At a glance. One lesson of about 60 minutes. Students have solved two-step equations in earlier years; this lesson adds brackets, a minus sign in front of a bracket, and the unknown on both sides, and names equations by degree.
Calculator: no calculator. Nothing here needs one. Every solution is a whole number or a short decimal, so the work is in the steps, not the arithmetic.
What students will learn
Know
- The degree of an equation is its highest power of the unknown after expanding: linear, quadratic, cubic, quartic.
- A number outside a bracket multiplies every term inside, including a negative term.
- To solve a linear equation do the same to both sides, and check by substituting.
Understand
- An equation is a balance with one value of the unknown that levels it.
- The same equation can be written in several equivalent forms and still be balanced by the same value.
- Collecting x terms on one side is not a rule to learn: it is taking the same amount from both sides.
Do
- Expand brackets, including with negative terms, and simplify.
- Solve linear equations with brackets and the unknown on both sides, and check the answer.
- Write an equation from a short context and solve it.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: expanding, collecting and solving multi-step linear equations (questions 1 to 8).
- Criterion B. Investigating patterns: change the numbers in one equation of the figure and say how the solution moves. Ask for a general statement.
- Criterion C. Communicating: show each line of working with the operation done to both sides written beside it.
- Criterion D. Applying mathematics in real-life contexts: the tea stall in the hook and the garden purchase in question 5.
The one thing to do with the figure
Ask for a prediction before anyone touches the slider. “Is x = 1, 3 or 5?” Let students test their pick, then ask them to read the expanded form on the panel and say why the sides cross at that value.
Then move to another equation and ask for the value before sliding. Insist on the left side first, then the right, so the comparison is a habit.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the figure, with predictions first. Include the three other equations. |
| 10 min | Key idea; the degree paragraph; students name the degree of three equations you write. |
| 10 min | Worked example; students do the faded example in pairs and compare working. |
| 5 min | Question 1, the check. If many choose A or C, go back to the figure and test x = 1. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 8, on paper or on screen. |
The answers
| Warm-up | x = 5; 3x + 10; 12. |
|---|---|
| Worked example | x = 7. |
| Faded example | x = 3. |
| 1. Expanding a bracket | B, 3x + 15. A multiplies only the x; C multiplies only the 5; D adds 3 and 5. |
| 2. Solve a one-bracket equation | 5. 2x + 8 = 18, 2x = 10. |
| 3. Degree of an equation | B. A expands to x² + 3x (quadratic); C is cubic; D is quadratic. |
| 4. Bracket on one side, x on both | 8. 5x − 10 = 3x + 6; 2x = 16. |
| 5. Write and solve from a context | 20. 3(x + 15) = 5x + 5; 3x + 45 = 5x + 5; 2x = 40. |
| 6. Brackets on both sides | 5. 8x − 12 = 3x + 13; 5x = 25. |
| 7. A bracket with a minus sign in front | 6. 15 − 2x = x − 3; 18 = 3x. |
| 8. Expanding both terms | B. A multiplies only the 2x; C drops the sign; D multiplies only the −3. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1; 7 or 14 in question 2 | Multiplied only the first term in the bracket, or stopped before dividing. | Test with a number: put x = 1 in 2(x + 4) and in 2x + 4 and compare. |
| A, C or D in question 3 | Called an equation linear by its look, not its expanded form. | Expand every bracket first and then name the highest power. |
| 4 or −2 in question 4 | Slipped on the sign of the number in the bracket, or multiplied only the x. | Write 5 × (−2) = −10 beside the line before moving on. |
| 5 in question 5 | Applied the number outside to the first term only. | Draw arrows from the outside number to each term inside the bracket. |
| 25 in question 5 | Moved a number across without changing its sign. | Show the operation on both sides: take 5 from both, and keep the equation balanced. |
| 3.2, 2.6 or 7 in question 6 | Expanded one bracket partly, or moved −5 the wrong way. | Simplify each side fully before moving any term. |
| 2, 3 or 4 in question 7 | Slipped on −2 × (−3), subtracted before expanding, or moved −3 the wrong way. | Order of operations: brackets, then multiplication, then subtraction. Write the sign of every term. |
| A, C or D in question 8 | Expanded one term only, or dropped a negative sign. | Substitute x = 1 in the original and in each option and see which agree. |
Questions worth asking
- Why must we multiply the 3 as well as the x? Listen for “the number outside multiplies everything inside” or a numerical check.
- Does the order you gather terms in change the answer? No. Both routes balance the same equation; compare moving the smaller or larger x term.
- Can an equation have no solution? Yes: 2(x + 3) = 2x + 5 gives 6 = 5. Expect this as a surprise and return to it in lesson 4.
- How do you know your answer is right without looking it up? Substitute into the original equation, both sides.
Differentiation
A way in for anyone stuck
- Give a worked example card with the operation written beside each line.
- Start with brackets on one side and unknowns on one side, then add both-sides cases.
- Let students test each answer on the figure before writing the check line.
Stretch
- Question 7, then: write your own equation with brackets on both sides whose solution is x = 4.
- Why does 2(x + 3) = 2x + 6 have every x as a solution? How is it different from 2(x + 3) = 2x + 5?
Students learning in English as an additional language
“Expand” here means “multiply out”, not “make bigger”. Say “do the same to both sides” each time you move a term.
| Word | Plain meaning | Example |
|---|---|---|
| equation | a statement that two expressions are equal | 3x + 4 = 19 |
| expand | multiply out a bracket | 3(x + 5) expands to 3x + 15 |
| linear | the unknown is only to the power 1 | 5x − 2 = 3x + 6 |
| degree | the highest power of the unknown | x² + 3x = 10 has degree 2 |
| solution | the value that makes both sides equal | x = 3 solves 2x + 1 = 7 |
| inverse operation | the operation that undoes another | subtract 3 undoes add 3 |
Sentence starters for explaining aloud or in writing:
- First I expand the bracket, which gives …
- Now I take … from both sides, so …
- I check my answer by putting x = … into the original equation: the left side is … and the right side is …
Two things not to say
“Move it across and change the sign.” Say what is really done: the same amount is taken from both sides.
“The bracket just goes away.” It multiplies; nothing disappears.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- ExpandWrite 4(x − 6) without a bracket.
4x − 24. - One bracketSolve 3(x + 2) = 21.
x = 5.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- Context3 boxes each hold x tea bags and 10 samples, so the total is 3(x + 10). That equals 4 boxes of x tea bags with 10 taken out: 4x − 10. Find x.
x = 40, because 3x + 30 = 4x − 10. - Earlier workSimplify 7x − 3 + 2x + 8.
9x + 5.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningWithout solving, say whether 3(x + 2) = 3x + 6 has a solution, and why.
Every x: both sides expand to 3x + 6. - Working backwardsWrite an equation with a bracket on each side whose solution is x = 4.
Any correct one, for example 2(x + 1) = x + 6.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both sliders work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under the figure, “Show me the steps” gives the same states in words.
No device? Write the two sides on separate cards and try values on paper.