Unit 1: Place value and operations with whole numbers and decimals

Unit 1: Place value and operations with whole numbers and decimals Content Area: Mathematics Course(s): Generic Course Time Period: 1st Marking Period Length: 10 Weeks Status: Published Unit Overview Students will understand the place value system. Students will perform operations with multi digit whole numbers and with decimals to hundredths. Benchmarks: During this unit students will be completing two benchmark assessments. The first benchmark covers Chapters 1 3 and should be given at the end of October. The second benchmark covers Chapters 4 6 and should be given at the end of December before winter break. Transfer Students will be able to independently use their learning to... Solve real world problems by performing operations with multi digit whole numbers and decimals to hundredths to demonstrate understanding of the place value system. For more information, read the following article by Grant Wiggins. http://www.authenticeducation.org/ae_bigideas/article.lasso?artid=60 Meaning Understandings Students will understand... Place value is an important tool for solving problems and checking that solutions make sense. The difference aspects of multiplication and stress that multiplication involves a logical progression of steps.

The different aspects and process of dividing both single and multi digit numbers. The different aspects of adding and subtracting decimals. The different aspects of multiplying and dividing decimals. Essential Questions Students will keep considering.. How does the position of a digit in a number relate to its value? What strategies can be used to multiply whole numbers? What strategies can be used to divide whole numbers? What strategies can I use to divide by a two digit number? How can I use place value and properties to add and subtract decimals? How is multiplying and dividing decimals similar to multiplying and dividing whole numbers? Application of Knowledge and Skill Students will know... Students will know... The place value system from the thousandths to the millions place. How to multiply multi digit whole numbers. The process for dividing with one and two digit divisors. The process for all decimal operations. The Associative, Commutative and Distributive Properties. That their knowledge of all four operations with whole numbers can be extended to decimals. Students will be skilled at...

Students will be skilled at... Determining the value of each digit in a decimal Reading, writing and recognizing decimals Comparing, ordering and rounding whole numbers and decimals. Solving problems with all four operations with whole numbers using the standard algorithm, equations, rectangular arrays and/or area models. Performing the four operations with decimals to the hundredths. Identifying and applying the properties to help solve problems. Academic Vocabulary Place Value Chart Period Place Place Value Standard form Expanded Form Decimal Decimal Point Decimals Equivalent Decimals Prime Factorization Exponent base

power squared cubed powers of 10 distributive property compatible numbers fact family unknown variable variable dividend divisor quotient remainder partial quotients commutative property of addition associate property of addition identity property of addition inverse operations associate property of multiplication commutative property of multiplication identity property of multiplication

Learning Goal 1- Place Value Students will be able to understand the place value system from the thousandths to the millions place. Target 1 (Level of Difficulty 2 Comprehension) SWBAT... Read and write whole numbers and decimals in standard form, expanded form and word form. Identify the place value of any given digit between the thousandths and millions places. 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. Use models to relate decimals to fractions. Examples: 1. 34.18 The following number can be written in standard form (shown before), expanded form: (3x10) + (4x1) + (1 x 1/10) + (8 x 1/100), and word form: thirty four and eighteen hundredths 2. 74,932.158: The underlined digit is in the hundreds place and the value of the 9 is 900. 3. See My Math Chapter 1, Lesson 3. MA.5.CCSS.Math.Content.5.NBT.A.1 MA.5.CCSS.Math.Content.5.NBT.A.3a MA.5.CCSS.Math.Content.5.NBT.A.4 MA.5.CCSS.Math.Content.5.NBT.B.6 MA.5.CCSS.Math.Content.5.NBT.B.7 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. Read and write decimals to thousandths using base ten 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). Use place value understanding to round decimals to any place. 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. 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. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course

if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. Target 2 ( Level of Difficulty 3 Analysis) SWBAT: Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. Order whole numbers and decimals from least to greatest and greatest to least order. Use place value to round decimals to any place. Examples: 1. Compare 4.4 to 4.44 2. Order the following numbers from least to greatest: 9.275;8.950;9.375 3. Round 9.848 to the nearest hundredths place. 9.85

MA.5.CCSS.Math.Content.5.NBT.A.3 MA.5.CCSS.Math.Content.5.NBT.A.3a MA.5.CCSS.Math.Content.5.NBT.A.3b MA.5.CCSS.Math.Content.5.NBT.A.4 Read, write, and compare decimals to thousandths. Read and write decimals to thousandths using base ten 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). Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. Use place value understanding to round decimals to any place. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered 7 5 + 7 3, in preparation for learning about the distributive property. In the expression x + 9 x + 14, older students can see the 14 as 2 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3( x y ) as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation ( y 2)/( x 1) = 3. Noticing the regularity in the way terms cancel when expanding ( x 1)( x + 1), ( x 1)( x + x + 1), and ( x 1)( x + x + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Learning Goal 2 - Whole Number Operations Students will be able to multiply and divide multi digit whole numbers. Target 1 (Level of Difficulty 2 Comprehension) SWBAT:

Find the prime factorization of numbers by creating a factor tree. Use powers and exponents in expressions. Explain that when you multiply a number by a multiple of 10 you attach the total number of zeroes of the factors you are multiplying. When multiplying a number by a power of ten, the decimal point moves to the right as many places as the number of zeroes in the multiple of ten. Also, when dividing a number by a power of ten, the decimal point moves to the left as many places as the number of zeroes in the multiple of ten. Use whole-number exponents to denote powers of 10. Examples: 1.Write 3x3x3x3 using an exponent: 3 4 2. Write 3 4 as a product. 3x3x3x3= 81 3.5 x 1/10 will relate well to subsequent work with operating with fractions. This example shows that when we divide by powers of 10, the exponent above the 10 indicates how many places the decimal point is moving (how many times we are dividing by 10, the number becomes ten times smaller). Since we are dividing by powers of 10, the decimal point moves to the left. Students need to be provided with opportunities to explore this concept and come to this understanding; this should not just be taught. MA.5.CCSS.Math.Content.5.NBT.A.1 MA.5.CCSS.Math.Content.5.NBT.A.2 MA.K 12.CCSS.Math.Practice.MP1 Target 2 (Level of Difficulty 3 Analysis) SWBAT... 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. 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. Make sense of problems and persevere in solving them. Use partial products to multiply two numbers. Use the Distributive Property to multiply whole numbers. Examples: 1. Use partial products to multiply two numbers like 3 and 24.

