Collegium Charter School Grade 5 Math Scope & Sequence. Global Vision. We Use Math in Our Everyday Lives

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1 Collegium Charter School Grade 5 Math Scope & Sequence Global Vision We Use Math in Our Everyday Lives Updated August 2015

2 Standards of Mathematical Practice (Habits of Mind) in 5th grade: 1. Make sense of problems and persevere in solving them. o Explain to his or herself the meaning of a problem and look for ways to conceptualize, solve & persevere in solving the problem. o Use concrete pictures or words to help his or herself and solve problems. o Check their thinking by asking his or herself, Does this make sense? and use another method to check their answers. 2. Reason abstractly and quantitatively. o Recognize that a number represents a specific quantity and connect the quantity to written symbols. o Use numbers, words, properties of operations and understandable representations to make sense of problems, paying attention to the meanings of numbers and units. o Write simple expressions, record calculations with numbers & represent or round numbers using place value concepts. o Extend this understanding from whole numbers to their work with fractions and decimals. 3. Construct viable arguments and critique the reasoning of others. o Make and present arguments using concrete referents, such as objects, diagrams, graphs and drawings, using examples & non-examples, and relate referents to context. o Analyze the reasoning of others by listening, asking and answering questions like how did you get that? & Why is that true? o Explain their thinking and make connections between models and equations. 4. Model with mathematics. o Recognize math in everyday life and use math they know to represent problem situations in multiple ways including estimations, numbers, words & manipulatives. o Connect the different representations and explain the connections. 5. Use appropriate tools strategically. o Know how and when to use the available tools when solving a mathematical problem (e.g. base-ten blocks, number lines, area models). o Use graph paper or a number line to represent and compare decimals & protractors to measure angles. o Use other measurement tools to understand the relative size of units within a system. o Express measurements given in larger units in terms of smaller units. 6. Attend to precision. o Be precise when solving problems and clear when communicating ideas by using: correct math vocabulary, context label, calculations that are accurate and efficient. o Specify units of measure and state the meaning of the symbols they choose. For instance, they use appropriate labels when creating line plots and other graphs. 7. Look for and make use of structure. o See and understand how numbers and spaces are organized and put together as parts and wholes. Examine numerical patterns and relate them to a rule. o Use properties of operations to explain calculations (partial products model). o Generate number or shape patterns that follow a given rule. 8. Look for and express regularity in repeated reasoning. o Notice when calculations are repeated in order to find general methods and short cuts. o Use models to explain calculations and understand how algorithms work. o Use models to examine patterns and generate their own algorithms. For instance students use visual models to show distributive property. Updated August

3 Understands the Problem & Uses Information Appropriately Investigation, Task & Problem Solving Rubric 15 points No attempt -0 Below Basic - 1 Basic - 2 Proficient - 3 No understanding Applies Appropriate Procedures Uses Representations Answers the Problem Communicates Effectively No procedures No representation No answer No explanation Doesn't understand enough to get started or make progress. Consistent support & redirection needed from T Uses information incorrectly. Applies procedures inappropriately. Uses a representation that gives little or no significant information about the problem. Wrong answer Minimal understanding regardless of teacher support & redirection. Explanation makes little sense Is difficult to follow. Understands enough to solve part of the problem or to get part of the solution. Partial support or redirection from Teacher. Uses some information correctly. Applies some procedures appropriately. Uses a representation that gives some important information about the problem. Incorrect answer Minor copying error, computational error that do not detract from demonstrating a general understanding. Or correct answer some work that is incomplete or unclear Or partial answer For problem with multiple answers, no answer statement, answer labeled incorrectly. Explanation makes partial sense. Provides partial explanation why procedures are appropriate to the problem. Some logical steps are described with partial connection between the essential mathematics and the representations in the solution. Understands the problem. Minimal support & little to no redirection needed from teacher. Feedback related to extended thinking. Uses all information correctly Applies procedures completely & appropriately. Uses a representation that clearly depicts the problem. Provides correct answer. Response may contain a minor blemish or omission in work that does not detract from demonstrating a thorough understanding. Explanation makes complete sense. Explains why procedures are appropriate to the problem. Uses logic and justifies conjectures when essential mathematics in the solution is connected to any work or representations used in solving. Updated August

