Research Theme: Grade 4 Mathematics Lesson Plan Date: June 27 (Wed.), 2007, Period 5 Teachers A: Masako Adachi (Classroom 1) B: Akihiko Suzuki (Classroom 2) C: Masako Koizumi (Math Open) Nurturing students to become people who can be trusted in an international society: Developing instruction that will foster students ability to communicate 1. Name of the unit: Let s think about how to divide 2. About the unit Flow of the contents Grade 3 Grade 4 Grade 5 (3) Division Meanings of division and the division sign Division that can be solved by using the basic multiplication facts (no remainder) (7) Division with remainders Division that can be solved by using the basic multiplication facts (with remainders) How to check answers Meaning of remainders This unit: The division algorithm (1) The division algorithm for a 2- or 3-digit number divided by a 1-digit number Methods of calculating math sentences with o both multiplication and division operations o consecutive division Extending the meanings of division and times as much (1 st and 3 rd uses of times as much) Division algorithm (2) Dividing by multiples of 10 Division algorithm for 2- or 3-digit numbers 1-digit numbers How to check answers for division Meaning of a tentative quotient and how to adjust it Properties of division (2) Multiplication and division of decimals (I) Division of whole numbers with decimal quotients and dividing decimals by whole numbers (4) Multiplication and division of decimals (II) Meaning of dividing by a decimal and how to calculate The division algorithm for dividing whole numbers and s by decimals (6) Fractions and Decimals Quotients of whole numbers can always be written as fractions for the Lesson Study Immersion Program in Japan, June 21- July 5, 2007.
Goals of the unit Students will understand the division algorithm for dividing 2- and 3-digit numbers by 1-digit numbers and be able to use it appropriately. [Interest, Desire, Attitude] Students will try to use their previous study of division to think about how to divide 2- and 3-digit numbers by 1-digit number. [Mathematical Thinking] Students will be able to clearly explain that the division algorithm proceeds from the tens digit and then the ones when you are divide 2-digit numbers by 1-digit numbers. Students will be able to clearly explain that the division algorithm for dividing 3-digit numbers by 1-digit numbers works in the same way as was learned previously (2-digit 1-digit). [Representations, Procedures] Students will be able to accurately calculate 2- and 3-digit numbers 1-digit numbers. Students will be able to mentally calculate 2- and 3-digit numbers 1-digit numbers with 2-digit quotients. [Knowledge, Understanding] Students will understand how to divide 2- and 3-digit numbers by 1-digit numbers using the division algorithm. Students will understand that they can use division to determine times as much. Students will understand that one math sentence may be used even when situations involve both multiplication and division or two consecutive divisions. Current state of the students According to the results of the readiness test, virtually all students can accurately calculate division problems that use the basic multiplication facts once, which was studied in Grade 3. Although a few students missed remainders on some problems, all problems were correctly answered by at least 97% of the students. There were 2 problems that involve the ideas to be studied in Grade 4: (1) 40 2, and (2) 600 3. Although it seems like students should be able to anticipate the answers, the success rates were (1) 85% and (2) 70%. Moreover, for both problems, 9% of the students left them blank. Given the success rates on the items involving previously learned topics, we would like to plan a lesson that will take advantage of their ability to read and interpret math sentences and further extend this ability. However, in the prior unit on circles and spheres, some students had difficulty using compasses appropriately, indicating that some children still lack fine motor skills. Moreover, some students understanding may still be rather superficial, and their computational skills are more advanced than their ability to think and reason. Therefore, we want to provide opportunities for students to enjoy solving problems by thinking carefully and manipulating objects.