3 24 = (3 20) + (3 4) Find partial products. = 60 + 12 Multiply. = 72 Add. 2. Find products like 4 65 mentally by using the Distributive Property. 4 65 = 4 (60 + 5) Write 65 as 60 + 5. = (4 60) + (4 5) Use the Distributive Property. = 240 + 20 Multiply. = 260 Add. MA.5.CCSS.Math.Content.5.NBT.A MA.5.CCSS.Math.Content.5.NBT.B.5 MA.K 12.CCSS.Math.Practice.MP7 Target 3 (Level of Difficulty 2 Comprehension) SWBAT... Understand the place value system. Fluently multiply multi digit whole numbers using the standard algorithm. Look for and make use of structure. Fluently multiply multi digit whole numbers using the standard algorithm. This standard refers to fluency which means accuracy (correct answer), efficiency (a reasonable amount of steps), and flexibility (using strategies such as the distributive property or breaking numbers apart also using strategies according to the numbers in the problem, 26 x 4 may lend itself to (25 x 4 ) + 4 where as another problem might lend itself to making an equivalent problem 32 x 4 = 64 x 2)). Students must be able to multiply a three digit factor by a two digit factor. Example: 1. Multiply numbers like 78 and 32. 78 32 156 + 2,340 2,496 Multiply the ones: 78 2. Multiply the tens: 78 30. Add.

MA.5.CCSS.Math.Content.5.NBT.B.5 Target 4 (Level of Difficulty 3 Analysis) SWBAT: Fluently multiply multi digit whole numbers using the standard algorithm. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered 7 5 + 7 3, in preparation for learning about the distributive property. In the expression x + 9 x + 14, older students can see the 14 as 2 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3( x y ) as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Find quotients of whole numbers with up to four digit dividends and two digit divisors by using multiple strategies including: area models, rectangular arrays, and the standard algorithm. This standard references various strategies for division. Division problems can include remainders. Even though this standard leads more towards computation, the connection to story context is critical. Make sure students are exposed to problems where the divisor is the number of groups and where the divisor is the size of the groups. In fourth grade, students experiences with division were limited to dividing by one digit divisors. This standard extends students prior experiences with strategies, illustrations, and explanations. When the two digit divisor is a familiar number, a student might decompose the dividend using place value. Examples: 1. Traditional Algorithm 1 5 23)3 4 5 2 3 1 1 5

1 1 5 0 2. Using a Model Solve a division problem like 268 4 using a bar diagram. 50 15 2 4 200 60 8 Divide each section by 4. 200 4 = 50 60 4 = 15 8 4 = 2 Add the quotients. 50 + 15 + 2 = 67 MA.5.CCSS.Math.Content.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. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the

situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered 7 5 + 7 3, in preparation for learning about the distributive property. In the expression x + 9 x + 14, older students can see the 14 as 2 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3( x y ) as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation ( y 2)/( x 1) = 3. Noticing the regularity in the way terms cancel when expanding ( x 1)( x + 1), ( x 1)( x + x + 1), and ( x 1)( x + x + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Learning Goal 3 - Decimal Operations Students will be able to add, subtract, multiply, and divide decimals. Target 1 (Level of Difficulty 4 Knowledge Utilization) Add, subtract, and multiply decimals to hundredths using multiple strategies. Divide numbers with decimals in the dividend AND divisor up to the hundredths. Example 1: Subtract 11.48 3.27 11.48 3.27 8.21 Example 2: Solve 3.82 0.4

So, 3.82 0.4 = 9.55 When dividing decimals you need to remember to bring up the decimal point into the quotient. Remember: when there is a decimal as your divisor, you need to move the decimal point to the right to make a whole number in the divisor and also move it in the dividend the same number of places to the right to make a whole number. Now you can divide. MA.5.CCSS.Math.Content.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. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the

results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered 7 5 + 7 3, in preparation for learning about the distributive property. In the expression x + 9 x + 14, older students can see the 14 as 2 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3( x y ) as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation ( y 2)/( x 1) = 3. Noticing the regularity in the way terms cancel when expanding ( x 1)( x + 1), ( x 1)( x + x + 1), and ( x 1)( x + x + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Target 2 (Level of Difficulty 2 Knowledge Utilization) Relate the strategy to a written method and explain their reasoning. In fifth grade, students begin adding, subtracting, multiplying and dividing decimals. This work should focus on concrete models and pictorial representations, rather than relying solely on the algorithm. The use of symbolic notations involves having students record the answers to computations (2.25 x 3= 6.75), but this work should not be done without models or pictures. This standard includes students reasoning and explanations of how they use models, pictures, and strategies. MA.5.CCSS.Math.Content.5.NBT.B.7 MA.K 12.CCSS.Math.Practice.MP1 MA.K 12.CCSS.Math.Practice.MP2 MA.K 12.CCSS.Math.Practice.MP3 MA.K 12.CCSS.Math.Practice.MP4 MA.K 12.CCSS.Math.Practice.MP6 MA.K 12.CCSS.Math.Practice.MP7 Summative Assessment Teacher Observation 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. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Attend to precision. Look for and make use of structure. Quizzes Common Assessment My Math Chapters 1 6 Assessments Projects Rubrics

Short & Extended Constructed Response Homework 21st Century Life and Careers WORK.5 8.9.1.8.1 WORK.5 8.9.1.8.A.1 WORK.5 8.9.1.8.B.1 WORK.5 8.9.1.8.1 WORK.5 8.9.1.8.C.1 WORK.5 8.9.1.8.C.3 WORK.5 8.9.1.8.1 The ability to recognize a problem and apply critical thinking and problem solving skills to solve the problem is a lifelong skill that develops over time. Develop strategies to reinforce positive attitudes and productive behaviors that impact critical thinking and problem solving skills. Use multiple points of view to create alternative solutions. Collaboration and teamwork enable individuals or groups to achieve common goals with greater efficiency. Determine an individual's responsibility for personal actions and contributions to group activities. Model leadership skills during classroom and extra curricular activities. Effective communication skills convey intended meaning to others and assist in preventing misunderstandings. Formative Assessment and Performance Opportunities Teacher observation Math journals Exit slips Quick checks during class Classwork activities and games Center work/ small group work Group work activities Homework MAP test Differentiation / Enrichment ELL Vocabulary 504 Accomodations IEP Modifications Leveled Centers Use of Manipulatives Place Value Reteach/ Enrich MyMath Chapter 1, Lesson 1 Compare Decimals Reteach/Enrich My Math Chapter 1, Lesson 7

Multiplication Reteach/Enrich My Math Chapter 2 Division Reteach/Enrich My Math Chapters 3 and 4 Decimal Operations Reteach/Enrich MyMath Chapters 5 and 6 Various Forms for Summative Assessment 3 Written Forms/ Oral Assessment per chapter Unit Resources MyMath Grade 5, Vol. 1Teacher Edition, Chapters 1 6 and Student Workbook: 2014 McGraw Hill Education My Math Online Portal

