What does a student learn in StateVirginia,GradeGrade 9,SubjectMathematics?

Students translate word problems into algebraic expressions, then plug in numbers to find the value. It connects everyday language like "three more than twice a number" to the math symbols that represent it.

Rational expressions are fractions with variables in them. Students add, subtract, multiply, and divide these fractions, then simplify the result the same way they would simplify a numeric fraction.

Adding, subtracting, multiplying, or dividing fractions that contain variables instead of just numbers. Students simplify each result the same way they would reduce a numeric fraction.

Students turn word problems into math expressions and math expressions back into plain sentences. They work with real situations, like calculating a phone bill or splitting a cost, not just abstract symbols.

Students simplify fractions that contain variables instead of plain numbers, factoring expressions like x squared plus 3x into pieces that cancel. The goal is to rewrite the fraction in its simplest form and explain why the two versions are equal.

Students substitute numbers (including fractions and negatives) into expressions with absolute value, square roots, and cube roots, then calculate the result. The denominator does not need to be simplified.

Students add, subtract, multiply, and factor polynomial expressions like x² + 5x + 6, breaking them down into simpler pieces or combining them into one. This is the algebra behind solving equations that show up throughout high school math.

A complex algebraic fraction has a fraction inside its numerator or denominator. Students rewrite it as a simpler multiplication or division problem and reduce it to lowest terms.

Students rewrite a fraction made with variables into an equivalent form, the same way they might reduce 6/8 to 3/4. The value stays the same; only the way it looks changes.

Students add, subtract, multiply, and simplify expressions that contain square roots or cube roots, keeping answers in simplest form.

Students add and subtract polynomials by combining like terms. They work with algebra tiles or diagrams before moving to purely symbolic work.

Students multiply two polynomials together, such as (4x + 2)(3x + 5), by distributing each term across the other expression. They practice this with area models, diagrams, and the standard algebraic method.

Students break down polynomial expressions into their simplest multiplied parts, starting by pulling out the largest shared factor. The expressions involve whole-number coefficients and cover both linear and quadratic forms.

Students simplify square roots and other radicals that contain numbers, variables, or both, rewriting them in their simplest form. This includes expressions like the square root of 50 or the cube root of a variable.

Dividing one polynomial by another, students split an expression like a long-division problem. The divisor is either a single term, a two-term binomial, or a fully factored expression.

Students add, subtract, multiply, and divide expressions that contain square roots or other radicals, then simplify the result. This sometimes means rewriting a fraction so no radical sits in the denominator.

Students show that two quadratic expressions are equal by representing them in more than one form, such as a graph, a table, or an equation, and confirming they produce the same values.

Students practice writing the same root (like the square root of 5) two ways: as a radical symbol and as a fraction in the exponent. The goal is to move fluently between both forms.

Students learn the rules for working with exponents, like why multiplying two powers with the same base means adding the exponents. They then use those rules to simplify and solve expressions.

Students add, subtract, multiply, and factor polynomial expressions, working with equations that have one or two variables. This is the algebra behind expanding and simplifying the kinds of expressions that show up throughout higher math.

Students figure out the rules for working with exponents by studying patterns, like what happens when you multiply or divide terms with the same base, or raise a power to another power.

Students add, subtract, and multiply polynomials, which are expressions like 3x² + 2x - 5, including ones that mix two variables such as x and y.

Students simplify expressions and fractions that mix multiple variables and integer exponents, applying exponent rules to reduce everything to its simplest form.

Students factor expressions like x² - 9 or x²y + 2xy into their simplest multiplied parts, working with up to four terms and whole numbers. No decimals, no fractions, just clean integer factors.

Students divide polynomials by expressions like a single term, a two-term expression, or a trinomial that factors cleanly. This is polynomial long division applied to expressions in one or two variables.

Students simplify square roots and cube roots, like rewriting the square root of 50 as 5 times the square root of 2. The goal is to recognize when two radical expressions are equal even if they look different.

Students check that two polynomial expressions are equal even when they look different, and confirm patterns like (a + b)² or a² - b² hold up by expanding and simplifying both sides.

Students simplify square roots by pulling out perfect squares, so something like the square root of 48 becomes 4 times the square root of 3. The goal is a cleaner, equivalent expression with no perfect squares left under the root symbol.

Students add, subtract, multiply, and divide complex numbers, which include a real part and an imaginary part written with the letter i. This builds on what students know about algebra to handle numbers that go beyond the regular number line.

