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Name ___________________________ Period _________ Date ___________ 8-4 STUDENT PACKET MATHLINKS GRADE 8 STUDENT PACKET 4 PATTERNS AND LINEAR FUNCTIONS 2 4.1 Growing Shapes • Use variables, parentheses, and exponents in expressions. • Use formulas to find perimeter and area of rectangles. • Describe geometric patterns numerically, symbolically, graphically, and verbally. • Plot ordered pairs that satisfy a specified condition. • Informally connect the slope of a line to its context in a graph. 1 4.2 Going to the Park • Solve time-distance problems. • Interpret time-distance graphs. • Explore rates of change on a graph. • Understand the meaning of the points of intersection of two graphs. • Informally connect the slope of a line to its context in a graph. 8 4.3 Stacking Cups • Use numbers, graphs, and symbols to represent data. • Understand and estimate a line that fits the data. • Draw conclusions based on data displays. 17 4.4 Skill Builders, Vocabulary, and Review MathLinks: Grade 8 (Student Packet 4) 21 Patterns and Linear Functions 2 WORD BANK Word or Phrase Definition or Explanation Example or Picture explicit rule (input-output rule) fourfold way inductive reasoning linear function point of intersection rate slope (or slope of a line) y-intercept MathLinks: Grade 8 (Student Packet 4) 0 Patterns and Linear Functions 2 4.1 Growing Shapes GROWING SHAPES Summary (Ready) Goals (Set) We will extend square and rectangle patterns. Then we will represent geometric measures in the patterns using inputoutput tables, a graphs, symbols, and words (the “fourfold way”). • Use variables, parentheses, and exponents in expressions. • Use formulas to find perimeter and area of rectangles. • Describe geometric patterns numerically, symbolically, graphically, and verbally. • Plot ordered pairs that satisfy a specified condition. • Informally connect the slope of a line to its context in a graph. Warmup (Go) Use inductive reasoning to complete each table. Write an explicit rule in words and symbols. Table 1 Table 2 Table 3 Input (x) Output (y) Input (x) Output (y) Input (x) Output (y) 1 1 1 2 1 2 2 2 2 3 2 4 3 3 3 4 3 6 4 4 4 5 5 5 Rule: If the input value is x, then the output value is: Rule: If the input value is x, then the output value is: Rule: If the input value is x, then the output value is: ________________ ________________ ________________ In symbols: In symbols: In symbols: y = ____________ y = ____________ y = ____________ MathLinks: Grade 8 (Student Packet 4) 1 Patterns and Linear Functions 2 4.1 Growing Shapes GROWING SQUARES 1. This is a pattern of growing squares built from unit squares. Continue the pattern for steps 4 and 5. Step # 1 2 3 4 5 2. Complete the tables. Table 1 Step Length of number side (n) (L) n Rule: L = _________ Table 2 Step number (n) Table 3 Perimeter (P) n Area (A) n Rule: P = _________ 3. What is the perimeter of the figure in step #10? Step number (n) Rule: A = _________ 4. If the perimeter of the figure is 84, what is the step number? rule: ___________________________ rule: ___________________________ substitute: ______________________ substitute: ______________________ perimeter: ______________________ step number: ___________________ 5. Use words or diagrams to explain how the length of the side and the perimeter of a square are related. 