A1 Algebraic expressions Maths Age 11-14 The aim of this unit is to teach pupils to: Use letter symbols and distinguish their different roles in algebra Know that algebraic operations follow the same conventions and order as arithmetic operations; use index notation and the index laws Simplify or transform algebraic expressions A1 Algebraic expressions

A1 Algebraic expressions Contents A1 Algebraic expressions A1 A1.1 Writing expressions A1 A1.2 Collecting like terms A1 A1.3 Multiplying terms A1 A1.4 Dividing terms A1 A1.5 Factorizing expressions A1 A1.6 Substitution

Using symbols for unknowns Look at this problem: + 9 = 17 The symbol stands for an unknown number. We can work out the value of . Introduce the idea of using symbols to represent unknowns in mathematics. = 8 because 8 + 9 = 17

Using symbols for unknowns Look at this problem: – = 5 The symbols stand for unknown numbers. and In this problem, and can have many values. Two examples are, 12 – 7 = 5 3.2 – –1.8 = 5 or and are called variables because their value can vary.

Using letter symbols for unknowns In algebra, we use letter symbols to stand for numbers. These letters are called unknowns or variables. Sometimes we can work out the value of the letters and sometimes we can’t. For example: We can write an unknown number with 3 added on to it as n + 3 This is an example of an algebraic expression.

Writing an expression Suppose Jon has a packet of biscuits and he doesn’t know how many biscuits it contains. He can call the number of biscuits in the full packet, b. If he opens the packet and eats 4 biscuits, he can write an expression for the number of biscuits remaining in the packet as: Talk through the example. Emphasize that b stands for the number of biscuits in the full packet. We could choose any letter to stand for the unknown number. We chose b because b stands for biscuit. We could also have n for number or any other letter. Point out that the expression b – 4 describes the relationship between the number of biscuits in the full packet and the number of biscuits in the packet after 4 biscuits have been eaten. Offer examples, such as if there were 20 biscuits in the original packet, then there are 20 – 4 = 16 biscuits after 4 biscuits have been eaten. b – 4

Writing an equation Jon counts the number of biscuits in the packet after he has eaten 4 of them. There are 22. He can write this as an equation: b – 4 = 22 We can work out the value of the letter b. Tell pupils the difference between an expression and an equation. An expression does not contain an equals sign. In an equation we can often work out the value of the letter symbol. In an expression we cannot. b = 26 That means that there were 26 biscuits in the full packet.

Writing expressions When we write expressions in algebra we don’t usually use the multiplication symbol ×. 5 × n or n × 5 is written as 5n. The number must be written before the letter. When we multiply a letter symbol by 1, we don’t have to write the 1. Introduce these algebraic conventions. When we write an algebraic expression we try to use as few numbers, letters and symbols as necessary. If we know that 5n means 5 lots of n, then we don’t need to write 5 × n. Mention that leaving out the multiplication sign × avoids confusing it with the letter symbol x which is often used in algebra. Also point out to pupils that when they write the letter x in algebra it should be written in script form to distinguish it from a multiplication sign. 1 × n or n × 1 is written as n.

Writing expressions When we write expressions in algebra we don’t usually use the division symbol ÷. Instead we use a dividing line as in fraction notation. n ÷ 3 is written as n 3 When we multiply a letter symbol by itself, we use index notation. Tell pupils that n² is read as ‘n squared’ or ‘n to the power of 2’. n squared n × n is written as n2.

Writing expressions Here are some examples of algebraic expressions: a number n plus 7 5 – n 5 minus a number n 2n 2 lots of the number n or 2 × n 6 n 6 divided by a number n 4n + 5 4 lots of a number n plus 5 Explain that algebra is like a language. It follows special rules, like the rules of grammar in a language. We have to keep to these rules so that any mathematician in the world can understand it. Algebra is very important in mathematics because it describes the relationships between numbers. Explain that there is a difference between an unknown and a variable. An unknown usually has a unique value which we can work out given enough information. A variable can have many different values. We can use any letter in the alphabet to stand for unknowns or variables but some letters are used more than others. For example, we often use a, b, n, x or y. But we try not to use o (because it looks like 0). Explain that in algebra we do not need to write the multiplication sign, ×, and so 2 lots of n is written as 2n. 3 lots of a, or 3 times a would be written as 3a. 5 lots of t, or 5 times t would be written as 5t. Give some examples of possible values for 2n. If n is worth 5 then 2n is equal to 10 (not 25). If n is worth 20 then 2n is worth 40. When we divide in algebra we write the number we are dividing by underneath, like a fraction. In this example, if n was worth 2, 6/n would be equal to 3. Tell pupils that n³ is read as ‘n cubed’ or ‘n to the power of 3’. Give some examples, such as if n is worth 2 then n³ is 2 × 2 × 2, 8. a number n multiplied by itself twice or n × n × n n3 3 × (n + 4) or 3(n + 4) a number n plus 4 and then times 3.

