represent the count of a small collection in more than one way.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Count
- say the number names in order, matching one number to one object
- Total
- how many there are altogether, the last number you say when counting
- Arranged
- objects placed in an order, such as a row, which makes them easier to count
- Scattered
- objects spread out with no order, which makes counting trickier
- One-to-one
- matching exactly one number word to exactly one object, no skips and no doubles
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. The last number you say is the total
ConcreteLine up some objects, say counters or blocks, in a row. Touch each one in turn and say the counting words: 1, 2, 3, 4, 5. Stop. The very last word you said, five, is not just naming the last counter, it tells you how many counters there are in the whole row. This is called the cardinality of the count: the last number said is the total.
Some children think counting is just a chant, a list of words said while pointing, without the last word meaning anything special. Make the total obvious: after counting, sweep your hand over the whole row and repeat, there are five, pointing at all of them together, not just the last one.
Try it with a different row length each time. Count 3 blocks, then 7 blocks, then 5 blocks again. Every time, the last number said names the size of the whole group.
- If I count a row of blocks and the last word I say is seven, how many blocks are there?
- Is the last number just the name of the last block, or is it something more?
2. It does not matter which one you start with
ConcreteHere is a test worth doing out loud. Count the same row of 6 counters starting from the left: 1, 2, 3, 4, 5, 6. Now count the very same 6 counters starting from the right: 1, 2, 3, 4, 5, 6. Same last number, six, even though you started from a different end and touched them in a different order.
This is the order-irrelevance principle: the order you count objects in does not change how many there are. It can feel surprising to a five-year-old, because the counting words came out attached to different objects each time. Prove it a few times with different starting points, and even by shuffling the counters between counts, until it stops feeling surprising.
This idea matters later too. When you add 3 + 5 by counting on, you will pick whichever number is bigger to start from, trusting that the total does not depend on which number you started with.
Count 6 counters starting from the left, then count the same 6 counters starting from the right. What do you get each time?
- Starting from the left: 1, 2, 3, 4, 5, 6.
- Starting from the right, same counters: 1, 2, 3, 4, 5, 6.
- Both counts end on the same last number.
Answer: Six both times. The order you count in does not change the total.
- If 8 blocks are counted left to right and get 8, what will counting them right to left give?
- Why does the order you count in not change the total?
3. Counting objects arranged in a line, up to 20
PictorialWhen objects are arranged neatly, in a row or a grid, counting is easier because you can see which ones are already counted. Point to each one in turn, moving in one direction, and do not skip back over ones already said. Arranged objects can be counted all the way up to 20.
Losing your place is the most common slip past ten, because the teen number words (eleven, twelve, thirteen) do not obviously connect to the numbers before them the way twenty-one obviously follows twenty. Slow down through the teens and keep touching one object per word.
Count the 14 squares arranged in the grid above. What is the total?
- Touch each square once, moving along the first row, then the second row.
- Say the counting words in order: ...11, 12, 13, 14.
- The last word said is the total.
Answer: 14 squares altogether.
- Counting an arranged row of 18 objects, what is the very last number you should say?
- Why is it easier to keep your place when objects are arranged in a row than when they are scattered?
4. Counting objects scattered with no order, up to 10
PictorialScattered objects, like buttons spilled on a table, are harder to count because there is no natural order to follow. The one-to-one rule still applies: touch each object exactly once and say exactly one number for it. The trick is to move each counted object to a new pile, or to point firmly and mentally cross it off, so nothing gets counted twice and nothing gets skipped. Scattered counting is asked for only up to 10, because keeping track without an order gets much harder past that.
A useful strategy: as you count a scattered group, physically slide each counted object to one side. What is left uncounted stays visually separate from what is already counted, so it is obvious what still needs a number.
9 buttons are spilled on the table with no order. How do you count them accurately?
- Touch one button, say one, and slide it to a counted pile.
- Keep touching one uncounted button at a time, sliding each one over as you say its number: two, three, four, five, six, seven, eight, nine.
- Once every button has been moved, the last number said is the total.
Answer: 9 buttons altogether.
- Why is sliding each counted object to a new pile a good strategy for scattered objects?
- Scattered counting only goes up to 10 in Kindergarten. Why might that be easier than scattered counting to 20?
Common misconceptions and how to address them
MisconceptionThe child treats counting as a memorised chant and cannot say how many objects there are after counting, because the last number was never connected to the total.
