What multiplication means
Some things come in groups that are all the same size, like muffins in pans or cans on shelves. How can you find how many there are in all without counting every one?
Where you are 16 in Multiplication and division
Why this exists
A child who can add can find how many cans are on three shelves by counting every can. That works, but it is slow, and it throws away something the child can see: every shelf holds the same number of cans. When the groups are all the same size, the total depends on only two numbers, how many groups there are and how many are in each group. Know those two and the total is decided.
This lesson is about that one idea, equal groups, and three ways to find the total that they make: adding the same number again and again, skip counting, and making equal hops along a number line. All three are the same thing seen from different sides. Together they are what multiplication means, long before the child meets a multiplication sign or learns a single multiplication fact. A child who knows that “three groups of four” means 4 + 4 + 4 will later see each fact as a picture, not a string of words.
The lesson runs over four days. Day 1 is what equal groups are and how adding finds the total. Day 2 finds the total faster by skip counting, including groups of ten. Day 3 shows the same groups as equal hops on a number line. Day 4 asks why the groups must be equal, and shows that three groups of four and four groups of three make the same total.
Read everything aloud with the child, and let them build it with counters, cubes, buttons or small cups.
Definition
A group is a set of things kept together: the muffins in one pan, the cans on one shelf, the counters inside one loop. Groups are equal groups when every group holds the same number of things.
“3 groups of 4” means 3 equal groups with 4 things in each group. It names two numbers, and they play different parts:
- the number of groups, here 3, says how many groups there are;
- the size of each group, here 4, says how many things are in every one of them.
The total is how many things there are in all the groups together. For 3 groups of 4, the total is 4 added once for each group:
Adding the same number again and again like this is called repeated addition. The number that is added is the size of each group, and the number of times it appears is the number of groups.
Intuition
Part 1: equal groups, and adding to find the total
Put 12 counters on the table and set them in three loops of string, four in each loop.
Ask the child three questions, always in the same order. How many groups are there? Three. How many are in each group? Four. How many are there in all? Children first answer the last question by counting every counter. Then show them a shorter way: the first group is 4; the second group brings 4 more, so 4 + 4 = 8; the third brings 4 more, so 8 + 4 = 12. The total is 12.
That running total, 4, then 8, then 12, is the heart of it. Each new group adds the same amount, so the child is never adding anything new, only 4 again. The adding facts used are ones the child already has from Adding and subtracting within 20, and as the totals grow past twenty, adding a one-digit number to a two-digit one.
Say it in words before writing it: “3 groups of 4 is 12.” Then write the adding sentence, one 4 for each group. Count the fours in the sentence and there are 3 of them, one for each loop. The sentence and the picture match.
The two numbers are not interchangeable in the picture. Three groups of four means three loops with four counters each. Four groups of three would be four loops with three in each. Those are different pictures. (Part 4 shows that they still have the same total, which is a surprise worth having.)
Equal groups are everywhere once you look. Wheels on a few tricycles, fingers on two hands, socks in pairs, eggs in boxes. Any time a child says “each one has”, there are equal groups. Groups set out in neat rows and columns make an array; arrays have their own lesson, arrays: rows and columns of objects.
Part 2: skip counting the total
Adding 4 + 4 + 4 + 4 + 4 one step at a time gives 4, 8, 12, 16, 20. Those are exactly the numbers said when counting by fours. So skip counting is repeated addition with the adding already done: say one number for each group and the last number said is the total.
The habit that makes this safe is keeping track of how many groups have been counted. A child counting 5 groups of 4 can touch each loop, or put up one finger for each group, while saying “four, eight, twelve, sixteen, twenty”. When the fifth finger goes up, the count stops, and the last number said is the total.
Skip counting is quickest with the counts the child knows best: twos, fives and tens.
Groups of ten are worth time of their own. 4 groups of 10 is
and counting by tens says “ten, twenty, thirty, forty”. The total is 4 tens, and 4 tens is 40, just as Tens and ones says. So a child who understands place value already knows every “so many groups of ten”: 3 groups of 10 is 30, 5 groups of 10 is 50. Bundles of ten sticks show this well: each bundle is a group of ten.
Part 3: equal hops on a number line
A number line shows equal groups as lengths. Start at 0 and make a hop of 4. Then another hop of 4, and another.
