Courses of Study : Mathematics

Ratios and Proportional Relationships
Understand ratio concepts and use ratio reasoning to solve problems.
Mathematics (2016)
Grade(s): 6
All Resources: 4
Learning Activities: 1
Lesson Plans: 3
1 ) Understand the concept of a ratio, and use ratio language to describe a ratio relationship between two quantities. [6-RP1]

Examples: "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."


NAEP Framework
NAEP Statement:
4NPO4a: Use simple ratios to describe problem situations.

NAEP Statement:
8NPO3a: Perform computations with rational numbers.

NAEP Statement:
8NPO4a: Use ratios to describe problem situations.

NAEP Statement:
8NPO4b: Use fractions to represent and express ratios and proportions.

NAEP Statement:
8NPO4d: Solve problems involving percentages (including percent increase and decrease, interest rates, tax, discount, tips, or part/whole relationships).


Mathematics (2016)
Grade(s): 6
All Resources: 3
Learning Activities: 2
Lesson Plans: 1
2 ) Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. [6-RP2]

Examples: "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup 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.)


NAEP Framework
NAEP Statement:
8NPO3a: Perform computations with rational numbers.

NAEP Statement:
8NPO4a: Use ratios to describe problem situations.

NAEP Statement:
8NPO4b: Use fractions to represent and express ratios and proportions.

NAEP Statement:
8NPO4d: Solve problems involving percentages (including percent increase and decrease, interest rates, tax, discount, tips, or part/whole relationships).


Mathematics (2016)
Grade(s): 6
All Resources: 4
Learning Activities: 1
Lesson Plans: 3
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-RP3]

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-RP3a]

b. Solve unit rate problems including those involving unit pricing and constant speed. [6-RP3b]

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'

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-RP3c]

d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. [6-RP3d]


NAEP Framework
NAEP Statement:
8M2b: Solve problems involving conversions within the same measurement system such as conversions involving square inches and square feet.

NAEP Statement:
8M2c: Estimate the measure of an object in one system given the measure of that object in another system and the approximate conversion factor. For example:
  • Distance conversion: 1 kilometer is approximately 5/8 of a mile.
  • Money conversion: U.S. dollars to Canadian dollars.
  • Temperature conversion: Fahrenheit to Celsius.


NAEP Statement:
8NPO3a: Perform computations with rational numbers.

NAEP Statement:
8NPO4a: Use ratios to describe problem situations.

NAEP Statement:
8NPO4b: Use fractions to represent and express ratios and proportions.

NAEP Statement:
8NPO4d: Solve problems involving percentages (including percent increase and decrease, interest rates, tax, discount, tips, or part/whole relationships).


The Number System
Apply and extend previous understandings of multiplication and division to divide by fractions.
Mathematics (2016)
Grade(s): 6
All Resources: 0
4 ) Interpret and compute quotients of fractions, and solve word problems involving division of fractions, e.g., by using visual fraction models and equations to represent the problem. [6-NS1]

Examples: 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 3/4 mi and area 1/2 square mi'


NAEP Framework
NAEP Statement:
8NPO1b: Model or describe rational numbers or numerical relationships using number lines and diagrams.

NAEP Statement:
8NPO3d: Describe the effect of multiplying and dividing by numbers including the effect of multiplying or dividing a rational number by:
  • Zero, or
  • A number less than zero, or
  • A number between zero and one,
  • One, or
  • A number greater than one.


NAEP Statement:
8NPO6b: Provide a mathematical argument to explain operations with two or more fractions.


Compute fluently with multi-digit numbers and find common factors and multiples.
Mathematics (2016)
Grade(s): 6
All Resources: 0
5 ) Fluently divide multi-digit numbers using the standard algorithm. [6-NS2]


NAEP Framework
NAEP Statement:
8NPO3a: Perform computations with rational numbers.


Mathematics (2016)
Grade(s): 6
All Resources: 1
Lesson Plans: 1
6 ) Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. [6-NS3]


NAEP Framework
NAEP Statement:
8NPO3a: Perform computations with rational numbers.


Mathematics (2016)
Grade(s): 6
All Resources: 0
7 ) 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. [6-NS4]

Example: Express 36 + 8 as 4(9 + 2).


NAEP Framework
NAEP Statement:
8NPO5b: Recognize, find, or use factors, multiples, or prime factorization.

NAEP Statement:
8NPO5e: Apply basic properties of operations.


Apply and extend previous understandings of numbers to the system of rational numbers.
Mathematics (2016)
Grade(s): 6
All Resources: 2
Learning Activities: 1
Lesson Plans: 1
8 ) 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, credits/debits, 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-NS5]


NAEP Framework
NAEP Statement:
8NPO1d: Write or rename rational numbers.

