Using a letter for the unknown number

When you reach for it

Until now a missing number in an equation has been marked with a box. In Finding the missing number in an equation the box sat in an adding or subtracting fact, such as 8 + \square = 15, and was found by asking what number makes both sides the same. In Equal group problems the box moved into multiplication and division: 6 \times 4 = \square, 8 \times \square = 56, \square \times 5 = 35.

This lesson changes only the mark. In place of the box, write a letter:

The letter does exactly the job the box did. It holds the place of one number that the story has not told us yet. Reach for a letter whenever a story has one number missing and you want to write the whole story down as a single equation before working anything out.

Why change at all, when the box worked? Three reasons, each of which shows up later.

• A letter can be chosen to remind the reader what it counts: c for the number of crayons, b for the number of baseball cards, m for the number of dollars still missing.
• A letter can be named in words. “Here c stands for the number of crayons in one box” is a sentence a reader can check against the story. Two boxes in one equation cannot be told apart.
• Letters are how all later mathematics writes an unknown. The thinking learned here carries straight into equations with more than one step.

The lesson takes three days. Part 1 puts a letter into multiplication and division stories. Part 2 puts a letter into adding and subtracting stories with numbers up to 1{,}000. Part 3 starts from a story alone and writes one equation that models it, with the letter wherever the unknown sits, in all four operations.

The procedure

Every equation in this lesson is a fact with one number hidden. The work has three parts, and each has a reason.

Write one equation that tells the story. Read the story and find its numbers: two are known, one is not. Write the equation the story describes, in the order the story happens, and put a letter where the unknown number sits. Then say in words what the letter stands for.

Find the number from a fact you know. The equation is a true fact with one number covered up. Uncover it with a fact you already know, or with the facts that come with it: every multiplication has two partner divisions and every addition has two partner subtractions (Fact families, Multiplication and division undo each other). For the numbers up to 1{,}000 in adding and subtracting, counting up (Counting up to find a difference) or subtracting (Subtracting three-digit numbers by place value, including across zero) finds it.

Check by putting the number back. Write the number in place of the letter and work out the equation. If both sides agree, the number is the one the story needs. Then answer the story in a sentence, saying what the number counts.

Part 1: a letter in place of the box

Naama has 8 crayon boxes with the same number of crayons in each. There are 64 crayons in all. How many crayons are in one box?

The number of boxes is known (8), the total is known (64), and the number in each box is not. With a box, this was 8 \times \square = 64. With a letter it is

and the sentence that goes with it is: c stands for the number of crayons in one box.

The bar shows why the letter appears once but means every part. All eight parts hold the same number, because the boxes are equal, so they all carry the same letter c. Eight c’s together make 64.

What number is c? The equation asks: eight times what number is 64? Example 1 answers it from a known fact and checks the answer by putting it back.

Two things about the letter are worth saying plainly, because the whole lesson rests on them.

A letter stands for a number. In this story c is the number of crayons in one box. It is not a short way to write the word “crayons”. The equation 8 \times c = 64 multiplies 8 by a number, and only a number can be multiplied. If a reader is asked “what is c?”, the answer is 8, a number, and the story then says what that number counts.

The same letter means the same number every time it appears in one equation. In 8 \times c = 64 the letter appears once, but in an equation such as c + c = 18 it appears twice, and both times it is the same number. Example 7 works through that.

A letter can be any letter. Most writers pick one that matches the story, but n or k works as well as c. One letter is best avoided in multiplication: the letter x looks very like the times sign \times, so 8 \times x is hard to read. Every equation in this lesson has exactly one letter in it.

The letter can sit in any place, just as the box could. Dividing 45 baseball cards into 5 equal piles and asking how many are in each pile gives 45 \div 5 = b, with the letter on the answer of the division. The partner multiplication, 5 \times b = 45, describes the same piles with the letter on a factor. Both are correct models of the story, and both give the same b.

Part 2: letters in adding and subtracting stories

Adding and subtracting stories have three numbers too: two parts and the whole they make. A letter can stand for any one of the three.

Yonit keeps her savings in an old pencil box. She has 237 dollars in it and wants 500 dollars for a bicycle. How many more dollars does she need?

The story is an addition with a part missing. What she has, plus what she still needs, makes 500:

Here m stands for the number of dollars she still needs. Finding m means finding the number that takes 237 up to 500, and counting up in friendly jumps does it.

The partner subtraction, 500 - 237, gives the same number, because adding a part and taking it away undo each other. Example 3 works it through, so m = 263.

The letter could have been in other places in a story with the same three numbers.

A subtraction story works the same way. When something is taken away, the letter can stand for the starting amount, the amount taken, or what is left. Finding it still comes back to one fact, because every subtraction is an addition read the other way: if 443 - 125 = 318, then 318 + 125 = 443.

Part 3: writing one equation for a story

Parts 1 and 2 each started from a story whose equation was easy to see. Part 3 makes that the skill itself: given only the story, write one equation that models it.

An equation models a story when each number in the equation is a number from the story, the operation is what happens in the story, and the letter sits where the story’s unknown number sits. Reading the equation aloud should retell the story.

Two habits make this reliable.

Write the story in the order it happens, letter and all. If the story starts with an unknown amount and then something happens to it, the letter goes first. “Naama shared some crayons equally among 4 friends, and each friend got 7” starts with an unknown number of crayons, which is then divided by 4:

There is no need to turn the story around before writing it. The equation can be written first and worked out second, which is the whole point of having a letter.

Name the letter in words. “Here c stands for the number of crayons Naama shared.” Saying it catches an equation that has the letter in the wrong place: if the sentence does not match what the story asks, the equation does not model the story.

