5 th Grade. Math Unit Guide

Transcription

1 5 th Grade Math Unit Guide

2 Jackson County School District Year At A Glance 5 th Grade Math Unit 1 Understanding volume 8 Days Unit 2 Developing multiplication and division strategies 12 Days Unit 3 Using equivalency to add and subtract fractions with unlike denominators 12 Days Unit 4 Expanding understanding of place value to decimals 10 Days Unit 5 Comparing and rounding decimals 10 Days Unit 6 Understanding the concept of multiplying fractions by fractions 12 Days Unit 7 Interpreting multiplying fractions as scaling 12 Days Unit 8 Developing the concept of dividing unit fractions 10 Days Unit 9 Solving problems involving volume 10 Days Unit 10 Performing operations with decimals 12 Days Unit 11 Classifying two-dimensional geometric figures 10 Days Unit 12 Solving problems with fractional quantities 12 Days Unit 13 Representing algebraic thinking 10 Days Unit 14 Exploring the coordinate plane 10 Days Unit 15 Finalizing multiplication and division with whole numbers 10 Days

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4 5 th Grade Math In previous grades, students learned various strategies for multiplication and division and demonstrated fluency in addition and subtraction. They developed understanding of structure of the place value system, and applied understanding of operations with whole numbers to begin developing computational strategies with fractions, paying special attention to unit fractions as the building blocks of all fractions. Students gained understanding that geometric figures can be analyzed and classified based on their properties. The Grade 5 year, as sequenced here, begins with developing conceptual understanding of volume as an attribute of solid figures. This is a new concept for Grade 5 and provides an engaging context that supports problem solving throughout the year. Students practice and refine their multiplication and division strategies, attaining fluency in multiplication with whole numbers by the end of the year. The domain of Number and Operations in Base Ten is finalized this year as students generalize their understanding of the base- ten system to include decimals. This document reflects the Dana Institute s current thinking related to the intent of the Common Core State Standards for Mathematics (CCSSM) and assumes 160 days for instruction, divided among 15 units. The number of days suggested for each unit assumes 45- minute class periods and is included to convey how instructional time should be balanced across the year. The units are sequenced in a way that we believe best develops and connects the mathematical content described in the CCSSM; however, the order of the standards included in any unit does not imply a sequence of content within that unit. Some standards may be revisited several times during the course; others may be only partially addressed in different units, depending on the focus of the unit. Strikethroughs in the text of the standards are used in some cases in an attempt to convey that focus, and comments are included throughout the document to clarify and provide additional background for each unit. Throughout Grade 5, students should continue to develop proficiency with the Common Core's eight Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. S. Use appropriate tools strategically. 2. Reason abstractly and quantitatively. 6. Attend to precision. 3. Construct viable arguments and critique the reasoning of others. 7. Look for and make use of structure. 4. Model with mathematics. 8. Look for and express regularity in repeated reasoning. These practices should become the natural way in which students come to understand and do mathematics. While, depending on the content to be understood or on the problem to be solved, any practice might be brought to bear, some practices may prove more useful than others. Opportunities for highlighting certain practices are indicated in different units in this document, but this highlighting should not be interpreted to mean that other practices should be neglected in those units. When using this document to help in planning your district's instructional program, you will also need to refer to the CCSSM document, relevant progressions documents for the CCSSM, and the appropriate assessment consortium framework.

5 Unit 1: Understanding volume Suggested number of days: 8 Unit Overview: Learning Targets Notes/Comments Unit Materials and Resources Students expand their understanding of geometric measurement and spatial structuring to include volume as an attribute of three- dimensional space. In this unit, students develop this understanding using concrete models to discover strategies for finding volume, whereas in unit 9 students generalize this understanding in real- world problems and apply strategies and formulas. Volume is addressed in two units (unit 1 and unit 9) because it is a major emphasis in Grade 5. The connection to multiplication and addition provides an opportunity for students to start the year off by applying the multiplication and addition strategies they learned in previous grades in a new, interesting context. Common Core State Standards for Mathematical Content Measurement and Data 5.MD C. Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition. 3. Recognize volume as an attribute of solid figures and understand concepts of volume measurement. a. A cube with side length 1 unit, called a "unit cube," is said to have "one cubic unit" of volume, and can be used to measure volume. b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units. 4. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. Common Core State Standards for Mathematical Practice 3. Construct viable arguments and critique the reasoning of others. 7. Look for and make use of structure. 5.MD.3a.1 5.MD.3b.1 5.MD MD.4.2 Understand that unit cubes are used to measure volume of solid figures. Understand that unit cubes cannot have gaps or overlap. Use a visual model to measure volume by counting unit cubes. Measure volume by counting unit cubes. Students decompose and recompose geometric figures to make sense of the spatial structure of volume (MP.7). In particular, students explain their thinking and analyze others' reasoning as they practice partitioning figures into layers and each layer into rows and each row into cubes (MP.3). Videos Math Fact Fluency Practice web/games/ encygame.html#/math- facts- addition- games ols.org/content/math_fact_flue ncy_practice_sheets.asp Lessons/Activities/Games tics.org/5 csdoc et/fifth+grade