Unit 2: Operations and Algebraic Thinking Content Area: Mathematics Course(s): Generic Course Time Period: 2nd Marking Period Length: 2 Weeks Status: Published Unit Overview In this unit, students will learn to use patterns and graphing to solve problems. Benchmarks: During this unit students will be completing one benchmark assessment. The end of Chapter 7 test will be the benchmark test. This test should be given at mid or end January. Transfer Students will be able to independently use their learning to... Write and interpret numerical expressions. Use parentheses, brackets, and or braces in numerical expressisons to evaluate them. Analyze patterns and relationships For more information, read the following article by Grant Wiggins. http://www.authenticeducation.org/ae_bigideas/article.lasso?artid=60 Meaning

Understandings Students will understand... That expressions are often used to represent patterns and to solve problems arising in everyday life. How to create and interpret a graph to answer questions, anazlyze relationships, and draw conclusions. Essential Questions Students will keep considering... How are patterns used to solve problems? Application of Knowledge and Skill Students will know... Students will know... How to write and interpret numerical expressions How to anazlye patterns and relationships Students will be skilled at... Students will be skilled at... Solving order of operations problems using parentheses and exponents Writing and interpreting numerical expressions Generating a numerical pattern(s) when given a rule Identifying and plotting coordinate pairs on a graph Writing a rule to describe a pattern Academic Vocabulary

numerical expression evaluate order of operations sequence term coordinate plane origin ordered pair x coordinate y coordinate Learning Goal 1- Operations and Algebraic Thinking Students will be able to write and interpret numerical expressions and analyze patterns and relationships. Target 1 Expressions (Level of Difficulty 3 Analysis) SWBAT: Use the order of operations to simplify expressions including parentheses and exponents. (My Math Chapter 7, Lesson 2) Write and interpret verbal phrases as numerical expressions. (My Math Chapter 7, Lesson 3) Examples: 1. 3 2 + (8+2) / 2 = 14 2. Phrase: subtract 2 from 8, then divide by 3 = (8 2) / 3 MA.5.CCSS.Math.Content.5.OA.A Write and interpret numerical expressions.

MA.5.CCSS.Math.Content.5.OA.A.1 MA.5.CCSS.Math.Content.5.OA.A.2 Target 2 Patterns (Level of Difficulty 3 Analysis) SWBAT: Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. Apply addition and subtraction to describe and extend a number patterns. (My Math Chapter 7, Lessons 5 and 6) Example: 1. 72, 67, 62, 57, 47, The next number in the pattern is 47 5, or 42. MA.5.CCSS.Math.Content.5.OA.B.3 Target 3 Graphing (Level of Difficulty 3 Analysis) SWBAT: 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. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. Name ordered pairs for points. (My Math Chapter 7, Lesson 8) Graph points on a coordinate plane and use the graph to solve real world and mathematical problems. (My

Math Chapter 7, Lesson 8) Compare numerical patterns graphically. (My Math Chapter 7, Lesson 9) Example: Connect these points in order on the coordinate grid below: (2, 2) (2, 4) (2, 6) (2, 8) (4, 5) (6, 8) (6, 6) (6, 4) and (6, 2). Example: Plot these points on a coordinate grid. Point A: (2,6) Point B: (4,6) Point C: (6,3) Point D: (2,3) Connect the points in order. Make sure to connect Point D back to Point A. 1. What geometric figure is formed? What attributes did you use to identify it? 2. What line segments in this figure are parallel? 3. What line segments in this figure are perpendicular? solutions: trapezoid, line segments AB and DC are parallel, segments AD and DC are perpendicular Example: Sara has saved $20. She earns $8 for each hour she works. If Sara saves all of her money, how much will she have after

working 3 hours? 5 hours? 10 hours? Create a graph that shows the relationship between the hours Sara worked and the amount of money she has saved. What other information do you know from analyzing the graph? MA.5.CCSS.Math.Content.5.OA.B.3 MA.5.CCSS.Math.Content.5.G.A MA.5.CCSS.Math.Content.5.G.A.1 MA.5.CCSS.Math.Content.5.G.A.2 MA.K 12.CCSS.Math.Practice.MP1 MA.K 12.CCSS.Math.Practice.MP2 MA.K 12.CCSS.Math.Practice.MP4 MA.K 12.CCSS.Math.Practice.MP5 MA.K 12.CCSS.Math.Practice.MP6 MA.K 12.CCSS.Math.Practice.MP7 MA.K 12.CCSS.Math.Practice.MP8 Summative Assessment Vocabulary Quiz 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. Graph points on the coordinate plane to solve real world and mathematical problems. 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). 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. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Skill Specific Quiz My Math Chapter 7 Assessment Teacher Created Project(s) 21st Century Life and Careers WORK.5 8.9.1.8.A WORK.5 8.9.1.8.B WORK.5 8.9.1.8.C WORK.5 8.9.1.8.D Critical Thinking & Problem Solving Creativity and Innovation Collaboration, Teamwork and Leadership Cross Cultural Understanding and Interpersonal Communication Formative Assessment and Performance Opportunities Teacher observation Classwork activities and games Center work/ small group work Group work activities

Homework MAP test Differentiation/Enrichment ELL Vocabulary 504 Accomodations/ Modifications IEP Accomodations/ Modifications Leveled Centers Use of Manipulatives Order of Operations Reteach/ Enrich MyMath Chapter 7, Lesson 2 ELL Word Web MyMath Chapter 7, Lesson 2 ELL Tiered Questions MyMath Chapter 7, Lesson 2 Writing Numerical Expressions Reteach/Enrich MyMath Lesson 3 Graphing Patterns Reteach/Enrich MyMath Chapter 7, Lesson 9 Various Forms for Summative Assessment 3 Written Forms/ Oral Assessment Unit Resources MyMath Grade 5, Vol. 1 Teacher Edition and Student Workbook: 2014 McGraw Hill Education My Math Online Portal

