Topics Total Teaching Days Dates Grade 5 Math Expressions 2017-2018 SY Benchmark Cycle 1 Benchmark Cycle 2 Benchmark Cycle 3 Cycle 4 September 5 - October 31 BM Window Nov 1-17 November 2 - January 26 BM Window Jan 29-Feb 13 January 30 - May 8 BM Window May 9-25 May 10 - June 12 Total Days: 39 Including 1 Half Day Total Days: 50 Days Including 4 Half Days Total Days Before PSSA: 50 Including 9 Half Days Total Days in Cycle: 60 Including 13 Half Days; Excluding Week of Math PSSA Total Days: 22 Including 1 Half Day 1 Addition and Subtraction with Fractions 2 Addition and Subtraction with Decimals 3 Multiplication and Division with Fractions 4 Multiplication with Whole Numbers and Decimals 5-1 5-5 Division with Whole Numbers 5-6 5-11 Division with Whole Numbers 6 Operations and Word Problems 7 Algebra, Patterns, and Coordinate Graphs 8-1 8-7 Measurement and Geometry 8-8 8-17 Measurement and Geometry Note: A Benchmark Cycle is defined as the time allotted to teach the content that is on each benchmark, and assumes the benchmark is taken on the first day of the window. This means that though it is fine to give the test later in the window, you should be moving on to new content as of the above listed dates, or you will fall behind. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 1
Table of Contents Benchmark Cycle 1 Standards... 3 Benchmark Cycle 1 Scope and Sequence... 5 Benchmark Cycle 2 Standards... 7 Benchmark Cycle 2 Scope and Sequence... 12 Benchmark Cycle 3 Standards... 15 Benchmark Cycle 3 Scope and Sequence... 21 Cycle 4 Standards... 26 Cycle 4 Scope and Sequence... 28 PA Core Standards and Eligible Content by Cycle... 30 Document Information Page... 33 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 2
Benchmark Cycle 1 Standards PA Common Core Standard CC.2.1.5.C.1 Use the understanding of equivalency to add and subtract fractions. CC.2.1.5.B.1 Apply placevalue concepts to show an understanding of operations and rounding as they pertain to whole numbers and decimals. PA Eligible Content M05.A-F.1.1.1 Add and subtract fractions (including mixed numbers) with unlike denominators. (May include multiple methods and representations.) Example: 2/3 + 5/4 = 8/12 + 15/12 = 23/12 M05.A-T.1.1.1 Demonstrate an understanding that in a multi-digit number, a digit in one place represents 1/10 of what it represents in the place to its left. Example: Recognize that in the number 770, the 7 in the tens place is 1/10 the 7 in the hundreds place. M05.A-T.1.1.3 Read and write decimals to thousandths using base-ten numerals, word form, and expanded form. Example: 347.392 = 300 + 40 + 7 + 0.3 + 0.09 + 0.002 = 3 100 + 4 10 + 7 1 + 3 (0.1) + 9 (0.01) + 2 (0.001) M05.A-T.1.1.4 Compare two decimals to thousandths based on meanings of the digits in each place using >, =, and < symbols. Common Core State Standard 5.NF.A.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.) 5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5+ 1/2 = 3/7, by observing that 3/7<1/2 5.NBT.A.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.A.3 Read, write, and compare decimals to thousandths. a. Read and write decimals to thousandths using baseten numerals, number names, and expanded form, e.g., 347.392 = 3 100 + 4 10 + 7 1 + 3 (1/10) + 9 (1/100) + 2 (1/1000). 5.NBT.A.3 Read, write, and compare decimals to thousandths. b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 3
CC.2.1.5.B.2 Extend an understanding of operations with whole numbers to perform operations including decimals. CC.2.4.5.A.1 Solve problems using conversions within a given measurement system. CC.2.4.5.A.4 Solve problems involving computation of fractions using information provided in a line plot. M05.A-T.1.1.5 Round decimals to any place (limit rounding to ones, tenths, hundredths, or thousandths place). M05.A-T.2.1.3 Add, subtract, multiply, and divide decimals to hundredths (no divisors with decimals). M05.D-M.1.1.1 Convert between different-sized measurement units within a given measurement system. A table of equivalencies will be provided. Example: Convert 5 cm to meters. M05.D-M.2.1.1 Solve problems involving computation of fractions by using information presented in line plots. Use place value understanding to round decimals to any place. 5.NBT.B.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.MD.A.1 Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems. 5.MD.B.2 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 4
