Using a Scatter Plot to Represent a Sequence. Goal Use scatter plots to represent number sequences.

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1 4.5 Using a Scatter Plot to Represent a Sequence Goal Use scatter plots to represent number sequences. STUDENT BOOK PAGES Guided Activity Prerequisite Skills/Concepts Use a table of values to solve patterning problems. Demonstrate an understanding of coordinates in a grid and plot points. Construct scatter plots. Assessment for Feedback Students will use scatter plots to represent number sequences Preparation and Planning Pacing (allow 5 min for previous homework) Materials Vocabulary/ Symbols 1 min Introduction 2 min Teaching and Learning 1 min Consolidation grid paper a ruler coloured pencils toothpicks a calculator Scaffolding for Lesson 4.5 (Master) pp scatter plot, coordinates, ordered pair, x-axis, y-axis Workbook pp Recommended 7, 8*, 9, 11 a) b) (Application of Practice Learning), Communication Related 1, 11 c) d) e) (Problem Solving/ Practice Thinking) Extending: 12 (Application of Learning, Knowledge and Understanding) Learning Skills Class Participation Independent Work Mathematical Reasoning and Proving, Processes Representing *Key Assessment of Learning Question (See chart on p. 39.) Expectations represent linear growing patterns (where the terms are whole numbers) using concrete materials, graphs[, and algebraic expressions] model real-life linear relationships graphically [and algebraically, and solve simple algebraic equations using a variety of strategies, including inspection and guess and check] represent linear growing patterns, using a variety of tools and strategies make predictions about linear growing patterns, through investigation with concrete materials model real-life relationships involving constant rates where the initial condition starts at, through investigation using tables of values and graphs What You Will See Students Doing When students understand Students will be able to use data from a table of values to represent ordered pairs (coordinates) and then plot the coordinates on the correct locations on a coordinate grid. They will be able to name each axis of the grid with the appropriate column name from the table of values. Meeting Individual Needs If students misunderstand Students may not readily recognize ordered pairs given a table of values. Carry out examples with students so that they see an ordered pair is a number from one column in the table of values and the corresponding number (same row) from a different column. Once the ordered pairs are derived, ensure that students understand how to plot the ordered pairs as coordinates on a grid. Some students may need help in naming the axes of the grid. Extra Challenge Suggest that students draw two scatter plots using the same set of data, but reversing the names on the axis for each. Ask them to determine whether it matters which variable is named for each axis. Some students may wish to research what the best-fit line on a scatter plot means. They can share their findings with their classmates. Extra Support Rather than having students directly plot coordinates onto the grid from the table of values, have students first list in writing the set of ordered pairs from the numbers in two columns in a table of values. Point out that the ordered pairs are written as (number of triangles, perimeter of path). Math Background In this lesson, scatter plots are used to help students visually see a relationship or pattern between two sets of numbers. The numbers are taken from a table of values. The axes on a grid are named from the titles of the table columns. The numbers in the columns then become sets of ordered pairs or coordinates that can be plotted on the grid. Each pair of numbers from the table, when plotted, gives a visual representation of the data. It may be easier for some students to determine relationships from the scatter plot than from lists of numbers. Copyright 26 by Thomson Nelson 35

Dealing with Homework (Pairs) about 5 min Students, working in pairs, can share and solve each other s problems they created for Question 13 of Lesson 4.

