Comparing numbers and groups
More, fewer and the same, comparing groups by matching and counting, and comparing written numbers
About three to four lessons of 30 to 40 minutes
Who has more?
Two children each grab a handful of counters and drop them in a pile. Everyone wants to know the same thing straight away: who has more? It is one of the first fair-share questions children ever ask, at the snack table, in the sandpit, everywhere.
Today we learn two ways to answer it for sure, not just by guessing. We can line the two piles up side by side and match them one to one, or we can count each pile and compare the numbers. By the end you will say which group has more, which has fewer, and when two groups have the same, and you will be able to prove it.
- 5 strawberries next to 3 strawberriesmatch them up: 5 has two left over, so 5 is more than 3
- A pile of socks matched into pairsif every sock finds a partner, the two groups are the same
- A row of 4 red counters and a row of 4 blueeach red lines up with a blue, none left over, so they are equal
- The written numbers 7 and 27 is further along when we count, so 7 is greater than 2
What students will be able to do
Students will decide whether one group has more, fewer, or the same number of objects as another by matching one to one and by counting, and will compare two written numbers from 1 to 10 to say which is greater, less, or equal.
- I can match two groups one to one to see which has more.
- I can count two groups and compare the totals.
- I can use the words more, fewer, and the same.
- I can tell when two groups are equal.
- I can look at two written numbers and say which is greater and which is less.
Standards this unit teaches
- K.CC.C.6Common Core (US)Compare groups of objects
Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g. by using matching and counting strategies.
- K.CC.C.7Common Core (US)Compare written numbers
Compare two numbers between 1 and 10 presented as written numerals.
- AC9MFN02Australian Curriculum v9 (ACARA)Count and compare to 20 (Foundation)
Count and compare collections of up to at least twenty objects, and explain the reasoning behind which group has more or fewer.
- AC9MFN06Australian Curriculum v9 (ACARA)Connect number names, numerals and quantities (Foundation)
Connect number names, numerals and quantities, and represent the count of a small collection in more than one way. Comparing two written numerals rests on this link between a numeral and its quantity.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- More (greater)
- a bigger amount, further along when you count
- Fewer (less)
- a smaller amount, not as many
- Equal (the same)
- the two groups have exactly the same number, none left over
- Match one to one
- pair each thing in one group with one thing in the other
- Leftover
- the extra things in the bigger group that have no partner
- Compare
- look at two amounts to see how they are different or the same
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. Matching to compare, without counting
ConcreteYou can find out which group has more before you can even count. Line the two groups up in two rows, one thing above one thing, and draw a line between each pair. If one row has things left over with no partner, that row has more. If they pair up perfectly, the groups are the same.
Matching is powerful because it answers the question directly: more, fewer, or the same, with no numbers at all. The leftovers are the proof. If two strawberries in the top row have no partner underneath, the top row has more, and it has more by exactly two.
Insist on lining the objects up neatly. If they are jumbled, a child might think a spread-out group of three is more than a bunched-up group of five. Comparing is about how many, not how much space they take up.
- If every thing in both rows finds a partner, what do we know?
- One row has two things with no partner. Which row has more?
- Does spreading a group out make it more?
2. Counting to compare
PictorialThe other way to compare is to count each group and then compare the two numbers. Count the first group and remember its number. Count the second group. Now you have two numbers, and the bigger number belongs to the group with more.
Counting works even when the groups are far apart or too big to line up. Count the red counters: 4. Count the blue counters: 6. Six is more than four, so there are more blue counters.
The number you count last tells how many, and when you count on, later numbers are bigger. Six comes after four when you count, so six is the greater amount. This links straight to the number line: further along means greater.
One plate has 7 grapes and another has 5. Which plate has more, and how many more?
- Count the first plate: 7.
- Count the second plate: 5.
- Compare: 7 comes after 5 when we count, so 7 is greater.
- Match to find how many more: 7 pairs with 5 and has 2 left over.
Answer: The first plate has more, by 2 grapes.
- One group counts to 8 and the other to 5. Which has more?
- If both groups count to 6, what do we say about them?
- How can matching tell us how many more, not just which is more?
3. When groups are the same
PictorialSometimes neither group has more. When you match them and every single thing finds a partner with none left over, the groups are equal. When you count them and get the same number, they are equal too. Equal means exactly the same amount.
Equal is not a leftover word for 'nearly the same'. It means the counts match exactly. Four red and four blue counters are equal because each red pairs with a blue and both counts are 4.
Show a nearly-equal pair too, like 4 and 5, so the class feels the difference. One leftover means not equal. Zero leftovers means equal.
- Both groups count to 5. Are they equal?
- If one thing has no partner, are the groups equal?
- What does equal mean in your own words?
4. Comparing written numbers
AbstractOnce you trust counting, you can compare two written numbers without any objects at all. Read the two numerals and picture how far along the count each one is. The number you would say later is the greater one. For 7 and 2, you say 7 much later than 2, so 7 is greater than 2.
