Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Unit 1, order of operations. What is 6 + 2 × 5?
Answer
16. Multiply first: 2 × 5 = 10, then 6 + 10 = 16. - From Unit 3, lesson 1, growing patterns. A pattern starts at 3 and adds 5 each time. What is the 4th term?
Answer
18. 3, 8, 13, 18: three steps of 5 after the first term. - From Unit 1, powers. What is 5²?
Answer
25. Two factors of 5: 5 × 5.
Words used in this lesson
- variable
- a letter that stands for a number that can change. For example: n in 3n + 2.
- expression
- numbers, letters and operations written together, with no equals sign. For example: 4h + 3.
- substitute
- put a number in the place of a letter. For example: put h = 6 into 4h + 3.
- sum, difference
- the result of adding, the result of subtracting. For example: the sum of 5 and 3 is 8.
- product, quotient
- the result of multiplying, the result of dividing. For example: the product of 5 and 3 is 15.
- term
- one part of an expression, separated by plus or minus. For example: 3n and 2 are the terms of 3n + 2.
A letter for a number that can change
A stall in a market sells oranges in bags. Every bag holds the same number, but you cannot see how many. Call that number n. Three bags hold 3 × n oranges, and we write that as 3n: the multiplication sign is left out when a number meets a letter. A letter that can take different values is a variable, and a sum such as 3n + 2 is an expression. It has no equals sign, because it is a quantity, not a statement.
Predict first: 3 bags and 2 loose oranges, with 4 in each bag. How many oranges? Then change n and see which part of the picture moves.
n = 4: 3 bags hold 4 each, so 3n + 2 = 3 × 4 + 2 = 14.
Show me the steps
- 3 bags with n = 4 in each, plus 2 loose: 3n + 2 = 3 × 4 + 2 = 14.
- Change n to 1: each bag has 1 counter. 3n + 2 = 3 × 1 + 2 = 5.
- Change n to 6: each bag has 6. 3n + 2 = 3 × 6 + 2 = 20. Only the bags changed; the 2 loose counters stayed.
- Make it 1 bag with no loose counters: the expression is just n, and its value is n.
- Make it 5 bags and 6 loose with n = 6: 5n + 6 = 5 × 6 + 6 = 36.
7h means 7 × h. Put a number in the letter's place and work it out with the order of operations: this is substitution. When n = 4, 3n + 2 is 3 × 4 + 2 = 14, because the multiplying is done before the adding.
So 3n is n + n + n, three lots of n. It is not n + 3 (a bag and 3 loose counters) and not 3 and n side by side.
Words and symbols go in both directions. Sum or more than means add. Difference or less than means subtract. Product means multiply and quotient means divide. “5 less than n” is n − 5. It is not 5 − n.
The order of the steps is part of the meaning
“7 added to a number, then the total multiplied by 4” is 4(n + 7). The brackets say the adding happens first. Without them, 4n + 7 means multiply first, then add. These are different expressions.
n = 3: add 7 then multiply by 4 gives 40; multiply by 4 then add 7 gives 19. The gap is 21.
Show me the steps
- n = 3 with ‘add 7, then multiply by 4’: 3 + 7 = 10, then 10 × 4 = 40. With ‘multiply by 4, then add 7’: 3 × 4 = 12, then 12 + 7 = 19.
- n = 0: 4(0 + 7) = 28, but 4 × 0 + 7 = 7. The gap is 21 again.
- n = 10: 4(10 + 7) = 68 and 4 × 10 + 7 = 47. The gap is still 21.
- Second pair, add 3 and multiply by 5, with n = 3: 5(3 + 3) = 30 and 5 × 3 + 3 = 18. The gap is 12.
- Third pair, add 5 and multiply by 2, with n = 4: 2(4 + 5) = 18 and 2 × 4 + 5 = 13. The gap is 5.
A common mistake: “4(n + 7) and 4n + 7 are the same, because 4 and 7 are both in there.” Try n = 3: they give 40 and 19. The brackets tell you which step to do first.
Worked example: from words to a value
Entry to a fort in Jaipur costs r rupees for each person, plus 50 rupees for a guide for the whole group. Write the cost for a group of 6, then find it when r = 30.
- Six people each pay r rupees, so that part is 6 × r = 6r.
- The guide is 50 rupees, once. The cost is 6r + 50 rupees.
- Put in r = 30: 6 × 30 + 50 = 180 + 50 = 230 rupees.
Your go. The first step is given. A group of 9 pays t rupees each, plus 80 rupees for parking the bus. Write the cost, then find it when t = 25.
- Nine people each pay t rupees, so that part is 9t.
- Now add the parking, then put in t = 25 yourself.
Check the answer
The cost is 9t + 80. With t = 25: 9 × 25 + 80 = 225 + 80 = 305 rupees. The multiplying comes before the adding, so 9 × 25 is worked out first.
Check before you go on
1. Which expression means “5 less than a number n”?
Your turn
Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. Work out 4h + 3 when h = 6.
3. A bag holds b mangoes. Aisha has 3 bags and 5 loose mangoes, which is 3b + 5. How many mangoes is that when b = 8?
Level 2 · Building
4. A taxi in Nairobi charges 80 shillings plus 30 shillings for each kilometre, so a trip of k km costs 80 + 30k shillings. How much does a 7 km trip cost, in shillings?
5. Work out 3(n + 4) − 2n when n = 5.
6. Which expression means “7 added to a number, then the total multiplied by 4”?
Level 3 · Stretch
7. Work out 5(n + 2) and 5n + 2 when n = 4. By how much is the first greater than the second?
8. When n = 4, n × n × n and n + n + n are not equal. What is the first minus the second?
Something to think about
Think of a number, double it, add 6, then halve the result. Write this as an expression in n. Try n = 10 and n = 7. What do you notice about the answers, and why?
One way to think about it
(2n + 6) ÷ 2. With n = 10: 26 ÷ 2 = 13. With n = 7: 20 ÷ 2 = 10. Each answer is 3 more than the number you started with. Halving undoes the doubling, and half of 6 is the 3 that is left over.
Exit ticket
9. Dilnoza writes “the sum of a number and 3, all multiplied by 2” as 2n + 3. Which reply is right?