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# Analysis of California Mathematics standards to Common Core standards-Grade 5

Strand Strand Number Sense 1.0 Number Sense CA Math Standard CA Math Standard 1.0 Students compute with very large and very small numbers, positive integers, decimals, and fractions and understand the relationship between decimals, fractions, and percents. They understand the relative magnitudes of numbers. 1.1 Estimate, round, and manipulate very large (e.g., millions) and very small (e.g., thousandths) numbers. Number Operations in Base Ten 5.NBT: (Cluster Statement) Perform operations with multi-digit whole numbers and with decimals to hundredths. Partial CCS Cluster Statement does not mention integers and percents. Domain Common Core Standard (CCS) Alignment Comments in Reference to CCS

## Number Operations in Base Ten

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. 5.NBT.3: Read, write, and compare decimals to thousandths. 5.NBT.3a: Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 x 100 + 4 x 10 + 7 x 1 + 3 x (1/10) + 9 x (1/100) + 2 x (1/1000). 5.NBT.3b: Compare two decimals to thousandths based on meaning of the digits in each place, using >, =, and < symbols to record the results of comparisons. 5.NBT.4: Use place value understanding to round decimals to any place.

Partial

CCS does not reference estimation directly. In the Mathematical Practice standards, CCS implies a thorough understanding of the concepts so students could develop strong estimation skills as a by-product of the depth of understanding. 6.RP.3: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

1.2 Interpret percents as a part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number.
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No

6.RP.3a: Make table of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use table to compare
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Strand

CA Math Standard

Domain

## Common Core Standard (CCS)

Alignment

Comments in Reference to CCS ratios. 6.RP.3b: solve unit rate problems including those involving unit pricing and constant speed. 6.RP.3c: Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity): solve problems involving finding the whole, given a part and the percent. 6.RP.3d: Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. 6.EE.1: Write and evaluate numerical expressions involving whole-number exponents.

1.3 Understand and compute positive integer powers of nonnegative integers; compute examples as repeated multiplication. 1.4 Determine the prime factors of all numbers through 50 and write the numbers as the product of their prime factors by using exponents to show multiples of a factor (e.g., 24 = 2 x 2 x 2 x 3 = 23 x 3).

No

No

1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers.

No

4.OA.4: Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range is prime or composite. 6.NS.6: Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in
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Strand

CA Math Standard

Domain

## Common Core Standard (CCS)

Alignment

Comments in Reference to CCS the plane with negative number coordinates. 6.NS.6a: Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3 and then 0 is its own opposite. 6.NS.6b: Write interpret, and explain statements of order for rational number in real-world contexts. 6.NS.6c: Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on the coordinate plane.

## 2.0 Number Sense

2.0 Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals.

Number and Operations Fractions Number and Operations Base Ten Number and Operations in Base Ten

5.NF: (Cluster Statement) Use equivalent fractions as a strategy to add and subtract fractions. Apply and extend previous understanding of multiplication and division to multiply and divide fractions. 5.NBT: (Cluster Statement) Perform Operations with multi-digit whole numbers and with decimals to hundredths. 5.NBT.7: Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; related the strategy to a written method and explain the reasoning used.

Yes

2.1 Add, subtract, multiply, and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify the reasonableness of the results.

Partial

6.NS.3: Fluently add, subtract, multiply and divide multi-digit decimals using the standard algorithm for each operation. 7.NS.1: Apply and extend previous understandings of
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Strand

CA Math Standard

Domain

## Common Core Standard (CCS)

Alignment

Comments in Reference to CCS addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

2.2 Demonstrate proficiency with division, including division with positive decimals and long division with multidigit divisors.

## Number Operations in Base Ten

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.NBT.7: Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. 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.

Yes

2.3 Solve simple problems, including ones arising in concrete situations, involving the addition and subtraction of fractions and mixed numbers (like and unlike denominators of 20 or less), and express answers in the simplest form. 2.4 Understand the concept of multiplication and division of fractions.

Yes

## Number and Operations Fractions

5.NF.4: Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. 5.NF.4a: Interpret the product (a/b) ! q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a ! q b. For example, use a visual fraction model to show (2/3) ! 4 = 8/3, and create a story context for this equation. Do the same with (2/3) ! (4/5) = 8/15. (In general, (a/b) ! (c/d) = ac/bd.)

Yes

*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.

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Strand

CA Math Standard

Domain

Common Core Standard (CCS) 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. 5.NF.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. 5.NF.7: Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions*. 5.NF.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. 5.NF.7: Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. 5.NF.7a: Interpret division of a unit fraction by a nonzero 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) ! 4 = 1/3. 5.NF.7b: 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 ! (1/5) = 4.

Alignment

## Comments in Reference to CCS

2.5 Compute and perform simple multiplication and division of fractions and apply these procedures to solving problems.

## Number and Operations Fractions

Yes

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Strand

CA Math Standard

Domain

Common Core Standard (CCS) 5.NF.7c: 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.

