Minnesota K–12 Academic Standards in Mathematics...

Minnesota K–12 Academic Standards in Mathematics Correlation to Eureka Math

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Grade 6 February 2016



Grade 6 Mathematics Many of the Grade 6 Minnesota K–12 Academic Standards in Mathematics are fully covered by the Grade 6 Eureka Math curriculum. The primary areas where the Grade 6 Minnesota K–12 Academic Standards in Mathematics and Eureka Math do not align are in the strands of Geometry and Measurement and Data Analysis and Probability. Standards from these strands will require use of Eureka Math content from other grade levels, along with supplemental materials to illustrate how a model of triangles can show the sum of the interior angles of a triangle is 180°. A detailed analysis of alignment is provided in the table below. With strategic placement of supplemental materials, Eureka Math can ensure students are successful in achieving the proficiencies of the Minnesota K–12 Academic Standards in Mathematics while benefiting from the coherence and rigor of Eureka Math.

Indicators

Green indicates the Minnesota standard is fully addressed in Eureka Math.

Yellow indicates the Minnesota standard may not be completely addressed in Eureka Math.

Red indicates the Minnesota standard is not addressed in Eureka Math.

Blue indicates there is a discrepancy between the grade level at which this standard is addressed in the Minnesota standards and in Eureka Math



Strand Standard Aligned Components of Eureka Math

Number and Operation

6.1.1.1 Read, write, represent and compare positive rational numbers expressed as fractions, decimals, percents and ratios; write positive integers as products of factors; use these representations in real-world and mathematical situations.

Benchmark: Locate positive rational numbers on a number line and plot pairs of positive rational numbers on a coordinate grid.

G5 M6 Topic A: Coordinate Systems

G6 M3 Topic C: Rational Numbers and the Coordinate Plane

Note: Grade 5 students locate and plot positive rational numbers on the coordinate grid. In Grade 6, the learning is extended to include locating and plotting positive and negative rational numbers in all four quadrants. The phrase rational number is first used in Grade 6.


Benchmark: Compare positive rational numbers represented in various forms. Use the symbols <, = and >.

For example: 12

> 0.36.

G6 M3 Lesson 10: Writing and Interpreting Inequality Statements Involving Rational Numbers





Benchmark: Understand that percent represents parts out of 100 and ratios to 100.

For example: 75% corresponds to the ratio 75 to 100, which is equivalent to the ratio 3 to 4.

G6 M1 Lesson 24: Percent and Rates per 100


Benchmark: Determine equivalences among fractions, decimals and percents; select among these representations to solve problems.

For example: If a woman making $25 an hour gets a 10% raise, she will make an additional $2.50 an hour, because $2.50 is 1

10 or 10% of $25.

G6 M1 Topic D: Percent





Benchmark: Factor whole numbers; express a whole number as a product of prime factors with exponents.

For example: 24 = 23 × 3.

G4 M3 Topic F: Reasoning with Divisibility

G6 M2 Lesson 18: Least Common Multiple and Greatest Common Factor

Note: Grade 4 students factor whole numbers, expressing a whole number as a product of two factors.


Benchmark: Determine greatest common factors and least common multiples. Use common factors and common multiples to calculate with fractions and find equivalent fractions.

For example: Factor the numerator and denominator of a fraction to determine an equivalent fraction.

G4 M5: Fraction Equivalence, Ordering, and Operations (Topics A–D)

G5 M3: Addition and Subtraction of Fractions

G6 M2 Lesson 18: Least Common Multiple and Greatest Common Factor

Note: Although finding the greatest common factor and least common multiple is addressed in Grade 6, students rename fractions and determine equivalent fractions using common factors and multiples in earlier grades.





Benchmark: Convert between equivalent representations of positive rational numbers.

For example: Express 107

as 7+37

= 77

+ 37

= 1 37.

G4 M5: Fraction Equivalence, Ordering, and Operations

G5 M3: Addition and Subtraction of Fractions

Note: The phrase rational number is not introduced until Grade 6.

6.1.2.1 Understand the concept of ratio and its relationship to fractions and to the multiplication and division of whole numbers. Use ratios to solve real-world and mathematical problems.

Benchmark: Identify and use ratios to compare quantities; understand that comparing quantities using ratios is not the same as comparing quantities using subtraction.

For example: In a classroom with 15 boys and 10 girls, compare the numbers by subtracting (there are 5 more boys than girls) or by dividing (there are 1.5 times as many boys as girls). The comparison using division may be expressed as a ratio of boys to girls (3 to 2 or 3: 2 or 1.5 to 1).