3. Abilities we would like to foster in this unit and special instructional strategies Target abilities for this unit 1. To make well-grounded explanations [Using prior understanding as the foundation for reasoning] Ability to use properties of division in explanations instead of just learning the algorithm procedurally. Ability to explain the reasons for the steps of the division algorithm instead of simply reciting, estimate, multiply, subtract, and bring-down. 2. To communicate effectively [Creating concise representations, diagrams and writing] Ability to describe situations involving times as much and division appropriately, utilizing diagrams and math sentences effectively. 3. To perceive and reason from multiple perspectives [Selecting more appropriate reasoning] Ability to interpret math sentences to know how others thought about problems involving both multiplication and division. Ability to select more appropriate reasoning using the criteria: efficient, simple, and accurate (ESA). Instructional strategies 1. Level-raising strategies The first 3 lessons of the unit will be used to discover the properties of division. To improve students ability to interpret math sentences, we will frequently have them sort shared solutions and strategies. Repeatedly ask students to use diagrams and math sentences as they explain. To make number lines a more familiar tool, we will frequently ask students to label the number lines. Use missing number problems to promote algebraic reasoning while exploring the algorithm and the properties of division. Employ small group instruction. 2. Effective learning tasks Use carefully selected numbers in the task to make it more realistic (problem involving both x and ). Use a familiar setting to make the task more interesting (times as much). Use a new type of problem to think deeply about the division algorithm and the properties of division (missing number). 3. Question posing In all lessons, include as many questions as possible to encourage students to look for more appropriate reasoning, foe example, Which operation is more ESA to calculate times as much/ or Which strategy is more ESA? Carefully sequence students strategies to be shared publicly, and ask students to share their observations so that they may recognize similarities in reasoning. 4. Interpreting students reasoning. Throughout every lesson, ask students to record their thinking in their notes to capture the change in their thinking. Have students write in a learning journal. Note: Arrows indicate how particular strategies may foster specific target abilitie
3. Group research examining the research theme (1) On times as much (2) On the use of number line model (3) Proposals on times as much and number line model
4. Plan of instruction Plan for Class 1 (Ms. Adachi) Class 2 Open # Goals Learning Task Evaluation (Suzuki) (Koizumi) (1) Properties of division (1) (1) (1) 1 (1) 1 1 Readiness test To discover patterns in division when the dividend stays constant. 2 To discover patterns in division when the divisor stays constant. 3 To discover patterns in division when the quotient stays constant. Administer the test. To discover patterns from the math sentence, 24 =. To discover patterns from the math sentence, 3 =. To discover patterns from the math sentence, = 3. Preassessment Are Ss trying to discover patterns in division? (Interest) Can Ss discover that when the dividend becomes times as much, the quotient also becomes times as much (constant divisor)? (Mathematical Thinking) Can Ss discover patterns and represent them clearly (constant quotient)? (Representation) (1) 2 (1) 2 (1) 3 (1) 3 (2) Dividing multiples of 10 and 100 (2) (2) 4/ (2) (2) 5 4/5 To understand how the division algorithm works when dividing multiples of 10 and 100 by 1-digit numbers. calculate 80 4. calculate 240 6. Can Ss consider the dividends using 10 as a unit and apply the basic multiplication facts to find the quotient? (Math Thinking) Can Ss divide multiples of 10 and 100 by a 1-digit number by using the relative sizes of the dividends? (3) Division algorithm (I) [2-digit 1-digit] (3) (3) 6/ (3) 7 6/7 To understand how the division algorithm works when dividing 2-digit numbers by 1-digit numbers (no remainder) 8 To understand how the division algorithm works when dividing 2-digit numbers by 1-digit numbers (with remainder, neither place is divisible). To understand the the problem situation and write a math sentence. calculate 52 4. To summarize how the algorithm works with 52 4. To check the answer for 52 4. To practice the algorithm. To understand the problem situation and write a math sentence. use the division algorithm for 76 3. To know what the quotient is. To check the answer Are Ss trying to use their prior learning to think about how to calculate 2-digit 1-digit? (Interest) Can Ss explain clearly that the division algorithm should proceed from the tens digit and then the ones when you are dividing 2-digit numbers by 1-digit numbers? (Mathematical Thinking) Can Ss. calculate 2-digit 1-digit using the division algorithm (with remainder and neither place is divisible)? Do students know how to divide 2-digit numbers by 1-digit numbers (with (3) 6/7 (3) 8 (3) 8