Alignment

## Strand Algebra Functions 1.0 Algebra Functions

CA Math Standard 1.0 Students use variables in simple expressions, compute the value of the expression for specific values of the variable, and plot and interpret the results. 1.1 Use information taken from a graph or equation to answer questions about a problem situation. Operations and Algebraic Thinking 5.OA: Write and interpret numerical expressions. Partial 6.EE: (Cluster Statement) Apply and extend previous understandings of arithmetic to algebraic expressions. 6.EE.9: Use variables to represent two quantities in a realworld problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and table, and relate these to the equation. 6.EE.2: Write, read, and evaluate expressions in which letters stand for numbers.

No

1.2 Use a letter to represent an unknown number; write and evaluate simple algebraic expressions in one variable by substitution. 1.3 Know and use the distributive property in equations and expressions with variables. 1.4 Identify and graph ordered pairs in the four quadrants of the coordinate plane.

No

No

Geometry

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 plan located by using an ordered pair of numbers, called its coordinates. Understand that the first number

Partial

6.EE.3: Apply the properties of operations to generate equivalent expressions. 6.NS.8: Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances
6

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Strand

CA Math Standard

Domain

1.5 Solve problems involving linear functions with integer values; write the equation; and graph the resulting ordered pairs of integers on a grid.

Common Core Standard (CCS) 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 xcoordinate, y-axis and y-coordinate). 5.G.2: 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. 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.

Alignment

Comments in Reference to CCS between points with the same first coordinate or the same second coordinate.

Partial

6.EE.9: Use variable to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variable using graphs and table and relate these to the equation.

## Strand Measurement and Geometry 1.0 Measurement and Geometry

CA Math Standard 1.0 Students understand and compute the volumes and areas of simple objects. 1.1 Derive and use the formula for the area of a triangle and of a parallelogram by comparing it with the formula for the area of a rectangle (i.e., two of the same triangles make a parallelogram with twice the area; a parallelogram is compared with a rectangle of the same area by cutting and pasting a right triangle on the parallelogram).
Measurement

and Data

5.MD: (Cluster Statement) Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

Yes

No

6.G.1: Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangle or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

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Strand

CA Math Standard 1.2 Construct a cube and rectangular box from twodimensional patterns and use these patterns to compute the surface area for these objects.

Domain

## Common Core Standard (CCS)

Alignment No

Comments in Reference to CCS 6.G.4: Represent threedimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real world and mathematical problems.

1.3 Understand the concept of volume and use the appropriate units in common measuring systems (i.e., cubic centimeter [cm3], cubic meter [m3], cubic inch [in3], cubic yard [yd3]) to compute the volume of rectangular solids.

Measurement

and Data

5.MD.3: Recognize volume as an attribute of solid figures and understand concepts of volume measurement. 5.MD.3a: 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. 5.MD.3b: 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. 5.MD.4: Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. 5.MD.5: Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. 5.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 edges lengths, equivalently by multiplying the heights by the area of the base. Represent threefold wholenumber products as volumes e.g., to represent the associative property of multiplication. 5.MD.5b: Apply the formulas V = l x w x h and V = b x h for rectangular prisms to find volumes of right rectangular with whole number edge lengths in the

Yes

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Strand

CA Math Standard

Domain

Common Core Standard (CCS) context of solving real world and mathematical problems. 5.MD.5c: Recognize volume as additive. Find volumes of solid figures composed of two nonoverlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems. 5.MD.4: Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

Alignment

## Comments in Reference to CCS

1.4 Differentiate between, and use appropriate units of measures for, two-and threedimensional objects (i.e., find the perimeter, area, volume).

Measurement

and Data

Partial

4.MD.2: Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. 4.MD.3: Apply the area and perimeter formulas for rectangles in real world and mathematical problems. CCS does not mention differentiate between appropriate units of measure.

## 2.0 Measurement and Geometry

2.0 Students identify, describe, and classify the properties of, and the relationships between, plane and solid geometric figures.

Geometry

5.G: (Cluster Statement) Classify two-dimensional figures into categories based on their properties. 5.G.3: Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. 5.G.4: Classify two-dimensional figures in a hierarchy based on properties.

Yes

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Strand

CA Math Standard 2.1 Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools (e.g., straightedge, ruler, compass, protractor, drawing software).

Domain
Measurement

and Data

Common Core Standard (CCS) 4.MD.5: Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement: 4.MD.5a: An angle is measure with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a one-degree angle and can be used to measure angles. 4.MD.5b: An angle that turns through n one-degree angles is said to have an angle measure of n degrees. 4.MD.6: Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure. 4.MD.1: Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

Alignment Partial

Comments in Reference to CCS 7.G: (Cluster Statement) Draw, construct and describe geometrical figures and describe the relationships between them.

2.2 Know that the sum of the angles of any triangle is 180 and the sum of the angles of any quadrilateral is 360 and use this information to solve problems.

