At a glance. One lesson of about 60 minutes. Students have solved multi-step equations in lessons 1 and 2 and seen a system as a crossing point in lessons 3 and 4. Substitution is the first algebraic method. The figure shows why the replaced equation is straight and crosses zero at the solution.
Calculator: no calculator. Nothing here needs one. Every solution is a whole number, so any slip is an algebra slip and not an arithmetic one.
What students will learn
Know
- Substitution replaces one unknown by an expression in the other.
- The replaced expression goes in brackets.
- The solution is checked in both original equations.
Understand
- The two equations are the same facts, restated so that one unknown is gone.
- Which unknown you replace is a choice; the solution is the same.
- A system is solved when you have a pair, and the pair is checked in both.
Do
- Solve a system by substitution, rearranging first when needed.
- Write a system from a context and solve it.
- Check a solution in both equations.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: substitution with brackets and negative terms (questions 1 to 8).
- Criterion B. Investigating patterns: use the figure to find what the gap line's slope and intercept are for a system, and why the crossing with zero is the solution.
- Criterion C. Communicating: lay out the working as replace, solve, back-substitute, check.
- Criterion D. Applying mathematics in real-life contexts: the school fair in the hook and the bake sale in question 5.
The one thing to do with the figure
Ask for the first line before sliding. “If y is 3x − 2, what does 2x + y become?” Students should say 2x + (3x − 2) before seeing it. Then slide x and ask what the gap is doing.
Ask them to find where the dot sits on the zero line, then to say what that means for the original two equations and for the graph from lesson 3.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and figure with predictions first. Include the other three systems. |
| 10 min | Key idea and worked example 1 (the fair). Students write each line in brackets. |
| 10 min | Worked example 2 (rearrange first) and the faded example in pairs. |
| 5 min | Question 1, the check. If many choose B, go back to the warning card. |
| 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; 8x − 12; y = 2x + 3. |
|---|---|
| The fair | A samosa is 5 pesos and a juice is 8 pesos. |
| Worked example 2 | (3, 5). |
| Faded example | (2, 7). |
| 1. Replace with brackets | A. B has no bracket; C mixes the equations; D loses the 3. |
| 2. Substitute and solve for x | 3. x + 2x + 1 = 10; 3x = 9. |
| 3. Find the second unknown | 5. x = 3, y = 2 × 3 − 1. |
| 4. Rearrange first | 2. y = 2x + 3; 7x + 6 = 20. |
| 5. Write the system and solve | 10. 2(s + 5) + 3s = 60; 5s + 10 = 60. |
| 6. A negative inside the replacement | 3. 5x − 12 + 8x = 27; 13x = 39. |
| 7. Substitute for x | 4. 6y + 3 − 5y = 7; y = 4. |
| 8. A solution is a pair | A. B swaps the pair; C has a sign slip in y; D satisfies only the first equation. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Left out brackets or mixed up the two equations. | Say aloud “in place of y I write…” and draw the bracket. |
| 9, 3.67 or 3.33 in question 2 | Stopped early or slipped on the constant. | Write each line and say which operation was done to both sides. |
| 3, 7 or 8 in question 3 | Gave x, a sign slip in y, or the total. | Underline the unknown the question asks for. |
| 14, 3.71 or 2.43 in question 4 | Stopped early, slipped when rearranging, or multiplied part of the replacement. | Rewrite y = … first, with a check by substituting a value. |
| 15, 11 or 50 in question 5 | Gave the drink, multiplied only one term, or stopped early. | Reread the question and underline what is asked. |
| −13, 2.31 or 39 in question 6 | Sign slip with a negative multiplier, partial expansion, or stopped early. | Expand with arrows and signs: −4 × 3 and −4 × (−2x). |
| 6, 9 or 10 in question 7 | Partial expansion, gave x, or a sign slip in x. | Replace the unknown the equation already isolates, and expand fully. |
| B, C or D in question 8 | Swapped the pair, a sign slip, or checked only one equation. | Test each option in both equations, crossing out as soon as one fails. |
Questions worth asking
- Why do the brackets matter? Listen for “the number outside multiplies the whole expression”.
- Could you have replaced x instead of y? Yes. Compare the work: pick the unknown with the easier expression.
- How is substitution related to the graph? The replaced equation is where the lines have the same y; it finds the shared x.
- How do you know the pair is right? It satisfies both original equations.
Differentiation
A way in for anyone stuck
- Give a template: “Replace y: ____ + (____) = ____” with blanks to fill.
- Start with systems where one equation is already y = … with positive terms.
- Let students check each answer on the figure before writing the final line.
Stretch
- Question 7, then: solve it by replacing y instead of x, and compare the work.
- Make up a system whose solution is (3, 1) in which one equation is y = … with a negative gradient.
Students learning in English as an additional language
“Substitute” means “put one thing in place of another”, like a substitute player. “Subject” here means the letter on its own on one side.
| Word | Plain meaning | Example |
|---|---|---|
| substitution | replacing a letter by an expression | replace y by 3x − 2 |
| subject | the letter on its own on one side | in y = 3x + 1 the subject is y |
| rearrange | rewrite so a different letter is the subject | y − 2x = 3 becomes y = 2x + 3 |
| brackets | symbols that show a whole expression is multiplied | 3(4x − 1) |
| back-substitute | put the answer in to find the other unknown | x = 5, so y = 5 + 3 |
| check | put the pair in both original equations | 10 + 8 = 18 |
Sentence starters for explaining aloud or in writing:
- First I make … the subject, which gives …
- Replacing y in the second equation gives …
- I check my answer in both equations: … and …
Two things not to say
“Just plug it in.” It must go in brackets, and in the equation not already used.
“You are done once you have x.” The pair is the solution, and it must be checked in both.
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.
- Replacey = x + 2 and 3x + y = 14. Write the equation in x only.
3x + (x + 2) = 14, so 4x + 2 = 14 and x = 3. - Back-substitutex = 4 and y = 2x − 3. Find y.
y = 5.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA notebook costs 4 pesos more than a pen. 3 notebooks and 2 pens cost 42 pesos. How much is a pen?
6 pesos: 3(p + 4) + 2p = 42, so 5p = 30. - Earlier workSolve (x + 3)/2 = 8.
x = 13.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningWhy does substitution into y = 2x + 1 and y = 2x + 5 end in a false statement?
2x + 1 = 2x + 5 gives 1 = 5, which is false for every x: parallel lines. - Working backwardsWrite a system, one equation of which is y = 3x − 2, whose solution is (2, 4).
Any correct one, for example x + y = 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? Work each line on paper with the replaced expression in a coloured pen.