What does a student learn in
New York, Grade 12, Mathematics ?

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 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.

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 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.