Each hop is one group, and its length is the size of the group. The hops land on 4, 8 and 12, the same numbers as the running total in Part 1 and the same numbers as counting by fours. After 3 hops of 4 the hop lands on 12, so 3 groups of 4 is 12 again.
The child reads the picture with the same three questions. How many hops? Three. How long is each hop? Four. Where does the last hop land? 12. The number of hops is the number of groups, the length of each hop is the size of each group, and the landing place is the total.
Two habits keep the number line honest. Start at 0: the first hop must begin where there is nothing yet. And make every hop the same length, counting the spaces, not the tick marks. A hop of 4 from 8 passes 9, 10, 11 and lands on 12: four spaces. Children who have read number lines with steps of two, five or ten will see that hops of 5 or 10 land on the labeled marks of such a line.
Part 4: why the groups must be equal, and turning the groups around
Here are 12 counters set out two ways.
On the left there are three groups too, holding 5, 4 and 3. Adding still finds the total, 5 + 4 + 3 = 12. But the left side is not “3 groups of” anything: there is no one number that says how many are in each group, so the two-number description fails. On the right, every group holds 4, and “3 groups of 4” says everything about the picture.
Unequal groups can sometimes be made equal by moving things from one group to another. Move one counter from the group of 5 to the group of 3, and the left side becomes 4, 4, 4. The total stays 12, because nothing was added or taken away. But not every total can be split into a given number of equal groups. 13 counters in three groups never come out equal: three groups of 4 make only 12, and three groups of 5 make 15, so one counter is always left over.
Now the surprise. Take three groups of four, made of four kinds of counter, one of each kind in every group.
Pull out the dark counter from each group: that is 3 dark counters, a new group of 3. Do the same for the shaded counters, the open ones and the ones with a dot. Four kinds, so four new groups, and each new group has one counter from each of the 3 old groups, so it holds 3. The same 12 counters now make 4 groups of 3:
Nothing was added and nothing was lost; the counters were only sorted a new way. So 3 groups of 4 and 4 groups of 3 always have the same total. This is the Properties of multiplication, seen here before it has a name.
The two are still different pictures and can be different stories. Three bags of four apples and four bags of three apples need different numbers of bags. The total is the same; the groups are not.
Results
Three ways to one total. For equal groups, these all give the same number:
- add the size of each group once for every group;
- skip count by the size of each group, saying one number per group;
- start at 0 on a number line and make one hop, as long as a group, for every group.
Two numbers decide it. The number of groups and the size of each group fix the total. Any equal groups with the same two numbers have the same total, whatever is in them.
Turning the groups around. a groups of b and b groups of a have the same total. For example, 3 groups of 4 and 4 groups of 3 are both 12.
Groups of ten. So many groups of ten is that many tens: 4 groups of 10 is 40.
Why it is true
Every thing in the picture sits in exactly one group. So counting all of them is the same as counting the first group, then carrying on through the second, then the third. Counting through a group of 4 adds 4. So the total is 4 added once for each group, which is the repeated addition.
Skip counting says the running totals of that addition: after the first group, after the second, and so on. The last one said is the whole sum.
A hop of 4 on the number line moves 4 spaces to the right, which adds 4 to where it started. Starting from 0, the first hop lands on 4, the second on 4 + 4, the third on 4 + 4 + 4. So the landing places are the running totals again, and the last landing place is the total.
For turning the groups around: give each thing in a group a place in its group, first, second, third, fourth. Sorting the things by their place makes one new group for each place. There are 4 places, so 4 new groups. Each old group gives exactly one thing to each new group, so each new group holds one thing per old group, 3 in all. Every thing is moved once and none is lost or added, so the total is unchanged. The same sorting works for any number of groups of any equal size. \blacksquare
Worked examples
Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, Examples 5 and 6 with Part 3, and Examples 7 and 8 with Part 4.
Muffins in pans
Rehana has 4 small muffin pans. Each pan holds 6 muffins, and every pan is full. How many muffins are there?
SolutionEach pan is a group, and every pan holds the same number, so these are equal groups. There are 4 groups, one for each pan, and 6 in each group. So this is 4 groups of 6, and the total is 6 added once for each pan:
Keep a running total, one pan at a time: 6, then 6 + 6 = 12, then 12 + 6 = 18, then 18 + 6 = 24. There are 24 muffins. Check the sentence against the story: it has four sixes, one for each pan.