No

2.3 Visualize and draw twodimensional views of threedimensional objects made from rectangular solids.

No

8.G.5: Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. 7.G.3: Describe the twodimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. CCS does not specify drawing two-dimensional views of three10

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Strand Strand Statistics, Data Analysis, and Probability 1.0 Statistics, Data Analysis, and Probability

CA Math Standard

Domain

Alignment

## Comments in Reference to CCS dimensional objects.

CA Math Standard 1.0 Students display, analyze, compare, and interpret different data sets, including data sets of different sizes. 1.1 Know the concepts of mean, median, and mode; compute and compare simple examples to show that they may differ.
Measurement

and Data

## 5.MD: (Cluster Statement) Represent and interpret data.

Yes

No

6.SP.5: Summarize numerical data sets in relation to their context, such as by: 6.SP.5a: Reporting the number of observations. 6.SP.5b: Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. 6.SP.5c: Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. 6.SP.5d: Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

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Strand

CA Math Standard 1.2 Organize and display singlevariable data in appropriate graphs and representations (e.g., histogram, circle graphs) and explain which types of graphs are appropriate for various data sets. 1.3 Use fractions and percentages to compare data sets of different sizes. 1.4 Identify ordered pairs of data from a graph and interpret the meaning of the data in terms of the situation depicted by the graph.

Domain

## Common Core Standard (CCS)

Alignment No

Comments in Reference to CCS 6.SP.4: Display numerical data in plots on a number line, including dot plots, histograms, and box plots. CCS does not mention circle graph.

Measurement

and Data

5.MD.2: Make a line plot to display a data set of measurements in fractions of a unit (1/2, ", 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots.

Partial

## CCS does not specify using percentages to compare data.

7.RP.2: Recognize and represent proportional relationships between quantities. 7.RP.2a: Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. 7.RP.2b: Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. 7.RP.2c: Represent proportional relationships by equations. 7.RP.2d: Explain that a point (x,y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0,0) and (1,r) where r is the unit rate.

No

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12

Strand

CA Math Standard 1.5 Know how to write ordered pairs correctly; for example, ( x, y ).

Domain Geometry

Common Core Standard (CCS) 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 plan 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 xcoordinate, y-axis and y-coordinate).

Alignment Yes

## Strand Mathematical Reasoning 1.0 Mathematical Reasoning

CA Math Standard 1.0 Students make decisions about how to approach problems: 1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, sequencing and prioritizing information, and observing patterns. 1.2 Determine when and how to break a problem into simpler parts. Mathematical Practice Standards Mathematical Practice Standards 5.MP1: Make sense of problems and persevere in solving them. 5.MP1: Make sense of problems and persevere in solving them. Yes

Yes

## Mathematical Practice Standards Mathematical Practice Standards Mathematical Practice Standards

5.MP1: Make sense of problems and persevere in solving them. 5.MP3: Construct viable arguments and critique the reasoning of others. 5.MP1: Make sense of problems and persevere in solving them.

Yes

## 2.0 Mathematical Reasoning

2.0 Students use strategies, skills, and concepts in finding solutions. 2.1 Use estimation to verify the reasonableness of calculated results.

Yes

Yes

CCS does not reference estimation directly. In the Mathematical Practice standards, CCS implies a thorough understanding of the concepts so
13

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Strand

CA Math Standard

Domain

## Common Core Standard (CCS)

Alignment

Comments in Reference to CCS students could develop strong estimation skills as a byproduct of the depth of understanding.

2.2 Apply strategies and results from simpler problems to more complex problems. 2.3 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. 2.4 Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work. 2.5 Indicate the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy. 2.6 Make precise calculations and check the validity of the results from the context of the problem. 3.0 Students move beyond a particular problem by generalizing to other situations. 3.1 Evaluate the reasonableness of the solution in the context of the original situation. 3.2 Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems.

## Mathematical Practice Standards Mathematical Practice Standards

5.MP1: Make sense of problems and persevere in solving them. 5.MP1: Make sense of problems and persevere in solving them.

Yes

Yes

## 5.MP6: Attend to precision.

Mathematical Practice Standards Mathematical Practice Standards Mathematical Practice Standards Mathematical Practice Standards

## 5.MP7: Look for and make use of structure.

5.MP8: Look for and express regularity in repeated reasoning. 5.MP7: Look for and make use of structure.

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14

Strand

CA Math Standard 3.3 Develop generalizations of the results obtained and apply them in other circumstances.

## Domain Mathematical Practice Standards

Common Core Standard (CCS) 5.MP7: Look for and make use of structure.

Alignment

## Comments in Reference to CCS

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15

Domain Operations and Algebraic Thinking Operations and Algebraic Thinking Number and Operations in Base Ten Number and Operations in Base Ten Number and Operations Fractions Common Core Standard 5.OA.1: Use parentheses brackets, or braces in numerical expressions, and evaluate these expressions with these symbols. 5.OA.2: Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. 5.NBT.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 10. 5.NBT.5: Fluently multiply multi-digit whole numbers using the standard algorithm. 5.NF.2: Solve word problems involving addition and subtraction of fractions referring to the same who, 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. 5.NF.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. 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. 5.NF.4b: Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriated 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. 5.NF.5: Interpret multiplications 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. b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater that the 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 = (n x a)/ (n x b) to the effect multiplying a/b by 1. 5.NF.7a: Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. 5.NF.7b: Interpret division of a whole number by a unit fraction, and compute such quotients. Found in CA Math Standards Yes Grade 4 AF1.2 Yes Grade 7 AF1.1 Partial Grade 7 NS1.1

## Number and Operations Fractions Number and Operations -

16

Domain Fractions

Common Core Standard For example, created a story context for 4 divided by (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain than 4 divided by (1/5) = 20 because 20 x (1/5) = 4. 5.MD.1: Convert among different-sized standard measurement units with a given measurement system (e.g., convert 5 cm to 0.05 m) and use conversions in solving multi-step, real world problems.

