Substituting into expressions

What does substitution mean? Ask pupils where they have heard the word substitution before. One example would be in team games when one player is replaced by another. In algebra, substitution means to replace letters with numbers. In algebra, when we replace letters in an expression or equation with numbers we call it substitution.

How can be written as an algebraic expression? Substitution 4 + 3 × How can be written as an algebraic expression? Using n for the variable we can write this as 4 + 3n. We can evaluate the expression 4 + 3n by substituting different values for n. When n = 5 4 + 3n = 4 + 3 × 5 = 4 + 15 Ask pupils to think about the earlier activity. What we were actually doing was a kind of substitution. We were replacing a symbol (the box) with a number each time. Ask pupils how we could write 4 + 3 ×  as an algebraic expression. It doesn’t matter what letter they use but do remind pupils that we don’t write the multiplication sign in algebra. Define the keyword, evaluate – to find the value of. Discuss the substitution and order of operations: When n is 5, what is 3n? (15) So what is 4 + 3n? (4 + 15 = 19) Suggest to pupils that they may wish to work out the value of 3n before writing anything down. This would avoid errors involving order of operations. = 19 When n = 11 4 + 3n = 4 + 3 × 11 = 4 + 33 = 37

Substitution 7 × 2 7n 2 can be written as We can evaluate the expression by substituting different values for n. 7n 2 7n 2 = When n = 4 7 × 4 ÷ 2 = 28 ÷ 2 Emphasize that when we are multiplying and dividing, it doesn’t matter what order we do it in. For example 7 × 4 ÷ 2 will always give the same answer as 4 ÷ 2 × 7. (The order is important when we combine multiplying and dividing with adding and subtracting. If there aren’t any brackets we always multiply or divide before we add or subtract.) = 14 7n 2 = When n = 1.1 7 × 1.1 ÷ 2 = 7.7 ÷ 2 = 3.85

Substitution 2 + 6 can be written as n2 + 6 We can evaluate the expression n2 + 6 by substituting different values for n. When n = 4 n2 + 6 = 42 + 6 = 16 + 6 = 22 Remind pupils that 4² is read as ‘4 squared’ and means ‘4 × 4.’ Pupils are less likely to make mistakes involving incorrect order of operations if they can be encouraged to square in their heads rather than write down the intermediate step of 4² + 6 = 4 × 4 + 6. In particular, expressions such as 3 + 2², may be written as 3 + 2 × 2 and then incorrectly evaluated to 10. When n = 0.6 n2 + 6 = 0.62 + 6 = 0.36 + 6 = 6.36

Substitution 2( + 8) can be written as 2(n + 8) 2( + 8) can be written as 2(n + 8) We can evaluate the expression 2(n + 8) by substituting different values for n. When n = 6 2(n + 8) = 2 × (6 + 8) = 2 × 14 = 28 Remind pupils again that when there are brackets we need to work out the value inside the brackets before we multiply. When n = 13 2(n + 8) = 2 × (13 + 8) = 2 × 21 = 42

Substitution exercise Here are five expressions. 1) a + b + c = 5 + 2 + –1 = 6 2) 3a + 2c = 3 × 5 + 2 × –1 = 15 + –2 = 13 3) a(b + c) = 5 × (2 + –1) = 5 × 1 = 5 4) abc = 5 × 2 × –1 = 10 × –1 = –10 Tell pupils that expressions can contain many different variables. Remember when we use a letter to represent a number in an expression it can have any value. The value can vary and so we call it a variable. If pupils are ready you may wish to use the above examples as a pupil exercise before revealing the solutions. Alternatively, talk through each example emphasizing the correct order of operations each time. Then set pupils an exercise made up of similar problems. Edit the slide to make the numbers being substituted more or less challenging. 5) a b2 – c 5 22 – –1 = = 5 ÷ 5 = 1 Evaluate these expressions when a = 5, b = 2 and c = –1.

Now try these:

Answers: 11. 24 12. 13 13. 3 14. -1 15. 17 16. -9 17. 30 18. 10 19. 3 20. -3 21. 45 22. 23 23. 41 24. 9 25. 4 26. 8 27. 21 28. -12 29. 14 30. 27 31. -7 32. 8 33. 9 34. 25