# Math Topics

Topics have a narrated lesson, virtual manipulative, question set, real world problem set, and challenge mode.

- See how fractions with common denominators can be added by summing their numerators. Discover why fractions with unlike denominators must be rewritten as equivalent fractions with a common denominator. Visualize fraction addition and subtraction using both fraction circle and number line models.
- 7.NS.A.1c
# Common Core - 7.NS.A.1c

Understand subtraction of rational numbers as adding the additive inverse,

*p*–*q*=*p*+ (–*q*). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. - 7.NS.A.3
# Common Core - 7.NS.A.3

Solve real-world and mathematical problems involving the four operations with rational numbers.

- 7.NS.A.1b
# Common Core - 7.NS.A.1b

Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

- Understand area as the amount of space a flat surface occupies. Derive the area formula for a circle by slicing the figure into wedges and forming a parallelogram. Lean how to find area given radius, or radius given area.
- 7.G.B.4
# Common Core - 7.G.B.4

Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

- 7.G.B.6
# Common Core - 7.G.B.6

Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

- Understand area as the amount of space a flat surface occupies. Derive the area formula for a parallelogram from the area formula for a rectangle. Learn to differentiate the height from the side. Recognize that parallelograms can have the same area, but different shapes.
- 6.G.A.1
# Common Core - 6.G.A.1

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.

- 7.G.B.6
# Common Core - 7.G.B.6

Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

- Understand area as the amount of space a flat surface occupies. Find the area of a rectangle by filling the rectangle with unit squares. Derive the area formula for a rectangle. Recognize that rectangles can have the same area, but different shapes.
- 6.G.A.1
# Common Core - 6.G.A.1

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.

- 7.G.B.6
# Common Core - 7.G.B.6

Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

- Understand area as the amount of space a flat surface occupies. Derive the area formula for a triangle from the area formula for a parallelogram. Learn to differentiate the height from the side. Recognize that triangles can have the same area, but different shapes.
- 6.G.A.1
# Common Core - 6.G.A.1

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.

- 7.G.B.6
# Common Core - 7.G.B.6

- Box and Whisker PlotsUse and interpret box plots as a way to summarize large amounts of data. Learn to find the extremes, quartiles, and median for a set of data. Practice constructing box plots. Interpret and use box plots to solve a real-world problem.
- 6.SP.B.4
# Common Core - 6.SP.B.4

Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

- 7.SP.B.3
# Common Core - 7.SP.B.3

Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.

- 7.SP.B.4
# Common Core - 7.SP.B.4

Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.

- Circumference of a CircleKnow the circumference is the distance around the edge of a circle. Discover that the ratio of the circumference to the diameter of a circle equals the constant PI. Learn how to use the radius or diameter of a circle to find the circumference.
- 7.G.B.4
# Common Core - 7.G.B.4

Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

- Classifying TrianglesUnderstand that triangles can be classified by their sides and angles. Investigate and explore scalene, equilateral, isosceles, obtuse, right, and acute triangles. Use properties to classify triangles. Realize that triangles can have more than one name.
- 7.G.A.2
# Common Core - 7.G.A.2

Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

- Understand that a common denominator is a common multiple of the denominators of two or more fractions. Learn how to find a common denominator for a pair of fractions. Understand the difference between a common denominator and the least common denominator (LCD). Use visual models to understand multiples, common denominators, and least common denominators.
- 6.NS.B.4
# Common Core - 6.NS.B.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.