9/18-10/5 1 Day Per Lesson + 4 Days 9/5 9/15 1 Day Per Lesson + 4 Days Benchmark Cycle 1 Scope and Sequence Suggested Dates Unit 1: Addition and Subtraction with Fractions Unit- Lesson Lesson Title Lesson Focus Eligible Content BIG IDEA 1: Equivalent Fractions 1-1 Introduce the MathBoard Use the MathBoard fraction bars to discuss basic fraction ideas. 1-2 Explain Equivalent Fractions Generate and explain simple equivalent fractions. 1-3 Equivalent Fractions and M05.A-F.1.1.1 Understand the role of the multiplier in equivalent fractions. Multipliers 1-4 Strategies for Comparing Fractions Use a variety of strategies to compare fractions. 1-5 Fractions Greater Than One Convert between fractions and mixed numbers. BIG IDEA 2: Addition and Subtraction with Fractions 1-6 Add and Subtract Like Mixed Numbers Add and subtract mixed numbers with like denominators. 1-7 Add Unlike Fractions Add fractions with different denominators. 1-8 Subtract Unlike Fractions Subtract fractions with different denominators. Solve with Unlike Mixed 1-9 Add and subtract mixed numbers with unlike denominators. Numbers M05.A-F.1.1.1 Practice with Unlike Solve problems involving the addition and subtraction of mixed 1-10 M05.D-M.2.1.1 Mixed Numbers numbers and unlike denominators. 1-11 Reasonable Answers Estimate sums and differences of fractions and mixed numbers and use estimates to decide whether answers for problems are reasonable. 1-12 Real World Problems Solve real world problems involving fractions and mixed numbers and use estimates to determine whether their answers are reasonable. 1-13 Focus on Mathematical Practices Solve real world problems involving fractions and mixed numbers. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 5
10/24-10/31 1 Day Per Lesson + 2.5 Days 10/13-10/23 1 Day Per Lesson + 3 Days 10/6-10/12 1 Day Per Lesson + 2 Days Suggested Dates Unit- Lesson 2-1 2-2 2-3 2-4 2-5 2-6 2-7 2-8 2-9 2-10 Unit 2: Addition and Subtraction with Decimals Lesson Title Lesson Objective Eligible Content Decimals as Equal Divisions Thousands to Thousandths Equate and Compare Thousandths Adding and Subtracting Decimals Add Whole Numbers and Decimals Subtract Whole and Decimal Numbers Properties and Strategies Round and Estimate with Decimals Graph with Decimal Numbers Focus on Mathematical Practices BIG IDEA 1: Read and Write Whole Numbers and Decimals Learn about decimals as equal divisions of a whole. Understand decimals to thousandths. Compare decimal numbers through thousandths. BIG IDEA 2: Addition and Subtraction Use models to add and subtract decimals. Add decimals by aligning their place values. Subtract whole and decimal numbers to hundredths. Use the Commutative, Associative and Distributive Properties to compute mentally. BIG IDEA 3: Round and Estimate with Decimals Estimate decimal sums and differences. Interpret and construct bar graphs that involve decimal numbers. Solve real world problems with decimals. Benchmark 1 Window: 11/1 11/17 M05.A-T.1.1.1 M05.A-T.1.1.3 M05.A-T.1.1.4 M05.A-T.2.1.3 M05.D-M.1.1.1 M05.A-T.1.1.4 M05.A-T.1.1.5 M05.A-T.2.1.3 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 6
Benchmark Cycle 2 Standards PA Common Core Standard CC.2.1.5.C.1 Use the understanding of equivalency to add and subtract fractions. CC.2.1.5.C.2 Apply and extend previous understandings of multiplication and division to multiply and divide fractions. PA Eligible Content M05.A-F.1.1.1 Add and subtract fractions (including mixed numbers) with unlike denominators. (May include multiple methods and representations.) Example: 2/3 + 5/4 = 8/12 + 15/12 = 23/12 M05.A-F.2.1.1 Solve word problems involving division of whole numbers leading to answers in the form of fractions (including mixed numbers). Common Core State Standard 5.NF.A.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.) 5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2. 5.NF.B.3 Interpret a fraction as division of the numerator by the denominator (a/b = a b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie? THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 7