6 h/search.html culum/mathematics_2/curriculu m- math- grade- five om/common- core- grade5.html andards/alignments/grade5.html 5mathteachingresources.com/5th - grade- number- activities.html m/ ex.php/common- core- standards als/stds.php#standard1159

7 Measurement Unit Unit Cubes Volume Vocabulary Essential Questions What tools and units are used to measure the attributes of an object? How are the units of measure within a standard system related? How do you decide which unit of measurement to use? How can I measure length, mass and capacity by using non- standard units? How do I choose the appropriate tool and unit when measuring? How do I estimate and measure? Formative Assessment Strategies Anecdotal Note Cards - The teacher can create a file folder with 5" x 7" note cards for each student for helpful tips and hints to guide students to remembering a process or procedure. Labels or Sticky Notes - Teachers can carry a clipboard with a sheet of labels or a pad of sticky notes and make observations as they circulate throughout the classroom. After the class, the labels or sticky notes can be placed in the observation notebook in the appropriate student's section and use the data collected to adjust instruction to meet student needs. Questioning - Asking questions that give students opportunity for deeper thinking and provide teachers with insight into the degree and depth of student understanding. Questions should go beyond the typical factual questions requiring recall of facts or numbers. Discussion - Teacher presents students with an open- ended question that build knowledge and develop critical and creative thinking skills. The teacher can assess student understanding by listening to responses and taking anecdotal notes.

8 Unit 2: Developing multiplication and division strategies Suggested number of days: 12 Learning Targets Notes/Comments Unit Materials and Resources Unit Overview: In this unit students build on their work from previous grade levels to refine their strategies for multiplication and division in order to reach fluency in multiplication by the end of the year. Students continue to develop more sophisticated strategies for division to become flexible and efficient with the standard algorithm in Grade 6. Students begin to find quotients with two- digit divisors early in the year to build strategies for accurate computations. Common Core State Standards for Mathematical Content Number and Operations in Base Ten - 5.NBT B. Perform operations with multi- digit whole numbers and with decimals to hundredths. 5. Fluently multiply multi- digit whole numbers using the standard algorithm. 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. Common Core State Standards for Mathematical Practice 1. Make sense of problem and persevere in solving them. 8. Look for and express regularity in repeated reasoning. 5.NBT NBT NBT NBT NBT.6.4 Fluently multiply multi- digit whole numbers. Determine the quotient of whole numbers with up to 3 digit dividends and 1 digit divisors, which are multiples of ten. Determine the quotient of whole numbers with up to 4 digit dividends and 1 digit divisors, which are multiples of ten. Determine the quotient of whole numbers with up to 4 digit dividends and 2 digit divisors. Illustrate and explain division using equations, rectangular arrays, and/or area models. In this unit 5.NBT.B.5 and 5.NBT.B.6 will focus on operations with whole numbers only. Operations with decimals will be introduced in unit 10. These standards will be finalized in unit 15, but should be practiced throughout the year to provide opportunities for students to develop proficiency with these operations. Students look for regularity in their work with multiplication and division use their understanding of the structure (MP.8) to make sense of their solutions and understand the approaches of other students (MP.1). Videos Math Fact Fluency Practice web/games/ encygame.html#/math- facts- addition- games ols.org/content/math_fact_flue ncy_practice_sheets.asp Lessons/Activities/Games tics.org/5 csdoc et/fifth+grade

