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More, fewer, the same

Numbers · level 1, lesson 3 of 6

A question to start

Here are two rows of buttons. Which row has more buttons?

Many people say "the top row, because it is longer". But look again. The top row is long because its buttons are spread out. The bottom row is short because its buttons are bunched up. Being longer is not the same as having more. In this lesson you will learn two sure ways to tell which group has more, and you will find the answer to this puzzle. (It is in worked example 1.)

The idea

Three words: more, fewer, the same

When we look at two groups, there are only three things that can be true:

  • the first group has more than the second;
  • the first group has fewer than the second (fewer means "not as many");
  • the two groups have the same number. We also say they are equal.

If Ali has more marbles than Bea, then Bea has fewer marbles than Ali. These say the same thing from the two sides.

Way 1: matching one to one

Here is a way to compare two groups without counting at all. Make pairs: one thing from the first group with one thing from the second group. Keep making pairs until one group runs out.

left over each cup has one saucer

Every cup has a saucer, but two saucers are left over. So there are more saucers than cups. And there are fewer cups than saucers. If nothing had been left over on either side, the groups would be the same.

Why does matching work? Each pair has exactly one cup and one saucer. So while we are making pairs, the cups and saucers are used up at the same speed. If the cups run out first, there must have been fewer cups. You do not even need to know the counting words to do this!

The story goes that long ago, shepherds who could not count to big numbers kept one pebble for each sheep. In the morning, as each sheep went out to the field, a pebble went into a bag. At night, as each sheep came home, a pebble came out. If a pebble was left in the bag, a sheep was missing. That is matching one to one.

Way 2: counting and comparing the numbers

Matching is sure, but it is slow if the groups are big or far apart. You cannot draw lines between the children in your class and the chairs in another room. So the second way is: count each group, then compare the two numbers.

The number that comes later when you count is the bigger number. 7 comes after 5 when we count, so 7 is more than 5. If Ali has 7 marbles and Bea has 5, Ali has more.

Why does this work? When you count to 7, you pass through 5 on the way: "one, two, three, four, five, six, seven". So 7 things are 5 things and some more.

Spread out or bunched up: the number stays the same

Look at these two rows. They have the same number of dots, 6 each. The bottom row has just been pulled apart.

6 still 6

Moving things does not add any or take any away. Spreading them out makes the row longer, but it does not make more. This is important, because our eyes like to trick us. If you are not sure, match or count. Do not trust how long a row looks.

The number line

A number line is a straight line with the numbers written along it, in counting order, with equal steps between them.

01234567891011121314151617181920 3 steps

The number line is like a road. You start at 0 on the left. Each step to the right is one more. Each step to the left is one less.

The number line makes comparing easy: the number further to the right is bigger. 15 is to the right of 12, so 15 is more than 12. And you can see how many more: from 12 to 15 is 3 steps. So 15 is 3 more than 12, and 12 is 3 less than 15.

Why are the steps equal? Because each step means "one more", and one more is always the same amount, whether you are going from 2 to 3 or from 17 to 18. If the steps were different sizes, the line would trick our eyes, just like the spread-out buttons.

One more, one less, and in between

The number line answers lots of questions at a glance:

  • One more than 9 is the number just to the right: 10.
  • One less than 14 is the number just to the left: 13.
  • A number between 6 and 9 is any number on the line after 6 and before 9: 7 or 8.

Worked examples

1. The buttons. Count the top row: 1, 2, 3, 4, 5, 6. Count the bottom row: 1, 2, 3, 4, 5, 6, 7. 7 comes after 6, so the bottom row has more, even though it looks shorter. It has 1 more button. The top row only looked bigger because its buttons are spread out.

2. Matching children and chairs. 8 children play musical chairs with 7 chairs. When the music stops, each child sits on one chair. Is everyone sitting?

Each chair takes one child: that is matching. 7 chairs make 7 pairs. 8 is 1 more than 7, so 1 child is left standing. (That is how the game works.)

3. Using the number line. Which is more, 11 or 18? How many more?

18 is further to the right than 11, so 18 is more. Count the steps from 11 to 18: 12, 13, 14, 15, 16, 17, 18. That is 7 steps, so 18 is 7 more than 11.

4. Same or not? Kim has 4 red cars and 5 blue cars. Lee has 9 cars, all lined up in a long row. Who has more?

Kim's cars all together: see 4, then count on "five, six, seven, eight, nine". She has 9. Lee has 9. They have the same number. The long row does not matter.

Try it

  1. Which is more: 6 or 9? Which is fewer?
  2. What number is one more than 13? One less than 13?
  3. Name all the numbers between 15 and 19.
  4. There are 5 dogs and 7 bones. If each dog gets one bone, how many bones are left over? Are there more dogs or more bones?
  5. On the number line, how many steps is it from 4 to 10? So 10 is how many more than 4?
  6. Sara spreads her 8 shells in a long line. Tim keeps his 10 shells in a little heap. Who has more shells? Why can the line fool you?
  7. I am a number. I am more than 12 and less than 14. What number am I?
  8. Two friends each have some stickers. They match them one to one and nothing is left over. What can you say?
Answers
  1. 9 is more (it comes later when counting, and it is further right on the number line). 6 is fewer.
  2. One more than 13 is 14. One less than 13 is 12.
  3. 16, 17 and 18. (15 and 19 themselves are not "between".)
  4. 5 bones go to the 5 dogs. 7 is 2 more than 5, so 2 bones are left over. There are more bones, and fewer dogs.
  5. 5, 6, 7, 8, 9, 10: 6 steps. So 10 is 6 more than 4.
  6. Tim, because 10 is more than 8. A line can fool you because spreading things out makes them take up more room, but it does not make more of them.
  7. 13. It is the only whole number after 12 and before 14.
  8. They have the same number of stickers. Every sticker found a partner, so neither of them has any extra.

Common mistake

"Longer means more." A long row, a tall tower or a big pile can have fewer things than a short row, a small tower or a little pile. The size of the things and the gaps between them change how big a group looks, but not how many are in it. When in doubt, match them or count them.

Counting the start when finding steps. From 12 to 15 is 3 steps, not 4. Put your finger on 12 and do not count it. Then hop: "13, 14, 15". Count the hops, not the numbers you touch.

Where it leads