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× 7","63"],["9 × 8","72"],["9 × 9","81"],["9 × 10","90"],["9 × 11","99"],["9 × 12","108"]]},{"source":"https://peabody.kids/flashcards/facts-times-all","id":"facts-times-all","title":"Multiplication Facts 0–12 (all 169)","category":"math","subject":"Arithmetic facts","levels":["3-5","6-8"],"description":"Every multiplication fact from 0 × 0 to 12 × 12.","tags":["facts","multiplication"],"frontLabel":"Problem","backLabel":"Answer","cards":[["0 × 0","0"],["0 × 1","0"],["0 × 2","0"],["0 × 3","0"],["0 × 4","0"],["0 × 5","0"],["0 × 6","0"],["0 × 7","0"],["0 × 8","0"],["0 × 9","0"],["0 × 10","0"],["0 × 11","0"],["0 × 12","0"],["1 × 0","0"],["1 × 1","1"],["1 × 2","2"],["1 × 3","3"],["1 × 4","4"],["1 × 5","5"],["1 × 6","6"],["1 × 7","7"],["1 × 8","8"],["1 × 9","9"],["1 × 10","10"],["1 × 11","11"],["1 × 12","12"],["2 × 0","0"],["2 × 1","2"],["2 × 2","4"],["2 × 3","6"],["2 × 4","8"],["2 × 5","10"],["2 × 6","12"],["2 × 7","14"],["2 × 8","16"],["2 × 9","18"],["2 × 10","20"],["2 × 11","22"],["2 × 12","24"],["3 × 0","0"],["3 × 1","3"],["3 × 2","6"],["3 × 3","9"],["3 × 4","12"],["3 × 5","15"],["3 × 6","18"],["3 × 7","21"],["3 × 8","24"],["3 × 9","27"],["3 × 10","30"],["3 × 11","33"],["3 × 12","36"],["4 × 0","0"],["4 × 1","4"],["4 × 2","8"],["4 × 3","12"],["4 × 4","16"],["4 × 5","20"],["4 × 6","24"],["4 × 7","28"],["4 × 8","32"],["4 × 9","36"],["4 × 10","40"],["4 × 11","44"],["4 × 12","48"],["5 × 0","0"],["5 × 1","5"],["5 × 2","10"],["5 × 3","15"],["5 × 4","20"],["5 × 5","25"],["5 × 6","30"],["5 × 7","35"],["5 × 8","40"],["5 × 9","45"],["5 × 10","50"],["5 × 11","55"],["5 × 12","60"],["6 × 0","0"],["6 × 1","6"],["6 × 2","12"],["6 × 3","18"],["6 × 4","24"],["6 × 5","30"],["6 × 6","36"],["6 × 7","42"],["6 × 8","48"],["6 × 9","54"],["6 × 10","60"],["6 × 11","66"],["6 × 12","72"],["7 × 0","0"],["7 × 1","7"],["7 × 2","14"],["7 × 3","21"],["7 × 4","28"],["7 × 5","35"],["7 × 6","42"],["7 × 7","49"],["7 × 8","56"],["7 × 9","63"],["7 × 10","70"],["7 × 11","77"],["7 × 12","84"],["8 × 0","0"],["8 × 1","8"],["8 × 2","16"],["8 × 3","24"],["8 × 4","32"],["8 × 5","40"],["8 × 6","48"],["8 × 7","56"],["8 × 8","64"],["8 × 9","72"],["8 × 10","80"],["8 × 11","88"],["8 × 12","96"],["9 × 0","0"],["9 × 1","9"],["9 × 2","18"],["9 × 3","27"],["9 × 4","36"],["9 × 5","45"],["9 × 6","54"],["9 × 7","63"],["9 × 8","72"],["9 × 9","81"],["9 × 10","90"],["9 × 11","99"],["9 × 12","108"],["10 × 0","0"],["10 × 1","10"],["10 × 2","20"],["10 × 3","30"],["10 × 4","40"],["10 × 5","50"],["10 × 6","60"],["10 × 7","70"],["10 × 8","80"],["10 × 9","90"],["10 × 10","100"],["10 × 11","110"],["10 × 12","120"],["11 × 0","0"],["11 × 1","11"],["11 × 2","22"],["11 × 3","33"],["11 × 4","44"],["11 × 5","55"],["11 × 6","66"],["11 × 7","77"],["11 × 8","88"],["11 × 9","99"],["11 × 10","110"],["11 × 11","121"],["11 × 12","132"],["12 × 0","0"],["12 × 1","12"],["12 × 2","24"],["12 × 3","36"],["12 × 4","48"],["12 × 5","60"],["12 × 6","72"],["12 × 7","84"],["12 × 8","96"],["12 × 9","108"],["12 × 10","120"],["12 × 11","132"],["12 × 12","144"]]},{"source":"https://peabody.kids/flashcards/math-2d-shapes","id":"math-2d-shapes","title":"Flat Shapes and Polygons","category":"math","subject":"Math","levels":["k-2","3-5","6-8"],"description":"Names and properties of 2D shapes, polygons by number of sides, and kinds of triangles and quadrilaterals.","tags":["shapes","geometry","polygons"],"frontLabel":"Shape","backLabel":"Facts","cards":[["Triangle","3 sides, 3 angles"],["Quadrilateral","4 sides, 4 angles"],["Pentagon","5 sides"],["Hexagon","6 sides"],["Heptagon","7 sides","Also called a septagon"],["Octagon","8 sides","A stop sign"],["Nonagon","9 sides"],["Decagon","10 