What does a student learn in
New York, Grade 10, Mathematics, any course ?

Adding, subtracting, or multiplying polynomials always produces another polynomial. Students practice combining and multiplying these expressions and learn why the result always stays in the same family of expressions.

Students find where a polynomial equation equals zero by breaking it into factors. This is the foundation for graphing parabolas and solving equations in Algebra II.

Students write equations or inequalities with one unknown to describe a real situation, like figuring out how many hours of work it takes to earn a target amount. The equation or inequality becomes a tool for solving the actual problem.

Students write equations and inequalities that model a real situation, like using two variables to describe cost and quantity at a store. The goal is to turn a word problem into math that can be graphed or solved.

Real problems come with limits: a budget, a time cap, a size requirement. Students translate those limits into equations or inequalities, then decide whether a solution actually works in the real situation or falls outside what's possible.

Students rearrange a formula to solve for a specific variable, the way they would solve a one-variable equation. For example, they take the formula for area and rewrite it to find height instead of area.

Solving for a single unknown: students work through equations and inequalities with one variable, including problems where some numbers are replaced by letters standing in for constants.

Students learn to solve equations where a variable is squared, using methods like factoring, the quadratic formula, and graphing. They also recognize when an equation has no real solution.

Every point on a line or curve in a graph is a solution to the equation it represents. Students learn to read graphs as a complete picture of all the answers, not just a few sample points.

Where two graphs cross, the x-value at that point solves the equation where both sides are equal. Students find those crossing points using a graphing tool or a table of values, then explain what the answer means in the problem's context.

Students shade the region of a graph that satisfies a linear inequality, then find where two shaded regions overlap to solve a system. A dashed boundary line means points on the line don't count as solutions.

Solving an equation isn't just getting the right answer. Students explain why each step is valid, showing that every move they make keeps both sides of the equation balanced.

Students find the point where two straight lines cross by solving the pair of equations with algebra and by graphing both lines to see where they meet.

Students find where a straight line and a U-shaped curve cross by solving them together with algebra and by graphing both on the same coordinate plane. Solutions are exact fractions or whole numbers.

Given an expression like 3x² + 5x - 7, students identify each part: the terms, their coefficients, the degree, and the constant. They also read chunks of an expression as a single unit to understand what the math is describing.

Students look at an expression like x² - 9 and spot patterns that let them rewrite it in a simpler or more useful form. Recognizing that structure is the first step to solving or simplifying harder problems.

Rewriting an exponential expression means using exponent rules to swap it for a simpler or more useful form. Students learn to spot what the rewritten version reveals about growth, decay, or rate that the original hides.

Dividing a polynomial by (x, a) leaves a remainder equal to p(a). If that remainder is zero, (x, a) is a factor, giving students a fast way to test whether a value is a root without doing full long division.

Students find where a polynomial equation crosses zero by breaking it into factors. This is the algebra behind locating where a curve touches or crosses the x-axis on a graph.

Students divide one polynomial by another, the way long division works with numbers, and rewrite what's left as a fraction tacked onto the whole-number part of the answer.

Students write an equation or inequality using one unknown to model a real situation, like figuring out how many hours of work it takes to afford a purchase. They set up the math before solving it.

Students solve equations that contain fractions with variables or square roots, then check whether each answer actually works in the original equation. Some answers look correct but break the math, and students learn to spot and explain those.

Students solve quadratic equations using several methods, from factoring to the quadratic formula to reading a graph. When solutions involve square roots of negative numbers, students write them in a + bi form.

Where two graphed lines or curves cross, the x-value at that crossing solves the equation. Students find those intersection points using a graphing calculator or table of values, then explain what the answer means in the real situation.

Solving an equation with fractions or square roots means every step has to follow logically from the one before it. Students explain why each move is valid and can defend their method if someone asks why it works.

Students find where a straight line and a curved parabola cross, using both algebra and a graph. They solve for the exact points where the two equations share the same x and y values.

Students look at an expression like x⁴ - 16 and spot a pattern that lets them rewrite it in a simpler or more useful form. Recognizing that structure is the skill, not just following steps.

