ALGEBRA PATTERNS AND RELATIONSHIPS What are the 2

ALGEBRA PATTERNS AND RELATIONSHIPS

What are the 2 missing shapes?

What are the 2 missing letters? A B A A X MAHTMATHMAHTM IVVIIIXIVVIIIXI VV IIIX A? B? A? C? A? D? A? E? A? F? A? G

Starter = = = 12 x 10 x ac 3 p d 14 x = ab = 8 cd = 6 fg = 2 ap = 8 ab = 6 pqr = 10 def = 6 pqr

Note 3: Patterns To complete a pattern, look for a rule to get from one term to another. Examples: Find the rule for these sequences and write down the next term: 3, 7, 11, 15, …… The rule is ‘start with 3 and add 4 each time’. The next term will be 15 + 4 = 19 1, 3, 9, 27, …… The rule is ‘start with 1 and multiply by 3 each time’. The next term will be 27 x 3= 81

Note 3: Patterns Sometimes it helps to find the rule if we look at the differences between each term. Examples: Find the next three terms in the following sequences 1, 3, 7, +2 +4 1, 3, +2 +4 13, +6 7, +8 15, +8 +16 21, 31, , 43, , +10 +12 31, +32 57, , +14 63, 127, 255, +64 +128

Note 3: Patterns A car salesman has said he wants to sell 10 cars in his first month of business. Every month after that, he aims to sell two more cars than the previous month. Calculate how many cars he aims to sell each month for the first 5 months of business. month = 10 2 nd month = 10 + 2 1 st = 12 3 rd month = 12 + 2 = 14 4 th month = 14 + 2 = 16 5 th month = 16 + 2 = 18 Alpha IWB Ex 12. 03 pg 312

Note 4: Finding terms from patterns Using the rule for the pattern, substitute values in for n using 1, 2, 3, 4, …. . etc until we have the required terms e. g. Find the first four terms in each of the following rules 2 n n = 1 (term 1) n = 2 (term 2) n = 3 (term 3) n = 4 (term 4) 2 2 x x (1) (2) (3) (4) Terms are: 2, 4, 6, 8, ……. . = = 2 4 6 8

Note 4: Finding terms from patterns Using the rule for the pattern, substitute values in for n using 1, 2, 3, 4, …. . etc until we have the required terms e. g. Find the first four terms in each of the following rules n + 2 n = 1 (term 1) n = 2 (term 2) n = 3 (term 3) n = 4 (term 4) 1 2 3 4 + + 2 2 Terms are: 3, 4, 5, 6, ……. . = = 3 4 5 6

Note 4: Finding terms from patterns Using the rule for the pattern, substitute values in for n using 1, 2, 3, 4, …. . etc until we have the required terms e. g. Find the first four terms in each of the following rules 2(n + 2) n = 1 (term 1) n = 2 (term 2) n = 3 (term 3) n = 4 (term 4) 2(1 2(2 2(3 2(4 + + 2) 2) Terms are: 6, 8, 10, 12 ……. . = = 6 8 10 12

Note 4: Finding terms from patterns Using the rule for the pattern, substitute values in for n using 1, 2, 3, 4, …. . etc until we have the required terms e. g. Find the first four terms in each of the following rules 3 n - 1 n = 1 (term 1) n = 2 (term 2) n = 3 (term 3) n = 4 (term 4) 3(1) 3(2) 3(3) 3(4) − − 1 1 Terms are: 2, 5, 8, 11, ……. . = 2 = 5 = 8 = 11

Task: Find the first four terms in each of the following rules by substituting 1, 2, 3 & 4 in the following rules. 3 n 3, 6, 9, 12, … n +5 6, 7, 8, 9, … n - 1 0, 1, 2, 3, … 4 n + 1 5, 9, 13, 17, … -2 + 7 n 5, 3, 1, − 1, … 5(n+2) n 2 + 4 n 2 +n 15, 20, 25, 30, … 1, 4, 9, 16, … 5, 8, 13, 20, … 2, 6, 12, 20, … Alpha IWB Ex 12. 03 pg 313 -314 PUZZLE pg 315 Ex 12. 04 pg 319 -323