6. Use words or diagrams to explain how the length of the side and the area of a square are related. MathLinks: Grade 8 (Student Packet 4) 2 Patterns and Linear Functions 2 4.1 Growing Shapes GROWING SQUARES GRAPHS 1. Graph the data from the tables on the previous page. Scale the graphs appropriately. One Graph on the Right (Use Table 3) Area in square units A (a third color) Lengths in units L (one color) Perimeter in units P (a second color) Two Graphs on the Left (Use Tables 1 and 2) Step number n Step number n 2. How is the graph on the right different from the two graphs on the left? 3. A linear function is a function whose graph is a line. Which of the rules describe linear functions? Explain. MathLinks: Grade 8 (Student Packet 4) 3 Patterns and Linear Functions 2 4.1 Growing Shapes GROWING RECTANGLES 1 1. This is a pattern of growing rectangles built from unit squares. Continue the pattern for steps 4 and 5. Step # 1 2 3 4 5 2. Complete the tables. Table 1 Step # base (n) (b) 1 2 n Rule: b = _________ Table 2 Step # height (n) (h) 1 1 n Table 3 Step # Perimeter (n) (P) 1 6 n Rule: h = _________ Rule: P = _________ 3. What is the base of the rectangle for step #12? 4. What is the height of the rectangle for step #14? 5. What is the perimeter of the rectangle for step #10? 6. If the base of the rectangle is 36, what is the step number? MathLinks: Grade 8 (Student Packet 4) 4 Patterns and Linear Functions 2 4.1 Growing Shapes GROWING RECTANGLES 1 (Continued) All the tables on the previous page compare the step number to a length. This is because base, height, and perimeter are all linear measurements. 7. Draw a vertical axis on the grid, label it “length,” and scale appropriately. Draw a horizontal axis on the grid, label it “step number,” and scale appropriately. 8. Draw graphs from the tables on the previous page using a different color and note the color used. a. Base vs. step # (color:_______________) b. Height vs. step # (color:_______________) c. Perimeter vs. step # (color:_______________) 9. Draw a trend line for each graph to show each linear pattern, and label each line by name. 10. Use words such as slope, parallel, flat, steep, and intersect to describe how the graphs are the same and how they are different. 11. Which rules above could describe a linear function? Explain. MathLinks: Grade 8 (Student Packet 4) 5 Patterns and Linear Functions 2 4.1 Growing Shapes GROWING RECTANGLES 2 1. This is another pattern of growing rectangles built from unit squares. Continue the pattern for steps 4 and 5. Step # 1 2 3 4 5 2. Complete the tables. Table 1 Table 2 Table 3 Step # (n) base (b) Step # (n) height (h) Step # (n) Perimeter (P) 1 3 1 2 1 10 n Rule: b = _________ n n Rule: h = _________ Rule: P = _________ 3. What is the base of the rectangle for step #18? 4. What is the height of the rectangle for step #24? 5. What is the perimeter of the rectangle for step #20? 6. If the base of the rectangle is 40, what is the step number? MathLinks: Grade 8 (Student Packet 4) 6 Patterns and Linear Functions 2 4.1 Growing Shapes GROWING RECTANGLES 2 (Continued) All the tables on the previous page compare the step number to a length. This is because base, height, and perimeter are all linear measurements. 7. Draw a vertical axis on the grid, label it “length,” and scale appropriately. Draw a horizontal axis on the grid, label it “step number,” and scale appropriately. 8. Draw graphs from the tables on the previous page using different colors and note the color used. a. Base vs. step # (color:_______________) b. Height vs. step # (color:_______________) c. Perimeter vs. step # (color:_______________) 9. Draw a trend line for each graph to show each linear pattern, and label each line by name. 10. Use words such as slope, parallel, flat, steep, and intersect to describe the graphs. How are they the same and how are they different? 11. Which rules above describe a linear functions? Explain. MathLinks: Grade 8 (Student Packet 4) 7 Patterns and Linear Functions 2 4.2 Going to the Park GOING TO THE PARK Summary (Ready) Goals (Set) • • • • We will use information about friends going to a park after school to help us understand time, distance, and rate of speed relationships using numbers, graphs, symbols, and words. Solve time-distance problems. Interpret time-distance graphs. Explore rates of change on a graph. Understand the meaning of the points of intersection of two graphs. • Informally connect the slope of a line to its context in a graph. Warmup (Go) Ginger and Rudy are racing. Use the graph below to answer the following questions. 