CC.2.1.5.C.2 Apply and extend previous understandings of multiplication and division to multiply and divide fractions. M05.A-F.2.1.2 Multiply a fraction (including mixed numbers) by a fraction. M05.A-F.2.1.3 Demonstrate an understanding of multiplication as scaling (resizing). Example 1: Comparing the size of a product to the size of one factor on the basis of the size of the other factor without performing the indicated multiplication. Example 2: Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number. 5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. a. Interpret the product (a/b) q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a q b. For example, use a visual fraction model to show (2/3) 4 = 8/3, and create a story context for this equation (In general, (a/b) (c/d) = (ac)/(bd). 5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. 5.NF.B.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. 5.NF.B.5 Interpret multiplication as scaling (resizing), by: a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 8
CC.2.1.5.C.2 Apply and extend previous understandings of multiplication and division to multiply and divide fractions. M05.A-F.2.1.4 Divide unit fractions by whole numbers and whole numbers by unit fractions. 5.NF.B.5 Interpret multiplication as scaling (resizing) by: b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n a)/(n b) to the effect of multiplying a/b by 1. 5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) 4 = 1/12 because (1/12) 4 = 1/3. 5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 (1/5) = 20 because 20 (1/5) = 4. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 9
CC.2.1.5.C.2 Apply and extend previous understandings of multiplication and division to multiply and divide fractions. CC.2.4.5.A.1 Solve problems using conversions within a given measurement system. CC.2.4.5.A.4 Solve problems involving computation of fractions using information provided in a line plot. CC.2.1.5.B.1 Apply placevalue concepts to show an understanding of operations and rounding as they pertain to whole numbers and decimals. M05.D-M.1.1.1 Convert between different-sized measurement units within a given measurement system. A table of equivalencies will be provided. Example: Convert 5 cm to meters. M05.D-M.2.1.1 Solve problems involving computation of fractions by using information presented in line plots. M05.A-T.1.1.1 Demonstrate an understanding that in a multi-digit number, a digit in one place represents 1/10 of what it represents in the place to its left. Example: Recognize that in the number 770, the 7 in the tens place is 1/10 the 7 in the hundreds place. M05.A-T.1.1.2 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 5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins? 5.MD.A.1 Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems. 5.MD.B.2 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally. 5.NBT.A.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.A.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. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 10
CC.2.1.5.B.1 Apply placevalue concepts to show an understanding of operations and rounding as they pertain to whole numbers and decimals. CC.2.1.5.B.2 Extend an understanding of operations with whole numbers to perform operations including decimals. 10. Example 1: 4 102 = 400 Example 2: 0.05 103 = 0.00005 M05.A-T.1.1.3 Read and write decimals to thousandths using base-ten numerals, word form, and expanded form. Example: 347.392 = 300 + 40 + 7 + 0.3 + 0.09 + 0.002 = 3 100 + 4 10 + 7 1 + 3 (0.1) + 9 (0.01) + 2 (0.001) M05.A-T.1.1.4 Compare two decimals to thousandths based on meanings of the digits in each place using >, =, and < symbols. M05.A-T.1.1.5 Round decimals to any place (limit rounding to ones, tenths, hundredths, or thousandths place). M05.A-T.2.1.1 Multiply multi-digit whole numbers (not to exceed three-digit by three-digit). M05.A-T.2.1.2 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors. M05.A-T.2.1.3 Add, subtract, multiply, and divide decimals to hundredths (no divisors with decimals). 5.NBT.A.3 Read, write, and compare decimals to thousandths. a. Read and write decimals to thousandths using baseten numerals, number names, and expanded form, e.g., 347.392 = 3 100 + 4 10 + 7 1 + 3 (1/10) + 9 (1/100) + 2 (1/1000). 5.NBT.A.3 Read, write, and compare decimals to thousandths. b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. 5.NBT.4 Use place value understanding to round decimals to any place. 5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm. 5.NBT.B.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.B.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. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 11
