Program Support Notes by: Janine Haeusler BEd, M.S in Ed Produced by: VEA Pty Ltd Commissioning Editor: Sandra Frerichs B.Ed, M.Ed. Executive Producer: Simon Garner B.Ed, Dip Management You may download and print one copy of these support notes from our website for your reference. Further copying or printing must be reported to CAL as per the Copyright Act 1968.
For Teachers Introduction This program takes students through step by step processes of finding a numerical solution to various types of linear equations, from the simplest to the more challenging (including linear inequalities). The program reviews a variety of methods for solving equations, including guess, check and improve, the balance method and backtracking, Throughout the program, examples are provided as word problems and translated into equations, and solutions are checked using substitution. Real life examples and applications from maths, science and the everyday, along with the use of modern technology provide an engaging context to this core section of the Australian Maths Curriculum. Timeline 00:00:00 GCI and backtracking 00:04:8 The balancing method 00:09:44 Working with fractions 00:12:49 Using formulae 00:18:13 Inequalities 00:23:02 Credits 00:23:41 End program Related Titles Algebra: The Basics Taking a Chance Key Probability Concepts Statistics Sampling, Surveying and Data Analysis Interest, Loans and Credit Recommended Resources http://www.mathsnet.net/algebra/balance.html http://www.flashymaths.co.uk/swf/equations.swf http://regentsprep.org/regents/math/algebra/math-algebra.htm http://teachers.henrico.k12.va.us/math/hcpsalgebra1/modules.html http://itech.pensacolastate.edu/falzone/course/0024handouts.htm 2
Student Worksheet Initiate Prior Learning Complete the following equation. Show your working out in the spaces provided. 1. 16 x? = 224 2. 216? = 12 3. 2 x? + 6 = 24 4. 23 x? = 11. x? 1 = 200 3
Active Viewing Guide GCI and Backtracking 1. Use the Guess, Check and Improve method to solve 23 x? = 161 2. a) Change the following sentence into an equation. 10 times a number plus 9, is 9. b) Draw a flow chart for your equation and back track to find the value of the number. c) Substitute your answer into the original equation to check. 3. a) A number divided by five subtract three equals sixteen. Using n to represent the original number, write this sentence as an equation. b) Draw a flow chart and backtrack to solve. c) Check your answer by substituting into the original equation. 4
Balancing Method 1. a) Five more than a number is the same as two times the number minus six. Write as an equation and use the balancing method to solve. b) Check your answer by substituting into the original equation. 2. Solve two times a number subtract five is the same as sixteen minus the number. Write an equation to represent this sentence then solve using the balancing method Remember to check your final answer by substituting into the original equation. (7 lines) Fractions 1. Find the value of n for the following equation by backtracking. 2n = 16 2. The difference between four fifths of a number and half a number is 3. a) Write as an equation and change to fractions with a common denominator. b) Find the value of the number.
Using Formula 1. Work out what temperature 18 degrees Celsius is in degrees Fahrenheit using the formula C = (F 32). 9 2. Rearranging the formula for density d= m, calculate the mass of 2L of milk if the density of milk is 1.03 kg/l v Inequalities 1. Draw a line to match the meanings to the correct symbols Meaning Less than Symbol More than < Is less than or equal to > Is more than or equal to 2. Write an inequality and solve for the sum of a number plus is at least 17. 3. Solve and draw a number line for 2x 6 < 2. 4. Solve -p 3 6
Extension Activities 1. Which is the best method to use to solve the following equations? a) 2g -6 = g + 3 b) x + 7 = 2 c) A number decreased by 20 equals 49. d) One third of a number is the same as two times the number subtract fifteen. 2. Form an equation from each statement and solve a) One quarter of a number is 2. b) When 3 is added to 3 times a certain number the result is -3. c) I think of a number, add 10 and divide by 3. This gives the same as doubling my number. d) Three times a number is less than twice the number added to 8. e) The opposite of 3 times a number is at least 48. 7
