Mathematics Success Level D

Transcription

1 T398 Mathematics Success Level D [OBJECTIVE] The student will identify, analyze, create, and extend numeric patterns. [PREREQUISITE SKILLS] multiples [MATERIALS] Student pages S135 S142 Transparencies T423, T425, T427 T429 and T431 Index cards (6 blank cards for each group of students) Teacher cards: Decks 1, 2, and 3 and 1 set of 6 blank index cards Student cards: 1 set of each group deck (6 total sets) Paper for foldable (1 sheet per student) [ESSENTIAL QUESTIONS] 1. What are the characteristics of a numeric pattern? 2. How can we extend a numeric pattern? 3. How can we find missing terms in a numeric pattern? [WORDS FOR WORD WALL] numeric pattern, sequence, term, rule, extending the pattern, missing 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, Graphic Organizer, Verbal Description, Pictorial Representation, Concrete Representation [WARM-UP] (5 minutes IP, I, WG) S135 (Answers on T422.) Have students turn to S135 in their books to begin the Warm-Up. Students will work with skip counting. 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 Description}

2 Mathematics Success Level D T399 [HOMEWORK] (5 minutes) Take time to go over the homework from the previous night. [LESSON] (60 minutes M, GP, I, IP, WG, CP) SOLVE Problem (3 minutes M, WG) T423, S136 (Answers on T424.) Have students turn to S136 in their books, and place T423 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 about numeric patterns. They will use this knowledge to complete this SOLVE problem at the end of the lesson. {SOLVE, Graphic Organizer} Numeric Patterns Concrete (3 minutes M, WG) 3 minutes M, WG: Use the following activities to introduce students to numeric patterns {Verbal Description, Concrete Representation} MODELING Numeric Patterns Concrete Step 1: Begin with the first student in the room and ask students to skip count by 2 s beginning with 2 until all students have had a turn. This will create a numeric pattern, or a pattern with numbers. Ask students the following questions. What did the group just do? (skip counted by 2 s around the room) Was there a pattern? (Yes, skipped every other number, skip counted by 2 s, etc.) How many students had a chance to respond? (Answers will vary.) How long was the list of numbers? (The same amount as the number of students.) What was the first number? (2) The second number or term? (4) Step 2: Begin with the first student again and ask students to skip count by 5 s, starting with 5, until 100 is reached. This will create a different numeric pattern from Step 1. Ask students the following questions. What did the group just do? (skip counted by 5 s around the room) Was there a pattern? (Yes.) How did it differ from the first pattern? (skip counted by 5 s rather than 2 s) How many students had a chance to respond? (20) How long was the list of numbers? (20 responses or to 100) What was the first number? (5) The second number or term? (10)

3 T400 Mathematics Success Level D Numeric Patterns Moving to Pictorial Terms (10 minutes M, GP, WG, IP, CP) Teacher Card Deck 1 and Group Cards, T423, S136 (Answers on T424.) 5 minutes M, CP, GP, WG: Have students turn to S136 in their books, and place T423 on the overhead. Have sets of student cards ready to distribute. Students will be working in small groups of 4-5. Assign the roles of Partner A and Partner B. {Verbal Description, Concrete Representation, Graphic Organizer} MODELING Numeric Patterns Moving to Pictorial Terms Step 1: Ask students to think about the activity they did with the multiples of 2 and ask the following questions. What did the group do when they counted by 2 s? (skip counted by 2 s around the room) Put Teacher Card Deck 1 on the chalk ledge or tape it to the board (NOT in order). Are these cards in the correct order to show the multiples of 2? (No.) Have a student rearrange the numbers in the correct order Partner A, explain the correct sequence to Partner B. Explain to students that the list of numbers creates a sequence for a numerical pattern. Record the definition and give an example for the sequence on the graphic organizer. NUMERIC PATTERNS Sequence Definition: the list of numbers for a numeric pattern Example: 2, 4, 6, 8, 10, 12

4 Mathematics Success Level D T401 Step 2: Explain to students that the number on each card is a term in the numeric pattern. Record the definition and give an example for the term on the Numeric Patterns graphic organizer. Partner B, identify the first term. (2) Partner A, identify the second term. (4) NUMERIC PATTERNS Sequence Definition: the list of numbers for a numerical pattern Example: 2, 4, 6, 8, 10, 12 Term Definition: each number in the sequence Example: The first term in the sequence is 2. Ask students to think about the activity they did with counting by 2 s. Partner B, explain what we added to the first term to get to the second term. (2) Partner A, explain what we added to the second term to get to the third term. (2) Partner B, explain how the pattern continued. (add 2)

