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Missing numbers

Algebra · level 1, lesson 4 of 5

A question to start

Here is a game to play at the kitchen table. You need 5 counters (buttons, coins or dried beans) and a cup.

Count the 5 counters together. Then the other player closes their eyes. You hide some of the counters under the cup and leave the rest out. The other player opens their eyes and sees this:

? 3 counters can be seen; the rest are under the cup

"There were 5 counters. I can see 3. How many are under the cup?"

Most children can answer this in a few seconds. The more interesting question is how they know. In this lesson you will find three different ways, and learn to write the game as a number sentence with a gap in it.

The idea

The hidden number

The counters under the cup are a mystery number, just like the bag on the balance scale. You cannot see them, but they are there, and there is exactly one right answer. The cup cannot hold 2 counters and 4 counters at the same time.

We know two things:

  • all together there are 5 counters (that is the whole), and
  • we can see 3 of them (that is one part).

The counters under the cup are the other part. The two parts together make the whole.

Writing it down

We can write the game as a number sentence. The 3 we can see, plus the ones under the cup, are the same amount as 5:

3 + ? = 5

The ? is the cup. It stands for the number we do not know yet. Remember from the last lesson that = means "is the same amount as". So the sentence says: "3 and the hidden counters is the same amount as 5."

Here is the same game as a picture of parts and a whole. The long bar is the whole. The two short bars under it are the two parts. Together the parts are exactly as long as the whole.

5 counters (the whole) 3 we can see ? under the cup the two parts together make the whole

Way 1: count on

Start at the part you know, 3, and count up to the whole, 5. Hold up a finger for each number you say:

"three... four, five."

You said two more numbers, and two fingers are up. So 2 counters are under the cup.

Why does this work? Counting on from 3 is like adding the hidden counters one at a time. You keep adding until you reach 5. The number of counters you added is the number that was hiding.

Way 2: know the partners

In Numbers you met number partners: two numbers that make another number. The partners of 5 are 0 and 5, 1 and 4, 2 and 3, 3 and 2, 4 and 1, 5 and 0. Which number goes with 3 to make 5? It is 2. If you know your partners well, you do not even need to count.

Way 3: take away

Imagine all 5 counters again. The cup took some of them. If you take away the 3 that you can see, what is left must be the ones under the cup:

5 − 3 = 2

So taking away gives the same answer. Why? The whole is the two parts put together. If you remove one part from the whole, the other part is left. Adding and taking away are opposites: one puts parts together, the other pulls them apart.

Always check: lift the cup

When you have an answer, check it. Put your number in the gap and see if the sentence is true. For 3 + ? = 5 with answer 2: 3 + 2 is 5, and 5 is the same as 5 ✓. Then lift the cup and count. If there are 2 counters, you were right.

The cup can be anywhere

The cup does not have to be second. Here it is first: the cup is on the left, and 2 counters are out, and all together there were 6.

? + = ? + 2 = 6

? + 2 = 6. Adding in either order gives the same total, so this is the same puzzle as 2 + ? = 6. Count on from 2: three, four, five, six. That is 4. Check: 4 + 2 = 6 ✓.

Taking away with a cup

Here is a different game. Start with 7 counters in a row. The other player closes their eyes. You put some of them away under the cup. Now only 4 are left on the table.

7 − ? = 4

"7, take away the hidden ones, leaves 4." How many did the cup take? The cup took the ones that are not on the table. Count up from 4 to 7: five, six, seven. That is 3. Check: 7 − 3 = 4 ✓.

Could the cup be empty?

Yes! If there were 5 counters and all 5 are still out, then the cup is hiding 0 counters. 5 + 0 = 5, which is true. Zero is a perfectly good answer. And if no counters are out at all, the cup is hiding all 5: 0 + 5 = 5.

Two cups, the same

Here is a harder game. There are 7 counters. Two cups each hide the same number of counters, and 1 counter is left out. How many are under each cup?

First take away the 1 that is out: 7 − 1 = 6, so the two cups hide 6 counters between them. The two cups hide the same number, so share 6 fairly between the two cups: 3 and 3. Each cup hides 3. Check: 3 + 3 + 1 = 7 ✓. This is the balance-scale idea again: take the same away from both sides, then share.

Worked examples

1. The starting game. 5 counters, 3 out. 3 + ? = 5. Count on from 3: four, five. That is 2. The cup hides 2. Check: 3 + 2 = 5 ✓.

2. A bigger whole. 9 counters, 6 out. 6 + ? = 9. Count on from 6: seven, eight, nine. Three fingers are up, so the cup hides 3. Check: 6 + 3 = 9 ✓. Or by taking away: 9 − 6 = 3.

3. Using a ten frame. 10 counters, 7 out. 7 + ? = 10. Put the 7 counters in a ten frame (a box of 10 squares in two rows of 5). Three squares are empty. The empty squares are where the hidden counters belong, so the cup hides 3. This is a partner of 10: 7 and 3.

4. Cup first. ? + 5 = 8. The same as 5 + ? = 8. Count on from 5: six, seven, eight. That is 3. Check: 3 + 5 = 8 ✓.

5. Taking away. 8 − ? = 6. 8 counters, some hidden, 6 left. Count up from 6 to 8: seven, eight. The cup took 2. Check: 8 − 2 = 6 ✓.

Try it

  1. 4 counters in all, 1 out. How many under the cup? Write the number sentence.
  2. 2 + ? = 6
  3. ? + 4 = 7
  4. 5 + ? = 10
  5. 6 − ? = 2
  6. 8 + ? = 8
  7. Ella says the answer to 4 + ? = 9 is 13, because 4 and 9 make 13. Check her answer. What is the right one?
  8. 9 counters. Two cups hide the same number each, and 3 counters are out. How many are under each cup?
Answers
  1. 3. The sentence is 1 + ? = 4. Count on from 1: two, three, four. That is 3. Check: 1 + 3 = 4 ✓.
  2. 4. Count on from 2: three, four, five, six. That is 4. Check: 2 + 4 = 6 ✓.
  3. 3. The same as 4 + ? = 7. Count on from 4: five, six, seven. That is 3. Check: 3 + 4 = 7 ✓.
  4. 5. 5 and 5 are partners of 10 (a double). Check: 5 + 5 = 10 ✓.
  5. 4. 6 counters, 2 left, so the cup took the rest. Count up from 2 to 6: three, four, five, six. That is 4. Check: 6 − 4 = 2 ✓.
  6. 0. The whole is 8 and 8 are already out, so nothing is under the cup. Check: 8 + 0 = 8 ✓.
  7. Check: 4 + 13 is 17, not 9. So 13 is wrong. Ella added the two numbers she could see, but 9 is the whole, not a part. Count on from 4 to 9: five, six, seven, eight, nine. That is 5. Check: 4 + 5 = 9 ✓.
  8. 3 each. Take away the 3 that are out: 9 − 3 = 6 under the two cups together. Share 6 fairly between two cups: 3 and 3. Check: 3 + 3 + 3 = 9 ✓.

Common mistake

Adding the two numbers you can see. In 4 + ? = 9, the 9 is the whole, the number of counters all together. It is not a part. Adding 4 and 9 gives a number bigger than the whole, which is impossible: the parts can never be more than the whole. Before you answer, ask "which number is the whole?"

Forgetting to check. A check takes a few seconds: put your answer in the gap and see if both sides are the same. It catches almost every mistake.

Where it leads