Comparing quantities

When you reach for it

To compare two groups, a child asks “which has more, which has fewer, or are they the same?” There are two honest ways to answer: match the things up one to one and see which group has some left without a partner, or count each group and see which number comes later in the counting words. Reach for it once a child can match and count one to one; it is the step before comparing two written numbers.

Keep every group at ten or fewer. Each day below takes about ten to fifteen minutes.

Day 1: Line them up

You need: plastic bottle caps in two colors (about ten dark and ten light), and the edge of a table or a strip of paper to line them along.

Put 7 dark caps in a row along the edge. Hand the child 5 light caps and say, “Put one light cap right under each dark cap.” When the light caps run out, ask, “Does every dark cap have a partner? Which color has more caps?” Then ask the other half: “Which color has fewer?”

Say both sentences together: “There are more dark caps. There are fewer light caps.” The two dark caps at the end with no partner are what make the dark group bigger. If every cap had a partner, you would say, “They are the same,” or “There are the same number.” No counting is needed today; the leftovers show the answer.

Practice. Line up the caps, one light cap under each dark cap.

1. 6 dark caps and 6 light caps. Answer: every cap has a partner, so there are the same number. No cap is left over.
2. 4 dark caps and 7 light caps. Answer: 3 light caps have no partner, so there are more light caps and fewer dark caps.
3. 8 dark caps and 2 light caps. Answer: 6 dark caps have no partner, so there are more dark caps and fewer light caps.
4. Line up 9 pasta shells and match each with a paper clip from a pile of 10 clips. Answer: 1 clip has no partner, so there are more clips and fewer shells.

Day 2: The longer row is not always more

You need: the caps, 5 shoes and 7 marbles (or other tiny things).

Lay 6 dark caps far apart in a long row. Under them, push 8 light caps close together in a shorter row. Ask, “Which row has more caps?” A child may point to the long row, because it looks bigger. Do not correct it; say, “Let’s check by matching.” Slide the light caps along so one sits under each dark cap.

Two light caps are left with no partner, so the light row has more, even though it was shorter. Ask, “Did we add any caps? Did we take any away?” No: only the spacing changed, and spacing does not change how many. Size works the same way. Lay out the 5 shoes and set one marble in each shoe. Ask, “Which group takes up more room? Which group has more things?”

Practice.

1. The rows in the picture, 6 dark caps and 8 light caps. Answer: 2 light caps have no partner, so there are more light caps, even though their row is shorter.
2. The 5 shoes and the 7 marbles, one marble in each shoe. Answer: 2 marbles are left, so there are more marbles and fewer shoes, even though the shoes take up more room.
3. Push 10 caps close together and spread 9 caps far apart. Answer: matching leaves 1 cap of the short row over, so the short row of 10 has more.
4. Spread 3 caps across the whole table and push 7 caps into a tight little row. Answer: 4 caps of the tight row have no partner, so the tight row has more.

Day 3: Count to compare

You need: two paper plates and about 20 caps.

Put a pile of caps on each plate and set the plates at opposite ends of the table, too far apart to line up. Ask, “Which plate has more caps? How can we tell without moving them?” Count one plate, touching each cap, and say the last number; then count the other. Now say the counting words slowly from “one”: “one, two, three, four, five, six, seven, eight, nine.” The number we say later names the bigger group. Seven comes before nine, so the pile of seven has fewer.

Last, pour both piles out and match them up to check. Matching and counting always agree, which is why either one may be used.

Practice.

1. The piles in the picture, 7 dark caps in A and 9 light caps in B. Answer: B has more, even though A is spread over more space; 9 comes after 7, and matching leaves 2 caps of B over.
2. One plate with 5 crackers and one with 8 crackers. Answer: 8 comes after 5, so the plate of 8 has more and the plate of 5 has fewer; 3 crackers would be left unmatched.
3. A jar with 10 pebbles and a jar with 6 pebbles. Answer: 10 comes after 6, so the jar of 10 has more; 4 pebbles would be left unmatched.
4. Ava counts 4 stickers and Eli counts 4 stickers. Answer: both counts end on the same word, so they have the same number.

Day 4: Make them the same

You need: the caps.

Lay out 3 dark caps and match them with a row of 6 light caps. Ask, “Which color has more? How many light caps have no partner?” Then ask, “What could we do so the two groups are the same?”

There are two good answers. Bring 3 more dark caps, one for each light cap with no partner, and now both colors have 6. Or take away the 3 light caps with no partner, and now both have 3. Either way, every cap has a partner, which is exactly what “the same” means. Count both colors afterward to see the two counts match.

Practice.

1. The rows in the picture, 3 dark caps and 6 light caps. Answer: 3 light caps have no partner. Add 3 dark caps (both groups 6) or take away 3 light caps (both groups 3).
2. 2 dark caps and 7 light caps. Answer: 5 light caps have no partner. Add 5 dark caps (both 7) or take away 5 light caps (both 2).
3. 9 dark caps and 8 light caps. Answer: 1 dark cap has no partner. Add 1 light cap (both 9) or take away 1 dark cap (both 8).
4. A tea party with 5 dolls and 4 plates. Answer: 1 doll has no plate, so there are fewer plates; one more plate makes them the same.

Exercises

1. Look at the picture for this exercise. Does pile A have more caps, does pile B have more, or are they the same?
2. Lay 4 toy trucks in a long row and put 10 dry pasta pieces in a small heap. Ask, “Which group has fewer? How can you check?”

Where this goes next

Comparing groups of things leads to comparing two written numbers, where 9 and 7 are compared without any caps in sight, and then to the signs that record the answer in Comparing numbers and the symbols >, < and =. Adding or taking away a single cap to change a group is the work of One more and one less. Asking “how many more?” and counting the caps with no partner grows into finding a difference by counting up.