GCSE Maths: Letters, expressions and formulae (Foundation)
Part of the GCSE Maths substitution and formulae chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Letters, expressions and formulae
Algebra uses letters to stand for numbers. A letter whose value can change is called a variable. If a bag holds n sweets, then n might be 20 today and 25 tomorrow. Writing a rule once with a letter saves us writing it again for every number.
Four words are used all the time in algebra. Exam questions expect you to know them without being told.
| Word | Meaning | Example |
|---|---|---|
| Expression | Numbers and letters joined by +, −, ×, ÷. There is no equals sign. | 3n + 5 |
| Equation | Two expressions joined by an equals sign. It is true for only some values of the letter. | 3n + 5 = 20 (true when n = 5) |
| Formula | A rule linking real quantities, written with an equals sign. You use it again and again with different numbers. | A = lw (area = length × width) |
| Identity | Two expressions that are equal for every value of the letter. We write ≡. | 2(n + 3) ≡ 2n + 6 |
A term is one part of an expression, separated from the next by a + or − sign. In 3n + 5 the terms are 3n and 5.
Writing algebra in short form
To keep algebra tidy we do not write the multiplication sign. The rules are:
Algebra shorthand
- 3 × n is written 3n: the number goes first and there is no × sign.
- n × n is written n² (say “n squared”), and n × n × n is n³.
- a × b is written ab, with the letters in alphabetical order.
- n ÷ 4 is written as a fraction: n4.
- 2 × (n + 1) is written 2(n + 1).
Turning words into algebra
Imagine Sam has n sweets and Tia has 5 more than Sam. Draw it as a bar. Tia has Sam’s amount n and then 5 extra, so Tia has n + 5 sweets. Drawing a bar like this is a good way to see what the words mean before you write any algebra.
Some words always point to the same operation. Use this table to translate, but always read the whole sentence.
| Words in the question | Operation | Example in algebra |
|---|---|---|
| more than, plus, sum, total, increased by | add (+) | 7 more than n is n + 7 |
| less than, minus, fewer, decreased by, take away | subtract (−) | 7 less than n is n − 7 |
| times, multiplied by, product, twice, double, triple | multiply (×) | double n is 2n |
| divided by, shared equally between, half of | divide (÷) | half of n is n2 |
Finally, an expression can be pictured as a function machine: you put a number in, the machine does the operations in order, and a number comes out. The expression 3n + 2 is the machine “multiply by 3, then add 2”.
Watch out
“Less than” swaps the order. “5 less than n” means start from n and take 5 away: n − 5. It is not 5 − n. Test it: if n = 8, then 5 less than 8 is 3, and 8 − 5 = 3. Also remember 2n (double n) is not the same as n² (n × n): with n = 5 they are 10 and 25.Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “7 less than” “Less than” means take away. The 7 is taken away from the number.− 7
- “n” The letter n stands for the number we start with. It comes first because we start from it.n
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.n − 7
Check. Test with a number. Let n = 10. In words: 7 less than 10 = 3. The expression gives 10 − 7 = 3. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “A pen costs p pence” We let the letter p stand for the price of one pen, in pence.p
- “4 pens” 4 pens cost 4 lots of p, which is 4 × p. In algebra shorthand that is 4p.4p
- “and a notebook costing 90 pence” “And” tells us to add the notebook. The notebook is a fixed 90 pence.+ 90
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.4p + 90
Check. Test with a number. Let p = 30. In words: 4 pens at 30p each is 120p, plus 90p = 210. The expression gives 4 × 30 + 90 = 120 + 90 = 210. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Method. For each one, look for an equals sign, then decide which word fits.
- (a) 5n − 2 There is no equals sign.It is an expression. It is a phrase made of the terms 5n and −2.
- (b) 5n − 2 = 18 There is an equals sign, but it is only true for one value of n. Try n = 1: 5 × 1 − 2 = 3, not 18. Try n = 4: 5 × 4 − 2 = 18, which works.It is an equation. Solving it gives n = 4.
- (c) 2(n + 3) ≡ 2n + 6 It uses ≡. Expanding the bracket: 2 × n = 2n and 2 × 3 = 6, so the left side is exactly 2n + 6 for every n.It is an identity.
Check. For (c) try n = 10: left = 2(10 + 3) = 26 and right = 2 × 10 + 6 = 26. They agree. ✓
Watch out. An equals sign alone does not make something an equation. Check whether it is true for all values (identity) or only some (equation).