- Comparing and Ordering DecimalsLearn how to compare and order positive and negative decimals. Understand place value using a visual model. Compare decimals by placing values on the number line. Discover how to read and write inequalities using greater than (>) and less than (<) symbols.
- Learn how to compare and order positive and negative fractions and mixed numbers. Compare fractions using multiple models: fraction chart, fraction bars, number line. Discover how to read and write inequalities using greater than (>) and less than (<) symbols.
- Congruent FiguresUnderstand that figures are congruent if one can be obtained from the other by a sequence of rotations, reflections, and translations. Understand how symbols are used to show congruency. Show whether figures are congruent by performing a sequence of slides, turns, and flips.
- 8.G.A.2
# Common Core - 8.G.A.2

Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

- Understand why fractions can be divided by inverting and multiplying. Represent fraction division with fraction strips. Discover that reciprocals result from switching the numerator and denominator. Divide fractions by inverting the second fraction and turning the division into multiplication.
- 6.NS.A.1
# Common Core - 6.NS.A.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.

- Discover that equivalent fractions represent the same part of a whole. Learn that equivalent fractions are found by multiplying the numerator and denominator by the same whole number. Understand when a fraction is in lowest terms. Investigate equivalent fractions using models and pattern blocks.
- ExponentsKnow that exponents indicate how many times a number or expression is used as a factor. Understand the meaning of positive and negative exponents. Discover the connection between exponents and repeated multiplication or division of the base.
- 6.EE.A.1
# Common Core - 6.EE.A.1

Write and evaluate numerical expressions involving whole-number exponents.

- 6.EE.A.2c
# Common Core - 6.EE.A.2c

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).

- 8.EE.A.1
# Common Core - 8.EE.A.1

Know and apply the properties of integer exponents to generate equivalent numerical expressions.

- 8.EE.A.3
# Common Core - 8.EE.A.3

Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.

- Understand that a fraction is a number that compares a part with a whole. Visualize and create different representations of a fraction (circle, bar, set, number line). Investigate improper fractions and mixed numbers. Discover fraction relationships using pattern blocks.
- Learn that a linear equation is an equation whose graph is a straight line. Understand how linear functions can be written in slope-intercept form. Learn how to calculate the slope of a linear function and discover the properties of different slope values. Investigate the y-intercept as the point where a graph intersects the y-axis.
- 8.EE.B.5
# Common Core - 8.EE.B.5

Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

- 8.EE.B.6
# Common Core - 8.EE.B.6

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation

*y*=*mx*+*b*for a line intercepting the vertical axis at*b*. - 8.F.A.3
# Common Core - 8.F.A.3

Interpret the equation

*y = mx + b*as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. - 8.F.B.5
# Common Core - 8.F.B.5

Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

- Understand that a histogram is a type of graph that shows the frequency distribution of data within equal intervals. Create a frequency table from a set of data. Construct a histogram from a frequency table. Understand how to analyze a histogram. Know that the size of the intervals impacts the shape of the histogram.
- 6.SP.B.4
# Common Core - 6.SP.B.4

Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

- 7.SP.B.4
# Common Core - 7.SP.B.4

Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.

- Learn that statistical questions are questions that have more than one answer. Understand that a measure of center summarizes a data set with a single number that represents a typical or center value. Learn how to find the mean, median and mode. Understand when each is most appropriate.
- 6.SP.A.1
# Common Core - 6.SP.A.1

Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.

- 6.SP.A.2
# Common Core - 6.SP.A.2

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.SP.A.3
# Common Core - 6.SP.A.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.B.4
# Common Core - 6.SP.B.4

Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

- 6.SP.B.5
# Common Core - 6.SP.B.5

Summarize numerical data sets in relation to their context, such as by:

- 6.SP.B.5a
# Common Core - 6.SP.B.5a

Reporting the number of observations.

- 6.SP.B.5b
# Common Core - 6.SP.B.5b

Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.

- 6.SP.B.5c
# Common Core - 6.SP.B.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.B.5d
# Common Core - 6.SP.B.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.

- Measures of VariabilityLearn that statistical questions are questions that have more than one answer. Understand that a measure of variability summarizes the spread of data values with a single number. Learn how to find the range, interquartile range, and mean absolute deviation. Understand when each is most appropriate.
- 6.SP.A.1
# Common Core - 6.SP.A.1

Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.

