Math 1 · Lesson 1 · Identity and inverse
The Do-Nothing Numbers and Their Undo Partners
Two numbers change nothing. Two kinds of partners undo everything. Learn the math words and the plain English, then catch the hidden 1s and 0s in math you already do.
Warm-up: change nothing, undo everything
Answer these four on paper before reading on.
Change nothing
(a) What can I add to 7 so it is still 7?
(b) What can I multiply 7 by so it is still 7?
Undo everything
(c) What do I add to 7 to get 0?
(d) What do I multiply 7 by to get 1?
Check your four answers
0, 1, -7, and 17. If you wrote -7 for (d), you are not alone. That mix-up is exactly what this lesson fixes.
Two of those numbers did nothing. Two of them undid everything. Today is about those four numbers.
The parallel: adding and multiplying
Say the math word first, then the plain English, every time.
Adding
Additive identity = the do-nothing number for adding. Add it and the number keeps its identity. a + 0 = a
Additive inverse = the opposite, the undo partner for adding. It brings you back to 0. a + (-a) = 0
Multiplying
Multiplicative identity = the do-nothing number for multiplying. Multiply by it and the number stays itself. a x 1 = a
Multiplicative inverse = the reciprocal, the undo partner for multiplying. It brings you back to 1. a x 1a = 1
| Term | Symbols | Plain English |
|---|---|---|
| Identity property | a + 0 = a and a x 1 = a | Adding 0 or multiplying by 1 never changes a number. Ever. |
| Inverse property | a + (-a) = 0 and a x 1a = 1 | Every number has an undo partner that brings it back to the do-nothing number. (Except 0 has no reciprocal.) |
Quick ones: the opposite of 6 is -6, of -2.5 is 2.5, of 12 is -12. The reciprocal of 6 is 16, of 25 is 52, of -4 is -14 (the sign stays, because -4 x (-14) = +1), and of 1 is 1.
Three quick questions
Which number is its own opposite? 0. Which numbers are their own reciprocal? 1 and -1. Why does 0 have no reciprocal? Nothing times 0 makes 1.
The one sentence to say out loud
An inverse is defined by the identity. The opposite of a number is whatever brings it back to zero. The reciprocal is whatever brings it back to one. If you know the do-nothing number for an operation, you already know what the undo partner has to do.
Which one is it?
For each statement, pick the property and the operation.
The hunt for hidden 1s and 0s
Three things you already know how to do. Where is the 1 or the 0 hiding?
Equivalent fractions
34 = 912. Hiding: we multiplied by 33, and 33 is 1. Nothing changed; it just looks different.
Unit conversions
2 feet = 24 inches. Hiding: 2 ft x 12 in1 ft. That fraction equals 1, because 12 inches and 1 foot are the same amount.
One-step equations
x + 5 = 12, so x = 7. Hiding: we added -5 to both sides. The left side became x + 0, and the 0 is why x is now alone.
The line to remember: every time you cancel, cross out, move something to the other side, or find a common denominator, you are secretly adding zero or multiplying by one. From now on, say the secret out loud.
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. While you work, say which do-nothing number you are aiming for and which undo partner gets you there.
Exit ticket
On paper or an index card. Use the math word and the plain English.
- What is the reciprocal of 38? What do you get when you multiply 38 by its reciprocal?
- Rewrite 56 as ?18. Circle the 1 you multiplied by.
- How are opposites and reciprocals alike? (One sentence.)
- Why is 1 the do-nothing number for multiplying, and not 0?
Check your exit ticket
- 83. Multiplying gives 2424 = 1, the multiplicative identity (the do-nothing number for multiplying).
- 56 x 33 = 1518. The circled 1 is 33.
- Both are undo partners that bring a number back to its do-nothing number (0 or 1).
- 7 x 1 = 7 (unchanged), but 7 x 0 = 0 (wiped out). Each operation has its own do-nothing number.
Printables
The worksheet mirrors this page (Parts A to E, with an answer key). The lesson plan has the 25-minute timeline, the vocabulary table, and the misconceptions to listen for.