4 Grade 5 critical areas of mathematical proficiency 1. Developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions) 2. Extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations. 3. Developing understanding of volume. Grade 5PA Core Standard Areas for Math M.05 A-T = Numbers & Operations in Base Ten M.05 A-F = Numbers & Operations Fractions M.05 B-O = Operations & Algebraic Thinking M.05 C-G = Geometry M.05 D-M = Measurement & Data Updated August

5 CMQ1 How do we use whole number and decimal place value operations in our everyday lives? Big Ideas: THE BASE TEN NUMERATION SYSTEM The base ten numeration system is a structure for recording numbers using digits 0-9, groups of ten, and place value. Each place value to the left of another is ten times greater than the one to the right. EQUIVALENCE: Numbers can be named in equivalent ways using place value. e.g. 2 hundreds 4 tens is equivalent to 24 tens. Numerical expressions can be named in an infinite number of different, but equivalent ways e.g. 24 x 6 = (20 + 4) x 6 PATTERNS: There are patterns when multiplying or dividing whole numbers and decimals by powers of ten. COMPARISON: Numbers, expressions, and measures can be compared by their relative values. Essential Questions: What are different ways to represent and name numbers? What do the place values of the digits to the right of a decimal point represent? How is decimal place value an extension of whole number place value? How can you use place value to prove 100= 10 x 10? What are some strategies for comparing numbers? How does a real-world situation determine how a remainder is to be interpreted when solving a problem? How are decimal quotients and quotients with remainders related? When are decimals used in real-world situations? Concepts & Competencies: Students develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of operations. They finalize fluency with multi-digit addition, subtraction, multiplication, and division. They apply their understandings of models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths. They develop fluency in these computations, and make reasonable estimates of their results. Students use the relationship between decimals and fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), to understand and explain why the procedures for multiplying and dividing finite decimals make sense. They compute products and quotients of decimals to hundredths efficiently and accurately. Assessment Anchor Descriptors: MO5.A-T.1.1 Demonstrate understanding of place value of whole numbers and decimals, and compare quantities or magnitudes of numbers. MO5.A-T.2.1 Use whole numbers and decimals to compute accurately (straight computation or word problems). Updated August

6 CMQ1 How do we use whole number and decimal place value operations in our everyday lives? CM1 Vocabulary: Teacher created assessment. Each word is worth one point. Teacher may differentiate the method of assessment. May be more than one test. Power of ten Decimal point Tenths Hundredths CM1FK1 Place Value Understandings Thousandths Quotient Divisor Dividend Decimal number Expanded form Place value M05.A-T.1.1.1Demonstrate an understanding that in a multi-digit number, a digit in one place represents 1/10 of what it represents in the place to its left. (5.NBT.1 ) Example: Recognize that in the number 770, the 7 in the tens place is 1/10 the 7 in the hundreds place. M05.A-T Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10. ( 5.NBT.2 ) Example 1: = 400 decimal moves to the right 2 places Example 2: = decimal moves to the left 3 places. M05.A-T Read and write decimals to thousandths using base-ten numerals, word form, and expanded form. (5.NBT.3. ) Example: = = (0.1) + 9 (0.01) + 2 (0.001) M05.A-T Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols. (5.NBT.3. ) M05.A-T Round decimals to any place (limit rounding to ones, tenths, hundredths, or thousandths place). (5.NBT.4. ) CM1FK2 Place Value Operations M05.A-T Multiply multi-digit whole numbers (not to exceed 3-digit by 3-digit). ( 5.NBT.5. ) M05.A-T Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors. ( 5.NBT.6 ) M05.A-T Add, subtract, multiply, and divide decimals to hundredths (no divisors with decimals) (5.NBT.7. ) Numbers & Operations Assessment CMQ1 Student completes task or investigation that addresses big ideas of this unit. Standards of Mathematical Practice are evaluated at the completion of the task. Task is graded using problem-solving rubric. Updated August

