What a letter is
35 min
Two hosts talk the lesson through. The voices are synthetic; the script was written from this lesson and checked against it, and asserts nothing the lesson does not.
- Say what a letter in algebra stands for, and what it does not stand for
- Evaluate an expression by substituting a number and doing the arithmetic in the right order
- Write an expression for a quantity described in words, and check it by putting a number in
Start with a sentence you can already do something with. Your phone plan costs $12 a month, plus 5 cents for every minute you talk.
You can work out this month's bill. If you talked for 100 minutes, it's $12 plus 100 lots of 5 cents, which is $12 plus $5, so $17. If you talked for 300 minutes it's $12 plus $15, so $27.
Notice what you just did twice. The same procedure, different number in. Algebra is what you get when you write down the procedure once instead of doing it again every time.
The letter is a number you have not fixed yet
Write the number of minutes as m. Then the bill, in dollars, is:
12 + 0.05m
That's the whole plan, written down. 0.05m means 0.05 times m, because in algebra we drop the multiplication sign between a number and a letter. And m is not the minutes; it's how many minutes, which is a number.
Put numbers in and it does what you already did by hand:
Minutes talked (m) |
12 + 0.05m |
Bill |
|---|---|---|
| 100 | 12 + 0.05 × 100 = 12 + 5 | $17 |
| 300 | 12 + 0.05 × 300 = 12 + 15 | $27 |
| 850 | 12 + 0.05 × 850 = 12 + 42.50 | $54.50 |
Three rows, one expression. The word variable means exactly that: the letter varies, and the expression holds anyway.
A different plan: nine dollars a month plus eight cents a minute, so the expression is 9 + 0.08m. Here is the working for 250 minutes, finished except for the last line. 9 + 0.08 × 250 = 9 + 20 = ? Finish it.
Show the answer
Twenty nine dollars. The point of doing the last step yourself rather than reading it is that it is the step you will have to do unaided, and reading someone else's completed working feels like understanding without producing any.
Same plan as before, twelve dollars and five cents a minute, and this month the bill was $44. How many minutes did you talk? You can work this out without any algebra at all, and it is worth doing that way once.
Show the answer
The $12 is there whatever happens, so $44 minus $12 is $32 of call charges. At 5 cents a minute, $32 buys 32 divided by 0.05, which is 640 minutes. You have just solved an equation by undoing it in your head, which is exactly what lesson 3 is about. The reason algebra is worth learning is not that this problem is hard; it is that the same undoing works when the numbers are horrible.
A letter stands for a number, not for a thing
This is the single most important sentence in the lesson, and it costs people years when they get it wrong.
If a bag holds a apples, then a is not apples. a is a number: how many. That distinction seems fussy until it isn't.
Suppose someone writes:
2a + 3b = 5ab
If you read a as "apples" and b as "bananas", this looks almost reasonable: two apples and three bananas make five pieces of fruit, so 5ab looks like "five fruits". It's wrong, and you can prove it's wrong in about ten seconds without knowing any rules at all. Put numbers in.
Say a = 2 and b = 3. Work out each side before you open this. Are they equal?
Show the answer
Left side: 2a + 3b = 2 × 2 + 3 × 3 = 4 + 9 = 13. Right side: 5ab = 5 × 2 × 3 = 30. Not equal, and nowhere near.
13 is not 30, so the two sides are not the same thing, so the statement is false.
Hold on to that move. Testing an algebraic claim by putting a number in is the most useful habit in this course, and we're going to use it in every lesson. It doesn't require you to remember a rule. It requires you to do arithmetic, which you can already do.
Expression, and the thing it isn't
Two words worth separating now, because lesson 2 turns on the difference.
An expression is a recipe for a number: 12 + 0.05m, 2a + 3b, x + 1. Give it values for its letters and it produces a number. On its own it doesn't claim anything, any more than a recipe claims anything.
An equation has an equals sign in it and does claim something: 12 + 0.05m = 44 says those two things are the same size. A claim can be true or false; a recipe can't.
For this lesson we stay with expressions. Lesson 2 is entirely about that equals sign, because almost everybody arrives at algebra with a wrong idea about it.
The conventions, which are just shorthand
Four of these and they cover nearly everything you'll meet. OpenStax's Elementary Algebra sets out the same conventions with more examples, free, if you want a second telling.
3xmeans 3 × x. The multiplication sign is dropped between a number and a letter, and between two letters:abmeans a × b.- A plain
xmeans1x. The 1 is there and invisible. x/4means x ÷ 4, and it's usually written as a fraction.x²means x × x. (Not 2x.x²at x = 3 is 9, and2xat x = 3 is 6.)
And one rule that isn't shorthand: the order of operations. Brackets first, then powers, then multiplication and division left to right, then addition and subtraction left to right.
What is 2 + 3 × 4?
Show the answer
14, not 20, because the multiplication happens first. If you got 20 you worked left to right, which is the natural way to read a sentence and the wrong way to read an expression.
That rule isn't a law of nature; it's an agreement, so that everybody reads the same string of symbols the same way. Without it, 2 + 3 × 4 would be ambiguous and algebra would not work at all.
Evaluating, with the negative that catches people
To evaluate an expression, substitute and compute. Two worked examples, and the second is the one worth slowing down for.