G6 M1 Lessons 1–2: Ratios

G6 M1 Lesson 10: The Structure of Ratio Tables—Additive and Multiplicative





Benchmark: Apply the relationship between ratios, equivalent fractions and percents to solve problems in various contexts, including those involving mixtures and concentrations.

For example: If 5 cups of trail mix contains 2 cups of raisins, the ratio of raisins to trail mix is 2 to 5. This ratio corresponds to the fact that the raisins are 2

5 of the total, or 40% of the total. And if one trail mix

consists of 2 parts peanuts to 3 parts raisins, and another consists of 4 parts peanuts to 8 parts raisins, then the first mixture has a higher concentration of peanuts.

G6 M1 Lessons 5–6: Solving Problems by Finding Equivalent Ratios

G6 M1 Lesson 7: Associated Ratios and the Value of a Ratio

G6 M1 Lesson 8: Equivalent Ratios Defined Through the Value of a Ratio

G6 M1 Lesson 9: Tables of Equivalent Ratios

G6 M1 Lesson 11: Comparing Ratios Using Ratio Tables


Benchmark: Determine the rate for ratios of quantities with different units.

For example: 60 miles for every 3 hours is equivalent to 20 miles for every one hour (20 mph.)

G6 M1 Lesson 16: From Ratios to Rates





Benchmark: Use reasoning about multiplication and division to solve ratio and rate problems.

For example: If 5 items cost $3.75, and all items are the same price, then 1 item costs 75 cents, so 12 items cost $9.00.

G6 M1: Ratios and Unit Rates

6.1.3.1 Multiply and divide decimals, fractions and mixed numbers; solve real-world and mathematical problems using arithmetic with positive rational numbers.

Benchmark: Multiply and divide decimals and fractions, using efficient and generalizable procedures, including standard algorithms.

G5 M2 Topic C: Decimal Multi-Digit Multiplication

G5 M2 Topic G: Partial Quotients and Multi-Digit Decimal Division

G5 M4: Multiplication and Division of Fractions and Decimal Fractions

G6 M2 Topic A: Dividing Fractions by Fractions

G6 M2 Topic B: Multi-Digit Decimal Operations—Adding, Subtracting, and Multiplying

G6 M2 Topic C: Dividing Whole Numbers and Decimals

Note: The work described by this benchmark begins in G5 M2 and is extended by G6 M2.





Benchmark: Use the meanings of fractions, multiplication, division and the inverse relationship between multiplication and division to make sense of procedures for multiplying and dividing fractions.

For example: Just as 124

= 3 means 12 = 3 × 4, 23

÷ 45

= 56 means

56

× 45

= 23.


G6 M2 Topic A: Dividing Fractions by Fractions

Note: Grade 6 focuses on dividing fractions. It is an extension of the work done with multiplying and dividing fractions in Grade 5.


Benchmark: Calculate the percent of a number and determine what percent one number is of another number to solve problems in various contexts.

For example: If John has $45 and spends $15, what percent of his money did he keep?

G6 M1 Topic D: Percent





Benchmark: Solve real-world and mathematical problems requiring arithmetic with decimals, fractions and mixed numbers.


G6 M2: Arithmetic Operations Including Division of Fractions


Benchmark: Estimate solutions to problems with whole numbers, fractions and decimals and use the estimates to assess the reasonableness of results in the context of the problem.

For example: The sum 13

+ 0.25 can be estimated to be between 12

and 1, and this estimate can be used to check the result of a more detailed calculation.

G5 M2: Multi-Digit Whole Number and Decimal Fraction Operations


G6 M2: Arithmetic Operations Including Division of Fractions

Note: Estimating the product or quotient with whole numbers is primarily addressed in Grade 5. It is extended by Grade 6.




Algebra 6.2.1.1 Recognize and represent relationships between varying quantities; translate from one representation to another; use patterns, tables, graphs and rules to solve real-world and mathematical problems.

Benchmark: Understand that a variable can be used to represent a quantity that can change, often in relationship to another changing quantity. Use variables in various contexts.

For example: If a student earns $7 an hour in a job, the amount of money earned can be represented by a variable and is related to the number of hours worked, which also can be represented by a variable.

G6 M4 Topic C: Replacing Letters and Numbers

6.2.1.2 Recognize and represent relationships between varying quantities; translate from one representation to another; use patterns, tables, graphs and rules to solve real-world and mathematical problems.

Benchmark: Represent the relationship between two varying quantities with function rules, graphs and tables; translate between any two of these representations.

