Chapter 4: Patterns and Relationships

Transcription

1 Chapter : Patterns and Relationships Getting Started, p. (a),, 9; rule: add fifteen, eighteen, twenty-one; rule: write out every third (c) n, q, t; rule: write every third letter (d) 55,, 77; rule: add (e),, ; rule: add a between the s (f),, 5; rule: the difference between each increases by. (a) E N S W The arrow points east, north, south, west and then it repeats.. (a),, ; pattern: subtract 5,, 5; pattern: add the next prime, subtract (c),, 9; pattern: subtract, add. output = (input ) + th output = ( ) + = + = 5 th output = ( ) + = + = 7 th output = ( ) + = + = 9 th output = ( ) + = + = 5th output = (5 ) + = + = 7th output = (7 ) + = + = 9th output = (9 ) + = 5 + = 55 Input Output (a) Label,,, as,,,. Every fourth after will be a. So the rd, 7th, th, 5th, 9th, rd, s are a. Therefore, the rd in the pattern is. For example, I labelled,,, and as,,, and. I know that every fourth after will be a. Since will be in that sequence of s, I know that the rd is.. (a) In each new figure cubes are added to the ends of the cross. (c) + = 5. 5 cubes are needed. 7. (a) Height of bean plant (mm) Height of Bean Plant vs. Days days I notice that the scatter plot has a pattern. Each day, the height of the bean plant increases by mm. On day 5, the bean plant is mm tall. I predict that on day, the bean plant will be mm + mm = mm tall.. Applying Pattern Rules, p.. (a),,, 9, 5. (a) pattern rule: add ;,, 5 pattern rule: add ;,, (c) pattern rule: add ; 7,, (d) pattern rule: multiply by ;,,. (a) The difference between the s increases by each time. 5, 79, 7 7. (a) 5,,, 5, 5 (c),, (d), 9, 7. (a) pattern rule: multiply by ;, 9, 9 pattern rule: multiply by ;,, 79 (c) pattern rule: multiply by ;,, (d) pattern rule: divide by ;,, 9 9., 7,, 59, 5., 7,, 7,. (a) 5, 5, 5, 5, 5, 57 9,,,, 5 7 Nelson Mathematics 7 Solutions -

2 . Each different colour of squares forms an L pattern. The s of squares in each of these L patterns represents an odd. The given diagram shows This is also a 5 5 square, so it contains 5 squares would form a square containing squares. So the sum of the first odd s must be.. For example, the rule for a sequence is, Start at 5, double the and subtract. 5, 7,, 9, 5, 7,. (a) 7,, 9,, 5. (a) term value = sum of the previous term values where the first terms are,, 5,,. Using a Table of Values to Represent a Sequence, p.. (a) (figure ) Picture ( of stars) 5 Start with and add each time. (c) = term + next sequential (d) th term = + 9 = 7 5. They are both right. For example, since the term s are sequential s from to 5, then starting with and adding to each term value is the same as multiplying the term by.. term value = previous term value + 5 th term value = rd term value + 5 = + 5 = (figure #) Picture 5th term value = th term value + 5 = + 5 = (# of tiles) 5 7. (a) term value = term 5th term value = 5 = 7th term value = 7 = th term value = = Picture (# of tiles) (figure #) 5 7 term value = ( term ) + 5th term value = (5 ) + = + = 7th term value = (7 ) + = + = 9 th term value = ( ) + = + = 5 Picture (figure ) ( of tiles) (a) Rule = ( term ) + ; I multiply the term by and then add. - Chapter : Patterns and Relationships

3 9. (a) cubes (c) = ( term ). Use a table of values to see how many figures she can draw using toothpicks and then draw the figure. Rule Therefore, she can draw the th figure.. Term Rule From my table of values, cubes are needed for the th figure in the pattern. Therefore, Matthew is right.. First I will write a rule to describe the pattern. Term Term value + = + = + = = 5 = sum of all term s 5th term value = I can see that 5 + = 5, + 9 = 5, + = 5 and so on. There are 5 such pairs of sums of 5. 5th term value = 5 5 = 75 Mid-Chapter Review, p.. For example, the nd diagonal has the pattern Start at, add each time. The 5th row has the pattern Start at 5, add 5 each time, and stop at 5. The th diagonal has the pattern, Start at, add each time..,,, 7, 5. For example, pattern rule: term value = (term + ) th term value = ( + ) = 7 = 9 Therefore, you would need 9 squares to build the th figure in this pattern.. Term Rule Term value Therefore, there are 5 toothpicks in the 5th figure in this sequence. 5. (a) For example, the term values increase by each time. Also, term value = ( term ) + previous term. Term Term value term value = ( term ) + th term value = ( ) + = + = 5th term value = (5 ) + = + = th term value = ( ) + = + = 7th term value = (7 ) + = + = th term value = ( ) + = + = 9th term value = (9 ) + = + = Nelson Mathematics 7 Solutions -

4 5 7 9 There are counters in the 9th term in this sequence. 7. (a) Rule (a) Rule term value = previous term value + 5 (c) term value = (term 5) (d) th term value = ( 5) = 5 = 7 9. Day Rule profit for th day profit for st day = ($) 5($) = $ $ = $5 Therefore, the difference between the profit for the st day and the profit for the th day is $5.. Solve Problems Using a Table of Values, p Figure toothpicks For example, for each figure, I added to the of toothpicks needed for the previous figure.. teams games required Therefore, Ravi would have to schedule 7 games for 9 teams and 9 games for teams. 7. Day students who donate food 5 9 I continued the pattern in a table of values until I got to 9 students donating food. Therefore, on the th day, 9 students will donate food.. people handshakes Therefore, there are 9 handshakes in total. - Chapter : Patterns and Relationships