- 6.SP.A.2
# Common Core - 6.SP.A.2

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.SP.A.3
# Common Core - 6.SP.A.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.B.4
# Common Core - 6.SP.B.4

Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

- 6.SP.B.5
# Common Core - 6.SP.B.5

Summarize numerical data sets in relation to their context, such as by:

- 6.SP.B.5a
# Common Core - 6.SP.B.5a

Reporting the number of observations.

- 6.SP.B.5b
# Common Core - 6.SP.B.5b

Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.

- 6.SP.B.5c
# Common Core - 6.SP.B.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.B.5d
# Common Core - 6.SP.B.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.

- Understand why fractions can be multiplied by just multiplying their numerators and multiplying their denominators. Represent fraction multiplication with area models. Discover that multiplying with a fraction can result in a smaller number. Learn how to multiply mixed numbers.
- Understand that Order of Operations gives the rules for the correct order when performing operations. Evaluate expressions by operating on parentheses first, then exponents, then performing multiplication or division from left to right, and finally performing addition or subtraction from left to right.
- 6.EE.A.1
# Common Core - 6.EE.A.1

Write and evaluate numerical expressions involving whole-number exponents.

- 6.EE.A.2c
# Common Core - 6.EE.A.2c

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.EE.A.3
# Common Core - 6.EE.A.3

Apply the properties of operations to generate equivalent expressions.

- 7.RP.A.1
# Common Core - 7.RP.A.1

Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.

- ProbabilityUnderstand that meaning of: "probability”, “experiment”, “outcome”, and “event”. Learn to find both theoretical and experimental probabilities. Discover that theoretical and experimental probabilities become closer as the number of experiments increases. Find probabilities of events using spinners, dice, coins, and cards.
- 7.SP.C.5
# Common Core - 7.SP.C.5

Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

- 7.SP.C.6
# Common Core - 7.SP.C.6

Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.

- 7.SP.C.7a
# Common Core - 7.SP.C.7a

Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.

- 7.SP.C.7b
# Common Core - 7.SP.C.7b

Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.

- ProportionsUnderstand that a proportion is an equation stating that two ratios are equivalent. Discover that the cross products of proportions are equal. Represent proportional relationships by equations. Use proportions to solve real-world problems.
- 7.RP.A.3
# Common Core - 7.RP.A.3

Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

- 7.RP.A.2a
# Common Core - 7.RP.A.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.

- 6.RP.A.3d
# Common Core - 6.RP.A.3d

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

- Pythagorean TheoremDiscover how the Pythagorean Theorem describes the relationship between the lengths of the sides of a right triangle. Understand visual and algebraic proofs of the Pythagorean Theorem. Apply the Pythagorean Theorem to find missing lengths and to calculate distances between points on the coordinate plane.
- 8.G.B.8
# Common Core - 8.G.B.8

Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

- 8.G.B.7
# Common Core - 8.G.B.7

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

- 8.G.B.6
# Common Core - 8.G.B.6

Explain a proof of the Pythagorean Theorem and its converse.

- RatesUnderstand that a rate is a ratio that compares two quantities in different units. Express rates as unit rates. Use unit rates to compare rates and find missing quantities in equivalent rates. Visualize rates and unit rates on a graph.
- 6.RP.A.2
# Common Core - 6.RP.A.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.RP.A.3b
# Common Core - 6.RP.A.3b

Solve unit rate problems including those involving unit pricing and constant speed.

- 7.RP.A.2b
# Common Core - 7.RP.A.2b

Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

- 7.RP.A.2d
# Common Core - 7.RP.A.2d

Explain what 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. - RatiosUnderstand that a ratio is a comparison of two numbers or quantities. Express ratios using words, colons, and fractions. View equivalent ratios using tables, double number lines, and coordinate plane. Find missing values for equivalent ratios.
- 6.RP.A.1
# Common Core - 6.RP.A.1

Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

- 6.RP.A.2
# Common Core - 6.RP.A.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.RP.A.3
# Common Core - 6.RP.A.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.A.3a
# Common Core - 6.RP.A.3a

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.

