What does a student learn in StateTennessee,GradeGrade 8,SubjectMathematics?

Irrational numbers like pi or the square root of 2 can't be written as simple fractions. Students learn to recognize them and find nearby fractions or decimals that get close enough to work with.

Some numbers, like pi or the square root of 2, never settle into a repeating or stopping decimal. Students learn to tell those irrational numbers apart from fractions and to convert repeating decimals back into fractions.

Students place numbers like the square root of 2 or pi on a number line by finding the two nearest fractions it falls between. They use that estimate to compare and order numbers that don't work out to clean decimals.

Radicals and integer exponents are shorthand for repeated multiplication or roots. Students read, write, and calculate expressions like 2 to the power of 8 or the square root of 64, and use the rules that govern how those expressions behave.

Exponent rules let students rewrite multiplication and division of powers into simpler forms. Students use those rules to show that two expressions with different-looking exponents are actually equal.

Students solve equations like x² = 25 or x³ = 8 by finding the number that was squared or cubed to get there. They work with square roots and cube roots of simple whole numbers they can calculate by hand.

Students use scientific notation to describe very large or very small numbers, like the distance to a star or the size of a cell. They also compare two of those numbers to see how many times bigger one is than the other.

Students write very large or very small numbers in scientific notation and pick units that make the size easy to grasp, like using millimeters instead of meters when measuring slow geological change.

Proportional relationships, straight-line graphs, and linear equations are three ways of describing the same pattern. Students learn to move between all three and explain what each one shows.

Students graph proportional relationships and identify the slope as the unit rate. They compare two proportional relationships, even when one is shown as a table and the other as a graph.

Similar triangles show why a straight line has the same steepness everywhere. Students use that idea to work with the equations y = mx and y = mx + b, connecting the slope and starting point to points on a graph.

Students write and solve equations with one unknown, compare quantities using inequality symbols, and find where two equations intersect. This is the algebra behind most real-world problems involving rates, prices, and unknowns.

Students solve equations with one unknown, like 3x + 5 = 20, by using inverse operations to get the variable alone. They also recognize when an equation has one solution, no solution, or is true for any number.

Students solve one-variable equations by simplifying them step by step until the answer becomes clear. That process reveals whether the equation has one solution, no solution, or is true for every number.

Solving equations where the numbers include fractions or decimals, and where students may need to simplify both sides first by distributing and combining similar terms before finding the answer.

Students find where two straight lines cross on a graph. That intersection point is the solution, the one pair of numbers that satisfies both equations at once.

When two straight lines are graphed on the same grid, the point where they cross is the solution to both equations at once. Students learn to read that intersection as the answer to a system of two equations.

Students plot two straight lines on a graph and find where they cross. That intersection point is the answer to both equations at once.

Students read or draw a line on a graph to find all the points that make an inequality true, then shade the region that fits.

Students learn what a function is, check its output for a given input, and compare how two functions behave. Most of this work uses tables, graphs, and equations.

A function is a rule where every input has exactly one output. Students read graphs and tables to check that each input value matches one output value, not two or more.

Students look at two functions shown in different forms, such as an equation and a graph, and compare what each one tells them, like which grows faster or which has a greater starting value.

Students learn that y = mx + b produces a straight line on a graph, making it a linear function. They also identify functions whose graphs curve or bend, which are not linear.

Students use equations and graphs to describe how one value changes as another changes, like how distance grows with time or cost rises with quantity.

Students find the starting value and rate of change for a straight-line relationship, then use those two numbers to write a rule connecting the variables. They read that information from a table, a graph, or a word problem.

Students read a line graph and explain in words whether a relationship is rising, falling, or curving. They also sketch a rough graph to match a description, like "the temperature climbed slowly, then dropped fast."

Students learn how sliding, flipping, and rotating shapes changes their position without changing their size. They also work out basic angle rules, like why angles in a triangle always add up to 180 degrees, using their own reasoning rather than formal proofs.

Translations, rotations, reflections, and dilations move, flip, turn, or resize a shape on a grid. Students describe exactly what happened to each corner of the shape by tracking how its coordinates changed.

When a shape is slid, flipped, or rotated, straight lines stay straight and equal-length segments stay equal. Students check this by comparing figures before and after the move.

When a shape is slid, flipped, or rotated, its angles stay the same size. Students check that a 90-degree corner, for example, is still 90 degrees after the move.

When a shape or figure slides, flips, or rotates, any parallel lines in it stay parallel. Students check this by tracing or folding to confirm the lines never meet.

Dilations are a type of resize: students learn how a shape grows or shrinks when multiplied by a scale factor, and why the result looks the same but at a different size.

Students learn the rules that govern angles in triangles and parallel lines, then use simple logical reasoning to explain why those rules work. They also use angle relationships to decide when two triangles have the same shape but different sizes.

Students use the Pythagorean Theorem to find missing side lengths in right triangles. They also apply it to find the straight-line distance between two points on a coordinate grid.

Students explain why a right triangle's three sides always follow the rule a² + b² = c², and why that rule works in reverse: if three side lengths fit the equation, the triangle must have a right angle.

Students use the rule that connects the three sides of a right triangle to find a missing side length. This comes up in real problems like finding the diagonal of a room or the distance between two points on a map.

Students use the Pythagorean Theorem to find the straight-line distance between two points on a grid. They treat the horizontal and vertical gaps as the two shorter sides of a right triangle, then solve for the diagonal.

Students calculate the volume of round 3-D shapes like cans, funnels, and balls. They apply the right formula to each shape and use those skills to solve practical problems.

Students use the volume formulas for cones, cylinders, and spheres to solve practical problems, like finding how much a can holds or how much space a ball takes up.

Students look at two sets of data at once to spot patterns, like whether taller students tend to score higher on a test. They use scatter plots and tables to describe what they find.

Students read scatter plots that show how two things relate, like height and shoe size. They describe what the pattern means: whether the dots cluster together, trend up or down, or form a curve instead of a line.

When a scatter plot shows points trending in a line, students draw a best-fit line through them by hand and judge how well it fits by checking how close the points are to that line.

Students use the equation of a trend line on a scatter plot to answer real questions, like predicting a person's height from their shoe size. They explain what the slope and starting point of the line mean in plain terms.

Students learn to predict how likely an event is, build simple probability models, and check whether those models match what actually happens when they run an experiment or collect data.

Students list every possible outcome for two or more combined events, such as flipping a coin and rolling a die, then use that list or a table to find the probability of a specific result.

Finding the probability of two events happening together works the same way as finding the probability of one event. Count how many outcomes in the full list match what you want, then divide by the total number of possible outcomes.

Students list every possible outcome for two-part events, like rolling two dice, using a table or a diagram. Then they circle the specific outcomes that match a given result, such as both dice landing on six.