CHAPTER. Linear patterns and relationships

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1 Linear patterns and relationships CHAPTER NCEA Level material included in this chapter is one section of Achievement Standard 902 (Mathematics and Statistics.) Investigate relationships between tables, equations and graphs. The chapter covers the following achievement objectives: Solve linear equations and inequations, and simultaneous equations with two unknowns. Relate graphs, tables and equations to linear relationships found in number and spatial patterns. Relate rate of change to the gradient of a graph. Sequences of numbers A sequence is a list of numbers, in a definite order, usually following a fixed rule or pattern. Some common sequences you should be filiar with, include: even numbers 2, 4, 6,, 0, 2,... odd numbers,,, 7, 9,,... square numbers, 4, 9, 6, 2, 6, 49,... cube numbers,, 27, 64, 2, 26,... In a sequence: the numbers in the sequence are called terms each term has a position in the sequence. For exple, in the sequence, 7,,,, the term in the first position is, the second term is 7, the third term is, and so on. Linear sequences The terms in a linear sequence increase (or decrease) steadily by the se ount. By continuing these patterns, you can work out more terms in the sequence. Q. Find the next two numbers in the following linear sequences:., 7,,, 9, , 22, 9, 6,... A.. Each number (after the first) in the sequence is formed by adding 4 to the previous number. The next two numbers are = 2 and = Each number (after the first) in the sequence is formed by subtracting from the previous number. The next two numbers are 6 = and = 0. The general rule for a linear sequence A more useful technique for finding terms in a sequence involves finding a general rule for the relationship between the position of a term and the value of the term.

2 Chapter : Linear patterns and relationships 7. Square concrete tiles are placed around a rectangular garden, as shown in the diagr. tiles garden Copy and complete the table. Number of tiles along the n length of the garden Total number of tiles used 0 96 m 6. A construction firm builds bridges using spans, as shown in the diagr. -span (6 struts) Copy and complete the table. 2-span (0 struts) -span (4 struts) Number of spans n Number of struts Ripeka created a chain by forming hexagons with matches. She predicts that the number of matchsticks needed to form a chain of 20 hexagons will be 2. a. Write a rule to find the total number, T, of matches needed to form a chain of n hexagons. b. Use the rule to test Ripeka s prediction.. Logs are stacked in layers that form patterns. The stacks can only be a maximum of four layers high. a. Draw the next stack of logs in this pattern. b. Write an equation relating the total number of logs in the stack, L, and n, the number of logs in the bottom row, where n is or more. c. Serena has 90 logs to stack. How many should she place along the bottom row? Plotting points on a graph Graphs are drawn on a Cartesian plane, which is formed by a pair of number lines called axes, drawn perpendicular to each other and intersecting at the origin, O. Points on the Chapter

3 Chapter 9: Linear graphs 27. y 0 x 2. y x Chapter 9 Line passes through (,2) and (4,2) m = y step x step = = 0 = 0 Line passes through (, ) and (,2) m = y step x step = 2 = 0 undefined The following diagr summarises the values of the gradient m for various straight line graphs. Positive gradient Negative gradient Horizontal gradient Vertical gradient Equal gradient Graph slopes uphill, m is positive Graph slopes downhill, m is negative Graph is horizontal, m is zero Graph is vertical, m is unidentified Lines are parallel, gradients are equal Activity 9B: The gradient of a straight line. For the following tables of points, state the rate of change of t with respect to n. a. n t b. n t c. n t d. n t Which lines in the diagr have: f y a b a. a positive gradient? b. a negative gradient? c. m = 0? d. a positive y-intercept? e. a negative y-intercept? f. an undefined gradient? 0 c d x e

4 Chapter 0: Quadratic patterns and relationships. The West Island schools are investigating the possibility of holding a basketball competition between schools. It is proposed that every te plays every other te once. A tick represents a ge played, a cross represents an impossible combination. A blank represents a ge combination already ticked. Copy and complete the table: Number of tes Total number of ges played 2 Tes Tes 4 Tes A B A B C A B C D 2 A A A B B B 4 6 C C D 0 n 6. a. Paul experimented with the cutting up of a pizza, using straight cuts. The following diagrs show the maximum number of pieces he got with, 2,, and 4 cuts. cut 2 cuts cuts 4 cuts Copy and complete Number of cuts Number of pieces the table alongside and 2 find a rule relating the 2 4 number of pieces (p) to the number of cuts (n). 7 4 b. If a circle is cut into 6 pieces, how many cuts were there? n 7. A quadrilateral has 2 diagonals, a pentagon has, and a hexagon has 9. a. How many diagonals will a polygon of sides have? b. What is the relationship between the number of sides (n) and the number of diagonals? Quadrilateral Pentagon Hexagon c. If a polygon has 44 diagonals, how many sides does it have? Chapter 0

