Focus Areas for Eighth Grade
- Formulating and reasoning about expressions and equations
- Understanding the concept of a function
- Analyzing two- and three-dimensional space and figures
- Working with irrational numbers, integer exponents, and scientific notation
Math Standards by Domain
The Number System (8.NS)
What Students Learn:
Students learn to distinguish between rational numbers (which can be expressed as fractions) and irrational numbers (like π and √2). They understand that all numbers fall into the real number system, and they learn to approximate irrational numbers using rational numbers for practical calculations.
Key Skills:
- Understand that irrational numbers cannot be expressed as simple fractions
- Know that the decimal expansion of irrational numbers is non-repeating and non-terminating
- Convert repeating decimals to fractions (e.g., 0.333... = 1/3)
- Use rational approximations for irrational numbers (√2 ≈ 1.414)
- Compare and order rational and irrational numbers on a number line
- Estimate square roots and cube roots of non-perfect squares/cubes
Rational: 3/4, 0.75, -2, 0.333... (all can be written as fractions)
Irrational: π (3.14159...), √2 (1.41421...), √5 (2.236...)
Estimate: √50 is between 7 and 8 (since 7² = 49 and 8² = 64), closer to 7
Order: -2 < √3 < 2 < √5 < 3 < π (approximately: -2 < 1.73 < 2 < 2.24 < 3 < 3.14)
Expressions and Equations (8.EE)
What Students Learn:
Students master the rules of exponents including negative and zero exponents, and learn to work with very large and very small numbers using scientific notation. They apply these skills to real-world problems involving exponential growth and extremely large or small quantities.
Key Skills:
- Apply exponent rules: x^m × x^n = x^(m+n), (x^m)^n = x^(mn), x^0 = 1, x^(-n) = 1/x^n
- Simplify expressions with integer exponents
- Work with square roots (√) and cube roots (∛)
- Write numbers in scientific notation (a × 10^n where 1 ≤ a < 10)
- Perform operations with scientific notation (add, subtract, multiply, divide)
- Compare numbers written in scientific notation
- Apply scientific notation to real-world contexts (astronomy, microscopic measurements)
Simplify: 3² × 3³ = 3^(2+3) = 3⁵ = 243
Simplify: (2³)² = 2^(3×2) = 2⁶ = 64
Simplify: 5^(-2) = 1/5² = 1/25
Example (Scientific Notation):Write 45,000,000 in scientific notation: 4.5 × 10⁷
Write 0.000032 in scientific notation: 3.2 × 10^(-5)
Multiply: (2 × 10³) × (3 × 10⁴) = 6 × 10⁷ = 60,000,000
Real-world: Distance to sun ≈ 9.3 × 10⁷ miles, size of bacteria ≈ 2 × 10^(-6) meters
What Students Learn:
Students deepen their understanding of proportional relationships by graphing them and interpreting the unit rate as the slope. They connect proportional relationships to similar triangles and understand how slope is derived from the ratio of vertical change to horizontal change.
Key Skills:
- Graph proportional relationships (y = kx) and identify the constant of proportionality (k)
- Interpret unit rate as the slope of the graph
- Compare two proportional relationships represented in different ways (equations, graphs, tables)
- Use similar triangles to explain why slope is constant between any two points on a line
- Understand that slope m = (y₂-y₁)/(x₂-x₁) represents the rate of change
- Apply proportional reasoning to solve real-world problems
Graph y = 3x: For every increase of 1 in x, y increases by 3 (slope = 3)
Unit rate: A car travels 120 miles in 2 hours → rate = 60 mph (the slope)
Example (Similar Triangles):On line y = 2x, triangle from (0,0) to (1,2) to (1,0) is similar to triangle from (0,0) to (3,6) to (3,0)
Both have slope = rise/run = 2/1 = 6/3 = 2
This shows slope is constant for any two points on the line
What Students Learn:
Students learn to solve increasingly complex linear equations including those with variables on both sides and with rational coefficients. They are introduced to systems of linear equations and learn multiple methods to solve them, understanding how the solution represents the intersection point of two lines.
