# CCSS Mathematics Implementation Guide Grade First Nine Weeks

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1 First Nine Weeks s The value of a digit is based on its place value. What changes the value of a digit? 5.NBT.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. MA Analyze the magnitude of a digit on the basis of its place value, using whole numbers and decimal numbers through thousandths., 6, Envision: Topic 3, Lesson : Our place value system is based on the power of ten patterns. : What pattern is our number system based on? Patterns are created when we multiply a number by powers of ten. 5.NBT. 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. MA Analyze the magnitude of a digit on the basis of its place value, using whole numbers and decimal numbers through thousandths., 6, Envision: Topic, Lesson 1 : What happens when we multiply a number by powers of ten? Page1

2 s : You can read and write decimals using base-ten numerals, number names and expanded form. : What are three ways you can express decimals? : Place value understanding is necessary to round a decimal. : How do you use place value to round a decimal? 5.NBT.3a. READ and WRITE decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., = 3 x x 10 + X X (1/10) + 9 x (1/100) + X (1/1000). MA Analyze the magnitude of a digit on the basis of its place value, using whole numbers and decimal numbers through thousandths. 5.NBT.4 USE place value understanding to ROUND decimals to any place. MA Analyze the magnitude of a digit on the basis of its place value, using whole numbers and decimal numbers through thousandths., 4, 5, 6, Envision: Topic 1, Lesson 3, 6, Envision: Topic, Lesson Page

3 s : There are multiple ways to find a quotient. : What are the ways to find a quotient with two-digit divisors? 5.NBT.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. 5.. Apply an algorithm to divide whole numbers fluently., 3, 4, 5, Envision: Topic 5 ALL : There are multiple ways to add, subtract, multiply and divide decimals Understand the relationship among the divisor, dividend, and quotient. 5.NBT. ADD, SUBTRACT, MULTIPLY, and DIVIDE decimals to hundredths, USING concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; RELATE the strategy to a written method and EXPLAIN the reasoning used., 3, 4, 5, Envision: Topic, Lesson 6 Envision: Topic, Lesson Envision: Topic, Lessson 8 Envision: Topic - ALL : What are some ways you can add, subtract, multiply and divide decimals? MA [Indicator] - Apply an algorithm to add and subtract decimals through thousandths. Page3

4 Second Nine Weeks s Common denominators to add or subtract fractions with unlike denominators. How do you add or subtract fractions with unlike denominators? 5.NF.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, /3 + 5/4 = 8/1 + 15/1 = 3/1. (In general, a/b + c/d = (ad + bc)/bd.) MA [Indicator] Classify numbers as prime, composite, or neither. MA.5.5. [Indicator] Generate strategies to find the greatest common factor and the least common multiple of two whole numbers. MA [Indicator] Generate strategies to add and subtract fractions with like and unlike denominators. MA [Indicator] Apply divisibility rules for 3, 6, and 9. 4 Envision: Topic 10, ALL Page4

5 s You can tell the validity of an answer by using benchmark fractions and number sense. 5.NF. 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 /5 + 1/ = 3/, by observing that 3/ < 1/. 1 3 Envision: Topic 10, ALL (Focus on word problems) How do you know when your answer is reasonable when MA [Indicator] Classify numbers as prime, composite, or neither. MA.5.5. [Indicator] Generate strategies to find the adding or subtracting fractions? greatest common factor and the least common multiple of two whole numbers MA [Indicator] Generate strategies to add and subtract fractions with like and unlike denominators. MA Apply divisibility rules for 3, 6, and 9. 8 Page5

6 s 5.OA WRITE simple expressions that record calculations with numbers, and INTERPRET numerical expressions WITHOUT EVALUATING 1 Envision: Topic 6, Lesson 1 Envision: Topic 6, Lesson Words and/or symbols can be used to describe numerical expressions. For example, express the calculation add 8 and, then multiply by as x (8+). Recognize that 3x(1893/91) is three times as large as 1893/91, without having to calculate the indicated sum or product. Envision: Topic 6, Lesson 3 Envision: Topic 6, Lesson 5 Envision: Topic 6, Lesson 6 What is a mathematical expression? MA [Indicator] Represent numeric, algebraic, and geometric patterns in words, symbols, algebraic expressions, and algebraic equations. How can you represent a mathematical expression? MA [Indicator] Analyze situations that show change over time. Page6

7 s A coordinate plane has an x and y axis. Coordinates are placed on this plane. What are coordinates and how are they used? 5.G.1 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). 4 6 Envision: Topic 1 ALL (focus on 1 st quadrant only/positive ordered pairs). Page