Unit 3 : Fractions Content Area: Mathematics Course(s): Generic Course Time Period: 2nd Marking Period Length: 10-12 Weeks Status: Published Unit Overview Extend understanding of fraction equivalence and ordering. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. Add, subtract, multiply, and divide fractions and mixed numbers with unlike denominators. Understand decimal notation for fractions, and compare decimals and fractions. Benchmarks: During this unit students will be completing one benchmark assessment. This is fifth grade common assessment which covers Chapters 8 10 (fractions) and should be given mid March. Transfer Students will be able to independently use their learning to... Compare, add, and subtract fractions with like and unlike denominators, as well as find and compare the decimal notation for the fractions. Students will be able to use their knowledge of fractions to assist with measurement of objects. For more information, read the following article by Grant Wiggins. http://www.authenticeducation.org/ae_bigideas/article.lasso?artid=60 Meaning Understandings Students will understand that... Computational fluency includes understanding not only the meaning, but also the appropriate use of numerical

operations. The magnitude of numbers affects the outcome of operations on them. In many cases, there are multiple algorithms for finding a mathematical solution, and those algorithms are frequently associated with different cultures. Fractions, decimals, and percents and how can fractions be modeled, compared, and ordered Essential Questions Students will keep considering... How are factors and multiples helpful in solving problems? How can equivalent fractions help me add and subtract fractions? What strategies can be used to multiply and divide fractions? Application of Knowledge and Skill Students will know... Students will know... How to compare and order fractions and mixed numbers How to add and subtract fractions and mixed numbers How to multiply and divide fractions and mixed numbers Students will be skilled at... Students will be skilled at... Finding common denominators and creating equivalent fractions in order to compare and order fractions. Adding and subtracting fractions with unlike denominators, including mixed numbers, by replacing given fractions with equivalent fractions. Choosing an operation involving fractions and mixed numbers to solve a real life problem. Finding area of a rectangle with fractional side lengths. Interpreting a fraction as division of the numerator by the denominator and solve word problems involving

division of whole numbers leading to answers in the form of fractions or mixed numbers. Solving real world problems involving division of fractions by non zero whole numbersand division of whole numbers by fractions. Academic Vocabulary benchmark fraction denominator estimate fraction mixed number reasonableness refer unlike whole word problem denominator fraction greater than interpret less than multiplication multiply number numerator

product real world unit fraction whole number data fraction measurement unit Learning Goal 1- Introduction to Fractions Students will explore the different aspects of fractions and decimals. Target 1 (Level of Difficulty 3 Analysis) SWBAT: Determine the commmon factors and the greatest common factor of a set of numbers. Generate equivalent fractions by writing a fraction in simplest form. Divisibility rules can be taught to assist in simplifying fractions. Examples: 1. Find the greatest common factor for 10 and 24. 10:,1,2,5,10 24: 1,2,3,4,6,8,12,24 The GCF is 2 2. Find the simplest for 10/12 10/12 in simplest form is 5/6

MA.5.CCSS.Math.Content.5.NF MA.5.CCSS.Math.Content.5.NF.A.1 MA.K 12.CCSS.Math.Practice.MP2 MA.K 12.CCSS.Math.Practice.MP3 MA.K 12.CCSS.Math.Practice.MP4 Target 2 (Level of Difficulty 3 Analysis) SWBAT: Number and Operations Fractions 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. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Determine the commmon multiples and least common multiples of a set of numbers. MyMath Chapter 8 lesson 5 Compare fractions by using the least common denominator MyMath Chapter 8 lesson 6 Examples: 1. Find the least common multiples of 8 and 12 8: 8,16,24 12: 12, 24, 2. Compare 1/2 and 2/5 1/2 can be turned into 5/10 2/5 can be turned into 4/10 1/2 > 2/5 MA.5.CCSS.Math.Content.5.NF MA.5.CCSS.Math.Content.5.NF.A.1 MA.K 12.CCSS.Math.Practice.MP1 MA.K 12.CCSS.Math.Practice.MP7 MA.K 12.CCSS.Math.Practice.MP8 Target 3 (Level of Difficulty 3 Analysis) SWBAT: Number and Operations Fractions 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. Make sense of problems and persevere in solving them. Look for and make use of structure. Look for and express regularity in repeated reasoning. Interpret a fraction as a division of the numerator by the denominator Use fraction equivalence to write fractions as decimals Examples: 1. 1/2 means 1 divided by 2 2. 1/2 can be written as 50/100 meaning.50 MA.5.CCSS.Math.Content.5.NF.B.3 MA.5.CCSS.Math.Content.5.NF.B.4a 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. 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.

Learning Goal 2- Adding and Subtracting Fractions Students will be able to add and subtract fractions and mixed numbers. Target 1 (Level of Difficulty 3 Analysis) SWBAT: Add and subtract fractions and mixed numbers with unlike denominators by finding a common denominator, creating equivalent fractions using the common denominator, solving, and simplifying. In fifth grade, the example provided in the standard has students find a common denominator by finding the product of both denominators. For 1/3 + 1/6, a common denominator is 18, which is the product of 3 and 6. This process should be introduced using visual fraction models (area models, number lines, etc.) to build understanding before moving into the standard algorithm. Students should apply their understanding of equivalent fractions and their ability to rewrite fractions in an equivalent form to find common denominators. They should know that multiplying the denominators will always give a common denominator but may not result in the least common denominator. Examples: 1. 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.) 2. (See Below) MA.5.CCSS.Math.Content.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. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them;

and knowing and flexibly using different properties of operations and objects. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered 7 5 + 7 3, in preparation for learning about the distributive property. In the expression x + 9 x + 14, older students can see the 14 as 2 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3( x y ) as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Target 2 (Level of Difficulty 3 Analysis) SWBAT: Apply their understanding of adding and subtracting fractions to solve word problems. (My Math Chapter 9 throughout) Use benchmark fractions and number sense to assess the reasonableness of answers. (My Math Chapter 9 throughout) Examples: 1. Matt is hiking a trail that is 11/12 mile long. After hiking 1/4 mile, he stops for water. How much farther must he hike to finish the trail? 11/12 1/4 = 8/12 = 2/3 2. 2/5 + 1/2 = 3/7, The student should recognize that this answer is unreasonable because 3/7 < 1/2. MA.5.CCSS.Math.Content.5.NF.A.1 MA.5.CCSS.Math.Content.5.NF.A.2 MA.K 12.CCSS.Math.Practice.MP1 MA.K 12.CCSS.Math.Practice.MP2 MA.K 12.CCSS.Math.Practice.MP3 MA.K 12.CCSS.Math.Practice.MP4 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. 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. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Learning Goal 3- Multiplying and Dividing Fractions Students will be able to apply and extend previous understanding of multiplication and division to multiply and divide fractions.