## Measurement and Data

17

Strand 1.0 Number Sense CA Math Standard 1.2 Interpret percents as a part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number. Found in Common Core Standards Yes 6.RP.3a: Make table of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use table to compare ratios. 6.RP.3b: solve unit rate problems including those involving unit pricing and constant speed. 6.RP.3c: Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity): solve problems involving finding the whole, given a part and the percent. 6.RP.3d: Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. Yes 6.EE.1: Write and evaluate numerical expressions involving wholenumber exponents. Yes 4.OA.4: Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range is prime or composite. Yes 6.NS.6: Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. 6.NS.6a: Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3 and then 0 is its own opposite. 6.NS.6b: Write interpret, and explain statements of order for rational number in real-world contexts. 6.NS.6c: Find and position integers and other rational numbers on a
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## 1.0 Number Sense 1.0 Number Sense

1.3 Understand and compute positive integer powers of nonnegative integers; compute examples as repeated multiplication. 1.4 Determine the prime factors of all numbers through 50 and write the numbers as the product of their prime factors by using exponents to show multiples of a factor (e.g., 24 = 2 x 2 x 2 x 3 = 23 x 3). 1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers.

## 1.0 Number Sense

Strand

CA Math Standard

Found in Common Core Standards horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on the coordinate plane. Yes 6.EE.9: Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and table, and relate these to the equation. Yes 6.EE.2: Write, read, and evaluate expressions in which letters stand for numbers. Yes 6.EE.3: Apply the properties of operations to generate equivalent expressions. Yes 6.G.1: Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangle or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. Yes 6.G.4: Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real world and mathematical problems. Yes 8.G.5: Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. Yes CCS does not specify drawing two-dimensional views of threedimensional objects. 7.G.3: Describe the two-dimensional figures that result from slicing threedimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

1.0 Algebra

1.1 Use information taken from a graph or equation to answer questions about a problem situation.

1.0 Algebra

1.2 Use a letter to represent an unknown number; write and evaluate simple algebraic expressions in one variable by substitution. 1.3 Know and use the distributive property in equations and expressions with variables. 1.1 Derive and use the formula for the area of a triangle and of a parallelogram by comparing it with the formula for the area of a rectangle (i.e., two of the same triangles make a parallelogram with twice the area; a parallelogram is compared with a rectangle of the same area by cutting and pasting a right triangle on the parallelogram). 1.2 Construct a cube and rectangular box from two-dimensional patterns and use these patterns to compute the surface area for these objects. 2.2 Know that the sum of the angles of any triangle is 180 and the sum of the angles of any quadrilateral is 360 and use this information to solve problems. 2.3 Visualize and draw two-dimensional views of threedimensional objects made from rectangular solids.

1.0 Algebra

19

## Strand 1.2 Statistics, Data Analysis, and Probability

CA Math Standard 1.1 Know the concepts of mean, median, and mode; compute and compare simple examples to show that they may differ.

Found in Common Core Standards Yes 6.SP.5: Summarize numerical data sets in relation to their context, such as by: 6.SP.5a: Reporting the number of observations. 6.SP.5b: Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. 6.SP.5c: Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. 6.SP.5d: Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. Yes 6.SP.4: Display numerical data in plots on a number line, including dot plots, histograms, and box plots. CCS does not mention circle graph. Yes 7.RP.2: Recognize and represent proportional relationships between quantities. 7.RP.2a: Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. 7.RP.2b: Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. 7.RP.2c: Represent proportional relationships by equations. 7.RP.2d: Explain that a point (x,y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0,0) and (1,r) where r is the unit rate.

## 1.2 Statistics, Data Analysis, and Probability

1.2 Organize and display single-variable data in appropriate graphs and representations (e.g., histogram, circle graphs) and explain which types of graphs are appropriate for various data sets. 1.4 Identify ordered pairs of data from a graph and interpret the meaning of the data in terms of the situation depicted by the graph.
The text passage is a detailed analysis comparing California Mathematics standards to Common Core standards for Grade 5. It lists various strands, standards, and their alignment, often including comments on the alignment.
no_question

the passage does not contain any mathematical questions or problems to be solved. it is a comparative analysis of educational standards, providing information on how specific california math standards align with common core standards.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
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# Analysis of California Mathematics standards to Common Core standards-Grade 5

Strand Strand Number Sense 1.0 Number Sense CA Math Standard CA Math Standard 1.0 Students compute with very large and very small numbers, positive integers, decimals, and fractions and understand the relationship between decimals, fractions, and percents. They understand the relative magnitudes of numbers. 1.1 Estimate, round, and manipulate very large (e.g., millions) and very small (e.g., thousandths) numbers. Number Operations in Base Ten 5.NBT: (Cluster Statement) Perform operations with multi-digit whole numbers and with decimals to hundredths. Partial CCS Cluster Statement does not mention integers and percents. Domain Common Core Standard (CCS) Alignment Comments in Reference to CCS

## Number Operations in Base Ten

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. 5.NBT.3: Read, write, and compare decimals to thousandths. 5.NBT.3a: Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 x 100 + 4 x 10 + 7 x 1 + 3 x (1/10) + 9 x (1/100) + 2 x (1/1000). 5.NBT.3b: Compare two decimals to thousandths based on meaning of the digits in each place, using >, =, and < symbols to record the results of comparisons. 5.NBT.4: Use place value understanding to round decimals to any place.

