Unit 4 Patterns and Algebra

Transcription

1 Unit 4 Patterns and Algebra In this unit, students will solve equations with integer coefficients using a variety of methods, and apply their reasoning skills to find mistakes in solutions of these equations. They will also investigate patterns and their equivalent linear relations, using tables of values, graphs, and formulas. Students will represent patterns and linear relations using a variety of tools and strategies and make connections between representations. They will use and improve their reasoning skills while analyzing patterns, formulas, and graphs. Materials In many lessons you will need a pre-drawn grid on the board. If such a grid is not already available, you can photocopy BLM 1-cm Grid Paper (p U-27) onto a transparency and project it onto the board. This will allow you to draw and erase points and lines on the grid without erasing the grid itself. Meeting Your Curriculum Teachers following the Ontario curriculum are not required to cover expressions of the form x b a +, so certain questions in Lessons PA8-16 through PA8-18 in the Workbook will be optional for Ontario students. Lesson PA8-23 is optional for students following the WNCP curriculum. Lesson PA8-31 takes the concepts learned in Grade 6 to a Grade 8 level, using expressions with more than one variable and integers. It prepares students for the work in PA8-32, so we recommend that both Ontario and WNCP teachers teach it. Teacher s Guide for Workbook 8.2 P-1

2 PA8-16 Substituting Integers for Variables Pages Curriculum Expectations Ontario: 8m2, 8m5, 8m6, 8m7, 8m62 WNCP: 8PR2, [C, CN, R] Vocabulary substitution variable integer evaluate Goals Students will substitute integers for variables in expressions that include one or more variables. PRIOR KNOWLEDGE REQUIRED Can substitute whole numbers for variables in an expression Can add, subtract, multiply, and divide integers Is familiar with variables Review basic operations with integers. You can use Workbook page 107 Questions 1 and 2 as a diagnostic test. Substituting integers for variables. Begin with expressions that require only one operation. EXAMPLES: a) 3x, x = 2 b) v 2, v = 7 c) w + 3, w = 11 Then continue with expressions that require two operations. EXAMPLES: a) 3t + 2, t = 4 b) 8m 1, m = 3 c) 5z + 1, z = 12 EXTRA PRACTICE: a) 9x, x = 2 b) r + 6, r = 3 c) w 5, w = 8 d) 4u 2, u = 56 e) 5w + 1, w = 2 f) 7r + 2, r = 2 g) 4v + 2, v = 56 h) 2 + 7m, m = 2 i) 2x + 7, x = 2 Process ExpectatioN Representing, Connecting Looking for equivalent answers. Tell students that among extra practice questions d) through i) there are two questions with the same answer, and ask students to find them. (f and h) Can students explain why the answers are the same? (7r + 2, r = 2 and 2 + 7m, m = 2 both give the answer 12 because the expressions have the same meaning and we substitute the same number into each one) Substituting integers for two variables. Have students substitute integers into expressions with two variables. Examples: Find the value of each expression for x = 3 and y = 2. a) 5x + 4y ( 23) b) 6x 2y ( 14) c) 5x 6y ( 3) Bonus 7xy (42) Evaluating expressions with negative coefficients. Have students evaluate 3a and 3a for positive and negative values of a by completing the chart below. What do students notice? (3a and 3a are opposite integers: 3a = a + a + a and 3a = 0 a a a) P-2 Teacher s Guide for Workbook 8.2

3 a a 3a a + a + a 0 a a a Process assessment 8m2, 8m7, [R, C] Workbook Question 9 Have students evaluate expressions with integer coefficients, first substituting positive numbers only, and then substituting both positive and negative numbers. Start with expressions with one operation, and continue to expressions with two operations. Finish with expressions with two variables. a) ( 5)x, x = 2 b) ( 6) r, r = 3 c) w ( 5), w = 8 d) ( 4)u 5, u = 6 e) ( 5)w + 1, w = 2 f) ( 7)r + ( 2), r = 3 g) ( 4)(v + 2), v = 5 h) 2 + ( 7) m, m = 6 i) ( 2) (x + 7), x = 12 j) ( 7)(8 n), n = 9 k) 2 (( 7) p), p = 4 l) ( 2) 6y, y = 11 m) ( 7)m 8n, m = 2, n = 3 n) y (( 4) p), y = 4, p = 2 o) ( 2)x ( 3)y, x = 5 y = 4 p) ( 2)(q n), q = 3, n = 9 q) ( 3)u p, u = 8, p = 2 r) ( 2)a + 6b, a = 6, b = 3 NOTE: Workbook Question 8 parts p) through u) are optional for Ontario students. Review the fractional notation for division before assigning the questions below. Students can convert fractions into division statements while evaluating the equations. Example: a) x, x = 12 (Solution: x 12 = = 12 3 = 4 ) b) d, d = 30 c) 5 e) z ( 6), z = 20 f) 5 b, b = 15 d) c + ( 5), c = t t + ( 6), 3 = 45 Patterns and Algebra 8-16 P-3

4 PA8-17 Solving Equations Guess and Check Page 109 Curriculum Expectations Ontario: 8m1, 8m7, 8m62, 8m64 WNCP: 8PR1, 8PR2, [C, CN, R] Vocabulary substitution variable integer evaluate solving for variable Process ExpectatioN Organizing data Goals Students will solve equations with integer coefficients by guessing integer values for x, checking by substitution, and then revising their answer. PRIOR KNOWLEDGE REQUIRED Can substitute integers for variables in an expression Can add, subtract, multiply, and divide integers Is familiar with variables NOTE: If you are following the WNCP curriculum, review using fractional notation for division before assigning Question 2 and part d) of Question 4. If you are following the Ontario curriculum, these questions are optional for your students. Use a chart to solve equations. Have students copy and complete the chart below in their notebooks. n ( 5)n ( 5)n Then have students use the chart to solve for n: a) 3 + ( 5)n = 23 b) 3 + ( 5)n = 7 c) 3 + ( 5)n = 17 d) 3 + ( 5)n = 8 Repeat with these charts and equations: x x 2 7 ( 4) (x 2) 28 a) ( 4) (x 2) = 20 b) ( 4) (x 2) = 8 c) ( 4) (x 2) = 12 d) ( 4) (x 2) = 8 a) a 6 a a ( 6 7) 12 a ( 6 7) = 6 b) a ( 6 7) = 8 c) a ( 6 7) = 10 d) a ( 6 7) = 3 P-4 Teacher s Guide for Workbook 8.2