- Scientific NotationLearn that scientific notation is a way to represent very large or very small numbers using powers of 10. Convert numbers between standard notation and scientific notation. Compare numbers expressed in scientific notation. Perform operations with numbers expressed in scientific notation.
- 8.EE.A.3
# Common Core - 8.EE.A.3

Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.

- 8.EE.A.4
# Common Core - 8.EE.A.4

Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

- Understand that expressions are simplified when replaced by equivalent expressions by combining all like terms and removing all parentheses. Learn to identify terms, like terms, and coefficients. Review and use the distributive property.
- 6.EE.A.2b
# Common Core - 6.EE.A.2b

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.EE.A.3
# Common Core - 6.EE.A.3

Apply the properties of operations to generate equivalent expressions.

- 6.EE.A.4
# Common Core - 6.EE.A.4

Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).

- 7.EE.A.1
# Common Core - 7.EE.A.1

Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

- Understand that solving an equation means to find values of a variable that make an equation true. Discover that you can make changes to an equation as long as you do the same thing to both sides. Learn to solve equations by isolating the variable. Solve one- and two-step equations.
- 6.EE.B.5
# Common Core - 6.EE.B.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.B.6
# Common Core - 6.EE.B.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.B.7
# Common Core - 6.EE.B.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 nonnegative rational numbers. - 7.EE.B.3
# Common Core - 7.EE.B.3

Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.

- 7.EE.B.4a
# Common Core - 7.EE.B.4a

Solve word problems leading to equations of the form

*px*+*q*=*r*and*p*(*x*+*q*) =*r*, where*p*,*q*, and*r*are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. - 8.EE.C.7b
# Common Core - 8.EE.C.7b

Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

- Triangle Angle Sum TheoremDiscover that the angles in a triangle always sum to 180°. Prove the Triangle Sum Theorem visually and algebraically. Review the Alternate Angle Theorem. Use the Triangle Sum Theorem to find missing angles in simple and compound triangles.
- 8.G.A.5
# Common Core - 8.G.A.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.

- Volume of Fractional Rectangular PrismsFill fractional rectangular prisms with appropriately sized cubes that are smaller than unit cubes. Show that the volume of the cubes used to fill the rectangular prism with fractional edges is the same as would be found using the formulas V = 𝓁 × w × and V = B × h. Practice using formulas to find the volume of rectangular prisms with fractional edges.
- Volume of Rectangular PrismsUnderstand that volume is a measure of the space inside a three-dimensional figure. Find volume by counting the number of unit cubes it takes to completely fill a figure. Derive and apple the formulas V = 𝓁 × w × h and V = B × h for rectangular prisms.
- Volume of Right PrismsLearn that a right prism is a figure with bases that are congruent and parallel to each other, and faces that are perpendicular to the bases. Understand and apply the formula V = B × h for any right prism.
- Volume of Special FiguresDerive the volume formulas for pyramids, cylinders, cones, and spheres from known formulas of simpler figures. Understand and apply the volume formulas for these special figures.

# Math Tools

Interactive learning tools and manipulatives to explore and understand math concepts.

- These versatile algebra tiles can be used to explore 1) modeling expressions, 2) simplifying expressions, 3) substitution, 4) adding and subtracting polynomials, 5) solving equations, 6) distribution, 7) multiplying and dividing polynomials, 8) factoring, and 9) completing the square.
- 6.EE.A.1
# Common Core - 6.EE.A.1

Write and evaluate numerical expressions involving whole-number exponents.

- 6.EE.A.2a
# Common Core - 6.EE.A.2a

Write expressions that record operations with numbers and with letters standing for numbers.

- 6.EE.A.2c
# Common Core - 6.EE.A.2c

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.EE.A.3
# Common Core - 6.EE.A.3

Apply the properties of operations to generate equivalent expressions.

