# SFUSD Mathematics Core Curriculum Development Project

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1 1 SFUSD Mathematics Core Curriculum Development Project Creating meaningful transformation in mathematics education Developing learners who are independent, assertive constructors of their own understanding

2 Grade 5 Unit 5.5: Multiplying and Dividing Decimals by Decimals 2 Number Lesson Reproducibles Number of Copies Materials of Days 1 Entry Task Farmer s Market 4 Lesson Series 1 Base Ten Activity Handout HW: Picturing Multiplication (3 pages) Multiplication Error Analysis Exit Slip Maze Scaling Multiplication HW: Multiplying Decimals (2 pages) Where s the Point HW: EM Study Link 2.7, 2.9 (2 pages) Missing Number Worksheet Sorting Products (3 pages) HW: Chris s Test EM Study Link base ten blocks class set calculators class set 1 Apprentice Task Apprentice Student Sheet Optional: Calculators. Digit cards 4 Lesson Series 2 Base Ten Division Student Sheet Division Error Analysis Exit Slip HW: Area Model of Division Maze Scaling Multiplication and Division HW: Another Method for Dividing (2 pages) EM Journal p 109 HW: Chris s Test #2 HW: EM Study Link 4.5 Estimate and Calculate Do You See an Error Recording Sheet HW: Unit Pricing 1 Expert Task Sewing A Quilt Student Sheet Base Ten Grid Paper EM Journal EM Study Link 1 per pair base ten blocks class set calculators class set base ten blocks class set calculators (optional)

3 3 3 Lesson Series 3 Road Trip Recording Sheet EM Grade 5 SRB p 388 U.S. Distance Map HW: Day 1 Word Problems HW: Day 2 Solve Each Problem HW: Day 3 Unit Pricing (2 pages) 1 Milestone Task Milestone Road Trip Milestone Constructed Response EM SRB Provided by AAO Provided by AAO calculators (optional) poster paper (optional)

4 Unit Overview Big Idea 4 The same rules and strategies that apply for multiplying and dividing whole numbers also apply for multiplying and dividing decimals. Because multiplication is scaling, when we multiply a quantity by a number larger than 1, it produces a larger quantity. When a quantity is multiplied by a number smaller than 1, it produces a smaller quantity. Unit Objectives Students will be able to multiply decimals by decimals with both factors and products to the hundredths. Use concrete models or drawings. Explain their thinking and strategies in numbers and words using understanding of place value and/or properties of operations. Answer questions about their strategies, including, but not limited to: How did you do it? What have you tried? What did you learn when that didn't work? Why does that make sense to you? Why do you think that's true? Is it always true? Under what conditions is it true? Students will be able to divide decimals by decimals with divisor, dividend, and quotient to the hundredths. Use concrete models or drawings. Explain their thinking and strategies in numbers and words using understanding of place value and/or properties of operations. Answer questions about their strategies, including, but not limited to: How did you do it? What have you tried? What did you learn when that didn't work? Why does that make sense to you? Why do you think that's true? Is it always true? Under what conditions is it true? Students will understand and be able to explain how the magnitude of the product relates to the magnitude of the factors. Unit Description This unit starts with students demonstrating their multiplication and division knowledge from previous units. This unit will expand upon their previous knowledge by continuing on to multiplication and division of decimals by decimals. In the first lesson series, students will use their knowledge of base ten blocks to learn how to model multiplication of decimals by decimals, they will also begin to notice how scaling is related to multiplication, i.e. the relationship between factors and products. A variety of activities will help students become proficient multiplying decimals by decimals, and placing the decimal in the correct place. The second lesson series will focus on division of decimals by decimals using base ten blocks to model the division. A variety of activities will continue to help students become proficient dividing decimals by decimals. Students should begin to realize how multiplying or dividing by decimals less than 1, or greater than 1 will affect the product or quotient of problems. This lesson series will mirror what was learned in the first lesson series. In these two lesson series, students will have opportunities to demonstrate their proficiency in both multiplication and division of decimals by decimals using the exit slips, as well as on the final activity on Day 4 of Lesson Series 2. During Day 3 of both Lesson Series 1 and 2, you will have an opportunity to demonstrate an algorithm to show multiplication and