7 CMQ2 How do we use fractions in our everyday lives? Big Ideas: EQUIVALENCE: Any fraction/ ratio can be represented by an infinite set of different fractions/ratios that have the same value. Numerical expressions can be named in an infinite number of different, 4 but equivalent ways. e.g. 2 = OPERATION MEANINGS & RELATIONSHIPS: The product of two positive fractions each less than one is less than either factor. The product of a fraction and a whole number (both positive) is less than the whole number factor but greater than the fraction factor Essential Questions: How are fractions and decimals related? Why use fractions when we have decimals? What strategies and models help us to understand fractions? What kind of real world situations use operations with fractions? What results in the product when you multiply two fractions less than one? How is dividing fractions related to multiplying fractions? How does knowing how to find equivalent fractions help you add or subtract? BASIC FACTS & ALGORITHMS: Basic facts and algorithms for operations with rational numbers use notions of equivalence to transform calculations into simpler ones. Concepts & Competencies: Students apply their understanding of fractions and fraction models to represent the addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators. They develop fluency in calculating sums and differences of fractions, and make reasonable estimates of them. Students also use the meaning of fractions, of multiplication and division, and the relationship between multiplication and division to understand and explain why the procedures for multiplying and dividing fractions make sense. (Note: this is limited to the case of dividing unit fractions by whole numbers and whole numbers by unit fractions.) Assessment Anchor Descriptors: M05.A-F.1.1 Solve addition and subtraction problems involving fractions (straight computation or word problems). M05.A-F.2.1 Solve multiplication and division problems involving fractions and whole numbers (straight computation or word problems). Updated August

8 CMQ2 How do we use fractions in our everyday lives? CM2 Vocabulary: Teacher created assessment. Each word is one point. Teacher may differentiate the method of assessment. May be more than one test. Common Denominator and LCD Denominator Equivalent fractions Improper fractions CM2FK1 Fraction Understandings, Sums and Differences Plot and order fractions on a number line (prerequisite to CM3) Mixed number Numerator Scale Simplest form Unit fraction M05.A-F Add and subtract fractions (including mixed numbers) with unlike denominators. (5.NF.1 ) (5.NF.2-word problems ) (May include multiple methods and representations.) Ex: 2/3 + 5/4 = 8/ /12 = 23/12 CM2FK2 Fraction Products and Quotients M05.A-F Multiply a fraction (including mixed numbers) by a fraction. Limit numerators to 2-10 (5.NF.6. ) M05.A-F Demonstrate an understanding of multiplication as scaling (resizing). (5.NF.5 ) Ex 1: Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. (e.g. ¼ is ½ of ½ ) Ex 2: Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number. M05.A-F Solve word problems involving division of whole numbers leading to answers in the form of fractions (including mixed numbers). (5.NF.3 ) M05.A-F Divide unit fractions by whole numbers and whole numbers by unit fractions. (unit fractions have 1 as the numerator and a whole number as a denominator) (5.NF.7.) Fractions Assessment CMQ2 Student completes task or investigation that addresses big ideas of this unit. Standards of Mathematical Practice are evaluated at the completion of the task. Task is graded using problem-solving rubric. Updated August

9 CMQ3 How do we use data in our everyday lives? Big Ideas: DATA COLLECTION: Some questions can be answered by collecting and analyzing data, and the question to be answered determines the data that needs to be collected and how best to collect it. DATA REPRESENTATION: Data can be represented visually using tables, charts, and graphs. The type of data determines the best choice of visual representation. Essential Questions: What is data? How can using graphs help us to solve problems and describe data we collect? When is it helpful to collect more than one set of data? How does the type of data we collect determine the type of graph we create? How does the scale of the graph influence how data is interpreted? What kind of different graphs are appropriate for different types of data? How can a graphic representation of data help us communicate? Concepts & Competencies: Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally. Assessment Anchor Descriptors: M05.D-M.2.1 Organize, display, and answer questions based on data. Updated August