A number line makes this certain. The number further from 0 is greater. Put a finger on each number: whichever is further to the right is more. If both numerals are the same, the numbers are equal.
Keep the range small at this grade, 1 to 10, and keep tying each numeral back to a quantity. Seven means seven things, and seven things is more than two things.
Which is greater, 8 or 5?
- Read both numbers: 8 and 5.
- Think how far along the count each one is. 8 comes after 5.
- The number you reach later is the greater one.
Answer: 8 is greater than 5. And 5 is less than 8.
- Which is greater, 9 or 4?
- Which is less, 3 or 6?
- Is 5 greater than, less than, or equal to 5?
Common misconceptions and how to address them
MisconceptionThe group that takes up more space has more, even if it has fewer things.
Why it happens: Young children judge by how much room a group fills, not by the count. A spread-out group looks bigger.
How to address it: Line both groups up in neat rows and match them. Then bunch the smaller group tight and spread it out again to show the count never changed.
MisconceptionThe bigger-sounding number is the last one said, so whatever you count last is more.
Why it happens: The child confuses 'the last number is how many' with 'the last group counted is bigger', regardless of the actual totals.
How to address it: Count both groups, write both numbers down, then compare the two numbers. Which group was counted second does not matter, only the totals do.
MisconceptionEqual just means close, so 4 and 5 are equal enough.
Why it happens: Everyday talk uses 'the same' loosely, and one leftover feels close.
How to address it: Match them one to one. The single leftover proves they are not equal. Equal means no leftovers and matching counts.
MisconceptionYou can only tell which is more by counting, never by matching.
Why it happens: Counting is taught first and hard, so matching is overlooked as a strategy.
How to address it: Compare two groups by matching alone, before counting. The leftovers answer 'which has more' directly and even show how many more.
MisconceptionA written 6 is less than a written 2 because 6 is drawn with fewer strokes, or for no clear reason.
Why it happens: The numeral is treated as a shape, disconnected from the quantity it stands for.
How to address it: Put both numerals on the number line and count up to each. The one you reach later, further from 0, is greater. Anchor 6 to six real things.
MisconceptionMore and fewer are the same kind of answer, so it does not matter which word you use.
Why it happens: The two comparison words are new and easy to swap.
How to address it: Use both words for the same pair every time: 7 is more than 5, and 5 is fewer than 7. The bigger group is more, the smaller group is fewer.
Guided practice (with answers)
1. Match these groups: which has more?
Answer: The top group. It sticks out past the bottom group by two, so it has more.
2. Count both groups and compare: 3 dots and 5 dots.
Answer: 5 is more than 3. The group of 5 has more, by 2.
3. Are 4 red and 4 blue counters equal?
Answer: Yes. They match one to one with none left over, and both count to 4.
4. Which is greater, 9 or 6?
Answer: 9. It comes later when you count and sits further from 0 on the number line.
5. Which is fewer, 2 or 8?
Answer: 2. It is the smaller amount.
6. One basket has 5 apples, another has 5 apples. Compare them.
Answer: They are equal. Both counts are 5.
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 comparing groups by matching, then compare written numbers.
Differentiation
- Compare by matching before counting, so the answer does not depend on a secure count.
- Keep both groups within 5 until the words more, fewer, and equal are confident.
- Use a two-row frame or a mat with paired spaces so the one-to-one match is built in.
- For written numbers, keep a labelled number line in front of the child to check which is further along.
- Ask how many more, not just which is more, so the leftover becomes a number.
- Compare three groups and order them from fewest to most.
- Compare written numbers up to 20, reading each off the hundred square.
- Have the child make a group that is equal to a given group, then one that has two more.
Assessment: exit ticket
A three-question exit check in the last five minutes. It samples comparing by matching, comparing by counting, and comparing written numbers.
1. Two groups: 6 and 4. Which has more?
Answer: 6 has more, by 2.
2. Are a group of 5 and a group of 5 equal?
Answer: Yes, they are equal.
3. Which is greater, 8 or 3?
Answer: 8.
Teacher notes and timings
- Rough timing across three to four short lessons: Lesson 1 matching one to one (section 1), Lesson 2 counting to compare (section 2), Lesson 3 equal groups (section 3), Lesson 4 comparing written numbers plus the exit check (section 4 and assessment).
- Language to keep saying: more, fewer, the same, match one to one, how many more. Use both comparison words for every pair.
- Keep two-row matching mats and small groups of counters on desks. When a child is unsure, have them line the groups up and look for leftovers.
- The comparison bars are drawn to scale on a shared scale, so the longer bar really is the bigger group. Use them to make more and fewer visible at a glance.
- Keep written-number comparisons within 1 to 10 (K.CC.C.7). Comparing two-digit numbers with the greater-than and less-than symbols is a Grade 1 step (1.NBT.B.3).
- Present mode and print both work: use the Print button for a clean teacher copy or a student handout, and project the bars and number line to compare together.