Exercise 8A Words, expressions and formulae
Non-calculatorKey words. Expression: no equals sign. Equation: equals sign, true for some values. Formula: a rule linking quantities. Identity: true for all values (≡).
Method. Look for the equals sign, then decide which of the four words fits.
- Look for an equals sign (or ≡). An expression never has one. Equations, formulae and identities always do.4n − 7
There is no equals sign. - Decide. An expression is a mixture of numbers and letters with no equals sign. It is a "phrase", not a full sentence.expression
Watch out. A formula and an equation both have an equals sign. The difference is the job: a formula is a rule for working things out; an equation is a puzzle to solve.
Key words. Expression: no equals sign. Equation: equals sign, true for some values. Formula: a rule linking quantities. Identity: true for all values (≡).
Method. Look for the equals sign, then decide which of the four words fits.
- Look for an equals sign (or ≡). An expression never has one. Equations, formulae and identities always do.4n − 7 = 21
There is a sign that means "equals" here. - Test the two sides with a number. If the sides are not equal for every number, it is an equation (true for only some values).Try n = 1: left = 4 × 1 − 7 = 4 − 7 = −3, right = 21. These are not equal, so it is only true for some values of n.
- Decide. An equation has an equals sign and is only true for some values of the letter. You can solve it to find them.equation
Watch out. A formula and an equation both have an equals sign. The difference is the job: a formula is a rule for working things out; an equation is a puzzle to solve.
Key words. Expression: no equals sign. Equation: equals sign, true for some values. Formula: a rule linking quantities. Identity: true for all values (≡).
Method. Look for the equals sign, then decide which of the four words fits.
- Look for an equals sign (or ≡). An expression never has one. Equations, formulae and identities always do.3(n + 2) ≡ 3n + 6
There is a sign that means "equals" here. - Test the two sides with a number. If they are always equal, it is an identity. Try n = 2 and n = 5.n = 2: left = 3 × (2 + 2) = 3 × 4 = 12, right = 3 × 2 + 6 = 6 + 6 = 12
n = 5: left = 3 × (5 + 2) = 3 × 7 = 21, right = 3 × 5 + 6 = 15 + 6 = 21
Both agree, and they always will: expanding the bracket proves it. - Decide. An identity uses ≡ and is true for every value of the letter, because the two sides are the same thing written in two ways.identity
Watch out. A formula and an equation both have an equals sign. The difference is the job: a formula is a rule for working things out; an equation is a puzzle to solve.
Key words. Expression: no equals sign. Equation: equals sign, true for some values. Formula: a rule linking quantities. Identity: true for all values (≡).
Method. Look for the equals sign, then decide which of the four words fits.
- Look for an equals sign (or ≡). An expression never has one. Equations, formulae and identities always do.2(l + w)
There is no equals sign. - Decide. An expression is a mixture of numbers and letters with no equals sign. It is a "phrase", not a full sentence.expression
Watch out. A formula and an equation both have an equals sign. The difference is the job: a formula is a rule for working things out; an equation is a puzzle to solve.
Method. A formula is a rule linking real quantities. Check for an equals sign and what the letters mean.
- Look for an equals sign. There is one.P = 2l + 2w
- Look at what the letters stand for. P is a perimeter, l a length and w a width. They are real quantities in a rule that works for any rectangle.This is a formula.
Check. Putting in l = 5 and w = 3 gives P = 2 × 5 + 2 × 3 = 16, a real perimeter. ✓
Watch out. A formula is for working things out; an equation is for solving.
Key words. Expression: no equals sign. Equation: equals sign, true for some values. Formula: a rule linking quantities. Identity: true for all values (≡).
Method. Look for the equals sign, then decide which of the four words fits.
- Look for an equals sign (or ≡). An expression never has one. Equations, formulae and identities always do.5 + 2x
There is no equals sign. - Decide. An expression is a mixture of numbers and letters with no equals sign. It is a "phrase", not a full sentence.expression
Watch out. A formula and an equation both have an equals sign. The difference is the job: a formula is a rule for working things out; an equation is a puzzle to solve.
Method. Write products and fractions in the shortest standard algebra form.
- Look at what is being multiplied or divided. A number multiplied by a letter: the number goes first and the × sign is left out.3 × n
- Write it in algebra shorthand. Numbers go before letters, we leave out the × sign, and letters go in alphabetical order.3n
Check. Put in a number to test: 3 × n and the answer should give the same value.