Partial

CCS does not reference estimation directly. In the Mathematical Practice standards, CCS implies a thorough understanding of the concepts so students could develop strong estimation skills as a by-product of the depth of understanding. 6.RP.3: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

1.2 Interpret percents as a part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number.
Analysis by Sacramento County Office of Education June 22, 2010 - Addendum A Sacramento County Office of Education, California, 2010. All rights reserved.

No

6.RP.3a: Make table of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use table to compare
1

Strand

CA Math Standard

Domain

## Common Core Standard (CCS)

Alignment

Comments in Reference to CCS ratios. 6.RP.3b: solve unit rate problems including those involving unit pricing and constant speed. 6.RP.3c: Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity): solve problems involving finding the whole, given a part and the percent. 6.RP.3d: Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. 6.EE.1: Write and evaluate numerical expressions involving whole-number exponents.

1.3 Understand and compute positive integer powers of nonnegative integers; compute examples as repeated multiplication. 1.4 Determine the prime factors of all numbers through 50 and write the numbers as the product of their prime factors by using exponents to show multiples of a factor (e.g., 24 = 2 x 2 x 2 x 3 = 23 x 3).

No

No

1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers.

No

4.OA.4: Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range is prime or composite. 6.NS.6: Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in
2

Analysis by Sacramento County Office of Education June 22, 2010 - Addendum A Sacramento County Office of Education, California, 2010. All rights reserved.

Strand

CA Math Standard

Domain

## Common Core Standard (CCS)

Alignment

Comments in Reference to CCS the plane with negative number coordinates. 6.NS.6a: Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3 and then 0 is its own opposite. 6.NS.6b: Write interpret, and explain statements of order for rational number in real-world contexts. 6.NS.6c: Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on the coordinate plane.

## 2.0 Number Sense

2.0 Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals.

Number and Operations Fractions Number and Operations Base Ten Number and Operations in Base Ten

5.NF: (Cluster Statement) Use equivalent fractions as a strategy to add and subtract fractions. Apply and extend previous understanding of multiplication and division to multiply and divide fractions. 5.NBT: (Cluster Statement) Perform Operations with multi-digit whole numbers and with decimals to hundredths. 5.NBT.7: Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; related the strategy to a written method and explain the reasoning used.

Yes

2.1 Add, subtract, multiply, and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify the reasonableness of the results.

Partial

6.NS.3: Fluently add, subtract, multiply and divide multi-digit decimals using the standard algorithm for each operation. 7.NS.1: Apply and extend previous understandings of
3

Analysis by Sacramento County Office of Education June 22, 2010 - Addendum A Sacramento County Office of Education, California, 2010. All rights reserved.

Strand

CA Math Standard

Domain

## Common Core Standard (CCS)

Alignment

Comments in Reference to CCS addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

2.2 Demonstrate proficiency with division, including division with positive decimals and long division with multidigit divisors.

## Number Operations in Base Ten

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.NBT.7: Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. 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.

Yes

2.3 Solve simple problems, including ones arising in concrete situations, involving the addition and subtraction of fractions and mixed numbers (like and unlike denominators of 20 or less), and express answers in the simplest form. 2.4 Understand the concept of multiplication and division of fractions.

Yes

## Number and Operations Fractions

5.NF.4: Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. 5.NF.4a: Interpret the product (a/b) ! q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a ! q b. For example, use a visual fraction model to show (2/3) ! 4 = 8/3, and create a story context for this equation. Do the same with (2/3) ! (4/5) = 8/15. (In general, (a/b) ! (c/d) = ac/bd.)

Yes

*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.

Analysis by Sacramento County Office of Education June 22, 2010 - Addendum A Sacramento County Office of Education, California, 2010. All rights reserved.

Strand

CA Math Standard

Domain

Common Core Standard (CCS) 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. 5.NF.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. 5.NF.7: Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions*. 5.NF.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. 5.NF.7: Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. 5.NF.7a: Interpret division of a unit fraction by a nonzero 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) ! 4 = 1/3. 5.NF.7b: 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 ! (1/5) = 4.

Alignment

## Comments in Reference to CCS

2.5 Compute and perform simple multiplication and division of fractions and apply these procedures to solving problems.

## Number and Operations Fractions

Yes

Analysis by Sacramento County Office of Education June 22, 2010 - Addendum A Sacramento County Office of Education, California, 2010. All rights reserved.

Strand

CA Math Standard

Domain

Common Core Standard (CCS) 5.NF.7c: 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.

Alignment

## Strand Algebra Functions 1.0 Algebra Functions

CA Math Standard 1.0 Students use variables in simple expressions, compute the value of the expression for specific values of the variable, and plot and interpret the results. 1.1 Use information taken from a graph or equation to answer questions about a problem situation. Operations and Algebraic Thinking 5.OA: Write and interpret numerical expressions. Partial 6.EE: (Cluster Statement) Apply and extend previous understandings of arithmetic to algebraic expressions. 6.EE.9: Use variables to represent two quantities in a realworld problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and table, and relate these to the equation. 6.EE.2: Write, read, and evaluate expressions in which letters stand for numbers.