Students simplify cube roots of integers, finding what number multiplied by itself three times equals the value under the root symbol. For example, the cube root of 27 is 3, because 3 x 3 x 3 = 27.

Students add, subtract, and multiply expressions like √12 and ∛5 by combining like roots and applying multiplication rules. No variables, just numbers under the radical sign.

Students learn that i stands for the square root of negative one, a value that does not exist on the regular number line. This idea opens up a new category of numbers used to solve equations that have no real solution.

Radicals with negative numbers under the root sign can be rewritten using *i*, the imaginary unit. Students rewrite those expressions in a + bi form and recognize when two forms represent the same value.

Students rewrite square roots and cube roots using fraction exponents, such as writing the square root of 5 as 5 to the one-half power, then show why both forms name the same value.

Students add, subtract, and multiply complex numbers the same way they work with polynomials, treating the imaginary unit i like a variable and simplifying using the fact that i squared equals negative one.

Students read "if-then" sentences in geometry, figure out what each part means, and use circle diagrams to show how groups of shapes or numbers overlap or stay separate.

Students learn to rewrite everyday "if-then" sentences and "and/or" statements using math symbols like arrows and tildes. This is the shorthand logicians use to connect ideas precisely.

Given an "if-then" statement, students flip, negate, or do both to build three new versions of it, then decide which versions are actually true.

Venn diagrams use overlapping circles to show how groups of things relate. Students read and draw these diagrams to show what two sets share, what belongs to only one, and what falls outside both.

Reading a Venn diagram means understanding what the overlapping circles show. Students identify what items share a common trait, what items don't, and what the diagram reveals about a real-world situation.

Students study what happens when a straight line crosses two parallel lines, then prove why certain angle pairs are always equal or supplementary. The focus is on building a logical argument, not just spotting the pattern.

When two parallel lines are crossed by a third line, specific angle pairs form predictable relationships. Students prove why those angles are equal or supplementary using logical steps and geometric rules.

Two parallel lines crossed by a third line create matching corners at each intersection. Students identify and prove that corresponding angles, the ones in the same position at each crossing, are equal.

When two parallel lines are crossed by a third line, the angles that fall between the parallel lines and on opposite sides of the crossing line are equal. Students identify and prove this relationship.

When two parallel lines are crossed by a third line, alternate exterior angles sit on opposite sides of that crossing line and on the outside of the parallel lines. Students prove these angles are always equal.

When two parallel lines are crossed by a third line, same-side interior angles sit between the parallel lines on the same side of the crossing line. Students prove that those two angles always add up to 180 degrees.

When two parallel lines are crossed by a third line, same-side exterior angles sit outside both parallel lines on the same side of the crossing line. Students identify these angle pairs and prove they add up to 180 degrees.

Students use angle measurements, sometimes with variables, to prove that two lines run parallel. If the angle relationships check out, the proof is complete.

Two parallel lines crossed by a third line create angle pairs with predictable relationships. Students use those relationships, like equal corresponding angles or supplementary same-side angles, to find missing angle measures.

Students identify lines of symmetry and apply rotations, reflections, and translations to figures. They solve real-world problems where shapes are flipped, turned, or slid across a plane.

Students find and draw the fold lines on a shape where both halves match exactly. This includes shapes that appear in real-world objects, like logos or tiles.

Students look at a shape and decide whether it has a line of symmetry, a center point of symmetry, both, or neither. This applies to real-world shapes too, like logos, tiles, or letters.

A shape has been moved, flipped, or rotated to a new position. Students look at the before and after, then name exactly which transformation (or sequence of transformations) produced the change.

Translations slide a shape to a new position without turning or flipping it. Students identify where each point lands after the move and confirm the shape's size and orientation stay the same.

Students reflect a shape across a horizontal line, a vertical line, or a diagonal line like y = x, and find where each point lands on the other side.

Students practice spinning a shape around the origin of a graph by 90, 180, 270, or 360 degrees, turning it either left or right, and finding where each point lands after the rotation.

Students scale a shape up or down on a coordinate grid by stretching or shrinking every point away from one fixed anchor point by the same factor.

Students learn to ask a testable question, decide what data to collect and where to get it, and identify any limits on what the data can actually show.

Students learn the steps of a statistical investigation: ask a question, collect data, analyze it, and draw conclusions. Each step shapes the next, so a weak question or messy data affects everything that follows.

Students look at real-world problems and decide which ones are best solved by collecting and analyzing data. They explain what makes a problem a good fit for a data-based approach.

Students turn a real situation into a question that data can actually answer. They practice deciding what to ask before they start collecting any numbers.