5 24 Achievement Standard 90 (Mathematics and Statistics.6) The trigonometric ratios In a right-angled triangle, each trigonometric ratio compares the length of one side with another. These rules are shown in full and in abbreviated form in the box below. The sine of an angle = length of opposite side length of hypotenuse sin θ = or length of adjacent side The cosine of an angle = length of hypotenuse cos θ = or length of opposite side The tangent of an angle = length of adjacent side tan θ = or These three trigonometric ratios must be learnt. SOH CAH TOA is a useful mnemonic for doing this: S O H Sine of angle Opposite over A Hypotenuse C H Cosine of angle Adjacent over O Hypotenuse T A Tangent of angle Opposite over Adjacent Using ΔUVW shown, the hypotenuse is UV (. cm). Relative to angle V, the adjacent is VW ( cm) and the opposite is UW ( cm). sin V = = 0.2 (2 dp) [sin = ] cos V = = 0.6 (2 dp) [cos = ] tan V = = 0.6 [tan = ] Relative to angle U, the opposite is VW ( cm) and the adjacent is UW ( cm). tan U = =.67 (2 dp) [tan = ] Activity 6C: Trigonometric ratios Chapter 6. Use the diagr of ABC to state: a. The length of the hypotenuse. b. The length of the side adjacent to angle B. c. The length of the side opposite angle C. d. sin C e. tan B f. cos B g. sin (90 C)

6 22 Achievement Standard 907 (Mathematics and Statistics.2) Fifty students were given a choice of sport to take part in for a phys-ed option. The results are shown in the following table: Sport Athletics Basketball Gym Swimming Weightlifting Number of students Calculating sector angles for each category (as for athletics) gives the following table, from which a pie graph can be drawn (using a protractor to measure each angle). Category Frequency Sector angle ( ) Athletics Basketball Gymnastics Swimming Weightlifting = Total Chapter 20 Using pie graphs to compare sets of data Since pie graphs show the proportion (or percentage) of the total quantity in each category, a pair of pie graphs can be used to compare proportions (the bigger the sector, the larger the proportion). Bar graphs A bar graph (sometimes called a column graph) is a common method of displaying discrete data. The frequencies of the data can be easily compared, because the length or height of the bars is proportional to the frequency of the data. The phys-ed choices from the above exple are shown in the graphs below:. 2. Note:. The frequency is usually on the vertical axis, but bar graphs may be turned sideways so that frequency is on the horizontal axis. 2. The bars should be the se width and should not be touching.

7 Chapter 20: Comparing variables using statistical displays 29 The seasonal variations are evident in the cyclical pattern: The graph shows that sales are highest in Summer (lowest in Winter). A short-term feature in this exple is that sales in Year appear to be lower than expected. For exple, summer sales in Year are lower than those of the previous two years, which is against the general long-term upward trend in sales. Comparing data using time-series graphs A pair of time-series graphs, plotted on the se axes, can be used to make comparisons between two data sets. Q. The following time-series graph shows the average age, at death, of New Zealanders over the years between 90 and Average age at death Females Males Year ended December Source: New Zealand Official Yearbook Comment on any trends in these graphs. 2. Compare the time-series graphs. A. Average age at death Females Males Year ended December Source: New Zealand Official Yearbook 2000 Age (years) Age (years) Chapter 20

8 296 Achievement Standard 907 (Mathematics and Statistics.2). Long-term trends There is a general long-term trend upwards in both graphs (shown dotted). Short-term features The male graph showed a lower than expected average age at death in 96 and between 97 and 979. The female graph showed a higher than expected average age at death between 962 and 967 and again in 96. Seasonal variations There are no regular seasonal variations in either graph. 2. The average age at death of females is higher than that for males. The average age at death of females is increasing at a greater rate (steeper trend line) than that for males. (Average age gap has grown from 2 years (90) to 6 years (2000).) Chapter 20 Activity 20C: Time-series graphs. Kirsty lives in Christchurch and one day recorded the temperature each hour. Her results are shown in the table. Time noon Temperature ( C) a. Draw a time-series graph for these data. b. Discuss any features of interest in Kirsty s graph. 2. The following time-series graph shows the total sales in the retail trade in New Zealand from June 990 to March $(thousand million) Retail trade total sales by quarter J S D M J S D M J S D M J S D M J S D M J S D M J S D M J S D M J S D M Source: New Zealand Official Yearbook 2000 What does the graph show about the total retail sales in the period from June 990 to March 999? You may wish to refer to long-term trends, seasonal factors, and short-term features.

