The Improving Mathematics Education in Schools (TIMES) Project NUMBER AND ALGEBRA Module 23 ALGEBRAIC EXPRESSIONS A guide for teachers - Year 7 June 2011 7YEAR Algebraic Expressions (Number and Algebra : Module 23) For teachers of Primary and Secondary Mathematics 510 Cover design, Layout design and Typesetting by Claire Ho The Improving Mathematics Education in Schools (TIMES) Project 2009‑2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations. The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations. © The University of Melbourne on behalf of the International Centre of Excellence for Education in Mathematics (ICE‑EM), the education division of the Australian Mathematical Sciences Institute (AMSI), 2010 (except where otherwise indicated). This work is licensed under the Creative Commons Attribution‑ NonCommercial‑NoDerivs 3.0 Unported License. 2011. http://creativecommons.org/licenses/by‑nc‑nd/3.0/ The Improving Mathematics Education in Schools (TIMES) Project NUMBER AND ALGEBRA Module 23 ALGEBRAIC EXPRESSIONS A guide for teachers - Year 7 June 2011 Peter Brown Michael Evans David Hunt Janine McIntosh Bill Pender Jacqui Ramagge 7YEAR {4} A guide for teachers ALGEBRAIC EXPRESSIONS ASSUMED KNOWLEDGE • Fluency with addition, subtraction, multiplication and division of whole numbers and fractions. • Ability to apply the any‑order principle for multiplication and addition (commutative law and associative law) for whole numbers and fractions. • Familiarity with the order of operation conventions for whole numbers. MOTIVATION Algebra is a fascinating and essential part of mathematics. It provides the written language in which mathematical ideas are described. Many parts of mathematics are initiated by finding patterns and relating to different quantities. Before the introduction and development of algebra, these patterns and relationships had to be expressed in words. As these patterns and relationships became more complicated, their verbal descriptions became harder and harder to understand. Our modern algebraic notation greatly simplifies this task. A well‑known formula, due to Einstein, states that E = mc2. This remarkable formula gives the relationship between energy, represented by the letter E, and mass, represented by letter m. The letter c represents the speed of light, a constant, which is about 300 000 000 metres per second. The simple algebraic statement E = mc2 states that some matter is converted into energy (such as happens in a nuclear reaction), then the amount of energy produced is equal to the mass of the matter multiplied by the square of the speed of light. You can see how compact the formula is compared with the verbal description. We know from arithmetic that 3 × 6 + 2 × 6 = 5 × 6. We could replace the number 6 in this statement by any other number we like and so we could write down infinitely many such statements. All of these can be captured by the algebraic statement 3x + 2x = 5x, for any number x. Thus algebra enables us to write down general statements clearly and concisely. The development of mathematics was significantly restricted before the 17th century by the lack of efficient algebraic language and symbolism. How this notation evolved will be discussed in the History section. The Improving Mathematics Education in Schools (TIMES) Project {5} CONTENT USING PRONUMERALS In algebra we are doing arithmetic with just one new feature – we use letters to represent numbers. Because the letters are simply stand‑ins for numbers, arithmetic is carried out exactly as it is with numbers. In particular the laws of arithmetic (commutative, associative and distributive) hold. For example, the identities 2 + x = x + 2 2 × x = x × 2 (2 + x) + y = 2 + (x + y) (2 × x) × y = 2 × (x × y) 6(3x + 1) = 18x + 6 hold when x and y are any numbers at all. In this module we will use the word pronumeral for the letters used in algebra. We choose to use this word in school mathematics because of confusion that can arise from the words such as ‘variable’. For example, in the formula E = mc2, the pronumerals E and m are variables whereas c is a constant. Pronumerals are used in many different ways. For example: • Substitution: ‘Find the value of 2x + 3 if x = 4.’ In this case the pronumeral is given the value 4. • Solving an equation: ‘Find x if 2x + 3 = 8.’ Here we are seeking the value of the pronumeral that makes the sentence true. • Identity: ‘The statement of the commutative law: a + b = b + a.’ Here a and b can be any real numbers. • Formula: ‘The area of a rectangle is A = lw.‘ Here the values of the pronumerals are connected by the formula. • Equation of a line or curve: ‘The general equation of the straight line is y = mx + c.’ Here m and c are parameters. That is, for a particular straight line, m and c are fixed. {6} A guide for teachers In some languages other than English, one distinguishes between ‘variables’ in functions and ‘unknown quantities’ in equations (‘incógnita’ in Portuguese/Spanish, ‘inconnue’ in French) but this does not completely clarify the situation. The terms such as variable and parameter cannot be precisely defined at this stage and are best left to be introduced later in the development of algebra. An algebraic expression is an expression involving numbers, parentheses, operation signs and pronumerals that becomes a number when numbers are substituted for the pronumerals. For example 2x + 5 is an expression but +) × is not. Examples of algebraic expressions are: 3x + 1 and 5(x2 + 3x) As discussed later in this module the multiplication sign is omitted between letters and between a number and a letter. Thus substituting x = 2 gives 3x + 1 = 3 × 2 + 1 = 7 and 5(x2 + 3x) = 5(22 + 3 × 2) = 30. In this module, the emphasis is on expressions, and on the connection to the arithmetic that students have already met with whole numbers and fractions. The values of the pronumerals will therefore be restricted to the whole numbers and non‑negative fractions. Changing words to algebra In algebra, pronumerals are used to stand for numbers. For example, if a box contains x stones and you put in five more stones, then there arex + 5 stones in the box. You may or may not know what the value of x is (although in this example we do know that x is a whole number). • Joe has a pencil case that contains an unknown number of pencils. He has three other pencils. Let x be the number of pencils in the pencil case. Then Joe has x + 3 pencils altogether. • Theresa has a box with least 5 pencils in it, and 5 are removed. We do not know how many pencils there are in the pencil case, so let z be the number of pencils in the box. Then there are z – 5 pencils left in the box. • There are three boxes, each containing the same number of marbles. If there are x marbles x marbles x marbles x marbles in each box, then the total number of marbles is 3 × x = 3x. The Improving Mathematics Education in Schools (TIMES) Project {7} n • If n oranges are to be divided amongst 5 people, then each person receives 5 oranges. (Here we assume that n is a whole number. If each person is to get a whole number of oranges, then n must be a multiple of 5.) The following table gives us some meanings of some commonly occurring algebraic expressions. x + 3 • The sum of x and 3 • 3 added to x, or x is added to 3 • 3 more than x, or x more than 3 x – 3 • The difference ofx and 3 (where x ≥ 3) • 3 is subtracted from x • 3 less than x • x minus 3 3 × x • The product of 3 and x • x is multiplied by 3, or 3 is multiplied by x x ÷ 3 • x divided by 3 • the quotient when x is divided by 3 2 × x – 3 • x is first multiplied by 2, then 3 is subtracted x ÷ 3 + 2 • x is first divided by 3, then 2 is added NOTATIONS AND LAWS Expressions with zeroes and ones Zeroes and ones can often be eliminated entirely. For example: x + 0 = x (Adding zero does not change the number.) 1 × x = x (Multiplying by one does not change the number.) Algebraic notation In algebra there are conventional ways of writing multiplication, division and indices. Notation for multiplication The × sign between two pronumerals or between a pronumeral and a number is usually omitted. For example, 3 × x is written as 3x and a × b is written as ab. We have been following this convention earlier in this module. It is conventional to write the number first. That is, the expression 3 ×a is written as 3a and not as a3. {8} A guide for teachers Notation for division The division sign ÷ is rarely used in algebra. Instead the alternative fraction notation for 24 division is used. We recall 24 ÷ 6 can be written as 6 . Using this notation, x divided by 5 x is written as 5 , not as x ÷ 5. Index notation x × x is written as x2 and y × y × y is written as y3. EXAMPLE Write each of the following using concise algebraic notation. a A number x is multiplied by itself and then doubled. b A number x is squared and then multiplied by the square of a second number y. c A number x is multiplied by a number y and the result is squared. SOLUTION a x × x × 2 = x2 × 2 = 2x2. b x2 × y2 = x2y2. c (x × y)2 = (xy)2 which is equal to x2y2 Summary • 2 × x is written as 2x • x1 = x (the first power ofx is x) • x × y is written as xy or yx • xo = 1 • x × y × z is written as xyz • 1x = x • x × x is written as x2 • 0x = 0 • 4 × x × x × x = 4x3 x • x ÷ 3 is written as 3 x ÷ (yz) is written as x • yz The Improving Mathematics Education in Schools (TIMES) Project {9} SUBSTITUTION Assigning values to a pronumeral is called substitution.