5 Bonus Fill in the chart to solve the equation ( 3)a + 5 = 2a + ( 10). a ( 3)a 18 ( 3)a a 12 2a + ( 10) 22 Process ExpectatioN Guessing, checking and revising Introduce the guess and check method to solve equations. Show the equation ( 7)h 2 = 51. Tell students that you are going to solve this equation by guessing and checking. Start by guessing h = 5. ASK: If h = 5, what is ( 7)h 2? ( 37) Then guess h = 6. ASK: If h = 6, what is ( 7) h 2? ( 44) What does this tell me? Which number is closer to 51: 37 or 44? ( 44) Should my next guess be smaller than 5 or larger than 6? (larger than 6) What would your next guess be? Continue in this way until students find that h = 7. Repeat with equation ( 2)(t 3) = 14. Start by guessing t = 3 and then t = 4. Make sure students realize that the next guess should be smaller than 3. Answer: t = 4. Repeat with equation ( 7)(t 23) = 28. Start by guessing t = 0 (which produces ( 7)(t 23) = 161) and then t = 1 (which produces ( 7) (t 23) = 154). ASK: Should my next guess be larger than 1 or smaller than 0? (larger than 1) Does it make sense to guess 2? (no) Why not? (The change from 161 to 154 is less than 10, and we need a change of about 120, so adding 1 to the possible answer will not change the answer enough.) Next, guess 10 (which produces ( 7)(t 23) = 91). ASK: Should the next guess be a lot larger than 10 or a little larger than 10? (a lot larger) Guess 20, which produces ( 7)(t 23) = 21. Should the next guess be larger than 20 or smaller than 20? (smaller) A lot smaller than 20 or a little smaller than 20? (a little) Have students continue guessing and checking until they find the answer, t = 19. Compare the two methods of solving equations. ASK: Which method takes less work? Which method is quicker? (the guess and check method is quicker) Have students practise solving equations by guessing and checking, always explaining the choice for each next guess. a) ( 3) (x 4) = 27 b) ( 4)y 2 = 14 c) 2 ( 4) z = 18 a d) 7 5u = 8 e) ( 5) (3 v) = 25 f) = 14 w p g) 2 = 17 h) 7 = ANSWERs: a) 5 b) 3 c) 4 d) 3 e) 8 f) 108 g) 45 h) 32 Patterns and Algebra 8-17 P-5

6 PA8-18 Solving Equations Page 110 Curriculum Expectations Ontario: 8m3, 8m6, 8m7, 8m62, 8m64 WNCP: 8PR2, [C, CN, R] Vocabulary substitution variable integer evaluate solving for a variable Goals Students will solve equations with integer coefficients using preservation of equality. PRIOR KNOWLEDGE REQUIRED Can substitute integers for variables in an expression Can add, subtract, multiply, and divide integers Is familiar with variables Can solve equations with whole number coefficients using preservation of equality NOTE: If you are following the WNCP curriculum, students need to be familiar with fractional notation for division to do Workbook Questions 1g), 3i) and j), and 6e) and f). These questions are optional for students in Ontario. Review solving equations with whole number coefficients using preservation of equality. Do the first several questions below as a class, and have students do a few more similar questions independently. a) 4x + 2 = 14 b) 6a 9 = 21 c) 10(x + 4) = 180 d) y = 105 e) 3b 4 = 14 f) 11c 9 = 57 g) d = 102 h) 2(m 4) = 18 ANSWERS: a) 3 b) 5 c) 14 d) 3 e) 6 f) 6 g) 6 h) 13 Solving equations with whole number coefficients and decimal answers. Present the next set of problems and have students solve them. Tell students that the answers will be fractions or decimals. a) 3x + 2 = 15 b) 6a 7 = 26 c) 10(x 4) = 18 d) 5 + 4y = 10 e) 3b 4 = 10 f) 10c 9 = 52 g) d = 19 h) 2(m + 4) = 27 ANSWERS: a) 13/3 b) 5.5 c) 5.8 d) 1.25 e) 14/3 f) 6.1 g) 1.4 h) 9.5 Have students check their answers by substitution. Process ExpectatioN Reflecting on what made the problem easy or hard Process ExpectatioN Reflecting on other ways to solve a problem Ask students to try to solve the equation 3x 4 = 15 using guess and check. Give students a few minutes to start solving the equation, then ASK: Are you finding it easy to solve this equation using guess and check? Are you having any problems? (We usually guess whole numbers, and 3x 4 equals 14 for x = 6 and 17 for x = 7. So the answer is between 6 and 7. Even if we try to guess 6 1, which is the correct answer, 3 substituting this number is hard work.) Then ask students to solve the same equation using preservation of equality. Which method is more convenient? Why? (preservation of equality because it let s us work with whole numbers to start fractions don t appear until the end and it s faster) P-6 Teacher s Guide for Workbook 8.2

7 Undoing operations with integers. Ensure that students are comfortable with undoing operations when integers are involved. Again, solve several questions together as a class, then have students solve equations independently. 1. Write the number that makes each equation true. a) a + ( 4) = a b) a ( 3) = a c) a ( 2) = a d) a ( 5) + = a a e) 2 = a f) ( 6) + a = a g) 6a = a h) ( 7)a = a i) ( 12) a = a 2. Write the operation and number that make each equation true. a) x + ( 2) = x b) y ( 2) = y c) ( 2) + z = z d) ( 2)u = u e) v ( 2) = v f) w ( 2) = w g) 2a = a h) m 2 i) ( 2) n = n = m Process assessment 8m6, [CN] Ask students to look at the equations in question 2 above and identify those that have the same answers. (b and h; a and c; d, e, g, i) Why do these questions have the same answers? (because the expressions on the left side of the equations in each group mean the same thing, so you need to perform the same operation to get back to the variable) Then have students identify the expressions in each of these groups that are always equal, for any number p: a) 5p p + 5 ( 5)p p ( 5) p 5 b) 7 + p p 7 p + ( 7) 7p 7 p p 3 c) p ( 3) p 3 ( 3) p p 3 p d) 8p 8 p ( 8) ( 8) ( 8 8) 8p 8 p 8 ( 8) Solving equations with one operation and integer coefficients. Examples: a) x + ( 2) = 12 b) y ( 2) = 34 c) ( 4) + z = 14 d) ( 3)u = 15 e) v ( 4) = 12 f) w ( 5) = 10 g) 6a = 216 h) m 2 = 35 i) ( 7) n = 35 j) x + ( 3) = 11 k) y ( 3) = 13 l) r ( 4) = 67 ANSWERS: a) 14 b) 68 c) 18 d) 5 e) 3 f) 5 g) 36 h) 70 i) 5 j) 8 k) 39 l) 71 3 p Patterns and Algebra 8-18 P-7