Watch out. 3n means 3 × n, not 30 + n.
Method. Write products and fractions in the shortest standard algebra form.
- Look at what is being multiplied or divided. A letter multiplied by itself is written as a square.n × n
- Write it in algebra shorthand. Numbers go before letters, we leave out the × sign, and letters go in alphabetical order.n2
Check. Put in a number to test: n × n and the answer should give the same value.
Watch out. 3n means 3 × n, not 30 + n.
Method. Write products and fractions in the shortest standard algebra form.
- Look at what is being multiplied or divided. Several things multiplied together are written side by side, with the number first.4 × a × b
- Write it in algebra shorthand. Numbers go before letters, we leave out the × sign, and letters go in alphabetical order.4ab
Check. Put in a number to test: 4 × a × b and the answer should give the same value.
Watch out. 3n means 3 × n, not 30 + n.
Method. Write products and fractions in the shortest standard algebra form.
- Look at what is being multiplied or divided. Put the number first and the letters in alphabetical order.b × a × 3
- Write it in algebra shorthand. Numbers go before letters, we leave out the × sign, and letters go in alphabetical order.3ab
Check. Put in a number to test: b × a × 3 and the answer should give the same value.
Watch out. 3n means 3 × n, not 30 + n.
Method. Write products and fractions in the shortest standard algebra form.
- Look at what is being multiplied or divided. A division is written as a fraction: the first thing on top, the second below.x ÷ 5
- Write it in algebra shorthand. Numbers go before letters, we leave out the × sign, and letters go in alphabetical order.x5
Check. Put in a number to test: x ÷ 5 and the answer should give the same value.
Watch out. 3n means 3 × n, not 30 + n.
Method. Write products and fractions in the shortest standard algebra form.
- Look at what is being multiplied or divided. A number multiplied by a bracket is written with the number just before the bracket.2 × (n + 4)
- Write it in algebra shorthand. Numbers go before letters, we leave out the × sign, and letters go in alphabetical order.2(n + 4)
Check. Put in a number to test: 2 × (n + 4) and the answer should give the same value.
Watch out. 3n means 3 × n, not 30 + n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “5 more than” “More than” means add.+ 5
- “n” n is the number we start from.n
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.n + 5
Check. Test with a number. Let n = 8. In words: 5 more than 8 = 13. The expression gives 8 + 5 = 13. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “6 times” “Times” means multiply.6 ×
- “n” n is the number.n
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.6n
Check. Test with a number. Let n = 4. In words: 6 times 4 = 24. The expression gives 6 × 4 = 24. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “9 less than” “Less than” means take away, and we start from n.− 9
- “n” It comes first.n
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.n − 9
Check. Test with a number. Let n = 20. In words: 9 less than 20 = 11. The expression gives 20 − 9 = 11. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “n” The total being shared.n
- “shared equally between 4” Sharing equally is dividing. We write a division as a fraction.÷ 4
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.n4
Check. Test with a number. Let n = 20. In words: 20 shared between 4 = 5. The expression gives 204 = 5. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “double n” Double means multiply by 2.2n
- “then add 3” Add 3 to the doubled number.+ 3
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.2n + 3
Check. Test with a number. Let n = 5. In words: double 5 is 10, then add 3 = 13. The expression gives 2 × 5 + 3 = 10 + 3 = 13. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “12” We start from 12.12
- “take n away” Taking away means subtract. This time 12 comes first.− n
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.12 − n
Check. Test with a number. Let n = 5. In words: 12 take away 5 = 7. The expression gives 12 − 5 = 7. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “Each pen costs 35 pence” The price of one pen is a fixed number.35
- “n pens” For n pens we multiply the price by the number of pens.35 × n
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.35n
Check. Test with a number. Let n = 4. In words: 4 pens at 35p = 140. The expression gives 35 × 4 = 140. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “n pens at 35p each” This is the cost of the pens.35n
- “and a ruler at 60p” Add the cost of the ruler.+ 60
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.35n + 60
Check. Test with a number. Let n = 4. In words: 4 pens (140p) and a 60p ruler = 200. The expression gives 35 × 4 + 60 = 140 + 60 = 200. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “She pays with a £5 note, which is 500p” Change the pounds to pence so the units match.500
- “takes away the cost” The change is what is left: 500 take away the cost. The cost is in brackets because the whole cost is taken away.− (35n + 60)
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.500 − (35n + 60)