6 Progression of Mathematical Ideas Prior Supporting Mathematics Current Essential Mathematics Future Mathematics 6 In fourth grade, students begin their learning of the decimal system with an introduction into decimal notations of the tenths or hundredths place. They learned how to compare decimals up to the hundredths place value, when they refer to the same whole. They are able to compare using <, >, = symbols and using visual models to justify their conclusions. In Unit 1 of fifth grade, students became fluent using an algorithm for multiplication and division of whole numbers, which can then be used for multiplication and division of decimals. Students will build on their work from Unit 5.4 by multiplying and dividing decimals by decimals. While deepening their sense of place value, students will also use their understanding of the relationship between multiplication and division to determine how decimal factors and divisors/dividends affect the magnitude of the product and quotients. Students will use concrete models and drawings to model these operations. In Unit 7 of fifth grade, students will extend their understanding of the properties of multiplication of non-whole numbers in the multiplication of fractions. In sixth grade, students will fluently calculate using all operations with decimals using the standard algorithm. In addition, students will extend their understanding of multiplication as scaling when they work with proportional relationships. In Unit 2 of fifth grade, students extended their learning by writing and comparing decimal to the thousandths place value. They round numbers to any place value, and begin to understand the relationship between place values and powers of 10. In Unit 4 of fifth grade, students began adding, subtracting, multiplying, and dividing decimals by whole numbers, using models or drawings, and written methods to explain their reasoning. Students learned how to multiply and divide by a power of 10.

7 Unit Design All SFUSD Mathematics Core Curriculum Units are developed with a combination of rich tasks and lessons series. The tasks are both formative and summative assessments of student learning. The tasks are designed to address four central questions: Entry Task: What do you already know? Apprentice Task: What sense are you making of what you are learning? Expert Task: How can you apply what you have learned so far to a new situation? Milestone Task: Did you learn what was expected of you from this unit? 7 1 day 4 days 1 day 4 days 1 day 3 days 1 day Total: 15 days

8 8 Entry Task Farmer s Market Apprentice Task Missing Numbers Expert Task Sewing a Quilt Milestone Task Road Trip! CCSS-M Standards 5.NBT.7 5NF.5a, 5.NF.5b 5.NBT.7 5.NF.5a, 5.NF.5b 5.NBT.7 5.NF.5a, 5.NF.5b 5.NBT.7 5.NF.5a, 5.NF.5b Brief Description of Task Students will calculate the total cost of apples, and also the total number of bananas they can purchase. They will write out how they know their answers are correct. Students find all the possibilities for missing numbers in a multiplication problem with decimals (. * 0.5 = 0. ). Students will write out all of the equations they come up with. Students notice patterns and justify their thinking around finding all possibilities. Students use their understanding of area models and the relationship between division and multiplication to find possible dimensions for a quilt with an area of 5.76 square meters. Students use multiplication and division of decimals to find kilometers traveled, gallons of gas used, and cost of gas in the context of a road trip. Students also show understanding of multiplication and division of decimals in context-free constructed response questions and in finding the errors in an example calculation. Source SFUSD Teacher Created (2014) SFUSD Teacher Created, modified from Georgia Department of Education, Fifth Grade Mathematics Unit 3, pp SFUSD Teacher Created (2014) SFUSD Teacher Created (2014)

11 11 Farmer s Market How will students do this? Focus Standards for Mathematical Practice: 4. Model with mathematics. 5. Use appropriate tools strategically Structures for Student Learning: Academic Language Support: Vocabulary: cost per pound, farmer s market Sentence frames: First I The way I figured out my answer was correct was to... Differentiation Strategies: Intervention: Have a discussion of how to multiply and divide decimals with whole numbers as learned in Unit 4. Extension: Have students write their own problems, switch with a partner, and solve theirs. Participation Structures (group, partners, individual, other): The students should work individually.