Target 1 (Level of Difficulty 3 Analysis) SWBAT: Understand a fraction as a division of two whole numbers Students need ample experiences to explore the concept that a fraction is a way to represent the division of two quantities. Students are expected to demonstrate their understanding using concrete materials, drawing models, and explaining their thinking when working with fractions in multiple contexts. They read 3/5 as three fifths and after many experiences with sharing problems, learn that 3/5 can also be interpreted as 3 divided by 5. Examples: Ten team members are sharing 3 boxes of cookies. How much of a box will each student get? When working this problem a student should recognize that the 3 boxes are being divided into 10 groups, so s/he is seeing the solution to the following equation, 10 x n = 3 (10 groups of some amount is 3 boxes) which can also be written as n = 3 10. Using models or diagram, they divide each box into 10 groups, resulting in each team member getting 3/10 of a box. Two afterschool clubs are having pizza parties. For the Math Club, the teacher will order 3 pizzas for every 5 students. For the student council, the teacher will order 5 pizzas for every 8 students. Since you are in both groups, you need to decide which party to attend. How much pizza would you get at each party? If you want to have the most pizza, which party should you attend? MA.5.CCSS.Math.Content.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. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered 7 5 + 7 3, in preparation for learning about the distributive property. In the expression x + 9 x + 14, older students can see the 14 as 2 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3( x y ) as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Target 2 (Level of Difficulty 4 Knowledge Utilization) SWBAT: Multiply a fraction by another fraction or a whole number. Multiply two fractions. Multiply length x width of a rectangle with fractional sides to find the area. Find the area of a rectangle with fractional lengths by using tiles. This standard references both the multiplication of a fraction by a whole number and the multiplication of two fractions. Visual fraction models (area models, tape diagrams, number lines) should be used and created by students during their work with this standard. As they multiply fractions such as 3/5 x 6, they can think of the operation in more than one way. 3 x (6 5) or (3 x 6/5) (3 x 6) 5 or 18 5 (18/5)

Students create a story problem for 3/5 x 6 such as, Isabel had 6 feet of wrapping paper. She used 3/5 of the paper to wrap some presents. How much does she have left? Every day Tim ran 3/5 of mile. How far did he run after 6 days? (Interpreting this as 6 x 3/5) Example: Three fourths of the class is boys. Two thirds of the boys are wearing tennis shoes. What fraction of the class are boys with tennis shoes? This question is asking what 2/3 of ¾ is, or what is 2/3 x ¾? What is 2/3 x ¾ in this case you have 2/3 groups of size ¾. ( a way to think about it in terms of the language for whole numbers is 4 x 5 you have 4 groups of size 5. The array model is very transferable from whole number work and then to binomials. Area Model The area model and the line segments show that the area is the same quantity as the product of the side lengths.

MA.5.CCSS.Math.Content.5.NF.B.4a MA.5.CCSS.Math.Content.5.NF.B.4b 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. 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. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able

to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered 7 5 + 7 3, in preparation for learning about the distributive property. In the expression x + 9 x + 14, older students can see the 14 as 2 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3( x y ) as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation ( y 2)/( x 1) = 3. Noticing the regularity in the way terms cancel when expanding ( x 1)( x + 1), ( x 1)( x + x + 1), and ( x 1)( x + x + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Target 3 (Level of Difficulty 3 Analysis) SWBAT: Interpret multiplication as scaling (resizing) by: Understanding that when the dimensions are increased the product increases proportionally. When both factors are greater than 1, your product will increase. When one factor is less than one, your product will decrease. Relate the principle of fraction equivalence Example: a/b = (n a)/(n b) to the effect of multiplying a/b by 1. This standard asks students to examine how numbers change when we multiply by fractions. Students should have ample opportunities to examine both cases in the standard: a) when multiplying by a fraction greater than 1, the number increases and b) when multiplying by a fraction less the one, the number decreases. This standard should be

explored and discussed while students are working with 5.NF.4, and should not be taught in isolation. Example: Mrs. Bennett is planting two flower beds. The first flower bed is 5 meters long and 6/5 meters wide. The second flower bed is 5 meters long and 5/6 meters wide. How do the areas of these two flower beds compare? Is the value of the area larger or smaller than 5 square meters? Draw pictures to prove your answer. MA.5.CCSS.Math.Content.5.NF.B.5a MA.5.CCSS.Math.Content.5.NF.B.5b 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. 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 = (na)/(nb) to the effect of multiplying a/b by 1. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are

sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered 7 5 + 7 3, in preparation for learning about the distributive property. In the expression x + 9 x + 14, older students can see the 14 as 2 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3( x y ) as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation ( y 2)/( x 1) = 3. Noticing the regularity in the way terms cancel when expanding ( x 1)( x + 1), ( x 1)( x + x + 1), and ( x 1)( x + x + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Target 4 (Level of Difficulty 4 Knowledge Utilization) SWBAT... Solve real world problems involving multiplication of fractions Solve real world problems involving mixed number fractions This standard builds on all of the work done in this cluster. Students should be given ample opportunities to use various strategies to solve word problems involving the multiplication of a fraction by a mixed number. This standard could include fraction by a fraction, fraction by a mixed number or mixed number by a mixed number. Example: There are 2 ½ bus loads of students standing in the parking lot. The students are getting ready to go on a field trip. 2/5 of the students on each bus are girls. How many busses would it take to carry only the girls?

MA.5.CCSS.Math.Content.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. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. Target 5 (Level of Difficulty 4 Knowledge Utilization) SWBAT... Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions using visual aids. Interpret division of a unit fraction by a non zero whole number. Solve real world problems involving division of unit fractions by non zero whole numbers and division of whole numbers by unit fractions Be able to use multiplication to check your division of fractions.

Be able to use, create, and solve word problems by using visual models. (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) Know the number of groups/shares and finding how many/much in each group/share Examples: 1. Four students sitting at a table were given 1/3 of a pan of brownies to share. How much of a pan will each student get if they share the pan of brownies equally? The diagram shows the 1/3 pan divided into 4 equal shares with each share equaling 1/12 of the pan. 5.NF.7b This standard calls for students to create story contexts and visual fraction models for division situations where a whole number is being divided by a unit fraction. Example: Create a story context for 5 1/6. Find your answer and then draw a picture to prove your answer and use multiplication to reason about whether your answer makes sense. How many 1/6 are there in 5? 5.NF.7c extends students work from other standards in 5.NF.7. Student should continue to use visual fraction models and reasoning to solve these real world problems. Example: How many 1/3 cup servings are in 2 cups of raisins?