9 To understand how the division algorithm works when dividing 2-digit numbers by 1-digit numbers (with remainder but the tens place is divisible). To be able to divide 2-digit numbers by 1-digit numbers by using the algorithm (with remainder and solvable by using the basic multiplication facts once). for 76 3. To read the Math Corner in the text and learn about the terms, sum, difference, and product. calculate 86 4 and 62 3. To practice dividing 2-digit numbers by 1-digit numbers (with remainder and solvable by using the basic multiplication facts once). remainder and neither place is divisible)? (Knowledge) Can Ss divide 2-digit numbers by 1-digit numbers (with remainder and solvable by using the basic multiplication facts once)? Do Ss understand how the division algorithm works when dividing 2-digit numbers by 1-digit numbers (with remainder but the tens place is divisible)? (Knowledge) (4) Patterns in the division algorithm and math sentences involving division 10 / To understand that Are Ss thinking logically 11 there are patterns in as they look for the digit the division algorithm in each? (Mathematical through examination of missing-digit (Lesson B in Class 2) By using the relationships, divisor > remainder and quotient x divisor < dividend, find the numerals that go in the. By thinking about the digit in the ones place of the dividend, explore the relationship between the quotient and the dividend. Thinking) Can Ss use the remainder to figure out the quotient? Can Ss discover the relationship between the dividend and the divisor by looking at the three problems? (3) 9 (3) 9 (5) 14 (4) 12 (5) - 15 (4) - 13 12 To understand that situations that involve both multiplication and division or two consecutive divisions may be written as single math sentences. (Lesson A in Class 1) Determine the number of pencils each person will receive when 4 dozen pencils are shared among 6 people. To understand that two-step problems involving both multiplication and division or two divisions can be written as single math sentences. Can Ss explain why situations that involve both multiplication and division or two consecutive divisions can be written as single math sentences? (Mathematical Thinking) Do Ss understand how to calculate math sentences that contain both multiplication and division or two consecutive divisions? (4) 10 [lesson A] (5) 14 [lesson C]
13 To master what they have been studying so far. To practice problems involving both multiplication and division and two consecutive divisions. To complete Let s master in the textbook. (5) Calculation involving times as much 14 To understand that division is used to determine how many times as much a given quantity is as the base quantity. (Lesson C in Open) To think about what operation should be used to determine how many times as long 15 m is as 3m by using diagrams. To learn that division can be used to determine how many times as much, and to work on application 15 To understand that division can be used to determine the base quantity given the compared quantity, and how many times it is as much as the base quantity. (6) Division (3) (3-digit 1-digit) 16 To understand how the division algorithm works when dividing 3-digit numbers by 1-digit numbers (no remainder, but the tens and hundreds places are not divisible). 17 To understand how the division algorithm works when dividing 3-digit numbers by 1-digit numbers (with 0 in the quotient). 