SMUSD Common Core Math Scope and Sequence Fifth Grade

Transcription

1 5 th Grade Revised April 2014 (pg. 1) SMUSD Common Core Math Scope and Sequence Fifth Grade Trimester One Trimester Two Trimester Three Unit 1 Unit 2 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 continued 7 weeks 4 weeks 4 weeks 7 weeks 2 weeks 6 weeks 4 weeks Place Value Decimals Converting and Whole The Four Operations with and Measurements Geometry Number Fractions Decimal and Finding Graphing Operations Operations Volumes OA.1 OA.2 NBT.1 NBT.2 NBT.5 NBT.6 NF.1 NF.2 NF.3 NF.4 NF.7 Mid-Unit fraction assessment for trimester one report card NF.1 NF.2 NF.3 NF.4 NF.5 NF.6 NF.7 End of unit posttest and performance task NBT.1 NBT.2 NBT.3 NBT.4 NBT.7 G.3 G.4 All units to contain the Standards for Mathematical Practice. MD.1 MD.3 MD.4 MD.5 NBT.5 G.1 G.2 MD.2 OA.3

2 5 th Grade Revised April 2014 (pg. 2) Standards for Mathematical Practice Explanations and Examples for Grade Five MP.1 Make sense of problems and persevere in solving them. In grade five, students solve problems by applying their understanding of operations with whole numbers, decimals, and fractions including mixed numbers. They solve problems related to volume and measurement conversions. Students seek the meaning of a problem and look for efficient ways to represent and solve it. For example, Sonia had 123 sticks of gum. She promised her brother that she would give him ½ of a stick of gum. How much will she have left after she gives her brother the amount she promised? Teachers can encourage students to check their thinking by asking themselves, What is the most efficient way to solve the problem?, Does this make sense?, and Can I solve the problem in a different way? MP.2 Reason abstractly and quantitatively. Students recognize that a number represents a specific quantity. They connect quantities to written symbols and create logical representations of problems, consider appropriate units and the meaning of quantities. They extend this understanding from whole numbers to their work with fractions and decimals. Teachers can support student reasoning by asking questions such as, What do the numbers in the problem represent? or What is the relationship of the quantities? Students write simple expressions that record calculations with numbers and represent or round numbers using place value concepts. For example, students use abstract and quantitative thinking to recognize that 0.5 (300 15) is 12 of (300 15) without calculating the quotient. MP.3 Construct viable arguments and critique the reasoning of others. In fifth grade students may construct arguments using visual models, such as objects and drawings. They explain calculations based upon models, properties of operations and rules that generate patterns. They demonstrate and explain the relationship between volume and multiplication. They refine their mathematical communication skills as they participate in mathematical discussions involving questions like How did you get that? and Why is that true? They explain their thinking to others and respond to others thinking. Students use various strategies to solve problems and they defend and justify their work with others. For example, two afterschool clubs are having pizza parties. The teacher will order 3 pizzas for every 5 students in the math club; and 5 pizzas for every 8 students in the student council. If a student is in both groups, decide which party he/she should to attend. How much pizza will each student get at each party? If a student wants attend the party with the most pizza (if divided equally between the students at the party), which party should he/she attend?