1. Who starts out faster? ________________________________________________ How do you know? ___________________________________________________ 2. Who starts out slower? ___________ Does she ever catch up? ________________ How do you know? ___________________________________________________ 3. Who wins the race? __________________________________________________ How do you know? ___________________________________________________ 4. Which of the graphs could represent a linear function? Ginger Distance goal Rudy Time MathLinks: Grade 8 (Student Packet 4) 8 Patterns and Linear Functions 2 4.2 Going to the Park GOING TO THE PARK: PART 1 INTRODUCTION A group of friends are going to meet at the park after school. They will all travel 90 meters straight down Euclid Street from the school to the park. Herbie got a new digital camera and wants to use it to take pictures of her friends’ journey. She will monitor their progress by taking nine pictures at six-second intervals from a building high above Euclid Street. She will lay the pictures down side-by-side, in order from the first picture to the last. She will then graph these images and analyze their movements. Herbie starts to take pictures at exactly 3:00:00 PM. Ellie is already walking and got a head start. At 3:00:00 PM, she is already 36 meters from school. Daisy is jumping rope and moving at a constant rate. She leaves the school at 3:00:00 PM. 1. Approximately how far is 90 meters? _____________________________________ 2. How many pictures is Herbie going to take?__________ How long will Herbie wait between snapping pictures? __________ 3. Beginning at 3:00:00 PM, record the first six times that Herbie will snap pictures. 3:00:00 ___________ ___________ ______________ ___________ __________ 4. Complete the table showing distances from school. Picture # Ellie’s distance from school (in meters) Daisy’s distance from school (in meters) 1 2 3 4 5 6 7 8 9 36 42 48 54 60 66 72 78 84 0 12 24 36 48 5. Does Ellie reach the park by the time Herbie finishes taking her pictures? __________ 6. Does Daisy reach the park by the time Herbie finishes taking her pictures? ________ MathLinks: Grade 8 (Student Packet 4) 9 Patterns and Linear Functions 2 4.2 Going to the Park GOING TO THE PARK: PART 1 RECORDING SHEET 1. Graph the information from the table for Ellie and Daisy. Show trend lines for each graph. 80 70 60 50 40 30 20 10 0 :00 :06 :12 :18 :24 :30 :36 :42 :48 (picture 1) (picture 2) (picture 3) (picture 4) (picture 5) (picture 6) (picture 7) (picture 8) (picture 9) Time (in seconds) elapsed since 3:00:00 2. If possible, mark and label the coordinate that shows the time when the girls are the same distance from school. 3. If possible, mark and label the coordinates that show when each of the girls reaches the park. MathLinks: Grade 8 (Student Packet 4) 10 Patterns and Linear Functions 2 4.2 Going to the Park DAISY’S JOURNEY Suppose that Daisy continues to jump rope through all 9 of Herbie’s pictures. 1. Complete the following information about Daisy’s journey. Picture # 1 2 3 4 5 Time (in seconds) 0 6 12 18 24 Distance (in meters) 0 12 24 6 7 8 9 2. Write a rule that describes the relationship between Daisy’s time and distance. In words: Daisy’s distance from school is ___________________________________ In symbols: D = _______________ 3. Could this rule represent a linear function? Explain 4. Use the information from problem 1 to complete the table below. Pictures Change in Distance (in meters) Elapsed Time (in seconds) Rate of change ⎛ meters ⎞ ⎜ second ⎟ ⎝ ⎠ 1 to 2 12 – 0 = 12 6–0=6 12m 2m = 6 sec 1 sec 1 to 3 2 to 4 5. In the last column in the table above, what do you notice about the rates of change? 