5 T402 Mathematics Success Level D Step 3: Explain to students that a rule was created to continue the numerical pattern. The rule is a relationship among the terms and should apply the same way to each term. Record the definition and give an example for the rule on the Numeric Patterns graphic organizer on S136. NUMERIC PATTERNS Sequence Definition: the list of numbers for a numerical pattern Example: 2, 4, 6, 8, 10, 12 Term Definition: each number in the sequence Example: The first term in the sequence is 2. Rule Definition: a relationship among the terms Example: The rule is + 2. What rule was followed to determine the numerical pattern? (+ 2) 3 minutes IP, CP: Distribute sets of student cards to groups of 4 5 students. Have students work with their groups to put the cards in a numerical sequence. Monitor closely to make sure students are using the appropriate vocabulary. {Verbal Representation, Concrete Representation} 2 minutes WG: Have students come back together as a class and share their sequences. As each numerical sequence is shared, students should record the sequence. Leave the cards in sequence by group for the next activity. {Concrete Representation, Verbal Description}

6 Mathematics Success Level D T403 Numeric Patterns Moving to Pictorial Rules (10 minutes M, GP, WG, IP, CP) Teacher Card Deck 2 and Group Cards, T423, S136 (Answers on T424.) 6 minutes M, GP, WG, CP: Students will use the sets of cards in their group. Have students work in partners and small groups. {Verbal Description, Concrete Representation, Graphic Organizer} MODELING Numeric Patterns Moving to Pictorial - Rules Step 1: Place the teacher Card Deck 2 on the front board. The cards should not be in the order of the numerical pattern. Ask students the following questions. How many cards are on the board? (6) Do you see a numerical pattern in the order? (No.) Partner A, identify the smallest term. (1) Place the term at the beginning of the sequence. 1 Partner B, identify the smallest term from the remaining cards. (2) Place the term next to the Partners, discuss what rule you could use to find the next term. (+1 or 2) Step 2: Direct attention to the terms. Partner A, identify the next smallest term from the remaining cards. (4) Place the term after the Did you add two or multiply by two to find the term of 4? (could do either one) Have we established a single rule? (No.) Partner A, identify the next smallest term from the remaining cards. (8) Place the term after the Did you add two or multiply by two to find the term of 8? (multiplied by 2) What is the rule of the numerical sequence? ( 2) What is the order of the remaining two terms? (16, 32) Place the terms in order in the numerical sequence

7 T404 Mathematics Success Level D Step 3: Direct students attention to the Finding a Rule graphic organizer on S136 and ask the following questions. What was the first step you used to find the numerical sequence of the terms on the cards? (Look at the smallest term.) Record on the graphic organizer. What was the next step? (Look at the 2nd term and determine rules that could apply from the 1st to 2nd term.) Record. What was the third step? (Look at the 3rd term and determine rules that could apply from the 2nd to 3rd term.) Record. What was the fourth step? (Continue to look at the order of the terms until all rules but one can be eliminated.) Record. 2 minutes IP, CP: Assign group numbers 1-6. Have students look at their card decks again and establish the rule for the deck using the graphic organizer. Then have them record the rule for their pattern sequence at the bottom of S136. Monitor closely to make sure students are using the appropriate vocabulary. {Verbal Representation, Concrete Representation, Graphic Organizer} 2 minutes WG: Have students come back together as a class and share their results. As each rule is shared, students should record it. Collect the card sets to be used again in another activity. {Concrete Representation, Verbal Description, Graphic Organizer} Extending Numerical Patterns Abstract (8 minutes M, GP, WG, IP, CP) T425, S137 (Answers on T426.) 3 minutes M, CP, WG, GP: Have students turn to S137 in their books, and place T425 on the overhead. Have students work in partners. Assign the roles of Partner A and Partner B for designated tasks. {Verbal Description, Graphic Organizer}

8 Mathematics Success Level D T405 MODELING Extending Numerical Patterns Abstract Step 1: Direct students attention to Problem 1 and ask the following questions. What do you notice about this numeric pattern? (It is not completed.) How many spaces are needed to complete the pattern? (2) Explain to students that this is called extending the pattern. Step 2: Direct attention to the terms. How many terms have been provided? (4) Partner A, identify the rule that was used from the first to the second term. (multiply by 3, or add 6) Partner B, identify which of these rules will work from the 2nd term to the 3rd term. (multiply by 3) Will the rule add 6 work from the 2nd term to the 3rd term? (No.) Will the rule multiply by 3 work from the 3rd term to the 4th term? (Yes.) Record the rule. If we use the rule multiply by 3 and multiply the 4th term by 3, what is the 5th term? (243) Record. How can we extend to the 6th term? (Use the rule multiply by 3 and multiply the 5th term by 3 to get a product of 729.) Record the term. Step 3: Direct students attention to Problem 2. Partner A, identify how many spaces are needed to complete the pattern. (2) Partner B, identify how many terms have been provided. (4) Partner A, explain the rule that was used from the first to the second term. (divide by 2, or subtract 64) Partner B, explain which of these rules will work from the 2nd term to the 3rd term. (divide by 2) Will the rule subtract 64 work from the 2nd term to the 3rd term? (No.) Will the rule divide by 2 work from the 3rd term to the 4th term? (Yes.) Record the rule. If we use the rule divide by 2 and divide the 4th term by 2, what is the 5th term? (8) Record. If we use the rule divide by 2 and divide the 5th term by 2, what is the 6th term? (4) Record.

