Minnesota Academic Standards for Mathematics 2007 Grade 6

A Correlation of, 2015 To the for Mathematics 2007

Number & Operation 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. 6.1.1.1 Locate positive rational numbers on a number line and plot pairs of positive rational numbers on a coordinate grid. Readiness 6-8: Comparing the Planets Lesson 8-1: Integers and the Number Line Lesson 8-2: Comparing and Ordering Integers Lesson 8-3: Absolute Value Lesson 8-4: Integers and the Coordinate Plane Lesson 8-5: Distance Lesson 8-6: Problem Solving Readiness 6-9: Baseball Stats Lesson 9-1: Rational and the Number Line Lesson 9-2: Comparing Rational Lesson 9-3: Ordering Rational Lesson 9-4: Rational and the Coordinate Plane Lesson 9-5: Polygons in the Coordinate Plane Lesson 9-6: Problem Solving 6.1.1.2 Compare positive rational numbers represented in various forms. Use the symbols <, = and >. For example: 1/2 > 0.36. Readiness 6-8: Comparing the Planets Lesson 8-2: Comparing and Ordering Integers Lesson 8-3: Absolute Value Readiness 6-9: Baseball Stats Lesson 9-2: Comparing Rational Lesson 9-3: Ordering Rational 6.1.1.3 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. Lesson 12-3: Introducing Percents Lesson 12-4: Using Percents Lesson 12-5: Problem Solving 1

6.1.1.4 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. Lesson 10-2: Exploring Equivalent Ratios Lesson 10-3: Equivalent Ratios Lesson 10-4: Ratios as Fractions Lesson 10-5: Ratios as Decimals Lesson 10-6: Problem Solving Readiness 6-11: School Fundraisers Lesson 11-1: Unit Rates Lesson 11-2: Unit Prices Lesson 11-3: Constant Speed Lesson 11-6: Problem Solving Readiness 6-12: Recycling Lesson 12-1: Plotting Ratios and Rates Lesson 12-2: Recognizing Proportionality Lesson 12-3: Introducing Percents Lesson 12-4: Using Percents Lesson 12-5: Problem Solving 6.1.1.5 Factor whole numbers; express a whole number as a product of prime factors with exponents. For example: 24 = 2 3 x 3 This standard is met in digits Grade 7. Please see: Lesson 2-4: Greatest Common Factor Lesson 2-5: The Distributive Property Lesson 2-6: Least Common Multiple Lesson 2-7: Problem Solving 6.1.1.6 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. Lesson 2-4: Greatest Common Factor Lesson 2-6: Least Common Multiple 2

6.1.1.7 Convert between equivalent representations of positive rational numbers. For example: Express 10/7 as (7 + 3)/7 = 7/7 + 3/7 = 13/7. Lesson 10-2: Exploring Equivalent Ratios Lesson 10-3: Equivalent Ratios Lesson 10-4: Ratios as Fractions Lesson 10-5: Ratios as Decimals Lesson 10-6: Problem Solving Readiness 6-11: School Fundraisers Lesson 11-1: Unit Rates Lesson 11-2: Unit Prices Lesson 11-3: Constant Speed Lesson 11-6: Problem Solving Readiness 6-12: Recycling Lesson 12-1: Plotting Ratios and Rates Lesson 12-2: Recognizing Proportionality Lesson 12-3: Introducing Percents Lesson 12-4: Using Percents Lesson 12-5: Problem Solving 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. 6.1.2.1 Identify and use ratios to compare Readiness 6-10: Working with Playlists quantities; understand that comparing Lesson 10-1: Ratios quantities using ratios is not the same as Lesson 10-2: Exploring Equivalent Ratios comparing quantities using subtraction. Lesson 10-3: Equivalent Ratios For example: In a classroom with 15 boys and Lesson 10-4: Ratios as Fractions 10 girls, compare the numbers by subtracting Lesson 10-5: Ratios as Decimals (there are 5 more boys than girls) or by Lesson 10-6: Problem Solving 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). 6.1.2.2 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. Readiness 6-11: School Fundraisers Lesson 11-1: Unit Rates Lesson 11-2: Unit Prices Lesson 11-3: Constant Speed Lesson 11-6: Problem Solving 3

