Math 1 · Lesson 3 · Step One
Step One: Translate Subtraction and Division
Only addition and multiplication follow all the properties. So before you solve anything, you turn every subtraction into adding the opposite, and every division into multiplying by the inverse. Do that first, and most of the mistakes disappear.
Warm-up: rewrite before you compute
Try these four on paper before reading on.
Rewrite it
(a) Write 7 − 3 using addition.
(b) Write 12 ÷ 3 using multiplication.
Then think balance
(c) What is −5 + (−2) if a negative means you owe?
(d) What is x + 2x?
Check your four answers
(a) 7 + (−3) = 4. (b) 12 × 13 = 4. (c) owe 5 and owe 2 is owe 7, so −7. (d) 1x + 2x = 3x.
Every one of those started by rewriting subtraction as addition or division as multiplication, or by seeing an invisible 1. That rewrite is the whole lesson.
Why we only keep + and ×
In Lesson 2 you tested the properties. Here is the punchline: the properties belong to addition and multiplication. You can add in any order, group either way, and distribute. Subtraction and division do not play fair (5 − 2 is not 2 − 5).
So we stop treating them as their own operations. Subtraction and division are just nicknames. Translate them into addition and multiplication, and every property is yours to use for the rest of the problem.
The one move, in two places
Subtraction and division translate the exact same way. Change the operation to its partner, and swap in the inverse from Lesson 1.
Subtraction → addition
a − b = a + (−b)
Change − to +, and use the opposite (the additive inverse). "Subtract 5" becomes "add negative 5."
Division → multiplication
a ÷ b = a × 1b
Change ÷ to ×, and use the reciprocal (the multiplicative inverse). "Divide by 2" becomes "multiply by one-half."
| Nickname | Real form | The partner you swap in |
|---|---|---|
| a − b | a + (−b) | the opposite of b |
| a ÷ b | a × 1b | the reciprocal of b |
Same idea both times. That is why Lesson 1 taught opposites and reciprocals together: they are the two undo partners, and this is where you use them.
The sentence to say out loud
Change the operation to its partner, and replace the number with its inverse. Minus becomes plus with the opposite. Divide becomes times with the reciprocal.
You will be multiplying fractions
Because "divide by 2" becomes "multiply by 12," you need to multiply fractions comfortably. It is the friendly one: multiply the tops, multiply the bottoms, then simplify.
The move
45 × 38 = 1240 = 310.
A whole number rides as itself over 1: 6 = 61. A mixed number turns improper first: 212 = 52.
The check that matters
A number times its reciprocal is 1: 23 × 32 = 66 = 1.
That is the whole point of the reciprocal, and it is why dividing by 23 is the same as multiplying by 32.
Negatives: think owe and have
Once every subtraction is "add the opposite," you are only ever adding. So you do not need a page of sign rules. Think of a balance: positive is money you have, negative is money you owe.
Same signs: add the amounts, keep the sign (owe and owe, or have and have). Different signs: take the difference, and keep the sign of the bigger pile. That is it. No other rules.
The invisible 1s
Because 1 is the do-nothing number for multiplying, math lets it hide. Writing it in is the second half of Step One.
Invisible coefficient
x means 1x. So x + 2x = 1x + 2x = 3x. And 5x − x = 5x + (−1x) = 4x, not 5. The lonely x was really 1x all along.
Invisible exponent
x means x¹. So x · x² = x¹ · x² = x³ (add the exponents, 1 + 2), not x². And −x is really (−1)x¹.
Why write them in early? The 1s never trip you on easy problems. They get you mid-problem, when your brain is busy. So reveal them at Step One, before the heavy lifting.
Step One, every single time
Before you solve, do two things: translate every subtraction to adding the opposite and every division to multiplying by the reciprocal, and write in the invisible 1s. Now the whole expression is addition and multiplication, and the properties do the rest.
Practice: try it, then get help one step at a time
Try each problem yourself first. Use Hint, Show next step, and Reveal answer only when you want them. Say the move out loud: change the operation to its partner, swap in the inverse.
Exit ticket
On paper or an index card. Say the move as you write.
- Rewrite 10 − 6 using addition, and 20 ÷ 5 using multiplication.
- Find −7 + (−3) and −7 + 3 using owe and have.
- Multiply 34 × 43. What number do you get, and why?
- Simplify 4x − x. Where was the invisible 1?
Check your exit ticket
- 10 + (−6) = 4, and 20 × 15 = 4.
- −7 + (−3) = −10 (owe and owe). −7 + 3 = −4 (owe 7, have 3, still owe 4).
- 1. A number times its reciprocal is the multiplicative identity, 1.
- 4x + (−1x) = 3x. The lonely x was 1x.
Printable study guide
One page of the method, worked examples, and a practice set with a full answer key. Print it, or work it on screen.