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Unknowns in stories

Algebra · level 2, lesson 6 of 6

A question to start

"Sam has some marbles. He wins 7 more. Now he has 12. How many marbles did Sam have at the start?"

The story has the word "more" in it, and many children add: 7 + 12 = 19. But think about it: if Sam started with 19 and won more, he would have more than 19, not 12. Something has gone wrong.

In this lesson you will learn to turn a story into a number sentence with a box, like ? + 7 = 12, so that the story tells you exactly what to do, and the word "more" can never trick you again. (Sam's answer is in the worked examples.)

The idea

A story with a hole in it

A maths story tells you some numbers and hides one. The hidden number is the unknown. In every lesson so far, the unknown has been a mystery number: in a cup, in a box, inside a machine. In a story it is the same. We write it as ?, and we find it.

Write the story in the order it happens

Here is the key idea: write the number sentence in the same order as the story, putting ? where the unknown is. Do not worry yet about how to solve it. First, just write what happens.

Sam's story:

The story says We write
Sam has some marbles ?
He wins 7 more + 7
Now he has 12 = 12

So the story is ? + 7 = 12. This is a missing-number sentence like the ones in lesson 1, and you already know how to solve those.

Three places the unknown can hide

A "getting more" story has three numbers: the start, the change (how many more), and the end. Any one of them can be the unknown.

Story Unknown Sentence Answer
Sam has 5 marbles. He wins 7 more. How many now? the end 5 + 7 = ? 12
Sam has 5 marbles. He wins some more. Now he has 12. How many did he win? the change 5 + ? = 12 7
Sam has some marbles. He wins 7 more. Now he has 12. How many did he have at the start? the start ? + 7 = 12 5

The same three numbers, 5, 7 and 12, belong to all three stories. They are one fact family, and the story only decides which one is hidden. When the end is unknown, you add. When the start or the change is unknown, you are looking for a part, and you take away (or count up).

A bar picture for any story

A bar model shows the story as lengths. The long bar is the whole (here, the marbles Sam has at the end). The bars under it are the parts (the marbles he had at the start, and the ones he won).

whole 12 marbles at the end parts ? at the start 7 won the start and the 7 he won make the 12 he has at the end

The picture shows what to do. The unknown is a part. The whole is 12, the other part is 7. So the unknown is 12 − 7, which is 5.

Why draw a picture? Because the picture does not care about the word "more". It only shows what is the whole and what are the parts. Once you can see that, you know whether to add or take away.

Taking-away stories

Stories about giving away, eating or losing things work the same way.

"Mia had some stickers. She gave 4 to her brother. Now she has 9 left. How many did she have at the start?"

In order: ? − 4 = 9. The unknown is what she started with, which is the whole: the 4 she gave away and the 9 she kept, put back together. 9 + 4 = 13.

whole ? stickers at the start parts 4 given away 9 left the start is the whole: the 4 given away and the 9 left

Check in the story: she had 13, gave away 4, and 13 − 4 = 9 are left ✓.

Here the story says "gave away", but you must add. The bar model shows why: the unknown is the whole.

Comparing stories

Some stories compare two people. "Ana has 4 more books than Ben. Ana has 11 books. How many does Ben have?"

Draw both people's books as bars. Ana's bar is Ben's bar and 4 more.

Ana same as Ben 4 more Ben Ben: ? Ana has 11: the same as Ben, and 4 more

Ana's 11 books are Ben's books and 4 more. So ? + 4 = 11, and Ben has 11 − 4 = 7. Check: 7 and 4 more is 11 ✓. Ana has more, so Ben must have fewer than 11. The word "more" is in the story, but the answer comes from taking away.

Five steps for any story

  1. Read the whole story. Picture it happening.
  2. Find the unknown. What is the question asking for? Mark it ?.
  3. Write the sentence in story order, with ? in the right place.
  4. Solve it: is the ? the whole or a part? Draw a bar model if you are not sure.
  5. Check in the story, and answer in words: "Sam had 5 marbles at the start."

Going the other way: a story for a sentence

You can also start with a sentence and make up a story. For 15 − ? = 9: "There were 15 birds on a fence. Some flew away. 9 were left. How many flew away?" The answer is 6, because 15 − 6 = 9. Making up stories is a great way to understand what a sentence means.

Worked examples

1. Sam's marbles. "Sam has some marbles. He wins 7 more. Now he has 12."

  • Unknown: the start. Sentence: ? + 7 = 12.
  • The ? is a part; 12 is the whole. 12 − 7 = 5.
  • Check: 5 marbles and 7 more is 12 ✓.
  • Answer: Sam had 5 marbles at the start. (Not 19: starting with 19 and winning more could never leave him with 12.)