8 Have students check their answers by substitution. Solving equations with two operations and integer coefficients. Examples: a) 3x + ( 5) = 13 b) ( 4)y ( 2) = 34 c) ( 4) + 9z = 14 d) 12 + ( 3)u = 15 e) v ( 4) 6 = 14 f) 11w ( 5) = 115 g) 12 6a = 216 h) m + ( 2 5) = 35 i) ( 7) n 16 = 33 j) ( 2)x + ( 3) = 11 k) y ( 3) ( 1) = 11 l) 2 r ( 4) = 66 ANSWERs: a) 6 b) 8 c) 2 d) 9 e) 5 f) 10 g) 34 h) 80 i) 7 j) 4 k) 36 l) 31 Have students check their answers by substitution. P-8 Teacher s Guide for Workbook 8.2

9 PA8-19 Concepts in Equations Page 111 Curriculum Expectations Ontario: 8m1, 8m3, 8m6, 8m7, 8m62, 8m64 WNCP: 8PR2, [C, CN, R, V] Goals Students will model equations with whole number coefficients and decimal or fractional answers. They will identify mistakes in solutions of equations with integer coefficients. PRIOR KNOWLEDGE REQUIRED Vocabulary substitution variable integer evaluate solving for a variable coefficient Can add, subtract, multiply, and divide integers Is familiar with variables Can substitute integers for variables in an expression Can solve equations with integer coefficients using preservation of equality and record the solution Can model equations with whole number coefficients Review modelling equations with whole number coefficients and solving equations given by models. Using the same notation as in the Workbook (triangle = variable, circle = 1), the expression 3a = 2 can be represented by three triangles on one scale balanced by two circles on the other. Have students draw the scales representing the following equations: a) 4x + 2 = 10 b) 6a + 3 = 9 c) x = 14 d) 1 + 3y = 10 Remind students of the steps to solve equations, below, and have students use Steps 2 and 3 to solve the equations they modelled. Step 1: Write the equation that represents the model. Step 2: Remove all circles from the side that has the triangle(s) and remove the same number of circles from the other side. Write the new equation. Step 3: Divide the circles into the number of groups given by the number of triangles. Keep only one group of circles and one triangle. Write the new equation. This will be the solution! Process ExpectatioN Modelling Solving equations that have fractional solutions. Present the model at left, and have students write the equation. SAY: To solve this equation, you would divide the circles into the number of groups given by the number of triangles, so you would need to divide the circles into three groups. ASK: How is this equation different from the previous ones? (you cannot evenly divide the circles into the number of groups given by the triangles) Have students solve the equation using preservation of equality. ASK: How is the answer different from the answers to the previous equations? (it s a fraction) How could we draw the solution using the model? (draw a triangle on one side of the balance and one and one third of a circle on the other side) Patterns and Algebra 8-19 P-9

10 Have students solve the equations with fractional solutions given by balance models. Examples: a) b) Review and use the distributive law. Remind students of the distributive law and how we can use it to rewrite expressions with variables. For example, we can rewrite the expression ( 2) (x + 5) as ( 2)x + ( 2) 5, which is equal to 2x 10. Have students rewrite the following expressions using the distributive law: a) ( 6) (x + 5) b) ( 3) (x 4) c) (x + 1) ( 3) d) 2(x + ( 8)) ANSWERs: a) 6x + ( 30) = 6x 30 b) ( 3) x ( 12) = 3x + 12 c) 3x + ( 3) = 3x 3 d) 2x + ( 16) = 2x 16 Then have students rewrite the constant term in several expressions as a product of the coefficient of the variable and another number. Example: ( 2)a + 14 = ( 2)a + ( 2)( 7) a) ( 2)x + 6 b) ( 3) x 24 c) 3x 6 d) 2x + ( 8) e) ( 3)x + ( 15) f) ( 4)x 4 g) 5x + ( 35) h) 2x ( 10) ANSWERs: a) ( 2)x + ( 2)( 3) b) ( 3) x ( 3)( 8) c) 3x 3 2 d) 2x + 2 ( 4) or 2x + 2( 4) e) ( 3)x + ( 3)( 5) f) ( 4)x ( 4)( 1) g) 5x + 5( 7) h) 2x 2( 5) Next, have students use the distributive law to rewrite the expressions above with brackets. Example: ( 2)a + ( 2)( 7) = ( 2)(a + ( 7)). Students can also rewrite this expression as ( 2)(a 7). ANSWERs: a) ( 2)(x + ( 3)) = ( 2)(x 3) b) ( 3)(x ( 8)) = ( 3)(x + 8) c) 3(x 2) d) 2(x + ( 4)) e) ( 3)(x + ( 5)) f) ( 4)(x ( 1)) g) 5(x + ( 7)) h) 2(x ( 5)) Have students practise rewriting expressions using the distributive law: a) ( 4)x + 12 b) ( 5) x 25 c) 7x 63 d) 4x + ( 8) e) ( 5)x + ( 15) f) ( 8)x 48 g) 15x + ( 135) h) 5x ( 10) ANSWERs: a) ( 4)(x 3) b) ( 5)(x + 5) c) 7(x 9) d) 4(x 2) e) ( 5)(x + 3) f) ( 8)(x + 6) g) 15(x 9) h) 5(x + 2) P-10 Teacher s Guide for Workbook 8.2