Students rewrite math expressions in a different but equal form to uncover useful information, like factoring a quadratic to find where its graph crosses zero, or using exponent rules to simplify a growth or decay formula.

Students write a rule, formula, or equation that captures how two quantities relate, like how total cost changes as more items are bought. They also describe a sequence by writing out the pattern or the steps needed to calculate each term.

Students learn how adding or multiplying a number to a function shifts, stretches, or flips its graph. They identify what changed, find the value that caused it, and write the updated function.

A function is a rule where every input has exactly one output. Students learn to read notation like f(x), connect it to a table or graph, and recognize that each x-value pairs with one y-value and no more.

Students learn to read and use function notation like f(x), plug in values to get an output, and explain what that output means in a real situation. Think of it as reading a rule: given an input, what does the function spit out, and what does that answer actually tell you?

A sequence is just a function with integers as its inputs. Students learn to see patterns like 2, 4, 6, 8 as a rule that assigns exactly one output to each counting number in order.

Students read a graph or table and explain what the peaks, valleys, and flat spots mean in real terms. Given a word description of a situation, they sketch a graph that shows the same story.

Reading a graph, students figure out which input values a function will actually accept. In a real-world problem, like distance over time, they also decide which values make sense for that situation.

Students find how fast a function's output is rising or falling over a given stretch of inputs, like checking how quickly a car's distance grows between two points in time. They also explain what that rate means in context.

Students sketch graphs of lines, curves, and other equations by hand and on a calculator, then label the key points, like where the graph crosses an axis or changes direction.

Students rewrite a quadratic equation by factoring or completing the square to find where the graph crosses zero, where it peaks or bottoms out, and what the results mean in a real situation.

Students compare two functions shown in different formats, such as one as an equation and one as a graph, to find which has a steeper slope, a higher starting value, or other key differences.

Students learn to tell the difference between linear and exponential patterns. A linear pattern adds the same amount each step, like saving five dollars a week. An exponential pattern multiplies by the same amount each step, like a population that doubles every year.

Students write the equation for a line or exponential curve using clues like a graph, a written description, or two points from a table. The skill is turning a visual or a few numbers into a usable formula.

Students compare how fast different types of equations grow by reading graphs and tables. A quantity that grows exponentially will always outpace one growing at a steady rate or even a curved rate, given enough time.

A linear or exponential equation has numbers baked into it that carry real meaning. Students figure out what those numbers represent in the actual situation, like what the starting amount was or how fast something is growing.

Students write equations that model real situations, like predicting profit based on costs and sales. They also build new functions by adding, subtracting, multiplying, or dividing simpler ones together.

Students write number patterns two ways: a rule that uses the previous term to find the next one, and a formula that jumps straight to any term in the sequence. They also use these patterns to model real situations like compound interest or steady salary increases.

Students learn to add up a sequence of numbers that follow a pattern, like a salary that grows by the same amount each year. They write that total using sigma notation, the shorthand mathematicians use to show a repeated sum.

Students work out where the formulas for adding up arithmetic and geometric sequences actually come from, then use those formulas to find sums quickly without adding every term by hand.

Students learn how changing a number inside or outside a function rule shifts, stretches, or flips its graph. They find that number from a graph, rewrite the function with it, and use graphing tools to check their work.

Students find the reverse rule of a function: the equation that undoes what the original does. On a graph, this means reflecting the original curve over the diagonal line y = x.

Students learn that exponents and logarithms undo each other, the way multiplication and division do. They work with equations and graphs to see how switching between the two forms reveals the same relationship from a different angle.

A sequence is just a function where the inputs are whole numbers (1, 2, 3, and so on). Students recognize that lists of numbers following a pattern are a type of function, not a separate idea.

Students read a graph or table and explain what the peaks, valleys, and flat spots mean in real terms. They also sketch a rough graph from a written description of how two quantities relate.

Students find how fast a function's output is rising or falling over a given interval, then explain what that rate means in plain terms. Think of it as finding the average speed of a graph between two points.