Students read a real-world situation and write a question worth investigating, then draw a conclusion once the data answers it.

Deciding whether a question calls for numbers (like test scores or heights) or categories (like favorite colors or yes/no answers) shapes how students collect and analyze data.

Students look at a real-world question and decide what kind of data would actually answer it. A question about school lunch choices needs different data than a question about traffic patterns.

Students learn to tell the difference between two quantities that change together and a fixed value that stays the same no matter what. They practice spotting which relationships shift and which ones hold constant.

Students learn the difference between studying an entire group (a population) and studying a smaller slice of it (a sample), and why the numbers from each can differ.

Students come up with a question they can actually test, then restate it in a way that points to real numbers. For example, "Do students who sleep more score higher on tests?" becomes something they can measure and check.

Students explain what limits a survey or study, such as a small sample, missing responses, or data collected from only one group. They describe how those limits affect what the results can and cannot show.

Students learn the difference between surveys, experiments, and direct observation, then choose the right method to collect real data. They practice planning a study before the first number is recorded.

Students learn the steps of working with data, from asking a question and collecting information to analyzing results and drawing conclusions. Each step feeds into the next, so a weak collection method, for example, can skew every finding that follows.

Students learn that data has limits. They practice spotting when a dataset is too small, biased, or missing key information to support a conclusion.

Students learn four ways to pick a sample from a large group, such as drawing names at random, splitting the group into categories first, or selecting every nth person. Each method affects how well the sample represents the whole population.

Students look at a real situation and decide which method of collecting a sample fits best. They explain why one approach gives more reliable results than another.

Students sketch out a full plan before collecting any data: what question they're asking, how they'll gather numbers, and how they'll analyze what they find. The plan follows the same steps a data scientist would use on a real project.

Students learn to spot the ways a survey or sample can quietly skew results, such as who gets left out of a poll or how a question's wording nudges people toward a particular answer.

Students pick a real question, then plan and run a study by collecting existing data rather than running an experiment. They follow each step of the process from forming the question to drawing a conclusion.

Students start a data project by writing down exactly what question they are trying to answer and why it matters in the real world.

Students identify who is affected by a data problem and what specific measurements or conditions matter before any collection begins.

Students plan and run an experiment the right way: choosing what to test, controlling what stays the same, and recording what changes. This is how scientists (and students) get results they can actually trust.

Students map out a project schedule showing what gets done and when, so the data work stays on track from start to finish.

Students learn what makes an experiment trustworthy: how to set up a fair test, control what you change, and decide who or what gets measured so the results actually mean something.

Students identify the specific, measurable targets a data project needs to hit before it counts as successful. Think of it like agreeing on the score that means you won before the game starts.

Students identify what tools and resources a data project will require before work begins, such as survey software, spreadsheets, or physical measuring equipment.

Students design an experiment by splitting a group into two parts: one that receives the thing being tested (the treatment) and one that does not (the control). Comparing the two groups shows whether the treatment made a real difference.

Students identify ways their data collection plan might produce skewed results, such as surveying only one group or asking a leading question, and adjust the plan to get a fairer picture.

Students learn why keeping participants unaware of which treatment group they're in, or giving some a fake treatment, helps produce honest results that aren't skewed by expectations.

Students identify what could go wrong or get in the way of a data project: a sample too small, missing data, or results that only apply to one group. Recognizing these limits is part of making the findings honest.

Experimental units are the people, animals, or objects being tested in an experiment. Students identify who or what receives each treatment so the results can be compared fairly.

Students look at a real-world problem and decide the best way to collect data to answer it. That means choosing who or what to sample and how to sample it so the results hold up.

Students learn two ways to set up an experiment: pairing similar subjects together to control for differences, or assigning every subject to a group purely by chance. Both approaches cut down on bias so the results actually mean something.

Designing a fair experiment means more than just running a test once. Students compare groups, assign participants randomly, repeat the test enough times to trust the results, and hold outside factors steady so one variable gets tested at a time.

Simple random sampling means every person or item in a group has an equal chance of being picked. Students learn to use this method when designing a data collection plan so results reflect the whole group, not just a convenient slice of it.

Students learn the difference between setting up a controlled experiment (where you change one thing on purpose) and simply watching what happens in real life. Each method limits what conclusions you can fairly draw.

Students plan data collection step by step before any numbers are recorded, choosing a method that removes guesswork and gives results others could repeat.

Students plan a real experiment from start to finish: they write a question, decide how to collect data, run the experiment, and use the results to draw a conclusion.

