Mathematics Lesson # _2 Grade 3 Lesson created by: Jenny Meis Name of Unit: Division Concepts and Facts

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1 Mathematics Lesson # _2 Grade 3 Lesson created by: Jenny Meis Name of Unit: Division Concepts and Facts Name of Lesson: Subtraction and Division Prerequisite Knowledge: Repeated subtraction is subtracting the same subtrahend multiple times from the minuend to find the difference. When using repeated subtraction to find the difference you are skip-counting backwards. Multiplication and division are inverse operations. If a x b = c than c a = b. (Hardcourt Math teacher edition, volume 2-3 rd grade, 2007) Learning Targets (Goals) (Understand that ) The students will understand that repeated subtraction may be used to solve a division problem. The students will understand when using repeated subtraction, start with the total and subtract equal groups until you reach zero. Count the number of times you subtracted to find the quotient. The students will understand that when using repeated subtraction to find the quotient, you are skip-counting backwards. The students will understand that division may be written in two different forms: with division house or as a division sentence. (Hardcourt Math teacher edition, volume 2-3 rd grade, 2007) Success Criteria (I can ) I can find the solution to a division fact by repeated subtraction. I can explain that when using repeated subtraction to find the quotient, you are skip-counting backwards. I can shoe two different ways division problems can be shown: with division house or as a division sentence. Process Standards (Problem Solving, Communication, Connections, Reasoning and Proof, Representations) Problem solving- The students will be problem solving in partners during Equal Groups activity, the exit slip, homework problems, and early finisher worksheets. The students will build new mathematical knowledge through problem solving worksheets and group activities. Throughout the worksheets, the students will apply and adapt a variety of appropriate strategies to solve problems including using subtraction multiple times, using a number line, and other student invented strategies. Communication- When the students solve problems and explain their answers during their partner activities they will be displaying communication to organize and consolidate their mathematical thinking. The students will communicate their mathematical thinking coherently and clearly to peers, teachers and others during the summarize portion of the lesson plan. The students will analyze and evaluate the mathematical thinking and 1

2 strategies of others during partner activities and through showing multiple representations for a solution through the finish the solution questioning strategy in their homework assignment. The students will use the language of mathematics to express mathematical ideas precisely by explaining their thinking when articulating their solution to a problem to partners and in the class discussions. Reasoning and Proof- The students will select and use various types of reasoning and methods of proof through completing the partner activities and using different methods of representation for repeated subtraction. They will also show reasoning and proof through their own invented strategies and using others. Representations- The students will create and use representations to organize, record, and communicate mathematical ideas through using number lines and invented strategies for repeated subtraction. The students will select, apply, and translate among mathematical representations to solve problems by selecting the one they are most comfortable with as well as using some outside their comfort zone in the Extra Practice homework problems. The students will use representations such as a number line and invented strategies to model and interpret physical, social, and mathematical phenomena through repeated subtraction. Materials and Preparation: Hardcourt Math teacher edition, volume 2 Worksheets and answer keys to each worksheet: (print out in different colors) (Attached) - Equal Groups - My Exit Slip -Relay Teams and Separating the Colors (early finisher) - Extra Practice! (homework) A group folder for each table that includes all of the worksheets listed above If using an overhead rather than a Smartboard- 3 transparency sheets for the warm-up Observation clipboard Warm-up/Opener: At your tables, draw one model as a group of how 18 can be divided into groups evenly and create a number sentence for it. 6x3=18/ 3x6=18 (they may also have written division problems) 1x 18=18/ 18x1=18 9x2=18/ 2x9=18 When you re finished, try to find additional ways 18 can be divided up evenly. Have groups hold up their model when teacher says 1, 2, 3, Reveal!! 2