NAEP Statement:
8NPO1e: Recognize, translate or apply multiple representations of rational numbers (fractions, decimals, and percents) in meaningful contexts.

NAEP Statement:
8NPO1h: Order or compare rational numbers (fractions, decimals, percents, or integers) using various models and representations (e.g., number line).

NAEP Statement:
8NPO5e: Apply basic properties of operations.


Mathematics (2016)
Grade(s): 6
All Resources: 2
Learning Activities: 1
Lesson Plans: 1
9 ) 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-NS6]

a. 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 that 0 is its own opposite. [6-NS6a]

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-NS6b]

c. 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 a coordinate plane. [6-NS6c]


NAEP Framework
NAEP Statement:
8A2c: Graph or interpret points represented by ordered pairs of numbers on a rectangular coordinate system.

NAEP Statement:
8NPO1d: Write or rename rational numbers.

NAEP Statement:
8NPO1e: Recognize, translate or apply multiple representations of rational numbers (fractions, decimals, and percents) in meaningful contexts.

NAEP Statement:
8NPO1g: Find or model absolute value or apply to problem situations.

NAEP Statement:
8NPO1h: Order or compare rational numbers (fractions, decimals, percents, or integers) using various models and representations (e.g., number line).

NAEP Statement:
8NPO5e: Apply basic properties of operations.


Mathematics (2016)
Grade(s): 6
All Resources: 0
10 ) Understand ordering and absolute value of rational numbers. [6-NS7]

a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. [6-NS7a]

Example: Interpret -3 > -7 as a statement that -3 is located to the right of -7 on a number line oriented from left to right.

b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. [6-NS7b]

Example: Write -3oC > -7oC to express the fact that -3oC is warmer than -7oC.

c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. [6-NS7c]

Example: For an account balance of -30 dollars, write |-30| = 30 to describe the size of the debt in dollars.

d. Distinguish comparisons of absolute value from statements about order. [6-NS7d]

Example: Recognize that an account balance less than -30 dollars represents a debt greater than 30 dollars.


NAEP Framework
NAEP Statement:
8NPO1d: Write or rename rational numbers.

NAEP Statement:
8NPO1e: Recognize, translate or apply multiple representations of rational numbers (fractions, decimals, and percents) in meaningful contexts.

NAEP Statement:
8NPO1g: Find or model absolute value or apply to problem situations.

NAEP Statement:
8NPO1h: Order or compare rational numbers (fractions, decimals, percents, or integers) using various models and representations (e.g., number line).

NAEP Statement:
8NPO5e: Apply basic properties of operations.


Mathematics (2016)
Grade(s): 6
All Resources: 3
Learning Activities: 2
Lesson Plans: 1
11 ) 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-NS8]


NAEP Framework
NAEP Statement:
8A2d: Solve problems involving coordinate pairs on the rectangular coordinate system.

NAEP Statement:
8NPO1d: Write or rename rational numbers.

NAEP Statement:
8NPO1e: Recognize, translate or apply multiple representations of rational numbers (fractions, decimals, and percents) in meaningful contexts.

NAEP Statement:
8NPO1h: Order or compare rational numbers (fractions, decimals, percents, or integers) using various models and representations (e.g., number line).


Expressions and Equations
Apply and extend previous understandings of arithmetic to algebraic expressions.
Mathematics (2016)
Grade(s): 6
All Resources: 0
12 ) Write and evaluate numerical expressions involving whole-number exponents. [6-EE1]

Mathematics (2016)
Grade(s): 6
All Resources: 3
Learning Activities: 2
Lesson Plans: 1
13 ) Write, read, and evaluate expressions in which letters stand for numbers. [6-EE2]

a. Write expressions that record operations with numbers and with letters standing for numbers. [6-EE2a]

Example: Express the calculation, "Subtract y from 5," as 5 - y.

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. [6-EE2b]

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.

c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used 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). [6-EE2c]

Example: Use the formulas V = s3 and A = 6s2 to find the volume and surface area of a cube with sides of length s = 1/2.


NAEP Framework
NAEP Statement:
4A3a: Use letters and symbols to represent an unknown quantity in a simple mathematical expression.

NAEP Statement:
4A3b: Express simple mathematical relationships using number sentences.


Mathematics (2016)
Grade(s): 6
All Resources: 0
14 ) Apply the properties of operations to generate equivalent expressions. [6-EE3]

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.

Mathematics (2016)
Grade(s): 6
All Resources: 0
15 ) Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). [6-EE4]

Example: The expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y represents.