- 6.EE.A.4
# Common Core - 6.EE.A.4

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.EE.B.5
# Common Core - 6.EE.B.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.B.6
# Common Core - 6.EE.B.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.B.7
# Common Core - 6.EE.B.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 nonnegative rational numbers. - 6.EE.C.9
# Common Core - 6.EE.C.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.

- 7.EE.A.1
# Common Core - 7.EE.A.1

Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

- 7.EE.B.4a
# Common Core - 7.EE.B.4a

Solve word problems leading to equations of the form

*px*+*q*=*r*and*p*(*x*+*q*) =*r*, where*p*,*q*, and*r*are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. - 8.EE.C.7a
# Common Core - 8.EE.C.7a

Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form

*x*=*a*,*a*=*a*, or*a*=*b*results (where*a*and*b*are different numbers). - 8.EE.C.7b
# Common Core - 8.EE.C.7b

Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

- Virtual base 10 blocks to model 1) number sense, 2) place value, 3) counting, 4) decimals, 5) addition with regrouping, and 6) subtraction with regrouping, and more. Includes stackable ones (units), tens (rods), hundreds (flats) and thousands (cubes).
- 6.NS.B.3
# Common Core - 6.NS.B.3

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

- 6.NS.C.7
# Common Core - 6.NS.C.7

Understand ordering and absolute value of rational numbers.

- Virtual colored cubes that link together to explore many math concepts: 1) counting, 2) representing numbers, 3) patterns, 4) place value, 5) adding and subtracting, 6) multiplying and dividing, 7) graphing
- Virtual hundred board to explore numbers, counting, number patterns, operations, multiples, rounding, place value, and more.
- Customizable number line includes interactive markers, arrows, inequalities, distances, and multiples. A powerful tool to teach all kinds of number and operations concepts.
- 6.EE.B.8
# Common Core - 6.EE.B.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.NS.C.7c
# Common Core - 6.NS.C.7c

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.NS.C.7a
# Common Core - 6.NS.C.7a

Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.

- 6.NS.C.7
# Common Core - 6.NS.C.7

Understand ordering and absolute value of rational numbers.

- 6.NS.C.6c
# Common Core - 6.NS.C.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 a coordinate plane.

- 6.NS.C.6a
# Common Core - 6.NS.C.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 that 0 is its own opposite.

- 6.NS.B.4
# Common Core - 6.NS.B.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.

- 7.NS.A.1
# Common Core - 7.NS.A.1

Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

- 7.NS.A.1c
# Common Core - 7.NS.A.1c

Understand subtraction of rational numbers as adding the additive inverse,

*p*–*q*=*p*+ (–*q*). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. - 7.EE.B.4b
# Common Core - 7.EE.B.4b

Solve word problems leading to inequalities of the form

*px*+*q*>*r*or*px*+*q*<*r*, where*p*,*q*, and*r*are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. - Number NudgetThis dynamic new manipulative develops conceptual understanding of numbers, counting, addition, subtraction, place value, regrouping and more.
- These versatile pattern blocks can be used to 1) Build and explore patterns, 2) Understand fractions, 3) Explore geometric shapes, 4) Investigate symmetry and transformations, 5) Model problem solving, 5) Discover tessellations and tiling, 6) Construct pictures.

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- Alpine CyclistDesign a mountain race course that meets length and slope specifications using Pythagorean Theorem, slope, and scale.
- 8.EE.B.5
# Common Core - 8.EE.B.5

Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

- 6.G.A.1
# Common Core - 6.G.A.1

- 8.EE.B.6
# Common Core - 8.EE.B.6

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation

*y*=*mx*+*b*for a line intercepting the vertical axis at*b*. - 8.G.B.6
# Common Core - 8.G.B.6

Explain a proof of the Pythagorean Theorem and its converse.

