Skip to content

Letters and shapes for unknowns

Algebra · level 3, lesson 1 of 6

A question to start

Here are two number sentences. The square stands for the same number in both of them, and so does the triangle.

\[□ + □ = 14 \qquad □ + △ = 10\]

What numbers are hiding behind the square and the triangle?

Think for a moment before you read on. The first sentence has two squares that add to 14. Since both squares hide the same number, each one must be 7. Now the second sentence says \(7 + △ = 10\), so the triangle is 3.

You just did algebra. You used shapes to stand for numbers you did not know yet, you followed a rule about what the shapes mean, and you found the hidden numbers. This lesson is about that rule, and about the letters that mathematicians use instead of shapes.

The idea

A box that holds a number

In level 2 you met sentences like \(8 + ? = 15\). The question mark is a hole in the sentence, a place where a number belongs. We can draw the hole in other ways:

  • with a box: \(8 + □ = 15\)
  • with a shape: \(8 + △ = 15\)
  • with a letter: \(8 + n = 15\)

All three say exactly the same thing: "8 and some number make 15". The symbol is just a name for the number we are looking for. A number we do not know yet is called an unknown. Finding it is called solving the sentence. Here the unknown is 7, because \(8 + 7 = 15\).

Why letters?

Shapes are friendly, but they are slow to draw, and you soon run out of easy ones. Letters are quick to write, there are lots of them, and you can choose one that reminds you what it stands for: \(c\) for the number of cats, \(t\) for tickets, \(m\) for minutes. Mathematicians have used letters for unknowns for about four hundred years, and \(x\) and \(n\) are the most popular.

A letter in algebra always stands for a number. In \(n \times 4 = 36\), the letter \(n\) is not "n for nine" or "n for nothing". It is a number in disguise, and our job is to find out which one.

The one big rule: same symbol, same number

In one sentence (or in one puzzle made of several sentences that go together), every copy of the same symbol stands for the same number. That is why the two squares in \(□ + □ = 14\) had to be equal: they are the same symbol, so they hide the same number.

Different symbols are allowed to hide different numbers. In \(□ + △ = 10\), the square and the triangle could be 7 and 3, or 4 and 6, or many other pairs, until another clue tells us which. Different symbols are even allowed to hide the same number: in \(□ + △ = 10\), the pair 5 and 5 is fine. The rule only works one way: same symbol means same number; different symbols might be different.

When you start a brand new puzzle, the symbols are fresh. The \(n\) in today's puzzle has nothing to do with the \(n\) in yesterday's puzzle, just like two different children can both be called Sam.

Solving with fact families

You already know fact families: three numbers that belong together, like 9, 14 and 23.

\[9 + 14 = 23 \qquad 14 + 9 = 23 \qquad 23 - 9 = 14 \qquad 23 - 14 = 9\]

A fact family tells you how to find a missing number. In \(□ + 9 = 23\), the two parts are □ and 9, and the whole is 23. To find a missing part, take the other part away from the whole: \(□ = 23 - 9 = 14\).

A bar model shows this as a picture. The long bar is the whole, 23. Underneath, it is cut into two parts: the unknown part and 9.

23 □ 9 the missing part is 23 take away 9

Times and divide have fact families too. The numbers 4, 9 and 36 belong together:

\[4 \times 9 = 36 \qquad 9 \times 4 = 36 \qquad 36 \div 4 = 9 \qquad 36 \div 9 = 4\]

So in \(n \times 4 = 36\), the letter \(n\) is one of two equal-sized factors, and \(n = 36 \div 4 = 9\). In a bar model, 36 is cut into 4 equal parts, and each part is \(n\).

36 nnnn

Why does "take away" find the missing part?

Look at the bar model again. The whole bar is made of two pieces that fit together with no gap. If you know the whole bar and you cover up the 9 piece, what is left showing is exactly the unknown piece. Covering up a piece of the bar is what "take away" means. The same thinking works for times: if 4 equal pieces make 36, then sharing 36 into 4 equal piles gives the size of one piece.

Always check

When you think you have found the unknown, put your number back into the sentence instead of the symbol. If the sentence is true, you are right. For \(n \times 4 = 36\): is \(9 \times 4 = 36\)? Yes. Checking takes a few seconds and catches most slips.