Then find the number with a fact: which number, divided into 4 equal shares, gives 7 in each? Example 5 finds it with the partner multiplication.

Often more than one equation models the same story, and that is fine. For “512 crayons were in the bin; after some were handed out, 384 were left”, both 512 - h = 384 and 384 + h = 512 are true to the story, since what was left and what was handed out make up the 512. Any equation that models the story gives the same number for the letter.

Worked examples

Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, and Examples 5 to 7 with Part 3.

Boundaries

• One letter, one number, inside one equation. Every time a letter appears in an equation it stands for the same number. A different equation for a different story may use the same letter for a different number: the c in Example 1 is 8 and the c in Example 5 is 28.
• A letter stands for a number, not a thing. The letter c means “the number of crayons in one box”, never “crayons”. This is what lets the letter be multiplied, divided, added and subtracted.
• There may be no whole number that works. In 6 \times r = 40 no whole number fits, because 6 \times 6 = 36 and 6 \times 7 = 42. In 300 + m = 200 no whole number fits either: adding to 300 can never give less than 300. A story that leads to such an equation has been read wrong or has a remainder, which belongs to Remainders and what they mean.
• There may be more than one number that works. In 0 \times n = 0 every number fits, because zero groups of anything is zero. Every equation in this lesson’s stories has exactly one number that works, but that is a fact about these stories, not about all equations.
• One step only. Each story here has one action and one unknown number. Stories that need two steps, written with a letter in the same way, are Problems using several operations.
• Within the facts and within 1{,}000. Multiplying and dividing here stay within the facts up to 10 \times 10 (and a one-digit number times a multiple of ten), and adding and subtracting stay within 1{,}000, so the letter can always be found from a known fact or by counting up.

Where this goes next

The next lesson, Problems using several operations, writes two-step stories with a letter for the unknown, using the same habits: one equation in the order of the story, the letter named in words, and a check by putting the number back. Later, What a letter stands for shows that a letter can also stand for any number at all, as in n + 0 = n, not only for one hidden number. Writing and reading numerical expressions from words turns phrases such as “five more than a number” into expressions, Evaluating expressions and formulas works out an expression once a letter’s number is known, and What a solution is, and checking one and One-step equations give the putting-back check its later name, a solution, and a general way to find it.

Exercises

Part 1: a letter in place of the box

1. Shira puts 9 baseball cards on each page of her album and fills 7 pages. Write an equation with a letter for the number of cards, and find it.
2. Yonit has 32 crayons and puts the same number into each of 4 crayon boxes. Write an equation with a letter, say what the letter stands for, and find it.
3. Naama sorts 54 baseball cards into piles of 6. How many piles does she make? Write an equation with the letter first.
4. Find the number each letter stands for. (a) 7 \times k = 56 (b) 81 \div 9 = m (c) r \times 4 = 36 (d) 3 \times 30 = t
5. Write a pencil box story for 6 \times n = 48, say what n stands for, and find n.
6. Put each number back into its equation to decide whether it makes the equation true. (a) 42 \div g = 7, with g = 6 (b) 9 \times w = 72, with w = 7
7. A pencil box holds 30 pencils. Yonit’s class has 6 full pencil boxes. Write an equation with a letter for the number of pencils, and find it.

Part 2: letters in adding and subtracting stories

8. Shira had 146 baseball cards and bought 78 more. Write an equation with a letter for the number she has now, and find it.
9. Naama had some dollars in her pencil box. She put in 125 more dollars and now has 410 dollars. Write an equation with the letter first, and find how many dollars she had at the start.
10. Shira has 700 baseball cards and gives 254 of them to Yonit. Write an equation and find how many cards Shira has left.
11. Yonit’s card album has room for 600 cards, and 428 cards are in it. Write an addition equation with a letter for the number of cards still needed to fill it, and find it.
12. Find the number each letter stands for, and check it by putting it back. (a) 360 + k = 900 (b) w - 200 = 475 (c) 813 - d = 600
13. Write a crayon story for 250 - n = 180, say what n stands for, and find n.

Part 3: writing one equation for a story

For each story in 14 to 17, write one equation that models it, say what the letter stands for, find the number, and check it.

14. Naama shared some crayons equally into 3 crayon boxes, and each box got 9 crayons. How many crayons did she share?
15. Shira had some baseball cards and sold 160 of them. She has 345 left. How many did she have at the start?
16. Yonit has 9 pencil boxes holding 54 pencils, the same number in each box. How many pencils are in each box?
17. Naama has 532 baseball cards. Some are in her album and 248 are in a shoebox. How many are in her album? Write two different equations that model the story.
18. Which equation models this story? “Shira puts 36 crayons into boxes, 4 in each box. How many boxes does she fill?” (a) 36 \times 4 = b (b) b \times 4 = 36 (c) 4 + b = 36. Find b.
19. Find the number each letter stands for. (a) n + n = 16 (b) k + k + k = 27
20. Yonit gives 8 crayons to each of some friends and gives away 48 crayons in all. Write an equation with a letter for the number of friends, and find it.

Explaining and reasoning

21. Explain why c cannot be 6 in one place and 8 in the other in c + c = 14, even though 6 + 8 = 14. What is c?
22. In 5 \times c = 40, for a story about crayon boxes, c stands for the number of crayons in one box. Explain why it is not enough to say that c stands for “crayons”.
23. Explain why 150 + m = 400 and 400 - 150 = m give the same number for m, and find it.
24. Is there a whole number r that makes 6 \times r = 40 true? Use multiplication facts to decide.
25. Write a baseball card story that b \div 7 = 6 models, say what b stands for, and find b.