6. How is the rate of 2 meters per second represented on the graph of Daisy’s trend line? 7. On the graph, we will call the vertical change per unit of horizontal change the __________ of the line. Daisy’s trend line is an example of a line with a _____________ MathLinks: Grade 8 (Student Packet 4) __________. 11 Patterns and Linear Functions 2 4.2 Going to the Park ELLIE’S JOURNEY Suppose that Ellie continues to walk through all 9 of Herbie’s pictures. 1. Complete the following information about Ellie’s journey. Picture # 1 2 3 4 5 Time (in seconds) 0 6 12 18 24 Distance (in meters) 36 42 48 54 6 7 8 9 2. Ellie thinks the rule to describe her time and distance is D = 36 + t. Is she correct? _______ Give two examples to support your answer. 3. Use the information from problem 1 to complete the table below. Pictures Change in Distance (in meters) Elapsed Time (in seconds) Rate of change ⎛ meters ⎞ ⎜ second ⎟ ⎝ ⎠ 1 to 2 42 – 36 = 6 6 –0 =6 6m 1m = 6 sec 1 sec 1 to 4 3 to 4 4. In the last column in the table above, what do you notice about the rates of change? 5. How is the rate of 1 meter per second represented on the graph of Ellie’s trend line? 6. On the graph, we will call the vertical change per unit of horizontal change the __________ of the line. MathLinks: Grade 8 (Student Packet 4) 12 Patterns and Linear Functions 2 4.2 Going to the Park GOING TO THE PARK: PART 2 Now, Herbie will take nine more pictures of Charlie and Duke going back to school, starting at 3:15:00. Again she will wait 6 seconds between snapping pictures. Charlie left earlier and is waiting 60 meters from the school. Duke is already at the park at 3:15:00. He leaves the park on his roller skates to go back to school for soccer practice at a steady rate. 1. Beginning at 3:15:00, record the first six times that Herbie will snap pictures. _________ __________ __________ ____________ ___________ __________ 2. Complete the table showing distances from school. Picture # (Picture 1 starts at 0 seconds. Herbie took a picture every 6 seconds.) Name Picture # 1 2 Charlie’s distance from school (in meters) 60 60 Duke’s distance from school (in meters) 90 72 3 4 5 6 7 8 9 3. Does Charlie get back to school by the time Herbie finishes taking her pictures? If so, when? How do you know? 4. Does Duke get back to school by the time Herbie finishes taking her pictures? If so, when? How do you know? MathLinks: Grade 8 (Student Packet 4) 13 Patterns and Linear Functions 2 4.2 Going to the Park GOING TO THE PARK: PART 2 RECORDING SHEET 1. Graph the information from the table about Charlie and Duke. 90 80 70 Meters from school 60 50 40 30 20 10 0 :00 :06 :12 :18 :24 :30 :36 :42 :48 (picture 1) (picture 2) (picture 3) (picture 4) (picture 5) (picture 6) (picture 7) (picture 8) (picture 9) Time (in seconds) elapsed since 3:15:00 2. If possible, mark and label the coordinate that shows when the boys are the same distance from school. 3. If possible, mark and label the coordinates that show when the boys reach the school. MathLinks: Grade 8 (Student Packet 4) 14 Patterns and Linear Functions 2 4.2 Going to the Park DUKE’S JOURNEY Suppose that Duke continues to roller skate through all 9 of Herbie’s pictures. 1. Complete the following information about Duke’s journey. Picture # 1 2 3 4 Time (in seconds) 0 6 12 18 Distance (in meters) 90 72 54 5 6 7 8 9 2. Circle the explicit rule that describes the relationship between Duke’s time and distance. D = 12t D = -3t + 90 D = 3t 3. Use the information from problem1 to complete the table below. Pictures Change in distance (in meters) Elapsed Time (in seconds) ⎛ meters ⎞ Rate ⎜ ⎟ ⎝ second ⎠ 1 to 2 72 – 90 = -18 6 –0 =6 -18 m -3 m = 6 sec 1 sec 1 to 3 2 to 4 4. What do you notice about the rates of change between pictures? 5. How is this represented on the graph of Duke’s trend line? 6. How long does it take Duke to roller skate to school? 