9 Algorithims Dividend Equal groups Factor Multiples Multiplication Partial products Product Multiplication symbols Vocabulary Essential Questions When given a problem with errors, how would you demonstrate and explain the process of multiplying multi- digit whole numbers using the standard algorithm? How can the methods and strategies you used to learn whole number multiplication help you to multiply decimals? Why does multiplying by a decimal result in a product less than one or both of the factors? How can you use what you know about place value to explain the relationship between expressions such as 3 x 5, 3 x 0.5, and 0.3 x 0.5? How can you use an area model to show the product of x? Formative Assessment Strategies Visual Representations/Drawings - Graphic organizers can be used as visual representations of concepts in the content areas. Many of the graphic organizers contain a section where the student is expected to illustrate his/her idea of the concept. The Mind Map - requires that students use drawings, photos or pictures from a magazine to represent a specific concept. Think/Pair/Share for Math Problem Solving - Place problem on the board. Ask students to think about the steps they would use to solve the problem, but do not let them figure out the actual answer. Without telling the answer to the problem, have students discuss their strategies for solving the problem. Then let them work out the problem individually and then compare answers. Math Center Fun- Practicing how to read large numbers, learning how to round numbers to various places, reviewing place value, solving word problems (as described above), recalling basic geometric terms, discussing the steps of division, discussing how to rename a fraction to lowest terms.

10 Unit 3: Using equivalency to add and subtract fractions with unlike denominators Suggested number of days: 12 Unit Overview: Learning Targets Notes/Comments Unit Materials and Resources In this unit students use what they've learned in Grades 3 and 4 about equivalency in terms of visual models and benchmarks to extend understanding of adding and subtracting fractions, including mixed numbers. They reason about size of fractions to make sense of their answers- e.g. they understand that the sum of 1/2 and 2/3 will be greater than 1. It is important to note that in some cases it may not be necessary to find least common denominator to add fractions with unlike denominators (any common denominator may apply). Students should be encouraged to use their conceptual understanding of fractions rather than just using the algorithm for adding fractions. In addition, there is no mathematical reason for students to write fractions in simplest form. Common Core State Standards for Mathematical Content Number and Operations- Fractions - 5.NF A. Use equivalent fractions as a strategy to add and subtract fractions. 1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/ /12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.) 5.NF NF NF.1.3 Rewrite fractions as equivalent fractions. Write my answer in simplest form. Add or subtract two fractions with unlike denominators (2, 4, 5, or 10). 5.NF.1.4 Add or subtract two mixed numbers with unlike denominators (2, 4, 5, or 10). 5.NF.1.5 Add and subtract two fractions with unlike denominators (any denominator). 5.NF.1.6 Add and subtract two mixed numbers with unlike denominators (any denominator). 5.NF.1.7 Add and subtract any number of fractions with unlike denominators (any denominator). 5.NF.1.8 Add and subtract any number of mixed numbers with unlike denominators (any denominator). 5.NF.1.9 Add and subtract fractions within the same expression In this unit, 5.NF.A.1 involves students using the same method from Grade 4 to generate equivalent fractions (4.NF.A.1). In unit 7 students will extend this understanding of equivalency to understand that multiplying by a fraction equivalent to 1 (e.g. 4/4) will result in an equivalent fraction (5.NF.B.5b). 4 Videos Math Fact Fluency Practice web/games/ encygame.html#/math- facts- addition- games ols.org/content/math_fact_flue ncy_practice_sheets.asp Lessons/Activities/Games tics.org/5 csdoc et/fifth+grade

11 2. Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2. 5.NF.2.1 Use benchmark numbers (0, ¼, ½, ¾, 1) to estimate sums and differences of fractions. 5.NF.2.2 Relate estimation to my answers to see if they make sense. 5.NF NF NF.2.5 Create a visual fraction model to represent the fractions in a word problem. Create an equation to represent a word problem. Create a word problem involving addition and subtraction of fractions. h/search.html culum/mathematics_2/curriculu m- math- grade- five om/common- core- grade5.html andards/alignments/grade5.html Common Core State Standards for Mathematical Practice 2. Reason abstractly and quantitatively. 4. Model with mathematics. Students use visual models and equations to solve problems involving the addition and subtraction of fractions, moving flexibly between the abstract and concrete representations (MP.2, MP.4). 5mathteachingresources.com/5th - grade- number- activities.html m/ ex.php/common- core- standards