11/29-12/11 1 Day Per Lesson + 3.5 Days 11/17-11/28 1 Day Per Lesson + 3 Half Days 11/2-11/16 1 Day Per Lesson + 3 Days Benchmark Cycle 2 Scope and Sequence Suggested Dates Unit- Lesson 3-1 Basic Multiplication Concepts Unit 3: Multiplication and Division with Fractions Lesson Title Lesson Objective Eligible Content BIG IDEA 1: Multiplication with Fractions Connect multiplying by 1/n to dividing by n, and extend this understanding to make multiplicative comparisons. Interpret a/b times a quantity as a of b equal parts of that quantity. Multiply a whole number by a fraction when the product is a fraction. 3-2 Multiplication with Non-Unit Fractions 3-3 Multiplication with Fractional Solutions 3-4 Multiply a Fraction by a Fraction Multiply any two fractions. 3-5 Multiplication Strategies Learn strategies to simplify multiplication of fractions. 3-6 Multiply Mixed Numbers Multiply with mixed numbers. BIG IDEA 2: Multiplication Links 3-7 Relate Fraction Operations Relate the operations with fractions to operations with whole numbers. 3-8 Solve Real-World Problems Add, subtract, multiply, and compare fractions to solve word problems. 3-9 Make Generalizations Predict how the size of a fractional factor will affect the size of the product relative to the other factor. BIG IDEA 3: Division with Fractions 3-10 When Dividing Is Also Multiplying Learn how division by a unit fraction or a whole number relates to multiplication. 3-11 Solve Division Problems Use diagrams to analyze division problems and learn to solve them. 3-12 Distinguish Multiplication from Discern whether word problems should be solved by Division multiplication or division equations. 3-13 Review Operations with Fractions Analyze and solve a variety of problems involving fractions and all the operations. 3-14 Focus on Mathematical Practices Solve real-world problems involving division with fractions. M05.A-F.2.1.3 M05.A-F.2.1.2 M05.A-F.1.1.1 M05.A-F.2.1.3 M05.A-F.2.1.2 M05.A-F.1.1.1 M05.A-F.2.1.1 M05.A-F.2.1.2 M05.A-F.2.1.3 M05.A-F.2.1.4 M05.D-M.2.1.1 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 12
1/3-1/18 1 Day Per Lesson + 4 Days 12/12-12/22 1 Day Per Lesson + 4 Days Suggested Dates Unit 4: Multiplication with Whole Numbers and Decimals Unit- Lesson Lesson Title Lesson Objective Eligible Content BIG IDEA 1: Multiplication with Whole Numbers 4-1 Shift Patterns in Multiplication Learn how the digits of a number shift their place value positions when multiplied by powers of 10. 4-2 Patterns with Fives and Zeros Learn how multiples of 5 affect the zeros pattern. 4-3 Sharing Methods for Multiplication 4-4 Multiply Two-Digit Numbers Use place value models to help solve multi-digit multiplication problems. Learn new methods, new Group Below Method, Place Value Rows Method, and Short Cut Method, to multiply multi-digit numbers. 4-5 Practice Multiplication Develop fluency multiplying multi-digit numbers. 4-6 Multiply Decimals by Whole Numbers BIG IDEA 2: Multiplication with Decimal Numbers Solve multiplication problems in which one factor is a decimal number. 4-7 Multiply by Decimals Multiply a whole or decimal number by a decimal number. 4-8 Multiply with Decimals Greater Than 1 4-9 Compare Shift Patterns 4-10 Estimate Products 4-11 Multiplication Practice 4-12 Focus on Mathematical Practices Multiply with decimal numbers greater than 1. Compare shift patterns when multiplying by whole numbers and when multiplying by decimals. Round whole numbers and decimal numbers to estimate a product. Solve a variety of problems that involve multiplying decimal numbers. Solve real world problems using multiplication of multi-digit decimal numbers. M05.A-T.1.1.1 M05.A-T.1.1.2 M05.A-T.2.1.1 M05.A-T.2.1.3 M05.A-F.2.1.3 M05.A-T.1.1.1 M05.A-T.1.1.2 M05.A-T.1.1.3 M05.A-T.1.1.4 M05.A-T.1.1.5 M05.A-T.2.1.1 M05.A-T.2.1.3 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 13