2. A bus. "There are 23 people on a bus. At the stop, some get on. Now there are 31."

  • Unknown: the change. Sentence: 23 + ? = 31.
  • Count up from 23: 7 more to 30, then 1 more to 31. That is 8.
  • Check: 23 and 8 more is 31 ✓. Answer: 8 people got on.

3. Biscuits. "Leo baked some biscuits. The family ate 14. There are 6 left."

  • Unknown: the start. Sentence: ? − 14 = 6.
  • The ? is the whole: the 14 eaten and the 6 left. 14 + 6 = 20.
  • Check: 20 − 14 = 6 ✓. Answer: Leo baked 20 biscuits.

4. Comparing. "A red ribbon is 9 cm shorter than a blue ribbon. The red ribbon is 16 cm long. How long is the blue ribbon?"

  • The blue ribbon is the longer one. It is the red ribbon and 9 cm more: 16 + 9 = ?.
  • 16 + 9: add 10 to get 26, then take 1 back: 25.
  • Check: 25 cm is 9 cm more than 16 cm ✓. Answer: the blue ribbon is 25 cm long. (Here "shorter" is in the story, but we add, because the unknown is the longer one.)

5. A Mars rover story. "A rover drove some metres on Monday. On Tuesday it drove 40 more metres. Altogether it drove 100 metres on the two days. How far did it drive on Monday?"

  • Sentence: ? + 40 = 100. The ? is a part. 100 − 40 = 60.
  • Check: 60 and 40 is 100 ✓. Answer: 60 metres on Monday. (Real rovers on Mars do drive slowly, often only a few metres to a few hundred metres in a day, because every move is planned carefully from Earth.)

Try it

For each story, write a number sentence with ?, solve it, and check it in the story.

  1. Ella has 8 shells. She finds some more. Now she has 15. How many did she find?
  2. Some children are in the park. 6 more come. Now there are 14. How many were there at the start?
  3. A baker has some loaves. He sells 12. He has 8 left. How many did he have at the start?
  4. There are 30 crayons in a box. Some are taken out. 18 are left. How many were taken out?
  5. Max is 5 years older than Zoe. Max is 13. How old is Zoe?
  6. A green pencil is 4 cm longer than a yellow pencil. The yellow pencil is 11 cm. How long is the green pencil?
  7. Write a story for ? + 9 = 20, and solve it.
  8. Tom says: "The story says 'more', so I always add." Use question 2 to explain to Tom why that does not always work.
Answers
  1. 8 + ? = 15. Count up from 8: 2 to 10, then 5 to 15, so 7. Check: 8 and 7 more is 15 ✓. She found 7.
  2. ? + 6 = 14. The ? is a part: 14 − 6 = 8. Check: 8 and 6 more is 14 ✓. 8 children at the start.
  3. ? − 12 = 8. The ? is the whole: 12 sold and 8 left, 12 + 8 = 20. Check: 20 − 12 = 8 ✓. 20 loaves.
  4. 30 − ? = 18. Count up from 18 to 30: 2 to 20, then 10 to 30, so 12. Check: 30 − 12 = 18 ✓. 12 were taken out.
  5. Zoe's age and 5 more is Max's age: ? + 5 = 13. 13 − 5 = 8. Check: 8 and 5 more is 13 ✓. Zoe is 8. (She is younger, so her age must be less than 13.)
  6. The green pencil is the yellow one and 4 cm more: 11 + 4 = ?, so 15. Check: 15 cm is 4 cm longer than 11 cm ✓. 15 cm.
  7. Many stories work. For example: "Kai had some stamps. He got 9 more for his birthday. Now he has 20. How many did he have before?" 20 − 9 = 11. Check: 11 and 9 more is 20 ✓. Kai had 11 stamps.
  8. In question 2 the story says "6 more come", but adding 6 and 14 gives 20, and 20 children plus 6 more would be 26, not 14. The 14 is the number at the end, the whole. The unknown is a part, so you take away: 14 − 6 = 8. The word "more" tells you what happened in the story, not what to do with the numbers. Writing the sentence in story order (? + 6 = 14) shows what to do.

Common mistake

Following a "key word" instead of the story. "More" does not always mean add, and "left" or "gave away" does not always mean take away. Write the sentence in story order and ask "is the unknown the whole or a part?"

Forgetting to check in the story. Put your answer back into the story and read it aloud. "Sam had 19, won 7 more, and now has 12" sounds wrong straight away.

Where it leads