12 Benchmark Fraction Common Denominator Denominator Difference Equivalent Fraction Fraction Bar Improper Fraction Like Denominator Mixed Number Number Lines Numerator Proper Fraction Sum Unlike Denominator Vocabulary Essential Questions How can you use a visual model to represent equivalent fractions? How can you add and subtract fractions with unlike denominators? In adding or subtracting fractions, when is it necessary to find a common denominator? Give an example to support your answer. How can you find a common denominator using equivalent fractions? How can benchmark fractions be used to estimate an answer? Give an example to support your thinking. Given the following model/picture/visual representation (fractions of the same whole with different denominator), what is a situation that could be represented by this model? Write and evaluate an expression to represent your situation. Formative Assessment Strategies Summaries and Reflections - Students stop and reflect, make sense of what they have heard or read, derive personal meaning from their learning experiences, and/or increase their metacognitive skills. These require that students use content- specific language. Lists, Charts, and Graphic Organizers - Students will organize information, make connections, and note relationships through the use of various graphic organizers. Visual Representations of Information - Students will use both words and pictures to make connections and increase memory, facilitating retrieval of information later on. This dual coding helps teachers address classroom diversity, preferences in learning style, and different ways of knowing. Collaborative Activities - Students have the opportunity to move and/or communicate with others as they develop and demonstrate their understanding of concepts. Do s and Don ts - List 3 Dos and 3 Don ts when using/applying/relating to the content (e.g., 3 Dos and Don ts for solving an equation). Example of Student Response: When adding fractions, DO find a common denominator, DO add the numerators once you ve found a common denominators, DON T simply add the denominators Three Most Common Misunderstandings - List what you think might be the three most common misunderstandings of a given topic based on an audience of your peers. Example of Student Response: In analyzing tone, most people probably confuse mood and tone, forget to look beyond the diction to the subtext as well, and to strongly consider the intended audience. Yes/No Chart - List what you do and don t understand about a given topic what you do on the left, what you don t on the right; overly- vague responses don t count. Specificity matters!

13 Unit 4: Expanding understanding of place value to decimals Suggested number of days: 10 Learning Targets Notes/Comments Unit Materials and Resources Unit Overview: In this unit students expand their previous understanding of place value to include decimal numbers. Grade 5 is the last grade in which the NBT domain appears in CCSSM. Later work in the base- ten system relies on the meanings and properties of operations. This also contributes to deepening students' understanding of computation and algorithms in the new domains that start in Grade 6. Common Core State Standards for Mathematical Content Number and Operations in Base Ten 5.NBT A. Understand the place value system. 1. Recognize that in a multi- digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. 2. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole- number exponents to denote powers of NBT NBT NBT NBT NBT NBT NBT.2.5 Represent place values of whole numbers through 100,000,000 and decimals to the thousandths with manipulatives or visual models. Recognize that in a multi- digit number, the digit to the left is 10x larger and the right is 1/10 smaller. Show repeated multiplication of tens as an exponent. Use manipulatives to explain patterns in the number of zeros of the product when multiplying a number by powers of 10. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10. Use manipulatives to explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Powers of 10 is a fundamental aspect of the base- ten system, thus 5.NBT.A.2 can help students extend their understanding of place value to incorporate decimals to hundredths. 6 Videos Math Fact Fluency Practice web/games/ encygame.html#/math- facts- addition- games ols.org/content/math_fact_flue ncy_practice_sheets.asp Lessons/Activities/Games tics.org/5 csdoc et/fifth+grade

14 3. Read, write, and compare decimals to thousandths. a. Read and write decimals to thousandths using base- ten numerals, number names, and expanded form, e.g., = 3 x x x x (1/10) + 9 x (1/100) + 2 x (1/1000). Common Core State Standards for Mathematical Practice 6. Attend to precision. 7. Look for and make use of structure. 5.NBT.3a.1 5.NBT.3a.2 5.NBT.3a.3 5.NBT.3a.1 Read and write decimals to the tenths place using numerals, number names, and expanded form. Read and write decimals to the hundredth place using numerals, number names, and expanded form. Read and write decimals to any place using numerals, number names, and expanded form. Read and write decimals to the tenths place using numerals, number names, and expanded form. [5.NBT.A.3a] Students will be reading and writing decimals in this unit. Comparing decimals (5.NBT.A.3b ) will be addressed in unit 6. Students use their understanding of structure of whole numbers to generalize this understanding to decimals (MP.7) and explain the relationship between the numerals (MP.6). h/search.html culum/mathematics_2/curriculu m- math- grade- five om/common- core- grade5.html andards/alignments/grade5.html 5mathteachingresources.com/5th - grade- number- activities.html m/ ex.php/common- core- standards als/stds.php#standard1159