Examples: Knowing how many in each group/share and finding how many groups/shares Angelo has 4 lbs of peanuts. He wants to give each of his friends 1/5 lb. How many friends can receive 1/5 lb of peanuts? A diagram for 4 1/5 is shown below. Students explain that since there are five fifths in one whole, there must be 20 fifths in 4 lbs. MA.5.CCSS.Math.Content.5.NF.B.7a MA.5.CCSS.Math.Content.5.NF.B.7b MA.5.CCSS.Math.Content.5.NF.B.7c Interpret division of a unit fraction by a non zero whole number, and compute such quotients. Interpret division of a whole number by a unit fraction, and compute such quotients. 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. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a

Summative Assessment Teacher Observation school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered 7 5 + 7 3, in preparation for learning about the distributive property. In the expression x + 9 x + 14, older students can see the 14 as 2 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3( x y ) as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation ( y 2)/( x 1) = 3. Noticing the regularity in the way terms cancel when expanding ( x 1)( x + 1), ( x 1)( x + x + 1), and ( x 1)( x + x + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Quizzes Tests/Common Assessment Projects Rubrics Short & Extended Constructed Response 21st Century Life and Careers WORK.5 8.9.1.8.1 The ability to recognize a problem and apply critical thinking and problem solving skills to solve the problem is a lifelong skill that develops over time.

WORK.5 8.9.1.8.A.1 WORK.5 8.9.1.8.1 WORK.5 8.9.1.8.B.1 WORK.5 8.9.1.8.1 WORK.5 8.9.1.8.C.1 WORK.5 8.9.1.8.C.3 Develop strategies to reinforce positive attitudes and productive behaviors that impact critical thinking and problem solving skills. Gathering and evaluating knowledge and information from a variety of sources, including global perspectives, fosters creativity and innovative thinking. Use multiple points of view to create alternative solutions. Collaboration and teamwork enable individuals or groups to achieve common goals with greater efficiency. Determine an individual's responsibility for personal actions and contributions to group activities. Model leadership skills during classroom and extra curricular activities. Formative Assessment and Performance Opportunities Teacher observation Math journals Exit slips Quick checks during class Classwork activities and games Center work/ small group work Group work activities Homework MAP test Differentiation / Enrichment ELL Vocabulary 504 Accomodations IEP Modifications Leveled Centers Use of Manipulatives Simplifying, Equivalents, LCM, GCF Reteach/ Enrich MyMath Chapter 8 Add/Subtract Fractions Reteach/Enrich My Math Chapter 9 Multiply/Divide Fractions Reteach/Enrich My Math Chapter 10 Various Forms for Summative Assessment 3 Written Forms/ Oral Assessment per chapter Unit Resources My Math Grade 5, Vol. 2 Teacher Edition, Chapters 8 10 and Student Workbook: 2014 McGraw Hill Education

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Unit 4 : Geometry Content Area: Mathematics Course(s): Generic Course Time Period: 3rd Marking Period Length: 9 Weeks Status: Published Unit Overview In this unit, students will graph points on the coordinate plane to solve real world and mathematical problems. Classify two dimensional figures into categories based on their properties. Understand concepts of volume and relate volume to multiplication and to addition. Benchmarks: During this unit students will be completing one benchmark assessment. The end of Chapter 12 test will be the benchmark test. This test should be given mid April. Transfer Students will be able to independently use their learning to... Understand the attributes belonging to a category of two dimensional figures and the subcategories of the category. Recognize volume as an attribute of solid figures and understand concepts of volume measurement. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. 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. Find volumes of solid figures composed of two non overlapping right rectangular prisms by adding the volumes of the non overlapping parts together. Students will be able to apply finding composite volume to solve real world problems. For more information, read the following article by Grant Wiggins. http://www.authenticeducation.org/ae_bigideas/article.lasso?artid=60

Meaning Understandings Students will understand that... Geometric properties can be used to construct geometric figures. Geometric relationships provide a means to make sense of a variety of phenomena. Everyday objects have a variety of attributes, each of which can be measured in many ways. How to classify polygons How to classify triangles How to classify quadrilaterals How to use attributes to describe two dimensional figures Find volume of prisms Find composite volumes of non overlapping figures Essential Questions Students will keep considering... How does geometry help me solve problems in everyday life? Application of Knowledge and Skill Students will know... Students will know... and understand concepts of volume and composite volume and relate volume to multiplication and to addition. and understand the attributes of two dimensional figures and classify them into categories based on their properties

how to apply their understandings of volume to solve real world problems Students will be skilled at... Students will be skilled at... Recognizing volume as an attribute Measuring volume by cubic units Relating volume to multiplication and addition in real world situations Classifying two dimensional figures in a hierarchy based on their properties Academic Vocabulary three Dimensional figure rectangular prism triangular prism face edge vertex prism base cube volume unit cube cubic unit

composite figure hierarchy property two Dimensional attribute centimeter cube polygon figure formula measure part relationship right rectangular prism solid sum Learning Goal 1- Polygons Students will be able to classify two dimensional figures into categories based on their properties. Target 1 (Level of Difficulty 2 Comprehension)

SWBAT: Classify polygons Examples: Regular polygons have all of their sides and angles congruent. Name or draw some regular polygons. If the opposite sides on a parallelogram are parallel and congruent, then rectangles are parallelograms A sample of questions that might be posed to students include: A parallelogram has 4 sides with both sets of opposite sides parallel. What types of quadrilaterals are parallelograms? MA.5.CCSS.Math.Content.5.G.B.3 Understand that attributes belonging to a category of two dimensional figures also belong to all subcategories of that category. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation ( y 2)/( x 1) = 3. Noticing the regularity in the way terms cancel when expanding ( x 1)( x + 1), ( x 1)( x + x + 1), and ( x 1)( x + x + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Target 2 Classifying ( Level of Difficulty 2 Comprehension) SWBAT: Classify triangles by its sides as isosceles, equilateral, and scalene triangles

Identify triangles by angles as right, acute and obtuse triangles. Examples: 1. A triangle has measurments of 5 cm, 8 cm and 10 cm. Identify the type of triangle this is based on its sides. This triangle is classified as scalene. 2. A triangle's angles are 90 degrees, 40 degrees, and 50 degrees. Identify the type of triangle based on its angles. This triangle is a right triangle based on its 90 degree angle.. MA.5.CCSS.Math.Content.5.G.B.4 Target 3 (Level of Difficulty 2 Comprehension) SWBAT: Classify two dimensional figures in a hierarchy based on properties. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation ( y 2)/( x 1) = 3. Noticing the regularity in the way terms cancel when expanding ( x 1)( x + 1), ( x 1)( x + x + 1), and ( x 1)( x + x + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Classify quadrilaterals based on a hierarchy of classifications Quadrilateral a four sided polygon Rectangle a quadrilateral with two pairs of congruent parallel sides and four right angles Rhombus a parallelogram with all four sides equal in length Square a parallelogram with four congruent sides and four right angles. Example: Why is a square always a rectangle, but a rectangle is not always a square? A square must have 4 right angles and 4 equal sides. A rectangle must only have 4 right angles and does not have to have 4 equal sides. List all of the names for a square. Quadrialteral, parallelogram, rhombus, and rectangle