9 T406 Mathematics Success Level D 3 minutes IP, CP: Have students work with their partners to complete Problems 3-8. Have Partner A complete Problems 3, 5, and 7; and Partner B complete Problems 4, 6, and 8. Have partners share their answers and discuss. Monitor closely to make sure students are using the appropriate vocabulary. {Verbal Representation, Graphic Organizer} 2 minutes WG: Have students come back together as a class and share their results. {Verbal Description, Graphic Organizer} Missing Terms (8 minutes M, GP, WG, IP, CP) T427, S138 Teacher Card Deck 2-3 and Group Cards, (Answers on T428.) 3 minutes M, WG, CP, GP: Have students turn to S138 in their books, and place T427 on the overhead. Remove the fourth or fifth card from the sequence in each group of student cards and pass one set of cards to each group of students. {Verbal Description, Graphic Organizer, Concrete Representation} MODELING Missing Terms Step 1: Place Teacher Card Deck 2 on the board, creating the pattern 1, 2, 4, 8, 32. Leave out the 16 card, but leave a space for it in the pattern. Ask students the following questions Partner A, explain the rule for the first two terms. (either add 1 or multiply by 2) Partner B, explain the rule for the second and third terms. (Multiply by 2.) Can add 1 be used? (No.) Does the rule multiply by 2 apply for the third and fourth terms? (Yes.) Partner A, explain the rule for the pattern? (Multiply by 2.) Look at the card pattern. Is the fifth term 32? (No, there is a space.) Partner B, explain what happened? (There is a missing term.) If the term 16 were added, would the last term be correct in the pattern sequence? Why? (Yes, because 16 multiplied by 2 is 32.) Add the card with the missing term

10 Mathematics Success Level D T407 Record the pattern sequence, including the missing term. Circle the missing term. Deck Pattern Teacher Deck 2: 1, 2, 4, 8, 16, 32, Step 2: Display Teacher Card Deck 3, leaving out the 12. Do not leave a space for the 12. Ask students the following questions. Partner A, explain the rule for the first two terms. (Either add 3 or multiply by 2.) Partner B, explain the rule for the second and third terms. (Add 3.) Partner A, explain the rule for the pattern. (Add 3.) Partner B, explain what the fourth term should be according to the sequence of the terms. (12) Look at the card pattern. Is the fourth term 12? (No, it is 15.) What do you think happened? (A term is missing.) Step 3: Direct students attention to the last two terms, 15 and 18. Partner A, explain how the rule add 3 applies? ( = 18) Partner B, determine what term is missing from the pattern sequence. (12) Add the card with the missing term. Partner A, what do you notice about the 1st, 3rd, and 5th terms? (3, 9 and 15 are odd numbers.) Partner B, what do you notice about the 2nd, 4th, and 6th terms? (6, 12, and 18 are even numbers.) What can you conclude from this? (When you add two odd numbers, the sum is even; when you add an even number and an odd number, the sum is odd.) Record the sequence, including the missing term. Circle the missing term. Deck Pattern Teacher Deck 2: 1, 2, 4, 8, 16, 32, Teacher Deck 3: 3, 6, 9, 12, 15, 18

11 T408 Mathematics Success Level D Step 4: Direct students attention to the teacher s blank set of index cards and ask the following questions. If this set of cards shows the number of terms in the pattern sequence, how many terms will be in the sequence? (6) What must we know before creating a pattern? (the rule) Partner A, if we use the rule multiply by 2 and begin the pattern with the term 5, what is the first term? (5) Partner B, determine the second term. (10) What are the third, fourth, fifth, and sixth terms? (20, 40, 80, 160) Do we have any missing terms in the pattern? (No.) If I remove the 80, can we determine the missing term? How? (Yes, we use the rule multiply by 2 and compare the terms.) 3 minutes IP, CP: Distribute 6 blank index cards to each group. Have students work with their groups to put the student cards in a numerical sequence, establish a rule, and determine the missing term. Then, have students create a pattern using the blank set of cards and share with another group, having students remove one card to find the missing term and then recording the correct sequence. Monitor closely to make sure students are using the appropriate vocabulary. {Verbal Representation, Concrete Representation, Graphic Organizer} 2 minutes WG: Have students come back together as a class and share their results. As each numerical sequence is shared, students should record the sequence on S138 (T428). Have students circle the missing term. {Concrete Representation, Verbal Description, Graphic Organizer} Working with Numeric Patterns (7 minutes M, GP, WG, IP, CP) T429, S139 (Answers on T430.) 3 minutes M, GP, WG, CP: Have students turn to S139 in their books, and place T429 on the overhead. Have students work in partners. {Verbal Description, Graphic Organizer}