15 Algorithim Equal Group Factor Multiples Multiplication Partial Products Product Symbols Vocabulary Essential Questions What effect does multiplying or dividing by ten or a power of ten have on a number? How do you determine which decimal is greater than, less than, or equal to another? How does expanded form help you understand the value of each digit in a number? When comparing numbers with decimals to the thousandths place, how does expanded form help you? What are the mathematical properties that govern addition and multiplication? How would you use them? How can multiples be used to solve problems? What strategies aid in mastering multiplication and division facts? How can numbers be broken down into its smallest factors? How can multiples be used to solve problems? How do you find the prime factors and multiples of a number? How can multiples be used to solve problems? How can I use the array model to explain multiplication? Formative Assessment Strategies Numbered Heads Together - Students sit in groups and each group member is given a number. The teacher poses a problem and all four students discuss. The teacher calls a number and that student is responsible for sharing for the group. Gallery Walk - After teams have generated ideas on a topic using a piece of chart paper, they appoint a person to stay with their work. Teams rotate around examining other team s ideas and ask questions of the person left at the paper. Teams then meet together to discuss and add to their information so the person there also can learn from other teams. Graffiti Groups receive a large piece of paper and felt pens of different colors. Students generate ideas in the form of graffiti. Groups can move to other papers and discuss/add to the ideas. One Question and One Comment - Students are assigned a chapter or passage to read and create one question and one comment generated from the reading. In class, students will meet in either small or whole class groups for discussion. Each student shares at least one comment or question. As the discussion moves student by student around the room, the next person can answer a previous question posed by another student, respond to a comment, or share their own comments and questions. As the activity builds around the room, the conversation becomes in- depth with opportunity for all students to learn new perspectives on the text.

16 Unit 5: Comparing and rounding decimals Suggested number of days: 10 Unit Overview: Learning Targets Notes/Comments Unit Materials and Resources In this unit students apply both their understanding of comparing fractions and their understanding of place value to compare decimals. Common Core State Standards for Mathematical Content Number and Operations in Base Ten - 5.NBT A. Understand the place value system. 3. Read, write, and compare decimals to thousandths. b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. 5.NBT.3b.1 5.NBT.3b.2 5.NBT.3b.3 5.NBT.3b.1 Compare decimals to the tenths place using inequality symbols (<,>,=). Compare decimals to the hundredths place using inequality symbols (<,>,=). Compare decimals to any place using inequality symbols (<,>,=). Compare decimals to the tenths place using inequality symbols (<,>,=). Videos Math Fact Fluency Practice web/games/ encygame.html#/math- facts- addition- games ols.org/content/math_fact_flue ncy_practice_sheets.asp Use place value understanding to round decimals to any place. 5.NBT.4.1 Use place value understanding to round decimals to any place. Lessons/Activities/Games tics.org/5 Common Core State Standards for Mathematical Practice 6. Attend to precision. 7. Look for and make use of structure. Students apply their understanding of the structure within the base- ten system and fraction- decimal equivalencies to precisely communicate their understanding of relative sizes of decimal numbers (MP.6, MP.7). csdoc et/fifth+grade