Stratified sampling means dividing a population into groups (like grade levels or neighborhoods) before randomly selecting from each one. Students use this method when they want their data to fairly represent every part of a group, not just whoever is easiest to reach.

Students group data points that are close together on a graph to spot natural patterns in the data. This helps them describe what a scatter plot is actually showing without relying on a single line or formula.

Students choose how to gather data for a given question, deciding whether a survey, measurement, observation, or experiment fits best.

Students name real problems (crime rates, hospital wait times, sports scores) that data and statistics can help solve or explain.

Students plan how to collect and analyze data before they start, making sure the results can be measured. They build the whole approach around the actual problem they're trying to solve.

Students compare three types of graphs (straight lines, U-shapes, and curves that grow rapidly) and explore how shifting or stretching each one changes its shape. The focus is on spotting what makes each graph family look and behave differently.

Students learn to recognize the simplest version of three function types: a straight line, a U-shaped curve, and a curve that rises or falls rapidly. These basic shapes are the starting point for every more complex graph they'll see in the unit.

Students look at an equation or graph and describe how it has been shifted, flipped, or stretched compared to the basic version of that function family.

Students look at a table, graph, or real-world situation and decide whether the pattern fits a straight line, a U-shaped curve, or exponential growth. Then they explain why that function type is the best match.

Students look at a graph and write the equation that describes it, starting from a basic parent curve and adjusting for any shifts, flips, or stretches they see.

Students use a graph or equation to answer real questions about a situation, like finding when a phone plan costs the same as a competitor's or predicting how a population grows over time.

Students start with a basic parent function and shift, flip, or stretch it to graph a new equation. A graphing calculator or software confirms the result.

Students look at the same relationship shown three ways (a graph, a table, and an equation) and explain how a straight-line pattern, a U-shaped curve, and a rapidly growing curve are alike and different.

Students read and interpret graphs of four function types: straight lines, U-shaped parabolas, exponential curves, and graphs built from separate pieces. They identify key features like slope, vertex, and where the graph crosses an axis.

Students look at a graph and identify which input values are allowed and which output values are possible, including cases where the real-world situation limits what makes sense.

Students look at a graph and identify which sections of the line or curve are rising, falling, or staying flat as the graph moves from left to right.

Reading a graph, students find the highest and lowest points a function reaches across its entire domain, then name both the input and output values at each of those peaks and valleys.

Students read a graph to find where a line or curve crosses the x-axis (the zeros) and where it crosses the y-axis (the intercepts). These points show where the function equals zero or starts its path across the graph.

Reading a graph tells you what a function is doing at any point. Students connect the coordinates of a point on a graph to its input and output values, explaining what each location on the curve or line actually means.

Students look at both ends of a graph and describe whether the line or curve rises, falls, or levels off as it stretches left and right toward infinity.

Students look at a graph and identify any lines the curve approaches but never crosses, whether those lines run left-right or up-down.

Students compare how the shapes of different graphs (straight lines, U-curves, rapid growth curves) connect to real-world situations, explaining what the curve's direction, steepness, or turning point actually means in context.

Students learn to find the best possible answer (like the highest profit or lowest cost) when real-world rules limit their choices. They set up and graph systems of linear equations or inequalities to see where those limits meet and read off the winning solution.

Students set up and solve real-world problems where two or more rules limit what's possible, like budgets and time constraints working together. They use systems of equations or inequalities to find the best outcome, then read what the solution means in context.

Students plot two to four lines or shaded regions on a graph to find where they overlap, showing the point or zone that satisfies every equation or inequality at once.

Students graph two or more inequalities on the same coordinate plane and shade the overlapping region where all the conditions are true at once. That overlapping area is the feasible region.

Students find the corner points of a shaded region on a graph formed by overlapping inequalities. Those corner points are where the best possible solution to a real-world problem often lives.

Students graph a shaded region bounded by two or more lines, then find which corner point of that region gives the highest or lowest value for a rule like profit or cost.

Students check whether answers to an optimization problem actually make sense, confirming the solution works in the graph, the equation, and the real-world situation it came from.

Students learn the six trig ratios (sine, cosine, tangent, and their three reciprocals) for angles in a right triangle, then use those ratios to find a missing side length or angle when only partial measurements are given.

Students learn the six trig ratios (sine, cosine, tangent, and their three reciprocals) by setting up fractions from the sides of a right triangle relative to one of its acute angles.

In a 30-60-90 triangle, the sides follow a fixed ratio every time. Students learn those ratios for both special right triangles so they can find a missing side without measuring or using a calculator.