Students graph advanced functions like polynomials, exponentials, and sine curves, marking where the graph crosses the axes, how it behaves at the far left and right, and (for wave functions) its height and spacing.

Rewriting an exponential function using exponent rules lets students see whether it models growth or shrinkage over time. Students identify key features like the rate and direction of change from the rewritten form.

Students compare two functions shown in different formats, such as reading one as an equation and the other as a graph or table, to figure out which grows faster, has a higher starting value, or behaves differently.

Students build a linear or exponential equation from a graph, a written description, or a table of values. The goal is to write the actual formula, not just describe the pattern.

Students use logarithms to solve equations where the unknown is an exponent, such as figuring out how long it takes money to double at a given interest rate. They use a calculator to find the final answer.

Students read a linear or exponential equation and explain what each number actually means in the situation. For example, they say why the starting value is 50 or why the growth rate is 1.03, not just what those numbers are.

Radians are a way to measure angles by asking how far you'd travel along the edge of a circle. Students learn that one radian means the arc length equals the circle's radius, giving angles a size grounded in distance rather than degrees.

Given an angle measured in radians, students use the unit circle to find the sine, cosine, tangent, and the other three trig ratios for that angle.

The unit circle is a circle with radius 1 used to define sine, cosine, and other trig values. Students use it to explain why some trig functions mirror themselves across an axis and why all trig functions repeat the same pattern at regular intervals.

Students pick a sine or cosine function that fits a repeating pattern, like a wave or a seasonal cycle, by setting the height of the peaks, how often the pattern repeats, and where it starts.

Students use the relationship sin²(θ) + cos²(θ) = 1 to find a missing trig value, like sine or cosine, when they already know one of the others. They also figure out which quadrant the angle sits in.

Students pick the right units for a problem (miles, dollars, seconds) and keep them consistent throughout. That includes reading scales and labels on graphs to make sure the numbers actually mean what they look like they mean.

When solving a real problem, students decide how precise their answer needs to be. A distance measured with a ruler doesn't need to be reported down to the millimeter if the context doesn't call for it.

Students add, subtract, multiply, and divide fractions, decimals, and square roots, then figure out whether the result is a clean number or one that goes on forever without repeating.

Algebra II introduces a special number called i, where i squared equals -1. From there, every complex number is written as a real number plus a real number multiplied by i, like 3 + 2i.

Adding, subtracting, and multiplying complex numbers works just like regular arithmetic, with one extra rule: i squared equals negative one. Students apply that rule to simplify expressions that mix real numbers with imaginary ones.

Rational exponents are fractions used as powers, like 9 to the one-half equaling 3. Students see how the rules for whole-number exponents stretch to cover these fractional ones, so expressions like x^(2/3) follow the same logic they already know.

Students rewrite expressions like the square root of x as x to a fractional power, and vice versa. The two forms mean the same thing, and knowing both makes it easier to simplify and solve problems.

Students take a set of data and display it as a dot plot, histogram, or box plot on a number line. The goal is to see how the values spread out and where they cluster.

Students compare two data sets by looking at where most values cluster and how spread out they are. They choose between mean or median based on the shape of the data, then measure spread using range or standard deviation.

Students look at two data sets side by side and explain what the differences in shape, center, and spread actually mean. They also check whether a single unusual value is skewing the picture.

Students read a two-way table that sorts data into two categories, like grade level and favorite subject, then figure out what the numbers reveal. They look at row totals, column totals, and individual cells to spot patterns or connections between the two categories.

Students plot two sets of real-world numbers on a graph to see if a pattern exists between them, then draw a line or curve that fits the data well enough to make predictions.

Students read a best-fit line on a scatter plot and explain what the slope and starting point actually mean in plain terms. For example, they might say the slope shows that each extra hour of practice adds two points to a score.

Students use a calculator or software to find the number (between -1 and 1) that shows how closely two things are related on a graph. They also explain what that number means in plain terms.

Correlation means two things tend to change together. Causation means one thing actually causes the other. Students learn why a pattern in data does not prove that one thing is making the other happen.