6.1.2.3 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). Readiness 6-11: School Fundraisers Lesson 11-1: Unit Rates Lesson 11-2: Unit Prices Lesson 11-3: Constant Speed Lesson 11-6: Problem Solving 6.1.2.4 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. Lesson 10-2: Exploring Equivalent Ratios Lesson 10-3: Equivalent Ratios Lesson 10-4: Ratios as Fractions Lesson 10-5: Ratios as Decimals Lesson 10-6: Problem Solving Readiness 6-11: School Fundraisers Lesson 11-1: Unit Rates Lesson 11-2: Unit Prices Lesson 11-3: Constant Speed Lesson 11-6: Problem Solving Readiness 6-12: Recycling Lesson 12-1: Plotting Ratios and Rates Lesson 12-2: Recognizing Proportionality Lesson 12-4: Using Percents Lesson 12-5: Problem Solving 4

Multiply and divide decimals, fractions and mixed numbers; solve real-world and mathematical problems using arithmetic with positive rational numbers. 6.1.3.1 Multiply and divide decimals and Readiness 6-5: Math in Music fractions, using efficient and generalizable Lesson 5-1: Multiplying Fractions and Whole procedures, including standard algorithms. Lesson 5-2: Multiplying Two Fractions Lesson 5-3: Multiplying Fractions and Mixed Lesson 5-4: Multiplying Mixed Lesson 5-5: Problem Solving Readiness 6-6: Making Pizzas Lesson 6-1: Dividing Fractions and Whole Lesson 6-2: Dividing Unit Fractions by Unit Fractions Lesson 6-3: Dividing Fractions by Fractions Lesson 6-4: Dividing Mixed Lesson 6-5: Problem Solving Readiness 6-7: Fast Food Nutrition Lesson 7-1: Adding and Subtracting Decimals Lesson 7-2: Multiplying Decimals Lesson 7-3: Dividing Multi-Digit Lesson 7-4: Dividing Decimals Lesson 7-5: Decimals and Fractions Lesson 7-6: Comparing and Ordering Decimals and Fractions Lesson 7-7: Problem Solving 6.1.3.2 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 12/4 = 3 means 12 = 3 4, 2/3 4/5 = 5/6 means 5/6 4/5 = 2/3. Readiness 6-5: Math in Music Lesson 5-1: Multiplying Fractions and Whole Lesson 5-2: Multiplying Two Fractions Lesson 5-3: Multiplying Fractions and Mixed Lesson 5-4: Multiplying Mixed Lesson 5-5: Problem Solving Readiness 6-6: Making Pizzas Lesson 6-1: Dividing Fractions and Whole Lesson 6-2: Dividing Unit Fractions by Unit Fractions Lesson 6-3: Dividing Fractions by Fractions Lesson 6-4: Dividing Mixed Lesson 6-5: Problem Solving 5

6.1.3.3 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? 6.1.3.4 Solve real-world and mathematical problems requiring arithmetic with decimals, fractions and mixed numbers. Lesson 12-3: Introducing Percents Lesson 12-4: Using Percents Lesson 12-5: Problem Solving Readiness 6-5: Math in Music Lesson 5-2: Multiplying Two Fractions Lesson 5-3: Multiplying Fractions and Mixed Lesson 5-4: Multiplying Mixed Lesson 5-5: Problem Solving Readiness 6-6: Making Pizzas Lesson 6-1: Dividing Fractions and Whole Lesson 6-2: Dividing Unit Fractions by Unit Fractions Lesson 6-3: Dividing Fractions by Fractions Lesson 6-4: Dividing Mixed Lesson 6-5: Problem Solving Readiness 6-7: Fast Food Nutrition Lesson 7-1: Adding and Subtracting Decimals Lesson 7-2: Multiplying Decimals Lesson 7-3: Dividing Multi-Digit Lesson 7-4: Dividing Decimals Lesson 7-5: Decimals and Fractions Lesson 7-6: Comparing and Ordering Decimals and Fractions Lesson 7-7: Problem Solving 6.1.3.5 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 1/3 + 0.25 can be estimated to be between 1/2 and 1, and this estimate can be used to check the result of a more detailed calculation. Lesson 5-1: Multiplying Fractions and Whole Lesson 6-1: Dividing Fractions and Whole Lesson 6-2: Dividing Unit Fractions by Unit Fractions Lesson 6-4: Dividing Mixed Lesson 6-5: Problem Solving Lesson 7-1: Adding and Subtracting Decimals Lesson 7-6: Comparing and Ordering Decimals and Fractions Readiness 6-10: Working with Playlists Lesson 11-2: Unit Prices Lesson 12-3: Introducing Percents 6