Students apply right triangle rules and trigonometric ratios to solve real-world problems, such as finding the height of a building or the distance across a lake, by working with side lengths and angle measures in any triangle.

Students apply right triangle ratios to solve real-world problems, such as finding the height of a building or the angle of a ramp. Problems often involve angles of elevation (looking up) or depression (looking down).

Students use two formulas, the Law of Sines and the Law of Cosines, to find missing side lengths, angles, and area in triangles that have no right angle.

Given a triangle with no right angle, students use two formulas to find missing side lengths or missing angles. Which formula they reach for depends on what measurements they already have.

When two sides and a non-included angle are known, the Law of Sines can produce two different valid triangles instead of one. Students learn to spot when this happens and check whether both solutions make sense.

Students use two triangle rules, the Law of Sines and the Law of Cosines, alongside an area formula to find missing side lengths, angles, and area in any triangle, including real-world problems where the triangle is not a right triangle.

Students build a logical argument by starting with what they know and showing, step by step, why a conclusion must be true. This is the foundation of formal math proofs and structured problem-solving.

Students use overlapping circles to sort groups, find what they share, and work out logic puzzles. Venn diagrams turn abstract "who fits where" questions into a picture students can reason through.

Students translate everyday "if-then" sentences into symbols like p → q. This is the shorthand mathematicians use to test whether an argument holds up.

Students take an "if-then" sentence and flip or negate it in three specific ways to create new statements. Understanding how those rewrites change (or preserve) the original meaning is the core of the skill.

Symbolic logic uses symbols like AND, OR, and NOT to represent rules and decisions. Students explore how computers use this same logic to run programs and make choices inside apps.

Students build a chart that lists every possible true/false combination for a logical statement and shows what the result is each time. It is the math version of mapping out every "what if" before making a decision.

Students learn the core rules that let you draw valid conclusions from true statements. That includes De Morgan's Law, which explains how "not (A and B)" and "not (A or B)" work when you flip them around.

Students take a real-world situation, build a logical argument around it, and use that argument to make a decision. The math here is about thinking clearly, not just calculating.

Students learn to build a logical argument step by step, showing why each conclusion follows from what came before. Think of it as a math proof: every claim needs a reason, and the reasoning has to hold together from start to finish.

Students use logical thinking to work through real-world problems, like predicting what a program will do next or figuring out a puzzle. The goal is a sound argument, not just a right answer.

Students learn to map out the skeleton of a formal math proof before writing it, deciding whether to build straight toward a conclusion, assume the opposite and find a contradiction, or step through an infinite chain of cases one at a time.

Students look at a math problem and decide which proof method fits best, such as working directly from known facts, assuming the opposite and finding a contradiction, or proving all cases at once.

Students learn two ways to prove something is true: building a chain of logical steps from what they know, or assuming the opposite is true and showing that leads to a contradiction. Both methods follow formal rules of inference.

Students write step-by-step proofs that show a pattern holds true for every number in a sequence, covering cases like adding up a series of numbers or showing one value stays larger than another.

Students build a truth table, a grid that tests every possible true/false combination, to show two statements always produce the same result. If the final columns match row for row, the statements are logically equivalent.

Students use true/false logic rules to describe how simple circuits decide between two outcomes. They trace how "and," "or," and "not" conditions combine to control whether a circuit switches on or off.

Students learn three core rules that simplify logic circuits: every Boolean rule has a mirror version (duality), every value has an opposite (complements), and expressions can be written in two standard formats that make circuit design predictable.

Students read a plain-English sentence ("the light turns on only when both switches are flipped") and rewrite it as a Boolean expression using AND, OR, and NOT. The goal is moving from words to the symbolic logic a circuit can follow.

Students use Boolean algebra rules to show two logic expressions are equivalent or to rewrite a complicated expression in a simpler form, the way you would simplify a fraction.

Students build truth tables that map out every possible true/false combination for two or more logical statements. The table shows exactly when the whole expression is true and when it is false.

Students learn how basic logic gates (AND, OR, NOT) control the flow of signals in a circuit. They explain what each gate does: which inputs produce a true output and which produce a false one.

Students learn how the same math that uses only 0s and 1s also controls how real circuits in computers and phones decide whether to switch on or off.

Students break down a logic circuit made of multiple gates, write the Boolean expression that describes what it does, and use algebra to simplify or evaluate the output.

Students build small logic circuits using three specialized gate types: NAND, NOR, and XOR. Each gate combines or inverts signals in a specific way, and students wire them together to produce a desired true-or-false output.