18 To understand how the division algorithm works when dividing 3-digit numbers by To think about what operation to use to determine the base quantity given 72 kg is 6 times as heavy as the base quantity. To learn that division can be used to determine the base quantity, and to work on application To understand the problem situation and write a math sentence. use the division algorithm for 734 5. To summarize how to use the division algorithm to calculate 734 5. use the division algorithm to calculate 843 4 and 619 3. To understand the problem situation and write a math sentence. (Knowledge) Are Ss appropriately using what they have learned so far to solve problems correctly? Are Ss trying to utilize diagrams such as number lines to grasp relationships among quantities? (Interest) Do Ss understand that division may be used to determine how many times as much? (Knowledge) Are Ss thinking about math sentences that will help them determine the base quantity? (Mathematical Thinking) Do Ss understand that division may be used to determine the base quantity? (Knowledge) Are Ss thinking about 3-digit 1-digit in the same manner as 2-digit 1-digit? (Mathematical Thinking) Can Ss accurately calculate 3-digit 1-digit=3-digit? Can Ss calculate 3-digit 1-digit = 3-digit (with 0 in the quotient)? Are Ss trying to relate to what they have learned previously? (Interest) Can Ss accurately (4) 11 (5) 15 (4) 12 (4) 10 (4) 13 (4) 11 (6) 16 (6) 16 (6) 17 (6) 17 (6) 18 (6) 18
1-digit numbers (2-digit quotient, i.e., the leading digit < divisor). (7) Mental computation 19 To understand how to mentally calculate 2-digit 1-digit = 1-digit and divisions of multiples of 10 and 100 by 1-digit numbers, and to be able to actually perform such calculations. use the division algorithm for 256 4. To summarize how to use the division algorithm to calculate 256 4. To check the answer for 256 4. calculate 74 2 mentally. calculate 740 2 mentally. (8) Summarizing [ 1 ~ 2 lessons ] 20 To review and assess students understanding of the contents. To complete the Let s check section of the textbook. calculate 3-digit 1-digit = 2-digit? Do Ss understand how to use long division to calculate 3-digit 1-digit = 2-digit? (Knowledge) Are students connecting mental calculation of 2- or 3-digit 1-digit to their prior learning by decomposing the dividends or considering the relative sizes of the dividends (i.e., using 10 or 100 as units)? (Mathematical Thinking) Can Ss mentally calculate 2- or 3-digit 1-digit? Do Ss understand the contents? (Knowledge) To deepen students understanding of the division algorithm by working on Challenge on p. 96 of the textbook. (7) 19 (7) 19 (8) 20 (8) - 20
Understand Examine Lesson Plan B (Class #1, Ms. Adachi) 1. Goals of the lesson: Students will be able to write math sentences for problems that require both multiplication and division operations. Students will understand that even when both multiplication and division operations are involved, the situation can be represented in a single math sentence. 2. Flow of the lesson Steps Questions ( ) 1. Understand the task. What operation do we need? (Why?) Learning activities ( ) Anticipated responses (C s) Read the problem. If you share dozen pencils among people, how many pencils will each person receive? 2. Solve the problem. Let s think about it in many different ways. T1 There are 8 dozen pencils. 6 people will share the pencils. C1 Maybe multiplication. C2 Must be division because we are sharing equally. C3 I think we need both because we don t know the number of pencils. C4 I have a question. Do we need to think about the number of pencils in the boxes? T2 The question says how many pencils, so we do have to think about the pencils in the boxes. C5 How many pencils are in the boxes? C6 It says 1 dozen, so there must be 12 pencils. C7 Should we draw a diagram? T3 It is up to you. If you think a diagram will be helpful, please draw it. Let s work on this problem. Pay close attention to what operation you will be using. Individual problem solving C1 12 x 8 6 = 16. Answer 16 pencils. (diagram) C2 12 x 8 = 96 96 6 = 16 C3 12 6 x 8 = 16 C4 12 6 = 2 2 x 8 = 16 C5 8 6 = 1 rem. 2 12 x 2 = 24 24 6 = 4 12 x 1 = 12 12 + 4 = 16 C6 8 6 = 1 rem. 2 (confuses 2 dozen and 2 pencils) 12 x 1 + 2 = 14 Answer 14 pencils. C7 When calculating the number of pencils, instead of (# of pencils/dozen) x (# of dozens), writes (# of boxes) x (# of pencils/dozen). C8 Makes computational errors while using one of the strategies for C1 ~ C4. C9 Only draws a diagram. C10 Cannot even get started. Points of consideration (*) Evaluation Strategies for improving communication ( ) Materials: 8 dozens of pencils Problem that focuses students attention on the quantities in the situations. Problem statement is similar to familiar division situations. * C1 ~ C5 Encourage them to think about other ways to solve the problem. * Write mainly math sentences on the whiteboards. * It is ok if not everyone solves the problem correctly. C7 Points out that the multiplicand should be the number of pencils in a box. C8 Asks student to check his/her calculation. C9 Encourage student to think about what math sentences can be used. C10 Provide concrete objects to use.
