How many more and how many fewer

When you reach for it

Some stories put two amounts side by side and talk about the gap between them. “Arlo has 3 more apples than Wren.” “Finn has 4 fewer crayons than Bea.” “The sunflower is taller than the tulip.” Nothing joins, nothing leaves; two amounts are simply compared. These are comparing stories.

A child has already met the gap itself in subtraction as the difference: when both amounts are known, the gap is the bigger take away the smaller. A story can hide any of the three numbers, though, not only the gap. It can give the gap and one amount and ask for the other amount. Then the question is whether to add or to subtract, and the words of the story do not settle it. “Tariq has 5 more marbles than Suki” can call for +5 or for -5, depending on whose marbles we already know. What settles it is a picture of who has more.

This lesson is about drawing that picture and reading the answer off it. Read every story aloud with the child, and let them build it with counters or cubes before any sum is written.

The procedure

Three numbers in every comparing story

Rosa has 8 stickers and Dev has 5. Line the stickers up in two rows, Rosa’s on top, and give each of Dev’s stickers a partner from Rosa’s row.

Rosa’s row now falls into two pieces. The first piece is the stickers that found a partner, and there are exactly as many of them as Dev has, 5. The second piece is the stickers with no partner, 3 of them: that is how many more Rosa has. So Rosa’s row is a whole made of two parts, just like the pictures in parts and the whole:

Every comparing story has these three numbers:

• the bigger amount, the whole of the longer row (Rosa’s 8),
• the smaller amount, which is also the matched part of the longer row (Dev’s 5),
• the gap, the part of the longer row with no partner (the 3).

The sentences “Rosa has 3 more stickers than Dev” and “Dev has 3 fewer stickers than Rosa” describe the same picture. One is read from the top row and one from the bottom row, but the rows and the gap are the same. Saying it both ways is good practice, because a story may use either.

The right half of the picture is the same story drawn as bars. The top bar stands for Rosa’s stickers and the bottom bar for Dev’s, lined up at the left just as the counters were. The piece of the top bar that sticks out past the end of the bottom bar is the gap. Bars are worth drawing once the numbers grow too big to lay out as counters, and they do not need to be drawn to size: the bar for the bigger amount just has to be longer. The bars are drawn here with the matched piece shaded, the extra piece plain, and anything still missing dashed.

Which number is missing

Because the bigger amount is the whole and the other two numbers are its parts, every comparing story is one of three kinds, and each kind has its sum.

The reason is the same reason as in every part and whole picture. Two parts make the whole, so when the whole is missing we put the parts together and add. When a part is missing we take the known part away from the whole, or count on from the known part up to the whole. Here the bigger amount is always the whole, so:

• if the missing number is the bigger amount, add;
• if the missing number is the smaller amount or the gap, subtract (or count on).

Reading who has more

To use that, a child has to know which amount is the bigger one before doing anything else. The comparing sentence tells them. In “Arlo has 3 more apples than Wren”, the name in front of “more” has more, so Arlo’s row is the long one. In “Finn has 4 fewer crayons than Bea”, the name in front of “fewer” has fewer, so Finn’s row is the short one and Bea’s is the long one.

A sentence can always be turned around without changing what it says. “Finn has 4 fewer than Bea” is the same fact as “Bea has 4 more than Finn”. Turning a sentence around until the bigger amount comes first is a good habit when a story feels muddled.

Then look at the amount the story gives. Is it the long row or the short row? That, and only that, decides between adding and subtracting. The word “more” says which row is longer; it does not say which row is missing. So “more” sometimes means add (Example 1) and sometimes means take away (Example 3), and “fewer” can mean add as well (Example 4). The word alone never says whether to add or take away, which is why the picture comes first.

“Less” and “fewer” mean the same thing here. Lengths and heights use their own words, “longer”, “shorter”, “taller”, but the picture is the same, with cubes or bars in place of counters.

Evening up

Some stories ask how to make the two amounts the same: “How many more does Kofi need to have as many as Lena?” That is the gap again. Giving the shorter row the gap fills it up until every counter in the longer row has a partner. Taking the gap away from the longer row works too, because then the unmatched counters are gone and what is left matches the shorter row exactly.

Checking an answer

Put the answer back into the story and make two checks. The one the story says has more must come out bigger. And the bigger take away the smaller must give the gap the story stated. A child who added when they should have taken away fails the first check at once, because their answer makes the wrong child the one with more.

Bigger numbers

Stories with numbers up to 100 use exactly the same picture. Only the working changes. To add, put on the tens and then the ones, as in Adding two-digit numbers without regrouping, making a new ten when the ones run past nine (Regrouping in addition). To subtract, either count up from the smaller number in friendly hops on an open number line or take away the tens and then the ones; use whichever is shorter for the numbers in front of you. Making a ten and Tens and ones do the rest.