- 8.G.B.7
# Common Core - 8.G.B.7

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

- 8.G.B.8
# Common Core - 8.G.B.8

Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

- Ancient PyramidBuild a pyramid that meets volume and surface area specifications.
- 6.G.A.1
# Common Core - 6.G.A.1

- 6.G.A.2
# Common Core - 6.G.A.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 = l w h*and*V = b h*to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. - 6.G.A.4
# Common Core - 6.G.A.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.

- 7.G.B.4
# Common Core - 7.G.B.4

Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

- 7.G.B.6
# Common Core - 7.G.B.6

- 8.G.C.9
# Common Core - 8.G.C.9

Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

- BisectionLearn how to use a compass, a protractor, and a ruler to bisect a segment and an angle.
- FREEFree topics can be used without an account, but have no reporting features enabled.
- Congruent Line Segments and AnglesExamine how congruent angles have the same measure and learn how to construct congruent angles.
- 7.G.A.2
# Common Core - 7.G.A.2

Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

- 8.G.A.1a
# Common Core - 8.G.A.1a

Lines are taken to lines, and line segments to line segments of the same length.

- 8.G.A.1b
# Common Core - 8.G.A.1b

Angles are taken to angles of the same measure.

- Diagonals of a PolygonExplore how diagonals of a polygon join two nonconsecutive vertices and how to calculate the number of diagonals in a polygon.
- Flight PlanningPlan an optimal flight route that minimizes time and delay using algebraic equations, measurement, and fractions.
- Healthy ForestPlant a forest that offsets a city\'s increasing carbon dioxide emissions using rates, graphs, and linear functions.
- 6.RP.A.2
# Common Core - 6.RP.A.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.RP.A.3
# Common Core - 6.RP.A.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.A.3b
# Common Core - 6.RP.A.3b

Solve unit rate problems including those involving unit pricing and constant speed.

- 6.EE.C.9
# Common Core - 6.EE.C.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.

- 7.RP.A.2b
# Common Core - 7.RP.A.2b

Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

- 8.EE.B.5
# Common Core - 8.EE.B.5

Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

- 8.F.B.4
# Common Core - 8.F.B.4

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (

*x, y*) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. - Integer Addition and SubtractionLearn how to add and subtract signed integers using counter and number line models. Discover how to rewrite addition as subtraction and subtraction as addition.
- 7.NS.A.3
# Common Core - 7.NS.A.3

Solve real-world and mathematical problems involving the four operations with rational numbers.

- 7.NS.A.1c
# Common Core - 7.NS.A.1c

Understand subtraction of rational numbers as adding the additive inverse,

*p*–*q*=*p*+ (–*q*). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. - 7.NS.A.1b
# Common Core - 7.NS.A.1b

Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

- Integer DivisionInvestigate division of integers using counters and directional line models on the number line. Explore how the sign of the divisor and dividend determines whether the quotient is positive or negative.
- 7.NS.A.2b
# Common Core - 7.NS.A.2b

Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If

*p*and*q*are integers, then –(*p*/*q*) = (–*p*)/*q*=*p*/(–*q*). Interpret quotients of rational numbers by describing real-world contexts. - 7.NS.A.3
# Common Core - 7.NS.A.3

Solve real-world and mathematical problems involving the four operations with rational numbers.

- Integer MultiplicationUnderstand integer multiplication as repeated addition or subtraction using visual models. Investigate how the sign of the factors determines whether the product is positive or negative.
- FREEFree topics can be used without an account, but have no reporting features enabled.
- 7.NS.A.2a
# Common Core - 7.NS.A.2a

Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.

- 7.NS.A.3
# Common Core - 7.NS.A.3

Solve real-world and mathematical problems involving the four operations with rational numbers.

- Package SortingProgram a scanner to recognize packages according to characteristic shape properties using sides, angles, and diagonals.
- Passenger BoardingDevelop a strategy to board passengers onto a plane as quickly as possible using data analysis, scatter plots, histograms, box plots, and probability.
- 6.SP.B.4
# Common Core - 6.SP.B.4

Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

- 7.SP.C.5
# Common Core - 7.SP.C.5

Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

- 7.SP.C.6
# Common Core - 7.SP.C.6

Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.