Deepen 3. Think about more appropriate strategies. Let s share our solutions. Look at all the solutions. Let s organize these solutions. Let s sort them according to their reasoning strategies. Let s explain. Listen to fellow classmates reasoning and compare solution methods. C1 Some of them start with 12 while others start with 8. C2 Some start with multiplication while others start with division. C3 Some have only one math sentence while others have 2. Example 1: C1, C2, C3, C4, C5, and C6 are all considered different. Example 2: Those with one math sentence and those with more than one. Example 3: Those that start with 8 and those that start with 12. Example 4: Those that use multiplication first and those that use division first. Example 5: Focus on the objects involved in the problem: those that shared pencils in each box first, those that shared the boxes first then dealt with the left over boxes, and those that figured out the total number of pencils first. C4 Some have different answers. The answer for 2-C6 is 14 pencils. There is no computation error, is there? So, why is it different? * There may not be enough space on the board to post all students work. Start with the students seated near the hallway, and have them move the whiteboards around to organize the solutions. * We will not sort the strategies based on the types of diagrams used. * If no one does 2-C6, suggest it as an example of an incorrect answer. (To understand the meaning of the remainder.) T1 Is there any other solution that used the same reasoning as 2-C6? C5 2-C5. C6 They both found how many dozen pencils each person will get. T2 So is the problem the ways they looked at the remainders? C7 When we do 8 6, we know that each person will receive 1 dozen, and there will be 2 dozen left. Since the remainder is not 2 pencils, I don t think we can add 2 to 12 pencils. T3 So, the remainder tells us that there are 2 more dozen, or 24 pencils, left to be shared. We calculated how many dozen pencils each person receives, so the remainder should also be in dozens, as 2-C5 did. Use a callout and write, First,find how many dozen pencils there are for each person.
Summarize Did you notice anything? 4. Summarize today s lesson and reflect. Which reasoning or math sentence is the best? Let s use ESA. T4 Now that we have sorted these solutions, let s hear the explanation for each. C8 I will explain 2-C2. First, we calculated the total number of pencils. There are 96 pencils altogether. Then, we shared them equally among 6 people. C9 I will explain 2-C1. First, we calculated the total number of pencils, then we shared them among 6 people. C10 Their reasoning is the same. T6 So can we call both of them the method that first calculates the total number of pencils? C11 Yes. T7 2-C3, can you explain 2-C4 s solution? C12 First, we shared the pencils in each box. Each person will get 2 pencils. Since there are 8 dozen pencils, each person will get 8 times as many as 2 pencils, or 16 pencils. C13 I will explain 2-C3. The reasoning is exactly the same as 2-C4, but it is written in one math sentence. T8 OK, so it is just like 2-C1 and 2-C2. One of them uses one math sentence while the other uses 2 math sentences. So, what is the difference between 2-C1/2 and 2-C3/4? C14 The difference is whether you calculate the total number of pencils first or the number of pencils each person receives from each box first. Students will see the merit of writing one single math sentence. Summarize today s lesson. C1 It is easy to make a mistake when you use reasoning that involved remainders. C2 When you write one math sentence, it looks simpler. T1 Is it easier to see the reasoning and solution in one math sentence than 2 or more math sentences? Please write down what you think is the best approach. T2 When both multiplication and division are in one math sentence, can we start with the last operation? (Example, 8 x 6 12.) C3 I don t think we can because we will get different answers. T3 Let s summarize. 2-C1 explains. Callout, Share the total number of pencils. 2-C2 explains. 2-C3 explains. Callout, Calculate the number of pencils for each box first. 2-C4 explains. Can Ss understand that the solution with one math sentence with both multiplication and division and the one with 2 math sentences are based on the same reasoning? (Math Thinking) Think using the criteria, ESA. Do Ss understand how to calculate math sentences with both multiplication and division? (Knowledge) We can write one math sentence even when both multiplication and division are involved. Math sentences with both multiplication and division must be calculated in order from left to right. Let s write a journal entry. C4 Writes his/her journal entry.