2 Dealing with Homework (Pairs) about 5 min Students, working in pairs, can share and solve each other s problems they created for Question 13 of Lesson 4.4. Those students who shared aloud in the closure of the previous lesson can ask their partner to solve any Practising question from Lesson Introduction (Small Groups/Whole Class) about 1 min Students should be familiar with scatter plots and plotting ordered pairs on a grid from earlier grades. On an overhead grid, have a number of students identify locations you point to by naming the ordered pair. Explain that another name for ordered pairs is coordinates. Next invite students to draw points on the grid according to the coordinates you give. Ensure that all students know that the first coordinate describes movement along the x-axis, and the second coordinate describes movement along the y-axis. What is a scatter plot? A scatter plot is a type of graph that has a whole bunch of points on a grid. How do you plot the points? The axes on the grid have names and when you plot points of ordered pairs, you move the first number on the x-axis and the second number on the y-axis, and that gets you to the point. Another name for ordered pairs is coordinates. 2. Teaching and Learning (Whole Class) about 2 min Learn about the Math Read through the problem and the central question together as a class. Go through prompts A to C together. Make sure that students understand that the two columns in the table of values represent the coordinates of the ordered pairs. On the overhead, copy and complete the table of values. Explain how the two numbers in the first row (1 and 3) can become the ordered pair (1, 3), which can be plotted on the grid. Show how to locate that point on the overhead grid. Repeat the process for the next two coordinates. In small groups ask students to complete the scatter plot for the final five points. Then have students work in small groups to complete prompts D and E. It is important that everyone in the small group sees how to turn numbers in a table of values into a scatter plot. Students can then begin the Reflecting questions. What are the coordinates from the table of values that need to be plotted? (1, 3), (2, 4), (3, 5), (4, 6), (5, 7), (6, 8), (7, 9), (8, 1) How can you use the pattern of points on the scatter plot to predict points that are not there? The points that are already plotted form a line. If you continue that line, you can predict other points. Answers to Learn about the Math A. Number of Perimeter of B. triangles path (units) C Perimeter of path (units) Number of triangles (Answers to Lesson 4.5, Learn About the Math continued on p. 7) Omar s Path 36 Chapter 4: Patterns and Relationships Copyright 26 by Thomson Nelson

Why do the coordinates in these two scatter plots form a line? The coordinates form a line because the numbers we are using are in a sequence. Answers to Reflecting 1.

3 Reflecting Questions 1 to 4 will extend students thinking about how to use scatter plots. How can you use a scatter plot to make predictions? You look at how the coordinates are placed on the graph and determine if there is a pattern. Then you can extend the pattern to make predictions. Why do the coordinates in these two scatter plots form a line? The coordinates form a line because the numbers we are using are in a sequence. Answers to Reflecting 1. They are the same because on both scatter plots the points form a line. They are different because the line on the second scatter plot is much steeper. 2. You extend the line formed by the points on the scatter plot. Then you look along the x-axis for the number of triangle stones and then go straight up to the line. When you reach the line you go straight left to the y-axis. That number will be the number of border pieces. 3. You can use the patterns from both a scatter plot and a table of values to predict. 4. When you draw the line formed by the point, it is not necessarily accurate. When you use a pattern rule from a table of values it is accurate. 3. Consolidation about 1 min Solved Examples (Pairs) Have students work in pairs to read through the examples. Have the first partner explain how the table of values was created. Then have the other partner explain how to make the scatter plot from the table of values. Students could make sure they understand how to predict points that are not given. A Work with the Math Checking (Pairs) For Questions 5 and 6, have students work in pairs. Ensure that each student knows how to do each of the following: make a table of values; create a scatter plot from the table of values; use the scatter plot to determine given values; and use the scatter plot to make predictions. Answers to Checking 5. Draw a scatter plot showing the pattern and extend the points to show 12 posts. Therefore, Mohammed will need 12 posts and 36 rails for a fence that is 12 sections long. Number of rails and posts Number of Rails and Posts vs. Number of Sections Number of sections Copyright 26 by Thomson Nelson 37

6. a) Number of toothpicks 3 28 26 24 22 2 18 16 14 12 1 8 6 4 2 Number of Toothpicks vs.