Writing expressions Miss Green is holding n number of cubes in her hand: Write an expression for the number of cubes in her hand if: She takes 3 cubes away. n – 3 Discuss the examples on the slides and give other examples from around the classroom. For example: Suppose Joe has p number of pencils in his pencil case. If Harry, sitting next to him, gives him two more pencils, what expression could we write for the number of pencils in his pencil case? (p + 2) We don’t know what p is (without counting) but we can still write an expression for the number of pencils in the pencil case. If p, the original number of pencils in the pencil case was 9, Joe would now have 11. If p was 15, Joe would now have 17. Now, Joe gives Harry back his pencils, so he has p pencils again. Suppose he shares his pencils equally between himself and two of his friends. How many pencils will they have each? (p ÷ 3) Tell pupils that if they are not sure whether or not an expression works, they should try using numbers in place of the letters to check. For example: If Joe had 18 pencils and shared them between himself and his 2 friends they would get 6 each, 18 ÷ 3, so our expression works. Suppose Mary has p pencils and Julia has q pencils. How many pencils do they have altogether? (p + q) She doubles the number of cubes she is holding. 2 × n or 2n

Equivalent expression match Match-up pairs of equivalent expressions

x + x + x is identically equal to 3x Identities When two expressions are equivalent we can link them with the  sign. x + x + x is identically equal to 3x For example: x + x + x  3x This is called an identity. In an identity, the expressions on each side of the equation are equal for all values of the unknown. Differentiate between the = sign meaning ‘is equal to’ and the  sign meaning ‘is identically equal to’. An equation is true for particular values of the unknown. An identity is true for all values of the unknown. The expressions are said to be identically equal.

A1.2 Collecting like terms Contents A1 Algebraic expressions A1 A1.1 Writing expressions A1 A1.2 Collecting like terms A1 A1.3 Multiplying terms A1 A1.4 Dividing terms A1 A1.5 Factorizing expressions A1 A1.6 Substitution

Like terms An algebraic expression is made up of terms and operators such as +, –, ×, ÷ and ( ). A term is made up of numbers and letter symbols but not operators. 3a + 4b – a + 5 is an expression. 3a, 4b, a and 5 are terms in the expression. Explain that we can think of algebra as the language of mathematics. 3a and a are called like terms because they both contain a number and the letter symbol a.

Collecting together like terms Remember, in algebra letters stand for numbers, so we can use the same rules as we use for arithmetic. In arithmetic: 5 + 5 + 5 + 5 = 4 × 5 In algebra: a + a + a + a = 4a In arithmetic, a number plus the same number plus the same number plus the same number = 4 × the number. Emphasize that a can be any number. In this example, we are using algebra to generalize the result. The a’s are like terms. We collect together like terms to simplify the expression.

Collecting together like terms Remember, in algebra letters stand for numbers, so we can use the same rules as we use for arithmetic. In arithmetic: (7 × 4) + (3 × 4) = 10 × 4 In algebra: 7 × b + 3 × b = 10 × b or In arithmetic, 7 × a number plus 3 × the same number = 10 × the number. Pupils may want an ‘answer’ to 7b + 3b = 10b. Explain that what this is telling us is that any number multiplied by 7 plus the same number multiplied by 3 is always equal to the number multiplied by 10. Emphasize that b can be any number. In this example, we are using algebra to generalize a result rather than to give an answer to a specific problem. 7b + 3b = 10b 7b, 3b and 10b are like terms. They all contain a number and the letter b.

