Solving Equations: Finding the Unknown Number
A 45-minute algebra-readiness lesson that treats an equation as a balance rather than a calculation, and gives students one repeatable move — undo it on both sides — for one-step and two-step equations alike.
Math
Subject
Grades 6-8
Grade Level
45 minutes
Duration
Equations and Variables
Topic
On this page: Materials · Warm-Up · Direct Instruction · Guided Practice · Independent Practice · Assessment · Closure · Related Resources
Learning Objectives
By the end of this lesson, students will be able to:
- Explain that a variable is a letter standing for a number that is not yet known
- Explain that the equals sign means both sides have the same value, not 'compute an answer'
- Solve a one-step equation by applying the inverse operation to both sides
- Solve a two-step equation by undoing addition or subtraction, then multiplication or division
- Check a solution by substituting it back into the original equation
Materials
- One-Step Equations Worksheet — one copy per student
- Writing Algebraic Expressions Worksheet, for the warm-up half of the lesson
- Two-Step Equations Worksheet, for independent practice
- A drawn or physical balance/seesaw, and small objects (counters or blocks) to represent x
- Index cards with one equation each, for guided practice stations
Vocabulary
- Variable — a letter standing for a number that is not yet known
- Equation — a statement that two expressions have the same value
- Solve — to find the value of the variable that makes the equation true
- Inverse operation — the opposite operation — subtraction undoes addition, division undoes multiplication
- Isolate — to get the variable alone on one side of the equation
Preparation
Draw a simple balance/seesaw on the board before class, or bring a real one if available. It needs to stay visible for the entire direct-instruction segment, not just the introduction.
Write the guided-practice equations onto index cards in advance, one per card, mixing one-step and two-step so groups do not settle into a single pattern.
Decide your substitution-check routine in advance — plugging the answer back into the ORIGINAL equation, not a simplified version of it, is the habit worth drilling from day one.
Warm-Up
5 minutes. Put x + 5 = 12 on the board and ask the class what x equals. Most will answer 7 correctly — this is not a hard equation.
Then ask the real question: how do you know, and what did you actually DO to get there? Answers will be vague — "I just knew it" or "I subtracted" without saying from where.
Say plainly where the lesson is going: today is not about equations you can solve in your head, it is about a method that still works once the numbers stop being obvious.
Direct Instruction
13 minutes. Draw the balance and put 5 counters plus an unmarked cup (standing for x) on one side, 12 counters on the other. Say plainly: this drawing IS the equation x + 5 = 12 — both sides weigh the same right now, and they have to keep weighing the same at every step.
Remove 5 counters from the left side. The balance tips — ask why, and let the class say it: you only took weight off one side. Remove 5 from the RIGHT side too, and the balance levels again, with the cup alone on the left and 7 counters on the right.
Name the rule directly: whatever is being done to x, do the opposite, to BOTH sides. In x + 5 = 12, 5 is being ADDED, so SUBTRACT 5 from both sides.
Do a second one-step example with multiplication — 3x = 15 — and ask what is being done to x this time (multiplied by 3) and what the opposite move is (divide both sides by 3). Same rule, different operation.
Now build a two-step equation on the balance: 2x + 3 = 11. Ask the class which operation to undo FIRST. Undo the addition first (subtract 3 from both sides, giving 2x = 8), then the multiplication (divide both sides by 2, giving x = 4). Say plainly: this is not two new rules, it is the same one-step move, done twice.
Finish with the check every solution gets: substitute 4 back into 2x + 3 = 11 and confirm both sides come out equal. A solution that has not been checked is a guess with confidence.
Guided Practice
12 minutes. Groups draw an equation card and solve it on a mini whiteboard, writing out every step rather than jumping to the answer — the step of NAMING which operation to undo is the one being practiced, not just the arithmetic.
For every card, the group must say out loud, before writing anything: what is being done to x, and what is the opposite? A group that cannot answer this has not earned the right to start solving.
Groups then check their own answer by substitution before showing it to you — a wrong answer caught by the group’s own check is worth more than one corrected by the teacher.
Circulate with one question: which side did you change first, and why? On a two-step card, the answer should always be addition or subtraction before multiplication or division.
Groups then start the One-Step Equations worksheet together.
Independent Practice
10 minutes. Students complete the One-Step Equations worksheet, then move to the Two-Step Equations worksheet.
Add one written item: solve 4x – 2 = 10 and explain, in a sentence, why the subtraction gets undone before the multiplication. Writing the reasoning out, not just the steps, is what shows the order was understood rather than memorized as a fixed sequence.
Assessment
3 minutes. Exit ticket, three items. Solve x – 6 = 9. Solve 5x = 35. Then: solve 3x + 4 = 19, and check your answer by substituting it back in.
The third item is the one that discriminates. The first two can be solved by a student who has only memorized a shortcut; the third requires the two-step order AND the check, which is the actual habit the lesson is building.
Closure
2 minutes. Point at the balance drawing one last time and ask the class to state the rule in their own words. Close on the sentence the lesson earns: an equation is a balance, and solving it means doing the same opposite move to both sides until x is standing there alone.
Differentiation and Accommodations
- Extra support: stay with one-step equations and the physical balance for today, and save two-step equations for a second lesson. Being able to name the operation and its opposite, reliably, is a real result worth protecting on its own.
- Extension: the writing algebraic expressions worksheet is the natural companion skill for students who found solving straightforward — translating a word phrase into the equation in the first place.
- Common difficulty: a student who applies the SAME operation to both sides instead of the OPPOSITE one — adding 5 to both sides of x + 5 = 12. Go back to the balance and ask what happens to the level if you add MORE weight to a side that is already too heavy.
- Watch for: a student who solves a two-step equation in the wrong order and still lands on a plausible-looking answer by coincidence. Always require the substitution check — it catches an out-of-order solve that a quick glance at the answer will not.
Extension Activities
Two-step equations with negative coefficients are the natural next difficulty step once the basic order is solid.
Have students write their own real-world situation that an equation could model — "I have some money, I spend $5, I have $12 left" — and then write and solve the equation for it. Generating the equation is a harder and different skill from solving one already given.
The integers lesson plan pairs naturally with this one: equations with a negative solution, or a negative coefficient, need both skills at once.
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