3 5 th Grade Revised April 2014 (pg. 3) MP.4 Model with mathematics In grade five, students experiment with representing problem situations in multiple ways such as using numbers, mathematical language, drawings, pictures, objects, charts, lists, graphs and equations. Teachers might ask, How would it help to create a diagram, chart or table? or What are some ways to represent the quantities? Students need opportunities to represent problems in various ways and explain the connections. Fifth graders evaluate their results in the context of the situation and they explain whether results to problems make sense. They evaluate the utility of models they see and draw and can determine which models can be the most useful and efficient to solve problems. MP.5 Use appropriate tools strategically. Students consider available tools, including estimation, and decide which tools might help them solve mathematical problems. For instance, students may use unit cubes to fill a rectangular prism and then use a ruler to measure the dimensions to find a pattern for volume using length of the sides. They use graph paper to accurately create graphs and solve problems or make predictions from real-world data. MP.6 Attend to precision. Students continue to refine their mathematical communication skills by using clear and precise language in their discussions with others and in their own reasoning. Teachers might ask, How do you know your solution is reasonable? Students use appropriate terminology when they refer to expressions, fractions, geometric figures, and coordinate grids. Teachers might ask, What symbols or mathematical notations are important in this problem? Students are careful to specify units of measure and state the meaning of the symbols they choose. For instance, to determine the volume of a rectangular prism, students record their answers in cubic units. MP.7 Look for and make use of structure. Students look closely to discover a pattern or structure. For instance, students use properties of operations as strategies to add, subtract, multiply and divide with whole numbers, fractions, and decimals. They examine numerical patterns and relate them to a rule or a graphical representation. Teachers might ask, How do you know if something is a pattern? or What do you notice when? MP.8 Look for and express regularity in repeated reasoning. Fifth graders use repeated reasoning to understand algorithms and make generalizations about patterns. Students connect place value and their prior work with operations to understand and use algorithms to extend multi-digit division from one-digit to two-digit divisors and to fluently multiply multidigit whole numbers. They use various strategies to perform all operations with decimals to hundredths and they explore operations with fractions with visual models and begin to formulate generalizations. Teachers might ask, Can you explain how this strategy works in other situations? or Is this always true, sometimes true or never true? (Adapted from Arizona Department of Education [Arizona] 2012 and North Carolina 97 Department of Public Instruction [N. Carolina] 2013)

4 5 th Grade Revised April 2014 (pg. 4) Number Talks to support mental computation and flexibility with numbers *All of the following information and problems have been adapted from Number Talks: Helping Children Build Mental Math and Computation Strategies by Sherry Parrish. For more detailed information refer to your personal or grade level copy of the book. During number talks, students should be accurate, efficient and flexible in their thinking. Many other areas can and should be addressed: number sense, place value, fluency properties, and connecting mathematical ideas. Unpacking each of these components and keeping them in the forefront during classroom strategy discussions will better prepare your students to be mathematically powerful and proficient. Number Sense: Every time an answer to problem in a number talk is elicited from a student, and you ask students to share whether the proposed solutions are reasonable, you are helping students build number sense. When students are asked to give an estimate before they begin thinking about a specific strategy, number sense is being fostered. Place Value: Students understand place value if they can apply their understanding in computation. Number talks provide the opportunity to confront and use place-value understanding on a continual basis. Fluency: Fluency is much more than fact recall. Fluency is knowing how a number can be composed and decomposed and using that information to be flexible and efficient with problem solving. Properties: As students invent their own strategies, they create opportunities to link their thinking to mathematical properties and understand how they work. Connecting Mathematical Ideas: Each time you compare student strategies, discuss how addition can be used to solve subtraction problems, and make links between arrays with multiplication and division you are grounding your students in the idea that mathematical concepts are related and make sense. Helping students connect mathematical ideas is a critical component of number talks. The following number talks are crafted to elicit specific strategies; however, you may find that students also share other efficient methods. Each strategy is labeled to help you better understand its foundation, however strategies are often named for the students who invented them. Keep in mind the overall purpose is to help students build a toolbox of efficient strategies based on numerical reasoning. Remember, the ultimate goal of number talks is for students to compute accurately, efficiently, and flexibly. Once number talks become a regular component of your math instruction, you may consider stopping showing how to solve computation problems. Instead, provide story problems that incorporate new operations and encourage students to make sense of the problems and numbers. Using real life context as a spring board for mathematical thinking engages students in mathematics that is relevant, attaches meaning to the numbers, and helps students access the mathematics.(see pp of Number Talks for further support.) As you begin to implement number talks in your classroom, start with small numbers. Using small numbers serves two purposes:

5 1) Students can focus on the nuances of the strategy instead of the magnitude of the numbers 2) Students are able to build confidence in their mathematical abilities. 5 th Grade Revised April 2014 (pg. 5) As students understanding of different strategies develops, you can gradually increase the size of the numbers. When the numbers become too large, students will rely on less efficient strategies, such as counting all, or resort to paper and pencil, thereby losing the focus on developing mental strategies. Trimester One Multiplication Number Talks: Instructions: The following number talks consist of three or more sequential problems. The sequence within a given number talk allows students to apply strategies from previous problems to subsequent problems. Strategy One: Making Landmark or Friendly Numbers A common error students make when changing one of the factors to a landmark number is to forget to adjust the number of groups. The problem 9 x 25 can help us consider the common errors children make when making this adjustment. If 9 had been changed to 10, then the product of 250 would need to be adjusted not just by 1 but by one group of 25. Number Talks pg. 267 Non- Example: X x = 249 Example: X x = 225

6 Category 1 Category 2 Category 3 5 x 5 5 x 10 5 x 30 5 x 29 4 x 5 4 x 10 4 x 50 4 x 49 3 x 10 3 x 50 3 x x x 50 6 x x x 60 5 x x x x x 60 5 x th Grade Revised April 2014 (pg. 6) 6 x x x x 19 3 x x x x x x 30 2 x x 29 Strategy Two: Doubling and Halving Halving and doubling is an excellent strategy to restructure a problem with multiple digits and make it easier to solve. Helping students to notice the relationship between the two factors and the dimensions of the accompanying array is important to understanding the strategy. An equally important idea in this strategy is that the factors can adjust while the area of the array stays the same. Number Talks pg x 16 2 x 8 4 x 4

7 Category 1 Category 2 Category 3 1 x 16 2 x 8 4 x 4 8 x 2 1 x 16 1 x 24 2 x 12 4 x 6 8 x 3 1 x 48 2 x 24 4 x 12 8 x 6 16 x 3 1 x 56 2 x 28 4 x 14 8 x 7 8 x x x x 5 42 x x x 8 70 x x x x x 2 Strategy Three: Breaking Apart Factors Breaking apart factors gives students the opportunity to apply the associative property of multiplication. Example: 8 x 25 = 2 x 4 x 25 or 8 x 25 = 8 x 5 x 5 or 8 x 25 = 2 x 4 x 5 x 5 5 th Grade Revised April 2014 (pg. 7) 3 x 60 6 x x 15 9 x x x 14 2 x x x x x 3 52 x 6 26 x 12 Category 1 Category 2 Category 3 4 x 3 x 4 3 x 4 x 25 2 x 2 x 12 3 x 5 x 4 5 x 12 x 5 8 x 3 x 2 2 x 15 x 2 5 x 2 x 25 2 x 2 x 3 x 4 15 x 4 12 x x 4

8 Breaking Factors Apart (cont.) 5 x 2 x 6 5 x 4 x 3 2 x 2 x 3 x 5 5 x 12 5 x 2 x 4 4 x 5 x 2 2 x 2 x 5 x 2 8 x 5 5 x 5 x 8 2 x 4 x 25 2 x 25 x 4 25 x 8 2 x 4 x 35 8 x 5 x 7 8 x 35 5 th Grade Revised April 2014 (pg. 8) 2 x 15 x 6 5 x 12 x 3 4 x 5 x 3 x 3 4 x 5 x 9 12 x 15 4 x 4 x 25 8 x 2 x 25 1 x 5 x 5 16 x 25 Strategy Four: Partial Products This strategy is based on breaking one or both factors into addends through using expanded notation and the distributive property. While both factors can be represented with expanded notation, keeping one number whole is often more efficient. Number Talks pg. 272 Example: 8 x 25 (4 + 4) x 25 = ( 4 x 25 ) + ( 4 x 25 ) ( ) x 25 = (2 x 25) + (2x25) + (4 x 25) 8 x (20 + 5) = (8 x 20) + (8 x 5) 8 x ( ) = (8 x 10) + (8 x 10) + (8 x 5) Category 1 Category 2 Category 3 2 x 7 4 x 7 4 x 8 3 x 8 8 x 7 2 x x 25 6 x x 4 6 x x x x x 5 13 x 15 3 x 8 2 x 6 6 x 8 2 x x x 10 5 x 50 5 x x x 1 10 x 11 5 x x 11