Cans on shelves
Akmal puts cans on 3 shelves. He puts 7 cans on each shelf. What are the groups, how many groups are there, how many are in each group, and how many cans are there in all?
SolutionEach shelf is a group. There are 3 groups, and each group holds 7 cans, so this is 3 groups of 7. Add one 7 for each shelf:
The first two shelves make 7 + 7 = 14, a double. The third shelf adds 7 more. To add 14 + 7, the 7 can be split into 6 and 1: 14 and 6 make 20, and one more is 21. There are 21 cans.
The two numbers in the question play different parts. The 3 is how many shelves there are and never appears in the adding sentence as a number being added; it shows up as how many sevens are written.
Blueberries on muffins
Abimbola puts 5 blueberries on top of each muffin. She tops 5 muffins. How many blueberries does she use?
SolutionEach muffin is a group of 5 blueberries, and there are 5 muffins, so this is 5 groups of 5. Skip count by fives, saying one number for each muffin and putting up one finger each time: “five, ten, fifteen, twenty, twenty-five”. The fifth finger goes up on “twenty-five”, so she uses 25 blueberries.
The skip count is the same as the repeated addition, with the running totals said out loud:
Here both numbers are 5, so it is easy to lose track of which five is which. The fingers settle it: they count muffins, and each number said counts blueberries.
Trays of ten
At a store, cans come in trays of 10. Rehana stacks 4 full trays on a shelf. How many cans are on the shelf? How many would there be with one more tray?
SolutionEach tray is a group of 10, and there are 4 trays, so this is 4 groups of 10. Count by tens, one number per tray: “ten, twenty, thirty, forty”. As an adding sentence,
There are 40 cans. Another way to see it: 4 groups of ten is 4 tens, and 4 tens is 40.
One more tray is one more group of 10, so the total grows by 10, from 40 to 50. That is 5 groups of 10, or 5 tens. Adding a group always adds the size of one group, never just one more can.
The grasshopper’s hops
A toy grasshopper starts at 0 and hops along a number line. Every hop is the same length. Read the picture: how many hops, how long is each, and where does it land? What equal groups does it show?
Count the arrows: there are 4 hops. Each arrow is labeled +5, and each goes from one dot to a dot 5 spaces further on, so every hop is 5 long. The last hop ends at 20. The landing places, 5, 10, 15, 20, are the count by fives, one number per hop.
So the picture is 4 groups of 5, and
Each hop is one group. The length of a hop is the size of the group, and where the last hop lands is the total.
How long is each hop?
Rehana makes 4 equal hops on a number line, starting at 0. Her last hop lands on 16. How long is each hop?
SolutionThis time the total and the number of groups are known, and the size of each group is missing. Try a length and see where 4 hops of it land.
Hops of 3 land on 3, 6, 9, 12. That stops at 12, short of 16, so 3 is too short.
Hops of 5 land on 5, 10, 15, 20. That passes 16, so 5 is too long.
Hops of 4 land on 4, 8, 12, 16. That is it: each hop is 4 long, and
Notice that the landing place grows as the hops grow. Longer hops always land further on, so once 3 is too short and 5 is too long, 4 is the only length left to try.
Making the bags equal
Akmal packs 18 cans into 3 bags. The bags hold 8, 4 and 6 cans. Are these equal groups? Can he move cans so that they are? With 19 cans, could 3 bags be equal?
SolutionThe bags do not all hold the same number, so they are not equal groups. They still have a total: 8 + 4 + 6 = 18 cans.
Moving cans from one bag to another never changes the total, so if the bags are made equal, they will be 3 equal groups that make 18. Count by sixes: 6, 12, 18. Three groups of 6 make 18, so each bag should hold 6. The bag of 6 is already right. Move 2 cans from the bag of 8 to the bag of 4: now the bags hold 6, 6 and 6. That is 3 groups of 6:
With 19 cans it cannot be done. Three equal bags of 6 hold 18, one short of 19, and three equal bags of 7 hold 21, two too many. No size in between is possible, so 19 cans always leave one bag holding more than another.
Two plates and five plates
Rehana puts 5 muffins on each of 2 plates. Abimbola puts 2 muffins on each of 5 plates. Who has more muffins? Show why without counting every muffin.
SolutionRehana has 2 groups of 5, and Abimbola has 5 groups of 2:
They have the same number, 10 each.