5 9. Day members seedlings planted Total of trees = = + = = + = = + = 5 = + = = + = 7 = = Therefore, the replanting will be complete on the 7th day.. Day peace cranes made Total of peace cranes = = = 5 + = = = = 9 + = = 55 By the th day, Heather makes a total of 55 peace cranes. Therefore she reaches her goal of 5 peace cranes.. (a) Round parents called 9 7 Therefore, 7 parents will receive a call in the third round of calls = 9 Therefore, 9 parents in total will receive a call by the end of the third round of calls. (c) Continue the table of values until parents have been called. Round parents called Total of parents called + 9 = 5 + = Therefore, 5 rounds of calls will be needed to call parents.. (a) Day tickets sold = Therefore, tickets are sold by the end of the 5th day. Day : $ = $ Day : ( + 9) $ = 5 $ = $7 Day : (5 + ) $ = 57 $ = $ Day : (57 + 5) $ = $ = $ Day 5: ( + ) $ = $ = $ Therefore, students will reach their goal on the 5th day of ticket sales.. For example, Jimmy wants to bake cookies. He gets cookies in each batch. How many batches of cookies does Jimmy need to bake to reach his goal? Batches cookies 7 5 Therefore, Jimmy must bake 5 batches of cookies to reach his goal of cookies..5 Using a Scatter Plot to Represent a Sequence, p. 5. Draw a scatter plot showing the pattern and extend the points to show posts. rails and posts Rails and Posts vs. Sections sections Therefore, Mohammed will need 9 posts and rails for a fence that is sections long.. (a) Toothpicks vs. toothpicks 5 If I follow the line past (5, ), I can see that I would need toothpicks in the 9th term of this sequence. (c) If I follow the line past (5, ) I can see that I can make the 7th term in the sequence with toothpicks. 7 9 Nelson Mathematics 7 Solutions -5

6 7. Term Value vs.. For example, T e r m v a l u e If I follow the line past (, ), I can fill the missing values in the table. Term rails and posts 7 5. sections (a) For sections, I must look at the scatter plot to see how many rails and posts are needed. Therefore, rails and posts are needed for sections. For sections, I must look at the scatter plot to see how many rails and posts are needed for posts. Therefore, 7 rails and posts are needed for sections. 9. (a) Figure Blue Purple Green 5 7 squares Rails and Posts vs. Sections Squares vs. Figure Number Figure (figure ) toothpicks ( of toothpicks) Toothpicks vs. Figure Number Figure. (a) Each row in the triangle increases by band members. 7 rows in the inner triangle. (c) Row Band Members Total I can see from the table that the total of band members is (row ). So for 5 rows you need 5 5 = = 7. 7 more band members are needed. (d) 7 players in the largest row means that 9 rows are needed. So 9 9 = total band members are required. 9 =. more band members are needed. (e) For example, How many rows of band members are needed if the inner triangle must contain people? (row ) = total (row ) = =, so rows are needed. - Chapter : Patterns and Relationships

7 . Cost triangle stones border pieces $5 5 $5 7 $95 9 $5 5 $55 $5 7 5 $5 7 $5 9 9 $75 $5 $5 5 $5 Therefore, Omar can afford to make a path triangle stones long. Chapter Self-Test, p.. (a), 7, ; pattern rule: add,, ; pattern rule: start by adding 7 and then double the difference each time (c),, ; pattern rule: each is the sum of the previous s (d) 9, 7, ; pattern rule: the difference increases by each time. (a) Position term value = term 5. (a) Input Output 5 5 For example, you could find a pattern in the points of the scatter plot and estimate what output would match an input of. (c) output = (input ). (a) 5 7 You can calculate the term value using the previous term or by using a relationship between the term and the term value. term value = previous term value + term value = ( term ) + 5. tables guests Therefore, guests can be seated at a row of tables.. members games Therefore, 7 games will be played in total. 7. (a) Hours cars washed = Therefore, they washed cars by the end of h. $5 = $ Therefore, they reached their goal. Nelson Mathematics 7 Solutions -7

8 Chapter Review, p. 5. (a), 7, ; pattern rule: add, 7, 5; pattern rule: the difference increases by each time (c), 7, ; pattern rule: add the previous terms. (a) Figure green counters red counters Total of counters (c) The difference between the of green counters increases by each time. The of red counters is equal to the figure squared. The difference between the total of counters increases by each time. (d) Add 7 to the total of counters for the 5th figure. total of counters for th figure = + 7 or 57 Therefore, you would need 57 counters in total to make the th figure. Add the of green counters and the of red counters. total of counters for th figure is 55 + = 55. Therefore, you would need 55 counters to make the th figure (55 green and red).. (figure ) (# of line segments) Therefore, it would take 5 line segments to join 5 dots to every other dot.. First, I will complete a table of values to show the pattern for days. Day pages = 5 Mike will read 5 pages in nights. Therefore, Mike is right. 5. (a) I could extend the line on the scatter plot until it passes through a term of.. (a) Figure Blue Purple Green squares Squares vs. Figure Number Figure - Chapter : Patterns and Relationships