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5th Grade Unit 2: Multi-Digit Whole Number and Decimal Fraction Operations (7 Weeks) Stage 1 – Desired Results Established Goals Unit Description Students will have a chance to practice and hone their skills at multiplying and dividing (decimal) numbers by 1-digit whole numbers. They will be able to generalize the 1-digit algorithms to the multi-digit whole number versions. The Mathematical Practices should be evident throughout instruction and connected to the content addressed in this unit. Students should engage in mathematical tasks that provide an opportunity to connect content and practices. Common Core Learning Standards 5.NBT.1: Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. 5.NBT.2: Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole number exponents to denote powers of 10. 5.NBT.5: Fluently multiply multi-digit whole numbers using the standard algorithm. 5.NBT.6: Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. 5.NBT.7: Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. 5.OA.1: Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols 5.OA.2: Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product Common Core Standards of Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ESL Language Standards Standard 1: Students will listen, speak, read, and write in English for information and understanding. 1.1. Identify and use reading and listening strategies to make text comprehensible and meaningful. 1.3 Select information appropriate to the purpose of the investigation, relate ideas from one written or spoken source to another, and exclude nonessential information. 1.5 Formulate, ask, and respond to various question forms to obtain, clarify, and extend information and meaning. 1.7 Present information clearly in a variety of oral and written forms for different audiences and purposes related to all academic content areas. 1.9 Convey and organize information, using facts, details, illustrative examples, and a variety of patterns and structures. 1.16 Apply learning strategies to acquire information and make texts comprehensible and meaningful. Big Ideas 1. Basic facts and algorithms for operations with rational numbers use notions of equivalence to transform calculations into simpler ones. Essential Questions 1. Numbers can be named in equivalent ways using place value 1. Decimal numbers can be named in an infinite number of equivalent but different forms 2. The same number sentence can be associated with different concrete or real-world situations, AND different number sentences can be associated with the same concrete or real-world situation. 2. The real world actions for addition and subtraction of whole numbers are the same for operations with fractions and decimals 2. Some real world problems involving joining groups, separating equal groups, comparison or combinations can be solved using multiplication, others can be solved with division 2. Different real world interpretations can be associated with multiplication and division calculations involving decimals Skills (Students will be able to…) A1. Determine the value of digits in a whole number and decimal number to the thousandths Content (Students will know….) A. A digit in one place represents ten times as much as it represents in the place to right and 1/10 of what it represents in the place to its left (5.NBT.1) B. There is a pattern in the number of zeros when multiplying or dividing by powers of ten. This is directly related to the base ten system (5.NBT.2) B1. Explain the patterns in the number of zeros when multiplying by powers of 10 B2. Explain the patterns in the decimal point when multiplying or dividing by powers of 10. B3. Use the understanding of the patterns above to efficiently multiply and divide by powers of 10. B4. Create equivalent expressions for multiples of 10 using exponents. For example: 36 x 10 = 36 x 101 = 360 36 x 10 x 10= 36 x 102 =3600 C. Standard algorithm for multiplying multi digit whole numbers (5.NBT.5) C1. Fluently (accurately, efficiently, and flexibly) multiply multi-digit whole numbers using the standard algorithm Note: should not exceed a two digit factor by three digit factor C2. Use alternative strategies such as partial products or an area model to build conceptual understanding of multiplication D. Division of whole numbers with up to four digit dividends by two digit divisors (5.NBT.6) D1. Find whole number quotients of whole numbers by whole numbers D2. Use strategies to find quotients based on place value, properties of operations, and the relationship between multiplication and division. D3. Illustrate and explain division by using equations, arrays and/or area model E. Operations with decimals to the hundredths (5.NBT.7) E1. Add, subtract, multiply (factors to hundredths) and divide decimals (quotients to thousandths) based on whole number operations E2. Use concrete models, drawings, and strategies based on place value, properties and/or the relationship between addition and subtraction E3. Relate the strategy chosen to a written method and explain the reasoning used F. Numerical expressions (5.OA.1) F1. Evaluate numerical expressions, including powers of ten, with parentheses ( ), brackets [ ] and braces { } using the conventional order of operations: 1. Grouping symbols: parentheses, then brackets then braces 2. Addition or subtraction (left to right) 3. Multiplication or division (left to right) G. Numerical expressions (5.OA.2) G1. Write simple numerical expressions given a verbal expression G2. Interpret numerical expressions without evaluating them G3. Describe the relationship between expressions without evaluating them Terms/ Vocabulary: place value, digit, decimal number, decimal point, tenths, hundredths, thousandths, power of ten, multiple, factor, product, divisor, dividend, quotient, algorithm, array, area model, decompose, compose, partition Stage 2 – Assessment Evidence Initial Task: Betty’s Bakery Final Performance Task: Thanksgiving Dinner Other Evidence Teacher observation, conferencing, teacher designed assessment pieces, student work, exit slips, journal entries Stage 3 – Learning Plan Everyday Mathematics /Impact Mathematic Lessons – The following lessons may support some of the CCLS & essential questions outlined in this unit map: 5.NBT.1-2.2, 2-3, 2-10, 7-2 5.NBT.2-1-1, 1-2, 1-5, 1-6, 1-8, 1-9, 2-1, 2-8, 2-9, 3-2, 3-5,3-8, 3-9, 4-1, 7-1, 7-2, 7-4, 7-7, 10-1, 10-3, 11-6 5.NBT.5-7-10, 9-2, 5.NBT.6-4-1, 4-2, 4-4, 4-6, 7-10 5.NBT.7-2.2, 2-3, 2-4, 2-5, 2-7, 2-8, 2-9, 4-5, 4-6, 5-11, 6-5, 6-7, 7-10, 9-8, 9-10, 10-6, 12-2, 5.OA.1 5.OA.2 Additional Resources: Unpacked standards from North Carolina http://www.ncpublicschools.org/docs/acre/standards/common-core-tools/unpacking/math/5th.pdf k-5 Math Teaching Resources – Activities listed by Common Core Standard http://www.k-5mathteachingresources.com/5th-grade-number-activities.html Adding and subtracting decimal tasks from Georgia Department of Education (5.NBT.1, 5.NBT.7) https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_5_Unit2FrameworkSE.pdf Multiplying and dividing decimal tasks from Georgia Department of Education (5.NBT.2, 5.NBT.7) https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_5_Unit3FrameworkSE.pdf Operations with Whole Numbers from Georgia Department of Education 95.NBT.1, 5.NBT.2, 5.NBT.5, 5.NBT.6) https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_5_Unit1FrameworkSE.pdf Additional Performance Task Assessment (5.NBT.5, 5.NBT.6) http://insidemathematics.org/common-core-math-tasks/5th-grade/5-2004%20Fruits%20&%20Vegetables.pdf Grade 5 Unit 2 Initial Performance Task: Betty’s Bakery Name_______________________ Date___________ 1. Betty’s Bake Shop makes 15 dozen sugar cookies and 288 chocolate chip cookies each day. For a – c, use any method to show your mathematical thinking. (1 dozen = 12 cookies) a. How many sugar cookies are made each day? b. How many sugar cookies are made in 2 weeks? c. How many dozen chocolate chip cookies are made each day? 2. Use the price table to answer questions. Use any method to prove your answers. Red Velvet Cupcakes $6.75 per box a. Price List Peanut butter chip cookies $4.80 per box Pumpkin pie $12.35 per pie A group of five friends order 2 boxes of cupcakes and 1 pumpkin pie for a party. The five friends will each pay an equal amount. How much will each friend pay? b. Sal wants to buy a box of peanut butter chip cookies, but he only has dimes! How many dimes will he need to pay for 1 box? c. The local supermarket placed a big Thanksgiving order for 10 boxes of cupcakes and 100 pumpkin pies. How much will the order cost? 3. Describe how the expression “double five and then add 26” is related to “10 + 26” 4. (5.OA.1) Alex and James evaluated the following equation: 9 + 2 x (10 - 4) = ? Alex thinks the solution is 21. James thinks the solution is 66. Who do you agree with? Why? Explain your thinking with words and numbers. Show your math thinking here: Grade 5: Initial Task Betty’s Bakery Scoring Guide Betty’s Bakery Scoring Guide The core elements of the performance required by this task are: Understand place value of whole numbers and decimal numbers to multiply and divide Represent their mathematical thinking through pictures, words and number models 1. (5NBT5, 5NBT6) a. Student uses any method to find correct answer of 180 sugar cookies (15 x 12) b. Student uses any method to find correct answer of 2, 520 sugar cookies (15 x 12 x 14) c. Student uses any method to find correct answer 24 dozen chocolate chip cookies (288 ÷ 12) 2. (5NBT1, 5NBT2, 