17 Decimal Division Hundredths Inequality Place Value Vocabulary Essential Questions How does understanding place value help you solve double digit addition and subtraction problems? How are place value patterns repeated in large numbers? What effect does multiplying or dividing by ten or a power of ten have on a number? How do you determine which decimal is greater than, less than, or equal to another? How does expanded form help you understand the value of each digit in a number? When comparing numbers with decimals to the thousandths place, how does expanded form help you? Formative Assessment Strategies Whip Around - The teacher poses a question or a task. Students then individually respond on a scrap piece of paper listing at least 3 thoughts/responses/statements. When they have done so, students stand up. The teacher then randomly calls on a student to share one of his or her ideas from the paper. Students check off any items that are said by another student and sit down when all of their ideas have been shared with the group, whether or not they were the one to share them. The teacher continues to call on students until they are all seated. As the teacher listens to the ideas or information shared by the students, he or she can determine if there is a general level of understanding or if there are gaps in students thinking. Word Sort - Given a set of vocabulary terms, students sort in to given categories or create their own categories for sorting Triangular Prism (Red/Green/Yellow)Students give feedback to teacher by displaying the color that corresponds to their level of understanding Take and Pass - Cooperative group activity used to share or collect information from each member of the group; students write a response, then pass to the right, add their response to next paper, continue until they get their paper back, then group debriefs. Student Data Notebooks - A tool for students to track their learning: Where am I going? Where am I now? How will I get there? Slap It - Students are divided into two teams to identify correct answers to questions given by the teacher. Students use a fly swatter to slap the correct response posted on the wall. Say Something - Students take turns leading discussions in a cooperative group on sections of a reading or video

18 Unit 6: Understanding the concept of multiplying fractions by fractions Suggested number of days: 12 Unit Overview: Learning Targets Notes/Comments Unit Materials and Resources In this unit students extend their understanding of multiplying a fraction by a whole number to multiplying fractions by fractions. In previous grades, students have developed understanding of fractions as numbers. In this grade level, students develop an understanding of the connection between fractions and division. They will use this understanding to explore the relationship of multiplication and division when multiplying fractions as explained in 5.NF.B.4a. Common Core State Standards for Mathematical Content Number and Operations- Fractions 5.NF B. Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 3. Interpret a fraction as division of the numerator by the denominator (a/b = a b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50- pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie? 5.NF NF NF NF.3.4 Interpret a fraction as a division problem. Ex. ¼ = 1 4 Interpret a division problem as a fraction. Ex. 1 4 = ¼ Solve division word problems and express the quotient as a fraction or mixed number by using visual fraction models. Solve division word problems and express the quotient as a fraction or mixed number by using equations. Videos Math Fact Fluency Practice web/games/ encygame.html#/math- facts- addition- games ols.org/content/math_fact_flue ncy_practice_sheets.asp Lessons/Activities/Games tics.org/5 csdoc et/fifth+grade

19 4. Apply and extend previous u nderstandings of multiplication to multiply a fraction or whole number by a fraction. a. Interpret the product (alb) x q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a x q b. For example, use a visual fraction model to show (2/3) x 4 = 8/3, and create a story context for this equation. Do the same with (2/3) x (4/5) = 8/15. (In general, (a/b) x (c/d) = ac/bd.) 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. Common Core State Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 4. Model with mathematics. 5. Use appropriate tools strategically. 5.NF.4a.1 5.NF.4a.2 5.NF.4a.3 5.NF.4a.4 5.NF.4a.5 5.NF.4b.1 5.NF.4b.2 5.NF.4b.3 5.NF.4b.4 Represent a whole number as a fraction. Multiply a fraction by a fraction. Multiply a fraction by a whole number. Use a visual fraction model to represent multiplication of fractions. Create a context for a problem involving multiplication of fractions. Find the area of a rectangle with fractional side lengths by tiling it with unit squares. Relate different strategies for calculating the area of a rectangle. (tiling vs. formula) Multiply fractional side lengths to find areas of rectangles. Apply my understanding of the area of rectangles to include fractional units. Representing multiplication of fractions with visual and concrete models is fundamental to this unit in order for students to make sense of multiplying fractions by fractions (MP.1, MP.4). Students select and use a variety tools to explore these concepts (MP.5). h/search.html culum/mathematics_2/curriculu m- math- grade- five om/common- core- grade5.html andards/alignments/grade5.html 5mathteachingresources.com/5th - grade- number- activities.html m/ ex.php/common- core- standards