12 Lesson Series #1 12 Lesson Series Overview: This lesson series begins with the use of base 10 blocks to model multiplication of decimals. It continues on with the practice of multiplication of decimals, while students begin to notice how factors affect an answer. Students will also practice decimal placement in multiplication problems, and see scaling effects on multiplication products. On daily exit slips, students work on noticing errors in example calculations. In preparation for the Apprentice Task, students work to find different ways to complete a CCSS-M Standards Addressed: 5.NBT.7, 5.NF.5a, 5.NF.5b Time: 4 days Lesson Overview Day 1 Description of Lesson: Students work in pairs to use base ten blocks to build rectangles that model the multiplication of decimals. To prepare to teach this lesson, read the BACKGROUND KNOWLEDGE: Representing decimal multiplication with base ten blocks portion of the Georgia Department of Education lesson (see Resources). When making rectangles to represent decimal multiplication, you are actually using the length and the width of each block to represent the factors. Therefore, a flat is actually 1 unit by 1 unit, a long is 1 unit by 0.1 unit, and a unit block is 0.1 unit by 0.1 unit (area model). The lesson should begin with a discussion of the area model of multiplication: the lengths of the sides of the rectangle are the factors in the problem; the area of the rectangle is the product. As a class, practice with the problem 3.2 * 2.4, filling in the area with base ten blocks and finding the product. Make sure to clarify that the area of the Then have students work in pairs to solve the other problems on the sheet. Some students may need to use the blocks all day. Encourage these students to draw the rectangles after building it. Others may be able to use drawings only. There may be some who notice a structure to the problems building off the work in Unit 5.4 who may not need to draw out the rectangles. Students should be able to support their answers with the area model whether or not they use it to solve every problem. Resources Georgia Department of Education, Fifth Grade Mathematics Unit 3, pp Base Ten Activity Handout Base ten blocks, enough for pairs of students Multiplication Error analysis exit slip Homework: Picturing Multiplication Key to Decimals, Book 2: Adding, Subtracting and Multiplying, pp. 25, 26, 34 Notes: The original lesson includes a couple of division problems. Do not use these today.

14 14 Have students do the Multiplication Error Analysis Exit Slip Hand out Exit Slips and have students show their work to correct the mistakes. Provide a brief note about what the student did wrong on the sample. Notes: Calculators are necessary for this activity. If you do not have enough calculators for each student, have them work in pairs or groups. Lesson Overview Day 3 Description of Lesson: Mini-lesson: Use of a multiplication algorithm Demonstrate and explain to students how to use a multiplication algorithm to multiply decimals by decimals. Where s the Point? (modified from Nimble with Numbers pp ) See Where s the Point? Direction sheet Students will work in small groups to figure out where to place the decimal point in decimal sentences. The first part of the activity students will place the decimal in the factor parts of a decimal sentence, given the correctly placed decimal in the product. The second part of the activity students will place the decimal in the product of a decimal sentence, given the correctly placed decimals in the factors. Resources Where s the Point? modified from Nimble with Numbers pp by Leigh Childs and Laura Choate Where s the Point? direction sheet Where s the Point? student handout Multiplication Error Analysis Homework: Everyday Mathematics Grade 5 Study Links 2.7 and 2.8 Multiplication Error Analysis Exit Slip Hand out second Exit Slips and have students show their work to correct the mistakes. Provide and a brief note about what the student did wrong on the sample. Notes: Intervention: Have students do fewer problems. Extension: Have students write out how they solved one problem from the first set of sentences. Have students create their own decimal sentence problems, and have partners solve them.

