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"For every"

Ratio, rates and money · level 1, lesson 2 of 5

A question to start

There are 3 bikes in the park. How many wheels are there?

Now here is a stranger one. The robot cars that drive on Mars, called rovers, have 6 wheels each. If 2 rovers were parked side by side, how many wheels would touch the red ground?

You could count every wheel, one by one. That works for 3 bikes. But what if there were 10 bikes? Or 100 chairs, and you wanted to know how many legs? Counting one by one gets slow, and it is easy to lose your place.

In this lesson you will learn a faster way. It uses two small words that are very big in math: for every.

The idea

Things that come in groups

Lots of things come in groups that are always the same size.

  • A bike has 2 wheels. Every bike. Big bike, small bike, red bike, blue bike: 2 wheels.
  • A chair has 4 legs.
  • A hand has 5 fingers.
  • A tricycle has 3 wheels.
  • A Mars rover such as Perseverance has 6 wheels.

We can say this with the words "for every":

There are 2 wheels for every bike.

There are 4 legs for every chair.

"For every" tells you what one group looks like. Once you know one group, you know everything.

Counting by groups

Here are 3 bikes. Each bike is one group of 2 wheels.

246

Instead of counting each wheel, we count the groups, and say the number of wheels after each group:

  • 1 bike: 2 wheels.
  • 2 bikes: 4 wheels.
  • 3 bikes: 6 wheels.

So 3 bikes have 6 wheels. Saying "2, 4, 6" is called counting by twos (some people call it skip counting, because you skip over the numbers in between).

Why it works

Why can we jump by 2 each time? Because every bike adds the same number of wheels. When one more bike rolls into the park, exactly 2 more wheels come with it. Never 1. Never 3. Always 2. So each new group makes the total jump up by 2.

That is the whole secret of "for every". The group never changes. So each new group adds the same jump.

A "for every" table

We can write the counting in a table. One column counts the groups. The other counts the things.

Chairs Legs
1 4
2 8
3 12
4 16
5 20

Look down the "Legs" column. Each row is 4 more than the row above, because each new chair brings 4 more legs. Counting by fours goes 4, 8, 12, 16, 20.

Look across a row too. The row "3 chairs, 12 legs" means 3 groups of 4. You can see the groups if you draw them:

4 legs8 legs12 legs

Going backwards

"For every" works backwards too. Suppose you see 10 wheels in the bike rack, and every bike has 2 wheels. How many bikes?

Count by twos until you reach 10, and keep track of how many jumps you make:

2 (one bike), 4 (two bikes), 6 (three bikes), 8 (four bikes), 10 (five bikes).

You made 5 jumps, so there are 5 bikes. Each jump is one group, so counting the jumps counts the groups.

Two kinds of counting

In "for every" problems there are always two things to count:

  • the groups (bikes, chairs, hands, rovers), and
  • the things in the groups (wheels, legs, fingers).

Before you start, say out loud which is which. "The groups are bikes. Each group has 2 wheels." That one sentence stops most mistakes.

Worked examples

Example 1. Mars rovers. Each Mars rover has 6 wheels. How many wheels do 2 rovers have?

  1. The groups are rovers. Each group has 6 wheels.
  2. Count by sixes: 1 rover, 6 wheels. 2 rovers, 12 wheels.
  3. Check by counting the wheels on each rover: 6 and 6 make 12. Yes.

So 2 rovers have 12 wheels. (Real Mars rovers are far apart, so they never park side by side. But the math is the same!)

Example 2. Fingers. How many fingers (counting thumbs) are on 4 hands?

  1. The groups are hands. Each group has 5 fingers.
  2. Count by fives: 5, 10, 15, 20.
  3. That is 4 jumps for 4 hands, so there are 20 fingers.

Here is a neat check. 4 hands is 2 people. You know one person has 10 fingers, and 10 and 10 make 20. The same answer two ways means you can be sure.

Example 3. Tricycles, backwards. At a playground there are some tricycles. You count 12 wheels. How many tricycles are there?

  1. The groups are tricycles. Each group has 3 wheels.
  2. Count by threes until you reach 12, counting the jumps: 3 (one), 6 (two), 9 (three), 12 (four).
  3. 4 jumps, so there are 4 tricycles.
  4. Check forwards: 4 tricycles with 3 wheels each: 3, 6, 9, 12. Yes.

Example 4. Two kinds together. In a shed there are 2 bikes and 1 tricycle. How many wheels?

  1. The bikes: 2 groups of 2 wheels. Count: 2, 4. So 4 wheels.
  2. The tricycle: 1 group of 3 wheels. So 3 wheels.
  3. Put them together: 4 and 3 make 7 wheels.

When there are different kinds of groups, count each kind by itself first, then put the totals together.

Try it

  1. There are 2 wheels for every bike. How many wheels do 4 bikes have?
  2. There are 4 legs for every chair. How many legs do 2 chairs have?
  3. A spider has 8 legs. How many legs do 2 spiders have?
  4. Count by fives to find how many fingers are on 6 hands (thumbs count as fingers).
  5. A farmer sees 8 legs walking past. They all belong to cows, and every cow has 4 legs. How many cows?
  6. A flower has 5 petals for every flower. You count 15 petals. How many flowers?
  7. A family has 3 people and 1 dog. How many legs are there in the family?
  8. Jo says: "One bike has 2 wheels, so 3 bikes have 2 and 3, which is 5 wheels." What did Jo do wrong?
Answers
  1. 8 wheels. Count by twos, one jump for each bike: 2, 4, 6, 8.
  2. 8 legs. Count by fours: 4, 8. Or: 4 and 4 make 8.
  3. 16 legs. Each spider is a group of 8: 8 and 8 make 16.
  4. 30 fingers. Count by fives, one jump for each hand: 5, 10, 15, 20, 25, 30. Check: 6 hands is 3 people, and 10, 20, 30.
  5. 2 cows. Count by fours until you reach 8: 4 (one cow), 8 (two cows). 2 jumps, so 2 cows.
  6. 3 flowers. Count by fives: 5 (one), 10 (two), 15 (three). 3 jumps, so 3 flowers.
  7. 10 legs. The people: 3 groups of 2 legs, so 2, 4, 6. The dog: 4 legs. Then 6 and 4 make 10.
  8. Jo added the number of bikes to the number of wheels on one bike. But 3 bikes is 3 groups of 2 wheels: 2, 4, 6. So there are 6 wheels, not 5. Jo mixed up the two kinds of counting (groups and things).

Common mistake

Adding the groups and the things. "3 bikes and 2 wheels make 5" mixes two different kinds of counting. The 3 counts bikes. The 2 counts wheels on one bike. To find all the wheels, count the 2 wheels again for every bike: 2, 4, 6.

Counting by ones in the middle. When counting by fours, some children say "4, 5, 6, 7, 8" and lose track. Use your fingers to count the groups (one finger per chair), and your voice to say the jumps (4, 8, 12).

Where it leads