Algebra 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. 6.2.1.1 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. Readiness 6-1: Rating Music Artists Lesson 1-2: Algebraic Expressions Lesson 1-3: Writing Algebraic Expressions Lesson 1-4: Evaluating Algebraic Expressions Lesson 1-5: Expressions with Exponents Lesson 1-6: Problem Solving Readiness 6-4: Working at an Amusement Park Lesson 4-1: Using Two Variables to Represent a Relationship Lesson 4-2: Analyzing Patterns Using Tables and Graphs Lesson 4-3: Relating Tables and Graphs to Equations Lesson 4-4: Problem Solving 6.2.1.2 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 t = 1, 4, 9, 16,... by using the rule t = n 2 for n = 1, 2, 3, 4,... Readiness 6-4: Working at an Amusement Park Lesson 4-1: Using Two Variables to Represent a Relationship Lesson 4-2: Analyzing Patterns Using Tables and Graphs Lesson 4-3: Relating Tables and Graphs to Equations Lesson 4-4: Problem Solving Use properties of arithmetic to generate equivalent numerical expressions and evaluate expressions involving positive rational numbers. 6.2.2.1 Apply the associative, commutative Readiness 6-4: Working at an Amusement and distributive properties and order of Park operations to generate equivalent expressions Lesson 4-1: Using Two Variables to Represent and to solve problems involving positive a Relationship rational numbers. For example: Lesson 4-2: Analyzing Patterns Using Tables 32/15 x 5/6 = 32x5/ 15x6 = 2x16x5 / and Graphs 3x5x3x2 = 16/9 x 2/2 x 5/5 = 16/9 Lesson 4-3: Relating Tables and Graphs to Another example: Use the distributive law to Equations write: ½+1/3(9/2 15/8) = ½+1/3 x 9/2 Lesson 4-4: Problem Solving 1/3 x 15/8 = ½ = 1/3 x 9/2 1/3 x 15/8 = ½ + 3/2 5/8 = 2 5/8 = 1 3/8. 7

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. 6.2.3.1 Represent real-world or mathematical situations using equations and inequalities involving variables and positive rational numbers. For example: The number of miles m in a k kilometer race is represented by the equation m = 0.62k. Readiness 6-2: Renting Movies Lesson 2-1: The Identity and Zero Properties Lesson 2-2: The Commutative Properties Lesson 2-3: The Associative Properties Lesson 2-4: Greatest Common Factor Lesson 2-5: The Distributive Property Lesson 2-6: Least Common Multiple Lesson 2-7: Problem Solving 6.2.3.2 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? Readiness 6-3: Video Game Economics Lesson 3-1: Expressions to Equations Lesson 3-2: Balancing Equations Lesson 3-3: Solving Addition and Subtraction Equations Lesson 3-4: Solving Multiplication and Division Equations Lesson 3-5: Equations to Inequalities Lesson 3-6: Solving Inequalities Lesson 3-7: Problem Solving Geometry & Measurement Calculate perimeter, area, surface area and volume of two- and three-dimensional figures to solve real-world and mathematical problems. 6.3.1.1 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. Readiness 6-14: Planning a Birthday Party Lesson 14-1: Analyzing Three-Dimensional Figures Lesson 14-2: Nets Lesson 14-3: Surface Areas of Prisms Lesson 14-4: Surface Areas of Pyramids Lesson 14-5: Volumes of Rectangular Prisms Lesson 14-6: Problem Solving 6.3.1.2 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. Readiness 6-13: Designing a Playground Lesson 13-1: Rectangles and Squares Lesson 13-2: Right Triangles Lesson 13-3: Parallelograms Lesson 13-4: Other Triangles Lesson 13-5: Polygons Lesson 13-6: Problem Solving 8

