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. [PREREQUISITE SKILLS] interpreting and creating patterns; addition, subtraction, multiplication, and division of whole numbers *Teacher Note: Have students complete Lesson 28 before this lesson. [MATERIALS] Student pages S137 - S149 Transparencies T415, T417, T419, T421, T423, T425, T427, and T431 Foldable from Lesson 13 Wall grid Beans (84 per student pair) Colored pencils [ESSENTIAL QUESTIONS] 1. How can you create a numerical pattern? 2. How can you generate an ordered pair using two patterns? 3. How do you graph an ordered pair on the coordinate plane? [WORDS FOR WORD WALL] ordered pair, pattern, sequence, coordinate plane, x-axis, y-axis, horizontal, vertical, scale, term [GROUPING] Cooperative Pairs (CP), Whole Group (WG), Individual (I) *For Cooperative Pairs (CP) activities, assign the roles of Partner A and Partner B to students. This allows each student to be responsible for designated tasks within the lesson. [LEVELS OF TEACHER SUPPORT] Modeling (M), Guided Practice (GP), Independent Practice (IP) [MULTIPLE REPRESENTATIONS] SOLVE, Graph, Verbal Description, Pictorial Representation, Concrete Representation, Graphic Organizer
T404 Mathematics Success Level E [WARM-UP] (5 minutes IP, I, WG) S137 (Answers are on T414.) Have students turn to S137 in their books to begin the Warm-Up. Students will extend patterns and write rules for given patterns. Monitor students to see if any of them need help during the Warm-Up. Give students 3 minutes to complete the problems and then spend 2 minutes reviewing the answers as a class. {Verbal Representation} [HOMEWORK] (5 minutes) Take time to go over the homework from the previous night. [LESSON] (60 minutes CP, WG, I, M, GP, IP) SOLVE Problem (3 minutes WG, GP) T415, S138 (Answers on T416.) Have students turn to S138 in their books, and place T415 on the overhead. The first problem is a SOLVE problem. You are only going to complete the S step with students at this point. Tell students that during the lesson they will learn how to analyze patterns and relationships with ordered pairs.they will use this knowledge to complete this SOLVE problem at the end of the lesson. {SOLVE, Graphic Organizer} Generating Patterns for Two Given Rules (15 minutes M, WG, GP, CP, IP) T415, T417, T419, S138, S139, S140 (Answers on T416, T418, T420.) 10 minutes M, WG, CP, GP: Have students turn to S138, and place T415 on the overhead. Students will be completing two patterns with different rules. Pass out beans and colored pencils to student pairs. Assign the roles of Partner A and Partner B to cooperative pairs. {Concrete Representation, Pictorial Representation, Verbal Description, Graphic Organizer}
T405 MODELING Generating Patterns for Two Given Rules Step 1: Direct students attention to Problem 1. Explain to students that they will be working with patterns and comparing relationships between two patterns. Partner A, identify the rule for the first pattern. (add 1) Partner B, explain what this means. (You add 1 to each term to find the next term.) Partner A, what is the first value in the pattern? (0) Partner B, if you add 1 to 0, what is the sum? (1) Model for students how to place 1 bean in the column above the 1. Partner A, if you follow the pattern of add 1 to the 1 bean, how many beans are in the next box? (2) Partner B, place 2 beans in the column above the 2. Model the complete pattern as partners continue with the pattern of add 1 by adding 1 bean to each prior value and writing the values in the graphic organizer. (Have students leave the beans on the chart.) Have students discuss in pairs their observations about the pattern of beans for add 1. (Possible answers: The number of beans increases by 1 in each column; the number of beans increases as the pattern increases.) Step 2: Have students look at the second pattern on S138. Partner B, identify the rule for the second pattern. (add 2) Partner A, explain what this means. (You add 2 to each term to find the next term.) Partner A, what is the first value in the pattern? (0) Partner B, if you add 2 to 0, what is the sum? (2) Model for students how to place 2 beans in the column above the 2. Partner A, if you follow the pattern of add 2 to the 2 beans, how many beans are in the next box? (4) Partner B, place 4 beans in the column above the 4. Model the complete pattern as partners continue with the pattern of add 2 by adding 2 beans to each prior value and writing the values in the graphic organizer. (Have students leave the beans on the chart.)
T406 Mathematics Success Level E Have students discuss in pairs their observations about the pattern of beans for add 2. (Possible answers: The number of beans increases by 2 in each column; the number of beans increases as the pattern increases.) Step 3: Have students look at the second value in each pattern and identify the values. (1 and 2) Partner B, identify the third pair of values. (2 and 4) Partner A, identify the fourth pair of values. (3 and 6) Have students discuss the relationship between the corresponding values. (The second pattern s value is always twice the first.) Have partners check and confirm this with the remaining number of beans in the graphic organizer. Is this relationship true for all of the corresponding values? (Yes.) Step 4: Direct students attention to Problem 2 on S139. Explain to students that they will use colored pencils to create the pictures of the patterns for the rules on S139. Partner A, identify the rule for the first pattern. (add 1) Partner B, explain what this means. (You add 1 to each term to find the next term.) Partner A, what is the first value in the pattern? (0) Partner B, if you add 1 to 0, what is the sum? (1) Model for students how to draw 1 circle for the 1 row. Partner A, if you follow the pattern of add 1 to the 1 circle, how many circles will you draw in the next box? (2) Record in both books. Model the complete pattern as partners continue with the pattern of add 1 by drawing an additional circle in each row and completing the graphic organizer. Record. Have students discuss in pairs their observations about the pattern for add 1. (Possible answers: The number of circles increases by 1 in each row; the number of circles increases as the pattern increases.)