9 Partial Products (cont.) 2 x 7 4 x 7 3 x 7 7 x 7 4 x x 6 8 x x 50 8 x 6 8 x th Grade Revised April 2014 (pg. 9) 4 x 22 6 x 11 3 x 22 6 x x x 22 Trimester Two Division Number Talks: Instructions: The following number talks consist of three or more sequential problems. The sequence within a given number talk allows students to apply strategies from previous problems to subsequent problems. Strategy One: Partial Quotients The Partial Quotients strategy maintains the integrity of place value and allows the students to approach the problem by building on multiplication problems with friendly multipliers such as 2, 5, 10, powers of 10, and so on. This strategy allows the student to navigate through the problem by building on what they know, understand, and can implement with ease. Number Talks pg. 288 Category 1 Category 2 Category

10 5 th Grade Revised April 2014 (pg. 10) Strategy Two: Multiplying Up Following the same principle as Adding Up to Subtract, Multiplying Up is an accessible division strategy that capitalizes on the relationship between multiplication and division. Similar to Partial Quotients, this strategy provides an opportunity for students to gradually build on multiplication problems they know until they reach the dividend. Number Talks pg. 292 Category 1 Category 2 Category 2 3x 10 3 x 20 3 x 3 3 x x 5 5 x 10 5 x x 10 6 x 5 6 x 6 6 x x 10 4 x 5 4 x 8 4 x x 25 4 x 50 4 x x x 50 8 x x 25 4 x x 10 4 x x x 50 8 x x x 50 3 x x 25 4 x 50 4 x x x 50 6 x 60 6 x x x 10 7 x 5 7 x Strategy Three: Proportional Reasoning When division is considered from a fractional perspective, this provides opportunities for students to explore the relationship of the part to the whole through proportional reasoning using equivalent fractions. Knowing that the divisor and dividend in share common factors, students can simplify the quantity to any of the following equivalent fractions:,, or. Number Talks pg. 299

11 th Grade Revised April 2014 (pg. 11) Trimester 3 Addition, Subtraction, Multiplication and Division Number Talks: Throughout trimester 3 students are encouraged to solve problems using a variety of operations with and without context. Throughout the trimester students select strategies to solve each problem and then discuss the efficiency of different strategies. The following sample problems are taken from different sections throughout the book Number Talks. For more samples, or if you would like to continue practice with only multiplication and division strategies see your grade level copy of the book Number Talks by Sherry Parrish x 5 35 x x 2 35 x x x x x x x x 8 18 x x x x x x 20 8 x 32 2 x x 15 x 6 5 x 12 x x 35

12 5 th Grade Revised April 2014 (pg. 12) Standards of Focus to be taught and practiced through problem solving: NF.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. NF.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual models or equations to represent the problem. Teaching Mathematics through Problem Solving (Adapted from Teaching Student-Centered Mathematics) Teaching mathematics through problem solving is a method of teaching mathematics that helps children develop relational understanding. With this approach, problem solving is completely interwoven with learning. As children do mathematics-make sense of cognitively demanding tasks, provide evidence or justification for strategies and solutions, find examples and connections, and receive and provide feedback about ideas-they are simultaneously engaged in the activities of problem solving and learning. Teaching mathematics through problem solving requires you to think about the types of tasks you pose to children, how you facilitate discourse in your classroom, and how you support children s variety of representations as tools for problem solving, reasoning, and communication. Teaching mathematics through problem solving promises to be an effective approach if our ultimate goal is deep (relational) understanding because it accomplishes these goals: 1) Focuses children s attention on ideas and sense making. When solving problems, children are reflecting on the concepts inherent in the problems. Emerging concepts are more likely to be integrated with existing ones, thereby improving understanding. 2) Emphasizes mathematical processes and practices. Children who are solving problems will engage in all five of the processes of doing mathematics-problem solving, reasoning, communication, connections, and representation, as well as the eight-mathematical practices outlined in the Common Core State Standards, resulting in mathematics that is more accessible, more interesting, and more meaningful.