3 Talk about all possible answers. Then, call on a student that thinks they can count on a number line to show this same number sentence. Have them show it on the overhead. (18 8=2) 2 1 (You skip-count backwards, or subtract 2 times to from 18 to 0. That means 18 8=2) Ask Students: Billy said division is like repeated addition. Do you agree or disagree? Explain why. Disagree. To find the quotient in a division problem you have to count how many times you skip-count backwards to find the quotient. Repeated subtraction if used in multiplication the product. That means you are skip counting forwards. Does anyone have any other suggestions for how we can show 18 8=2? Ask Students: How is division like subtraction? They both separate objects into equal groups. Other Possible answers: They both subtract numbers. If this is said, reply by suggesting you were talking about objects rather than just numbers. Ask students to show as a group two different ways or symbols division can be written. (Use 1, 2, 3, Reveal!!) and (Questions and teaching adapted by (Meis, 2001) from Hardcourt Math teacher edition, volume 2-3 rd grade, 2007) Launch: When I say go, I want everyone to get with their summer partners.(they will know what this means. If someone does not have a partner due to an absent student, place them with another group to make a group of 3). Right now, I need everyone to grab their (color) equal Groups worksheet sheet and the (color) Early finisher worksheet from your group folder. What you re going to do once you find your partner, is complete the equal groups worksheet together. Make sure both partners can explain how they found the solutions to the problems. When you get to #4, try to come up with a strategy we have not talked about already. 3

4 When you and your partner are finished, I would like you to work on your Early Finisher Worksheets. Before heading to your groups, you need to take with you your (color) Equal Groups worksheet sheet, and your Relay teams and Separating the Colors (Early Finisher activity), and a pencil with you. Ok, get with your summer partner. GO! Explore: A.) Anticipated student questions, teacher questions, and answers can be found on the worksheets. (see attached) B.) I will be doing my assessment through anecdotal notes on my teacher clipboard. I will be observing and taking notes about what students are struggling, which students need further challenged, common questions and misconceptions I see throughout the activity that need to be discussed when we come back together as a class. C.) Formative assessment: Agree/ Disagree questioning strategy in the early finisher(see warmup) Fit the condition questioning strategy (See Exit Slip) Finish the Solution questioning Strategy (See Extra Practice!) Add in color strategy to add anything they think is important to their array and variable worksheet when we go over it in class The teacher will be using formative assessment by taking anecdotal notes during the Equal Groups partner worksheet to see what students aren t getting about division. These concerns or misconceptions will be addressed during the summative portion of the lesson. D.) Differentiated Questions/Tasks: The advanced partnerships that finish the Equal Groups partner activity quickly will have an early finisher activity that will challenge them further. Since they are working with partners, if they end up being at two different levels in division, they will help each other. The advanced student will gain deeper understanding by explaining things in different ways and the lower level partner will get things explain to them in a way they might grasp better understanding. E.) Early Finisher ( Relay Teams and Separating the Colors See attached) Summarize: A.) Students will share the results and understanding of the Equal Groups worksheet together as a class discussion. If they hear something they 4

5 think is important they will be instructed to add it to their Equal Groups sheet in colored pencil. B.) C.) Key mathematical ideas that must be discussed include: Repeated subtraction may be used to solve a division problem. When using repeated subtraction, start with the total and subtract equal groups until you reach zero. Count the number of times you subtracted to find the quotient. When using repeated subtraction to find the quotient, you are skipcounting backwards. Division may be written in two different forms: with division house or as a division sentence. Check for Understanding- They will show their understanding of the lesson by completing a fit the condition questioning strategy worksheet and showing a model that represents the division problem. (Exit Slipattached) Application: Students will apply what they learned through the lesson by doing homework problems that require knowledge of division as repeated subtraction, ways to model it, and number sense. (Extra practice!- See attached) 5