7. Duke is NOT traveling at a negative rate of speed, which is impossible. Rather, Duke is traveling in the opposite direction as the others, at a rate of 3 meters per second. Duke’s trend line is an example of a line with a _______________ MathLinks: Grade 8 (Student Packet 4) __________. 15 Patterns and Linear Functions 2 4.2 Going to the Park CHARLIE’S JOURNEY Suppose that Charlie remains 60 meters from school through all 9 of Herbie’s pictures. 1. Complete the following information about Charlie’s journey. Picture # 1 2 3 4 Time (in seconds) 0 6 12 18 Distance (in meters) 60 60 60 5 6 7 8 9 2. Write a rule that describes the relationship between Charlie’s time and distance. __________________________________________________________________ 3. Use the information from problem 1 to complete the table below. Pictures Change in distance (in meters) Elapsed Time (in seconds) ⎛ meters ⎞ Rate ⎜ ⎟ ⎝ second ⎠ 1 to 2 60 – 60 = 0 6 –0 =6 0m 0m = 6 sec 1 sec 1 to 4 3 to 4 4. What do you notice about the rates of change between pictures? 5. How is this represented on the graph of Charlie’s trend line? 6. Charlie’s trend line is an example of a line with a _______________ MathLinks: Grade 8 (Student Packet 4) __________. 16 Patterns and Linear Functions 2 4.3 Stacking Cups STACKING CUPS Summary (Ready) Goals (Set) We will measure and record heights of stacked cups, and represent the data using numbers, graphs, and algebraic symbols. Then we will estimate a line that fits our data. • Use numbers, graphs, and symbols to represent data. • Understand and estimate a line that fits the data. • Draw conclusions based on data displays. Warmup (Go) Graph each set of ordered pairs, connect them in order to a form shapes, and name each shape. 1. (Quadrant I) (2.5, 1), (2.5, 5), (6.5, 5), (6.5, 1) shape: _______________ y 2. (Quadrant II) (-3, 3), (-7, 3), (-8, 6), (-5, 8), (-2, 6) shape: _______________ 3. (Quadrant III) (-4, -3 ), (-7, -7 ), (-1, -7 ) x shape: _______________ 4. (Quadrant IV) (3, -2), (1, -4), (1, -6), (3, -8), (5, -8), (7, -6), (7, -4), (5, -2) shape: _______________ MathLinks: Grade 8 (Student Packet 4) 17 Patterns and Linear Functions 2 4.3 Stacking Cups STACKING CUPS: TABLE, GRAPH, AND RULES You will measure heights of cups as you stack them together and then analyze the data. 1. What is an appropriate unit of measure? _________________________, rounded to the nearest _________________________. 2. Measure and record the height of the first cup. Place a second cup inside the first, and measure and record the new height. Continue this process a few more times with different numbers of cups. Number of cups (x) 1 2 3 4 Height (y) 3. Graph the ordered pairs and sketch a line that fits the data. Scale the axes as needed. 4. We might say that the height of the stack depends on the number of cups in the stack, so the independent variable is _____________________________, and the dependent variable is _____________________________. 5. What is a good approximation for a recursive rule (in words)? 6. What is a good approximation for an explicit rule (using symbols)? 7. Though you sketched a line to fit the data, why do the points located in between the ones you graphed NOT make sense in this context? MathLinks: Grade 8 (Student Packet 4) 18 Patterns and Linear Functions 2 4.3 Stacking Cups STACKING CUPS: ANALYSIS 1. How did you approximate a line to fit your data? 2. How did you determine the recursive rule? 3. What part of the cup represents the difference between the heights in successive cup measurements? 4. How does this difference relate to the numbers in the table? 5. Where is this difference visible on the graph? 6. Estimate the slope of this line. 7. Estimate the height (in centimeters) of: a. A stack of 10 cups b. A stack of 100 cups 8. If you extended your line to x = 0, approximately where would it intersect the y-axis? In other words, name the y-intercept. 