9 ANSWERS Achievement Standard 9026 (Mathematics and Statistics.) Activity A: Order of operations and integers (page 7). a. 0 b. 4 c. 2 d. e. 0 f. 2 g. 2 h. 6 i. 9 j. 9 k. 4 l. m. 6 n. o. 2 p. 2 q. 2 r. 0 s. Undefined t. Undefined 2. $0. kg 4. a. 4 b. 7 c. 2 d. e. 20 f. 2 g. 24 h. 2 i. 0 j. k. 7 l. 2 m. n. o. 2 p. q. 2 r. 7 s. t.. a. 4 b. 4 c. d. 6. a. 2 C b. 0 C c. 2 C 7. a. i. 7 C ii. 7 C b. C c. 2 C. $4.90, i.e. $4.90 overdrawn a. Each month Paul increases his balance by = $76. Current balance ( $ 7) is $ 07 below $ 200 balance. Number of months to achieve $ 200 balance is = In months his balance will be at least $ 200 (at 4 months the balance was only = $ 97). b. To achieve $ 200 in months, monthly increase needs to be 07 = $ 02. This is = $27 more than current monthly increase. His spending needs to reduce to = $4. Activity B: Factors, multiples and primes (page ). a. 96 b. 0 c. 20 d. 04 e. 99 f a. 0 b. 4 c. d. 7. a. 7 b c. 2 2 d. 2 2 e. 7 f. 2 g a. 60 sec b. 2 times. 2, 4, = 7 9 so 2 has 4 factors:, 7, 9, 2 7. a. = 2 2 and 2 and 2 are both factors of the number (since 2 divides 4 exactly and 2 divides 2 exactly). b. 40 = 2 and 2 and are both factors of the number. c. 9 = 7 and divides 4 exactly and 7 divides 7 2 exactly.. Because 2 = 2 is not a factor of Answers 0. Substituting n = 4 gives P = which factorises to 4(4 ) which simplifies to 4 9 = 4 which is not a prime number (more than 2 factors). Activity C: Powers and roots of numbers (page 4). a. 2 b. c. 62 d. 49 e. 6 f. g. 6 h. 9 i..60 j..2 k. 29. l m. n. 4 o. p. 9 q. r. 20 s t. 2. a. 6 b. c. 6 d. 7 e. 7 f. 0. a. 2 b. 9 0 c. 6 d a. 7 cm b..26 cm longer. a. m 2 b. 9 m 6. a. 2 b. 6 6 c. 24 d e. f. g. 4 h. 2 i..4 j. k..479 l m. 2 n. o. 7 p. 4 q. 6 r. 6 s. Undefined. t a. 6 b. c..96 d. e. 2 f.. a. 2 cm b. 6. cm 9. a..4 cm b. 7.6 cm c. so k = cm long short = 400 = Activity D: Integer and fractional powers (page 7). a. b. c. d e. f. 2. a. b. 6 c. 2 d. e f a. 0 b. 2 c. d. 2 e. 9 f a. 9 b. c. d. 2 e. 2 f. 0. a. 2 b. 26 c. 6 d e. 72 f Activity 2A: Equivalent fractions and mixed numbers (page 22). a. b. c. 2. a. 4. a. 2 g. 7 7 b. b c. 2 7 c. 7. a. d. 4 d. 2 0 = 2 0 b. 0 0 = e. 6 c. f. 0 = h. i. 7 0 j. k. 7

10 INDEX A adjacent side 2 algebraic equation algebraic expressions 6, 6 algebraic fractions 70 algebraic terms 6 alternate angles 9 alternate segment 224 ount analysis 247, 24 angle 7 angle of depression 24 angle of elevation 24 arc 2 arithmetic mean 27 average 26 axes 7 axis of symmetry, 97 B bar graph 22 base 67 base angles 97 BEMA, 64 benchmark fractions 20 biased 27 bi-modal 24 bivariate data 2 box 29 box-and-whisker plot 2, 26 brackets 7 C cancelling 20 categorical data 247, 20 census 249, 27 centre 2 certain event changing the subject 92 chord 26 circle 2 class interval 277 cluster 24, 290 cluster spling 20 coefficient 6, 7 co-interior angles 92 column graph 22 common factor 9 comparing fractions 20 comparison question 247, 249 complement of an event 22 composite bar graphs 2 composite numbers 0 compound interest 9 concertina of axis 24 conclusions (statistical) 24 concyclic points 22 conditional probability 2 constant ratio 74 constraints 79 continuous data 247, 24 coordinates corresponding angles 90 cosine 2