Lesson plan A (Class # 2, Mr. Suzuki) 2 lessons (today s lesson / ) 1. Goals of the lesson By thinking about the missing digits, students will reaffirm their understanding of the properties of division. (First half: By thinking about the missing digits, students will recognize the relationship, b > e.) (Second half: By thinking about the missing digits, students will understand that the properties of division can be observed in the procedure of the division algorithm.) 2. Flow of the lesson Steps Questions ( ) 1. Understand the task. Learning activities ( ) Anticipated responses (C s) Read the problem. T1 We have been studying division of 2-digit numbers with remainders. Today s lesson is based on that idea. Write the numerals from 1 to 4 in the boxes labeled A to D so that the division problem shown is correct. Points of consideration (*) Evaluation * Materials: Numeral cards (1 ~ 4 and 1 ~ 9), several sets and several copies of the problems (shown on the left) on cards. Understand In the boxes A ~ D, you need to write numerals 1 ~ 4. C1 Can we use them many times? C2 What should go in the other boxes? T2 You can use any numeral, but not everything will make the division problem correct. C3 I wonder if we can just pick numerals. T3 For each numeral, there is reason why it has to be in a specific box. For example, if you think about it very carefully, you will know that 1 can only go into C. You can use the numerals 1 4 once and only once. If you are not sure, you can try putting a numeral in a box and see what happens. * Some students may be hesitant getting started. It is important that students actually start trying.
2. Solve the problem. Individual problem solving. C1 (correct) C2 C3 * Explain how Ss can use the hint card (when you are not sure, you can open it up little by little). Hint card Examine Deepen Summarize 3. Think about why. I thought like this (incorrect answer). What do you think? How about these? They all have B > D. C1 Correct answer. C2 Understands Divisor > Remainder. C3 Does not understand Divisor > Remainder. C4 Tries putting numerals in the boxes. C5 Cannot get started. Listen to other students ideas and think about how to get the correct answer. Show 2-C3 on the board. C1 That s not correct. C2 Since the remainder is greater than the divisor, I don t think that is correct. C3 It has to be B > D. T1 Show answers like 2-C2 (show several of the same pattern at once). C1 If you do B x C, it will be greater than A in all of them. C2 You can t subtract from A. T2 So what do we need to do? C3 Explain by moving the cards around: Since we can use each card once and only once, the only combinations that will make it possible to subtract is B = 3 and D = 2. So, the answer is, A = 4, B = 3, C = 1, and D = 2. T1 We used the rule that the remainder must be less than the divisor and solved the problem systematically. (Blackboard writing, too.) C1 What numerals go in the other boxes? Will 5, 6, 7, 8 and 9 work? 1. What number can go into the remainder, D? (Let s think about the remainder first.) 2. Which is larger, B or D? ( C3) 3. There is a numeral that cannot go into B. ( B "1) 4. Try putting the numerals in any way. 5. Which is it, C < A or C > A? 6. B = 3 * The answer to be posted on the board will be prepared by the teacher. * Have a student who found the same answer as 2-C3 explain. (Point to focus.) Blackboard: B > D * All of them are incorrect answers. Are Ss reasoning logically to determine what numerals will go in the boxes? (Math Thinking) * Acknowledge students interest to explore further.
Extension: Understand 1. Understand the task. That is a very good question. What numerals will go in E and F? Think about a new problem. T1 We can use the numerals 1 to 9 in E and F, and we can use the same numeral at most twice. What patterns can you find when the division problem is correct? We are not changing 1, 2, 3, 4 that are already shown. E F * We are not changing the numerals in A to D. Are Ss trying to examine whether their previous reasoning can be extended to determine the numerals for boxes E and F? (Interest) Extension: Examine 2. Individual problem solving (4) E F I J G H K L Solve the problem individually. C1 Cannot get started. C2 I = 3 and G = 1. C3 Could do the same as C2, but after that just uses random trial and error. C4 Keeps trying to put different numerals in E and F. C5 I found one. By trial and error. C6 I found two. C7 I figured it out. It s easy. There are 3 kinds. C8 Understand that the number (4), 1H, must be two more than a multiple of 3, and use that idea to find E and F. Answer A Answer B Answer C Hint card 1. What should be I? J = 0, isn t it? 2. What should be G? 3. What can you say about KL? (K = 0 or 1). 4. How many more is the number GH than the number KL? What numeral can go into E? Use whiteboards. Can Ss figure out the quotient by focusing on the remainder? 3. Look for patterns. Look for patterns from the solutions to the missing digit problem. Display the three correct solutions. Extension: Deepen Look at E (in the quotient) and the divisor, 4F. Answer A Answer B Answer C (see above) C1 The number, 1H, must be 3 more than a multiple of 3. So, 9 + 2 = 11, 12 + 2 = 14, 15 + 2 = 17 are the possibilities. So, F must be 1, 4, or 7, and E must be 3, 4, or 5. C2 The quotients are increasing by 1. C3 The dividends are increasing by 3. C4 There are other patterns. 9, 12 and 15 are the 3 s facts in the multiplication table. C5 The numbers, (4), are also increasing by 3. Can Ss discover the pattern between the dividend and the divisor from the three correct answers? (Math Thinking)
Extension: Summarize 4. Summarize the lesson. Summarize the patterns of division. T1 41 3 = 13 rem. 2 44 3 = 14 rem. 2 47 3 = 15 rem. 2 (dividend) (divisor) = quotient + remainder C1 We did this before. C2 Even in the division algorithm, we have the same pattern we saw in the 3 s facts when one number increases by 1 the other increases by 3. <Summary> When the quotient increases by 1, the quotient will Let s write today s journal entry. increase by the divisor, 3. C3 Write his/her journal entry.