11 A shortcut for solving equations. Remind students that sometimes people don t write out every step when they solve equations. For example, the solution 3x + ( 12) = 33 ( 3)(x + 4) = 33 ( 3)(x + 4) ( 3) = 33 ( 3) x + 4 = 11 x = 11 4 x = 15 Rewrite the left side using the distributive law Divide both sides by 3 Rewrite both sides Subtract 4 from both sides Rewrite both sides can be shortened to 3x + ( 12) = 33 3(x + 4) = 33 x + 4 = 11 x = 15 Rewrite the left side Divide both sides by 3 Subtract 4 from both sides Instead of rewriting both sides, we can do that step mentally. Put the following solution on the board and ASK: How would you rewrite this solution? 3x + ( 12) = 33 3x + ( 12) ( 12) = 33 ( 12) Subtract 12 from both sides 3x = 21 Rewrite both sides 3x 3 = 21 3 Divide both sides by 3 x = 7 Rewrite both sides Answer: 3x + ( 12) = 33 3x = 21 Subtract 12 from both sides x = 7 Divide both sides by 3 Emphasize that if students can subtract 12 from both sides and rewrite them mentally, then students can skip writing that step. However, they might be more likely to make mistakes when they don t write down a step, so they have to be more careful. Have students write the missing steps in each of these solutions: a) 3x + 4 = 23 b) 2(x + 5) = 24 c) ( 4)x 3 = 25 d) 5x 3 = 18 3x = 27 x + 5 = 12 ( 4)x = 28 5x = 15 x = 9 x = 17 x = 7 x = 3 Sample answer: a) 3x + 4 = 23 3x = 23 4 Subtract 5 from both sides 3x = 27 Rewrite both sides 3x 3 = 27 3 Divide both sides by 3 x = 9 Rewrite both sides Tell students that, when solving problems, they will never be expected to skip steps in fact, they should always write out all steps so that you can Patterns and Algebra 8-19 P-11

12 check their understanding. Even if they tend to skip steps with equations involving whole numbers, integers are trickier, and writing out all the steps helps to avoid mistakes. Also, students need to know how to read solutions that have skipped steps. Identifying mistakes by using substitution. Tell students that you saw two different solutions to the same equation. Write these on the board. Solution 1 Solution 2 3x + 6 = 18 3x + 6 = 18 3x = (x + 2) = 18 3x = 12 3(x + 2) ( 3) = 18 ( 3) 3x ( 3) = 12 ( 3) x + 2 = 6 x = 4 x = 6 2 x = 8 Process ExpectatioN Reflecting on the reasonableness of an answer ASK: These students are both trying to find the x that satisfies 3x + 6 = 18, so they should get the same answer. Did they? (no) How can we decide who is right? (we could check each step of both solutions) Is there a way to know if the answer is right before checking so that we can know ahead of time that we are looking for a mistake? (yes, substitute the answer into the original expression and check if you get 18) Have students substitute both of the answers into the equation to determine which answer is correct. ( 3( 4) + 6 = = 18, but 3( 8) + 6 = = 30, so x = 4 is correct and x = 8 is incorrect) Have students individually describe each step of Solution 1. To find the mistake in Solution 2, have students substitute the right answer into every step of the solution. They will see that the second line is the first line where the correct answer does not work. ( 3( 4 + 2) = 6 18) Process ExpectatioN Reflecting on other ways to solve the problem Discuss with students what other methods could be used to find the mistake (e.g., perform each step of the solution correctly and compare it to the given solution until you find the discrepancy). Then present the following incorrect solution: Process ExpectatioN Working backwards Solution 3 3x + 6 = 18 3(x 2) = 18 3(x 2) ( 3) = 18 ( 3) x 2 = 6 x = x = 8 Suggest that students work backwards. Substituting 8 into the second-last line gives = 8 and = 8, so the mistake is above that. Continue to the line before it: 8 2 = 6, and not 6. ASK: What does this mean? (the mistake was made between this line and the next one) Have students correct the mistake. (x = 6 + 2) Have students solve the equation correctly from that point. Point out that when a mistake was made close to the beginning of the solution and you know the right answer, substituting the right answer from the beginning is the quicker way to find P-12 Teacher s Guide for Workbook 8.2

13 the mistake. If the mistake was made close to the end and you do not know what the right answer is, working backwards will give you the answer faster. Now show students these solutions to the same problem. Solution 4 Solution 5 3x + 6 = 18 3x + 6 = 18 3(x 2) = 18 3(x + 6) = 18 3(x 2) ( 3) = 18 ( 3) 3(x + 6) ( 3) = 18 ( 3) x 2 = 6 x + 6 = 6 x = x = 6 6 x = 4 x = 12 Solution 6 Solution 7 3x + 6 = 18 3x + 6 = 18 3x = (x 2) = 18 3x = 12 x 2 = 18 3x ( 3) = 12 3 x = x = 4 x = 20 ASK: Which solution is correct? (Solution 4) How do you know? (it gets the right answer) Have students go through each incorrect solution line by line to find the mistake. ANSWERS: Solution 5: wrote 3x + 6 as 3(x + 6) this is not true since 3(x + 6) = 3x + ( 18), not 3x + 6 Solution 6: divided the left side by 3 but the right side by 3, so the two sides are no longer equal Solution 7: if 3 (x 2) is 18, then x 2 isn t equal to 18 forgot to divide 18 by 3 Have students identify and correct any mistakes below. If the solution is correct, they should write correct. If no correct solution is given for one of the equations (and indeed there is none for 5(x + 2) = 30), students should also produce a correct solution. a) 5(x + 2) = 30 5x + 2 = 30 5x = x = 32 5x 5 = 32 5 x = 6.4 c) 5(x + 2) = 30 5(x + 2) 5 = 30 5 x + 2 = 6 x = 6 2 x = 4 b) 5(x + 2) = 30 5(x + 2) 5 = 30 x + 2 = = 30 2 x = 32 d) 2x + ( 8) = 22 2x = x = 30 2x 2 = 30 2 x = 15 Patterns and Algebra 8-19 P-13