To see why without adding, start from Rehana’s plates. Take one muffin from each of her 2 plates and put the pair on a new plate. Do it again, and again, until her plates are empty. Each time, the new plate gets 2 muffins, one from each of her plates, and it can be done 5 times because each of her plates started with 5. So her muffins end up as 5 plates of 2, which is exactly Abimbola’s arrangement. No muffin was added or eaten, so the totals match.
The stories are still different: Rehana uses 2 plates and Abimbola uses 5. Only the total is the same.
Boundaries
- The groups must be equal. “So many groups of so many” describes a picture only when every group holds the same number. Groups of 5, 4 and 3 can be added, but they are not “3 groups of” any number, and skip counting by one number does not count them.
- The hops start at 0. Hops of 4 that start at 2 land on 6, 10, 14. The last landing place is then 2 more than the total of the groups. The landing place is the total only when the first hop starts at zero.
- Every thing is counted once. Each thing must be in exactly one group. If two loops share a counter, the counter is counted twice and the total comes out too big.
- Things are not cut up here. Every group holds a whole number of things, so some totals cannot be split into a given number of equal groups: 13 into 3 groups, or 19 into 3 groups, always leaves something over. Sharing those leftovers fairly needs cutting things up, which is a much later idea.
- Zero and one. 1 group of 4 is just 4. 4 groups of 0 are four empty loops, so the total is 0 + 0 + 0 + 0 = 0. 0 groups of 4 means there are no groups at all, and the total is 0 too.
- Same total, different groups. 3 groups of 4 and 4 groups of 3 have the same total, but they are not the same picture or the same story. A question that asks how many bags, or how many in each bag, needs the right one.
Where this goes next
Arrays set equal groups out in rows and columns, which makes them easy to count and easy to turn around; they come next in Arrays: rows and columns of objects. Later, “3 groups of 4” gets a short way to write it, 3 \times 4, read “three times four”, and arrays are counted that way in Arrays. Every multiplication fact in Multiplication facts is a number of equal groups, and a child who can still picture the groups can rebuild any fact they forget.
The turning-around of Part 4 becomes the Properties of multiplication. Story problems about bags, boxes and rows, in Equal group problems, use exactly the three numbers of this lesson: the number of groups, the size of each group and the total. When the size of each group is missing, as in Example 6, or the number of groups is missing, the question is a division, which is where what division means begins. Equal hops also return in times as many, and a picture graph where one picture stands for several things, a scaled picture graph, is equal groups again.
Exercises
Exercises 1 to 6 go with Part 1, 7 to 12 with Part 2, 13 to 17 with Part 3, and 18 to 21 with Part 4. The last group draws on all four.
Part 1: equal groups, and adding to find the total
Use the groups for Exercises 1 to 3. For each row, say how many groups there are, how many are in each group, and the total. Write the adding sentence.
- Row A
- Row B
- Row C
- Write 3 groups of 8 as an adding sentence and find the total.
- 6 + 6 + 6 = \square. How many groups is this, and how many are in each group?
- Akmal has 2 shelves with 8 cans on each shelf. How many cans are there?
Part 2: skip counting the total
- Skip count by fours to find the total of 5 groups of 4.
- How many is 3 groups of 10?
- Rehana stacks 5 trays of 10 cans. Count by tens to find how many cans there are.
- Abimbola has 2 bowls with 7 plums in each bowl. How many plums are there?
- A store shelf has 2 full trays of 10 cans and 3 loose cans. How many cans are on the shelf?
- Skip count by threes. How many groups of 3 make 18?
Part 3: equal hops on a number line
Use the number lines for Exercises 13 and 14. For each line, say how many hops there are, how long each hop is, and where the last hop lands. Write the adding sentence.
- Line A
- Line B
- Draw a number line from 0 to 20. Make 2 hops of 10 from 0. Where does the last hop land?
- A grasshopper starts at 0 and lands on 2, 4, 6, 8, 10, 12 and 14. How many hops did it make, and how long was each hop? Write the adding sentence.
- Akmal makes 2 equal hops from 0 and lands on 22. How long is each hop?
Part 4: equal groups, and turning them around
- Rehana puts 3, 3, 3 and 4 muffins on four plates. Are these equal groups? How many muffins are there?
- Five shelves hold 2, 4, 1, 3 and 5 cans. Move cans so that every shelf holds the same number. How many cans go on each shelf?