Students learn to prove that a math formula works for every number by showing it holds for a starting case, then proving it must carry forward to the next. It's the logic behind why some rules apply infinitely.

Inductive reasoning draws a general rule from specific examples. Deductive reasoning starts with a known rule and proves a specific case must be true. Students learn how mathematicians use both to build and test arguments.

Weak induction proves a statement holds for the next step by assuming it works for one specific case. Strong induction assumes it works for all previous cases. Students learn when each approach is the right tool for the proof.

Students learn to prove that a number expression is always divisible by a given value, for every counting number, by showing the rule holds at the start and carries forward step by step.

Students use step-by-step logical proof to show why the formula for expanding something like (a + b)⁵ always works, no matter how large the exponent gets.

Students learn to read and write numbers in different formats, like binary (the 1s and 0s computers use) and the base-10 system people use every day, and convert between them.

Students learn to write the same number three ways: as the everyday digits we all use, as a string of 1s and 0s the way a computer stores it, and as the shorthand code programmers use to read large binary values quickly.

Students practice switching numbers between the counting systems computers use, such as converting a binary number like 1010 into its decimal form (10) or into a hexadecimal symbol.

Students learn to tell apart different kinds of data a computer can store, such as whole numbers, decimals, and text. Each type has its own rules for how it behaves and how much space it takes up.

Variables in a program can hold different kinds of data: whole numbers, decimals, letters, or true/false values. Students learn what makes each type different and when to use one over another.

Students learn that a Boolean holds only one of two possible values: true or false. It is the simplest data type in programming, used whenever code needs to make a yes-or-no decision.

A character is a single letter, digit, or symbol stored in memory as one unit. Students learn that a word like "dog" is actually three separate characters the computer stores one at a time.

An integer is a whole number with no decimal point, like -5, 0, or 42. When writing code, students choose this data type to store whole numbers and save memory compared to types that allow decimals.

Decimal numbers store values with a fractional part, like 3.14 or 99.9. Students learn when to use this data type instead of a whole-number type, and why the extra decimal places matter for accuracy in calculations.

A string is a data type that stores text rather than numbers. Students learn to recognize when a value like a name, address, or password should be stored as a string instead of as an integer or decimal.

Picking the right data type means choosing whether to store a number, a decimal, a single letter, or a word before writing code. Students learn why that choice matters and how to match the data to the job it needs to do.

Students choose how to organize and store information, such as picking a list, a table, or a grid based on what the data needs. The choice affects how easy it is to find, update, or use that information later.

Students look at a programming task and choose the right way to store the data for it. A shopping list calls for a different structure than a student record or a grid of numbers.

Students work with lists and grids of data stored in code, adding items, removing them, or pulling out specific values. Think of a one-dimensional list as a single row of lockers and a two-dimensional one as a full grid, like a spreadsheet.

Students learn to create a list or grid of stored values in code, giving a program a named place to hold a collection of related information, like a column of scores or a table of data.

Students decide whether a list of values needs whole numbers, decimals, or text, then pick the matching data type before writing the code. Getting this right keeps the program from storing data in a format that breaks later.

Students practice loading values into a list or array, slot by slot, until the structure holds a complete set of data ready to use in a program.

Students find a specific item inside a stored list and change its value, like updating one score in a column of grades without touching the rest.

Students use built-in object types (like a record or struct) to group related pieces of data together. For example, a single student object might hold a name, an ID number, and a grade all at once.

Students learn to recognize different types of equations, such as those with fractions, absolute values, or square roots, describe how each one behaves, and draw its graph.

Students look at an equation or graph and name what type of function it is, such as one with absolute values, fractions, or roots.

Given a graph, table, or equation, students identify key features of different function types: where the graph rises or falls, where it crosses the axes, and any gaps or breaks in the line.

Students identify which input values a function will accept and which output values it can produce. For a graph, that means reading the x-axis for domain and the y-axis for range.

Finding the roots of a polynomial means locating where its graph crosses the x-axis. Some polynomials also have complex roots, values involving the square root of a negative number that don't show up on the graph but still help explain the full solution set.

Students find where a function's graph crosses the x-axis and y-axis. Those crossing points, called intercepts, reveal where the output equals zero and where the graph begins its run across the coordinate plane.

Students identify whether a graph is symmetric across the y-axis (even) or the origin (odd), then connect that visual pattern to the function's equation.

Rational functions can have invisible boundary lines their graphs approach but never cross. Students identify where those lines are, whether they run side to side, up and down, or at a diagonal.