13 5 th Grade Revised April 2014 (pg. 13) 3) Develops children s confidence and identities. Every time teachers pose a problem-based task and expect a solution, they implicitly say to children, I believe you can do this. When children are engaged in problem solving and discourse in which the correctness of the solution lies in justification of the processes, they begin to see themselves as capable of doing mathematics and that mathematics make sense. 4) Provides a context to help children build meaning for the concept. Using a context facilitates mathematical understanding, especially when the context is grounded in an experience familiar to children and when the context uses purposeful constraints that potentially highlight the significant mathematical ideas 5) Allows entry and exit points for a wide range of children. Good problem-solving based tasks have multiple paths to the solution, so each child can make sense of and solve the task by using his or her own ideas. Furthermore, children expand their ideas and grow in their understanding as they hear, critique, and reflect on the solution strategies of others. 6) Allows for extensions and elaborations. Extensions and what if questions can motivate advanced learners or quick finishers, resulting in increased learning and enthusiasm for doing mathematics. 7) Engages students so that there are fewer discipline problems. Many discipline issues in a classroom are the result of children becoming bored, not understanding the teacher directions, or simply finding little relevance in the task. Most children like to be challenged and enjoy being permitted to solve problems in ways that make sense to them, giving them less reason to act out or cause trouble. 8) Provides formative assessment data. As children discuss ideas, draw diagrams, or use manipulatives, defend their solutions and evaluate those of others, and write reports or explanations, they provide the teacher with a steady stream of valuable information that can be used to inform subsequent instruction. 9) It s a lot of fun! Children enjoy the creative process of problem solving and sharing how they figured something out. After seeing the surprising and inventive ways that children think and how engaged children become in mathematics, very few teachers stop using a teaching-through-problem solving approaching.

14 5 th Grade Revised April 2014 (pg. 14) A Three-Phase Lesson Format Before: In the before phase of the lesson you are preparing children to work on the problem. As you plan for the before part of the lesson, analyze the problem you will give to children in order to anticipate children s approaches and possible misinterpretations or misconceptions. This can inform questions you ask in the before phase of the lesson to clarify children s understanding of the problem (i.e., knowing what it means rather than how they will solve it.) During: In the during phase of the lesson children explore the problem (alone, with partners, or in small groups). This is one of two opportunities you will get in the lesson to find out what the children know, how they think, and how they are approaching the task you have given them (the other is in the discussion period of the after phase). You want to convey a genuine interest in what the children are thinking and doing. This is not the time to evaluate or to tell children how to solve the problem. When asking whether a result or method is correct, ask children, How can you decide? or Why do you think that might be right? or How can we tell if that makes sense? Use this time in the during phase to identify different representations and strategies children used, interesting solutions, and any misconceptions that arise that you will highlight and address during the after phase of the lesson. After: In the after phase of the lesson your children will work as a community of learners, discussing, justifying, and challenging various solutions to the problem that they have just worked on. It is critical to plan for and save ample time for this part of the lesson. Twenty minutes is not at all unreasonable for a good class discussion and sharing of ideas. It is not necessary to wait for every child to finish. Here is where much of the learning will occur as children reflect individually and collectively on ideas they have explored. This is the time to reinforce precise terminology, definitions, or symbols. After children have shared their ideas, formalize the main ideas of the lesson, highlighting connections between strategies or different mathematical ideas.

15 5 th Grade Revised April 2014 (pg. 15) Sample Problem: (NF.2) Jenny was making two different types of cookies. One recipe needed 3/4 cup of sugar and the other needed 2/3 cup of sugar. How much sugar did he need to make both recipes? Solutions: Mental Estimation (MP.2): A student may say that Jenny needs more than 1 cup of sugar but less than 2 cups, since both fractions are larger than ½, while at the same time both fractions are less than 1. Area Model to Show equivalence (MP.5): A student may choose to represent each partial cup of sugar using an area model, find equivalent fractions, and then add: I see that 3/4 of a cup of sugar is equivalent to 9/12 of a cup, while 2/3 of a cup is equivalent to 8/12 of a cup. Altogether, I have 17/12 of a cup. This is more than one cup since = + = 1 Sample Problem: (5.NF.6)