9. Although zero cups would have zero height, the line does not pass through the point (0, 0). Why? MathLinks: Grade 8 (Student Packet 4) 19 Patterns and Linear Functions 2 4.3 Stacking Cups STACKING BOWLS Maxine measured and recorded the height of a bowl in centimeters. Then she placed a second bowl inside the first, measured and recorded the new height, and continued this process a few more times. 1. Write in reasonable heights for 5 and 6 bowls in the table below. Number of bowls (x) Height in cm (y) 1 4.2 2 6.3 3 8.2 4 10.3 Height in Centimeters (y) 2. Graph the ordered pairs below and draw a line of best fit. 5 6 3. The independent variable is ____________________, and the dependent variable is __________________. Number of Bowls (x) 4. What is a good approximation for a recursive rule (in words)? 5. What is a good approximation for an explicit rule (using words and/or symbols)? 6. Though you sketched line that fits the data, why do the points located in between the ones you graphed not make sense in this context? MathLinks: Grade 8 (Student Packet 4) 20 Patterns and Linear Functions 2 4.4 Skill Builders, Vocabulary, and Review SKILL BUILDERS, VOCABULARY, AND REVIEW SKILL BUILDER 1 Compute. 1. 4− 1 3 2. 2 2 + 6 5 3. 1 5 2 +3 4 8 4. 5 3 2 +1 6 4 5. 3 2 5 −2 4 3 6. 1 1 6 −3 5 4 Place parentheses in the equations below to make true statements. Write “none needed” if the equation is already true. 7. 48 – 6 ÷ 3 • 2 = 7 8. -12 • 6 – 4 + 2 = -48 10. -6 – 9(2 – 6) 12. x – 10 = -2 4 Simplify. 9. 75 ÷ 3 • 52 Use a mental strategy to solve each equation. 11. -6(x + 1) = -48 MathLinks: Grade 8 (Student Packet 4) 21 Patterns and Linear Functions 2 4.4 Skill Builders, Vocabulary, and Review SKILL BUILDER 2 Compute. 1. (-7) + (9) 2. 4. (-50) – (-20) 5. 7. (10) – (-16) 8. 10. (-7) • (-9) 11. (16) + (-12) 3. (-32) + (-42) (50) – (-20) 6. (10) – (16) (16) – (-10) 9. (0) – (73) 12. -3(-7)(-4) -56 -8 Solve each inequality mentally. Then graph the solutions by drawing a ray. Check a point on this ray. 13. 6x > -30 14. x + (-4) ≤ -2 Solution: Solution: Graph: Graph: Check a point on the ray. Check a point on the ray. Place parentheses in the equations below to make true statements. Write “none needed” if the equation is already true. 15. 24 ÷ 2 + 1 • 3 = 24 MathLinks: Grade 8 (Student Packet 4) 16. 24 ÷ 2 + 1 • 3 = 39 22 Patterns and Linear Functions 2 4.4 Skill Builders, Vocabulary, and Review SKILL BUILDER 3 Compute. 1. 3 3 • 5 4 2. 5 9 • 12 10 3. 2 3 1 •2 3 4 4. 9 3 ÷ 10 5 5. 3 9 ÷ 5 10 6. 2 3 1 ÷2 3 4 Simplify. 7. 18 ÷ 2 • 3 ÷ 4 9. 5 – 10 ÷ (-2) + 3(-4 – 2) 11. -3( 2 − 6)2 3•5 +1 MathLinks: Grade 8 (Student Packet 4) 8. 18 ÷ (2 • 3) ÷ 4 10. 5 – 10 ( -2) + 3 ( -4 – 2) 12. -3(2 − 6)2 ÷ 3 • 5 +1 23 Patterns and Linear Functions 2 4.4 Skill Builders, Vocabulary, and Review SKILL BUILDER 4 Compute. 1. 3.08 + 1.91 2. 10.55 – 2.3 3. 0.8 • 0.3 4. 1.3 • 3.4 5. 0.48 ÷ 1.2 6. 6.4 ÷ 0.02 Write the expressions using symbolic notation and then evaluate for the given variables. 7. expression in words The cost of green shirts is g, and the cost of pink shirts is p. What is the cost of 6 green shirts and 8 pink shirts? expression in symbols evaluate for g = $8.50, p = $11.00 8. expression in words The number of DVDs is d, and they are being equally shared by a group of girls (g) and boys (b). What is the number of DVDs each person gets? expression in symbols evaluate for d = 56, g = 3, b = 4 MathLinks: Grade 8 (Student Packet 4) 24 Patterns and Linear Functions 2 4.4 Skill Builders, Vocabulary, and Review SKILL BUILDER 5 Draw the next step suggested by this pattern. Then complete the table and find a rule for the number of dots at step n. step1 Step # step 2 0 1 step 3 2 step 4 3 4 5 50 n Number of dots Break apart numbers 1. Label the horizontal and vertical axes and graph the data points. 2. Recursive Rule: 3. Explicit Rule: 4. How many dots are in step #100? 