Worked examples

Read each story aloud. Counters, cubes or two strips of paper are enough to act out every one.

Boundaries

• Two of the three numbers are needed. “Ivo has 6 more marbles than Nell” gives the gap and nothing else. Ivo could have 6 and Nell none, or Ivo 10 and Nell 4, or many other pairs. Without one of the amounts, the story has no single answer.
• The gap cannot be bigger than the bigger amount. With counted things nobody has fewer than none. “Tariq has 4 marbles, which is 6 more than Suki” cannot happen, since Suki would need 4 - 6 marbles, and there is no such number of marbles. The gap can equal the bigger amount only when the smaller amount is 0.
• Both amounts must count the same kind of thing. 3 bags of apples and 8 apples cannot be compared until we know how many apples are in a bag. Lengths must be measured in the same unit, cubes against cubes or centimeters against centimeters.
• A comparing story is not a change story. If a child gives some marbles to another child, both amounts change and the story is about joining and taking away, as in Problems with the start, change or result missing. In a comparing story nothing moves; the gap is a description, not an event.
• “More” here means added on, not times as many. “3 more than Wren” means Wren’s amount and 3 extra. “Three times as many as Wren” is a different kind of comparison, taken up much later in Times as many.

Where this goes next

Comparing stories are one family among the addition and subtraction stories. The others put amounts together, take them apart, or change them, and the same question decides each one: is the missing number a whole or a part? Stories with two questions in a row, such as “Arlo has 3 more apples than Wren, who has 4. How many apples do they have together?”, need one answer before the next and belong to Two-step addition and subtraction problems.

The gap questions here are what a child asks of a bar graph or pictograph (“how many more children chose dogs than cats?”), and of two lengths in Measuring length. Each story also gives a whole fact family: 4 + 3 = 7, 7 - 4 = 3 and 7 - 3 = 4 all describe Arlo and Wren. The two lined-up bars grow into the tape diagrams of later years, and comparing by “times as many” in place of “more than” becomes Times as many.

Exercises

Find the gap

1. Kofi has 7 pencils. Rosa has 10 pencils. How many more pencils does Rosa have than Kofi?
2. There are 11 ducks and 5 geese on a pond. How many fewer geese are there than ducks?
3. Lena’s tower has 16 blocks. Hugo’s tower has 9 blocks. How many more blocks are in Lena’s tower?
4. Wren has 6 stamps. Arlo has 13 stamps. How many more stamps does Wren need to have as many as Arlo?
5. Marco has 15 stickers. Ellie has 9 stickers. How many stickers would Marco have to put away to have as many as Ellie?

Find the bigger amount

6. Juno has 6 grapes. Ravi has 4 more grapes than Juno. How many grapes does Ravi have?
7. A red ribbon is 9 cubes long. A blue ribbon is 5 cubes longer than the red ribbon. How many cubes long is the blue ribbon?
8. Nell has 8 marbles. She has 7 fewer marbles than Ivo. How many marbles does Ivo have?
9. There are 12 red cups. There are 5 fewer red cups than blue cups. How many blue cups are there?

Find the smaller amount

10. Bea has 10 cherries. Finn has 3 fewer cherries than Bea. How many cherries does Finn have?
11. Tariq picked 14 apples. He picked 6 more apples than Suki. How many apples did Suki pick?
12. There are 16 children on a bus. There are 9 more children than grown-ups. How many grown-ups are on the bus?
13. Marco has 20 crayons. Ellie has 8 fewer crayons than Marco. How many crayons does Ellie have?

Read the bars

Use the bars for Exercises 14 to 16. What number goes where the question mark is?

14. Bars A.
15. Bars B.
16. Bars C.

Bigger numbers

17. The blue team has 34 cans. The green team has 25 more cans than the blue team. How many cans does the green team have?
18. A jar has 63 beads. A box has 28 fewer beads than the jar. How many beads are in the box?
19. Pablo read 47 pages. He read 19 more pages than Ines. How many pages did Ines read?
20. A sunflower is 85 centimeters tall. A tulip is 38 centimeters tall. How much taller is the sunflower?
21. Zara has 26 stickers. She has 18 fewer stickers than Hugo. How many stickers does Hugo have?

Explain

22. Kofi has 12 pens. He has 4 more pens than Rosa. Lena says: “The story says more, so I add. Rosa has 16 pens.” Is Lena right? Draw the rows to show why.
23. Do “Ravi has 5 more cards than Juno” and “Juno has 5 fewer cards than Ravi” say the same thing? Show it with counters.
24. Ivo has 6 more marbles than Nell. Can you tell how many marbles Ivo has? What else would you need to know?
25. Tell two stories that both use the numbers 9 and 4 and the word “more”. In the first the answer should be 13, and in the second it should be 5.