Teaching unit · Grade 6 (ages 11 to 12)

Understanding and solving one-step equations

What it means for a value to be a solution, and solving one-step equations from real-world problems

About three lessons of 45 to 60 minutes

Start here · hook

What number makes x + 3 = 10 actually true?

An equation is a question in disguise: which value of the unknown makes both sides equal? Test x = 5 in x + 3 = 10: 5 + 3 = 8, which is not 10, so 5 is not the solution. Test x = 7: 7 + 3 = 10, which matches, so 7 IS the solution. Solving an equation is really just finding, efficiently, the one value that passes this test.

Rather than guessing and checking every time, you can solve directly by undoing whatever was done to x. If 3 was added to x, subtract 3 from both sides to undo it. This 'undo' idea works the same way for every one-step equation, whatever operation is used.

x + 3 = 10undo the +3 by subtracting 3 from both sides: x = 7
Checking x = 7substitute back in: 7 + 3 = 10, true, so 7 is the solution
5x = 35undo the x5 by dividing both sides by 5: x = 7
Tickets cost $5 each; the total was $355x = 35 models the real problem, and x = 7 tickets

Learning objective

What students will be able to do

Students will understand that solving an equation means finding the value or values of the variable that make the equation true when substituted, and will write and solve one-step equations of the form x + p = q, x - p = q, px = q, or x / p = q, arising from real-world and mathematical problems.

Success criteria

I can check whether a given value is a solution to an equation by substituting it in.
I can solve a one-step addition or subtraction equation by undoing the operation.
I can solve a one-step multiplication or division equation by undoing the operation.
I can write a one-step equation to model a real-world problem.
I can check my solution by substituting it back into the original equation.

Curriculum anchor

Standards this unit teaches

6.EE.B.5Common Core (US)
Solutions of equations
Understand that solving an equation means finding the values that make it true when substituted.
6.EE.B.7Common Core (US)
Solve one step equations
Solve real world problems by writing and solving simple one step equations of the form x plus p equals q.
AC9M6A02Australian Curriculum v9 (ACARA)
Unknowns with multiplication (Year 6)
Find unknown values in equations involving multiplication and division using the properties of numbers and operations.
AC9M7A03Australian Curriculum v9 (ACARA)
Solve linear equations (Year 7)
Solve one-variable linear equations with whole-number solutions and check each answer by substitution. Australia's formal equation-solving descriptor sits at Year 7, so this unit runs about a year ahead of that placement, building on the Year 6 unknowns-with-multiplication descriptor.

Before you start

Prior knowledge

This unit builds on skills students should already have met. Revisit any that are shaky first.

Key vocabulary

Words to teach and display

Equation: a mathematical statement that two expressions are equal, joined by an equals sign
Solution: the value of the variable that makes an equation true
Solve: to find the solution of an equation
Inverse operation: the operation that undoes another, such as subtraction undoing addition, or division undoing multiplication
Substitute: to replace a variable with a specific number to check or evaluate

Teaching sequence

Teach it: concrete, pictorial, abstract

The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.

1. What it means to be a solution

Concrete

Before solving anything, understand what a solution actually is: a value that, once substituted for the variable, makes the equation a true statement. Test several candidate values for x + 3 = 10 by substitution, one at a time, and see that only one of them, x = 7, makes both sides equal.

This test-by-substitution idea is also exactly how you check a solved answer: after solving any equation, substitute your answer back in. If both sides match, the solution is correct; if they do not match, a mistake was made somewhere in solving.

Worked example

Is x = 4 a solution to 3x + 2 = 14?

Substitute x = 4 into the left side: 3 x 4 + 2.
3 x 4 = 12. Then 12 + 2 = 14.
The left side equals 14, which matches the right side.

Answer: Yes, x = 4 is a solution to 3x + 2 = 14, because substituting it makes both sides equal.

Check for understanding, ask

What does it mean for a value to be 'the solution' of an equation?
How do you check whether a given value is a solution?

2. Solving one-step equations by undoing the operation

Pictorial

Rather than testing values one by one, solve directly by undoing whatever operation was done to x, doing the same undo to both sides so the equation stays balanced. Addition is undone by subtraction, subtraction by addition, multiplication by division, and division by multiplication.

For x + 9 = 15, the +9 is undone by subtracting 9 from both sides: x + 9 - 9 = 15 - 9, giving x = 6. For 5x = 35, the x5 is undone by dividing both sides by 5: 5x / 5 = 35 / 5, giving x = 7.

Worked example

Solve x - 8 = 12 and x / 4 = 9.

x - 8 = 12: undo the -8 by adding 8 to both sides: x - 8 + 8 = 12 + 8, so x = 20.
x / 4 = 9: undo the /4 by multiplying both sides by 4: (x / 4) x 4 = 9 x 4, so x = 36.

Answer: x - 8 = 12 gives x = 20. x / 4 = 9 gives x = 36.

Check for understanding, ask

What inverse operation undoes multiplication?
Why must you do the same undo operation to both sides of the equation?

3. Writing and solving equations from real problems

Abstract

Real problems rarely arrive as a ready-made equation, so the first job is translating the story into one. Identify the unknown, name it with a letter, and write an equation that matches the relationship described, then solve and check it in context.