Finding General Rules for number patterns Think of an integer, multiply it by 3, and subtract 2. You may have thought of many different numbers: e. g. 3 × 4 – 2 = 10 3 × 2 – 2 = 4 3 × − 7 − 2 = − 23 3 × 9 − 2 = 25 What is the same about each sentence? What is the different about each sentence? How could you write a sentence that describes all possible sentences? How could you write that sentence in maths language? 3×n− 2 3 n − 2

Note 5: Finding General Rules for number patterns A variable is a letter or symbol that is used to describe infinitely many numbers. We write rules in the form of y = dn ± x d and x are numbers that we calculate To do this: 1. ) Find the common difference between the terms. This number is d. 2. ) Subtract the first term from the common difference. This number is x. e. g. What is the rule for 3, 5, 7, 9, ….

Note 5: y = dn ± x Finding General Rules for number patterns d = common difference x = Term 1 − common difference e. g. What is the rule for 3, 5, 7, 9, …. Term number Term (y) Common Difference 1 3 2 5 2 x = Term 1 − Common Difference = 3 − 2 =1 3 7 2 4 9 2 d=2 y = dn ± x The equation is y = 2 n + 1

Note 5: y = dn ± x Finding General Rules for number patterns d = common difference x = Term 1 − common difference e. g. What is the rule for 10, 14, 18, 22 …. Term number Term (y) 1 10 Common Difference 2 14 4 x = Term 1 − Common Difference = 10 − 4 =6 3 18 4 4 22 4 d=4 y = dn ± x The equation is y = 4 n + 6

Note 5: y = dn ± x Finding General Rules for number patterns d = common difference x = Term 1 − common difference e. g. What is the rule for 35, 30, 25, 20 …. Term number Term (y) 1 35 Common Difference 2 30 -5 x = Term 1 − Common Difference =35 −− 5 = 35 + 5 = 40 3 25 -5 4 20 -5 d = -5 y = dn ± x The equation is y = -5 n+40

Note 5: y = dn ± x Finding General Rules for number patterns d = common difference x = Term 1 − common difference e. g. What is the rule for 4, 1, -2, -5, …. Term number Term (y) Common Difference 1 4 2 1 -3 x = Term 1 − Common Difference =4 −− 3 =4 + 3 =7 3 -2 -3 4 -5 -3 d = -3 y = dn ± x The equation is y = -3 n+7

Now its your turn! Find d and x first Write rules in the form of y = dn ± x e. g. What is the rule for : 2, 4, 6, 8, … d = 2, x = 0 y = 2 n + 0 17, 12, 7, 2, … d = − 5, x = 22 y = − 5 n + 22 18, 27, 36, 45, … d = 9, x = 9 y = 9 n + 9 37, 29, 21, 13, … d = − 8, x = 45 y = -8 n + 45 Alpha IWB Ex 12. 04 pg 319 -323

Spatial Patterns

Spatial Patterns 4 7 10 How many matches are there in each pattern? How many matches will there be in a pattern with 4 squares? 10 + 3 = 13

Spatial Patterns 7 4 Number of Squares (s) Number of matches 10 1 4 s = Term 1 − Common Difference = 4 − 3 =1 13 2 7 3 10 4 13 d=3 y = dq ± s The equation is y = 3 q+ 1

Spatial Patterns 3 9 6 Draw the next 2 shapes in this pattern Number of Triangles (t) Number of matches s = Term 1 − Common Difference = 3 − 3 =0 1 3 2 6 3 9 4 12 d=3 y = dt ± s The equation is y = 3 t + 0

Spatial Patterns 6 11 y = dh ± x 16 Draw the next 2 shapes in this pattern Number of Houses (h) Number of matches x = Term 1 − Common Difference = 6 − 5 =1 1 6 2 11 3 16 4 21 d=5 y = dh ± x The equation is y = 5 h + 1

Note 6: Using rules, tables & graphs Sarah babysits on the weekend. She is paid $5. 00 each time, plus $6 per hour that she is there. Complete the table to show her earnings for 1 -5 hours at a time. 1 2 3 4 5 Pay $11 $17 $23 $29 $35 Plot the coordinates from the table to a graph Rule: 5 + 6 h Wages earned $ Hours worked 40 30 20 10 0 0 2 4 Hours worked 6