Lesson Plan C (Mathematics Open Room, Ms. Koizumi small group for mathematics) Goals of the lesson: Using a number line or tape diagram, students will think about methods to determine how many times as much the base number is as the given number. Flow of the lesson Steps Questions ( ) 1. Understand the task for the lesson. Learning activities ( ) Anticipated responses (C s) I can swim 15 m without taking a breath. How many times as far as can I swim without taking a breath? Points of consideration (*) Evaluation Strategies for improving communication ( ) At first, display the problem statement that does not include the base amount. Pose a problem that is appropriate for the season. Understand This is today s problem. That s true. can swim 3m without taking a breath. C1 Something is wrong. C2 We need to know how many meters can swim without taking a breath. C3 We can do this. * Help students pay attention to the base amount. * Make sure students understand that the base amount is 3 m. I can swim 15 m without taking a breath. can swim 3 m without taking a breath. How many times as far can I swim without taking a breath as? Examine 2. Solve the problem. Let s use what we have already studied to figure out how many times as far as I can swim without taking a breath. Solve the problem using own strategies. Cs Write their own strategies on the worksheet. Use multiplication to determine how many times. 3 x [5] = 15 5 times as far. 3 x 1 = 3 3 m 3 x 2 = 6 6 m 3 x 3 = 9 9 m 3 x 4 = 12 12 m 3 x 5 = 15 15 m 5 times as far. Think, how many sets of 3 m will make 15 m? 15 3 = 5, 5 times as far. * Prepare the worksheet. * For those students who cannot even get started, provide tapes that are 3 cm and 15 cm so that they can draw a tape diagram. Are students trying to solve the problem? (Interest) Can Ss solve the problem using their own ideas and represent it in a diagram, a math sentence, or a chart? (Mathematical Thinking) * Have students write their math sentences and diagrams on their whiteboards.
3. Read other students ideas. Let s look at all the solutions. I wonder if we can sort them? Compare their own ideas with other students ideas. Explain their own reasoning. Multiplication methods Division methods Number line methods Acknowledge and praise that they were able to solve the problem using many different ideas. Have students explain other students ideas, not their own. Deepen There is no space to show times as much on a number line, is there? It shows only m, doesn t it? Where can we show times as much? * If there is any strategy that cannot be grouped with others, or other students cannot figure out the strategy, have the student who came up with the strategy explain it. Summarize 4. Summarize. When you are figuring out how many times as much, what operation should we use? Think ESA. Using the idea from today s lesson, let s try this problem. A mother whale is 24 m long. A baby whale is 6 m long. Summarize today s lesson on the worksheet. To find how many times as much, we can use division. Draw a number line model. Students will attempt the problem using what they learned in today s lesson. How many times as long is the mother whale as the baby whale? Based on students ideas, orchestrate the discussion so that students will recognize that different ideas can be summarized using number line models. * Make sure that students understand the importance of drawing a number line correctly and using 1 as the base of comparison. * Review how to draw a number line. Can Ss draw a number line correctly? (Mathematical Thinking) 5. Reflect on today s lesson. Please write what you noticed or realized in today s lesson. Write a journal entry. * Encourage students to write about the base amount in their journal entries.