Dressing a teddy

Counting and combinations · level 1, lesson 1 of 6

A question to start

Teddy has 2 shirts: an orange one and a blue one. Teddy has 3 pairs of pants: green ones, striped ones and spotty ones. Every morning Teddy puts on one shirt and one pair of pants.

How many different outfits can Teddy wear? Make a guess before you read on.

A lot of people say 5, because 2 and 3 make 5. But that is not right. The true answer is bigger, and by the end of this lesson you will know it for sure, and you will know why you have found every outfit and none twice.

The idea

An outfit is a pair

An outfit here means one shirt and one pair of pants, worn together. So an outfit is a pair: a shirt with some pants.

Adding 2 and 3 counts the clothes in the cupboard. That is 5 pieces of clothing. But we did not ask how many clothes there are. We asked how many outfits there are, and each outfit uses two clothes at once. Counting clothes and counting outfits are different questions, so they have different answers.

Lay them out with cut-out clothes

The best way to start is with your hands. Cut out 2 paper shirts and 3 paper pants (or use real dolls' clothes, or just draw them). Now make outfits on the table.

Here is the trick that keeps you tidy: pick one shirt and keep it still. Put the orange shirt down. Now try each pair of pants under it, one after another: green, then striped, then spotty. That is 3 outfits with the orange shirt.

Then pick up the next shirt, the blue one, and do the same thing: green, striped, spotty. That is 3 more outfits.

Put them in rows, one row for each shirt:

Count them: 3 in the top row and 3 in the bottom row. 3 and 3 make 6. Teddy has 6 outfits.

How do we know none is missing?

This is the most important question in this whole topic. Anyone can find some outfits. The hard part is being sure you have them all.

Look at the picture again. It is a neat box with rows (going across) and columns (going down).

• Every row is one shirt. There are 2 shirts, so there are 2 rows. No shirt is left out.
• Every column is one pair of pants. There are 3 pants, so there are 3 columns. No pants are left out.
• Every row has one outfit in every column, so every shirt meets every pair of pants.

Any outfit you can think of, like "blue shirt with spotty pants", has a home in this box: go to the blue row, then go along to the spotty column. It is there. So nothing is missing.

How do we know none is counted twice?

Each place in the box holds a different outfit. Two outfits in the same row have the same shirt but different pants. Two outfits in the same column have the same pants but different shirts. Two outfits in different rows and different columns differ in both. So no outfit appears twice.

That is a real mathematical reason, and it is called a proof: a reason that shows something must be true, not just that it seems true.

A box of outfits always has rows of the same length

Here is the pattern. If Teddy had 3 shirts, there would be 3 rows of 3: that is 3 and 3 and 3, which makes 9. If Teddy had 2 shirts and 4 pants, there would be 2 rows of 4: 4 and 4 make 8.

So to count outfits: make one row for each shirt, put every pair of pants in each row, and add up the rows. Because every row is the same length, you can count one row and then count how many rows there are. Grown-ups write "2 rows of 3" as 2 × 3, and say "two times three". You will meet that in level 2.

Why not just 5?

Look at the box one more time. To get 5, you would have to count the 2 shirts down the side and the 3 pants along the bottom: that is counting the labels of the box, not the outfits inside it. The outfits live inside the box. There are 6 of them.

Worked examples

1. Hats and scarves. A snowman has 3 hats (red, yellow, purple) and 2 scarves (long, short). How many hat-and-scarf outfits?

Make one row for each hat:

• Red hat: with the long scarf, with the short scarf. 2
• Yellow hat: with the long scarf, with the short scarf. 2
• Purple hat: with the long scarf, with the short scarf. 2

2 and 2 and 2 make 6. Notice: this is 3 rows of 2, while Teddy had 2 rows of 3. Both make 6, just turned round.

2. Only one shirt. Teddy loses the blue shirt. Now Teddy has 1 shirt and 3 pants. How many outfits?

There is only one row: orange with green, orange with striped, orange with spotty. 3 outfits. One row of 3 is just 3.

3. A new pair of pants. Teddy gets a fourth pair of pants, a yellow pair, and finds the blue shirt again. How many outfits now?

We do not need to start again. The yellow pants make a new column: orange with yellow, blue with yellow. That is 2 new outfits. So 6 and 2 more make 8. Check with rows: 2 rows of 4, and 4 and 4 make 8. Both ways agree.

4. Socks too! Teddy has 2 shirts, 3 pants, and now 2 pairs of socks (white and red). An outfit is a shirt, pants and socks. How many?

We already know there are 6 shirt-and-pants outfits. Each one can be worn with white socks or with red socks. So each of the 6 turns into 2 outfits:

• 6 outfits with white socks,
• 6 outfits with red socks.

6 and 6 make 12 outfits. Adding one more kind of clothing made the number of outfits double.

Try it

1. A doll has 2 dresses and 2 hats. How many dress-and-hat outfits? Draw the box.
2. A bear has 3 shirts and 3 pants. How many outfits?
3. A puppet has 4 shirts and only 1 pair of pants. How many outfits?
4. At a party, each child gets one cup (red or blue) and one straw (green, yellow, pink or white). How many different cup-and-straw pairs are there?
5. Teddy has 6 outfits from 2 shirts and 3 pants. Teddy's friend has 5 shirts and 1 pair of pants. Who has more outfits?
6. A robot toy can have one head (round or square) and one body (big or small) and one pair of feet (wheels or legs). How many different robots?
7. Sam says: "I have 3 shirts and 2 pants, so I have 5 outfits." Explain kindly to Sam what went wrong.

Answers

1. 4. Two rows (one for each dress), each with both hats: 2 and 2 make 4.
2. 9. Three rows of 3: 3 and 3 and 3 make 9.
3. 4. There is one column (the only pants) and 4 rows: each shirt with the same pants.
4. 8. Red cup with each of the 4 straws makes 4. Blue cup with each of the 4 straws makes 4 more. 4 and 4 make 8.
5. Teddy, with 6. The friend has 5 rows of 1, which is only 5 outfits. More clothes in the cupboard (6 for the friend, 5 for Teddy) does not always mean more outfits.
6. 8. Round head and square head, each with a big or small body, makes 4 head-and-body robots. Each of those can have wheels or legs, so 4 turns into 4 and 4, which is 8.
7. Sam counted the clothes, not the outfits. Each shirt can go with either pair of pants, so every shirt makes 2 outfits. 3 shirts make 2 and 2 and 2, which is 6 outfits.

Common mistake

Adding the clothes. "2 shirts and 3 pants make 5" counts the clothes, not the outfits. Before you count, ask: what is one thing I am counting? Here one thing is an outfit, a shirt with pants.

Jumping around. If you pick outfits in any order ("blue and green, orange and spotty, blue and striped..."), it is very easy to miss one or say one twice. Keep one shirt still and go through all the pants before you change shirts.

Where it leads

• Next: Ice-cream towers, where we find out what happens when the order of two things matters.
• Level 1 page · Counting topic page
• The box of rows and columns is the same idea as an array of dots, and it is how multiplying works: Numbers.
• Later, in level 2, the box grows into a tree diagram, which can handle 3 or more kinds of clothes at once.