Check. Test with a number. Let n = 4. In words: 500p take away 200p = 300. The expression gives 500 − (35 × 4 + 60) = 500 − (140 + 60) = 500 − 200 = 300. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “Ali is a years old” We use the letter a for Ali’s age.a
- “Ben is 3 years older” Older means add.+ 3
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.a + 3
Check. Test with a number. Let a = 10. In words: 3 years older than 10 = 13. The expression gives 10 + 3 = 13. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “Cara is half of Ali’s age” Half means divide by 2. We write the division as a fraction.a ÷ 2
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.a2
Check. Test with a number. Let a = 10. In words: half of 10 = 5. The expression gives 102 = 5. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “Ali” Ali’s age.a
- “Ben” Ben’s age from part (a).+ (a + 3)
- “Cara” Cara’s age from part (b).+ a/2
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.a + (a + 3) + a2
Check. Test with a number. Let a = 10. In words: 10 + 13 + 5 = 28. The expression gives 10 + (10 + 3) + 102 = 10 + 13 + 102 = 10 + 13 + 5 = 23 + 5 = 28. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Method. Test Dan’s expression with a number and compare it with what the words really give.
- Work out what the words give for n = 5. “4 more than” means add 4.4 more than 5 = 5 + 4 = 9
- Work out Dan’s expression for n = 5. 4n means 4 × n.4 × 5 = 20
- Compare. 9 is not 20, so Dan is wrong.Dan multiplied instead of adding.
- Write the correct expression. “More than” means add.n + 4
Check. With n = 5: n + 4 = 9, the same as the words give. ✓
Watch out. A number next to a letter always means multiply. To add 4, you need a + sign.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “£3 to start” This is a fixed charge, paid whatever the distance.3
- “£2 for every mile, m miles” The cost per mile (2) is multiplied by the number of miles.2m
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.C = 2m + 3
Check. Test with a number. Let m = 5. In words: £3 plus 5 miles at £2 = 13. The expression gives 2 × 5 + 3 = 10 + 3 = 13. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.
Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.
- Write down the formula. This is the rule we are going to use.C = 2m + 3
- Substitute: m = 5. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.C = 2 × 5 + 3
- Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.2 × 5 = 10
C = 10 + 3 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.10 + 3 = 13
C = 13 - Write the answer. The answer should say what was being worked out.C = £13
Check. Put the numbers straight into a calculator: 2 × 5 + 3 gives 13. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “Add 3 to n, then double the result” Add first, so the addition must be in a bracket before doubling.2(n + 3)
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.2(n + 3)
Check. Test with a number. Let n = 4. In words: 4 + 3 = 7, doubled = 14 = 14. The expression gives 2 × (4 + 3) = 2 × 7 = 14. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “Double n, then add 3” Double first, then add 3.2n + 3
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.2n + 3
Check. Test with a number. Let n = 4. In words: double 4 = 8, plus 3 = 11 = 11. The expression gives 2 × 4 + 3 = 8 + 3 = 11. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “Multiply n by itself, then add 3” Multiplying a number by itself is squaring it.n2 + 3
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.n2 + 3
Check. Test with a number. Let n = 4. In words: 4 × 4 = 16, plus 3 = 19 = 19. The expression gives 42 + 3 = 16 + 3 = 19. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “Add 3 to n, then multiply the result by itself” The addition happens first, so it goes in brackets before squaring.(n + 3)2
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.(n + 3)2
Check. Test with a number. Let n = 4. In words: 4 + 3 = 7, 7 × 7 = 49 = 49. The expression gives (4 + 3)2 = 72 = 49. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Method. The perimeter is the total distance around the outside, so add the three sides, then collect like terms.
- Write the sum of the three sides. Perimeter means add all the sides.x + (x + 3) + (2x − 1)
- Remove the brackets. There is a plus sign in front of each bracket, so nothing changes sign.x + x + 3 + 2x − 1
- Collect the x terms. x + x + 2x = 4x.4x
- Collect the numbers. 3 − 1 = 2.+ 2
- Write the answer. 4x + 2 cm
Check. Try x = 5: the sides are 5, 8 and 9. Sum = 22. And 4 × 5 + 2 = 22. ✓
Watch out. Only add terms that have the same letter part: xs with xs, and numbers with numbers.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “Bea has £5 more than Amir” More than means add.a + 5
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.a + 5
Check. Test with a number. Let a = 10. In words: £5 more than £10 = 15. The expression gives 10 + 5 = 15. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “Cal has twice as much as Amir” Twice means multiply by 2.2a
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.2a
Check. Test with a number. Let a = 10. In words: twice £10 = 20. The expression gives 2 × 10 = 20. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Method. Add the three amounts, then collect like terms.