Worked example 1. Evaluate 5x - 2 when x = 3.
- Substitute:
5(3) - 2 - Multiply first:
15 - 2 - Subtract: 13
Worked example 2. Evaluate x² + 2x when x = -1.
- Substitute, and use brackets around the negative number, which is the step people skip:
(-1)² + 2(-1) - Powers first.
(-1)²is(-1) × (-1), which is +1, because a negative times a negative is positive. - Then the multiplication:
2(-1)is-2. - So:
1 + (-2)= -1
The brackets in that second line are not decoration. Without them, -1² reads as "the negative of 1 squared", which is -1, and you get the wrong answer while doing everything else correctly.
Evaluate 3x - x² when x = 4, before reading on. Write out the substitution line before you compute anything.
Show the answer
Substitute first: 3(4) - (4)². Then powers: (4)² is 16. Then multiplication: 3(4) is 12. So it's 12 - 16, which is -4. If you got 4, you read x² as 2x, so you computed 12 minus 8. If you got 64, you subtracted before squaring, so you did (12 - 4)². Writing the substitution line first is what prevents both, which is why it is a step rather than a formality.
Going the other way: words into symbols
Most of the value in algebra is in this direction, and it's harder than evaluating. The lesson on turning sentences into equations is entirely about the hard cases. The building blocks:
| Words | Expression |
|---|---|
| five more than a number | n + 5 |
| twice a number | 2n |
| a number decreased by 3 | n - 3 |
| half a number | n / 2 |
| three less than twice a number | 2n - 3 |
the cost of t tickets at $9 each |
9t |
$20 minus the cost of t tickets at $9 each |
20 - 9t |
Two of those are worth staring at.
"Three less than twice a number" is 2n - 3, not 3 - 2n. The words arrive in the order "three... less than... twice a number", and if you write the symbols in the order the words came, you get it backwards. That is the word-order habit at work, writing the symbols in the order the words arrived. It is the same habit behind the best-documented mistake in all of algebra, which Clement measured in 1982 and which the lesson on turning sentences into equations is entirely about.1
And the check is the same check as before. If the number is 10, "three less than twice a number" should be 17. Test 2n - 3: 2(10) - 3 = 17. ✓. Test 3 - 2n: 3 - 20 = -17. ✗. Ten seconds, no rules.
Work these on paper, and write the substitution line before you compute. If you want more of the same afterwards, Khan Academy's algebra basics has an unlimited supply.
- Evaluate
4x - 3whenx = 5. - Evaluate
2a - bwhena = 2andb = 7. - Evaluate
x² - 5xwhenx = -2. (Brackets around the -2.) - A taxi charges $3.50 to get in and $1.20 a mile. Write an expression for the fare for
dmiles. - Use your expression to find the fare for 7 miles.
- Write "seven more than three times a number" as an expression, then check it with the number 4.
- Someone claims
3x + 2x = 6x. Test it withx = 2and say whether they're right.
Answers to the practice set, once you have done all seven
Show the answer
- -3. 3.
(-2)² - 5(-2)= 4 + 10 = 14. 4.3.50 + 1.20d. 5. 3.50 + 8.40 = $11.90. 6.3n + 7; at n = 4 that gives 19, and "seven more than twelve" is 19. 7. Left side: 6 + 4 = 10. Right side: 12. Not equal, so no. The right answer is5x, and at x = 2 that gives 10.
- -3. 3.
What people get wrong
"The letter is the thing." a is not apples, it's how many apples. Every "fruit salad" error comes from here, and every one of them dies when you put a number in.
"x² and 2x are about the same." They're equal only when x is 0 or 2. At x = 10 they're 100 and 20.
Losing the minus sign when substituting. -3 substituted into x² is (-3)², which is +9. Write the brackets.
Writing symbols in the order the words came. Sometimes right, often wrong, and always worth checking with a number.
Connections
Lesson 2 takes the equals sign apart, which is the other half of what you need before you can solve anything.
Lesson 3 solves equations, and the "undoing" you did in the first predict block, taking off the $12 and dividing by 0.05, is exactly the method, written down properly.
Lesson 4 draws 12 + 0.05m as a line, and the $12 and the 5 cents both turn out to be visible in the picture.
The lesson on turning a sentence into an equation is the words-into-symbols direction taken seriously, including the mistake in the table above.
Sources
[1] John Clement, "Algebra word problem solutions: thought processes underlying a common misconception", Journal for Research in Mathematics Education, 1982, and Clement, Lochhead and Monk (1981). In the student-professor problem, 37% of engineering freshmen wrote the relationship backwards, and 68% of the incorrect responses took the same form. The lesson on turning sentences into equations treats this properly.
Go deeper
- OpenStax, Elementary Algebra 2e, chapter 1, free online under a Creative Commons licence. Section 1.2, "Use the language of algebra", covers this lesson at more length and with many more exercises.
- Khan Academy, Algebra basics. Practice is the one thing a written lesson cannot give you, and their exercise engine is free and better than anything we could build.
Check your understanding
This lesson has a 6-question quiz. Pass it and the questions come back on a schedule in Review, so what you learned stays learned. Your progress is saved in your browser; no account needed.