T407 Step 5: Have students look at the second pattern on S139. Partner B, identify the rule for the second pattern. (add 3) Partner A, explain what this means. (You add 3 to the term to find the next term.) Partner A, what is the first value in the pattern? (0) Partner B, if you add 3 to 0, what is the sum? (3) Model for students how to draw 3 circles in the row labeled 3. Partner A, if you follow the pattern of add 3 to the 3 circles, how many circles will you draw in the next row? (6) Record in both books. Model the complete pattern as partners continue with the pattern of add 3 by drawing the circles for each value of the pattern and completing the graphic organizer. Have students discuss in pairs their observations about the pattern of circles for Add 3. (Possible answers: The number of circles increases by 3 in each row; the number of circles increases as the pattern increases.) Step 6: Have students look at the second value in each pattern and identify the values. (1 and 3) Partner B, identify the third pair of values. (2 and 6) Partner A, identify the fourth pair of values. (3 and 9) Have students discuss the relationship between the corresponding values. (The second pattern s value is always three times the first.) Have partners check and confirm this pattern with the remaining values in the graphic organizer. Is this relationship true for all of the corresponding values? (Yes.)
T408 Mathematics Success Level E 4 minutes IP, CP: Have partners work together to complete the patterns for Problems 3-5 on S140. {Verbal Description, Graphic Organizer} 1 minute WG: Go over the answers to Problems 3-5 as a whole group. {Verbal Description, Graphic Organizer} Identifying Relationships Between Corresponding Terms (12 minutes M, WG, GP, CP, IP) T421, T423, S141, S142 (Answers on T422, T424.) 5 minutes M, WG, CP, GP: Have students turn to S141, and place T421 on the overhead. Students will be working to identify the relationship between corresponding terms from two patterns. {Verbal Description, Graphic Organizer} MODELING Identifying Relationships Between Corresponding Terms Step 1: Direct students attention to Problem 1. Explain to students that they will be examining two patterns and looking for any relationships between the terms in the patterns. Partner A, identify the first set of terms in the sequence. (0, 0) Partner B, explain the possible relationship between the two values. (They are equal.) Step 2: Have students look at the second row in the pattern table. Partner A, identify the second set of terms. (2, 4) Partner B, describe any possible relationship between the terms. (add 2 or multiply by 2) Record. Step 3: Have students look at the third row in the pattern table. Partner B, identify the third set of terms. (4, 8) Partner A, describe any possible relationship between the terms. (add 4 or multiply by 2) Record.
T409 Step 4: Have students look at the fourth row in the pattern table. Partner A, identify the fourth set of terms. (6, 12) Partner B, describe any possible relationship between the terms. (add 6 or multiply by 2) Record. Step 5: Have students look at the last row in the pattern table. Partner B, identify the fifth set of terms. (8, 16) Partner A, describe any possible relationship between the terms. (add 8 or multiply by 2) Record. Step 6: Have partners discuss the relationship between the terms in the two patterns. Is there a relationship that is the same for all of the pairs? (Yes.) Partner A, identify the relationship. (The terms in the add 4 sequence are always twice the terms in the add 2 sequence.) Record. Partner B, explain why. [The value in the second rule (add 4) is always twice the value of the first rule (add 2).] Record. Step 7: Complete the chart for Problem 2 using Steps 1-6 to model. 5 minutes IP, CP: Have partners work together to complete Problems 3-6 on S141 and S142. {Verbal Description, Graphic Organizer} 2 minutes WG: Go over the answers to Problems 3-6 as a whole group. Make sure that students understand the relationship between the corresponding terms in each set of patterns. {Verbal Description, Graphic Organizer}
T410 Mathematics Success Level E Graphing and Forming Ordered Pairs from Pattern Pairs (18 minutes M, WG, GP, CP, IP) T425, T427, S143, S144, S145, S146 (Answers on T426, T428, T429, T430.) 10 minutes M, WG, CP, GP: Have students turn to S143, and place T425 on the overhead. Students will be working to identify the ordered pairs from patterns and graphing them on the coordinate plane. Post the wall grid to plot the ordered pairs. {Verbal Description, Graph, Concrete Representation, Pictorial Representation, Graphic Organizer} MODELING Graphing and Forming Ordered Pairs from Pattern Pairs Step 1: Have students look at the coordinate plane for Problem 1. Explain to students that the values in patterns can be used to identify points on the coordinate plane. Partner A, identify the label on the horizontal axis. (x-axis) Partner B, identify the label on the vertical axis. (y-axis) Partner A, identify the first two corresponding values in the chart below the coordinate plane. (0, 0) Explain to students that those values, (0, 0), are a point on the coordinate plane. Ask students to find the point where the zeros meet and mark it with a dot on page S143 as you model on the wall grid. Step 2: Have students look at the pattern in the graphic organizer. Partner A, identify the first pattern. (add 2) Explain to students that we are going to use add 2 to mark a pattern on the coordinate plane. Model on the wall grid as students work on the coordinate grid on S143. Partner B, identify the starting point of the pattern. (0, 0) Partner A, identify the scale on the x-axis. (0-20 by 2s) This means that the value of each space is 2. Model for students how to add 2 by drawing a horizontal arrow from the 0 to the 2. Write in the value of +2 on the arrow. Explain that the first coordinate always tells us how many to move to the right.