12 Mathematics Success Level D T409 MODELING Working with Numeric Patterns Step 1: Direct students attention to Problem 1 and ask the following questions. Partner A, explain what this problem asks us to do. (Extend the pattern.) What are the steps we will use to do this? (Look at the first two terms and decide what rule might be used to get from the first term to the second term. Decide which of the rules would work for the third term. Look at the next term to establish the rule. Use the rule to extend the terms.) Partner B, determine the rule. (+ 8) Partner A, identify the first missing term to be extended. (33) Record. Partner B, identify the second missing term to be extended. (41) Record. Partner A, identify the third missing term to be extended. (49) Record. Partner B, make a prediction about all the terms. (They will be odd.) Partner A, explain why. (Adding odd + even = odd.) 1. Extend the pattern. 1, 9, 17, 25, 33, 41, 49 Rule: Add 8 Step 2: Direct students attention to Problem 5 and ask the following questions. Partner B, explain what this problem asks us to do. (Find the missing term.) What are the steps we will use to do this? (Look at the first two terms and decide what rule might be used to get from the first term to the second term. Decide which of the rules would work for the third term. Look at the next term to establish the rule. Use the rule to decide which term is missing.) Partner A, determine the rule. (Subtract 3.) Partner B, identify the missing term. (21) Rewrite the pattern sequence, including the missing term. Partner A, make a prediction about the next term. (even number) Partner B, explain why. (Subtracting odd odd = even; subtracting even odd = odd.)

13 T410 Mathematics Success Level D Step 3: Direct students attention to Problem 7 and ask the following questions. Partner A, identify what this problem asks us to do. (Create a 6 term pattern sequence.) What are the two things we must do to create a pattern sequence? (Determine a rule and what the first term will be.) Partner B, if the rule is +6 and the first term is 9, what is the second term? (15) Record. What are the last 4 terms? (21, 27, 33, 39) Record. 2 minutes IP, CP: Have students work with their partners to complete the rest of the problems. Monitor closely to make sure students are using the appropriate vocabulary. {Verbal Representation, Graphic Organizer} 2 minutes WG: Have students come back together as a class and share their results. {Graphic Organizer, Verbal Description} Foldable (4 minutes GP, WG) Use the following activity to help students make a numeric pattern foldable. {Verbal Representation, Graphic Organizer} MODELING Numeric Pattern Foldable Pass out one sheet of colored paper to each student. Use the following activity to model for students how to fold the piece of paper. Together with the students, complete the foldable. Step 1: Fold piece of paper horizontally, hamburger style. Then fold vertically, hot dog style. Step 2: On the corner with the fold, fold the corner down 1 inch to form a triangle. Step 3: Open the paper to show the following:

14 Mathematics Success Level D T411 Step 4: In the center diamond write Numeric Patterns. In the top left hand corner of this section, write What Does It Look Like? In the top right hand corner of this section, write Finding the Rule. In the bottom left hand corner of this section, write Extending the Pattern. In the bottom right hand corner of this section, write Finding a Missing Term. Step 5: Complete the rest of the foldable, filling in all sections. You can create a transparency to model for students what should be written on it. Use your foldable to reference what you want written in the foldable. SOLVE Problem (5 minutes GP, WG) T431, S140 (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. (Students will be working with numeric patterns.) {SOLVE, Verbal Description, Graphic Organizer} If time permits (10 minutes IP, I) S141 (Answers on T433.) Have students complete Problems 1 10 on S141. [CLOSURE] (2 minutes) To wrap up the lesson, go back to the essential questions and discuss them with students. What are the characteristics of a numeric pattern? (It is a group of numbers that follows a pattern. It has a sequence that consists of terms.) How can we extend a numeric pattern? (Determine the rule and apply it to the terms to be extended.) How can we find missing terms in a numeric pattern? (Determine the rule and apply it to the terms in the pattern sequence. Determine where the missing term would be.) [HOMEWORK] Assign S142 for homework. (Answers on T434.) [QUIZ ANSWERS] T435 T436 The quiz can be used at any time as extra homework or to assess how students progress on understanding numeric patterns.