- Do 2 groups of 4 and 4 groups of 2 have the same total? Find both.
- Which is more, 5 groups of 3 or 2 groups of 8?
Explain
- What is the total of 4 groups of 0? Of 0 groups of 4? Explain each.
- Hops of 3 start at 0. Explain why the fifth hop lands on the fifth number said when counting by threes.
- Lines A and B in Exercises 13 and 14 land on the same number. Explain with counters why 5 groups of 3 and 3 groups of 5 must have the same total.
- Can 11 counters be put into equal groups, with more than one group and more than one counter in each group? Explain.
Answers
- 2 groups, 6 in each, total 12: 6 + 6 = 12.
- 4 groups, 2 in each, total 8: 2 + 2 + 2 + 2 = 8.
- 3 groups, 3 in each, total 9: 3 + 3 + 3 = 9.
- 8 + 8 + 8 = 24. Two eights make 16, and 16 + 8 = 24.
- \square = 18. It is 3 groups of 6: three sixes, one for each group. Running total 6, 12, 18.
- 16 cans. Each shelf is a group of 8, so 8 + 8 = 16.
- 20. “Four, eight, twelve, sixteen, twenty”, one number for each of the 5 groups: 4 + 4 + 4 + 4 + 4 = 20.
- 30. “Ten, twenty, thirty”, or 3 tens is 30: 10 + 10 + 10 = 30.
- 50 cans. “Ten, twenty, thirty, forty, fifty”, one number for each tray: 5 tens is 50.
- 14 plums. 7 + 7 = 14, a double.
- 23 cans. The trays are 2 groups of 10, which is 20, and the 3 loose cans add 3 more: 20 + 3 = 23. The loose cans are not a group of ten, so they are added on their own.
- 6 groups. “Three, six, nine, twelve, fifteen, eighteen” is six numbers, so 3 + 3 + 3 + 3 + 3 + 3 = 18.
- 5 hops, each 3 long, landing on 15: 3 + 3 + 3 + 3 + 3 = 15.
- 3 hops, each 5 long, landing on 15: 5 + 5 + 5 = 15.
- On 20. The first hop lands on 10, the second on 20: 10 + 10 = 20.
- 7 hops, each 2 long. There are seven landing places after 0, and each is 2 more than the one before: 2 + 2 + 2 + 2 + 2 + 2 + 2 = 14.
- 11. Two equal hops make a double, and 11 + 11 = 22. (Hops of 10 land on 20, too short; hops of 12 land on 24, too far.)
- No, one plate has 4 and the others have 3. There are 13 muffins: 3 + 3 + 3 + 4 = 13.
- 3 cans on each shelf. The total is 2 + 4 + 1 + 3 + 5 = 15, and 5 groups of 3 make 15: 3 + 3 + 3 + 3 + 3 = 15. Moving cans does not change the total. For example, move 1 can from the shelf of 4 to the shelf of 2, and 2 cans from the shelf of 5 to the shelf of 1.
- Yes. 4 + 4 = 8 and 2 + 2 + 2 + 2 = 8. Taking one counter from each group of 4 makes a group of 2, and this can be done 4 times.
- 2 groups of 8. 3 + 3 + 3 + 3 + 3 = 15 and 8 + 8 = 16, and 16 is more than 15.
- Both are 0. Four groups of 0 are four empty groups, 0 + 0 + 0 + 0 = 0. Zero groups of 4 means there are no groups, so there is nothing to count.
- Each hop of 3 adds 3 to where it starts. From 0, the hops land on 3, 6, 9, 12, 15, and each landing place is the running total of the threes so far. Counting by threes says those same running totals, “three, six, nine, twelve, fifteen”, so the fifth hop and the fifth number are both 15.
- Build 5 groups of 3 so that each group has one counter of each of three kinds. Pull the counters of each kind into a group of their own: there are 3 kinds, so 3 new groups, and each takes one counter from each of the 5 old groups, so each holds 5. The same 15 counters now make 3 groups of 5, with none added or lost.
- No. Groups of 2 make 2, 4, 6, 8, 10, 12, skipping 11. Groups of 3 make 3, 6, 9, 12, skipping 11. Groups of 4 make 4, 8, 12, and groups of 5 make 5, 10, 15, both skipping 11. Groups of 6 or more make at least 12 with just two groups. So 11 counters can only be 1 group of 11 or 11 groups of 1.