3. Solve the following a) 12 y = -2 b) d+1 = 3 c) y 3 = 6 3y d) 3x 13 + x 4. The amount of interest charged on a loan can be calculated using the formula: I = PRT P = principal 100 R = Interest rate T = time of loan a) Calculate the time of the loan when I = $26, P = $400 and R = 9.%. Using the formula C = (F-32) solve for F when C equals 4 9 6. Given that the average 70kg adult male s body contains L of blood, using the formula d = m calculate the mass of blood (m) if the density of blood (d) is 1060 kg/m 3. v Hint: Don t forget to change v to m 3. (1L = 0.001 m 3 ) 8
Suggested Student Responses Initiate Prior Learning Complete the following equation. Show your working out in the spaces provided. 1. 16 x 14 = 224 2. 216 18 = 12 3. 2 x 9 + 6 = 24 4. 23 x = 11. x 43 1 = 200 9
Active Viewing Guide GCI and Backtracking 1. Use the Guess, Check and Improve method to solve 23 x 7 = 161 2. a) Change the following sentence into an equation. 10 times a number plus 9, is 9. 10n + 9 = 9 b) Draw a flow chart for your equation and back track to find the value of the number. a) Substitute your answer into the original equation to check. 10n + 9 = 9 10 x + 9 = 9 9 = 9 3. a) A number divided by five subtract three equals sixteen. Using n to represent the original number, write this sentence as an equation. n 3 = 16 b) Draw a flow chart and backtrack to solve. 9 c) Check your answer by substituting into the original equation. n - 3 = 16 9-3 = 16 19 3 = 16 16 = 16 Balancing Method 1. a) Five more than a number is the same as two times the number minus six. Write as an equation and use the balancing method to solve. n + = 2n - 6, n = 11 b) Check your answer by substituting into the original equation. n + = 2n 6 11 + = 22 6 16 = 16 2. Solve two times a number subtract five is the same as sixteen minus the number. Write an equation to represent this sentence then solve using the balancing method Remember to check your final answer by substituting into the original equation. (7 lines) 2n = 16 n, n = 7 10
Fractions 1. Find the value of n for the following equation by backtracking. 2n = 16 40 2. The difference between four fifths of a number and half a number is 3. a) Write as an equation and change to fractions with a common denominator. 4n n = 3, 8n n = 3 2 10 10 b) Find the value of the number. 10 Using Formula 1. Work out what temperature 18 degrees Celsius is in degrees Fahrenheit using the formula C = (F 32). 9 64.4 0 F 2. Rearranging the formula for density d= m, calculate the mass of 2L of milk if the density of milk is 1.03 kg/l v 2.06 kg Inequalities 1. Draw a line to match the meanings to the correct symbols Meaning Less than Symbol More than < Is less than or equal to > Is more than or equal to 2. Write an inequality and solve for the sum of a number plus is at least 17. n + 17, n 12 3. Solve and draw a number line for 2x 6 < 2. x < 4, 4. Solve -p 3 p -3 11
Extension Activities 1. Which is the best method to use to solve the following equations? a) 2g -6 = g + 3 Balance b) x + 7 = 2 GCI or backtrack c) A number decreased by 20 equals 49. GCI or backtrack d) One third of a number is the same as two times the number subtract fifteen. Balance 2. Form an equation from each statement and solve a) One quarter of a number is 2. n = 2, n = 8 4 b) When 3 is added to 3 times a certain number the result is -3. 3n + 3 = -3, n = -2 c) I think of a number, add 10 and divide by 3. This gives the same as doubling my number. n+10 = 2n, n = 2 3 d) Three times a number is less than twice the number added to 8. 3n < 2n + 8, n < 8 e) The opposite of 3 times a number is at least 48. n 48, n 144 3 3. Solve the following a) 12 y = -2 14 b) d+1 = 3 12 c) y 3 = 6 3y 1.12 d) 3x 13 + x x -2 12
4. The amount of interest charged on a loan can be calculated using the formula: I = PRT P = principal 100 R = Interest rate T = time of loan a) Calculate the time of the loan when I = $26, P = $400 and R = 9.% 6 years. Using the formula C = (F-32) solve for F when C equals 4 9 113 0 F 6. Given that the average 70kg adult male s body contains L of blood, using the formula d = m calculate the mass of blood (m) if the density of blood (d) is 1060 kg/m 3. v Hint: Don t forget to change v to m 3. (1L = 0.001 m 3 ).3kg 13