1/19-1/26 1 Day Per Lesson + 1 Day Suggested Dates Unit 5: Division with Whole Numbers and Decimals Unit- Lesson Lesson Title Lesson Objective Eligible Content BIG IDEA 1: Division with Whole Numbers 5-1 Divide Whole Numbers by One Digit Divide multi-digit numbers by single-digit divisors. 5-2 Explore Dividing by Two-Digit Whole Numbers Divide when the divisor has two digits. 5-3 Too large, Too Small, or Just Adjust estimated digits that are too high or too low and M05.A-T.2.1.2 Right? recognize how to handle each case. 5-4 Interpret Remainders Interpret remainders for a variety of problem types. 5-5 Division Practice Increase competency in division of whole numbers by providing numerical and word problems. Benchmark 2 Window: 1/29 2/13 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 14
Benchmark Cycle 3 Standards PA Common Core Standard CC.2.1.5.B.1 Apply placevalue concepts to show an understanding of operations and rounding as they pertain to whole numbers and decimals. CC.2.1.5.B.2 Extend an understanding of operations with whole numbers to perform operations including decimals. PA Eligible Content M05.A-T.1.1.2 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. Example 1: 4 102 = 400 Example 2: 0.05 103 = 0.00005 M05.A-T.1.1.4 Compare two decimals to thousandths based on meanings of the digits in each place using >, =, and < symbols. M05.A-T.1.1.5 Round decimals to any place (limit rounding to ones, tenths, hundredths, or thousandths place). M05.A-T.2.1.1 Multiply multi-digit whole numbers (not to exceed three-digit by three-digit). M05.A-T.2.1.2 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors. M05.A-T.2.1.3 Add, subtract, multiply, and divide decimals to hundredths (no divisors with decimals). Common Core State Standard 5.NBT.A.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.A.3 Read, write, and compare decimals to thousandths. b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. 5.NBT.4 Use place value understanding to round decimals to any place. 5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm. 5.NBT.B.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.B.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 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 15
CC.2.1.5.C.1 Use the understanding of equivalency to add and subtract fractions. M05.A-F.1.1.1 Add and subtract fractions (including mixed numbers) with unlike denominators. (May include multiple methods and representations.) Example: 2/3 + 5/4 = 8/12 + 15/12 = 23/12 and subtraction; relate the strategy to a written method and explain the reasoning used. 5.NF.A.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.) CC.2.1.5.C.2 Apply and extend previous understandings of multiplication and division to multiply and divide fractions. M05.A-F.2.1.1 Solve word problems involving division of whole numbers leading to answers in the form of fractions (including mixed numbers). 5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5+1/2=3/7, by observing that 3/7<1/2 5.NF.B.3 Interpret a fraction as division of the numerator by the denominator (a/b = a b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie? THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 16
CC.2.1.5.C.2 Apply and extend previous understandings of multiplication and division to multiply and divide fractions. CC.2.1.5.C.2 Apply and extend previous understandings of multiplication and division to multiply and divide fractions. M05.A-F.2.1.2 Multiply a fraction (including mixed numbers) by a fraction. M05.A-F.2.1.3 Demonstrate an understanding of multiplication as scaling (resizing). Example 1: Comparing the size of a product to the size of one factor on the basis of the size of the other factor without performing the indicated multiplication. Example 2: Explaining why multiplying a given number by a fraction greater than 1 5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. a. Interpret the product (a/b) q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a q b. For example, use a visual fraction model to show (2/3) 4 = 8/3, and create a story context for this equation. Do the same with (2/3) (4/5) = 8/15. (In general, (a/b) (c/d) = (ac)/(bd). 5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. 5.NF.B.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. 5.NF.B.5 Interpret multiplication as scaling (resizing), by: a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 17