MA.5.CCSS.Math.Content.5.G MA.5.CCSS.Math.Content.5.G.B MA.5.CCSS.Math.Content.5.G.B.3 MA.5.CCSS.Math.Content.5.G.B.4 Learning Goal Geometry Classify two dimensional figures into categories based on their properties. Understand that attributes belonging to a category of two dimensional figures also belong to all subcategories of that category. Classify two dimensional figures in a hierarchy based on properties. Level 4 ( Level of Difficulty 2 Comprehension for Volume and 3 Analysis for Composite Volume) SWBAT: Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. Recognize volume as an attribute of solid figures and understand concepts of volume measurement. 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. Find volumes of solid figures composed of two non overlapping right rectangular prisms by adding the volumes of the non overlapping parts together. Students will be able to apply finding composite volume to solve real world problems. Extend their knowledge of Area to understanding the concept of volume as an attribute of solid figures and relate the measure to the cubic unit. The following standards represent the first time that students begin exploring the concept of volume. In third grade, students begin working with area and covering spaces. The concept of volume should be extended from area with the idea that students are covering an area (the bottom of cube) with a layer of unit cubes and then adding layers of unit cubes on top of bottom layer (see picture below). Students should have ample experiences with concrete manipulatives before moving to pictorial representations. Students prior experiences with volume were restricted to liquid volume. As students develop their understanding volume they understand that a 1 unit by 1 unit by 1 unit cube is the standard unit for measuring volume. This cube has a length of 1 unit, a width of 1 unit and a height of 1 unit and is called a cubic unit. This cubic unit is written with an exponent of 3. Students connect this notation to their understanding of powers of 10 in our place value system. Models of cubic inches, centimeters, cubic feet, etc are helpful in developing an image of a cubic unit. Students estimate how many cubic yards would be needed to fill the classroom or how many cubic centimeters would be needed to fill a pencil box. Know that solid figures have volume Measure volume

Understand a cubic unit Measure volume with a cubic unit Examples: base 10 blocks How many units in a 1000 cube? (3 x 2) represented by first layer (3 x 2) x 5 represented by number of 3 x 2 layers (3 x 2) + (3 x 2) + (3 x 2) + (3 x 2)+ (3 x 2) = 6 + 6 + 6 + 6 + 6 + 6 = 30 6 representing the size/area of one layer Multiply volume Adding volume Problem solving with volume Find volume using the equation V = l x w x h and whole numbers Find volume using cubic units Compare the volume found with the equation and the volume found with maniplulatives Use the formulas for rectangular prisms to problem solve V = l x w x h and V = b x h Find the volume of an irregular shape by breaking the shape into smaller rectangular shapes and adding your answers. Problem solve

MA.5.CCSS.Math.Content.5.MD.C MA.5.CCSS.Math.Content.5.MD.C.3 MA.5.CCSS.Math.Content.5.MD.C.3a MA.5.CCSS.Math.Content.5.MD.C.3b MA.5.CCSS.Math.Content.5.MD.C.4 MA.5.CCSS.Math.Content.5.MD.C.5a MA.5.CCSS.Math.Content.5.MD.C.5b MA.5.CCSS.Math.Content.5.MD.C.5c Summative Assessment Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition. Recognize volume as an attribute of solid figures and understand concepts of volume measurement. 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. 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. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. Find the volume of a right rectangular prism with whole number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently 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. 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. Recognize volume as additive. Find volumes of solid figures composed of two non overlapping right rectangular prisms by adding the volumes of the non overlapping parts, applying this technique to solve real world problems. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. Common Assessments homework performance based tasks problem based learning tasks Projects Quizzes Short and extended constructed responses Tests 21st Century Life and Careers........

WORK.5 8.9.1.8.1 WORK.5 8.9.1.8.A.1 WORK.5 8.9.1.8.A.2 WORK.5 8.9.1.8.1 WORK.5 8.9.1.8.B.1 WORK.5 8.9.1.8.1 WORK.5 8.9.1.8.C.1 WORK.5 8.9.1.8.C.2 WORK.5 8.9.1.8.2 WORK.5 8.9.1.8.C.3 The ability to recognize a problem and apply critical thinking and problem solving skills to solve the problem is a lifelong skill that develops over time. Develop strategies to reinforce positive attitudes and productive behaviors that impact critical thinking and problem solving skills. Implement problem solving strategies to solve a problem in school or the community. Gathering and evaluating knowledge and information from a variety of sources, including global perspectives, fosters creativity and innovative thinking. Use multiple points of view to create alternative solutions. Collaboration and teamwork enable individuals or groups to achieve common goals with greater efficiency. Determine an individual's responsibility for personal actions and contributions to group activities. Demonstrate the use of compromise, consensus, and community building strategies for carrying out different tasks, assignments, and projects. Leadership abilities develop over time through participation in groups and/or teams that are engaged in challenging or competitive activities. Model leadership skills during classroom and extra curricular activities. Formative Assessment and Performance Opportunities Homework Classwork Map Test Performance Based Test Projects Teacher Observations See Chapter 12 for Formative Assessment Differentiation / Enrichment Accomodations as per 504 clickers computers manipulatives modifications as per IEP small group instruction Smarboard/Interactive White board Unit Resources McGraw Hill My Math Volume 2 Chapter 12....... Connected.ed.mcgraw hill.com

Unit 5 : Measurement and Data Content Area: Mathematics Course(s): Generic Course Time Period: 4th Marking Period Length: 2 Weeks Status: Published Unit Overview In this unit, students will learn to convert like measurement units within the customary and metric systems and how to represent and interpret data using a line plot. Benchmarks: During this unit students will be completing one benchmark assessment. The Benchmark 4 test is a cumulative end of the year test, with a heavy emphasis on last quarter of the year. This test should be given in May. Transfer Students will be able to independently use their learning to solve real world multi step problems involving unit conversions. Students will also be able to apply their ability to interpret and create line plots when solving real world word problems. For more information, read the following article by Grant Wiggins. http://www.authenticeducation.org/ae_bigideas/article.lasso?artid=60 Meaning Understandings Students will understand... How to convert among different sized standard measurement units.

