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MATH · GRADE 7Algebra

Introducing Algebra

Feed the machine 2 and it returns 7. Feed it 5 and it returns 13. What's the rule?

Grade 7
n →?252in7137out?what's the rule…
In this lesson

The case of the missing number

A friend tells you: "I'm thinking of a number. Double it, add three, and I get eleven. What's the number?"

You're a detective now. Give the unknown number a short name, x, and the whole case fits on one line:

2x+3=112x + 3 = 11

The letter x is called a variable. There's exactly one number you can put in for it that makes the line true: x = 4. Finding it is called solving, and it's most of what algebra does.

One thing about the = sign before we go on. Read it as "is the same value as," not "compute the answer." The sign points both ways: 12 = 8 + 4 is fine, and so is 2(x + 3) = 2x + 6. Both sides agree at every x. An equation isn't a calculation; it's a claim that two expressions name the same value.

Guess the rule

Your friend's number trick is really a machine: a number goes in, a rule runs, a number comes out. Below are three machines with their rules hidden. Feed each one numbers of your choosing, then call the rule — written in algebra, where n means "whatever you feed in."

A warning from every detective who's worked this case: one test is never enough. Two different rules can agree on one input and disagree everywhere else. The machine knows this, and it will use it against you.

Try it

Machine 1 of 3 — its rule is hidden

A hidden-rule machine with three rounds. The student feeds the machine whole numbers of their choosing, building an input-output table, then picks the machine's rule from four algebraic expressions. Rule chips that fit every test so far but are not the true rule prompt the machine to name a counterexample input to feed. Each round is sealed by predicting the machine's output for a fresh input before running it.
n =

What's the rule? n stands for whatever you feed in.

The machine wants at least two tests before you guess its rule.

Words into algebra

Most algebra puzzles start as a sentence. Pick a letter for the unknown, then translate.

In words

Three more than a number

Pick a letter

n = the number

As an expression

n + 3

In words

Twice a number, less five

Pick a letter

x = the number

As an expression

2x − 5

In words

The cost of t-shirts at $12 each

Pick a letter

t = number of t-shirts

As an expression

12t

In words

A taxi: $4 flat plus $2 per kilometre, for d km

Pick a letter

d = kilometres

As an expression

4 + 2d

Two habits keep this clean. Name what the letter stands for: "d is kilometres," not just "let d be d." Read the operation off the words: "more than" is +, "less" is −, "of" or "each" is usually ×.

Worksheet

Try these before moving on. The goal is to tell the difference between a variable, an expression, an equation, and a solution.

Practice · Not graded

MA.7.ALG.1

Practice the idea

01 / 05

A taxi costs $4 flat plus $2 per kilometre. If d is kilometres, which expression matches the cost?

Multiple choice: choose the expression for a taxi that costs four dollars flat plus two dollars per kilometre for d kilometres.

What's next

The strand has five lessons. Each one adds a new move you can make.

  1. Simplifying Algebraic Expressions. Rewrite without changing the value.
  2. Equations as Balance. Solve one- and two-step equations with positive numbers.
  3. When the Balance Breaks. What to do when negatives show up.
  4. Variables on Both Sides. Solve harder equations, including the three special cases.
  5. Modelling Word Problems with Equations. Take a real situation, name the letter, write the equation, solve.