14 Process assessment 8m3, [R] Workbook Question 3 e) 2x + ( 8) = 22 2(x 4) = 22 x 4 = 11 x = x = 7 g) 3x + 9 = 27 3(x + 3) = 27 x + 3 = 9 x = 6 x = 6 i) 3x + 9 = 27 3(x + ( 3)) = 27 x + ( 3) = 9 x + ( 3) ( 3) = 12 x = 12 f) 2x + ( 8) = 22 2(x + ( 4)) = 22 x + ( 4) = 11 x + ( 4) ( 4) = 11 x = 11 h) 3x + 9 = 27 3x = x = 18 3x ( 3) = 18 ( 3) x = 6 ANSWER: Only e) and i) are correct, and the right answer for 5(x + 2) = 30 is x = 8. Solving problems, checking the answer using substitution, and identifying and correcting their own mistakes. Have students use the distributive law (see Method 2 from Workbook page 111 Question 2) to solve these equations and say what they do at each step. Students should check their answer by substituting it into the original expression. If incorrect, have students find their own mistake! a) 3x + ( 9) = 15 b) 4x + 20 = 36 c) 2x + 8 = 20 d) ( 3)x + 12 = 27 e) 2x + 4 = 12 f) 3x 6 = 15 g) 5x + ( 10) = 25 h) 5x ( 5) = 20 ANSWERS: a) x = 8 b) x = 4 c) x = 14 d) x = 5 e) x = 8 f) x = 7 g) x = 3 h) x = 3 Word problems practice: Remind the students how to find the mean of a set of numbers before assigning this question. a) Find the average temperature in Yellowknife, NWT, over a week in October: 1 C, 2 C, 1 C, 0 C, 2 C, 2 C, 3 C (Answer: 1 C) b) Find the average temperature in Labrador City, NL, over the same week in October: 1 C, 0 C, 1 C, 0 C, 0 C, 1 C, 0 C (Answer: 1/7 C 0.14 C) c) Which city was colder on average during that week? (Yellowknife was colder) P-14 Teacher s Guide for Workbook 8.2

15 PA8-20 Formulas Pages Curriculum Expectations Ontario: 7m60, 7m61, 7m62; 8m1, 8m7, 8m56, 8m58, 8m60 WNCP: 6PR1, 6PR2, 6PR3; 7PR1, 7PR2; 8PR1, [C, R, PS] Vocabulary T-table formula equation variable coefficient input output substitute Goals Students will create tables of values for linear relations, produce formulas such as n + a or a n for patterns and tables of values, and predict terms of patterns using the formulas they produced. PRIOR KNOWLEDGE REQUIRED Can create and extend a T-table for a pattern Is familiar with variables Can extend a linear increasing sequence Can translate a statement into an algebraic expression How formulas help with patterns. Draw a simple geometric design, like the one in the margin. Ask: How many pentagons did I use? How many triangles? How many triangles and how many pentagons will I need for two such designs? For three designs? Remind your students that in earlier grades they used T-tables to solve this type of question (see also PA8-3). Ask students to draw a T-table and to fill it in for five designs. ASK: I want to make 20 such designs. Should I continue the table to check how many pentagons and triangles I need? Can you think of a more efficient way to find the number of pentagons and triangles? How many triangles are needed for each pentagon in the design? (5) What do you do to the number of pentagons to find the number of triangles in any number of designs? (multiply by 5) Remind students that mathematicians often use letters instead of numbers to represent a changing quantity. These letters are called variables, because the quantity they represent can vary (change). The quantity that does not change (the number the variable is multiplied by) is called a coefficient. For example, mathematicians could use p for the number of pentagons and t for the number of triangles. We have a verbal rule for the number of triangles used in a design: Multiply the number of pentagons by 5 to get the number of triangles. What algebraic equation does this rule produce? (5 p = t or 5p = t) Explain that an equation that shows how to calculate one quantity from another is called a formula. Write the term on the board beside the formula itself. ASK: In which other contexts have we seen and used formulas? (area, volume) For example, a formula for the area of a circle (A = πr 2 ) shows us how to find the area of a circle from its radius. Both the area and the radius are represented by variables. Producing a table of values for a formula. Write another formula, such as 8 s = t. Explain that s represents the number of squares and t is the number of triangles in a pattern, as before. What rule does the formula express? (The number of triangles is 8 times the number of squares, or multiply the number of squares by 8 to get the number of triangles.) Remind students how to make a table of values that matches the formula Patterns and Algebra 8-20 P-15

16 by substituting actual numbers of squares for s and multiplying. The result of the multiplication is the number of triangles. Draw the table of values: # of Squares (s) Formula (8 s = t) # of Triangles (t) = = = Ask students to copy the table and to fill in the missing numbers. Then have them add two more rows to the table. ASK: Do we need to extend the table to find how many triangles will be needed for 25 squares? (no) How will you find how many triangles are needed for 25 squares? (substitute 25 for s) Ask students to find the number of triangles for 25 squares (200). Let your students practise drawing tables for more formulas, such as 3 t = s, 6 s = t (t = number of triangles, s = number of squares) Producing a formula of the type t = a s for a T-table. Tell students that the T-tables below were created using a formula of the same type as above. Now, instead of creating the table for a formula, students will do the opposite: produce a formula for the table. First identify the type of formula. ASK: How are all the formulas for the tables below the same? (all formulas are of the sort number s = t ) How could you find the coefficient for each formula from the table? (e.g., look at the number of triangles in the row with s = 1 or divide the number of triangles in any row by the number of squares) Point out to students that it is essential to check that the formula they produced works for all rows of the table. Include several more challenging examples, such as the last two below. Squares (s) Triangles (t) Squares (s) Triangles (t) Squares (s) Triangles (t) Squares (s) Triangles (t) Squares (s) Triangles (t) Squares (s) Triangles (t) Students can practise creating T-tables for multiplicative rules (rules that involve only multiplication) and finding formulas for them using the Activity below. Formulas for patterns of the type n + a. Start with a simple problem: Rose invites some friends to a party. She needs one chair for each friend and one for herself. Can you give Rose a formula or equation for the number of chairs she will need? Ask your students to suggest a variable for the number of friends and a variable for the number of chairs. Given the number of friends, how do you find the number of chairs? Ask students to write a formula for the number of P-16 Teacher s Guide for Workbook 8.2