- 7.SP.C.7
# Common Core - 7.SP.C.7

Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.

- 7.SP.C.7a
# Common Core - 7.SP.C.7a

Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.

- 7.SP.C.7b
# Common Core - 7.SP.C.7b

Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.

- 8.SP.A.1
# Common Core - 8.SP.A.1

Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

- 8.SP.A.2
# Common Core - 8.SP.A.2

Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

- Patterns I: Finding and ExtendingLearn how to identify a pattern and predict the next term. Understand that there can be multiple ways to analyze and view a pattern.
- Patterns II: Function RulesRealize that a number pattern can be described by a rule using term numbers as input. Learn how to analyze a number pattern and write a rule for the pattern using function notation.
- PercentUnderstand numbers expressed as percents and be able to translate from rates and decimals.
- PerimeterExamine how perimeter is the distance around a shape or figure and investigate perimiter formulas for some common shapes.
- Perpendicular LinesExplore how perpendicular lines intersect to form right angles and how the slopes of perpendicular lines equals negative one.
- Phone PlanDesign a phone plan to maximize revenue using data analysis, linear functions, averages, and problem solving.
- 6.EE.C.9
# Common Core - 6.EE.C.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.

- 6.SP.A.2
# Common Core - 6.SP.A.2

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.

- 8.F.A.3
# Common Core - 8.F.A.3

Interpret the equation

*y = mx + b*as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. - 8.F.B.4
# Common Core - 8.F.B.4

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (

*x, y*) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. - 8.F.B.5
# Common Core - 8.F.B.5

Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

- 8.SP.A.1
# Common Core - 8.SP.A.1

Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

- 8.SP.A.2
# Common Core - 8.SP.A.2

Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

- Place ValueLearn how place value determines the value of digits based on their position in a number.
- RadiusUnderstand how radius is the distance from the center of a circle to any point on the circle.
- Regular PolygonsUnderstand that regular polygons are plane figures that are equilateral and equiangular.
- RoundingDiscover how rounding is a process which produces another number that is close to the original number. Learn how to round numbers up or down to the nearest rounding number.
- FREEFree topics can be used without an account, but have no reporting features enabled.
- Scatter PlotExamine scatter plots as a way to graphically summarize how two sets of data are related and understand how to interpret positive, negative, weak and strong correlations.
- 8.SP.A.1
# Common Core - 8.SP.A.1

Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

- 8.SP.A.2
# Common Core - 8.SP.A.2

Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

- SequencesExplore the characteristics of two special sequences: arithmetic and geometric sequences.
- Similar FiguresExplore the angle and side congruency of similar shapes.
- 8.G.A.4
# Common Core - 8.G.A.4

Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

- 8.G.A.5
# Common Core - 8.G.A.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.

- Sports EmpireStart and grow a Segway Polo league into an international phenomena with billions of fans.
- Stem and Leaf PlotsUnderstand how stem and leaf plots display the frequency of the values in a data set by groups. Learn how to construct stem and leaf plots.
- Surface AreaInvestigate surface area as the total area of the surface of a space figure. Derive surface area formulas for common three dimensional figures.
- FREEFree topics can be used without an account, but have no reporting features enabled.
- 6.G.A.4
# Common Core - 6.G.A.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.

- 7.G.B.6
# Common Core - 7.G.B.6

- VolumeExamine volume as the interior space of a three dimensional figure and derive volume formulas for common objects.
- 6.G.A.2
# Common Core - 6.G.A.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 = l w h*and*V = b h*to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. - 7.G.B.6
# Common Core - 7.G.B.6

- 8.G.C.9
# Common Core - 8.G.C.9

Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

- X- and Y-InterceptLearn how x- and y-intercepts are points at which a graph intercepts the x- and y-intercepts as well as how to determine these points.