15 15 Lesson Overview Day 4 Description of Lesson: Part 1: Missing Numbers Lesson Introduce the concept of filling in the blanks with numbers that make the number sentence true. Show the blank equation, and hand out the Missing Number Worksheet. x. =. For this question, they should only use the numbers 0 5. Allow them to find as many options as possible to fill the blanks. Encourage students to seek out rules or patterns that will help them find multiple possibilities and determine whether they have all the possibilities. Probe with questions like: How did you get your answer? How do you know your answer is correct? What patterns are you noticing? Do you have all the possibilities? How do you know? When students believe they have all the possibilities (you may want to check), they should write a paragraph answering these questions: How did you find your solutions? How do you know you have all the possibilities? Have students share out their solutions and how they proved the number sentence made sense. Students should try to find as many solutions as possible and to share strategies with the class as to how they came up with their solution. Resources Missing Number Worksheet SFUSD Teacher Created Multiplication Error Analysis Sorting Products - Number Sense Homework: Key to Decimals, Book 2: Adding, Subtracting and Multiplying, p. 38 Part 2: Sorting Products If you have time after the Missing Numbers work, or as a homework assignment, have students do the first half of Activity 3 of the Sorting Products lesson. In this activity, students order products without calculating, only looking at the factors. If sorted correctly, the letters that correspond with the products will spell a word. Make sure to do the first step of the Using the Activities section as a class so that students know how to complete the activity. Activities 1 and 2 and the second half of 3 can be used as extensions. Multiplication Error Analysis Exit Slip Hand out the third Exit Slips and have students show their work to correct the mistakes. Provide a brief note about what the student did wrong on the sample.

17 17 Missing Numbers How will students do this? Focus Standards for Mathematical Practice: 2. Reason abstractly and quantitatively. 7. Look for and make use of structure. Structures for Student Learning: Academic Language Support: Vocabulary: factors, product Sentence frames: I started by worked because didn t work because I know we got all the possibilities because Differentiation Strategies: Extension: Students can explore further problems like this, finding missing numbers when multiplying by a number with decimals. For example: o. * 1.5 =. Intervention: Provide students with examples with one whole number before using two decimal factors. Include some examples that only have one solution before moving to multiple solutions. Also, calculators can be used for this task. Students can use digit cards to generate random numbers to try. Participation Structures (group, partners, individual, other): Students work in pairs.

18 Lesson Series #2 18 Lesson Series Overview: This lesson series begins with the use of base ten blocks to model division of decimals. It continues on with practice of both multiplication and division of decimals, using their knowledge of scaling to find the highest answer. Students will also practice magnitude estimates before calculating quotients. Finally, students will show their proficiency in both multiplication and division of decimals by identifying errors and correcting them in student work samples. CCSS-M Standards Addressed: 5.NBT.7, 5.NF.5a, 5.NF.5b Time: 4 days Lesson Overview Day 1 Description of Lesson: Students work in pairs to use base ten blocks to model division with decimals. This lesson mirrors the first lesson of Lesson Series #1. To prepare to teach this lesson, read the BACKGROUND KNOWLEDGE: Representing decimal division with base ten blocks portion of the Georgia Department of Education lesson (see Resources). When making rectangles to represent decimal multiplication, you are actually using the length and the width of each block to represent the factors. Therefore, a flat is actually 1 unit by 1 unit, a long is 1 unit by 0.1 unit, and a unit block is 0.1 unit by 0.1 unit (area model). Resources Georgia Department of Education, Fifth Grade Mathematics Unit 3, pp Base Ten Division Base Ten Division Student Sheet Division Error Analysis Exit Slip Homework Day 1 (SFUSD Teacher Created, 2014) The lesson should begin with a discussion of the rectangle area model of division. This focuses on finding a missing factor in a multiplication problem. As a class, practice with the two problems demonstrated in the Background Knowledge section. The first, 3.6 : 1.2, has students make groups of 1.2 and figure out how many of those can be made with a set of 3.6. The second, 4.83 : 2.1, has students make a rectangle with one side of 2.1 and find the length of the other side. Then have students work in pairs to solve the other problems on the sheet. Some students may need to use the blocks all day. Encourage these students to draw the rectangle or groups after building. Others may be able to use drawings only. There may be some who notice a structure to the problems building off the work in Unit 5.4 who may not need to draw out the models. Students should be able to support their answers