5NBT7) a. Student uses any viable method to find the correct answer of $5.17 per person. $6.75 x 2 = $13.50 $13.50 + 12.35 = $25.85 $25.85 ÷ 5 = $5.17 b. Student correctly answers “48 dimes” using any viable method c. Student correctly answers $1,302 using any viable method ( $6.75 x 10) + ($12.35 x 100) 3. (5OA2) Student is able to explain that the two expressions are equivalent because “doubling 5” is the same as 10 AND “+ 26” remains the same in both expressions“. Note: Student may evaluate both expressions but it is not required in this standard to evaluate 4. (5OA1) Student earns 1 point for correct mathematical answer and 2 points for either of following correct explanation: Alex is correct because he follows the order of operations correctly: 9 + 2 x (10 – 4) 9+2x6 9 + 12 21 James is incorrect because he evaluates from left to right without regard to the order of operations: 9 + 2 x (10 – 4) 11 x (10 – 4) 11 x 6 66 Total Points Novice 0-3 Apprentice 4-7 Practitioner 8 - 10 Rubric Points Section Points 1 3 1 1 2 4 1 1 2 2 1 2 3 12 12 Expert 11- 12 Grade 5 Unit 2 Final Performance Task: Thanksgiving Dinner Name: __________________________ Date: _______________ The teachers at PS 276 are planning a Thanksgiving meal for their students. Use the price chart to answer the following questions. Use any method to show your mathematical thinking. Potatoes .89 per pound Grocery Store Price List Turkey Cranberries Stuffing .75 per pound $2. 39 per bag $1.25 per box String Beans $ 1.66 per pound 1. The teachers buy 14 pounds of potatoes and 4 bags of cranberries. What is the total cost? 2. The teachers have $21 for a turkey. What is the heaviest turkey they can buy (in pounds)? 3. The 5th grade teachers buy 3 boxes of stuffing and 4 pounds of string beans. The 4th grade teachers buy 2 boxes of stuffing and 3 pounds of string beans. How much more did the 5th grade teachers spend than the 4th grade teachers? The local supermarket places orders with the farm. Use the chart to answer the following questions. Use any method to show your mathematical thinking. Apples $14 per crate Walnuts $156 per crate Pumpkins ? per crate 4. The supermarket orders 12 crates of pumpkins. The farm charges $456. What is the price per crate? 5. The supermarket spends a total of $650 on a combination of apples and walnuts. They order 13 crates of apples. How many crates of walnuts did they order? 6. Evaluate the following expressions. Show all steps. a. 3 + 104 ÷ 10 x (32 ÷ 8) = b. 42 + 7 x 5 + (70 x 800) = 7. Write an expression that means “triple the sum of twelve and seventeen”. How is it different from the expression “the sum of three times twelve and seventeen”? Grade 5: Final Task Thanksgiving Dinner Scoring Guide Thanksgiving Dinner Scoring Guide The core elements of the performance required by this task are: Understand place value of whole numbers and decimal numbers to multiply and divide Represent their mathematical thinking through pictures, words and number models 1. (5.NBT.7) Student correctly answers $22.02 using any method to show their work such as: ( 14 x $.89) + (2 x $2.39) = $22.02 2. (5.NBT.7) Student correctly answers 28 pounds and uses any viable method to prove the answer such as using algorithm, drawing a diagram or recording in a t-chart 3. (5.NBT.7) Student correctly answers “the 5th grade teachers spend $2.91 more” OR “the 4th grade teachers spend $2.91 less” using any viable solution method. Note: the most efficient method is to add 1 box of stuffing to 1 pound of string beans since this is the difference between the two orders. 4. (5.NBT.6) Student correctly answers $38. Student may choose any method to show their thinking. 5. (5.NBT.5, 5.NBT.6) Student correctly answers “3 crates of pumpkins” and shows correct work such as algorithm, diagram or chart. $650 – 13($14) = $ on walnuts $650 - $182 = $468 on walnuts $468 ÷ $156 = 3 crates 6. (5.OA.1) a) Student correctly evaluates expression and follows correct order of operations 3 + 104 ÷ 10 x (32 ÷ 8) 3 + 104 ÷ 10 x (4) 3 + 10,000 ÷ 10 x (4) 3 + 10,000 ÷ 10 x (4) 3 + 1,000 x 4 3 + 4,000 4,003 b) Student correctly evaluates expression and follows correct order of operations 42 + 7 x 5 + (70 x 800) 42 + 7 x 5 + 56,000 42 + 35 + 56,000 77 + 56,000 56,077 Rubric Points Section Points 2 2 1 1 2 2 1 1 2 2 2 4 2 7. (5.OA.2) Student correctly translates the verbal expression into a numerical expression 1 3 Student is able to explain the difference between the two expressions and articulate 2 how the order of the operations is different. In the first expression, 12 is added to 17 first: 3(12 + 17) then multiplied by 3. But in the second expression the 3 and 12 are multiplied first then added to 17 ie: (3 x 12) + 17 Total Points 15 Novice 0-3 Apprentice 4-7 Practitioner 9 - 12 Expert 13- 15 15