CC.2.1.5.C.2 Apply and extend previous understandings of multiplication and division to multiply and divide fractions. results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number. M05.A-F.2.1.4 Divide unit fractions by whole numbers and whole numbers by unit fractions. 5.NF.B.5 Interpret multiplication as scaling (resizing), by: b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n a)/(n b) to the effect of multiplying a/b by 1. 5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) 4 = 1/12 because (1/12) 4 = 1/3. 5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 (1/5) = 20 because 20 (1/5) = 4. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 18
CC.2.1.5.C.2 Apply and extend previous understandings of multiplication and division to multiply and divide fractions. CC.2.2.5.A.1 Interpret and evaluate numerical expressions using order of operations. CC.2.2.5.A.4 Analyze patterns and relationships using two rules. M05.A-F.2.1.4 Divide unit fractions by whole numbers and whole numbers by unit fractions. M05.B-O.1.1.1 Use multiple grouping symbols (parentheses, brackets, or braces) in numerical expressions and evaluate expressions containing these symbols. M05.B-O.1.1.2 Write simple expressions that model calculations with numbers and interpret numerical expressions without evaluating them. Example 1: Express the calculation add 8 and 7, then multiply by 2 as 2 (8 + 7). Example 2: Recognize that 3 (18,932 + 921) is three times as large as 18,932 + 921 without having to calculate the indicated sum or product. M05.B-O.2.1.1 Generate two numerical patterns using two given rules. Example: Given the rule add 3 and the starting number 0 and given the rule add 6 and the starting number 0, generate terms in the resulting sequences. M05.B-O.2.1.2 Identify apparent relationships between corresponding terms of two patterns with the same starting numbers that follow different rules. Example: Given two patterns in which the first pattern follows the rule add 8 and the second pattern follows the rule add 2, observe that the terms in the first pattern are 4 times the size of the terms in the second pattern. 5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins? 5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. 5.OA.A.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. 5.OA.B.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 19
CC.2.3.5.A.1 Graph points in the first quadrant on the coordinate plane and interpret these points when solving real world and mathematical problems. CC.2.4.5.A.1 Solve problems using conversions within a given measurement system. CC.2.4.5.A.4 Solve problems involving computation of fractions using information provided in a line plot. M05.C-G.1.1.1 Identify parts of the coordinate plane (xaxis, y-axis, and the origin) and the ordered pair (xcoordinate and y-coordinate). Limit the coordinate plane to quadrant I. M05.C-G.1.1.2 Represent real-world and mathematical problems by plotting points in quadrant I of the coordinate plane and interpret coordinate values of points in the context of the situation. M05.D-M.1.1.1 Convert between different-sized measurement units within a given measurement system. A table of equivalencies will be provided. Example: Convert 5 cm to meters. M05.D-M.2.1.1 Solve problems involving computation of fractions by using information presented in line plots. 5.G.A.1 Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x- coordinate, y-axis and y-coordinate). (Not exact match) 5.G.A.2 Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. 5.MD.A.1 Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems. 5.MD.B.2 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 20