19 19 with the models whether or not they use it to solve every problem. Division Error Analysis Exit Slip Hand out the first Exit Slips and have students show their work to correct the mistakes, and a brief note about what the student did wrong on the sample. Notes: This should build on the work students did on the first day of the first lesson series. Lesson Overview Day 2 Description of Lesson: Students work individually or in pairs to try to make the largest number possible by going through a number maze. They use the properties of multiplication and division to make their number larger, at the same time noticing how different factors and divisors affect the result. Be sure to do all three portions of the lesson to help students get full understanding from this activity. Resources Maze Scaling Directions Maze Scaling Multiplications and Division.pdf Division Error Analysis Exit Slip Homework: Another Method for Dividing Key to Decimals, Book 3: Dividing pp. 27, Whole Class: Write the problem (as described next) on the chalkboard or overhead. Ask students to discuss what they notice. 4.5 : 0.9 = 0.50 Lead a discussion that focuses on these key points: In computing the quotient of 4.5 and 0.9, a student calculated that the answer is Reflection on the answer should have caused the student to realize the product was too small. Dividing 4.5 by a number slightly less than 1 produces an answer a little more than 4.5. Instead, this student applied an incorrect procedure (making the number of digits behind the decimal point correspond) and did not reflect on whether the resulting answer was reasonable. Remind students of the activity in the previous lesson series where they had to make the largest number using the multiplication maze. Explain that they will do the same today using multiplication and division to modify their original number. Encourage students to trace several paths through the maze while always looking for the path that will yield the greatest increase in the calculator's display. 2. Individual/pairs/groups: (depending on how many calculators you have available) Give each student a calculator and a copy of the Maze Scaling Activity - Multiplication and Division activity sheet.

20 20 Students are to choose a path through the maze. To begin, have the students enter 100 on their calculator. For each segment chosen on the maze, the students should key in the assigned operation and number. The goal is to choose a path that results in the largest value at the finish of the maze. Students may not retrace a path or move upward in the maze. In pairs or in groups of three, students should discuss their strategies (after playing the game) and what worked best for them. 1. Whole Class: As a class, discuss what happens to your total when you divide by various numbers on the maze, greater and less than one. Guide them to notice when the quotient is larger or smaller than the dividend. How is this similar to and different from multiplication? Possible follow-up activities include finding the path that leads to the smallest finish number or finding a path that leads to a finish number as near the start number (100) as possible. 2. Individual: Division Error Analysis Exit Slip Handout the second exit slip and have students show their work to correct the mistakes, and a brief note about what the student did wrong on the sample. *Homework day 2 Notes: Calculators are necessary for this activity. If you do not have enough calculators for each student, have them work in pairs or groups. Lesson Overview Day 3 Description of Lesson: Mini-lesson: Division of Decimals by Decimals Demonstrate and explain to students how to use a division algorithm to divide decimals by decimals. Magnitude Estimates before Calculating Quotients - Everyday Mathematics Grade 5, Lesson 4.5 Division Error Analysis Exit Slip Hand out the third Exit Slips and have students show their work to correct the mistakes, and a brief note about what the student did wrong on the sample. Resources Everyday Mathematics Grade 5, Lesson 4.5 pp Division Error Analysis Exit Slip Homework: Division Error Analysis Key to Decimals, Book 3: Dividing p. 41 Everyday Mathematics Grade 5 Study Links 4.5

