Grade 8 Math: build it by talking it through

A taste of a lesson on The Pythagorean Theorem and Its Converse

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The 31 topics in Grade 8 Math

Each topic builds on the ones before it. The tutor checks what you already know and starts there.

1. Rational Numbers as Repeating Decimals

   Every rational number has a decimal expansion that either terminates or repeats eventually. Teach converting a fraction to a decimal by long division and recognizing the repeating block, and converting a repeating decimal back to a fraction using an algebraic trick (let x equal the decimal, multiply by a power of 10 to align repeats, subtract). Students should explain why denominators with only factors 2 and 5 terminate.

2. Irrational Numbers and Decimal Expansions

   A number that is not rational is irrational; examples include square roots of non-perfect squares and pi. Every real number has a decimal expansion; irrational ones never repeat. Students should be able to explain why a non-repeating decimal cannot be a fraction and classify numbers as rational or irrational.

   Builds on Rational Numbers as Repeating Decimals

3. Comparing and Locating Irrational Numbers

   Use rational approximations (truncated or rounded decimals) to compare sizes of irrational numbers such as sqrt(2) and sqrt(7) or pi, place them approximately on a number line, and estimate values of expressions like pi squared or sqrt(50). Students should refine approximations stepwise until the order is clear.

   Builds on Irrational Numbers and Decimal Expansions

4. Integer Exponents and Exponent Rules

   Define powers with zero and negative integer exponents (a^0 = 1, a^-n = 1/a^n). Apply the product, quotient, power-of-a-power, and power-of-a-product rules to generate equivalent numerical expressions, such as 3^2 times 3^-5 = 3^-3 = 1/27. Students should be able to justify each rule from the meaning of exponents.

5. Square Roots and Cube Roots

   A square root of p is a number whose square is p; a cube root is a number whose cube is p. Use the radical symbols to solve x^2 = p and x^3 = p for positive rational p, knowing that x^2 = p has two solutions (positive and negative) when p is positive. Evaluate roots of small perfect squares and cubes and estimate roots of non-perfect squares between consecutive integers.

   Builds on Integer Exponents and Exponent Rules

6. Scientific Notation

   Write numbers as a single nonzero digit times an integer power of 10. Convert between standard and scientific form for very large and very small numbers, and use it to estimate quantities and compare how many times larger one is than another (for example, comparing the mass of Earth to the Moon). Students should choose a sensible unit for measurements at very large or very small scales.

   Builds on Integer Exponents and Exponent Rules

7. Operations with Scientific Notation

   Add, subtract, multiply, and divide numbers in scientific notation, and mix decimal and scientific notation in one problem. Multiply or divide powers of 10 with the exponent rules, then renormalize the coefficient to a single digit before the power of 10. Students should check that answers are reasonable and written in proper scientific notation.

   Builds on Scientific Notation

8. Graphing Proportional Relationships and Unit Rate as Slope

   A proportional relationship y = kx has a graph that is a straight line through the origin, and the constant k is the unit rate and the slope. Graph proportional relationships from tables, equations, or descriptions, and compare two proportional relationships given in different forms (for example, a distance-time graph versus a distance-time equation) to decide which has the greater rate.

9. Slope and Similar Triangles

   Use similar right triangles drawn between two points on a non-vertical line to show that the ratio of rise to run is the same for any two distinct points, so slope m is constant. Derive y = mx for lines through the origin and y = mx + b for lines with y-intercept b, by reasoning from similar triangles and the slope formula.

   Builds on Graphing Proportional Relationships and Unit Rate as Slope

10. Solving Linear Equations in One Variable

    Solve equations of the form ax + b = cx + d with rational coefficients by inverting operations and collecting variable terms on one side. Use the distributive property to expand expressions and combine like terms. Students should check solutions by substitution.

    Builds on Graphing Proportional Relationships and Unit Rate as Slope

11. Number of Solutions of Linear Equations

    Classify linear equations as having one solution, infinitely many solutions, or no solution by transforming them step by step into simpler forms. Give examples of each type; for instance, 2(x+1) = 2x+2 is always true (infinitely many) and x+1 = x+3 is never true (no solution). Explain what the final form means.

    Builds on Solving Linear Equations in One Variable

12. Systems of Linear Equations as Intersecting Lines

    A solution to a system of two linear equations in two variables is an ordered pair that satisfies both equations, which corresponds to the point where the two graphs intersect. Explain why, graph pairs of lines to find the intersection, and recognize parallel lines (no solution) and identical lines (infinitely many solutions).

    Builds on Slope and Similar Triangles, Number of Solutions of Linear Equations

13. Solving Systems Algebraically and by Inspection

    Solve systems of two linear equations using substitution and elimination, and estimate solutions by graphing. Solve simple cases by inspection, such as recognizing that 3x+2y=5 and 3x+2y=6 cannot both hold. Check solutions in both original equations.

    Builds on Systems of Linear Equations as Intersecting Lines

14. Modeling Problems with Systems of Equations

    Translate real-world and mathematical situations into two linear equations in two variables and solve them, including deciding whether the line through one pair of points intersects the line through another pair. Interpret the solution in context and check that it is reasonable.

    Builds on Solving Systems Algebraically and by Inspection

15. Functions: Input, Output, and Graphs

    A function assigns to each input exactly one output; a relation with one input giving two outputs is not a function. The graph of a function is the set of ordered pairs (input, output). Use the vertical line test and tables or mappings to decide whether a relation is a function.

