Free / Grades 6-8, with signed-number support
Equation Workshop
How can I show that an equation solution really works?
4 flexible sessions / about 145 minutes including practice / algebra, explanation and visual design
Suggested rhythm: two sessions a week for two weeks. Week 1: equality and two-step equations. Week 2: signed solutions and an original puzzle. Pause for integer practice whenever needed.
Reading, watching and paper activities are free. Parents and instructors can save a collection; scheduling it for a student requires Plus or Lifetime. No upload is required.
Before you begin
Add, subtract, multiply and divide whole numbers; begin signed-number work with support. A letter stands for a number. Start with positive examples and return to negative examples after integer practice.
A paper workshop, not a weighing experiment. Use written numbers and drawings only. No weights, tools, purchases or real balance are required. A balance is a limited analogy: negative numbers are not negative physical weights. Read, point or dictate instead of handling small objects.
Materials
- 4 sheets per learner: Project journal
Print four pages or copy the headings into a notebook.
- 2 sheets per learner: First explanation and final puzzle
Use clean scrap paper and retain the first explanation. Dictation is valid.
- 1 per learner: Pencil
Point, dictate or use an accessible writing tool.
- 1 shared: Browser with audio or transcript
Read the transcript together. Separate worksheet printing is optional.
Choose the support that fits
- More support: label left and right, use one operation per line and say each change aloud. Work with positive values first; do not treat a drawing as proof about negative weights.
- More challenge: design an equation with a fractional solution. Explain why multiplying both sides by zero loses information and why dividing by zero is not permitted.
- Access options: use large written expressions, point to a line or dictate a check. No upload, recording, public presentation or purchase is needed.
What good evidence looks like
- Read the equals sign as two expressions with the same value.
- Name the operation on both entire sides and preserve the solution.
- Keep negative signs attached to their numbers.
- Substitute into the original equation and compare both values.
- Retain a first explanation and justify a repair or an improvement to the final puzzle.
These are discussion criteria, not a new automatic score. Existing lesson and worksheet records keep their own subjects. Checking off a planned task does not demonstrate mastery or add a second grade.
Session 1 / about 30 minutes
The equals sign has two sides
Goal: Explain equality and check a proposed value without assuming it is a solution.
Preparation / about 6 minutes of adult support: Have journal 1 and a first-explanation sheet ready. Read the expressions aloud and preserve the first explanation for session 4.
An equation says two expressions have the same value. The equals sign is not an instruction to put an answer only on the right. In a separate example, x + 4 = 9 and 9 = x + 4 describe the same condition. Testing x = 5 gives 9 on both sides; testing x = 2 gives 6 and 9. A guess becomes a checked solution only when both sides match. The equals sign is a referee, not a conveyor belt for answers.
Ask whether 8 + 2 = 5 + 5 is a meaningful equation even though neither side is a single answer. What would you calculate to check it?
- Keep both sides equal / about 5 minutes
- Test a proposed value on both sides / about 15 minutes
- Keep my first equality explanation / about 10 minutes
Fun fact: An equation can be written with its sides swapped without changing which values satisfy it. Nine equals x plus four asks the same question as x plus four equals nine. Source
S3U / Equation Workshop / Journal 1 of 4
Does this value really work?
For every row, test x = 4. Calculate the left and right sides separately, then say whether that proposed value satisfies the equation. Keep a first explanation of the equals sign on a separate sheet.
| Original equation | Left value | Right value | Does x = 4 work? |
|---|---|---|---|
| x + 7 = 11 | |||
| 2x = 8 | |||
| x - 3 = 4 | |||
| 6 = x + 2 |
Explain 8 + 2 = 5 + 5 without calling the equals sign "the answer comes next."
Is testing one value the same as proving that every value works? Explain.
Session 2 / about 35 minutes
Undo the extra, then the groups
Goal: Solve two-step equations by applying and naming the same operation on both sides.
Preparation / about 7 minutes of adult support: Have journal 2 ready. Use one line for each new equation. Keep the original visible so that the final check uses it, not a later mistaken line.
For a separate example, 2x + 3 = 11, subtract 3 from both sides: 2x = 8. Divide both sides by 2: x = 4. Check 2 times 4 plus 3 against 11. Subtraction removes the extra amount; division finds one equal group. You can choose another valid route, but it must act on both entire sides. A term does not teleport across the equals sign and change its outfit by magic. State what you did.
A learner changes 2x + 5 = 17 to 2x = 17. Ask what operation was applied on each side. Can a check of the original expose the problem?