8 s a. Different rules create different patterns of numbers. 5.OA.3 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. Envision: Topic 1, Lesson 4 Envision: Topic 18, Lesson 3 (Extend to include graphing two equations on same coordinate plane). b.numerical patterns can be compared on a coordinate plane. a.what causes numerical patterns to change? MA [Indicator] Represent numeric, algebraic, and geometric patterns in words, symbols, algebraic expressions, and algebraic equations. MA [Indicator] Analyze patterns and functions with words, tables, and graphs. MA [Indicator] Match tables, graphs, expressions, equations, and verbal descriptions of the same problem situation. For example, 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, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so. b.how can numerical patterns be compared? MA [Indicator] Analyze situations that show change over time. Page8

9 Third Nine Weeks s Attributes are used to define two dimensional shapes. What are the attributes that define two dimensional shapes? 5.G.3 Understand figures also belong to all subcategories of that category. MA [Indicator] Apply the relationships of quadrilaterals to make logical arguments about their properties. MA Compare the angles, side lengths, and perimeters of congruent shapes. MA Classify shapes as congruent. MA Translate between two dimensional representations and three dimensional objects. MA Predict the results of multiple transformations on a geometric shape when combinations of translation, reflection, and rotation are used. MA Analyze shapes to determine line symmetry and/or rotational symmetry, 6, Envision: Topic 8, Lesson 3 Envision: Topic 8, Lesson 4 Envision: Topic 8, Lesson 5 Envision: Topic 8, Lesson 6 Envision: Topic 13, Lesson 1 Envision: Topic 13, Lesson Envision: Topic 19, Lesson 1 Envision: Topic 19, Lesson Envision: Topic 19, Lesson 3 Envision: Topic 19, Lesson 4 Envision: Topic 19, Lesson 5 For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.that attributes belonging to a category of two dimensional Page9

10 s 5.G.4 Classify two dimensional figures in a hierarchy based on properties. MA Apply the relationships of quadrilaterals to Two dimensional shapes can be make logical arguments about their properties. characterized based on their properties. MA Compare the angles, side lengths, and perimeters of congruent shapes. How can you classify twodimensional shapes? MA Classify shapes as congruent. MA Translate between two dimensional representations and three dimensional objects. MA Predict the results of multiple transformations on a geometric shape when combinations of translation, reflection, and rotation are used. MA Analyze shapes to determine line symmetry and/or rotational symmetry, 3, 5, 6, Envision Topic 8, Lesson (no measuring) Envision: Topic 8, Lesson 3 Envision: Topic 8, Lesson 4 Envision: Topic 8, Lesson 5 Envision: Topic 8, Lesson 6 Envision: Topic 13, Lesson 1 Envision: Topic 13, Lesson Envision: Topic 19, Lesson 1 Envision: Topic 19, Lesson Envision: Topic 19, Lesson 3 Envision: Topic 19, Lesson 4 Envision: Topic 19, Lesson 5 Page10

11 s a. To solve real world problems, you may need to convert measurements. b. Each measurement system has its own set of conversions. a. Why are measurement conversions necessary? b. How can we use conversions to solve multi step, real world problems? 5.MD.1 Convert among different sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi step, real world problems. MA [Indicator] Use equivalencies to convert units of measure within the metric system: converting length in millimeters, centimeters, meters, and kilometers; converting liquid volume in milliliters, centiliters, liters, and kiloliters; and converting mass in milligrams, centigrams, grams, and kilograms. MA [Indicator] Recall equivalencies associated with length, liquid volume, and mass: 10 millimeters = 1 centimeter, 100 centimeters = 1 meter, 1000 meters = 1 kilometer; 10 milliliters = 1 centiliter, 100 centiliters = 1 liter, 1000 liters = 1 kiloliter; and 10 milligrams = 1 centigram, 100 centigrams = 1 gram, 1000 grams = 1 kilogram. 1,, 5, 6 Envision: Topic 14, Lesson 1 Envision: Topic 14, Lesson Envision: Topic 14, Lesson 3 Envision: Topic 14, Lesson 4 Envision: Topic 14, Lesson 5 Page11

12 s Line plots can be used to add, subtract, multiply and divide fractional data sets. How can fractional data sets be analyzed using line plots? Italicized and Bolded standards indicate support 5.MD. Make a line plot to display a data set of measurements in fractions of a unit (1/, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. Envision: Topic 18, Lesson 1 (extend to include fractions) 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. You can count cubic units to measure volume. 5.MD.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. 4 5 Envision: Topic 13, Lesson 4 Envision: Topic 13, Lesson 5 Envision: Topic 13, Lesson 6 How can cubic units be used to measure volume? MA [Indicator] Apply strategies and formulas to determine the volume of rectangular prisms. MA [Indicator] Identify applications of commutative, associative, and distributive properties with whole numbers. 6 Envision: Topic 13, Lesson Page1