6 Mathematics Lesson # _3 Grade 3 Lesson created by: Jenny Meis Name of Unit: Division Concepts and Facts Name of Lesson: Multiplication and Division Prerequisite Knowledge: Know how to find a quotient by using an array by count the number in each row. Use a related multiplication fact to find a quotient. Multiplication and division are inverse operations. If a x b = c than c a = b. (Hardcourt Math teacher edition, volume 2-3 rd grade, 2007) Learning Targets (Goals) (Understand that ) The students will understand what it means to relate division to multiplication The students will understand and use multiplication and division fact families The students will understand what a variable is and how to find it s solution through using fact families and an array (Hardcourt Math teacher edition, volume 2-3 rd grade, 2007) Success Criteria (I can ) I can find the solution to a division fact by relating it to multiplication. I can find the solution to a division problem using fact families and an array I can tell you that a variable is an unknown number in a number sentence and I can find the solution for the variable by using fact families or an array Process Standards (Problem Solving, Communication, Connections, Reasoning and Proof, Representations) Problem solving- The students will be build mathematical knowledge through problem solving working with partners during the Break It In Parts activity, the worksheet on arrays and variables, the exit slip, and the homework problems. Throughout the worksheets, the students will apply and adapt a variety of appropriate strategies to solve problems including the array model and student invented strategies. The students will be problem solving in partners during the Equal Groups activity, early finisher worksheets. The students will build new mathematical knowledge through problem solving worksheets individually as well as in group activities. Throughout the worksheets, the students will apply and adapt a variety of appropriate strategies to solve problems including using break apart cubes, array models, and student invented strategies. Connections- The students will recognize and use connections among mathematical ideas by relating multiplication and division. This will also help them understand how mathematical ideas interconnect and build on one another to produce a coherent whole. The students will recognize and apply mathematics in contexts outside of mathematics through story problems in their worksheets. Communication- When the students solve problems and explain their answers during their 6

7 partner activities they will be displaying communication to organize and consolidate their mathematical thinking. The students will communicate their mathematical thinking coherently and clearly to peers, teachers and others during the warm-up and summarize portion of the lesson plan. The students will analyze and evaluate the mathematical thinking and strategies of others during partner activities and through showing multiple representations for a solution through warm up, and division with Arrays and variables worksheet. The students will use the language of mathematics to express mathematical ideas precisely by explaining their thinking when articulating their solution to a problem to partners as well as in the class discussions. They will also explain their mathematical language n an individual basis by completing the exit slip. Reasoning and Proof- When the students are completing the partner activities they, they will be reasoning and proving why they chose the answers they did. This will help them determine if their solutions make sense. Another time they are using reasoning and proof skills is by answering the reasonable/unreasonable questioning strategy during warm-up. The students will select and use various types of reasoning and methods of proof through completing the partner activities and using different methods of representation multiplication. They will also show reasoning and proof through their own invented strategies and the use others. Representations- The students will create and use representations to organize, record, and communicate mathematical ideas through using Break Apart manipulates during the Break It In Parts activity. The students will select, apply, and translate among mathematical representations to solve problems by selecting the one they are most comfortable with as well as using some outside their comfort zone in the Extra Practice homework problems. The students will use representations such as Break Apart cubes and invented strategies to model and interpret physical, social, and mathematical phenomena by dividing through multiplication. Materials and Preparation: Hardcourt Math teacher edition, volume 2 Worksheets and answer keys to each worksheet: (print out in different colors) (Attached) - Build it in Parts - Arrays and Variables - What s the Meaning of 0 (early finisher) - Extra Practice! (homework) A group folder for each table that includes all of the worksheets listed above Break apart cubes (enough for each pair of students can have at least 12 in two different colors If using an overhead rather than a smartboard- 3 transparency sheets for the 7

8 warm-up Observation clipboard Warm-up/Opener: Draw an array showing 18 3= 6. Explain how you can look at your array 2 different ways and still see 18 3=6. ( 6 groups of 3; 3 groups of 6) Teacher Questions: Think 3x6=18. What do you know about 3x6? (3x4=12 is the same thing as having 4 groups of 3 to equal 12 altogether) In this division problem, what number is the divisor? Dividend? Quotient? 18 3= 6 Dividend Divisor Quotient (Write on overhead or smartboard) Ask Students: Amy Says multiplication and division are opposite or inverse operations. Do you think this is reasonable or unreasonable? Why? Reasonable. The operations are opposite because multiplication puts together equal groups while division separates equal groups. (Questions and teaching adapted by (Meis, 2001) from Hardcourt Math teacher edition, volume 2-3 rd grade, 2007) Launch: When I say go, I want everyone to get with their fall partners. (They will know what this means. If someone does not have a partner due to an absent student, place them with another group to make a group of 3). But first, I need everyone to grab their (color) Build It in Parts worksheet sheet and the (color) Arrays and Variables worksheet, and the (color) Early finisher worksheet from your group folder. What you re going to do once you find your partner, is have one person grab a box of break-a-part cubes. Then you and your partner will decide on a number between 5 and 12. I m going to choose 8 to demonstrate. I know that 4x2=8, so 4 green cubes and another 4 blue cubes will make 8 Now, I am going to color on the grid paper that 4 green plus 4 blue equal 8 and write the equation for it underneath my array. (Ask if anyone knows any other facts that make 8. Should get the answer 8x1=8) (Demonstrate drawing the arrays as well) (Demonstrate on overhead or smartboard) When you think you have found all the possible combinations to make the first number you and your partner has chosen, choose a number larger than the first to challenge yourselves. Do this with different 3 numbers and then 8