17 chairs. Suggest that students make a T-table similar to the one they used for multiplicative rules. They should start at 1 and fill the table in for a few rows. Examples: a) r = 3 + t; t = 1, 2, 3 b) r = t 5, t = 10, 11, 12 c) t = r 6, r = 12, 15, 18 Have students practise producing T-tables for formulas using addition or subtraction. Tell students that the following tables were made by adding or subtracting a number from the variable, and have students find the formulas that show how to get the second number (t) from the first number (s). Examples: a) s t b) s t c) s t d) s t e) s t f) s t g) s t h) s t ANSWERS: a) t = s + 5 b) t = s 1 c) t = s + 11 d) t = s + 9 e) t = s 6 f) t = s 4 g) t = s + 4 h) t = s 6 Tables with input and output. Explain to students that the number that you put into a formula in place of a letter is often called the input. The result that the formula provides the number of chairs, for instance is called the output. Write these terms on the board and ask volunteers to circle the input and underline the output in the formulas you have written on the board. Example: r = 3 + t Draw several T-tables with headings Input and Output on the board, provide a rule for each and the input numbers, and ask your students to find the output numbers. Start with simple inputs like 1, 2, 3 or 5, 6, 7 and continue to more complicated combinations like 6, 10, 14. Provide all types of rules: additive (add 4 to the input), multiplicative (multiply the input by 5), and subtractive (subtract 3 from the input). Suggest that students try a more complicated task: produce a rule and a formula for a given table. Ask students to think about what was done to the input to get the output. Remind them to check that the formula works for all rows. For example, for the first table below, if you look only at the first row, the formula could be Output = Input 5 or Output = Input + 4, so you need to check the other rows. Give students several simple tables to work with. Examples: Input Output Input Output Input Output Input Output Output = Input + 4 Output = Input 2 Output = Input 3 Output = Input 3 Patterns and Algebra 8-20 P-17

18 Activity Process ExpectatioN Looking for a pattern Subtract Add Multiply Students work in pairs. Each student decides on a formula (such as s = 3 t) and makes a T-table of values for it, with three rows. Students exchange their T-tables. They have to find the formula the T-table was made with and then check each other s answers. They can also try to produce a design that will go with the formula they found. Variation: Use a spinner as shown and a die to randomize the formulas students produce. Students spin the spinner and roll the die. They should write a formula for the rule given by the spinner and the die. For example, if a student spins Multiply and rolls 3, the rule is Multiply the input by 3 and the formula is 3 Input = Output. Extension Tell students that a family is having a party. The formula for the number of chairs they will need for the party is g + 4 = c. ASK: If g is the number of guests and c is the total number of chairs needed, how many people are in the family? (4) Point out that any change in the number of guests produces a change in the total number of chairs needed. For example, if there are two guests, g = 2 and the family will need 6 chairs; if there are three guests, g = 3 and the family will need 7 chairs; and so on. The number of family members is always 4, and it does not change. Process ExpectatioN Organizing data Next, show a different formula for the number of chairs: g + f 1 = c. Say that f represents the number of family members, and 1 represents a baby in the family who does not need a chair. This time, the number of family members can change, too. What other quantities can change? (the number of guests, the number of chairs) If the family has 10 chairs, how many guests and how many family members could be at this party? (There are different solutions to this problem. Students should find them systematically.) Have students write a formula that shows how to find the number of chairs if, among the guests and the family, there are b babies that do not need chairs. (Answer: c = f + g b) P-18 Teacher s Guide for Workbook 8.2

19 PA8-21 Formulas for Patterns Pages Curriculum Expectations Ontario: 7m60, 7m62; 8m1, 8m7, 8m56, 8m58, 8m63 WNCP: 6PR3; 7PR1; 8PR2, [C, R, V] Vocabulary T-table variable input output term number direct variation Goals Students will identify sequences (presented numerically or geometrically) that vary directly with the term number and find formulas for such sequences. PRIOR KNOWLEDGE REQUIRED Can create and extend a T-table for a pattern Is familiar with variables Can identify increasing and decreasing sequences Can find the gaps in a sequence Review finding formulas of the type t = an for tables and patterns. Show students several sequences made of blocks with a multiplicative rule, such as: ASK: How do we obtain each new figure from the previous one? (by adding two squares and four triangles) Which rule is this rule similar to, Start at 2 and add 2 each time or Multiply the term number by 2. (Start at 2 and add 2 each time) Can you describe how to get a next term in a different way, using only the first figure? (repeat the first figure n times to get the nth figure: 1 for the first term, 2 for the second term, etc.) Point out that the first figure is repeated the number of times that is equal to the figure number. Have students draw T-tables for the number of triangles and the number of squares, and fill in the numbers for 3 figures in the sequence. Figure Number (f) Number of Squares (s) Figure Number (f) Number of Triangles (t) Number of Squares (s) Number of Triangles (t) Ask your students to write a formula for each table. (If necessary, prompt students to use the general rule they figured out for the whole pattern: the first figure is repeated term number times. There are 2 squares in the first figure, so the nth figure there will have 2n squares.) ANSWERs: s = 2 f, t = 4 f, t = 2 s. Direct variation. Remind students of the meaning of the terms input and output. Explain to students that when the rule is Multiply the input by, we say that the output varies directly with the input. So in this pattern the Number of Squares varies directly with the Figure Number, and the Number of Triangles varies directly with both the Figure Number and the Number of Squares. Patterns and Algebra 8-21 P-19

20 Present several tables and have students say whether the output varies directly with the input or not. Examples: Input Output Input Output Input Output Input Output Bonus Input Output Draw a sequence of squares with side lengths 1, 2, 3, etc. Ask students to find the areas and the perimeters of the squares. Ask them to make a T-table for both and to check which quantity varies directly with the side length. (perimeter) Ask students to write a formula for the perimeter and for the area of the square. Bonus a) The number of feet, f, varies directly with the number of people, p. (2 people, 4 feet; 3 people, 6 feet; 4 people, 8 feet; f = 2 p). Does the number of paws vary directly with the number of cats? What is the formula? (c = 4p) b) A cat has five claws on each front paw and four claws on each back paw. Make a T-table showing the number of cats and the number of claws and another T-table showing the number of paws (add one paw at a time in the same order, e.g., right front, left front, right hind, left hind then go to the next cat right front, left front, and so on) and the number of claws. Does the total number of claws vary directly with the number of paws or with the number of cats? (The total number of claws varies directly with the number of cats (claws = 18 cats), but not with the number of paws.) Formulas that do not vary directly with the figure number. Draw the sequence of figures and shade the growing towers of blocks in each figure, as shown. Figure 1 Figure 2 Figure 3 Ask students to draw two T-tables, one for the figure number and the number of shaded blocks, and the other for the figure number and the total number of blocks. ASK: In which T-table does the number of blocks P-20 Teacher s Guide for Workbook 8.2