Represent and interpret data. Essential Questions Students will keep considering... How can I use measurement conversions to solve real world problems? Application of Knowledge and Skill Students will know... Students will know... How to convert between different units of measurements. How to create and interpret a line plot Students will be skilled at... Students will be skilled at... Measuring objects length, height, and capacity in the customary unit of measurement. Converting among different sized standard measurement units within the customary and metric systems. Creating and interpreting line plots to display a data set of measuremts in fractions of a unit. Redistributing data from a line plot into equal groups. Academic Vocabulary capacity

centimeter (cm) convert cup (c) customary system fair share fluid ounce (fl oz) foot (ft) gallon (gal) gram (g) inch (in.) kilogram (kg) kilometer (km) length liter (L) mass meter (m) metric system mile (mi) milligram (mg) milliliter (ml) ounce (oz) pint (pt) pound (lb) quart (qt)

ton (T) weight yard (yd) Learning Goal 1 - Converting Measurments The students will be able to convert like measurement units within a given measurement system. Target 1 (Level of Difficulty 2 Comprehension) SWBAT: Convert customary units of length. Convert customary units of weight. Convert customary units of length. Example: 1. 45 ft = yds; Since 3 feet = 1 yard, and feet are smaller than yards, divide by 3. 45/3=15; Answer = 15 yds MA.5.CCSS.Math.Content.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. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use

technological tools to explore and deepen their understanding of concepts. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered 7 5 + 7 3, in preparation for learning about the distributive property. In the expression x + 9 x + 14, older students can see the 14 as 2 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3( x y ) as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Target 2 (Level of Difficulty 2 Comprehension) SWBAT: Convert metric units of measurement using muliplication or division. Convert metric units of measurement by moving the decimal point right or left the correct amount of places. Example: 1. 12kg = g ; Since there are 1,000 grams in 1 kilogram, and kilograms are larger than grams, multiply by 1,000. 12 x 1000 = 12,000; Answer = 12,000 g 2. K H D B d c m ; Since kg to g is 3 places to the right, move the decimal point at the end of the 12, 3 places to the right. This leaves 3 empty spaces after the 12. Add zeros in these spaces and you get 12,000. 12 kg = 12000 g MA.5.CCSS.Math.Content.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. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to

decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered 7 5 + 7 3, in preparation for learning about the distributive property. In the expression x + 9 x + 14, older students can see the 14 as 2 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3( x y ) as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Target 3 (Level of Difficulty 3 Analysis) SWBAT: Use conversions in solving multi step, real world problems. Example: 1. A slice of apple pie weighs 4 ounces. Do 5 slices weigh more than 1 pound? 2. Three friends each measured their height. Their heights are 4 feet 10 inches, 4 feet 9 inches, and 4 feet 7 inches. Use the clues to determine the height, in inches, of each person. a) Elliot is taller than Jorge. b) Nicole is 3 inches taller than the shortest person. c) Elliot is 57 inches tall. MA.5.CCSS.Math.Content.5.MD.A MA.5.CCSS.Math.Content.5.MD.A.1 Convert like measurement units within a given measurement system. 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. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition

equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. Learning Goal 2 - Line Plots Students will be able to represent and interpret data. Target 1 (Level of Difficulty 3 Analysis) SWBAT... Make a line plot Display fractional measurements on a line plot Problem solve using a line plot This standard provides a context for students to work with fractions by measuring objects to one eighth of a unit. This includes length, mass, and liquid volume. Students are making a line plot of this data and then adding and subtracting fractions based on data in the line plot. Example: Students measured objects in their desk to the nearest ½, ¼, or 1/8 of an inch then displayed data collected on a line plot. How many object measured ½? ¼? If you put all the objects together end to end what would be the total length of all the objects?

Example: Ten beakers, measured in liters, are filled with a liquid. The line plot above shows the amount of liquid in liters in 10 beakers. If the liquid is redistributed equally, how much liquid would each beaker have? (This amount is the mean.) Students apply their understanding of operations with fractions. They use either addition and/or multiplication to determine the total number of liters in the beakers. Then the sum of the liters is shared evenly among the ten beakers. MA.5.CCSS.Math.Content.5.MD.B.2 MA.K 12.CCSS.Math.Practice.MP1 MA.K 12.CCSS.Math.Practice.MP4 MA.K 12.CCSS.Math.Practice.MP5 Summative Assessment Vocabulary Quiz 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. Make sense of problems and persevere in solving them. Model with mathematics. Use appropriate tools strategically. Skill Specific Quiz (Conversion or Line Plots) My Math Chapter 11 Assessment Teacher Created Project(s) Rubrics Short & Extended Constructed Response 21st Century Life and Careers

WORK.5 8.9.1.8.1 WORK.5 8.9.1.8.A.1 WORK.5 8.9.1.8.A.2 WORK.5 8.9.1.8.B.1 WORK.5 8.9.1.8.B.2 WORK.5 8.9.1.8.1 WORK.5 8.9.1.8.C.1 WORK.5 8.9.1.8.C.2 WORK.5 8.9.1.8.2 WORK.5 8.9.1.8.C.3 The ability to recognize a problem and apply critical thinking and problem solving skills to solve the problem is a lifelong skill that develops over time. Develop strategies to reinforce positive attitudes and productive behaviors that impact critical thinking and problem solving skills. Implement problem solving strategies to solve a problem in school or the community. Use multiple points of view to create alternative solutions. Assess data gathered to solve a problem for which there are varying perspectives (e.g., cross cultural, genderspecific, generational), and determine how the data can best be used to design multiple solutions. Collaboration and teamwork enable individuals or groups to achieve common goals with greater efficiency. Determine an individual's responsibility for personal actions and contributions to group activities. Demonstrate the use of compromise, consensus, and community building strategies for carrying out different tasks, assignments, and projects. Leadership abilities develop over time through participation in groups and/or teams that are engaged in challenging or competitive activities. Model leadership skills during classroom and extra curricular activities. Formative Assessment and Performance Opportunities Teacher observation Math journals Exit slips Quick checks during class Classwork activities and games Center work/ small group work Group work activities Homework MAP test Differentiation / Enrichment ELL Vocabulary 504 Accomodations IEP Modifications Leveled Centers Use of Manipulatives Converting Customary Units of Length Reteach/ Enrich MyMath Chapter 11, Lesson 2 Problem Solving Investigation My Math Chapter 11, Lesson 3 Display Measurement Data on a Line Plot Reteach/Enrich MyMath Chapter 11, Lesson 8 Various Forms for Summative Assessment 3 Written Forms/ Oral Assessment

Unit Resources MyMath Grade 5, Vol. 2 Teacher Edition Chapter 11 and Student Workbook: 2014 McGraw Hill Education My Math Online Portal