9 choose your favorite example you have made to share with your table group after the partner activities. After your example is chosen, double checks to make sure you have found all the possible solutions. Then, work on the (color) Arrays and Variables worksheet with your partner. If you get finished with both of those activities work on your Early Finisher worksheet. You need to take with you, your (color) Build It in Parts worksheet sheet, the (color) Variables worksheet, The meaning of 0 (Early Finisher activity), a pencil, and your colored pencils with you. Ok, get with your fall partner. GO! Explore: F.) Anticipated student questions, teacher questions, and answers can be found on the worksheets. (see attached) G.) I will be doing my assessment through anecdotal notes on my teacher clipboard. I will be observing and taking notes about what students are struggling, which students need further challenged, common questions and misconceptions I see throughout the activity that need to be discussed when we come back together as a class. H.) Formative assessment: Reasonable/unreasonable strategy in the warm up (see warm-up) Agree/ Disagree strategy in the early finisher (see attached) Fit the condition questioning strategy (See Extra Practice!) Add in color strategy to add anything they think is important to their array and variable worksheet when we go over it in class The teacher will be using formative assessment by taking anecdotal notes during the Break It in Parts activity and the Arrays and Variable worksheet with their partners I.) Differentiated Questions/Tasks: Build It in Parts - This activity is designed in a way that can be easily tiered. For the struggling students I will have them try to demonstrate smaller numbers and for the advanced student I will have them work on numbers above 12 that will be more challenging for those individuals. Once the students have mastered a smaller number they are instructed to break a part a bigger number that will have more factors and therefor more challenging. The advanced students who finish the partner activities quickly also will have an early finisher activity that will challenge them. J.) Early Finisher ( What s the Meaning of 0? Early finisher- Attached) 9

10 Summarize: D.) E.) F.) Students will share the results from their break it in parts activity with their table groups. Then, we will go over and finish the variables worksheet together as a class. If they hear something they think is important they will be instructed to add it to their sheet in colored pencil. Key mathematical ideas that must be discussed include: How division and multiplication are related. How fact families are used when using multiplication to divide. How a variable and array can be used for division. Check for Understanding- They will explain how to use an array to multiply and divide. They will write the division equation for the provided array and an additional equation that includes a variable. (Exit Slipattached) Application: Students will apply what they learned through the lesson by doing homework problems that require knowledge of division, fact families, and number sense. (Extra practice!- See attached) 10

11 Mathematics Lesson # _4 Grade 3 Lesson created by: Jenny Meis Name of Unit: Division Concepts and Facts Name of Lesson: Fact Families Prerequisite Knowledge: Multiplication facts through 10 Know how to make equal groups A fact family is a set of related multiplication and division sentences. Think of the quotient or missing divisor as one of the missing factors in the related multiplication sentence. To use a related multiplication table and related multiplication fact to find a quotient in the top row, look down to find the dividend, and then look to the left to find the missing number. (Hardcourt Math teacher edition, volume 2-3 rd grade, 2007) Learning Targets (Goals) (Understand that ) The students will understand they can use related multiplication facts to find quotients or missing divisors in division sentences. The students will understand you cannot write a division sentence using a smaller dividend than the divisor and still get a whole number. The students will know that they can use a multiplication table to find the missing quotient or divisor in a division sentence. (Hardcourt Math teacher edition, volume 2-3 rd grade, 2007) Success Criteria (I can ) I can use multiplication and division fact families to help me find the missing variable in a division or multiplication problem. If I write a division sentence using a smaller dividend than the divisor I will not get a whole number. I will get a fraction. I can find the missing quotient or divisor by using a multiplication table. Process Standards (Problem Solving, Communication, Connections, Reasoning and Proof, Representations) Problem solving- The students will build new mathematical knowledge through solving problems with variables, writing fact families with their Triangular Fact Family cards, and completing the Extra Practice! homework problems to reinforce their new skills. Students will apply and adapt a variety of appropriate strategies to solving problems with missing variables. This will be demonstrated through the Triangle Family Fact cards, the variable worksheet, the early finisher, as well as the Extra Practice! homework problems. The students will monitor and reflect on the process of mathematical problem solving by using the smile face self-assessment tool (putting a happy, uncertain, or sad face by any math problem they do during this lesson to let themselves as well as me know what they are doing well with and what they need more help with. 11