Key Skills:
- Solve linear equations with rational number coefficients (fractions and decimals)
- Solve equations with variables on both sides (3x + 5 = 2x + 9)
- Solve equations requiring distribution and combining like terms
- Identify equations with one solution, no solution, or infinitely many solutions
- Solve systems of two linear equations algebraically (substitution and elimination)
- Solve systems graphically by finding the intersection point
- Understand that the solution to a system is the (x, y) pair that satisfies both equations
- Apply systems of equations to real-world problems
Solve: 3x + 7 = 2x + 12 → x = 5
Solve: 2(x - 3) + 4 = 10 → 2x - 6 + 4 = 10 → 2x - 2 = 10 → x = 6
No solution: 2x + 3 = 2x + 5 (impossible, 3 ≠ 5)
Example (Systems):Solve: y = 2x + 1 and y = -x + 7
Substitution: 2x + 1 = -x + 7 → 3x = 6 → x = 2, y = 5. Solution: (2, 5)
Real-world: Two phone plans - Plan A: $20 + $0.10/min, Plan B: $10 + $0.15/min. When are they equal?
Functions (8.F)
What Students Learn:
Students are introduced to the formal concept of a function as a rule that assigns exactly one output to each input. They learn to identify functions from tables, graphs, and equations, and to compare different functions by analyzing their rates of change and initial values.
Key Skills:
- Understand that a function assigns exactly one output (y-value) to each input (x-value)
- Use function notation: f(x) means "the function f of x"
- Identify whether a relationship is a function (vertical line test for graphs)
- Compare two functions represented in different forms (table, graph, equation, verbal description)
- Identify rate of change (slope) and initial value (y-intercept) from different representations
- Interpret f(3) = 7 as "when input is 3, output is 7"
- Sketch graphs of functions given verbal descriptions of relationships
Is {(1,2), (2,4), (3,6)} a function? Yes - each input has exactly one output
Is {(1,2), (1,3), (2,4)} a function? No - input 1 has two outputs (2 and 3)
Function notation: If f(x) = 3x + 2, then f(4) = 3(4) + 2 = 14
Example (Comparing Functions):Function A: y = 2x + 3 (rate of change = 2, initial value = 3)
Function B from table: (0,5), (1,8), (2,11) (rate of change = 3, initial value = 5)
Function B grows faster (rate of 3 vs. 2) but starts higher
What Students Learn:
Students learn to create function models for real-world linear relationships and interpret graphs qualitatively. They describe how functions behave (increasing, decreasing, constant) and relate graph features to real-world contexts like time, distance, and speed.
Key Skills:
- Construct functions to model linear relationships between quantities
- Determine rate of change (slope) and initial value from context
- Write equations in y = mx + b form from word problems
- Describe qualitative features of graphs: increasing, decreasing, constant intervals
- Identify where a function increases or decreases most rapidly
- Relate features of graphs to the context (e.g., flat line = no change)
- Sketch graphs from verbal descriptions of situations
- Interpret x-intercepts and y-intercepts in context
A tank contains 50 gallons and drains at 5 gallons per minute. Model: y = 50 - 5x
y = amount remaining, x = minutes, initial value = 50, rate = -5 (decreasing)
When is tank empty? Set y = 0: 0 = 50 - 5x → x = 10 minutes
Example (Qualitative Features):Distance-time graph of a car trip: Increasing slope = accelerating, flat = stopped, decreasing = returning
Temperature graph: Increases from midnight to 3pm (warming), decreases from 3pm to midnight (cooling)
Steeper slope = faster rate of change
Geometry (8.G)
What Students Learn:
Students explore transformations (rotations, reflections, translations, and dilations) and understand how these transformations relate to congruence and similarity. They learn that congruent figures result from rigid transformations (preserving size and shape) while similar figures result from dilations combined with rigid transformations.
Key Skills:
- Perform and describe rotations, reflections, and translations on the coordinate plane
- Verify properties of transformations using geometric tools and software
- Understand that two figures are congruent if one can be obtained from the other by rigid transformations
- Describe a sequence of transformations that maps one figure onto another
- Understand that two figures are similar if one can be obtained from the other by dilations and rigid transformations
- Verify that corresponding sides of similar figures are proportional
- Use similarity transformations to establish the AA criterion for triangle similarity
- Explain angle relationships in parallel lines cut by a transversal using transformations
Triangle ABC with vertices A(1,1), B(4,1), C(1,3) is reflected over y-axis to get A'(-1,1), B'(-4,1), C'(-1,3)
The triangles are congruent - same size and shape, just flipped
Example (Similarity):Dilate triangle with vertices (2,2), (4,2), (2,6) by scale factor 2 from origin → (4,4), (8,4), (4,12)
New triangle is similar to original - same shape but twice as large
Corresponding sides: 2/4 = 4/8 = 4/8 = 1/2 (proportional)
Example (Transformations):To map triangle ABC to A'B'C': Rotate 90° counterclockwise, then translate right 3 units
What Students Learn:
Students learn the Pythagorean Theorem (a² + b² = c²) and understand why it works through geometric proofs. They apply this fundamental theorem to solve real-world problems involving right triangles, find distances on the coordinate plane, and determine whether triangles are right triangles.