    Builds on Graphing Proportional Relationships and Unit Rate as Slope

16. Linear Functions: y = mx + b

    Interpret y = mx + b as defining a linear function whose graph is a straight line, with m the rate of change and b the initial value. Construct a linear function from a description, from two (x, y) values, or from a table, and identify rate of change and initial value. Give examples of nonlinear functions such as A = s^2 for the area of a square.

    Builds on Slope and Similar Triangles, Functions: Input, Output, and Graphs

17. Comparing Functions in Different Representations

    Compare two functions each given as an equation, a graph, a table, or a verbal description. For example, decide which of two linear functions has the greater rate of change or the greater initial value when one is given by a table and the other by an equation.

    Builds on Linear Functions: y = mx + b

18. Qualitative Features of Function Graphs

    Describe the relationship between two quantities from a graph: intervals where the function is increasing or decreasing, whether it is linear or nonlinear, and what the graph says about the situation. Sketch a graph that exhibits qualitative features described in words, such as a bike ride that speeds up, stops, then returns.

    Builds on Linear Functions: y = mx + b

19. Rigid Transformations: Translations, Reflections, Rotations

    Translations slide, reflections flip across a line, and rotations turn about a point. Experimentally verify that these preserve lines (lines map to lines), segment lengths, angle measures, and parallelism. Describe a transformation's effect on coordinates, such as (x,y) to (x+a, y+b) for a translation, reflection across the x-axis to (x,-y), and rotation of 90 degrees about the origin.

20. Dilations and Coordinate Effects of Transformations

    A dilation with center at the origin and scale factor k maps (x,y) to (kx,ky); it preserves angles but changes lengths by factor k. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates, and apply sequences of these to figures on the coordinate plane.

    Builds on Rigid Transformations: Translations, Reflections, Rotations

21. Congruence Through Rigid Motions

    Two figures are congruent if one can be obtained from the other by a sequence of rotations, reflections, and translations. Given two congruent figures, describe a sequence of rigid motions that maps one onto the other, matching corresponding vertices and sides.

    Builds on Dilations and Coordinate Effects of Transformations

22. Similarity Through Transformations and AA

    Two figures are similar if one can be obtained from the other by a sequence of rigid motions and dilations. Given similar figures, describe such a sequence. Use informal arguments about angles to show that two triangles with two pairs of equal angles are similar (AA criterion).

    Builds on Congruence Through Rigid Motions

23. Angle Sums, Exterior Angles, and Parallel Lines

    Informally argue that the angles of a triangle sum to 180 degrees and that an exterior angle equals the sum of the two remote interior angles. When parallel lines are cut by a transversal, alternate interior, alternate exterior, and corresponding angles are congruent, and same-side interior angles are supplementary. Apply these to find unknown angles.

    Builds on Rigid Transformations: Translations, Reflections, Rotations

24. Using Angle Relationships to Establish Similarity

    Use the triangle angle sum and parallel-line angle facts to justify that two triangles are similar by the AA criterion, and apply similarity to find unknown side lengths. Connect this to the slope argument that similar right triangles on a line have equal rise-to-run ratios.

    Builds on Similarity Through Transformations and AA, Angle Sums, Exterior Angles, and Parallel Lines

25. The Pythagorean Theorem and Its Converse

    For a right triangle with legs a and b and hypotenuse c, a^2 + b^2 = c^2. Explain a proof, such as rearranging squares or using similar triangles, and state the converse: if a^2 + b^2 = c^2 then the triangle is right. Use the theorem to find unknown sides.

    Builds on Using Angle Relationships to Establish Similarity

26. Applying the Pythagorean Theorem in Two and Three Dimensions

    Use the Pythagorean Theorem to find unknown side lengths in real-world problems, including ladders, diagonals of rectangles, and diagonals of rectangular boxes in three dimensions. Students should identify the right triangle inside a figure before computing.

    Builds on The Pythagorean Theorem and Its Converse

27. Distance Between Points in the Coordinate Plane

    Apply the Pythagorean Theorem to find the distance between two points (x1, y1) and (x2, y2) by forming a right triangle with horizontal and vertical legs, giving d = sqrt((x2-x1)^2 + (y2-y1)^2).

    Builds on Slope and Similar Triangles, Applying the Pythagorean Theorem in Two and Three Dimensions

28. Volume of Cones, Cylinders, and Spheres

    Know and apply V = pi r^2 h for cylinders, V = (1/3) pi r^2 h for cones, and V = (4/3) pi r^3 for spheres. Explain the 1/3 relationship between cones and cylinders with equal base and height, and solve real-world problems involving these solids, including composite and inverse problems.

    Builds on Applying the Pythagorean Theorem in Two and Three Dimensions

29. Scatter Plots and Patterns of Association

    Construct scatter plots from bivariate measurement data and describe patterns: clustering, outliers, positive or negative association, linear or nonlinear association, and no association. Students should describe each pattern in context.

    Builds on Linear Functions: y = mx + b

30. Fitting and Interpreting a Line of Best Fit

    For a scatter plot with linear association, informally fit a straight line by eye, assess fit by how closely points cluster around it, and use the equation of the model to predict. Interpret slope and intercept in context, for example a slope of 1.5 cm per hour meaning the plant grows 1.5 cm each hour and the intercept as the starting height.

    Builds on Scatter Plots and Patterns of Association

31. Two-Way Tables for Categorical Data

    Construct and interpret two-way tables summarizing two categorical variables, with frequencies and relative frequencies (by row or column). Use them to look for patterns of association, such as whether one group is more likely to have a given response than another.

    Builds on Scatter Plots and Patterns of Association

Practice questions in Grade 8 Math

A few of the kinds of questions you will work through, with answers.