1/30-2/9 1 Day Per Lesson + 5 Days Italicized (5-7 and 5-8 go beyond the standard; optional and not counted in days) Benchmark Cycle 3 Scope and Sequence Suggested Dates Unit 5 Continued: Division with Whole Numbers and Decimals Unit- Lesson Lesson Title Lesson Objective Eligible Content BIG IDEA 2: Division with Decimal Numbers 5-6 Divide Decimal Numbers by Whole Expand knowledge of division to include division of decimal Numbers numbers by whole numbers. 5-7 Divide Whole Numbers by Decimal Divide when the divisor is a decimal number. Numbers M05.A-T.1.1.2 5-8 Divide a Decimal Number by a Solve problems where both the dividend and the divisor are M05.A-T.1.1.4 Decimal Number decimal numbers. M05.A-T.2.1.1 5-9 Division Practice Solve a variety of division problems involving whole M05.A-T.2.1.2 numbers and decimal numbers. M05.A-T.2.1.3 5-10 Distinguish Between Multiplication Decide whether a word problem involving decimals can be M05.A-F.2.1.3 and Division solved by using multiplication or by using division. 5-11 Focus on Mathematical Practices Solve real world problems involving multiplication and division of decimal numbers. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 21
2/28-3/6 1 Day Per Lesson + 1 Day 2/22-2/27 1 Day Per Lesson + 1 Day 2/12-2/21 1 Day Per Lesson + 3 Half Days Suggested Dates Unit 6: Operations and Word Problems Unit- Lesson Lesson Title Lesson Objective Eligible Content BIG IDEA 1: Equations and Problem Solving Situation and Solution Equations Write situation and solution equations for addition and 6-1 M05.A-T.1.1.5 for Addition and Subtraction subtraction situations. M05.A-T.2.1.1 Situation and Solution Equations Write situation and solution equations for multiplication 6-2 M05.A-T.2.1.2 for Multiplication and Division and division situations. M05.A-T.2.1.3 Write word problems for equations involving fractions and 6-3 Write Word Problems M05.A-F.1.1.1 decimals and model the problem situation. M05.A-F.2.1.2 Learn a variety of methods to determine if answers are 6-4 Determine Reasonable Answers M05.A-F.2.1.4 reasonable. BIG IDEA 2: Comparison Word Problems 6-5 Language of Comparison Problems Apply comparison language to solve additive and M05.A-T.2.1.1 multiplicative comparison problems. M05.A-T.2.1.2 6-6 Multiplicative Comparison Model and solve problems involving multiplicative M05.A-T.2.1.3 Problems comparisons. M05.A-F.1.1.1 6-7 Types of Comparison Problems Distinguish additive and multiplicative comparisons. M05.A-F.2.1.3 M05.A-F.2.1.2 6-8 Equations and Parentheses 6-9 Multi-step Word Problems BIG IDEA 3: Problems with More Than One Step Use parentheses to write equations for word problems that require two steps. Practice writing equations and solving multi-step word problems. 6-10 Practice Problem Solving Solve multi-step problems. 6-11 Focus on Mathematical Practices Solve multi-step real world problems. M05.B-O.1.1.1 M05.A-T.2.1.1 M05.A-T.2.1.2 M05.A-T.2.1.3 M05.A-F.1.1.1 M05.A-F.2.1.1 M05.A-F.2.1.2 M05.A-F.2.1.4 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 22
3/19-3/22 1 Day Per Lesson + 2.5 Days 3/7-3/13 1 Day Per Lesson + 2 Days Suggested Dates Unit- Lesson Topic 7: Algebra, Patterns, and Coordinate Graphs Lesson Title Lesson Objective Eligible Content BIG IDEA 1: Algebraic Reasoning and Expressions 7-1 Read and Write Expressions Read and write numerical expressions. 7-2 Simplify Expressions Simplify numerical expressions. 7-3 Evaluate Expressions Write and evaluate expressions that contain variables. 7-4 Patterns and Relationships 7-5 The Coordinate Plane BIG IDEA 2: Patterns and Graphs Generate and extend numerical patterns and identify relationships of corresponding terms. Locate and plot points in the first quadrant of the coordinate plane. 7-6 Graph Ordered Pairs Solve real world problems by graphing ordered pairs. 7-7 Focus on Mathematical Practices Solve real world problems involving graphing and the coordinate plane. M05.B-O.1.1.1 M05.B-O.1.1.2 M05.B-O.1.1.1 M05.B-O.1.1.2 M05.B-O.2.1.1 M05.B-O.2.1.2 M05.C-G.1.1.1 M05.C-G.1.1.2 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 23
3/23-4/13 1 Day Per Lesson + 1 Day + ELA PSSA Week Suggested Dates Unit- Lesson Unit 8: Measurement and Geometry Lesson Title Lesson Objective Eligible Content BIG IDEA 1: Measurements and Data 8-1 Metric Units of Length Convert among metric units of length. 8-2 Customary Units of Length Convert among customary units of length. 8-3 Perimeter and Area of Rectangles Use a formula to find the perimeter and area of a rectangle with fractional side lengths. 8-4 Cubic Units and Volume Use a formula to find the volume of a rectangular prism. 8-5 Visualize Volume Compute the volume of a rectangular prism. M05.D-M.1.1.1 M05.D-M.2.1.1 8-6 Introduce Volume Formulas Use a formula to find the volume of a rectangular prism. 8-7 Relate Length, Area, and Volume Identify whether a situation involves length, area, or volume. 8-8 Volume of Composite Solid Figures Find the volume of a composite solid figure. 4/16 4/20 MATH PSSA THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 24