No

1.2 Use a letter to represent an unknown number; write and evaluate simple algebraic expressions in one variable by substitution. 1.3 Know and use the distributive property in equations and expressions with variables. 1.4 Identify and graph ordered pairs in the four quadrants of the coordinate plane.

No

No

Geometry

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 plan located by using an ordered pair of numbers, called its coordinates. Understand that the first number

Partial

6.EE.3: Apply the properties of operations to generate equivalent expressions. 6.NS.8: Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances
6

Analysis by Sacramento County Office of Education June 22, 2010 - Addendum A Sacramento County Office of Education, California, 2010. All rights reserved.

Strand

CA Math Standard

Domain

1.5 Solve problems involving linear functions with integer values; write the equation; and graph the resulting ordered pairs of integers on a grid.

Common Core Standard (CCS) 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 xcoordinate, y-axis and y-coordinate). 5.G.2: 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. 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.

Alignment

Comments in Reference to CCS between points with the same first coordinate or the same second coordinate.

Partial

6.EE.9: Use variable to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variable using graphs and table and relate these to the equation.

## Strand Measurement and Geometry 1.0 Measurement and Geometry

CA Math Standard 1.0 Students understand and compute the volumes and areas of simple objects. 1.1 Derive and use the formula for the area of a triangle and of a parallelogram by comparing it with the formula for the area of a rectangle (i.e., two of the same triangles make a parallelogram with twice the area; a parallelogram is compared with a rectangle of the same area by cutting and pasting a right triangle on the parallelogram).
Measurement

and Data

5.MD: (Cluster Statement) Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

Yes

No

6.G.1: Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangle or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

Analysis by Sacramento County Office of Education June 22, 2010 - Addendum A Sacramento County Office of Education, California, 2010. All rights reserved.

Strand

CA Math Standard 1.2 Construct a cube and rectangular box from twodimensional patterns and use these patterns to compute the surface area for these objects.

Domain

## Common Core Standard (CCS)

Alignment No

Comments in Reference to CCS 6.G.4: Represent threedimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real world and mathematical problems.

1.3 Understand the concept of volume and use the appropriate units in common measuring systems (i.e., cubic centimeter [cm3], cubic meter [m3], cubic inch [in3], cubic yard [yd3]) to compute the volume of rectangular solids.

Measurement

and Data

5.MD.3: Recognize volume as an attribute of solid figures and understand concepts of volume measurement. 5.MD.3a: 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. 5.MD.3b: 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. 5.MD.4: Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. 5.MD.5: Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. 5.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 edges lengths, equivalently by multiplying the heights by the area of the base. Represent threefold wholenumber products as volumes e.g., to represent the associative property of multiplication. 5.MD.5b: Apply the formulas V = l x w x h and V = b x h for rectangular prisms to find volumes of right rectangular with whole number edge lengths in the

Yes

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Strand

CA Math Standard

Domain

Common Core Standard (CCS) context of solving real world and mathematical problems. 5.MD.5c: Recognize volume as additive. Find volumes of solid figures composed of two nonoverlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems. 5.MD.4: Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

Alignment

## Comments in Reference to CCS

1.4 Differentiate between, and use appropriate units of measures for, two-and threedimensional objects (i.e., find the perimeter, area, volume).

Measurement

and Data

Partial

4.MD.2: Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. 4.MD.3: Apply the area and perimeter formulas for rectangles in real world and mathematical problems. CCS does not mention differentiate between appropriate units of measure.

## 2.0 Measurement and Geometry

2.0 Students identify, describe, and classify the properties of, and the relationships between, plane and solid geometric figures.

Geometry

5.G: (Cluster Statement) Classify two-dimensional figures into categories based on their properties. 5.G.3: Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. 5.G.4: Classify two-dimensional figures in a hierarchy based on properties.

Yes

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Strand

CA Math Standard 2.1 Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools (e.g., straightedge, ruler, compass, protractor, drawing software).

Domain
Measurement

and Data

Common Core Standard (CCS) 4.MD.5: Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement: 4.MD.5a: An angle is measure with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a one-degree angle and can be used to measure angles. 4.MD.5b: An angle that turns through n one-degree angles is said to have an angle measure of n degrees. 4.MD.6: Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure. 4.MD.1: Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

Alignment Partial

Comments in Reference to CCS 7.G: (Cluster Statement) Draw, construct and describe geometrical figures and describe the relationships between them.

2.2 Know that the sum of the angles of any triangle is 180 and the sum of the angles of any quadrilateral is 360 and use this information to solve problems.

No

2.3 Visualize and draw twodimensional views of threedimensional objects made from rectangular solids.

No

8.G.5: Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. 7.G.3: Describe the twodimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. CCS does not specify drawing two-dimensional views of three10

Analysis by Sacramento County Office of Education June 22, 2010 - Addendum A Sacramento County Office of Education, California, 2010. All rights reserved.