Graphs of some functions have gaps or holes where the line suddenly breaks or a single point sits isolated from the rest. Students find those break points and explain what happens to the function there.

Students identify the sections of a graph where the line rises, falls, or stays flat as it moves from left to right.

End behavior describes what happens to the y-values of a graph as x moves far to the left or far to the right. Students identify whether the graph rises or falls at each end.

Students find the peaks and valleys of a graph, identifying where a function reaches its highest or lowest point overall, and where it dips or rises locally before continuing in another direction.

Students sketch the graphs of several function types, including ones with sharp corners, breaks, or curves, showing how each equation's shape changes based on its rule.

Students find what value a function is heading toward as the input gets closer and closer to a specific number, even if the function never actually reaches that value.

Students use a graphing calculator to check whether their guess about where a function's output is heading actually holds up as the input gets close to a specific value.

Students find what value a function approaches as it gets closer to a specific input, using algebra to work it out and a graph to confirm the answer.

Students find what value a function is approaching as the input gets close to a certain number, using a table of values to check and a graph to confirm.

Students learn to write limits using formal notation, including how to describe what happens to a function as x grows very large or very small. This connects symbolic shorthand to the shape of a graph.

Students find what value a function gets closer and closer to as the input approaches a specific number, even if the function never actually reaches that value.

Students find the limit of a function by plugging a number directly into the equation and reading the result. This works when the function produces a clean, defined output at that point.

Students find the value a function approaches as the input gets closer to a number, using algebra to simplify or factor the expression when direct substitution does not work.

Students read a table of values to figure out what a function is approaching as the input gets close to a specific number, even if the function never actually reaches it.

Students read a graph to find the value a function is approaching as it gets closer to a specific point. This is called the limit, and reading it from a picture is often the first step before using algebra.

Students look at what happens to a function's output as the input grows toward positive or negative infinity. The goal is to describe where the graph is headed at the far left and far right edges.

Students examine a graph to see whether a function flows smoothly from one point to the next or breaks apart at gaps, holes, or jumps.

Students learn to spot whether a function's graph can be drawn in one smooth stroke or whether it breaks, jumps, or has a gap at some point.

Students describe whether a function has breaks, jumps, or holes, then write that observation using formal notation. They apply this to graphs and equations across several function types, from smooth curves to step-shaped graphs.

Students prove a function is continuous at a point by showing the limit exists there, equals the actual function value, and matches from both sides. This is the formal checklist behind the intuition that a graph has no breaks or holes.

Students look at a break in a graph and name what kind it is: a hole, a jump, or an infinite gap. Each type breaks continuity in a different way, and students explain which rule the function failed.

Students learn to measure angles two ways (degrees and radians), draw those angles on a coordinate grid, and calculate the six trig ratios for any angle using a point on its terminal side or one known trig value.

Students learn what a radian is: one way to measure an angle using the circle itself. They practice converting between radians and degrees and find how an angle's size connects to the length of the arc it cuts along the circle's edge.

Students learn two ways to measure how far an angle rotates around a circle: degrees (like on a protractor) and radians. They practice both positive and negative rotations, meaning turns that go clockwise or counterclockwise.

Students find other angles that land on the same position on a circle as a given angle, by adding or subtracting full rotations. They find at least one angle going forward and one going backward.

Students look at an angle drawn from the center of a coordinate plane and name which of the four sections (or which axis line) the angle's outer ray lands in.

Students plot a point on a graph, draw lines to form a right triangle, and use that triangle to measure the angle. It connects a point's location to the angle it makes with the horizontal axis.

Students start with a single trig ratio (like sine or cosine) and sketch the right triangle that fits it. From that triangle, they can read off all six trig values for the angle.

Given a point on a graph, students find all six trig ratios for the angle that opens to that point. This connects x and y coordinates to sine, cosine, tangent, and their reciprocals.

Given one trig ratio for an angle, students find the remaining five. They use right triangle relationships and the unit circle to move from a single known value to a complete picture of the angle.

Students find how long a curved piece of a circle's edge is by using the angle at the center, measured in radians, and the size of the circle.

Students find the area of a pie-slice portion of a circle using the circle's radius and the angle at its center.

Students learn how a circle with radius 1 works as a reference tool for measuring angles in both degrees and radians. They use it to find exact sine, cosine, and tangent values at key angles.

Students learn to switch between two ways of measuring angles: degrees (the familiar 0 to 360 scale) and radians (a scale based on the radius of a circle). They practice the conversions for the key angles on the unit circle from memory, without a calculator.

