Maryland College and Career-Ready Standards 6th Grade Math Activities

MD.MA.6.RP. Ratios and Proportional Relationships (RP)

Understand ratio concepts and use ratio reasoning to solve problems.

6.RP.1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

6.RP.1.1. Knowledge of ratio as a comparison of any two quantities.

6.RP.1.2. Knowledge of a ratio is not always a comparison of part-to-whole; Can be part-to-part or whole-to-whole.

6.RP.2. Understand the concept of a unit rate a/b associated with ratio a:b with b ≠ 0 (b not equal to zero), and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cups of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” (Expectations for unit rates in this grade are limited to non-complex fractions.)

6.RP.2.1. Knowledge that a unit rate emphasizes finding an equivalent ratio with a denominator of 1.

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.

6.RP.3.a. Make tables 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 tables to compare ratios.

6.RP.3.a.2. Ability to use multiplicative relationships to extend an initial ratio to equivalent ratios; When working backward, use the inverse operation (division).

6.RP.3.b. Solve unit rate problems including those involving unit pricing and constant speed. For example: If it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

6.RP.3.b.1. Ability to use division to determine unit rate.

6.RP.3.c. 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.3.c.1. Ability to introduce percent as a special rate where a part is compared to a whole and the whole always has a value of 100.

6.RP.3.c.2. Ability to solve problems using equivalent ratios. (NOTE: Proportions are not introduced until Grade 7.) This is developing proportional reasoning without formal proportions.

6.RP.3.d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

6.RP.3.d.1. Ability to expand ratio reasoning to units of measurement.

MD.MA.6.NS. The Number System (NS)

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

6.NS.1. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3)÷(3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3)÷(3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b)÷(c/d) = ad/bc. How much chocolate will each person get if 3 people share 1/2 lb. of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4mi and area 1/2square mi?

6.NS.1.1. Ability to explore the concept that division breaks quantities into groups.

6.NS.1.2. Ability to emphasize that when dividing by a value that is less than one, the quotient is greater than the dividend.

6.NS.1.3. Ability to explore both the measurement concept and the partition concept of division of fractions.

6.NS.1.4. Ability to introduce the fact that the measurement concept uses repeated subtraction or equal groups.

6.NS.1.5. Ability to explore the common denominator algorithm as a method of repeated subtraction.

6.NS.1.6. Knowledge of partition concept focuses on “How much is one?”

6.NS.1.7. Knowledge of algorithm a/b ÷ c/d = a/b x d/c = ad/bc (invert and multiply) is an extension of the partition concept.

Compute fluently with multi-digit numbers and find common factors and multiples.

6.NS.2. Fluently divide multi-digit numbers using the standard algorithm.

6.NS.2.1. See the skills and knowledge that are stated in the Standard.

6.NS.3. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

6.NS.3.1. See the skills and knowledge that are stated in the Standard.

6.NS.4. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36+8 as 4(9+2).

6.NS.4.1. Ability to build on student knowledge of and differentiate between factors and multiples (CC.4.OA.4.).

6.NS.4.2. Ability to build on student knowledge of factor pairs of whole numbers (CC.4.OA.4).

6.NS.4.3. Ability to identify and differentiate between common factors and common multiples of 2 whole numbers.

Apply and extend previous understandings of numbers to the system of rational numbers.

6.NS.5. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, debits/credits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

6.NS.5.1. See the skills and knowledge that are stated in the Standard.

6.NS.6. 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.6.b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

6.NS.6.b.1. Ability to introduce and define coordinate plane terminology, including coordinate plane and quadrants I, II, III, and IV.

6.NS.7. Understand ordering and absolute value of rational numbers.

6.NS.7.b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3°C > – 7°C to express the fact that –3°C is warmer than –7°C.

6.NS.7.b.1. See the skills and knowledge that are stated in the Standard.

6.NS.7.d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.

6.NS.7.d.1. Ability to develop understanding within real-world contexts.

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 between points with the same first coordinate or the same second coordinate.