13 s You can find volume of a rectangular prism by packing it with unit cubes, multiplying the edge lengths, and by multiplying the height by the area of the base. How can volume of a rectangular prism be measured? MD.5a. Find the volume of a right rectangular prism with whole number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole number products as volumes, e.g., to represent the associative property of multiplication. MA [Standard] The student will demonstrate through the mathematical processes an understanding of the units and systems of measurement and the application of tools and formulas to determine measurements. MA [Indicator] Identify applications of commutative, associative, and distributive properties with whole numbers. 1,, 3, 4, 5, 6,, 8 Envision: Topic 13, Lesson 4 Envision: Topic 13, Lesson 5 Envision: Topic 13, Lesson 6 Envision: Topic 13, Lesson Page13

14 s You can find the volume of a rectangular prism by multiplying the l x w x h OR b (base area) x h. 5.5b Apply the formulas V = l x w x h and V = b x h for rectangular prisms to find volumes of right rectangular prisms with whole number edge lengths in the context of solving real world and mathematical problems. 1,, 3, 4, 5, 6,, 8 Envision: Topic 13, Lesson 4 Envision: Topic 13, Lesson 5 Envision: Topic 13, Lesson 6 Envision: Topic 13, Lesson What are the two formulas to find the volume of a right rectangular prism? MA [Standard] The student will demonstrate through the mathematical processes an understanding of the units and systems of measurement and the application of tools and formulas to determine measurements. MA [Indicator] Identify applications of commutative, associative, and distributive properties with whole numbers. Page14

15 s You can find the volume by adding the volume of each prism. 5.5c Recognize volume as additive. Find volumes of solid figures composed of two non overlapping parts, applying this technique to solve real world problems. 1,, 3, 4, 5, 6,, 8 Envision: Topic 13, Lesson 6 How can you find the volume for a prism that is not a standard shape? MA [Standard] The student will demonstrate through the mathematical processes an understanding of the units and systems of measurement and the application of tools and formulas to determine measurements. MA [Indicator] Identify applications of commutative, associative, and distributive properties with whole numbers. Page15

16 Fourth Nine Weeks s You can use a sequence of operations to solve equations. If an equation has more than one operation, how do you solve it? 5.NF.4a. Interpret the product (a/b) 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 (/3) x 4 = 8/3, and create a story context for this equation. Do the same with (/3) x (4/5) = 8/15. (In general, (a/b) x (c/d) = ac/bd.) Envision: Topic 11, Lesson 1 You can find the area of rectangles with fractional side lengths using tiles and multiplication. 5.NF.4b 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 Envision: Topic 11, Lesson How can you find areas of rectangles? Page16

17 s a.if you multiply a number a number by a fraction that is greater (lesser) than one the product will be bigger (lesser) than the number. b. If a fraction is multiplied by one (4/4), the quantity is unchanged. 5.NF.5b 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 Envision: Topic 11, Lesson 1 Envision: Topic 11, Lesson 3 Envision: Topic 11, Lesson 5 a. What causes the product of a given number to be greater or lesser than the given number when multiplied by a fraction? b. How does the identity property relate to multiplication of fractions? Page1

18 s A fraction divided by a non zero, whole number will be a smaller fraction. 5.NF.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 = 1/1 because (1/1) x 4 = 1/ No Envision lesson for dividing a fraction by a whole number. What happens to a fraction when you divide it by a non zero, whole number? When you divide a whole by a fraction, the quotient will be a larger whole number. What happens to a whole number when you divide it by a fraction? 5.NF.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) = 0 because 0 x (1/5) = Envision: Topic 11, Lesson 4 Envision: Topic 11, Lesson 5 Page18

19 s When you divide a whole by a fraction, the quotient will be a larger whole number. What happens to a whole number when you divide it by a fraction? These standards are necessary for PASS. 5.NF.c Solve real world problems involving division of unit fractions by non zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/ lb of chocolate equally? How many 1/3 cup servings are in cups of raisins? MA [Indicator] Use appropriate tools and units to measure objects to the precision of one eighth inch. MA [Indicator] Use a protractor to measure angles from 0 to 180 degrees. MA [Indicator] Apply formulas to determine the perimeters and areas of triangles, rectangles, and parallelograms Envision: Topic 11, Lesson 4 (Does not include division of unit fractions by a non zero number). Envision: Topic 1, Lesson 1 Envision: Topic 8, Lesson Envision: Topic 1, Lesson 3 Envision: Topic 1, Lesson 4 Envision: Topic 1, Lesson 5 Envision: Topic 1, Lesson 6 Envision: Topic 14, Lesson 6 Envision: Topic 14, Lesson Envision: Topic 14, Lesson 8 Envision: Topic 14, Lesson 9 MA Apply procedures to determine the amount of elapsed time in hours, minutes, and seconds within a 4 hour period. MA [Indicator] Understand the relationship between the Celsius and Fahrenheit temperature scales. Page19

20 s These standards are necessary for PASS. MA [Indicator] Design a mathematical investigation to address a question. MA [Indicator] Analyze how data collection methods affect the nature of the data set. MA [Indicator] Represent the probability of a singlestage event in words and fractions. MA [Indicator] Conclude why the sum of the probabilities of the outcomes of an experiment must equal 1. Page0

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