Strand Strand Statistics, Data Analysis, and Probability 1.0 Statistics, Data Analysis, and Probability

CA Math Standard

Domain

Alignment

## Comments in Reference to CCS dimensional objects.

CA Math Standard 1.0 Students display, analyze, compare, and interpret different data sets, including data sets of different sizes. 1.1 Know the concepts of mean, median, and mode; compute and compare simple examples to show that they may differ.
Measurement

and Data

## 5.MD: (Cluster Statement) Represent and interpret data.

Yes

No

6.SP.5: Summarize numerical data sets in relation to their context, such as by: 6.SP.5a: Reporting the number of observations. 6.SP.5b: Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. 6.SP.5c: Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. 6.SP.5d: Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

Analysis by Sacramento County Office of Education June 22, 2010 - Addendum A Sacramento County Office of Education, California, 2010. All rights reserved.

11

Strand

CA Math Standard 1.2 Organize and display singlevariable data in appropriate graphs and representations (e.g., histogram, circle graphs) and explain which types of graphs are appropriate for various data sets. 1.3 Use fractions and percentages to compare data sets of different sizes. 1.4 Identify ordered pairs of data from a graph and interpret the meaning of the data in terms of the situation depicted by the graph.

Domain

## Common Core Standard (CCS)

Alignment No

Comments in Reference to CCS 6.SP.4: Display numerical data in plots on a number line, including dot plots, histograms, and box plots. CCS does not mention circle graph.

Measurement

and Data

5.MD.2: Make a line plot to display a data set of measurements in fractions of a unit (1/2, ", 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots.

Partial

## CCS does not specify using percentages to compare data.

7.RP.2: Recognize and represent proportional relationships between quantities. 7.RP.2a: Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. 7.RP.2b: Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. 7.RP.2c: Represent proportional relationships by equations. 7.RP.2d: Explain that a point (x,y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0,0) and (1,r) where r is the unit rate.

No

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12

Strand

CA Math Standard 1.5 Know how to write ordered pairs correctly; for example, ( x, y ).

Domain Geometry

Common Core Standard (CCS) 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 plan 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 xcoordinate, y-axis and y-coordinate).

Alignment Yes

## Strand Mathematical Reasoning 1.0 Mathematical Reasoning

CA Math Standard 1.0 Students make decisions about how to approach problems: 1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, sequencing and prioritizing information, and observing patterns. 1.2 Determine when and how to break a problem into simpler parts. Mathematical Practice Standards Mathematical Practice Standards 5.MP1: Make sense of problems and persevere in solving them. 5.MP1: Make sense of problems and persevere in solving them. Yes

Yes

## Mathematical Practice Standards Mathematical Practice Standards Mathematical Practice Standards

5.MP1: Make sense of problems and persevere in solving them. 5.MP3: Construct viable arguments and critique the reasoning of others. 5.MP1: Make sense of problems and persevere in solving them.

Yes

## 2.0 Mathematical Reasoning

2.0 Students use strategies, skills, and concepts in finding solutions. 2.1 Use estimation to verify the reasonableness of calculated results.

Yes

Yes

CCS does not reference estimation directly. In the Mathematical Practice standards, CCS implies a thorough understanding of the concepts so
13

Analysis by Sacramento County Office of Education June 22, 2010 - Addendum A Sacramento County Office of Education, California, 2010. All rights reserved.

Strand

CA Math Standard

Domain

## Common Core Standard (CCS)

Alignment

Comments in Reference to CCS students could develop strong estimation skills as a byproduct of the depth of understanding.

2.2 Apply strategies and results from simpler problems to more complex problems. 2.3 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. 2.4 Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work. 2.5 Indicate the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy. 2.6 Make precise calculations and check the validity of the results from the context of the problem. 3.0 Students move beyond a particular problem by generalizing to other situations. 3.1 Evaluate the reasonableness of the solution in the context of the original situation. 3.2 Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems.

## Mathematical Practice Standards Mathematical Practice Standards

5.MP1: Make sense of problems and persevere in solving them. 5.MP1: Make sense of problems and persevere in solving them.

Yes

Yes

## 5.MP6: Attend to precision.

Mathematical Practice Standards Mathematical Practice Standards Mathematical Practice Standards Mathematical Practice Standards

## 5.MP7: Look for and make use of structure.

5.MP8: Look for and express regularity in repeated reasoning. 5.MP7: Look for and make use of structure.

Analysis by Sacramento County Office of Education June 22, 2010 - Addendum A Sacramento County Office of Education, California, 2010. All rights reserved.

14

Strand

CA Math Standard 3.3 Develop generalizations of the results obtained and apply them in other circumstances.

## Domain Mathematical Practice Standards

Common Core Standard (CCS) 5.MP7: Look for and make use of structure.

Alignment

## Comments in Reference to CCS

Analysis by Sacramento County Office of Education June 22, 2010 - Addendum A Sacramento County Office of Education, California, 2010. All rights reserved.