6.NS.8.1. See the skills and knowledge that are stated in the Standard.

MD.MA.6.EE. Expressions and Equations (EE)

Apply and extend previous understandings of arithmetic to algebraic expressions.

6.EE.1. Write and evaluate numerical expressions involving whole-number exponents.

6.EE.1.1. Ability to develop understanding of a whole-number exponent as shorthand for repeated multiplication of a number times itself.

6.EE.1.2. Ability to introduce squares and cubes first because they can be represented geometrically.

6.EE.1.3. Ability to extend understanding of order of operations to include exponents.

6.EE.2. Write, read, and evaluate expressions in which letters stand for numbers.

6.EE.2.a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5 – y.

6.EE.2.a.1. Ability to define what a variable is.

6.EE.2.b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2(8+7) as a product of two factors; view (8+7) as both a single entity and a sum of two terms.

6.EE.2.b.1. Ability to introduce and define coefficient and term.

6.EE.2.b.2. Ability to read expressions aloud to explore the concept of quantities.

6.EE.2.c. Evaluate expressions at specific values for their variables. Include expressions that arise from formulas in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V=s^3 and A=6s^2 to find the volume and surface area of a cube with sides of length S = 1/2.

6.EE.2.c.1. See the skills and knowledge that are stated in the Standard.

6.EE.3. Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2+x) to produce the equivalent expression 6+3x; apply the distributive property to the expression 24x+18y to produce the equivalent expression 6(4x+3y); apply properties of operations to y+y+y to produce the equivalent expression 3y.

6.EE.3.1. Ability to use properties of operations to simplify expressions, therefore producing equivalent expressions.

Reason about and solve one-variable equations and inequalities.

6.EE.5. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

6.EE.5.1. See the skills and knowledge that are stated in the Standard.

6.EE.6. Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

6.EE.6.1. See the skills and knowledge that are stated in the Standard.

6.EE.7. Solve real-world and mathematical problems by writing and solving equations of the form x+p = q and px = q for cases in which p, q and x are all non-negative rational numbers.

6.EE.7.1. Ability to reinforce that solving equations is finding values of the variable that make the equation true.

6.EE.7.2. Ability to develop conceptual understanding of inverse operations.

6.EE.8. Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

6.EE.8.1. Ability to develop conceptual understanding of representing solutions to inequalities on a number line diagram.

6.EE.8.2. Knowledge of ≤ and ≥.

6.EE.8.3. Knowledge of symbolic components of the graph of an inequality; Specifically, open circle vs. closed circle, direction of shading.

6.EE.8.4. Knowledge that an open circle represents a value that is NOT actually part of the solution set.

6.EE.8.5. Knowledge that solutions to x>c or x<c are not just integers but also fractions and decimals.

Represent and analyze quantitative relationships between dependent and independent variables.

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 tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

6.EE.9.3. Knowledge of terminology associated with graphing ordered pairs (5.OA.3).

6.EE.9.4. Ability to write an equation based on a graph or a table.

MD.MA.6.G. Geometry (G)

Solve real-world and mathematical problems involving area, surface area, and volume.

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

6.G.1.3. Knowledge of the base and height of a right triangle are the length and width of a rectangle to discover the formula A = bh/2.

6.G.2. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = lwh and V = bh to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

6.G.2.1. See the skills and knowledge that are stated in the Standard.

6.G.3. Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

6.G.3.1. See the skills and knowledge that are stated in the Standard SC 6.

6.G.4. Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

6.G.4.1. See the skills and knowledge that are stated in the Standard.

MD.MA.6.SP. Statistics and Probability (SP)

Develop understanding of statistical variability.

6.SP.1. Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.

6.SP.1.1. Ability to introduce and develop statistical reasoning.

6.SP.3. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

6.SP.3.1. Knowledge of median and mean as measures of center.

6.SP.3.2. Knowledge of range as a measure of variation.

6.SP.3.3. Ability to look at a set of data and estimate the measures of center.