16 5 th Grade Revised April 2014 (pg. 16) Essential Learning for the Next Grade Adapted from: The Mathematics Framework was adopted by the California State Board of Education on November 6, 2013 Fifth Grade To be prepared for grade six mathematics students should be able to demonstrate they have acquired certain mathematical concepts and procedural skills by the end of grade four and have met the fluency expectations for the grade. For fifth graders, the expected fluency is to multiply multi-digit whole numbers with up to four-digits using the standard algorithm (5.NBT.5). These fluencies and the conceptual understandings that support them are foundational for work in later grades. Of particular importance at grade five are concepts, skills, and understandings needed to understand the place value system (5.NBT.1-4 ); perform operations with multi-digit whole numbers and with decimals to hundredths (5.NBT.5-7 ); use equivalent fractions as a strategy to add and subtract fractions (5.NF.1-2 ); apply and extend previous understandings of multiplication and division to multiply and divide fractions. (5.NF.3-7 ); geometric measurement: understand concepts of volume and relate volume to multiplication and to addition (5.MD.3-5 ). In addition graphing points on the coordinate plane to solve real-world and mathematical problems (5.G.1-2) is an important part of a students progress to algebra. Fractions Student proficiency with fractions is essential to success in algebra at later grades. By the end of grade five students should be able to add, subtract and multiply any two fractions and understand how to divide fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions). Students should understand fraction equivalence and use their skills to generate equivalent fractions as a strategy to add and subtract fractions, with unlike denominators including mixed fractions. Students should use these skills to solve related word problems. This understanding brings together the threads of fraction equivalence (grades three through five) and addition and subtraction (kindergarten through grade four) to fully extend addition and subtraction to fractions. By the end of grade five students know how to multiply a fraction or whole number by a fraction. Based on their understanding of the relationship between fractions and division, students divide any whole number by any nonzero whole number and express the answer in the form of a fraction or mixed number. Work with multiplying fractions extends from students understanding of the operation of multiplication. For example, to multiply a/b x q (where q is a whole number or a fraction), students can interpret x q as meaning a parts of a partition of q into b equal parts. This interpretation of the product leads to a product that is less than, equal to or greater than q, depending on whether a/b<1, = 1 or >1, respectively. For a/b<1, this result of multiplying contradicts earlier student experience with whole numbers, so this result needs to be discussed, explained, and emphasized. Grade five students divide a unit fraction by a whole number or a whole number by a unit fraction. By the end of grade five students should know how to multiply fractions to be prepared for division of a fraction by a fraction in grade six.

17 5 th Grade Revised April 2014 (pg. 17) Decimals In grade five students will integrate decimal fractions more fully into the place value system as they learn to read, write, compare, and round decimals. By thinking about decimals as sums of multiples of base-ten units, students extend algorithms for multi-digit operations to decimals. By the end of grade five students understand operations with decimals to hundredths. Students should understand how to add, subtract, multiply, and divide decimals to hundredths using models, drawings, and various methods including methods that extend from whole numbers and are explained by place value meanings. The extension of the place value system from whole numbers to decimals is a major accomplishment for a student that involves both understanding and skill with base-ten units and fractions. Skill and understanding with adding, subtracting, multiplying, and dividing multi-digit decimals will culminate in fluency with the standard algorithm in grade six. Fluency with Whole Number Operations In grade five the fluency expectation is to multiply multi-digit whole numbers (one-digit numbers times a number with up to four-digits and two-digit numbers times two-digit numbers) using the standard algorithm. Students also extend their grade four work in finding wholenumber quotients and remainders to the case of two-digit divisors. Skill and understanding of division with multi-digit whole numbers will culminate in fluency with the standard algorithm in grade six. Volume Students in grade five work with volume as an attribute of a solid figure and as a measurement quantity. Students also relate volume to multiplication and addition. Students understanding and skill with this work supports a learning progression leading to valuable skills in geometric measurement in middle school.