6.3.1.3 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. Readiness 6-13: Designing a Playground Lesson 13-2: Right Triangles Lesson 13-4: Other Triangles Lesson 13-5: Polygons Lesson 13-6: Problem Solving Understand and use relationships between angles in geometric figures. 6.3.2.1 Solve problems using the For related content, please see digits Grade 7: relationships between the angles formed by Lesson 10-2: Adjacent Angles intersecting lines. Lesson 10-3: Complementary Angles For example: If two streets cross, forming Lesson 10-4: Supplementary Angles four corners such that one of the corners Lesson 10-5: Vertical Angles forms an angle of 120, determine the Lesson 10-6: Problem Solving measures of the remaining three angles. Another example: Recognize that pairs of interior and exterior angles in polygons have measures that sum to 180. 6.3.2.2 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. 6.3.2.3 Develop and use formulas for the sums of the interior angles of polygons by decomposing them into triangles. For related content, please see digits Grade 7: Lesson 10-2: Adjacent Angles Lesson 10-3: Complementary Angles Lesson 10-4: Supplementary Angles Lesson 10-5: Vertical Angles Lesson 10-6: Problem Solving This standard is met in digits Grade 7. Please see: Lesson 10-2: Adjacent Angles Lesson 10-3: Complementary Angles Lesson 10-4: Supplementary Angles Lesson 10-5: Vertical Angles Lesson 10-6: Problem Solving Choose appropriate units of measurement and use ratios to convert within measurement systems to solve real-world and mathematical problems. 6.3.3.1 Solve problems in various contexts Readiness 6-11: School Fundraisers involving conversion of weights, capacities, Lesson 11-1: Unit Rates geometric measurements and times within Lesson 11-3: Constant Speed measurement systems using appropriate units. Lesson 11-6: Problem Solving 9

6.3.3.2 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. Data Analysis & Probability Use probabilities to solve real-world and mathematical problems; represent probabilities using fractions, decimals and percents. 6.4.1.1 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 6x6 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. This standard is met in digits Grade 7. Please see: Lesson 17-2: Sample Spaces Lesson 17-3: Counting Outcomes Lesson 17-4: Finding Theoretical Probabilities Lesson 17-7: Problem Solving 6.4.1.2 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. This standard is met in digits Grade 7. Please see: Readiness 7-16: Basketball Stats Lesson 16-1: Likelihood and Probability Lesson 16-2: Sample Spaces Lesson 16-3: Relative Frequency and Experimental Probability Lesson 16-4: Theoretical Probability Lesson 16-5: Probability Models Lesson 16-6: Problem Solving Readiness 7-17: Games and Probability Lesson 17-1: Compound Events Lesson 17-2: Sample Spaces Lesson 17-3: Counting Outcomes Lesson 17-4: Finding Theoretical Probabilities Lesson 17-5: Simulation With Random Lesson 17-6: Finding Probabilities by Simulation Lesson 17-7: Problem Solving 10

6.4.1.3 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. 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 This standard is met in digits Grade 7. Please see: Lesson 17-5: Simulation With Random Lesson 17-6: Finding Probabilities by Simulation Lesson 17-7: Problem Solving 6.4.1.4 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. This standard is met in digits Grade 7. Please see: Readiness 7-16: Basketball Stats Lesson 16-3: Relative Frequency and Experimental Probability Lesson 17-5: Simulation With Random Lesson 17-6: Finding Probabilities by Simulation Lesson 17-7: Problem Solving 11