21 in the output column vary directly with the figure number: the T-table that shows the number of shaded blocks or the one that shows the total number of blocks? Have students produce a formula for the number of shaded blocks and explain their reasoning. Then ASK: Does the number of unshaded blocks change from figure to figure? To get the total number of blocks, what do you have to add to the number of shaded blocks? (the number of unshaded blocks) Have students write the formula for the total number of blocks. Then have students practise finding formulas for the number of shaded blocks and the total number of blocks using Question 4 on Workbook page 116. Process assessment 8m1, 8m7, [R, C] Have students compare the formulas they produced. How are the formulas (and rules) that show direct variation different from the formulas that do not show direct variation? (the formulas for direct variation are all of the type Multiply the figure number by a number, the formulas that do not show direct variation involve addition as well) Then ask students whether the formulas below show direct variation. Have students signal yes and no to assess the whole class at a glance. Then ask students to pick a formula that does not show direct variation and explain how they know it does not show direct variation. Which part of the formula does show direct variation? Examples: a) 3 Figure Number b) 3 Figure Number 5 c) Figure Number 4 5 d) Figure Number 6 e) Figure Number 7.5 f) Figure Number Word problems practice: A cab charges a flat rate of $4 (you pay this just for using the cab) and $2 for every minute of the ride. Write a formula for the price of a cab ride. How much will you pay for a 4-minute cab ride? For a 5-minute ride? Extensions What does each formula below help you to find? Which of the formulas show direct variation? Which quantity varies directly with which? a) C = 2πr b) Area = width length c) Perimeter = 4 side length Bonus Area = πr 2 ANSWERS: a) Circumference of a circle varies directly with the radius of a circle. b) Area of a rectangle does not vary directly with width or length, because neither of these is a constant. c) Perimeter of a square or a rhombus varies directly with side length. Bonus: Area of a circle varies directly with the square of the radius, but not with the radius itself. Find a sequence of rectangles in which area will vary directly with length. Does the perimeter vary directly with the length? (Answer: Use rectangles of the same width, say 5. The area will be 5l, so it will vary directly with the length. The perimeter in this case will be 2l + 10 and will not vary directly with the length.) Patterns and Algebra 8-21 P-21

22 PA8-22 Formulas for Patterns Advanced Page 117 Curriculum Expectations Ontario: 7m60, 7m62; 8m1, 8m7, 8m56, 8m58, 8m63 WNCP: 6PR3; 7PR1; 8PR2, [C, R, V] Vocabulary variable term number direct variation Goals Students will find formulas for patterns represented geometrically, predict further terms of such patterns, and identify term numbers from term values in patterns. PRIOR KNOWLEDGE REQUIRED Is familiar with variables Can find a simple formula for a pattern (such as n + a, an) Can identify a pattern or part of a formula that shows direct variation Can perform operations with variables using the correct order of operations Can apply the distributive law to expressions with variables Can substitute whole numbers into expressions with variables and evaluate the expressions Can solve equations of the form ax + b = c Review finding formulas for patterns of the type n + a. Present several geometric patterns where a fixed number of blocks is added to the term number of blocks to produce each term, and have students identify the formulas for the patterns. Examples: a) b) ANSWERS: a) n + 3 b) n + 2 Finding formulas for patterns in various ways. Present the following pattern and explain that you would like to find a formula for it, but you want to use a different method from the one you used in the previous lesson. Ask students to look at the unshaded blocks in each figure. What do they notice? (there are 6 in all figures) Does the number of unshaded blocks vary directly with the term number? (no) Ask students to find the formula for the number of shaded blocks in each circled group of shaded blocks. (Figure Number + 1) Then ask them to find the formula for the total number of shaded blocks. (The formula is 4 (Figure Number + 1). PROMPT: How many times does the circled group of shaded blocks repeat in each figure?) Finally, ask students to find the formula for the total number of blocks in the pattern. (4 (Figure Number + 1) + 6) Redraw the pattern on the board but don t shade or circle any of the blocks. Ask students to copy the pattern and to pretend that they are making the P-22 Teacher s Guide for Workbook 8.2

23 figures from cubes. They should build the first figure, then add some blocks to it to make the second figure, then add more blocks to make the third figure from the second, and so on. How many blocks will students need to add each time? (4) Ask them to shade the blocks that were added in the second figure. In the third figure, ask them to shade both the blocks that were added when making the second figure and the blocks added to make the third figure. Have students find the formula for the number of shaded blocks in each figure. If students need a prompt, shade the blocks on the board, so that there are four separate groups of shaded blocks in both second and third figures, and circle one of the groups in each. Figure 1 Figure 2 Figure 3 Have students find the formula for the number of shaded blocks in the circle first. (Figure Number 1 in the circle, so the number of shaded blocks is 4 (Figure Number 1)) ASK: What is the number of unshaded blocks in each figure? (the number of blocks in the first figure, 14) What is the formula for the total number of blocks? (4 (Figure Number 1) + 14) Process ExpectatioN Reflecting on the reasonableness of the answer Process assessment 8m7, [C] Workbook Question 1 Comparing the formulas. Say: We found the number of blocks in two patterns in two different ways, and obtained two different formulas. What should we expect from the formulas we got? (they should be the same) Have students use the distributive law to write both formulas without the brackets, and compare the results. Do both formulas give the same result? Yes, they do: 4 (Figure Number + 1) + 6 = 4 Figure Number = 4 Figure Number (Figure Number 1) + 14 = 4 Figure Number = 4 Figure Number + 10 Have students compare all three formulas: 4 (Figure Number + 1) + 6, 4 (Figure Number 1) + 14, and 4 Figure Number How are the formulas the same? (all have coefficient 4, neither shows direct variation) How are they different? (different expressions are multiplied by 4, the number added to the multiple of 4 is different) Ask students to draw another copy of the pattern and to shade 4 Figure Number blocks in each figure. What is the number of unshaded blocks in each figure? (10) Does this fit the last formula? (yes) Have students look at the part of each formula that represents the number of shaded blocks. How is the number of shaded blocks the same in each formula? (it shows direct variation, with coefficient 4) What does the number of shaded blocks vary directly with in each formula? (Figure Number + 1 in the first, Figure Number 1 in the second, and Figure Number in the third) Point out that the fact that the number of shaded blocks was different and varied directly with different quantities is responsible for the different additive constant (the number of unshaded blocks) in each formula. Emphasize that all three Patterns and Algebra 8-22 P-23