Reason about and solve one-variable equations and inequalities.
Mathematics (2016)
Grade(s): 6
All Resources: 1
Lesson Plans: 1
16 ) 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-EE5]


NAEP Framework
NAEP Statement:
4NPO2c: Verify solutions or determine the reasonableness of results in meaningful contexts.

NAEP Statement:
8A4a: Solve linear equations or inequalities (e.g., ax + b = c or ax + b = cx + d or ax + b > c).

NAEP Statement:
8A4b: Interpret "=" as an equivalence between two expressions and use this interpretation to solve problems.

NAEP Statement:
8A5a: Make, validate, and justify conclusions and generalizations about linear relationships.

NAEP Statement:
8NPO2c: Verify solutions or determine the reasonableness of results in a variety of situations, including calculator and computer results.


Mathematics (2016)
Grade(s): 6
All Resources: 2
Learning Activities: 1
Lesson Plans: 1
17 ) 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-EE6]


NAEP Framework
NAEP Statement:
8A4a: Solve linear equations or inequalities (e.g., ax + b = c or ax + b = cx + d or ax + b > c).

NAEP Statement:
8A4b: Interpret "=" as an equivalence between two expressions and use this interpretation to solve problems.


Mathematics (2016)
Grade(s): 6
All Resources: 0
18 ) 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 nonnegative rational numbers. [6-EE7]


NAEP Framework
NAEP Statement:
8A4a: Solve linear equations or inequalities (e.g., ax + b = c or ax + b = cx + d or ax + b > c).

NAEP Statement:
8A4b: Interpret "=" as an equivalence between two expressions and use this interpretation to solve problems.

NAEP Statement:
8A4c: Analyze situations or solve problems using linear equations and inequalities with rational coefficients symbolically or graphically (e.g., ax + b = c or ax + b = cx + d).


Mathematics (2016)
Grade(s): 6
All Resources: 0
19 ) 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-EE8]


NAEP Framework
NAEP Statement:
8A4a: Solve linear equations or inequalities (e.g., ax + b = c or ax + b = cx + d or ax + b > c).

NAEP Statement:
8A4b: Interpret "=" as an equivalence between two expressions and use this interpretation to solve problems.

NAEP Statement:
8A4c: Analyze situations or solve problems using linear equations and inequalities with rational coefficients symbolically or graphically (e.g., ax + b = c or ax + b = cx + d).


Represent and analyze quantitative relationships between dependent and independent variables.
Mathematics (2016)
Grade(s): 6
All Resources: 0
20 ) 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. [6-EE9]

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.


NAEP Framework
NAEP Statement:
4A3a: Use letters and symbols to represent an unknown quantity in a simple mathematical expression.

NAEP Statement:
8A4c: Analyze situations or solve problems using linear equations and inequalities with rational coefficients symbolically or graphically (e.g., ax + b = c or ax + b = cx + d).

NAEP Statement:
8A4e: Use and evaluate common formulas (e.g., relationship between a circle's circumference and diameter [C = pi d], distance and time under constant speed).


Geometry
Solve real-world and mathematical problems involving area, surface area, and volume.
Mathematics (2016)
Grade(s): 6
All Resources: 0
21 ) Find the 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-G1]


NAEP Framework
NAEP Statement:
8A4c: Analyze situations or solve problems using linear equations and inequalities with rational coefficients symbolically or graphically (e.g., ax + b = c or ax + b = cx + d).

NAEP Statement:
8G2d: Predict results of combining, subdividing, and changing shapes of plane figures and solids (e.g., paper folding, tiling, cutting up and rearranging pieces).

NAEP Statement:
8G3b: Apply geometric properties and relationships in solving simple problems in two and three dimensions.

NAEP Statement:
8G3c: Represent problem situations with simple geometric models to solve mathematical or real-world problems.

NAEP Statement:
8G3f: Describe or analyze simple properties of, or relationships between, triangles, quadrilaterals, and other polygonal plane figures.

NAEP Statement:
8M1f: Solve mathematical or real-world problems involving perimeter or area of plane figures such as triangles, rectangles, circles, or composite figures.


Mathematics (2016)
Grade(s): 6
All Resources: 0
22 ) 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-G2]


NAEP Framework
NAEP Statement:
8A4c: Analyze situations or solve problems using linear equations and inequalities with rational coefficients symbolically or graphically (e.g., ax + b = c or ax + b = cx + d).

NAEP Statement:
8A4e: Use and evaluate common formulas (e.g., relationship between a circle's circumference and diameter [C = pi d], distance and time under constant speed).

NAEP Statement:
8G3b: Apply geometric properties and relationships in solving simple problems in two and three dimensions.

NAEP Statement:
8G3c: Represent problem situations with simple geometric models to solve mathematical or real-world problems.