10 CMQ3 How do we use data in our everyday lives? CM3 Data FK Assessments CM3 Vocabulary: Teacher created assessment. Each word is one point. Teacher may differentiate the method of this assessment. May be more than one test. Bar graph Interval Line graph CM3FK1 Data Line plot Pictograph Scale Table Tally Tally chart M05.D-M Solve problems involving computation of fractions by using information presented in line plots. (5.MD.2 ) M05.D-M Display and interpret data shown in tallies, tables, charts, pictographs, bar graphs, and line graphs, and use a title, appropriate scale, and labels. A grid will be provided to display data on bar graphs or line graphs. (NoCC) Data Assessment CMQ3 Student completes task or investigation that addresses big ideas of this unit. Standards of Mathematical Practice are evaluated at the completion of the task. Task is graded using problem-solving rubric. Updated August

11 CMQ4 How do we use measurement in our everyday lives? Big Ideas: MEASUREMENT: Some attributes of objects are measurable and can be quantified using unit amounts. Standard units of measurement provide common language for communicating. Context helps us decide the appropriate degree of accuracy in measurement. Concepts & Competencies: Essential Questions: Measurement involves a selected attribute of an object (length, area, mass, volume, capacity) and a comparison of the object being measured against a unit of the same attribute. The longer the unit of measure, the fewer units it takes to measure the object. The magnitude of the attribute to be measured and the accuracy needed determines the appropriate measurement unit. Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems. Students recognize volume as an attribute of three-dimensional space. They understand that volume can be measured by finding the total number of same-size units of volume required to fill the space without gaps or overlaps. They understand that a unit cube is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving problems that involve estimating and measuring volume. They decompose three-dimensional shapes and find volumes of right rectangular prisms by viewing them as decomposed into layers of arrays of cubes. They measure necessary attributes of shapes in order to determine volumes to solve real world and mathematical problems. Assessment Anchor Descriptors: M05.D-M.1.1 Solve problems using simple conversions (may include multistep, real-world problems). M05.D-M.3.1 Use, describe and develop procedures to solve problems involving volume. Updated August

12 CMQ4 How do we use measurement in our everyday lives? CM4 Measurement FK Assessments CM4 Vocabulary: Teacher created assessment. Each word is worth one point. Teacher may differentiate the method of assessment. May be more than one test. Area Capacity Celsius ( C) Convert Customary units Degree Elapsed time Fahrenheit ( F) Fluid oz g, kg Gallon Quart Pint Cup in, ft, yd CM4FK1 Converting Measures Length Linear Mass Metric units ml and L mm, cm, m and km oz, lb Perimeter Temperature Thermometer Weight Face Height of prism Length of prism Polyhedron Prism Rectangular prism (right) Right angle Solid M05.D-M Convert among different-sized measurement units within a given measurement system. (5.MD.1) A table of equivalencies will be provided. Example: Convert 5 cm to meters. CM4FK2 Volumes of Solids Vertex Volume Width of prism Base Congruent Cubic units Edges M05.D-M Apply the formulas V = l w h and V = B h for rectangular prisms to find volumes of right rectangular prisms with wholenumber edge lengths in the context of solving real-world and mathematical problems. (5.MD5.) Formulas will be provided. M05.D-M Find volumes of solid figures composed of two non-overlapping right rectangular prisms. (5.MD5.) Measurement Assessment CMQ4 Student completes task or investigation that addresses big ideas of this unit. Standards of Mathematical Practice are evaluated at the completion of the task. Task is graded using problem-solving rubric. Updated August

13 CMQ5 How do we use geometry in our everyday lives? Big Ideas: ORIENTATION & LOCATION: Objects in space can be oriented in an infinite number of ways, and an object s location in space can be described quantitatively. SHAPES & SOLIDS: Two- and three-dimensional objects with or without curved surfaces can be described, classified, and analyzed by their attributes. Essential Questions: How do the values of ordered pairs give a location for a point? How do ordered pairs (coordinates) relate to situations on the real world? What are some ways to classify different shapes and solids? How can you classify different types of quadrilaterals? Concepts & Competencies: Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y- axis and y-coordinate). Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles. Classify two-dimensional figures in a hierarchy based on properties Assessment Anchor Descriptors: M05.C-G.1.1 Identify parts of a coordinate grid, and describes or interprets points given an ordered pair. M05.C-G.2.1 Use basic properties to classify two-dimensional figures. Updated August