15

Domain Operations and Algebraic Thinking Operations and Algebraic Thinking Number and Operations in Base Ten Number and Operations in Base Ten Number and Operations Fractions Common Core Standard 5.OA.1: Use parentheses brackets, or braces in numerical expressions, and evaluate these expressions with these symbols. 5.OA.2: Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. 5.NBT.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 10. 5.NBT.5: Fluently multiply multi-digit whole numbers using the standard algorithm. 5.NF.2: Solve word problems involving addition and subtraction of fractions referring to the same who, 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. 5.NF.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. 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. 5.NF.4b: Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriated 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. 5.NF.5: Interpret multiplications 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. b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater that the 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 = (n x a)/ (n x b) to the effect multiplying a/b by 1. 5.NF.7a: Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. 5.NF.7b: Interpret division of a whole number by a unit fraction, and compute such quotients. Found in CA Math Standards Yes Grade 4 AF1.2 Yes Grade 7 AF1.1 Partial Grade 7 NS1.1

## Number and Operations Fractions Number and Operations -

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Domain Fractions

Common Core Standard For example, created a story context for 4 divided by (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain than 4 divided by (1/5) = 20 because 20 x (1/5) = 4. 5.MD.1: Convert among different-sized standard measurement units with a given measurement system (e.g., convert 5 cm to 0.05 m) and use conversions in solving multi-step, real world problems.

## Measurement and Data

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Strand 1.0 Number Sense CA Math Standard 1.2 Interpret percents as a part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number. Found in Common Core Standards Yes 6.RP.3a: Make table of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use table to compare ratios. 6.RP.3b: solve unit rate problems including those involving unit pricing and constant speed. 6.RP.3c: Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity): solve problems involving finding the whole, given a part and the percent. 6.RP.3d: Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. Yes 6.EE.1: Write and evaluate numerical expressions involving wholenumber exponents. Yes 4.OA.4: Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range is prime or composite. Yes 6.NS.6: Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. 6.NS.6a: Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3 and then 0 is its own opposite. 6.NS.6b: Write interpret, and explain statements of order for rational number in real-world contexts. 6.NS.6c: Find and position integers and other rational numbers on a
Analysis by Sacramento County Office of Education June 22, 2010 Addendum C Sacramento County Office of Education, California, 2010. All rights reserved. 18

## 1.0 Number Sense 1.0 Number Sense

1.3 Understand and compute positive integer powers of nonnegative integers; compute examples as repeated multiplication. 1.4 Determine the prime factors of all numbers through 50 and write the numbers as the product of their prime factors by using exponents to show multiples of a factor (e.g., 24 = 2 x 2 x 2 x 3 = 23 x 3). 1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers.

## 1.0 Number Sense

Strand

CA Math Standard

Found in Common Core Standards horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on the coordinate plane. Yes 6.EE.9: Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and table, and relate these to the equation. Yes 6.EE.2: Write, read, and evaluate expressions in which letters stand for numbers. Yes 6.EE.3: Apply the properties of operations to generate equivalent expressions. Yes 6.G.1: Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangle or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. Yes 6.G.4: Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real world and mathematical problems. Yes 8.G.5: Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. Yes CCS does not specify drawing two-dimensional views of threedimensional objects. 7.G.3: Describe the two-dimensional figures that result from slicing threedimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

1.0 Algebra

1.1 Use information taken from a graph or equation to answer questions about a problem situation.

1.0 Algebra

1.2 Use a letter to represent an unknown number; write and evaluate simple algebraic expressions in one variable by substitution. 1.3 Know and use the distributive property in equations and expressions with variables. 1.1 Derive and use the formula for the area of a triangle and of a parallelogram by comparing it with the formula for the area of a rectangle (i.e., two of the same triangles make a parallelogram with twice the area; a parallelogram is compared with a rectangle of the same area by cutting and pasting a right triangle on the parallelogram). 1.2 Construct a cube and rectangular box from two-dimensional patterns and use these patterns to compute the surface area for these objects. 2.2 Know that the sum of the angles of any triangle is 180 and the sum of the angles of any quadrilateral is 360 and use this information to solve problems. 2.3 Visualize and draw two-dimensional views of threedimensional objects made from rectangular solids.

1.0 Algebra

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## Strand 1.2 Statistics, Data Analysis, and Probability

CA Math Standard 1.1 Know the concepts of mean, median, and mode; compute and compare simple examples to show that they may differ.

Found in Common Core Standards Yes 6.SP.5: Summarize numerical data sets in relation to their context, such as by: 6.SP.5a: Reporting the number of observations. 6.SP.5b: Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. 6.SP.5c: Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. 6.SP.5d: Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. Yes 6.SP.4: Display numerical data in plots on a number line, including dot plots, histograms, and box plots. CCS does not mention circle graph. Yes 7.RP.2: Recognize and represent proportional relationships between quantities. 7.RP.2a: Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. 7.RP.2b: Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. 7.RP.2c: Represent proportional relationships by equations. 7.RP.2d: Explain that a point (x,y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0,0) and (1,r) where r is the unit rate.

## 1.2 Statistics, Data Analysis, and Probability

1.2 Organize and display single-variable data in appropriate graphs and representations (e.g., histogram, circle graphs) and explain which types of graphs are appropriate for various data sets. 1.4 Identify ordered pairs of data from a graph and interpret the meaning of the data in terms of the situation depicted by the graph.
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The text passage is a detailed analysis comparing California Mathematics standards to Common Core standards for Grade 5. It lists various strands, standards, and their alignment, often including comments on the alignment.

TYPE: NO_QUESTION

The passage does not contain any mathematical questions or problems to be solved. It is a comparative analysis of educational standards, providing information on how specific California Math Standards align with Common Core Standards.