5. In what step number are there exactly 626 dots? 6. Is this a linear function? Explain. MathLinks: Grade 8 (Student Packet 4) 25 Patterns and Linear Functions 2 4.4 Skill Builders, Vocabulary, and Review SKILL BUILDER 6 For each explicit rule, complete the input-output table. 1. x 0 1 y 2. 0 x 0 0 y = 2 2 0 x–2 Rule: 1 1 y y = 2 2 0 x Rule: 1 1 y 3. 2 2x + 4 Rule: 1 y = 2 1 x 2 Find the missing values in each input-output table, and write an explicit rule for the data. 4. 5. 6. x y x y x y 1 7 -3 -15 -5 2 9 4 -24 0 3 11 12 -72 9 3 13 0 0 42 14 5 -6 -1 6 6 1 Rule: y = ___________ MathLinks: Grade 8 (Student Packet 4) Rule: y = ___________ Rule: y = ___________ 26 Patterns and Linear Functions 2 4.4 Skill Builders, Vocabulary, and Review SKILL BUILDER 7 Rick is saving to buy a pair of tickets to the UCLA football game against USC at the end of the season. Since the demand is very high, this will cost $240. He has $60 saved already. He will save $30 per month. 1. Use a table to determine how long it will take Rick to save enough money to buy the tickets. # of months (x) total saved in $ (y) 0 1 60 2. Write an equation of the form y = mx + b that can be used to find the total amount saved. Let m = the amount of money Rick is going to save each month Let b = the amount of money Rick already has saved Let x = the number of months he has been saving Let y = total amount saved 3. Use your equation to determine how long it will take Rick to save enough money to buy the tickets. 4. Find all combinations of nickels, dimes, and quarters that can make exactly $0.65. MathLinks: Grade 8 (Student Packet 4) 27 Patterns and Linear Functions 2 4.4 Skill Builders, Vocabulary, and Review SKILL BUILDER 8 Pat went jogging at the park. Use the graph to complete the table. Distance 600 yd 400 yd 0 1 min 4 min Graph is not drawn to scale. Time Time period 1. From 0 minutes to 1 minute 2. From 1 minute to 4 minutes 3. From 0 minutes to 4 minutes Distance traveled Average rate of speed 4. In what part of the jog did Pat run faster, the initial one minute or the last three minutes? Explain by referencing numbers and the shape of the graph. 5. Could this graph represent a linear function? Explain. Write an equation to match each statement. Then use a mental strategy to solve for the unknown value. 6. Paul has 36 strawberries, which is 3 times the number of strawberries that Barry has. How many strawberries does Barry have? 7. Kim ate 24 grapes on Tuesday. On Tuesday, she ate 13 less grapes than she did on Monday. How many grapes did she eat on Monday? MathLinks: Grade 8 (Student Packet 4) 28 Patterns and Linear Functions 2 4.4 Skill Builders, Vocabulary, and Review FOCUS ON VOCABULARY Across Down 2 an input-output rule 1 rate of change of a linear function 4 y-coordinate at which a graph crosses the y-axis 3 where two lines meet 6 altitude of a figure 5 _____ reasoning is reasoning based on examples 7 tables, graphs, symbols, words (2 words) 8 distance 9 function whose graph is a line 10 ratio with units attached 11 side of a figure 12 measure of size, expressed in square units 13 distance around a polygon MathLinks: Grade 8 (Student Packet 4) 29 Patterns and Linear Functions 2 4.4 Skill Builders, Vocabulary, and Review SELECTED REPONSE Show your work on a separate sheet of paper and choose the best answer(s). 1. The first three steps of a growing shape are shown below. Which choice gives the correct output value for the input value x = 5, and a correct explicit rule? Step # (x) 1 2 3 Number of dots (y) 1 3 5 x A. 7; y = 2x + 1 B. 9; y = 2x + 1 C. 7; y = 2x – 1 D. 9; y = 2x – 1 2. Jesse starts running at noon at a constant rate. The table shows the distance he travels at several 5-second intervals past noon. Time (in seconds past noon) Distance (in meters) 0 5 10 15 20 25 30 35 40 0 10 20 30 40 50 60 70 80 What is a rule for the relationship between the distance and time traveled by Jesse? A. d=t+5 B. d = 2t C. d = t + 10 D. d = 5t 3. In front of the grocery