14 CMQ5 How do we use geometry in our everyday lives? CM5 Geometry FK Assessments CM5 Vocabulary: Teacher created assessment. Each word is one point. Teacher may differentiate the method of this assessment. May be more than one test. Ordered pair Coordinate CM5FK1 X-axis y-axis The Coordinate Plane polygon quadrilateral parallelogram 2-dimensional figure Point Quadrant M05.C-G Identify parts of the coordinate plane (x-axis, y-axis, and the origin) and the ordered pair (x-coordinate and y-coordinate). (5.G.1 ) Limit the coordinate plane to quadrant I. M05.C-G Represent real-world and mathematical problems by plotting points in quadrant I of the coordinate plane, and interpret coordinate values of points in the context of the situation. (5.G.2 ) CM5FK2 Classifying Polygons M05.C-G Classify two-dimensional figures in a hierarchy based on properties. (5.G.4.) Ex 1: All polygons have at least 3 sides, and pentagons are polygons, so all pentagons have at least 3 sides. Ex 2: A rectangle is a parallelogram, which is a quadrilateral, which is a polygon; so, a rectangle can be classified as a parallelogram, as a quadrilateral, and as a polygon. Geometry Assessment CMQ5 Student completes task or investigation that addresses big ideas of this unit. Standards of Mathematical Practice are evaluated at the completion of the task. Task is graded using problem-solving rubric. Origin Updated August

15 CMQ6 How do we use algebra in our everyday lives? Big Ideas: PATTERNS: Relationships can be described and generalizations made for mathematical situations that have numbers or objects that repeat in predictable ways. OPERATION MEANINGS & RELATIONSHIPS: The same number sentence (e.g = 8) can be associated with different concrete or real-world situations, AND different number sentences can be associated with the same concrete or real-world situation. Essential Questions: What kind of number patterns can be generated by skip-counting? How can you prove that the difference between successive terms in some sequences is constant? How can you use known elements or terms in a pattern to predict other elements or terms? How can grouping symbols be used to write numerical expressions to represent real life situations? How can parenthesis and other grouping symbols affect the order of operations when evaluating a numerical expression? Concepts & Competencies: 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. 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. Assessment Anchor Descriptors: M05.B-O.1.1 Analyze and complete calculations by applying the order of operations. M05.B-O.2.1 Create, extend and analyze patterns Updated August

16 CMQ6 How do we use algebra in our everyday lives? CM6 Algebra FK Assessments CM6 Vocabulary: Teacher created assessment. Each word is worth one point. Teacher may differentiate the method of assessment. May be more than one test. Algebraic expression Braces Bracket CM6FK1 Writing and Evaluating Expressions Corresponding (like) terms Evaluate Numerical expression Parenthesis Pattern Sequence M05.B-O Use multiple grouping symbols (parentheses, brackets, or braces) in numerical expressions, and evaluate expressions containing these symbols. (5.OA.1 ) *Match expressions to real life situations and evaluate (assessed, but not explicitly stated in standards) M05.B-O Write simple expressions that model calculations with numbers, and interpret numerical expressions without evaluating them. (5.OA.2.) Ex 1: Express the calculation add 8 and 7, then multiply by 2 as 2 (8 + 7). Ex 2: Recognize that 3 (18, ) is three times as large as 18, , without having to calculate the indicated sum or product. CM6FK2 Numerical Sequences M05.B-O Generate two numerical patterns using two given rules. (5.OA.3 ) Ex: Given the rule Add 3 and the starting number 0, and given the rule Add 6 and the starting number 0, generate terms in the resulting sequences. M05.B-O Identify apparent relationships between corresponding terms of two patterns with the same starting numbers that follow different rules.(5.oa.3 ) Ex: Given two patterns in which the first pattern follows the rule add 8 and the second pattern follows the rule add 2, observe that the terms in the first pattern are 4 times the size of the terms in the second pattern. Algebra Assessment CMQ6 Student completes task or investigation that addresses big ideas of this unit. Standards of Mathematical Practice are evaluated at the completion of the task. Task is graded using problem-solving rubric. Updated August

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