sides"],["Hendecagon","11 sides","Also called an undecagon"],["Dodecagon","12 sides"],["Circle","All points the same distance from the center; no sides or corners"],["Oval (ellipse)","A stretched circle"],["Square","4 equal sides and 4 right angles"],["Rectangle","4 right angles; opposite sides equal"],["Rhombus","4 equal sides; opposite angles equal"],["Parallelogram","Both pairs of opposite sides are parallel and equal"],["Trapezoid","A quadrilateral with at least one pair of parallel sides"],["Kite","A quadrilateral with two pairs of equal sides that meet side by side"],["Equilateral triangle","All 3 sides equal; all angles 60 degrees"],["Isosceles triangle","At least 2 equal sides and 2 equal angles"],["Scalene triangle","No equal sides"],["Right triangle","Has one 90-degree angle"],["Acute triangle","All three angles are less than 90 degrees"],["Obtuse triangle","Has one angle greater than 90 degrees"],["Regular polygon","A polygon with all sides equal and all angles equal"],["Irregular polygon","A polygon whose sides or angles are not all equal"],["Convex polygon","A polygon with no angle pointing inward"],["Concave polygon","A polygon with at least one angle pointing inward"],["Sum of angles in a triangle","180 degrees"],["Sum of angles in a quadrilateral","360 degrees"],["Sum of angles in a pentagon","540 degrees"],["Sum of angles in a hexagon","720 degrees"],["Sum of angles in an n-sided polygon","(n − 2) × 180 degrees"],["Each angle of a regular hexagon","120 degrees"],["Each angle of a regular pentagon","108 degrees"],["Each angle of a regular octagon","135 degrees"],["Number of diagonals of a quadrilateral","2"],["Number of diagonals of a pentagon","5"],["Number of diagonals of a hexagon","9"],["Diagonal","A segment joining two corners of a polygon that are not next to each other"],["Is a square a rectangle?","Yes. A square is a rectangle with all sides equal."],["Is a square a rhombus?","Yes. A square is a rhombus with right angles."],["Is a rectangle a parallelogram?","Yes. Its opposite sides are parallel."],["Do the diagonals of a rectangle have equal lengths?","Yes"],["Do the diagonals of a rhombus meet at right angles?","Yes, and they bisect each other"],["Crescent","A moon shape formed by two arcs"],["Semicircle","Half of a circle"],["Quarter circle","A quarter of a circle; a 90-degree sector"],["Star shape (pentagram)","A five-pointed star drawn with five straight lines"]]},{"source":"https://peabody.kids/flashcards/math-3d-solids","id":"math-3d-solids","title":"Solid Shapes: Faces, Edges, Vertices","category":"math","subject":"Math","levels":["k-2","3-5","6-8"],"description":"3D shapes and their parts, including the Platonic solids and Euler's formula.","tags":["solids","3d shapes","geometry"],"frontLabel":"Solid","backLabel":"Facts","cards":[["Cube","6 square faces, 12 edges, 8 vertices"],["Rectangular prism","6 rectangular faces, 12 edges, 8 vertices","A shoebox"],["Triangular prism","5 faces, 9 edges, 6 vertices"],["Square pyramid","5 faces, 8 edges, 5 vertices"],["Triangular pyramid (tetrahedron)","4 faces, 6 edges, 4 vertices"],["Pentagonal prism","7 faces, 15 edges, 10 vertices"],["Hexagonal prism","8 faces, 18 edges, 12 vertices"],["Cylinder","2 flat circular faces and 1 curved surface; no vertices"],["Cone","1 flat circular face, 1 curved surface, 1 vertex"],["Sphere","A perfectly round solid; every surface point is the same distance from the center"],["Hemisphere","Half of a sphere"],["Face","A flat surface of a solid"],["Edge","A line where two faces meet"],["Vertex","A corner where edges meet"],["Base","The face a solid sits on, or the matching ends of a prism"],["Prism","A solid with two