store, Darryl notices the shopping carts stacked together, and measures the lengths of the stacks of different numbers of carts. Which explicit rule below illustrates the relationship (shown in the table)? A. Number of carts (x) 1 2 3 4 5 Length in meters (y) 1.8 2.1 2.4 2.7 3.0 y = x + 0.3 B. MathLinks: Grade 8 (Student Packet 4) y = x + 1.8 C. y = 0.3x + 1.5 D. y = 0.3x + 1.8 30 Patterns and Linear Functions 2 4.4 Skill Builders, Vocabulary, and Review KNOWLEDGE CHECK Show your work on a separate sheet of paper and write your answers on this page. 4.1 Growing Shapes 1. Write a numerical expression for step 4. 5 + (2 • 1) 5 + (2 • 2) 5 + (2 • 3) step 1 step 2 step 3 step 4 2. Write a variable expression for the pattern illustrated above for step n. 4.2 Going to the Park Jamal starts rollerblading at 8:00:00 AM. The table below shows the distance Jamal travels at four-second intervals after 8:00:00 AM. 3. Complete the table using the established pattern. Time 0 4 8 (seconds past 8:00:00 AM) Distance 0 20 40 (meters traveled) 12 16 20 24 28 60 4. Write a rule about the relationship between Jamal’s time and distance. In words: Distance = ____________________________________________ In symbols: D = __________________________________________________ 4.3 Stacking Cups Marisol made a table showing the height of a stack of plates for a given number of plates. Number of Plates (x) Height of Plates (y) 1 2 3 4 5 2.2 cm 2.4 cm 2.6 cm 2.8 cm 3.0 cm 5. Make a graph of the data. Be sure to label the axes and give your graph a title. 6. Write a rule to determine the height of the plates given the number of plates. 7. What will be the height of 27 plates? MathLinks: Grade 8 (Student Packet 4) 31 Patterns and Linear Functions HOME-SCHOOL CONNECTION Here are some questions to review with your young mathematician. 1. Use inductive reasoning to complete the table and find an explicit rule. The first three steps are given. step2 step3 Step # (x) 1 2 3 4 Perimeter (y) 4 6 8 10 2. Two dogs, Wrigley and Squibbles, are running in the park to retrieve a ball. Based on the graph, who is running faster? How can you tell? 5 6 x Wrigley Distance step1 Squibbles Time Yuki made a table showing the heights of stacks of dishes. Number of dishes (x) Height of dishes (y) 1 2 3 4 5 2.25 in 2.75 in 3.25 in 3.75 in 4.25 in 3. Make a graph of Yuki’s data. 4. Write an explicit rule that best fits the data. 5. Use your rule to predict the height of a stack of 10 dishes. 6. Explain why the y-intercept of this graph is NOT 0, even though the height of zero dishes is 0. Parent (or Guardian) Signature ____________________________ MathLinks: Grade 8 (Student Packet 4) 32 Patterns and Linear Functions This page was intentionally left blank. MathLinks: Grade 8 (Student Packet 4) 33 Patterns and Linear Functions COMMON CORE STATE STANDARDS – MATHEMATICS STANDARDS FOR MATHEMATICAL CONTENT 8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. 8.EE.8a Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. 8.EE.8c Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. 8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. 8.F.3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give 2 examples of functions that are not linear. For example, the function A = s giving the area of a square as a function of its side length is not linear because its graph contains the points (1, 1), (2, 4) and (3, 9), which are not on a straight line. 8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. 8.F.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. 8.SP.2 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. 8.SP.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. STANDARDS FOR MATHEMATICAL PRACTICE MP2 Reason abstractly and quantitatively MP4 Model with mathematics. MP6 Attend to precision. MP7 Look for and make use of structure. © 2013 Center for Math and Teaching MathLinks: Grade 8 (Student Packet 4) 34