Students sort possible outcomes into groups, then combine or compare those groups using "or," "and," and "not." For example, rolling an even number "or" a number greater than four covers outcomes that meet either condition.

A two-way table sorts data into rows and columns to compare two categories at once, like grade level and favorite subject. Students use those counts to figure out whether two categories are related and to calculate the probability of one thing happening given that another already has.

Students use a formula to find the chance that at least one of two events happens. They add the two individual probabilities, then subtract the overlap so it isn't counted twice.

Students look at survey or experiment results and decide whether a given average or percentage is a reasonable outcome or a surprising one.

Surveys ask people questions, experiments change one thing to see what happens, and observational studies just watch without interfering. Students learn why researchers choose each method and how random selection makes results more trustworthy.

Students run a simulation, then use the results to build a confidence interval showing where the true population value likely falls. They decide whether a given claim about that value is reasonable or not.

Students read a data summary (a mean, a chart, or a percentage) and use it to draw a real conclusion, not just report the number. The work is turning statistics into a claim that holds up.

Students read a news article or study and pick apart the claim. They identify whether the data actually proves cause or just shows a pattern, and whether the sample or methods could be skewed.

Students plot two related sets of data on a graph, look for a pattern, and draw a curve or line that fits the data. Then they use that line or curve to answer real questions about the data.

Students look at a set of data and decide whether it fits the classic bell-curve shape, where most values cluster in the middle and fewer appear at the edges.

Students use a graphing calculator to find what percentage of a population falls within a range on a normal bell curve, when the data fits that shape.

Students show why every circle, no matter its size, is the same shape as every other circle. They do this by showing that any circle can be scaled up or down to match another exactly.

Students use ratios to find a missing measurement in a circle. Given any two of these four values, a central angle, an arc length, a radius, or a sector's area, students calculate the one that's missing.

Students learn how the angles formed inside or around a circle connect to the arcs they cut across. They use those relationships to find missing angle measures and arc lengths.

Students learn how lines and segments interact with a circle: when a line just grazes the edge, cuts through it, or connects two points on it. They use those relationships to solve problems about angles and lengths.

Students learn the exact definitions of basic shapes and relationships in a flat plane: what makes two lines parallel, what a circle is, and how angles and line segments are precisely described. These definitions are the foundation for every proof and theorem that follows.

Students learn how moving, flipping, or rotating a shape maps each point to a new location. Some moves keep the shape the same size and angles intact; others stretch or distort it.

Students look at a shape and figure out which turns and flips would land it back in exactly the same position. They do this for both regular shapes like squares and irregular ones that don't have equal sides.

Rotations, reflections, and translations each have precise geometric definitions built from angles, circles, and lines. Students learn exactly what makes each move tick, not just what it looks like.

Students take a shape and move, flip, or turn it, then draw where it lands. They also work backward, figuring out the exact steps needed to slide one shape onto another.

Students slide, flip, or rotate a shape and predict exactly where it lands. Then, given two shapes, they decide if one can be moved onto the other perfectly to confirm they are congruent.

Two triangles are congruent when you can slide, flip, or rotate one to land exactly on the other. Students prove this by showing that every matching side and every matching angle between the two triangles are equal.

Students explain why two triangles are identical in size and shape by showing that one can be flipped, slid, or rotated onto the other using side lengths and angle measures as the proof.

Students prove and apply rules about how lines and angles behave, such as why vertical angles are equal or why parallel lines cut by a third line create predictable angle pairs.

Triangles follow rules about their angles and sides, and students prove why those rules are true using logic and prior knowledge. They then apply those rules to solve problems about unknown angles or lengths.

Students use a compass and straightedge to draw precise geometric figures: copying angles, bisecting line segments, and constructing perpendicular or parallel lines. The focus is on following exact steps and explaining why each construction works.

Students use only a compass and straightedge to draw a triangle, square, or six-sided shape that fits perfectly inside a circle, then explain why each construction works.

Students explain why the area and volume formulas for circles, cylinders, pyramids, and cones actually work, not just how to use them. They connect the shape of each figure to the math behind its formula.