NAEP Statement:
8G3f: Describe or analyze simple properties of, or relationships between, triangles, quadrilaterals, and other polygonal plane figures.

NAEP Statement:
8M1h: Solve problems involving volume or surface area of rectangular solids, cylinders, prisms, or composite shapes.

NAEP Statement:
8M1i: Solve problems involving rates such as speed or population density.


Mathematics (2016)
Grade(s): 6
All Resources: 0
23 ) 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-G3]


NAEP Framework
NAEP Statement:
4G4d: Construct geometric figures with vertices at points on a coordinate grid.

NAEP Statement:
8A4c: Analyze situations or solve problems using linear equations and inequalities with rational coefficients symbolically or graphically (e.g., ax + b = c or ax + b = cx + d).

NAEP Statement:
8G1d: Draw or sketch from a written description polygons, circles, or semicircles.

NAEP Statement:
8G1f: Demonstrate an understanding about the two-and three-dimensional shapes in our world through identifying, drawing, modeling, building, or taking apart.

NAEP Statement:
8G3b: Apply geometric properties and relationships in solving simple problems in two and three dimensions.

NAEP Statement:
8G3c: Represent problem situations with simple geometric models to solve mathematical or real-world problems.

NAEP Statement:
8G3f: Describe or analyze simple properties of, or relationships between, triangles, quadrilaterals, and other polygonal plane figures.

NAEP Statement:
8G4d: Represent geometric figures using rectangular coordinates on a plane.

NAEP Statement:
8M1i: Solve problems involving rates such as speed or population density.


Mathematics (2016)
Grade(s): 6
All Resources: 2
Learning Activities: 1
Lesson Plans: 1
24 ) 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. [6-G4]


NAEP Framework
NAEP Statement:
8A4c: Analyze situations or solve problems using linear equations and inequalities with rational coefficients symbolically or graphically (e.g., ax + b = c or ax + b = cx + d).

NAEP Statement:
8G1c: Identify, define, or describe geometric shapes in the plane and in three-dimensional space given a visual representation.

NAEP Statement:
8G3b: Apply geometric properties and relationships in solving simple problems in two and three dimensions.

NAEP Statement:
8G3c: Represent problem situations with simple geometric models to solve mathematical or real-world problems.

NAEP Statement:
8G3f: Describe or analyze simple properties of, or relationships between, triangles, quadrilaterals, and other polygonal plane figures.


Statistics and Probability
Develop understanding of statistical variability.
Mathematics (2016)
Grade(s): 6
All Resources: 1
Lesson Plans: 1
25 ) Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. [6-SP1]

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.

Mathematics (2016)
Grade(s): 6
All Resources: 1
Lesson Plans: 1
26 ) Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. [6-SP2]


NAEP Framework
NAEP Statement:
4DASP1a: Read or interpret a single set of data.

NAEP Statement:
4DASP2b: Given a set of data or a graph, describe the distribution of data using median, range, or mode.

NAEP Statement:
8DASP1a: Read or interpret data, including interpolating or extrapolating from data.


Mathematics (2016)
Grade(s): 6
All Resources: 1
Lesson Plans: 1
27 ) 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-SP3]


NAEP Framework
NAEP Statement:
8DASP2a: Calculate, use, or interpret mean, median, mode, or range.


Summarize and describe distributions.
Mathematics (2016)
Grade(s): 6
All Resources: 2
Learning Activities: 2
28 ) Display numerical data in plots on a number line, including dot plots, histograms, and box plots. [6-SP4]


NAEP Framework
NAEP Statement:
4DASP1b: For a given set of data, complete a graph (limits of time make it difficult to construct graphs completely).

NAEP Statement:
8DASP1b: For a given set of data, complete a graph and then solve a problem using the data in the graph (histograms, line graphs, scatterplots, circle graphs, and bar graphs).


Mathematics (2016)
Grade(s): 6
All Resources: 1
Learning Activities: 1
29 ) Summarize numerical data sets in relation to their context, such as by: [6-SP5]

a. Reporting the number of observations. [6-SP5a]

b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. [6-SP5b]

c. 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-SP5c]

d. 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. [6-SP5d]


NAEP Framework
NAEP Statement:
4DASP1c: Solve problems by estimating and computing within a single set of data.

NAEP Statement:
4DASP2b: Given a set of data or a graph, describe the distribution of data using median, range, or mode.

NAEP Statement:
8DASP2a: Calculate, use, or interpret mean, median, mode, or range.

NAEP Statement:
8DASP2b: Describe how mean, median, mode, range, or interquartile ranges relate to distribution shape.

NAEP Statement:
8DASP2c: Identify outliers and determine their effect on mean, median, mode, or range.