4 6. a) Number of toothpicks Number of Toothpicks vs. Term Number Term number b) If I follow the line past (5, 16), I can see that I would need 28 toothpicks in the 9th term of this sequence. c) If I follow the line past (5, 16) I can see that I can make the 7th term in the sequence with 22 toothpicks B Practising (Individual) Creating scatter plots is the focus of these questions. For the first question, a table of values is provided. For the remaining questions students must first make the table of values and then the scatter plot. Answers to Key Assessment Question 8. Number of rails and posts Number of Rails and Posts vs. Number of Sections Number of sections a) For 8 sections, I must look at the scatter plot to see how many rails and posts are needed. Therefore, 41 rails and posts are needed for 8 sections. b) For 14 sections, I must look at the scatter plot to see how many rails and posts are needed for 14 posts. Therefore, 71 rails and posts are needed for 14 sections. C Extending (Individual) For Question 12, students must recognize that more columns are needed for the table of values. Key Assessment of Learning Question (See chart on p. 39.) Closing (Whole Class) Have students write a short note to a classmate who has missed this lesson and the previous lessons. The note should explain how to use a table of values to make a scatter plot for a sequence of numbers and how to make predictions about the sequence using the scatter plot. Chapter Project Link: In this lesson, students learn about using a scatter plot to represent a sequence and make predictions. They can apply and extend their understanding of these concepts in the Chapter Project p. 64. Follow-Up and Preparation for Next Class Ask students to observe and think about any patterns they have encountered in nature. That is, not patterns of numbers, but patterns in numbers of things. As next day s lesson is the Chapter Self-Test, have students make a list of additional questions they might have about patterns and relationships in this chapter and bring these questions to the next class. 38 Chapter 4: Patterns and Relationships Copyright 26 by Thomson Nelson

5 STUDENT BOOK PAGE 142 Curious Math: The Fibonacci Sequence Using Curious Math The sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, is known as the Fibonacci sequence. Any new term can be determined by the sum of the previous two terms. The Fibonacci sequence occurs in nature, art, music, and mathematics. The number of leaves that plants tend to have and the number of petals flowers tend to have are both Fibonacci numbers. In objects such as seeds in a sunflower head or sea shells, the number of the spirals that curve to the left and the number of spirals that curve to the right will be adjacent numbers in the Fibonacci sequence. Encourage students to look for real-life examples of the Fibonacci sequence. Answers to Curious Math 1. a) 8 spirals go to the left and 13 spirals go to the right. b) The numbers are consecutive Fibonacci numbers. 2. a) i) 1 ii) 3 iii) 8 b) The numbers are all numbers in the Fibonacci sequence. 3. a) For example, 1, 1, 2, 3; 1 3 and 1 2; the products have a difference of 1 b) For example, 2, 3, 5, 8; 2 8 and 3 5; the products have a difference of 1 5, 8, 13, 21; 5 21 and 8 13; the products have a difference of 1 c) For four sequential Fibonacci numbers, if you multiply the first and last numbers and then multiply the two middle numbers, the difference between the two products is always 1. Assessment of Learning What to Look for in Student Work Assessment Strategy: Written Answer Application of Learning, Communication Key Assessment Question 8 Mohammed is building another rail fence that has sections like this. Use a scatter plot to determine the number of posts and rails Mohammed will need to build each fence. a) 8 sections long b) 14 sections long Application of Learning Demonstrates limited ability to transfer mathematical knowledge has difficulty using scatter plots to determine the number of posts and rails needed) Demonstrates some ability to transfer mathematical knowledge demonstrates some ability to use scatter plots to determine the Demonstrates considerable ability to transfer mathematical knowledge uses scatter plots to determine the Demonstrates sophisticated ability to transfer mathematical knowledge demonstrates sophisticated ability to use scatter plots to determine the Communication Few conventions (e.g., scales, Some conventions (e.g., scales, Most conventions (e.g., scales, Almost all conventions (e.g., scales, Copyright 26 by Thomson Nelson 39

Mathematics Success Level E

Mathematics Success Level E T403 [OBJECTIVE] The student will generate two patterns given two rules and identify the relationship between corresponding terms, generate ordered pairs, and graph the ordered pairs on a coordinate plane.