12 Communication- The students will solve problems and explain their answers during the activities (Triangle Fact Family Cards, and multiplication table worksheet) and displaying communication to organize and consolidate their mathematical thinking. The students will communicate their mathematical thinking coherently and clearly to peers, teachers and others during the summarize portion of the lesson plan and the warm-up by discussing and agreeing on a common solution. The students will analyze and evaluate the mathematical thinking and strategies of others during the warm-up group discussion and the summarize portions of the lesson plan. Advanced students and struggling learners will show multiple representations for a division sentence with the same quotient during the Extra Practice! homework problems. The students will use the language of mathematics to express mathematical ideas precisely by explaining their thinking when articulating their solution to problems during group and whole class discussions as well as individually on the Extra Practice! and multiplication table worksheets. Connections- The students will be making connections in this lesson by relating fact families (multiplication problems) to division problems to help them find out a missing variable. They will also be making connections and building number sense through using a multiplication table in their multiplication table worksheet. Working with a multiplication table and making their own fact family cards will allow the student interconnect and build on existing mathematical knowledge to produce a coherent whole. Reasoning and Proof- The students will select and use various types of reasoning and methods of proof through using a multiplication table to find variable in a division number sentence as well as through using their fact family cards. Representations- The students will create and use representations to organize, record, and communicate mathematical ideas through making their own triangular fact family cards during the explore phase. The students will select, apply, and translate among mathematical representations to solve problems by selecting the strategies that work the best for them for a certain problem during their Extra Practice! homework problems. The students will use representations such as a multiplication table and their fact family cards to develop strategies to model and interpret physical, social, and mathematical phenomena through multiplication. Materials and Preparation: Hardcourt Math teacher edition, volume 2 Worksheets (1 for each student)and answer keys to each worksheet: 12

13 (print out in different colors) -Directions to the Triangle Fact Family activity and 1 piece of colored paper - Multiplication Table and worksheet - Fact Families(early finisher) - Extra Practice! (homework) A group folder for each table that includes all of the worksheets listed above If using an overhead rather than a smartboard- 3 transparency sheets for the warm-up. Observation clipboard Students will need their scissors Warm-up/Opener: This error analysis questioning strategy will be displayed or written on the board. Students will first work on it by themselves and then discuss it with their table groups. Victoria says 3 4 = 12. What error is she making? Victoria is writing the problem wrong because she is switching around the divisor and the dividend. She has the dividend as 3 and the divisor as 4. Teacher Questions: Think about the parts of a division problem that we learned yesterday. Divisor, dividend, and quotient. What does the divisor mean in an array? Dividend? Divisor-The total number of rows and columns Dividend- The number of rows. Once you discuss as a whole group what she is doing wrong have each group write the division problem the correct way on whiteboards, one per group. Have them label each part of the problem = 4 Dividend Divisor Quotient (Write all answers from groups on the overhead or smartboard. Then have them choose as a group which one is correct) Ask Students: Can anyone come up with an equation for Victoria s problem by switching the operation? Explain your thinking. 3 x 4 = 12 because multiplication is the inverse operation of division. You can put the little number first in multiplication because it doesn t matter what comes first. 3x4 is the same as 4x3. But in division, if you divide a smaller number from a larger number you will get a fractional number. Like 2 4 = ½. Demonstrate the 2 4 = ½ issue with a pizza if students can t explain why 13

