At a glance. One lesson of about 60 minutes. Students meet a letter as a number that can vary, write expressions from situations and words, substitute values using the order of operations, and see why brackets change the meaning.
Calculator: no calculator. Nothing here needs one. The numbers are small, and the point is the structure of the expression, not the arithmetic.
What students will learn
Know
- A letter can stand for a number that varies.
- 7h means 7 × h; a number next to a letter is a multiplication.
- The words sum, difference, product and quotient, and more than and less than.
Understand
- The same expression gives different values for different letters, and the order of operations decides the value.
- Brackets are part of the meaning, not decoration.
- An expression is a quantity; it is not an equation and has no answer to find.
Do
- Substitute a value into an expression with one or two operations.
- Write an expression from words or a situation.
- Compare two expressions that use the same numbers in a different order.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: substituting and reading expressions with the order of operations (questions 2 to 5, 7 and 8).
- Criterion B. Investigating patterns: the second figure. Ask students to find what is the same about the gap for every n, and then to predict the gap for a new pair of numbers before they test it.
- Criterion C. Communicating: translating between words and symbols and justifying the brackets (questions 6 and 9).
- Criterion D. Applying mathematics in real-life contexts: fares, entry fees and goods in bags, where one part is paid once and another part for each (questions 3 and 4).
The one thing to do with the figure
Change only n and ask what moves. The bags change, the loose counters do not. Ask students to say it aloud: “3n is three bags, so when n changes all three bags change together.”
Then use the second figure to show that the gap is the same for every n. Ask why before telling them: 4(n + 7) is 4n + 28, which is 21 more than 4n + 7.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the first figure, with predictions first. |
| 10 min | Key idea and vocabulary; the second figure on brackets. |
| 10 min | Worked example on fares; students finish the faded one in pairs. |
| 5 min | Question 1, the check. If many choose A, go back to the words list. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 16; 18; 25. |
|---|---|
| Worked example | 6r + 50, which is 230 rupees when r = 30. |
| Faded example | 9t + 80, which is 305 rupees when t = 25. |
| 1. Words to expression | B, n − 5. A reverses the order, which is the usual slip with “less than”; C multiplies; D adds. |
| 2. Substituting | 27. 4 × 6 + 3 = 24 + 3. |
| 3. Expression for a situation | 29. 3 × 8 + 5 = 24 + 5. |
| 4. Substituting in a context | 290. 80 + 30 × 7 = 80 + 210. |
| 5. Substituting with brackets | 17. 3 × 9 − 10 = 27 − 10. |
| 6. Words to expression with brackets | B. A multiplies first; C has no brackets, so the multiply goes first; D swaps which number goes with the letter. |
| 7. Comparing two expressions | 8. 30 − 22. In general the gap is 2 × (5 − 1) = 8 for every n. |
| 8. n times n times n | 52. 64 − 12. n × n × n is written n³, n + n + n is 3n, and n + 3 is different again. |
| 9. Brackets and meaning | B. A and D accept the unbracketed form; C puts the multiply on the wrong number. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A in question 1 | Reversed the order with “less than”. | Pick n = 12 and ask what 5 less than 12 is. Then write what you did. |
| 36 or 49 in question 2 | Applied the multiplication to the 3 as well, or read 4h as the digits 4 and 6 side by side. | Draw 4 bags of 6 and then put 3 loose. Write 4 × 6 with the multiplication sign. |
| 39 or 16 in question 3 | Multiplied the loose mangoes, or just added everything. | Redraw: 3 bags and 5 loose. Which part is multiplied? |
| 770 or 117 in question 4 | Multiplied the fixed charge, or added all three numbers. | Ask which charge is paid once and which is paid for every kilometre. |
| 9 or 125 in question 5 | Ignored the bracket, or subtracted before multiplying. | Mark the bracket, then the multiplication, then the subtraction. Work through in that order. |
| A or C in question 6 | Did not see that the brackets control which step goes first. | Substitute n = 2 in each option and compare with the words. |
| 22 or 30 in question 7 | Gave the value of one expression, not the difference between the two. | Write both values on one line before the subtraction. |
| 12 or 64 in question 8 | Gave one of the two values, not their difference. | Write both values on one line, then subtract. |
Questions worth asking
- Why do we leave out the multiplication sign in 3n? So that it is not confused with the letter x, and because it is shorter. Listen for the idea that 3n means three lots of n.
- Is n always the same number in a question? Within one question, yes. Between questions it can be anything. A variable varies from case to case.
- What is the difference between 4n + 7 and 4(n + 7)? The bracket makes the adding happen first. Ask for n = 3: 19 and 40.
- Is 3n + 2 an equation? No: there is no equals sign. It has a value for each n, but nothing to solve yet. Equations come in lesson 6.
Differentiation
A way in for anyone stuck
- Use bags and counters, or paper cups and beans, before the figure. Let n be a number the student does not know until the cup is opened.
- Write a mini table of n = 1, 2, 3 and the value of the expression, so the pattern can be seen.
- Give them the words list on a card (sum, difference, product, quotient, more than, less than).
Stretch
- Show that 4(n + 7) is always 21 more than 4n + 7. What is the gap for 6(n + 3) and 6n + 3? (15, since 6 × 3 − 3.)
- Write two different expressions for “add 3 to a number and double the result”, and show with a table that they agree. (2(n + 3) and 2n + 6.)
Students learning in English as an additional language
Word and symbol pairs are the main load. Give each phrase beside its symbol. “Less than” and “subtracted from” reverse the order and need extra care.
| Word | Plain meaning | Example |
|---|---|---|
| variable | a letter that stands for a number that can change | n in 3n + 2 |
| expression | numbers, letters and operations written together, with no equals sign | 4h + 3 |
| substitute | put a number in the place of a letter | put h = 6 into 4h + 3 |
| sum, difference | the result of adding, the result of subtracting | the sum of 5 and 3 is 8 |
| product, quotient | the result of multiplying, the result of dividing | the product of 5 and 3 is 15 |
| term | one part of an expression, separated by plus or minus | 3n and 2 are the terms of 3n + 2 |
Sentence starters for explaining aloud or in writing:
- The letter n stands for …, so … means …
- When n is …, the expression is … × … + … = …
- The brackets mean … is done first, so …
Two things not to say
“n is always the unknown.” In an expression n varies; nothing is being solved. It is only in an equation that a letter can be a particular unknown.
“Letters are just a trick to hide numbers.” They give one description that covers every value.
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.
- SubstitutionWork out 5a − 2 when a = 7.
33. 5 × 7 = 35, then 35 − 2. - WordsWrite “a number n divided by 4” as an expression, and find its value when n = 20.
n ÷ 4, and 5.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA water tank holds 300 litres. It loses 12 litres each minute, so after m minutes it holds 300 − 12m litres. How much is left after 9 minutes?
192 litres. 12 × 9 = 108, then 300 − 108. - Earlier work, order of operationsWork out 36 ÷ (2 + 4) × 3.
18. The bracket is 6, then 36 ÷ 6 = 6, then 6 × 3.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningFind a value of n that makes n + 6 and 3n + 2 equal. Test it.
n = 2: both are 8. (Try n = 1, 2, 3 and watch the values.) - Earlier work, powersWhat does n² mean when n = 6? Is that the same as 2n?
36, and 2n is 12, so no. They agree only for n = 2 (4 and 4) or n = 0.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Use paper cups with a number hidden in each, and loose counters beside them.