Why it happens: Reciting number words in order (the stable-order principle) can be learned before the cardinality principle, that the last word said is the total, is understood.
How to address it: After every count, sweep a hand over the whole group and restate the total: there are six, pointing at all six together, not just the last object touched. Ask 'so how many are there?' right after every count to force the connection.
MisconceptionWhen objects are rearranged or the same group is recounted from a different starting point, the child believes the total has changed.
Why it happens: Without a secure grasp of the order-irrelevance principle, a different sequence of counting words attached to different objects feels like a different result.
How to address it: Count the same fixed group twice, from two different starting points, side by side, out loud. Land on the same last number both times and say clearly: same objects, same total, no matter where we started.
MisconceptionWhen counting scattered objects, the child skips one object or counts one object twice, most often when a group is spread out rather than lined up.
Why it happens: Without a fixed order to follow, it is easy to lose track of which objects are already counted, especially with irregular arrangements.
How to address it: Slide each object into a counted pile as it is touched, or line scattered objects up into a row before counting. One touch, one number, no exceptions.
MisconceptionCounting arranged objects into the teens, the child skips or mangles a number word, such as going ...10, 11, 20 or ...15, 16, 18.
Why it happens: The teen number names (eleven to nineteen) do not follow the same clean pattern as the decade names, so the sequence has to be memorised rather than derived.
How to address it: Practise the teen sequence on its own, slowly, with a number line or number track in view so the child can see the count is unbroken from ten to twenty.
Guided practice (with answers)
1. Count 8 blocks arranged in a row. What is the last number you say?
Answer: 8. The last number said when counting is the total.
2. 9 counters are counted left to right and the total is 9. If the same 9 counters are counted right to left, what is the total?
Answer: Still 9. The order you count in does not change the total.
3. Count the 12 squares arranged in a grid of 2 rows of 6. What is the total?
Answer: 12, because 2 rows of 6 is 12 squares altogether.
4. 7 buttons are scattered on a table with no order. What strategy keeps the count accurate?
Answer: Touch and count each button exactly once, sliding each counted button aside so it is not counted again.
5. Count arranged objects up to 20: what is the last number after nineteen?
Answer: 20.
6. A child counts 5 toys and says the total is 5, then recounts starting from a different toy. What should the new total be?
Answer: Still 5, because rearranging the starting point never changes how many objects there are.
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 the counting set, then use number lines and comparing to keep reinforcing that a count has one fixed total.
Differentiation
- Stay within 5 until one-to-one touching is completely reliable, then stretch to 10 and eventually 20.
- Always arrange objects in a row before counting until scattered counting is secure.
- Use a number track alongside the objects so the child can point to each number as well as each object.
- Slow the teen numbers right down, one at a time, with plenty of repetition.
- Count backward from 20 to check the same total in reverse.
- Estimate a scattered group first, then count to check the estimate.
- Count a group of up to 20 that is scattered, not arranged, as a challenge.
- Ask which of two counted groups has more without recounting, using the totals already found.
Assessment: exit ticket
A three-question exit ticket for the last few minutes, sampling the total idea, order-irrelevance, and scattered counting.
1. Count 6 blocks arranged in a row. What is the total?
Answer: 6, the last number said when counting.
2. If counting a group from the left gives 10, what will counting the same group from the right give?
Answer: 10, the order of counting does not change the total.
3. 8 counters are scattered with no order. What should you do so none is missed or double counted?
Answer: Touch each one exactly once, sliding each counted counter aside.
Teacher notes and timings
- Rough timing across three lessons: Lesson 1 the last-number-is-the-total idea (section 1), Lesson 2 order-irrelevance (section 2), Lesson 3 arranged counting to 20 and scattered counting to 10 (sections 3 to 4 and the exit ticket).
- This is genuinely one of the harder ideas in early number sense despite looking simple: children can recite numbers fluently long before they grasp that the last number said is the total (the cardinality principle) and that counting order does not matter (order-irrelevance). Both are named research findings (Gelman and Gallistel's counting principles), not just classroom folklore, so do not rush past them.
- Curriculum note: US Kindergarten (K.CC.B.4, K.CC.B.5) states both the cardinality and the arranged-versus-scattered distinction explicitly. ACARA Foundation (AC9MFN02, AC9MFN06) covers counting and comparing to 20 and connecting names to quantities, a close match without separately distinguishing arranged from scattered objects.
- Present mode and print both work: use the Print button for a student worksheet, or project the page and count the on-screen groups together as a class.