- Undo a Two-Step Equation - Practice 2 / about 10 minutes
- Write a reversible solution chain / about 15 minutes
- Explain a one-sided-operation error / about 10 minutes
Fun fact: Adding the same number to both sides can undo subtracting it. The operation name and the number both matter: removing a negative is not the same as removing a positive. Source
S3U / Equation Workshop / Journal 2 of 4
Every line earns its equals sign
Solve 3x + 5 = 23 and 5x - 4 = 16. Use the table to name matching operations, then check in the originals. Flawed work to repair: 2x + 5 = 17; 2x = 17; x = 8.5.
| Original | First operation and new equation | Second operation and solution |
|---|---|---|
| 3x + 5 = 23 | ||
| 5x - 4 = 16 | ||
| Repair 2x + 5 = 17 |
Show the original left and right values for each of your three proposed solutions.
Which is the first invalid line in the flawed work, and why?
Session 3 / about 40 minutes
A minus sign is not a decoration
Goal: Solve equations with negative values and verify signs through substitution.
Preparation / about 8 minutes of adult support: Have journal 3 ready and review signed arithmetic if needed. Write multiplication with parentheses when substituting a negative number.
In a separate example, -2x + 3 = 11, subtract 3 from both sides to get -2x = 8. Divide both sides by -2, giving x = -4. The check is (-2) times (-4) plus 3 = 11. Dividing by positive 2 would not undo multiplication by negative 2. A drawing of a balance can help with equality, but it is not a literal model of negative weights. If the signs start doing acrobatics, slow down and give each one its own line.
Ask why a negative solution can be valid even when the right side is positive. Compare the coefficient with the unknown rather than declaring that one sign belongs everywhere.
- Undo a Two-Step Equation - Practice 1 / about 10 minutes
- Solve and check signed equations / about 20 minutes
- Name the sign error / about 10 minutes
Fun fact: A valid divisor in an equation can be negative. It cannot be zero: dividing by zero is undefined, so it cannot be used as a solving step. Source
S3U / Equation Workshop / Journal 3 of 4
Keep each sign with its number
Solve each equation. Put a proposed negative value in parentheses when multiplying, and calculate the original sides separately. These are number relationships, not measurements of negative weights.
| Original equation | Solution steps | Original check |
|---|---|---|
| 4x + 9 = -7 | ||
| 2x - 7 = -1 | ||
| -3x + 2 = 17 |
Someone gets x = 4 from -3x = 12 by dividing the right side by 3. Explain the operation that both sides actually need.
Why is a negative answer not automatically a wrong answer? Use one original equation as evidence.
Session 4 / about 40 minutes
Build a puzzle that passes its own check
Goal: Create an equation with a chosen solution and justify the complete solution chain.
Preparation / about 8 minutes of adult support: Bring the first explanation, all journals and one new puzzle sheet. An available adult can discuss the check, or the learner can rehearse privately.
Choose a solution first, then a nonzero whole-number coefficient and an added whole number. For a separate example, choosing x = 4, coefficient 3 and extra -2 gives 3x - 2 = 10. Hide the chosen value on your puzzle front, and keep the operations and original check on the back or in the journal. A test value that works is evidence for that equation, not proof that every number works. Compare your first explanation with the finished one; improve a real gap rather than inventing a mistake.
Ask a partner, or yourself, to challenge one line: what happened to the whole left side and the whole right side? Can the check find an intentionally wrong candidate?
- Design an original equation puzzle / about 15 minutes
- Prove my puzzle solution works / about 15 minutes
- Revise my equality explanation / about 10 minutes
Fun fact: Multiplying both sides by zero can turn a useful equation into 0 = 0 and hide which value was required. That is why undoable steps matter when solving. Source
S3U / Equation Workshop / Journal 4 of 4
My puzzle and its evidence
Choose an integer solution from -5 through 5. Choose a coefficient from 2 through 5 and an integer extra from -9 through 9. Compute the right side using your solution. Make the puzzle on a separate sheet and keep your first explanation.
| Part of my puzzle | My record |
|---|---|
| Chosen solution, coefficient and extra | |
| Original equation and matching operations | |
| Substitution check in the original | |
| A different candidate and why it fails |
Explain one improvement to my first explanation, with evidence. Do not invent an error if the first version was already sound.
Why did this puzzle require a nonzero coefficient? What information would multiplying everything by zero lose?
What changed in your explanation?
Show your evidence, explain one revision, and choose a question to investigate next. You can keep everything on paper.
Optional: record actual offline learning in Learning Records. Keep reported minutes separate from website time. For an assigned pack, use your existing daily tasks; this page does not award additional completion credit.