- Write the three amounts. Amir has a, Bea has a + 5 and Cal has 2a.a + (a + 5) + 2a
- Collect the a terms. 1a + 1a + 2a = 4a.4a + 5
- Write the answer. Pounds, so it is money.£(4a + 5)
Check. Let Amir have £10. Then Bea has £15, Cal has £20 and the total is £45. And 4 × 10 + 5 = 45. ✓
Watch out. Make sure you include all three people in the total, including Amir.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “the next whole number after n” The next whole number is 1 more.n + 1
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.n + 1
Check. Test with a number. Let n = 7. In words: the next after 7 = 8. The expression gives 7 + 1 = 8. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “the whole number before n” The number before is 1 less.n − 1
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.n − 1
Check. Test with a number. Let n = 7. In words: the number before 7 = 6. The expression gives 7 − 1 = 6. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “the next even number after the even number 2n” Even numbers go up in twos, so add 2.2n + 2
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.2n + 2
Check. Test with a number. Let n = 7. In words: the even number after 14 = 16. The expression gives 2 × 7 + 2 = 14 + 2 = 16. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Method. Substitute the numbers into both expressions and compare. One mismatch is enough to show she is wrong.
- Test n = 1 in 3(n + 2). Brackets first.3 × (1 + 2) = 3 × 3 = 9
- Test n = 1 in 3n + 2. Multiply first.3 × 1 + 2 = 3 + 2 = 5
- Compare. 9 and 5 are different, so the expressions are not the same.Tia is not right.
- Say what is wrong. The 3 must multiply every term in the bracket.3(n + 2) = 3n + 6, not 3n + 2
Check. Test n = 4: 3 × (4 + 2) = 3 × 6 = 18 and 3 × 4 + 2 = 12 + 2 = 14. Different again. ✓
Watch out. Multiplying a bracket: the number outside multiplies every term inside. See Chapter 7.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “3 adult tickets at £a each” Three lots of a.3a
- “2 child tickets at £c each” Two lots of c, added on.+ 2c
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.3a + 2c
Check. Test with a number. Let a = 8, c = 5. In words: 3 × £8 + 2 × £5 = 24 + 10 = 34. The expression gives 3 × 8 + 2 × 5 = 24 + 10 = 34. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Method. An even number is a multiple of 2 (it can be shared into two equal groups with none left). Use that to describe each expression.
- Think about 2n. 2n means 2 × n, which is 2 multiplied by a whole number.So 2n is a multiple of 2. Multiples of 2 are even.
- Think about 2n + 1. Adding 1 to a multiple of 2 leaves 1 left over when you try to share into twos.So 2n + 1 is always one more than an even number, which is odd.
- Try some numbers. It helps to see it.n = 3: 2n = 6 (even) and 2n + 1 = 7 (odd).
n = 10: 2n = 20 (even) and 2n + 1 = 21 (odd).
Check. Both examples agree with the explanation. ✓
Watch out. Examples are not a proof. The explanation using “multiple of 2” works for every whole number n.
Method. Find the length first, then add the four sides, then simplify.
- Write the length. Twice the width is 2w. Then 3 more.length = 2w + 3
- Write the perimeter. Perimeter = 2 lengths + 2 widths.P = 2(2w + 3) + 2w
- Expand the bracket. Multiply both terms in the bracket by 2.2 × 2w = 4w and 2 × 3 = 6, so P = 4w + 6 + 2w
- Collect like terms. 4w + 2w = 6w.P = 6w + 6
- Explain. Factorise: both terms divide by 6.6w + 6 = 6(w + 1), a multiple of 6 when w is a whole number.
Check. Try w = 4: length = 11, perimeter = 2 × 11 + 2 × 4 = 30. And 6 × 4 + 6 = 30. ✓
Watch out. The length is 3 more than twice the width: 2w + 3, not 2(w + 3).