Worked examples

1. A missing part. Solve \(□ + 9 = 23\).

  • The parts are □ and 9; the whole is 23.
  • Missing part = whole take away the other part: \(23 - 9 = 14\).
  • Check: \(14 + 9 = 23\). ✓ So □ is 14.

2. A missing factor. Solve \(n \times 4 = 36\).

  • 4 equal groups of \(n\) make 36, so share 36 into 4: \(36 \div 4 = 9\).
  • Check: \(9 \times 4 = 36\). ✓ So \(n = 9\).

3. A tricky take-away. Solve \(50 - k = 18\).

  • This one fools many people. Here 50 is the whole, and it is split into \(k\) and 18.
  • So \(k = 50 - 18 = 32\).
  • Check: \(50 - 32 = 18\). ✓ So \(k = 32\). (The answer is not \(50 + 18 = 68\): check it, \(50 - 68\) is not 18.)

4. A missing divisor. Solve \(45 \div m = 5\).

  • "45 shared into \(m\) equal groups gives 5 in each group." That means \(m\) groups of 5 make 45.
  • \(m \times 5 = 45\), so \(m = 9\).
  • Check: \(45 \div 9 = 5\). ✓

5. Two shapes, two clues. Solve \(△ + △ + △ = 24\) and \(△ + ○ = 15\).

  • Three equal triangles make 24, so each is \(24 \div 3 = 8\).
  • Then \(8 + ○ = 15\), so ○ is \(15 - 8 = 7\).
  • Check both: \(8 + 8 + 8 = 24\) ✓ and \(8 + 7 = 15\) ✓.

6. The starting puzzle. In \(□ + □ = 14\), the two squares are equal, so each is \(14 \div 2 = 7\). Then \(7 + △ = 10\) gives △ \(= 3\).

Try it

  1. Solve \(□ + 17 = 40\).
  2. Solve \(a \times 6 = 54\).
  3. Solve \(30 - y = 12\).
  4. Solve \(72 \div p = 8\).
  5. Solve \(n + n + n + n = 28\).
  6. In one puzzle, \(□ + △ = 20\) and \(□ - △ = 4\). Find □ and △. (Hint: try pairs that add to 20.)
  7. Ella says: "In \(x + y = 10\), \(x\) and \(y\) can't both be 5, because they are different letters." Is she right?
  8. Make up your own two-shape puzzle like worked example 5, and give it to someone to solve.
Answers
  1. □ \(= 40 - 17 = 23\). Check: \(23 + 17 = 40\). ✓
  2. \(a = 54 \div 6 = 9\). Check: \(9 \times 6 = 54\). ✓
  3. 30 is the whole, split into \(y\) and 12, so \(y = 30 - 12 = 18\). Check: \(30 - 18 = 12\). ✓
  4. \(p\) groups of 8 make 72, so \(p = 72 \div 8 = 9\). Check: \(72 \div 9 = 8\). ✓
  5. Four equal numbers make 28, so \(n = 28 \div 4 = 7\). Check: \(7 + 7 + 7 + 7 = 28\). ✓
  6. Pairs that add to 20: 19 and 1 (difference 18), 15 and 5 (difference 10), 12 and 8 (difference 4). So □ is 12 and △ is 8. Check: \(12 + 8 = 20\) ✓ and \(12 - 8 = 4\) ✓.
  7. No. Different letters are allowed to stand for different numbers, but they don't have to. \(x = 5\) and \(y = 5\) make \(x + y = 10\) true, so it is a correct answer. Only the same letter must mean the same number.
  8. Any puzzle works if you check it. For example: \(○ + ○ = 18\) and \(○ + □ = 12\) gives ○ \(= 9\) and □ \(= 3\).

Common mistake

Adding when you should take away. In \(50 - k = 18\), the numbers 50 and 18 are both in the sentence, and the word "minus" makes some people add them. Ask: which number is the whole? Here it is 50, so the unknown is a part, and a part is always smaller than the whole. A quick check stops the mistake every time.

Giving the same symbol two different values. In \(□ + □ = 14\), the answer "□ is 10 and □ is 4" is not allowed. Two copies of the same symbol in one puzzle must hide the same number.

Where it leads