A pack of 5 identical tickets cost $35 in total. Let x be the price of one ticket: 5x = 35. Solve: x = 35 / 5 = 7. Check in context: 5 tickets at $7 each is $35, which matches the story, and $7 is a sensible price for one ticket.

Worked example

A number plus 12 equals 20. Write and solve an equation, then check your answer.

Let x be the unknown number: x + 12 = 20.
Solve by undoing the +12: x = 20 - 12 = 8.
Check: 8 + 12 = 20, which matches.

Answer: x = 8.

Check for understanding, ask

What is the first step in turning a word problem into an equation?
Why does checking a solution in the context of the story matter, not just algebraically?

Watch for

Common misconceptions and how to address them

MisconceptionAny value that appears in the equation, such as x = 3 in x + 3 = 10, must be the solution.

Why it happens: Students confuse a number that appears in the equation's text with the actual unknown value that makes the equation true.

How to address it: Test it by substitution: if x = 3, then 3 + 3 = 6, not 10, so 3 is not the solution. The solution has to be found (or checked) by substitution, not just picked out of the equation's numbers.

MisconceptionTo solve x + 9 = 15, do the same operation (add 9) to both sides, since 'you keep doing the same thing'.

Why it happens: Students remember 'do the same to both sides' but forget it must be the inverse (undoing) operation, not the same operation already in the equation.

How to address it: You undo what was done, using the opposite operation. Since 9 was added to x, subtract 9 from both sides to undo it: x + 9 - 9 = 15 - 9, giving x = 6. Adding 9 again would move further from an isolated x, not closer.

MisconceptionYou only need to solve the equation; checking the answer afterward is an optional extra step.

Why it happens: Once a numeric answer appears, students treat the problem as finished without verifying it.

How to address it: Make substituting the answer back into the original equation a required last step every time, since it catches arithmetic slips immediately and for a real-world problem, also confirms the answer makes sense in context.

Do it together

Guided practice (with answers)

1. Solve x + 12 = 20.
Answer: x = 8, since 20 - 12 = 8.
2. Solve 6x = 54.
Answer: x = 9, since 54 / 6 = 9.
3. Solve x - 15 = 5.
Answer: x = 20, since 5 + 15 = 20.
4. Solve x / 3 = 8.
Answer: x = 24, since 8 x 3 = 24.
5. Is x = 4 a solution to 3x + 2 = 14?
Answer: Yes, since 3 x 4 + 2 = 12 + 2 = 14.

On their own

Independent practice worksheets

Set the matching ChalkBee algebra worksheets for independent work. The answer keys are computed in code, so they are never wrong. Start with checking whether a given value is a solution, then move to solving and writing one-step equations.

Grade 6 algebra worksheet (Set A)solving simple equations for x Grade 6 algebra worksheet (Set B)more equation-solving practice Algebra: Equations & Expressions (Grade 6, Set B)one-step and two-step equations, evaluating expressions and function tables Grade 6 word problems worksheet (Set A)real problems to translate into one-step equations

Reach every student

Differentiation

Support

Use a physical balance-scale model (or a drawn balance) so 'do the same to both sides to keep it balanced' has a concrete image.
Keep an inverse-operations reference card (addition undoes subtraction, multiplication undoes division, and so on) visible during early practice.
Practise the substitution-check step as a separate, required skill before combining it with solving.
Start real-world problems with the equation already written, before requiring students to write the equation themselves.

Extension

Solve one-step equations with negative numbers, such as x + 8 = 3.
Write and solve an original one-step equation to model a real situation of the student's choosing, then swap with a partner.
Preview two-step equations by solving a problem like 3x + 2 = 14 with guided support, discussing which operation to undo first.
Explore why an equation like x + 3 = x + 5 has no solution, extending the meaning of 'a value that makes it true'.

Check it stuck

Assessment: exit ticket

A three-question exit ticket for the last five minutes, sampling checking a solution and solving both directions.

1. Solve x + 7 = 19.
Answer: x = 12, since 19 - 7 = 12.
2. Solve 9x = 63.
Answer: x = 7, since 63 / 9 = 7.
3. Is x = 10 a solution to x - 4 = 5? If not, find the correct solution.
Answer: No, since 10 - 4 = 6, not 5. The correct solution is x = 9, since 9 - 4 = 5.

For the teacher

Teacher notes and timings

Rough timing across three lessons: Lesson 1 what a solution means (section 1), Lesson 2 solving by undoing (section 2), Lesson 3 real-world equations plus the exit ticket (section 3 and assessment).
Language to keep saying: substitute to check, undo with the inverse operation, do the same to both sides, check the answer in the story too. These target the three main misconceptions directly.
This unit assumes fact families (multiplication and division as inverses) and comfort with negative-free whole-number arithmetic; if either is shaky, a quick warm-up on inverse operations pays off before this unit starts.
Curriculum note: ACARA v9 finds unknown values in multiplication and division equations at Year 6 (AC9M6A02), a partial match to this US Grade 6 standard's scope. Australia's fuller formal linear-equation-solving descriptor, including substitution checks, sits at Year 7 (AC9M7A03), so the general one-step-equation skill here runs about a year ahead of that Year 7 placement.
Present mode and print both work: use Present to model the balance-scale undoing live with the class, then print for independent solving and checking practice.

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