20 Area Division Fraction Mixed Numbers Side length Tile Unit Square Visual Model Whole Number Vocabulary Essential Questions How is fraction notation related to division? How can you use a visual model, such as a fraction bar, to represent division? How would you represent a fraction as division? How do you find the area of a rectangle using tiles? How can I use fractions in real life? How can decimals be rounded to the nearest whole number? How can models be used to compute fractions with like and unlike denominators? How many ways can we use models to determine and compare equivalent fractions? How are models used to show how fractional parts are combined or separated? How do I identify and record the fraction of a whole or group? How do I identify the whole? How do I explain the meaning of a fraction and its numerator and denominator, and use my understanding to represent and compare fractions? How do I explain how changing the size of the whole affects the size or amount of a fraction? Formative Assessment Strategies Fill In Your Thoughts - Written check for understanding strategy where students fill the blank. (Another term for rate of change is or.) Circle, Triangle, Square - Something that is still going around in your head (Triangle) Something pointed that stood out in your mind (Square) Something that Squared or agreed with your thinking. ABCD Whisper - Students should get in groups of four where one student is A, the next is B, etc. Each student will be asked to reflect on a concept and draw a visual of his/her interpretation. Then they will share their answer with each other in a zigzag pattern within their group. Onion Ring - Students form an inner and outer circle facing a partner. The teacher asks a question and the students are given time to respond to their partner. Next, the inner circle rotates one person to the left. The teacher asks another question and the cycle repeats itself.

21 Unit 7: Interpreting multiplying fractions as scaling Suggested number of days: 12 Unit Overview: Learning Targets Notes/Comments Unit Materials and Resources In this unit students build on their work with "compare" problems in Grade 4 (4.0A.A.1) to develop a foundational understanding of multiplication as scaling. They interpret, represent, and explain situations involving multiplication of fractions. Students apply their whole number work with multiplication to develop conceptual understanding of multiplying a fraction by a fraction. Scaling is foundational for developing an understanding of ratios and proportion in future grade levels. Common Core State Standards for Mathematical Content Number and 0perations- Fractions - 5.NF B. Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 5. Interpret multiplication as scaling (resizing), by: a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. 5.NF.5a.1 5.NF.5a.2 5.NF.5a.3 5.NF.5a.4 Use visual models or manipulatives to interpret multiplication scaling and correctly perform the indicated multiplication. Interpret multiplication scaling by performing the indicated multiplication where one factor is a fraction. Interpret multiplication scaling without performing the indicated multiplication where one factor is a mixed number. Interpret multiplication scaling without performing the indicated multiplication where both factors are fractions. In this unit, 5.NF.B.5a and 5.NF.B.5b involve only multiplication by fractions. Division by unit fractions will be introduced in unit 8. Videos Math Fact Fluency Practice web/games/ encygame.html#/math- facts- addition- games ols.org/content/math_fact_flue ncy_practice_sheets.asp Lessons/Activities/Games tics.org/5 csdoc et/fifth+grade

22 b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (nxa)/(nxb) to the effect of multiplying a/b by 1. 5.NF.5b.1 5.NF.5b.2 5.NF.5b.3 5.NF.5b.4 Predict the size of the product based on the size of the factors. Ex: fraction x fraction = smaller fraction, fraction x whole number = a fraction of the whole number. Use visual models or manipulatives to explain when multiplying by a fraction greater than one, the number increases and when multiplying by a number less than one, the number decreases. Explain when multiplying by a fraction greater than one, the number increases and when multiplying by a number less than one, the number decreases. Explain that when multiplying the numerator and denominator by the same number is the same as multiplying by one. h/search.html culum/mathematics_2/curriculu m- math- grade- five om/common- core- grade5.html andards/alignments/grade5.html 5mathteachingresources.com/5th - grade- number- activities.html 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. Common Core State Standards for Mathematical Practice 2. Reason abstractly and quantitatively. 4. Model with mathematics. 6. Attend to precision. 5.NF.6.1 Solve real- world problems involving multiplication of fractions and mixed numbers. In 5.NF.B.6 students should have opportunities to work with all problem types. Students reason abstractly and practice communicating their thinking in real world situations (MP.2, MP.6). They use number lines and other visual models to interpret situations involving multiplication by numbers larger than one (when the result will be larger than the original quantity) and involving multiplication by a fraction smaller than 1 (when the result will be smaller than the original quantity) (MP.4). m/ ex.php/common- core- standards

