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Algebra Patterning and Graphs

Algebra Patterning and Graphs. Exploring Patterns. e.g. Write the next two numbers in the following patterns and describe the pattern. a) 2, 5, 8, 11, 14,. 17, 20. b) 40, 34, 28, 22, 16,. 10, 4. Add 3. Subtract 6. c) 1, 3, 9, 27, 81,. 243, 729. d) 1, 4, 9, 16, 25,.

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Algebra Patterning and Graphs

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  1. Algebra Patterning and Graphs

  2. Exploring Patterns e.g. Write the next two numbers in the following patterns and describe the pattern. a) 2, 5, 8, 11, 14, 17, 20 b) 40, 34, 28, 22, 16, 10, 4 Add 3 Subtract 6 c) 1, 3, 9, 27, 81, 243, 729 d) 1, 4, 9, 16, 25, 36, 49 Multiply by 3 Square Numbers e.g. Draw the next shape in the pattern and add the number below 1 3 6 10

  3. Finding a Rule for Linear Patterns - Linear number patterns are sequences of numbers where the difference between terms is always the same (constant) • Rule generating a linear pattern is: • Difference × n ± a constant e.g. Write a rule (using n) to describe the following number patterns. 3×1= 3 4×1= 4 + 3 + 4 3 = 1 - 2 4 = 5 + 1 + 3 + 4 + 3 + 4 3×4 – 2 4×4 + 1 + 3 + 4 Rule: s = 3×n - 2 Rule: d = 4×n + 1 1. Find the difference between terms and if the same multiply by n 2. Substitute to find constant 3. Check if rule works

  4. e.g. To make these squares, the amount of matches below are needed. 4 7 10 13 These results are shown on the table below a) Draw the next set of squares + 3 b) Create a rule for the pattern and use it to help fill in the gaps in the table + 3 13 M = 3×n + 1 3×10 + 1 31 c) Write the rule in words The number of matches equals three times the number of squares plus one. 3×150 + 1 451 3×1 = 3 3 = 4 + 1 3×4 + 1 = 13

  5. Simple Quadratic Patterns - Quadratic number patterns are sequences of numbers where the difference between terms is not the same - You need to look at the differences of the differences. If it is a ‘2’, then the rule contains ‘n2’ e.g. Write a rule for the following pattern Rule: T = n2 + 3 + 3 12 = 1 + 2 + 5 1 = 4 + 3 + 2 + 7 + 2 42 + 3 = 19 + 9 1. Find the difference between terms 3. If the 2nd difference is a ‘2’, the rule contains n2 2. If difference is not the same, find the difference of the differences! 4. Substitute to find constant 5. Check if rule works

  6. Co-ordinates - Are two references used to identify places of interest - The horizontal reference is written first with the vertical second e.g. Maps MathematicalCo-ordinates - Are used to describe positions of points e.g. Plot the following points: A = (1, 3), B = (4, 2), C = (3, -4), D = (-5, 1) y 4 A 1. Add in axes (if needed) 3 B 2. Label and number axes x = horizontal, y = vertical 2 D 1 3. Plot points using first number as the x co-ordinate and the second the y co-ordinate. -5 -4 -3 -2 -1 1 2 3 4 5 x -1 4. Label points -2 -3 -4 C

  7. Co-ordinate Patterns - Involves plotting rules that link the x co-ordinate to the y co-ordinate e.g. Complete the tables below and plot the following rules a) y = 2x b) y = ½x – 1 c) y = -3x + 2 2 x -2 -4 ½ x -2 – 1 -2 2 x -1 -2 ½ x -1 – 1 -1 ½ 0 -1 2 -½ 4 0 -3 x -2 + 2 8 -3 x -1 + 2 5 2 -1 -4

  8. Scatterplots - Show relationships between two quantities - Has two axes, each showing a different quantity with its own scale e.g. Height (m) Tom Jane Bob Mary Age (years) a) Who is the tallest? Tom b) Who is the same age? Bob and Mary c) Who is the oldest? Jane

  9. Line Graphs - Show how one quantity changes as another one does e.g. a) What is the room temperature? 20ºC b) How long does it take for the sweetcorn to cool down to room temperature? 80 – 15 = 65 mins

  10. e.g. Draw a graph to show how the water level may change in the following situation. - The sink is filled ¾ full of water - All of the dishes are put into the sink at once - The dishes are washed and removed separately - The water is then drained

  11. Distance/Time Graphs - Are line graphs with time on horizontal and distance on vertical axis. - If the line is horizontal the object is not moving - The steeper the line, the faster the movement e.g. Steepest a) How far out from the harbour did the yacht travel? 5 km b) What happened while the graph was horizontal? The yacht was stationary c) Which part of the journey was quickest? The return journey

  12. Applications - When using graphs to read off values and explain rules e.g. A student lends out his scooter for a fee of $3 and $2 for every km travelled. Complete the table and plot on graph below. 1 × 2 + 3 5 2 × 2 + 3 7 9 11 13 a) What will be the charge for a journey of 2 ½ km? $8

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