Collecting together like terms Remember, in algebra letters stand for numbers, so we can use the same rules as we use for arithmetic. In arithmetic: 2 + (6 × 2) – (3 × 2) = 4 × 2 In algebra: x + 6x – 3x = 4x In arithmetic, a number plus 6 × the same number minus 3 × the same number = 4 × the number. Discuss the algebraic equivalent of this. Emphasize that x can be any number. Again, we are using algebra to generalize a result rather than to give an answer to a specific problem. x, 6x, 3x and 4x are like terms. They all contain a number and the letter x.

Collecting together like terms When we add or subtract like terms in an expression we say we are simplifying an expression by collecting together like terms. An expression can contain different like terms. 3a + 2b + 4a + 6b = 3a + 4a + 2b + 6b = 7a + 8b Explain the meanings of each keyword and phrase. In the example 3a + 2b + 4a + 6b, explain that it is helpful to write like terms next to each other. (Remember that we can add terms in any order.) Point out that we cannot simplify 7a + 8b any further. We cannot combine a’s and b’s. This says 7 times one number plus eight times another, possibly different, number. This expression cannot be simplified any further.

Collecting together like terms Simplify these expressions by collecting together like terms. 1) a + a + a + a + a = 5a 2) 5b – 4b = b 3) 4c + 3d + 3 – 2c + 6 – d = 4c – 2c + 3d – d + 3 + 6 = 2c + 2d + 9 Whenever possible make comparisons to arithmetic by substituting actual values for the letters. If it is true using numbers then it is true using letters. For example, in 1) we could say that 7 + 7 + 7 + 7 + 7 is equivalent to 5 × 7. For example 1) and example 2), remind pupils that in algebra we don’t need to write the number 1 before a letter to multiply it by 1. 1a is just written as a and 1b is just written as b. For example 3), explain that when there are lots of terms we can write like terms next to each other so that they are easier to collect together. The numbers without any letters are added together separately. In example 4) emphasize that n² is different from n, this can be demonstrated by thinking about possible values for n, and because they are different they cannot be collected together. 4n – 3n is n and n² stays as it is. If we can’t collect together any like terms, as in example 5), we write ‘cannot be simplified’. 4) 4n + n2 – 3n = 4n – 3n + n2 = n + n2 5) 4r + 6s – t Cannot be simplified

Algebraic perimeters Remember, to find the perimeter of a shape we add together the lengths of each of its sides. Write algebraic expressions for the perimeters of the following shapes: 2a 3b Perimeter = 2a + 3b + 2a + 3b = 4a + 6b For the first example, remind pupils that we need to add together the length of each side. The lengths of two of the sides have not been written on. Since this is a rectangle we can deduce that the length of the side opposite the side of length 2a is also 2a and the length of the side opposite the side of length 3b is also 3b. Ask pupils: How could the longer side be represented by 2a and the shorter side by 3b? Deduce that the letter a must represent a bigger number than the letter b. 5x 4y x Perimeter = 4y + 5x + x + 5x = 4y + 11x

Algebraic pyramids Use the algebraic pyramid to practise collecting together like terms. Start by revealing all of the expressions along the bottom row of the pyramid. Find the expression in each brick by adding the two expressions below it. Modify the activity by revealing expressions in the bricks above and one of the expressions in the bricks below it. Pupils must then subtract expressions to find those that are hidden.

Algebraic magic square Remind pupils that in a magic square each row, column and diagonal has the same sum, called the ‘magic total’. Start by working out the ‘magic total’ by revealing three expressions in any row column or diagonal. This can also be found by multiplying the expression in the centre by three. Reveal the expressions in one or two more squares so that pupils have enough information to work out the missing expressions. Clicking on each cell will reveal the missing expression.