Students learn to define all six trig functions, sine, cosine, tangent, and their three reciprocals, using angles plotted on the unit circle. The x and y coordinates of a point on the circle become the building blocks for every function.

Students find the sine, cosine, and tangent of common angles like 30, 45, and 60 degrees by hand, using what they know about right triangles and the unit circle. No calculator allowed.

Solving multi-step equations and inequalities with one unknown variable, such as finding the value of x in an equation with several steps. Students also rearrange formulas, like solving a distance formula for time instead of distance.

Students solve equations and inequalities that use absolute value, then explain what the answer means. This includes finding one solution, two solutions, or a range of values on a number line.

Reading a word problem and turning it into an equation or inequality that fits the situation. Students figure out what the variable represents, then write a math sentence that matches the real-world setup.

Students read a real-world scenario and write an absolute value equation that captures it. This usually means describing a situation where a number can land a set distance above or below a target value.

Solving a multi-step equation means working through several operations in order to find the one value of a variable that makes both sides balance. Students solve these using rules like combining like terms and performing the same operation on both sides.

Solving an absolute value equation means finding every value of x that makes the equation true. Students solve it with algebra, then check their answers by graphing both sides and confirming where the lines meet.

Solving a multistep inequality means finding all the values of a variable that make it true, then marking that range on a number line. Students work through several steps of algebra, applying the same rules they use for equations, including flipping the inequality sign when multiplying or dividing by a negative number.

Rearranging a formula means isolating one variable, like solving the distance formula for time instead of distance. Students rewrite equations by applying the same operation to both sides until the target variable stands alone.

Students read a real-world situation and write an absolute value inequality to model it, such as finding all temperatures within a certain range of a target value.

Students figure out whether a linear equation has exactly one answer, no answer at all, or an endless number of answers, and explain why.

Students solve inequalities that use absolute value, then show the solution three ways: in set notation, in interval notation, and as a shaded graph on a number line.

Students check their answers to multi-step equations and inequalities by plugging values back in, reading a graph, or using a calculator. Then they explain in plain language why the answer makes sense for the problem.

Students check their answers to absolute value equations by plugging values back into the equation, plotting them on a graph, and using a calculator. Then they explain in plain language what those answers mean in the original problem.

Solving quadratic equations sometimes produces answers that aren't on the number line. Students learn to handle those results using complex numbers, and they solve quadratic inequalities to find ranges of values that make a statement true.

Students find where two lines cross on a graph, figure out which side of a line satisfies an inequality, or solve pairs of inequalities at once. They explain what those solutions mean in plain terms.

Students write a quadratic equation or inequality to match a real-world situation, like finding when a thrown ball hits the ground or when a cost stays under a budget.

Students take a real-world situation, such as comparing two phone plans or splitting a bill, and write two equations that together describe it. Both equations use the same two unknowns.

Students solve two equations at once to find the one point where both lines cross. They work through the algebra step by step and check the answer on a graph.

Students solve quadratic equations where the answer might be a real number or an imaginary one. They use algebra to find every possible solution, including complex numbers that involve the square root of a negative number.

Two straight lines drawn on the same graph can cross at one point, run parallel and never meet, or sit perfectly on top of each other. Students figure out which of those three situations a pair of equations describes.

Students solve quadratic inequalities like x² - 5x + 4 < 0 by finding where the expression is positive or negative, then writing the answer as a range of values on a number line.

Students write an inequality like 2x + 3y < 20 to model a real situation, such as spending limits or time constraints, where two quantities together must stay above or below a certain value.

Students solve quadratic equations, then check their answers by plugging values back in, reading a graph, or using a calculator. They also explain why their answer makes sense for the real situation the problem describes.

Students solve pairs of equations where one involves a squared variable, finding the points where two curves (or a curve and a line) cross on a graph.

Students graph a linear inequality on a coordinate plane, shading the region that shows all the points making the inequality true. This connects an algebraic rule to a visual picture of every solution at once.

Students write a pair of inequality rules (like "spend no more than $50" and "buy at least 3 items") that together capture the limits of a real-world situation using two unknowns.

Students set up a pair of equations, at least one of which is curved (like a parabola), to describe a real situation, such as when two objects meet or when a price and a demand curve intersect.

Students draw two boundary lines on a graph and shade the overlapping region where both inequalities are satisfied at once. That shaded overlap is the solution.

Students check whether a point actually solves a pair of equations or inequalities by testing it with math, a graph, and a calculator. Then they explain what that answer means in the real situation the problem describes.