For example: Describe the terms in the sequence of perfect squares 𝑡𝑡 = 1, 4, 9, 16, ... by using the rule 𝑡𝑡 = 𝑛𝑛2 for 𝑛𝑛 = 1, 2, 3, 4, ...

G6 M1 Lesson 13: From Ratio Tables to Equations Using the Value of a Ratio

G6 M1 Lesson 14: From Ratio Tables, Equations, and Double Number Line Diagrams to Plots on the Coordinate Plane

G6 M4 Lesson 31: Problems in Mathematical Terms

G6 M4 Lesson 32: Multi-Step Problems in the Real World




6.2.2.1 Use properties of arithmetic to generate equivalent numerical expressions and evaluate expressions involving positive rational numbers.

Benchmark: Apply the associative, commutative and distributive properties and order of operations to generate equivalent expressions and to solve problems involving positive rational numbers.

For example: 3215

× 56

= 32×515×6

= 2×16×53×5×3×2

= 169

× 22

× 55

= 169

.

Another example: Use the distributive law to write:

12

+ 13�92− 15

8� = 1

2+ 1

3× 9

2− 1

3× 15

18= 1

2+ 3

2− 5

8= 1 3

8.

G6 M4 Topic A: Relationships of the Operations

G6 M4 Lesson 6: The Order of Operations

G6 M4 Topic D: Expanding, Factoring, and Distributing Expressions

6.2.3.1 Understand and interpret equations and inequalities involving variables and positive rational numbers. Use equations and inequalities to represent real-world and mathematical problems; use the idea of maintaining equality to solve equations. Interpret solutions in the original context.

Benchmark: Represent real-world or mathematical situations using equations and inequalities involving variables and positive rational numbers.

For example: The number of miles 𝑚𝑚 in a 𝑘𝑘 kilometer race is represented by the equation 𝑚𝑚 = 0.62 𝑘𝑘.

G6 M4 Topic G: Solving Equations

G6 M4 Topic H: Applications of Equations




6.2.3.2 Understand and interpret equations and inequalities involving variables and positive rational numbers. Use equations and inequalities to represent real-world and mathematical problems; use the idea of maintaining equality to solve equations. Interpret solutions in the original context.

Benchmark: Solve equations involving positive rational numbers using number sense, properties of arithmetic and the idea of maintaining equality on both sides of the equation. Interpret a solution in the original context and assess the reasonableness of results.

For example: A cellular phone company charges $0.12 per minute. If the bill was $11.40 in April, how many minutes were used?

G6 M4 Topic G: Solving Equations

Geometry and Measurement

6.3.1.1 Calculate perimeter, area, surface area and volume of two- and three-dimensional figures to solve real-world and mathematical problems.

Benchmark: Calculate the surface area and volume of prisms and use appropriate units, such as cm2 and cm3. Justify the formulas used. Justification may involve decomposition, nets or other models.

For example: The surface area of a triangular prism can be found by decomposing the surface into two triangles and three rectangles.

G5 M5 Topic A: Concepts of Volume

G5 M5 Topic B: Volume and the Operations of Multiplication and Addition

G6 M5 Topic C: Volume of Right Rectangular Prisms

G6 M5 Topic D: Nets and Surface Area

Note: Volume is introduced in Grade 5 through concrete exploration and, ultimately, the development of the formula for right rectangular prisms. This learning is extended in Grade 6 when students find the volume of right rectangular prisms with fractional edge lengths.




6.3.1.2 Calculate perimeter, area, surface area and volume of two- and three- dimensional figures to solve real-world and mathematical problems.

Benchmark: Calculate the area of quadrilaterals. Quadrilaterals include squares, rectangles, rhombuses, parallelograms, trapezoids and kites. When formulas are used, be able to explain why they are valid.

For example: The area of a kite is one-half the product of the lengths of the diagonals, and this can be justified by decomposing the kite into two triangles.

G3 M4: Multiplication and Area

G4 M3 Topic A: Multiplicative Comparison Word Problems

G5 M5 Topic C: Area of Rectangular Figures with Fractional Side Lengths

G6 M5 Topic A: Area of Triangles, Quadrilaterals, and Polygons

6.3.1.3 Calculate perimeter, area, surface area and volume of two- and three-dimensional figures to solve real-world and mathematical problems.

Benchmark: Estimate the perimeter and area of irregular figures on a grid when they cannot be decomposed into common figures and use correct units, such as cm and cm2.

G3 M7 Lesson 10: Decompose quadrilaterals to understand perimeter as the boundary of a shape.

G3 M7 Lesson 11: Tessellate to understand perimeter as the boundary of a shape.