A1 Algebraic expressions Contents A1 Algebraic expressions A1 A1.1 Writing expressions A1 A1.2 Collecting like terms A1 A1.3 Multiplying terms A1 A1.4 Dividing terms A1 A1.5 Factorizing expressions A1 A1.6 Substitution

Multiplying terms together In algebra we usually leave out the multiplication sign ×. Any numbers must be written at the front and all letters should be written in alphabetical order. 4 × a = 4a 1 × b = b We don’t need to write a 1 in front of the letter. b × 5 = 5b We don’t write b5. Ask pupils why they think we try not to use the multiplication symbol ×. One reason is that it is easily confused with the letter, x. Another reason is that when we use algebra we try to write things as simply as possible, only writing what is absolutely necessary. It’s simpler to write 2n than 2 x n. It is also unnecessary to write a 1 in front of a letter to multiply it by 1. Multiplying by 1 has no effect so we can leave it out altogether. 3 × d × c = 3cd We write letters in alphabetical order. 6 × e × e = 6e2

Using index notation Simplify: x + x + x + x + x = 5x x to the power of 5 Simplify: x × x × x × x × x = x5 This is called index notation. Similarly: Start by asking pupils to simplify x + x + x + x + x. This is 5 lots of x, which we have seen is written as 5x. Now, ask pupils how they might simplify x × x × x × x × x. Impress upon pupils the difference between this, and the previous expression, as they are often confused. If x is equal to 2, for example, x + x + x + x + x equals 10, and x × x × x × x × x equals 32. Some pupils may suggest writing xxxxx. This is not strictly incorrect, however, it should be discouraged in favour of using index notation. When we write a number or term to the power of another number it is called index notation. The power, or index, is the raised number, in this case 5. The plural of index is indices. The number or letter that we are multiplying successive times, in this case, x, is called the base. x² is read as ‘x squared’ or ‘x to the power of 2’. x³ is read as ‘x cubed’ or ‘x to the power of 3’. x4 is read as ‘x to the power of 4’. x × x = x2 x × x × x = x3 x × x × x × x = x4

Using index notation We can use index notation to simplify expressions. 3p × 2p = 3 × p × 2 × p = 6p2 q2 × q3 = q × q × q × q × q = q5 3r × r2 = 3 × r × r × r = 3r3 Discuss each example briefly. In the last example 2t × 2t the use of brackets may need further clarification. We must put a bracket around the 2t since both the 2 and the t are squared. If we wrote 2t², then only the t would be squared. Give a numerical example, if necessary. If t was 3 then 2t would be equal to 6. We would then have 6², 36. If we wrote 2t², that would mean 2 × 32 or 2 × 9 which is 18. Remember the order of operations - BIDMAS. Brackets are worked out before indices, but indices are worked out before multiplication. 2t × 2t = (2t)2 or 4t2

A1 Algebraic expressions Contents A1 Algebraic expressions A1 A1.1 Writing expressions A1 A1.2 Collecting like terms A1 A1.3 Multiplying terms A1 A1.4 Dividing terms A1 A1.5 Factorizing expressions A1 A1.6 Substitution

Work it out! 4 + 3 × 0.6 –7 43 8 5 The aim of this exercise is to revise the correct order of operations and to introduce the concept of substitution. The following values are inserted in turn: 8 5 43 0.6 –7 Pupils could be asked to write the answers down on paper or write them down on individual whiteboards. = 133 = –17 = 5.8 = 28 = 19

Work it out! 7 × 0.4 –3 22 6 9 2 The following values are inserted in turn: 6 9 22 0.4 –3 = –10.5 = 31.5 = 1.4 = 21 = 77

Work it out! 0.2 –4 12 9 3 2 + 6 The following values are inserted in turn: 3 9 12 0.2 –4 = 6.04 = 150 = 22 = 87 = 15

Work it out! 2( + 8) –13 3.6 69 18 7 The following values are inserted in turn: 7 18 69 3.6 –13 = 23.2 = 154 = –10 = 30 = 52

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 How can be written as an algebraic expression? 4 + 3 × 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.

Noughts and crosses – substitution Divide the class into two teams, red and blue. The starting team chooses an expression from the board. Click on the expression to highlight it. The number that appears in the large rectangle must be substituted into the expression. Everyone in the team must try to work out the answer mentally within 5 seconds. Select a pupil from the team to give you their answer. Check their answer by clicking on the notelet. If the answer is correct, select the teams’ symbol. If the answer is incorrect mark the cell with the opposing team’s symbol. Then it is the turn of the opposing team. The game is over when one of the teams gets three of their symbols in a row, horizontally, vertically, or diagonally. (Or when the board is full, in which case, the game ends in a draw).