Students use formulas to find the volume of shapes like cylinders, cones, and spheres. Think of figuring out how much water fills a cone-shaped cup or how much space is inside a ball.

Slice through a 3-D shape like a cone or cylinder and name the flat shape the cut reveals. Students also picture what solid a flat shape would sweep out if spun around an axis.

Students use equations and coordinates on a graph to prove why geometric rules work, such as showing that two sides of a shape are parallel or that a point sits exactly on a circle.

Students use slope to decide whether two lines on a graph are parallel, perpendicular, or neither. They apply that relationship to solve geometry problems involving coordinates.

Given two points on a graph, students find the exact location that splits the line between them at a specific ratio, such as one-third of the way from start to finish.

Students use the x- and y-coordinates of a shape's corners to calculate how far around the outside it measures and how much space it covers inside.

Students use the Pythagorean Theorem to build the equation of a circle from its center point and radius. They also work backwards, reading a circle's equation to find where it's centered and how wide it is.

Students read an equation for a circle and plot that circle accurately on a coordinate grid.

Students look at real objects like buildings, bridges, or packaging and describe them using shapes, angles, and measurements. It connects classroom geometry to things students can see and touch.

Students figure out how much of something (people, trees, mass) fits into a given area or space. They use area and volume to solve real-world problems like population density or material weight.

Students use geometry to solve real-world design problems, such as figuring out how much material a structure needs or whether a shape fits a given space.

Students explore what happens when a figure is scaled up or scaled down from a fixed point. They check that lines shift but stay parallel, and that lengths grow or shrink by the exact same ratio as the scale factor.

Students look at two shapes and decide whether one is a scaled version of the other. For similar triangles, they explain why the matching angles are equal and why the matching sides grow or shrink by the same ratio.

Two triangles can be declared similar without measuring every side and angle. Students use shortcut rules, matching two angles, three proportional sides, or two proportional sides with the angle between them, to prove triangles have the same shape. Wait, I used an em dash. Let me fix that. Two triangles can be declared similar without measuring every side and angle. Students use shortcut rules (matching two angles, three proportional sides, or two proportional sides with the angle between them) to prove triangles have the same shape. Checking: that's two sentences, roughly 44 words, no em dashes, no triads (the parenthetical lists three items but it's a parenthetical clarification, not a rhythmic triad of sentences). Actually, let me reconsider the three-part list inside the parenthetical, the rule says no three-part rhythms. Let me trim. Two triangles can be declared similar without measuring every side and angle. Students use shortcut rules (matching two angles, or three proportional sides) to prove the triangles have the same shape without checking every measurement. That's cleaner but loses SAS. Let me keep two of the three shortcuts and note there are a few. Two triangles can be declared similar without measuring every side and angle. Students use a set of shortcut rules, like matching two angles or checking that sides are proportional, to prove two triangles have the same shape. Good. 39 words, no em dashes, no triads, plain language. Two triangles can be declared similar without measuring every side and angle. Students use a set of shortcut rules, like matching two angles or checking that sides are proportional, to prove two triangles have the same shape.

Similarity theorems let students prove that two triangles have the same shape even when their sizes differ. Students use those proofs to find missing side lengths, work with proportions, and solve multi-step problems involving scaled figures.

Students use rules about matching sides and angles to show two triangles are identical or scaled copies of each other, then apply those rules to find missing measurements or prove why a geometric relationship must be true.

Sine, cosine, and tangent come from the ratios of sides in a right triangle. Students learn that those ratios stay the same for any triangle with the same angles, which is why they can be used to find unknown sides and angles.

Sine and cosine are connected: the sine of any angle equals the cosine of its complement, and vice versa. Students use this shortcut to find missing trig values when two angles in a right triangle add up to 90 degrees.

Students use the Pythagorean Theorem and basic trig ratios (sine, cosine, tangent) to find missing side lengths and angles in right triangles. The problems are grounded in real situations, like finding the height of a ramp or the distance across a field.

Students find the area of any triangle, including ones without a right angle, by dropping a perpendicular line from one corner to the opposite side. That extra line unlocks a formula using two side lengths and the angle between them.