24 ways of finding the number of blocks in each pattern should produce the same formula when it is simplified. Process ExpectatioN Reflecting on the reasonableness of the answer Using the formula to find terms or term numbers. Ask students how they could use the formula to find which figure in the pattern will have 50 blocks.(by using equations) Have students solve the equation 4 Figure Number + 10 = 50. (Figure Number = 10) Then have students predict which figurewill have: 150 blocks (35 th figure), 210 blocks (50 th figure), 1010 blocks (250 th figure). Then ask which figure will have blocks. Have students solve the equation 4 Figure Number + 10 = (Figure Number = ) ASK: What does this mean? (no figure in this pattern can have this number of blocks) Have students check whether there is a figure in this pattern that will have the number of blocks given, and, if yes, which figure it will be: a) 325 (none) b) 326 (79 th figure) c) (none) d) (1 417 th figure) EXTRA PRACTICE: Use the patterns in Questions 2 and 3 on Workbook page 117. In each pattern, is there a term that has 100 shapes? 1000 shapes? shapes? If yes, which term is that? ANSWERS: 100 shapes: 2a) no 2b) 32 nd 3) 47 th shapes: 2a) no 2b) 332 nd 3) 497 th shapes: 2a) th 2b) no 3) no P-24 Teacher s Guide for Workbook 8.2

25 PA8-23 Stepwise Rules Page 118 Curriculum Expectations Ontario: 6m60; 7m60, 7m61, 7m63, 7m68; 8m1, 8m2, 8m3, 8m6, 8m7, 8m60, 8m61, 8m63 WNCP: optional, [C, CN, R] Vocabulary formula expression substitution variable term T-table term number general rule stepwise rule sequence gap coefficient Goals Students will compare stepwise rules, general rules, and formulas for patterns. PRIOR KNOWLEDGE REQUIRED Can create and extend a T-table for a pattern Is familiar with variables Can identify increasing and decreasing sequences Can identify sequences that vary directly with the term number Can find the gaps in a sequence Can produce a sequence using a rule such as Start at 3. Multiply by 2 each time. Can perform basic operations with integers Review using formulas for linear relations. Tell students that you have to pay $30 a month for up to 600 text messages plus $2 for each additional text message. ASK: How much do each of the first 600 text messages cost? ($ = $0.05 = 5 ) If I send 610 text messages this month (i.e., 10 additional text messages), how much will my cellphone bill be? Have students write the expression using the quantities $30, $2, and 10. Answer: $30 + $2 10. ASK: Which of these amounts is most likely to change from month to month? (the 10 additional text messages) What do we use to represent an amount that changes? (a variable) What number in the expression $30 + $2 10 should we replace with a variable? (10) Why? (because that is what changes) Write on the board: monthly cost of cellphone = n, where n is the number of additional text messages above 600 Remind students that an expression of the sort they have just written can be converted into a verbal rule, such as Multiply the number of text messages above 600 by 2 and add 30. ASK: How much would it cost if I sent 625 text messages? ($30 + $2 25 = $30 + $50 = $80) Remind students that replacing a variable in an expression with a number and evaluating it is called substitution. They substituted n = 25 into the formula for the cost, n. Making a sequence from a formula. Remind students that a formula tells you how to calculate the term from the term number. That is, you would substitute 1 into the formula to get the first term, substitute 2 into the formula to get the second term, and so on. Demonstrate with this formula: Term = 2 Term Number + 1 Term Number Term Patterns and Algebra 8-23 P-25

26 Model substituting 1 and 2 into the expression and add the results to the table of values, then have students do 3, 4, and 5 individually. Leave the table on the board (to refer to during the discussion below). Point out that the terms now form a sequence. Have students convert these formulas to sequences: a) 4 Term Number 4 b) 15 2 Term Number c) Multiply the Term Number by 3 and add 5 d) Subtract the Term Number from 12 e) 3n + 1 f) ( 2)n + 7 g) 2n 2 h) 12 4n Bonus Subtract the Term Number multiplied by ( 3) from 21 ANSWERS: a) 0, 4, 8, 12, 16 b) 13, 11, 9, 7, 5 c) 8, 11, 14, 17, 20 d) 11, 10, 9, 8, 7 e) 4, 7, 10, 13, 16 f) 5, 3, 1, 1, 3 g) 0, 2, 4, 6, 8 h) 8, 4, 0, 4, 8 Bonus 24, 27, 30, 33, 36 Point out that we substitute numbers into the formula in place of Term Number. This means Term Number is a quantity that changes. What do we call a quantity that changes in a formula? (a variable) So Term Number is a variable. Converting a formula into a general rule and vice versa. Tell students that the formula Term = 2 Term Number + 1 can be converted to a verbal rule: Multiply the term number by 2 and add 1. Then look at an example with a negative coefficient. For example, the formula Term = 3 2 Term Number can be converted to a verbal rule in two ways. Multiply the Term Number by 2 and subtract the result from 3 is one way. The other way is to think of subtraction as adding a negative number: 3 2 Term Number = 3 + ( 2) Term Number = ( 2) Term Number + 3. Process ExpectatioN Representing Now we can translate the formula to a verbal rule slightly differently: Multiply the Term Number by ( 2) and add 3. Have students check that both rules produce the same sequence. Then ask students to convert several formulas to verbal rules and vice versa. Use the same formulas as above: b) 15 2 Term Number f) ( 2)n + 7 h) 12 4n ANSWERS: b) Multiply the term number by ( 2) and add 15, f) Multiply the term number by ( 2) and add 7, h) Multiply the term number by ( 4) and add 12. General and stepwise rules. Write the two types of rules for the sequence 3, 5, 7, 9, 11 next to the table of values already on the board. (Start at 3 and add 2 each time; Multiply the term number by 2 and add 1) Look at the rules and the table of values side by side, and ASK: How are these P-26 Teacher s Guide for Workbook 8.2