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

Students learn what makes two shapes identical or proportionally scaled versions of each other. They test this by moving, flipping, and resizing shapes to see what stays the same and what changes.

Students test what happens to a line, angle, or shape after it is flipped, slid, or turned. They confirm that the size and shape stay the same, only the position changes.

When a shape is flipped, slid, or rotated, straight lines stay straight and line segments keep their exact length. The move changes position, not size.

When a shape is flipped, slid, or rotated, its angles stay exactly the same size. Students learn that moving a shape around never stretches or squishes its corners.

When two shapes are the same size and shape, their parallel lines stay parallel after any flip, slide, or rotation. Moving a figure never turns parallel lines into lines that eventually meet.

Two shapes are congruent when one can be flipped, turned, or slid to land exactly on the other. Students identify those moves and describe the steps that show why two shapes match perfectly.

Students learn what happens to a shape's coordinates when it's slid, spun, flipped, or stretched on a grid. They practice predicting where each corner lands after the move.

Two shapes are similar when one can be flipped, slid, turned, or scaled up and down to match the other. Students identify those moves and describe the steps that show how the two shapes line up.

Students use reasoning (not formal proofs) to explain why triangle angles always add up to 180 degrees, why certain angle pairs match when a line crosses two parallel lines, and why two triangles with the same two angles must be the same shape.

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

Students explain why the Pythagorean Theorem works, not just how to use it. They also show that if a triangle's sides fit the a² + b² = c² relationship, the triangle must have a right angle.

Students use the Pythagorean Theorem to find a missing side of a right triangle, whether the problem involves a flat diagram or a real object like a ramp or a box.

Students find the straight-line distance between two points on a grid by treating the gap between them as the long side of a right triangle. They use the Pythagorean Theorem to calculate that length.

Students use formulas to find the volume of rounded 3-D shapes like cans, ice cream cones, and balls. They apply those formulas to real problems, not just practice exercises.

Students learn the volume formulas for cones, cylinders, and spheres, then use those formulas to figure out how much space a real object holds, like a soup can, an ice cream cone, or a basketball.

Some numbers, like the square root of 2 or pi, cannot be written as a simple fraction. Students learn to recognize these irrational numbers and find close decimal approximations for them.

Some numbers, like 1/3, turn into decimals that repeat forever (0.333...). Others, like the square root of 2, never settle into a pattern. Students learn to tell these two types apart and convert repeating decimals back into fractions.

Students learn to place numbers like pi or square roots at the right spot on a number line by finding the closest fraction or decimal that fits. They also use those estimates to compare two irrational numbers or calculate rough values of expressions.

Students practice two related skills: rewriting numbers using exponents (like 10³ for 1,000) and working with square roots and cube roots. Both tools show up in science class and real measurements.

Multiplying or dividing numbers written with exponents follows a short set of rules. Students use those rules to rewrite expressions like 3 to the 4th divided by 3 squared into a simpler, equal form.

Students learn what it means to "undo" a square or a cube. They find the square root of numbers like 25 or 64, recognize that the square root of 2 cannot be written as a simple fraction, and use root symbols to solve basic equations.

Students write very large or very small numbers using scientific notation, like 3 x 10^6 for three million, then compare two of them to see how many times bigger one is than the other.

Students add, subtract, multiply, and divide numbers written in scientific notation, such as 3.2 x 10^8, and read those numbers correctly when a calculator displays them. They also pick units that make large or small measurements easier to work with.

Proportional relationships, straight-line graphs, and linear equations all describe the same kind of change. Students learn to move between those three forms and explain what each one shows.

Students graph proportional relationships and read the slope as the unit rate. They also compare two proportional relationships that may be shown in different ways, like one as a graph and another as a table.

Students use matching triangle shapes plotted on a graph to show why a straight line rises or falls at a steady rate. From that idea, they write the equation that describes any straight line.

Students solve equations with one unknown, then work up to pairs of equations solved together. The goal is finding the exact value or values that make both equations true at once.

Students solve equations with one unknown, like finding the value of x that makes both sides of an equation balance. This includes equations that may need several steps to simplify before solving.

Students sort one-variable equations into three types: one answer, no answer, or every number works. They simplify the equation step by step until the result makes the category obvious.

Students solve equations that include fractions or decimals, using distribution and combining like terms to find the value of the unknown.

Two equations, two unknowns. Students find the one pair of numbers that makes both equations true at the same time, using graphs, substitution, or elimination.

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

Students solve two equations at once to find the single point where both are true, using algebra or a graph. Some simpler pairs can be solved just by looking at them.

Students solve everyday problems that require two equations at once, like figuring out prices or distances when two unknowns are involved. They find the one pair of numbers that makes both equations true.

Students look at two sets of data together (like height and shoe size) to find out if a pattern connects them. They use scatter plots and tables to decide whether a relationship exists and how strong it is.

Students plot two measurements on a graph to see if they move together, such as height and shoe size. They look for patterns, like whether the dots cluster together, trend upward or downward, or whether a few dots sit far from the rest.

When a scatter plot's dots seem to follow a straight path, students draw a line through the middle of them and judge how well it fits by checking how close the dots are to that line.

Students use a line drawn through a scatter plot to make predictions, then explain what the steepness of the line means in real terms. For example, they might say how much taller a plant grows for each extra day of sunlight.

Students read a two-way table that sorts the same group of people by two categories at once, like pet ownership and grade level, then use the percentages to say whether the two categories seem connected.

Students learn what a function is, practice finding its output for a given input, and compare how two functions behave. Think of it as reading and comparing two vending machines: put a number in, get a number out.

A function is a rule where every input has exactly one output. Students read graphs and tables to see how inputs and outputs pair up, like how one address maps to exactly one house.

Students compare two functions shown in different forms, such as an equation and a graph, to figure out which has a steeper slope or a higher starting value.

Slope-intercept form (y = mx + b) produces a straight line on a graph. Students identify what makes a function linear and recognize functions whose graphs curve or bend instead.

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

Students find the starting value and rate of change for a straight-line relationship, whether the information comes from a table, a graph, or a word problem, and explain what those numbers mean in context.

Students read a graph to describe how two things change together, such as whether a value is rising, falling, or leveling off. They also sketch a rough graph from a written description of that relationship.