21 21 Notes: In Class: Use Part 1 (the math message, making magnitude estimates before calculating quotients). Part 2 (playing Division Top-It, math boxes 4.5 (box 4 and 5) Homework: Study Link 4.5 (Math Masters p.113) Lesson Overview Day 4 Description of Lesson: Do You See An Error? Students will work individually to analyze the student samples to determine decimal division errors. Encourage students to be sure they have discovered all errors in the samples. Focus student thinking by asking: What happens when we divide a decimal by a decimal? How can we check for errors in division of decimals? When students have found all errors, they should find the correct solution using two methods. Then they should write a paragraph explaining what the student did wrong, and the steps they need to take next time to find the correct solution. Resources Georgia Department of Education Fifth Grade Mathematics Unit 3 pp Homework: Unit Pricing Key to Decimals, Book 3: Dividing p. 40 Notes: Students should work individually on this activity.

23 23 Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 4. Model with mathematics. Materials Base Ten Blocks Base Ten Grid Paper Calculators (optional) Structures for Student Learning: Academic Language Support: Vocabulary: array, area, length, width, dimensions Sentence frames: First I tried I noticed that and it helped me worked because didn t work because... Sewing A Quilt How will students do this? Differentiation Strategies: Extension: Have students find as many possible dimensions as they can. Intervention: Calculators can be used for this task. Provide students with Divisibility Rules (Student Reference Book p 11) Participation Structures (group, partners, individual, other): Students can work in pairs or triads.

25 25 Notes: Students should only select 1 city per day, do not allow them to travel through cities to get to their final day s destination. (e.g., Do not go S.F. to Denver. Day 1; they should go S.F. to Las Vegas. Then Day 2 they would go Las Vegas to Denver.) Modifications: Have students make calculations for one city at a time. Allow groups to select less than five cities as appropriate. Have students use calculators to check their calculations. Give students copies of tables provided on teacher reference sheet. Extension: Have groups who finish all questions create a poster representing their road trip. *A car would usually have a higher rate than 9.6 kilometers per gallon. However, in order to keep the divisor a two-digit number, a low rate was selected. Description of Lesson: See Day 1 description of the lesson. Notes: See Day 1 notes. Description of Lesson: See Day 1 description of lesson. Notes: See Day 1 notes. Lesson Overview Day 2 Lesson Overview Day 3 See resources for Day 1. Resources Homework: Solve Each Problem Key to Decimals, Book 3: Dividing p. 12 See resources for Day 1. Resources Homework: Unit Pricing Key to Decimals, Book 3: Dividing pp

26 Milestone Task CLA2: Road Trip! and Constructed Responses What will students do? 26 Mathematics Objectives and Standards Math Objectives: Students will efficiently solve multiplication problems with decimals. CCSS-M Standards Addressed: 5.NBT.7, 5.NF.5a, 5.NF.5b Potential Misconceptions When multiplying the product is always larger than both factors: This misconception should be familiar to students from their work multiplying fractions by whole numbers in fourth grade and from work during Lesson Series 1. When one (or more) of the factors is less than one, the product will be less than the other factor. When dividing the quotient is always smaller than the dividend: this misconception should be familiar to students from their work in lesson series 2. When the divisor is smaller than 1, the quotient will be larger than the dividend. Framing Student Experience Launch: Remind students of their work in Lesson Series 3 calculating the number of kilometers traveled, the gas consumed, and the cost of gas. Today they will solve one set of problems using this context, plus three stand alone problems: a multiplication problem, a division problem, and a find the error problem (as they did in lesson series 1 and 2). During: Students work independently to solve the problems. They may use any of the tools that they worked with during this unit. Closure/Extension: When all students have finished, you may decide to review the answers as a group.

27 27 Focus Standards for Mathematical Practice: 2. Reason abstractly and quantitatively. 6. Attend to precision. Structures for Student Learning: Academic Language Support: Vocabulary: miles, kilometers, gallons Sentence frames: none CLA2: Road Trip! and Constructed Responses How will students do this? Differentiation Strategies: Students may use any of the manipulatives, drawings, or tools they have used during this unit, other than calculators. Participation Structures (group, partners, individual, other): Individual

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