14 they will get a fraction. (Questions and teaching adapted by (Meis, 2001) from Hardcourt Math teacher edition, volume 2-3 rd grade, 2007) Launch: The students will Make their own set of triangle fact cards for each of these products: 12, 15, 18, 20, 24, 25, and 30. They will be doing this in their table groups. Before the students do it on their own, make a triangle family fact card for the product of 15 together on the overhead or smartboard. Factor Factor Product A. First, you will fold each paper in half 3 times. Open up the paper and cut along the folds to make triangle cards. B. Make triangle facts for each of these products: 12, 15, 18, 20, 24, 25, and 30. C. Write fact families for at least 3 triangle fact cards. When you have completed your Triangle Fact Family cards work on your Multiplication Table worksheet/activity. You will find the directions and the paper you need in your group folder. Get that out now along with the Multiplication table worksheet and the Early Finisher. You will also need scissors from your desk. If you get done early work on the Early finisher activity. Explain to the students that today you would also like them to put a :l (happy, middle, or sad face by any math problem they complete today to let me know what you understand and what you aren t so sure about. Explore: K.) Anticipated student questions, teacher questions, and answers can be found on the worksheets. (see attached) 14

15 L.) Formative Assessment Strategies and questions: I will be doing my assessment through anecdotal notes on my teacher clipboard. I will be observing and taking notes about what students are struggling, which students need further challenged, common questions and misconceptions I see throughout the activity that need to be discussed when we come back together as a class. I will also have the students self-assess by way of happy, uncertain, or sad face smiles by any math problems they do to let themselves as well as me know what they are going great with and what they need a little extra help with. Error Analysis questioning strategy in the warm up (see warm-up) The teacher will be using formative assessment by taking anecdotal notes during the Triangle Fact Family activity Reasonable/ unreasonable questioning strategy in Extra Practice! Homework assignment Revise in color if they have a wrong factor for a Triangle fact family card or have written a wrong fact family equation (See the Summarize portion of the lesson) M.) Differentiated Questions/Tasks: Triangle Fact Family activity- This activity has been designed to be tiered easily. For the struggling student, I will have them create fact family cards for smaller quotients (landmark numbers) If the struggling students still don t seem to be getting a grasp on division, I will introduce them to counting by multiples. 20 5= (4) (count by fives up to 20. Keep track of how many times you count to get to the dividend) For the advanced student (I will notice them because they will be working on the different colored Early Finisher worksheet)who might get done relatively quickly, I will have them write division sentences that has the same quotient as the division sentences below: 24 3=8 Possible Solutions: 16 2=8; 40 5=8 18 2=9 27 3=9; 36 4=9 The early finisher can also be used as a challenge for the advanced student. N.) Early Finisher ( Fact families - Attached) Summarize: G.) When finished with the Triangle fact family activity, I will have all of the students come together and go over their fact Triangle cards as a class. If they have wrong factors or something wrong in a fact family equation, they will be instructed to revise in color. 15

16 H.) I.) Key mathematical ideas that must be discussed include: Each Triangle fact family card has 2 multiplication equations and 2 division equations. They can use related multiplication facts to find quotients or missing divisors in division sentences. Multiplication tables and how to use them to find variables. Check for Understanding (Exit Slip- attached) Application: Students will apply what they learned through the lesson by doing homework problems that require them to practice the skills they learned today such as how a division sentence can use the same numbers as a multiplication sentence and how to use a multiplication table to find solutions to division problems. (Extra practice!- See attached) 16

17 Name: Number Lesson 4- Quiz Write the fact family. 1. 2, 7, , 5, , 9, , 23 Hardcourt Math teacher edition, volume 2-3 rd grade, 2007

18 Name: Number Write the fact family. 1. 2, 7, 14 2 x 7 = 14 7 x 2 = 14 14/2=7 14/7=2 2. 6, 9, 59 6 x 9 = 54 9 x 6= 54 54/9=6 54/6=9 3. 4, 8, 23 4 x 8 = 32 8 x 4 = 32 32/4=8 32/8=4 4. 4, 5, 20 4 x 5 = 20 5 x 4 = 20 20/5=4 20/4=5 Hardcourt Math teacher edition, volume 2-3 rd grade, 2007