23 Area Division Fraction Mixed Numbers Side length Tile Unit Square Visual Model Whole Number Vocabulary Essential Questions How can I use fractions in real life? How can decimals be rounded to the nearest whole number? How can models be used to compute fractions with like and unlike denominators? How many ways can we use models to determine and compare equivalent fractions? How are models used to show how fractional parts are combined or separated? How do I identify and record the fraction of a whole or group? How do I identify the whole? How do I explain the meaning of a fraction and its numerator and denominator, and use my understanding to represent and compare fractions? How do I explain how changing the size of the whole affects the size or amount of a fraction? How is fraction notation related to division? How can you use a visual model, such as a fraction bar, to represent division? How would you represent a fraction as division? Formative Assessment Strategies Quick Write - The strategy asks learners to respond in 2 10 minutes to an open- ended question or prompt posed by the teacher before, during, or after reading. Direct Paraphrasing - Students summarize in well- chosen (own) words a key idea presented during the class period or the one just past. RSQC2 - In two minutes, students recall and list in rank order the most important ideas from a previous day's class; in two more minutes, they summarize those points in a single sentence, then write one major question they want answered, then identify a thread or theme to connect this material to the course's major goal. I have the Question, Who has the Answer? - The teacher makes two sets of cards. One set contains questions related to the unit of study. The second set contains the answers to the questions. Distribute the answer cards to the students and either you or a student will read the question cards to the class. All students check their answer cards to see if they have the correct answer. A variation is to make cards into a chain activity: The student chosen to begin the chain will read the given card aloud and then wait for the next participant to read the only card that would correctly follow the progression. Play continues until all of the cards are read and the initial student is ready to read his card for the second time.

24 Unit 8: Developing the concept of dividing unit fractions Suggested number of days: 10 Unit Overview: Learning Targets Notes/Comments Unit Materials and Resources In this unit students will use their understanding of the relationship of multiplication and division to develop a conceptual understanding of division with fractions (division of a whole number by a unit fraction or a unit fraction by a whole number). Common Core State Standards for Mathematical Content Number and Operations- Fractions - 5.NF B. Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. a. Interpret division of a unit fraction by a non- zero whole number, and compute such quotients. For example, create a story context for (1/3) 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) 4 = 1/12 because (1/12) x 4 = 1/3. 5.NF.7 5.NF.7a.1 5.NF.7a.2 Understand the relationship between multiplication and division. Use a visual fraction model to divide a unit fraction by a whole number. Create a context for a problem involving division of a unit fraction by a whole number. Videos Math Fact Fluency Practice web/games/ encygame.html#/math- facts- addition- games ols.org/content/math_fact_flue ncy_practice_sheets.asp Lessons/Activities/Games tics.org/5 csdoc et/fifth+grade

25 b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 (1/5) = 20 because 20 x (1/5) = 4. NOTE: Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade. Common Core State Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 5.NF.7b.1 5.NF.7b.2 Use a visual fraction model to divide a whole number by a unit fraction. Create a context for a problem involving division of a whole number by a unit fraction. In this unit it is critical for students to use concrete objects or pictures to help conceptualize, create, and solve problems (MP.1, MP.2). h/search.html culum/mathematics_2/curriculu m- math- grade- five om/common- core- grade5.html andards/alignments/grade5.html 5mathteachingresources.com/5th - grade- number- activities.html m/ ex.php/common- core- standards als/stds.php#standard1159

26 Division Fraction Mixed Numbers Multiplication Unit Fraction Visual Model Whole Number Vocabulary Essential Questions How can you use a visual model to divide a whole number by a unit fraction? What is the relationship between multiplication and division? How can I use fractions in real life? How can decimals be rounded to the nearest whole number? How can models be used to compute fractions with like and unlike denominators? How many ways can we use models to determine and compare equivalent fractions? How are common and decimal fractions alike and different? What strategies can be used to solve estimation problems with common and decimal fractions? How are models used to show how fractional parts are combined or separated? How do I identify and record the fraction of a whole or group? How do I identify the whole? Formative Assessment Strategies Journal Entry - Students record in a journal their understanding of the topic, concept or lesson taught. The teacher reviews the entry to see if the student has gained an understanding of the topic, lesson or concept that was taught. Choral Response - In response t o a cue, all students respond verbally at the same time. The response can be either to answer a question or to repeat something the teacher has said. A- B- C Summaries - Each student in the class is assigned a different letter of the alphabet and they must select a word starting with that letter that is related to the topic being studied. Debriefing - A form of reflection immediately following an activity. Idea Spinner - The teacher creates a spinner marked into 4 quadrants and labeled Predict, Explain, Summarize, Evaluate. After new material is presented, the teacher spins the spinner and asks the students to answer a questions based on the location of the spinner. For example, if the spinner lands in the Summarize quadrant, the teacher might say, List the key concepts just presented.