4/23-5/8 1 Day Per Lesson + 1 Whole and 4 Half Days The rest of Unit 8 will not be on the benchmark, so its standards are included in Cycle 4. To ensure you have time in Cycle 4 to finish the unit and to review and extend content before the end of the year, it is recommended that you continue Unit 8 now. Unit 8 Continued: Measurement and Geometry Suggested Dates Unit- Lesson Lesson Title Lesson Objective Eligible Content BIG IDEA 2: Liquid Volume, Mass, and Weight 8-9 Metric Units of Liquid Volume Convert among metric units of liquid volume. 8-10 Metric Units of Mass Convert among metric units of mass. 8-11 Customary Units of Liquid Volume Convert among customary units of liquid volume. 8-12 Customary Units of Weight Convert among customary units of weight. 8-13 Read and Make Line Plots Make and analyze line plots. PA-1 Display Data in Different Ways PA-2 Interpret Data Displays M05.D-M.3.1.1 M05.D-M.3.1.2 M05.D-M.2.1.2 Benchmark 3 Window: 5/9 5/25 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 25
PA Common Core Standard CC.2.4.5.A.2 Represent and interpret data using appropriate scale. CC.2.4.5.A.5 Apply concepts of volume to solve problems and relate volume to multiplication and to addition. PA Eligible Content Cycle 4 Standards M05.D-M.2.1.2 Display and interpret data shown in tallies, tables, charts, pictographs, bar graphs, and line graphs, and use a title, appropriate scale, and labels. A grid will be provided to display data on bar graphs or line graphs. M05.D-M.3.1.1 Apply the formulas V = l w h and V = B h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real-world and mathematical problems. Formulas will be provided. Common Core State Standard 5.MD.C.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement. a. A cube with side length 1 unit, called a "unit cube," is said to have "one cubic unit" of volume, and can be used to measure volume. (Not exact match) 5.MD.C.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement. b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units. (Not exact match) 5.MD.C.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. 5.MD.A.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. a. Find the volume of a right rectangular prism with whole-number side lengths by packing with unit cubes, and show that volume is the same as would be found by multiplying the edge lengths, or by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 26
CC.2.4.5.A.5 Apply concepts of volume to solve problems and relate volume to multiplication and to addition. CC.2.4.5.A.5 Apply concepts of volume to solve problems and relate volume to multiplication and to addition. CC.2.3.5.A.2 Classify twodimensional figures into categories based on an understanding of their properties. M05.D-M.3.1.1 Apply the formulas V = l w h and V = B h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real-world and mathematical problems. Formulas will be provided. M05.D-M.3.1.2 Find volumes of solid figures composed of two non-overlapping right rectangular prisms. M05.C-G.2.1.1 Classify two-dimensional figures in a hierarchy based on properties. Example 1: All polygons have at least three sides, and pentagons are polygons, so all pentagons have at least three sides. Example 2: A rectangle is a parallelogram, which is a quadrilateral, which is a polygon; so, a rectangle can be classified as a parallelogram, as a quadrilateral, and as a polygon. 5.MD.A.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. b. Apply the formulas V = l w h and V = b h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems. 5.MD.A.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the nonoverlapping parts, applying this technique to solve real world problems. 5.G.B.3 Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles. (Not exact match) 5.G.B.4 Classify two-dimensional figures in a hierarchy based on properties. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY 2017-2018 27