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Pre-algebra

A letter in an equation is not a new kind of number. It is a name for the one you do not know yet, and "n + 8 = 15" asks exactly what a box with 8 added to it would ask. What changes is the method: instead of counting up until the box is full, you undo whatever was done to the letter, and you do it to both sides so the equation stays true.

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Undoing an addition

n + 8 = 15

  1. Something was added to n: eight of it.
  2. The opposite of adding eight is taking eight away.
  3. Do it to both sides: n + 8 - 8 = 15 - 8.
  4. The left side leaves n on its own, and the right side is 7. So n = 7.

Worth knowing. Checking is quick and worth the ten seconds: put the 7 back into the original equation and see whether it is true. 7 + 8 is 15, so it is.

When the letter is on the right of a take-away

20 - n = 7

  1. This one cannot be undone in a single step, because n is being taken FROM something rather than having something taken from it.
  2. Rearrange first: if 20 take n leaves 7, then n and 7 together make 20.
  3. So n + 7 = 20.
  4. Undo the addition: n = 20 - 7, which is 13.

Worth knowing. This is the shape that catches people out. A child who can solve "n - 20 = 7" without thinking will often subtract here as well and answer 27. It is worth meeting on its own, which is why the levels have it as a separate step.

Where it usually goes wrong

  • Doing the same operation as the equation instead of the opposite one — seeing "n + 8" and adding 8.
  • Changing one side and not the other, which quietly turns the equation into a different equation.
  • Treating "20 - n = 7" like "n - 20 = 7". The first is 13 and the second is 27, and they look almost the same on the page.

The Professor asks himself what has been done TO the letter, and then does the opposite. Eight was added, so he takes eight away. From both sides, every time — an equation is a balance, and a balance you only touch on one side stops being one.

Practise it

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Questions teachers and parents ask

Why use a letter instead of a box?

They mean the same thing, and the box comes first for good reason. The letter takes over when equations get long enough that you need to talk about the unknown — and you cannot say "the box" once there are two of them.

Does it matter which letter I use?

Not to the mathematics. x is the habit rather than the rule, and n, a and t all work the same way. What matters is that you use the same letter throughout one equation.

How do I know if my answer is right?

Put it back into the equation you started with. If n = 7 turns "n + 8 = 15" into "15 = 15", it is right. This is the one check in mathematics that costs almost nothing and catches almost everything.

When is this taught?

Using a letter to stand for a number is grade 6 (6.EE.B.6), and solving one-step equations of the form x + p = q and px = q is 6.EE.B.7. Equations that take two steps to solve come a year later and sit in the algebra topic.

Where this shows up

Standards

  • 6.EE · Expressions and equations
  • CCSS.MATH.CONTENT.1.OA.D.8 — Determine the unknown number in an addition or subtraction equation.
  • CCSS.MATH.CONTENT.3.OA.A.4 — Determine the unknown whole number in a multiplication or division equation.
  • CCSS.MATH.CONTENT.6.EE.A.2.C — Evaluate expressions, including those arising from formulas, using the conventional order of operations.
  • CCSS.MATH.CONTENT.6.EE.B.6 — Use variables to represent numbers and write expressions when solving a problem, understanding that a letter can stand for any number in its set.
  • CCSS.MATH.CONTENT.6.EE.B.7 — Solve real-world and mathematical problems by writing and solving equations of the form x plus p equals q and px equals q for non-negative rational numbers.

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