G3 M7 Lesson 16: Use string to measure the perimeter of various circles to the nearest quarter inch.

G3 M7 Topic E: Problem Solving with Perimeter and Area

G7 M6 Lesson 22: Area Problems with Circular Regions

Note: The use of a grid will require supplemental materials.




6.3.2.1 Understand and use relationships between angles in geometric figures.

Benchmark: Solve problems using the relationships between the angles formed by intersecting lines.

For example: If two streets cross, forming four corners such that one of the corners forms an angle of 120°, determine the measures of the remaining three angles.

Another example: Recognize that pairs of interior and exterior angles in polygons have measures that sum to 180°.

G6 M4 Lesson 30: One-Step Problems in the Real World

G7 M3 Lessons 10–11: Angle Problems and Solving Equations

G7 M6 Topic A: Unknown Angles


Benchmark: Determine missing angle measures in a triangle using the fact that the sum of the interior angles of a triangle is 180°. Use models of triangles to illustrate this fact.

For example: Cut a triangle out of paper, tear off the corners and rearrange these corners to form a straight line.

Another example: Recognize that the measures of the two acute angles in a right triangle sum to 90°.

G8 M2 Lesson 13: Angle Sum of a Triangle





Benchmark: Develop and use formulas for the sums of the interior angles of polygons by decomposing them into triangles.

G4 M4 Lesson 9: Decompose angles using pattern blocks.

6.3.3.1 Choose appropriate units of measurement and use ratios to convert within measurement systems to solve real-world and mathematical problems.

Benchmark: Solve problems in various contexts involving conversion of weights, capacities, geometric measurements and times within measurement systems using appropriate units.

G4 M2: Unit Conversions and Problem Solving with Metric Measurement

G4 M7 Topic C: Investigation of Measurements Expressed as Mixed Numbers

G5 M1 Lesson 4: Use exponents to denote powers of 10 with application to metric conversions.

G5 M2 Topic D: Measurement Word Problems with Whole Number and Decimal Multiplication

G5 M2 Topic H: Measurement Word Problems with Multi-Digit Division

G5 M4 Topic E: Multiplication of a Fraction by a Fraction




6.3.3.2 Choose appropriate units of measurement and use ratios to convert within measurement systems to solve real-world and mathematical problems.

Benchmark: Estimate weights, capacities and geometric measurements using benchmarks in measurement systems with appropriate units.

For example: Estimate the height of a house by comparing to a 6-foot man standing nearby.

G2 M2: Addition and Subtraction of Length Units (Topics B–C)

G3 M2 Topic B: Measuring Weight and Liquid Volume in Metric Units

G5 M4 Lesson 1: Measure and compare pencil lengths to the nearest 1

2, 14, and 1

8 of an inch, and analyze the data

through line plots.

G5 M4 Lesson 9: Find a fraction of a measurement, and solve word problems.

G5 M4 Lesson 19: Convert measures involving whole numbers, and solve multi-step word problems.

Data Analysis and Probability

6.4.1.1 Use probabilities to solve real-world and mathematical problems; represent probabilities using fractions, decimals and percents.

Benchmark: Determine the sample space (set of possible outcomes) for a given experiment and determine which members of the sample space are related to certain events. Sample space may be determined by the use of tree diagrams, tables or pictorial representations.

For example: A 6 × 6 table with entries such as (1,1), (1,2), (1,3),…, (6,6) can be used to represent the sample space for the experiment of simultaneously rolling two number cubes.

G7 M5 Topic A: Calculating and Interpreting Probabilities





Benchmark: Determine the probability of an event using the ratio between the size of the event and the size of the sample space; represent probabilities as percents, fractions and decimals between 0 and 1 inclusive. Understand that probabilities measure likelihood.

For example: Each outcome for a balanced number cube has probability 1

6, and the probability of rolling an even number is 1

2.


G7 M5 Topic B: Estimating Probabilities


Benchmark: Perform experiments for situations in which the probabilities are known, compare the resulting relative frequencies with the known probabilities; know that there may be differences.

For example: Heads and tails are equally likely when flipping a fair coin, but if several different students flipped fair coins 10 times, it is likely that they will find a variety of relative frequencies of heads and tails.








Benchmark: Calculate experimental probabilities from experiments; represent them as percents, fractions and decimals between 0 and 1 inclusive. Use experimental probabilities to make predictions when actual probabilities are unknown.

For example: Repeatedly draw colored chips with replacement from a bag with an unknown mixture of chips, record relative frequencies, and use the results to make predictions about the contents of the bag.



Minnesota K–12 Academic Standards in Mathematics...

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