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


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1 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. 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.
2 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...
3 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
4 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
5 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: 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, : The underlined digit is in the hundreds place and the value of the 9 is 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., = (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
6 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 Order the following numbers from least to greatest: 9.275;8.950; Round to the nearest hundredths place. 9.85
7 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., = (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 , 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 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:
8 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 wholenumber exponents to denote powers of 10. Examples: 1.Write 3x3x3x3 using an exponent: Write 3 4 as a product. 3x3x3x3= 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.
9 3 24 = (3 20) + (3 4) Find partial products. = Multiply. = 72 Add. 2. Find products like 4 65 mentally by using the Distributive Property = 4 (60 + 5) Write 65 as = (4 60) + (4 5) Use the Distributive Property. = 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 ,340 2,496 Multiply the ones: Multiply the tens: Add.
10 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 , 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 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 )
11 Using a Model Solve a division problem like using a bar diagram Divide each section by = = = 2 Add the quotients = 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
12 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 , 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 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 Example 2: Solve
13 So, = 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
14 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 , 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 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
15 Short & Extended Constructed Response Homework 21st Century Life and Careers WORK WORK A.1 WORK B.1 WORK WORK C.1 WORK C.3 WORK 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
16 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
17 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. Meaning
18 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
19 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: (8+2) / 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.
20 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
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