Key Skills:
- Understand and explain a proof of the Pythagorean Theorem
- Apply the Pythagorean Theorem: a² + b² = c² (where c is the hypotenuse)
- Find the length of an unknown side in a right triangle
- Apply the theorem to find distances between points on the coordinate plane
- Determine whether a triangle is a right triangle using the converse of the theorem
- Solve real-world problems: heights, distances, diagonal measurements
- Apply the theorem in three dimensions (diagonal of a rectangular prism)
- Understand Pythagorean triples (3-4-5, 5-12-13, 8-15-17)
Right triangle with legs a = 6 and b = 8. Find hypotenuse c.
6² + 8² = c² → 36 + 64 = c² → 100 = c² → c = 10
Example (Distance on Coordinate Plane):Distance from (1,2) to (4,6): Form right triangle with legs 3 (horizontal) and 4 (vertical)
d² = 3² + 4² = 9 + 16 = 25 → d = 5 units
Example (Real-World):A ladder is 13 feet long, leaning against a wall. The base is 5 feet from the wall. How high does it reach?
5² + h² = 13² → 25 + h² = 169 → h² = 144 → h = 12 feet
Example (Checking Right Triangle):Is a triangle with sides 7, 24, 25 a right triangle? Check: 7² + 24² = 49 + 576 = 625 = 25². Yes!
What Students Learn:
Students learn the volume formulas for three-dimensional curved shapes: cylinders, cones, and spheres. They apply these formulas to solve real-world problems and understand the relationships between these shapes (e.g., a cone has 1/3 the volume of a cylinder with the same base and height).
Key Skills:
- Know and apply the volume formula for cylinders: V = πr²h
- Know and apply the volume formula for cones: V = (1/3)πr²h
- Know and apply the volume formula for spheres: V = (4/3)πr³
- Understand the relationship between cylinder and cone volumes (cone = 1/3 cylinder)
- Solve problems involving composite figures (combining shapes)
- Use volume formulas to find missing dimensions when volume is known
- Apply formulas to real-world contexts: containers, buildings, planets
- Work with both exact answers (in terms of π) and approximations (using 3.14 for π)
Find volume of cylinder with radius 3 cm and height 10 cm
V = πr²h = π(3²)(10) = 90π cm³ ≈ 282.7 cm³
Example (Cone):Find volume of cone with radius 4 inches and height 9 inches
V = (1/3)πr²h = (1/3)π(4²)(9) = 48π in³ ≈ 150.8 in³
Example (Sphere):Find volume of sphere with radius 6 meters
V = (4/3)πr³ = (4/3)π(6³) = (4/3)π(216) = 288π m³ ≈ 904.3 m³
Example (Real-World):An ice cream cone (cone shape) is 12 cm tall with diameter 6 cm (radius 3 cm). How much ice cream fits?
V = (1/3)π(3²)(12) = 36π cm³ ≈ 113.1 cm³
Statistics and Probability (8.SP)
What Students Learn:
Students learn to analyze relationships between two quantitative variables using scatter plots. They identify patterns of association (positive, negative, or no correlation), fit lines to data, and use these models to make predictions. They also work with two-way tables to analyze categorical data.
Key Skills:
- Construct scatter plots for bivariate measurement data (data with two variables)
- Describe patterns in scatter plots: positive association, negative association, or no association
- Identify linear and non-linear associations from scatter plots
- Describe clusters and outliers in data
- Informally fit a straight line to data showing linear association
- Use the equation of a linear model to solve problems and make predictions
- Interpret slope and y-intercept of fitted lines in context
- Construct and interpret two-way tables to analyze categorical data
- Calculate relative frequencies and use them to identify associations between variables
Plot height vs. shoe size for students: Shows positive association (taller students tend to have larger shoe sizes)
Plot study time vs. test scores: Positive linear association (more study time → higher scores)
Plot temperature vs. hot chocolate sales: Negative association (warmer weather → fewer sales)
Example (Line of Best Fit):For study time (x) vs. test score (y), fitted line: y = 5x + 60
Interpretation: Starting score of 60, each hour studied adds 5 points
Prediction: If study 8 hours, expect score of 5(8) + 60 = 100
Example (Two-Way Table):Survey: 100 students - preference for cats vs. dogs by gender
Boys: 30 prefer dogs, 20 prefer cats; Girls: 25 prefer dogs, 25 prefer cats
Relative frequencies show boys prefer dogs more than girls (60% vs. 50%)