T411 Step 3: Have students look back at the pattern in the graphic organizer. Partner A, identify the second pattern. (add 4) Explain to students that they will start from the position after completing the first pattern and this time they will go up a value of 4. Partner B, identify the value of each square. (2) Partner A, how many spaces do you go up for add 4? (Two because each space represents a value of 2.) Model on the wall grid as students work on the coordinate grid on S143. Model for students how to add 4 by drawing a vertical arrow up to the 4. Write in the value of +4 next to the arrow. Explain that the second coordinate always tells us how many to move up. Model how to draw a point at (2, 4). Step 4: Continue with the pattern (add 2, add 4) by drawing the arrows and placing a point on the coordinate plane to represent the pattern. Partner A, identify the next point in the (add 2, add 4) pattern. (4, 8) Partner B, identify the next point in the (add 2, add 4) pattern. (6, 12) Partner A, identify the last point in the (add 2, add 4) pattern. (8, 16) Step 5: Have students look at the graphic organizer below the coordinate plane. Have students discuss what they notice about the relationship between the values in the chart and the points that they graphed on the coordinate plane. (The values in the chart are the same as the coordinates of the points.) Step 6: Explain to students that any time we have a pattern, the values of the pattern can be written as an ordered pair which corresponds to a position on the coordinate plane. Partner A, identify the first set of terms in each pattern. (0, 0) Partner B, identify the second ordered pair. (2, 4) Record. Partner A, identify the third ordered pair. (4, 8) Record.
T412 Mathematics Success Level E Partner B, identify the fourth ordered pair. (6, 12) Record. Partner A, identify the last order pair. (8, 16) Record. Step 7: Use Steps 1-6 to model Problem 2 on S144 (T427). Model all the steps showing the arrows for the values from the pattern so that students can see how the pattern is created before creating the ordered pairs. 6 minutes IP, CP: Have partners work together to complete Problems 3-4 on S145 and S146. Students will mark the pattern on the coordinate plane and then create the ordered pairs as in Problems 1 and 2. {Verbal Description, Graphic Organizer, Graph, Pictorial Representation} 2 minutes WG: Go over the answers to Problems 3-4 as a whole group. Use the wall grid and give students the opportunity to come up and graph the points on the grid for each problem. {Verbal Description, Graphic Organizer, Graph, Pictorial Representation} Expressions Foldable (5 minutes M, GP, WG) MODELING Expressions Foldable Step 1: Have students take out their Expressions Foldable from Lesson 13. Create a transparency to model for students what should be written on each page. Step 2: On the third flap of the Expressions Foldable, model for students how to label the section Patterns and Relationships. Discuss with students the steps for patterns and relationships and then write the appropriate steps under the flap. Use your foldable to reference what you want written in the foldable.
T413 SOLVE Problem (5 minutes GP, WG ) T431, S147 (Answers on T432.) Remind students that the SOLVE problem is the same one from the beginning of the lesson. Complete the SOLVE problem with your students. Ask them for possible connections from the SOLVE problem to the lesson. (analyzing patterns and relationships with ordered pairs) {SOLVE, Verbal Description, Graphic Organizer} If time permits (10 minutes IP, I, CP) S148 (Answers on T433.) Have students complete the pattern chart and graph the points on page S148. [CLOSURE] (2 minutes) To wrap up the lesson, go back to the essential questions and discuss them with students. How can you create a numerical pattern? (Use the given rule and apply it to each value in order to determine the next value.) How can you generate an ordered pair using two patterns? (Form the ordered pair by using the corresponding terms from two patterns.) How do you graph an ordered pair on the coordinate plane? [Graph the first value (x-value) on the horizontal axis and the second value (y-value) on the vertical axis.] [HOMEWORK] Assign S149 for homework. (Answers on T434.) [QUIZ ANSWERS] T435- T438 The quiz can be used at anytime as extra homework or to assess how students progress on understanding analyzing patterns and relationships with ordered pairs.