parallel identical bases and flat rectangular sides"],["Pyramid","A solid with one base and triangular sides meeting at an apex"],["Apex","The top point of a pyramid or cone"],["Euler's formula for polyhedra","Vertices − Edges + Faces = 2","V − E + F = 2"],["Polyhedron","A solid with only flat faces"],["Platonic solid","A polyhedron whose faces are all the same regular polygon, with the same number meeting at each vertex"],["How many Platonic solids are there?","Five"],["Tetrahedron","Platonic solid with 4 triangular faces"],["Cube (hexahedron)","Platonic solid with 6 square faces"],["Octahedron","Platonic solid with 8 triangular faces"],["Dodecahedron","Platonic solid with 12 pentagonal faces"],["Icosahedron","Platonic solid with 20 triangular faces"],["Net of a cube","A flat pattern of 6 connected squares that folds into a cube"],["Lateral face","A side face of a prism or pyramid (not a base)"],["Slant height","The distance along the sloped side of a cone or pyramid from apex to base edge"],["Which solid rolls and slides?","A cylinder"],["Which solid rolls but does not stack?","A sphere"],["Cuboid","Another name for a rectangular prism"],["Torus","A donut shape"],["Ellipsoid","A stretched sphere, like a rugby ball"],["Frustum","A cone or pyramid with its top sliced off parallel to the base"]]},{"source":"https://peabody.kids/flashcards/math-algebra-rules","id":"math-algebra-rules","title":"Algebra Rules and Properties","category":"math","subject":"Math","levels":["6-8","9-12"],"description":"Properties of numbers, rules for solving equations and inequalities, and common factoring patterns.","tags":["algebra","properties","rules"],"frontLabel":"Rule","backLabel":"Statement","cards":[["Commutative property of addition","a + b = b + a"],["Commutative property of multiplication","ab = ba"],["Associative property of addition","(a + b) + c = a + (b + c)"],["Associative property of multiplication","(ab)c = a(bc)"],["Distributive property","a(b + c) = ab + ac"],["Additive identity","a + 0 = a"],["Multiplicative identity","a × 1 = a"],["Additive inverse","a + (−a) = 0"],["Multiplicative inverse","a × (1/a) = 1, for a not zero"],["Zero property of multiplication","a × 0 = 0"],["Reflexive property","a = a"],["Symmetric property","If a = b, then b = a"],["Transitive property","If a = b and b = c, then a = c"],["Addition property of equality","If a = b, then a + c = b + c"],["Multiplication property of equality","If a = b, then ac = bc"],["Substitution property","If a = b, then b can replace a in any expression"],["Rule for multiplying two negatives","The product is positive"],["Rule for multiplying a negative and a positive","The product is negative"],["Rule for dividing two negatives","The quotient is positive"],["Subtracting a negative","a − (−b) = a + b"],["Solve x + 7 = 12","x = 5"],["Solve 3x = 21","x = 7"],["Solve 2x − 5 = 9","x = 7"],["Solve x/4 = 6","x = 24"],["Solve 5x + 3 = 2x + 15","x = 4"],["Rule for inequalities","Multiplying or dividing by a negative number flips the inequality sign"],["Solve −2x > 6","x < −3"],["Solve |x| = 5","x = 5 or x = −5"],["Solve |x| < 3","−3 < x < 3"],["Solve |x| > 3","x < −3 or x > 3"],["FOIL","First, Outer, Inner, Last: multiplying two binomials"],["Expand (x + 3)(x + 2)","x² + 5x + 6"],["Expand (x + 4)²","x² + 8x + 16"],["Expand (x − 5)²","x² − 10x + 25"],["Expand (x + 3)(x − 3)","x² − 9"],["Factor x² + 7x + 12","(x + 3)(x + 4)"],["Factor x² − 9","(x − 3)(x + 3)"],["Factor x² − 5x + 6","(x − 2)(x − 3)"],["Factor x² + 2x + 1","(x + 1)²"],["Factor 2x² + 6x","2x(x + 3)"],["Sum of cubes","a³ + b³ = (a + b)(a² − ab + b²)"],["Difference of cubes","a³ − b³ = (a − b)(a² + ab + b²)"],["Binomial cube","(a + b)³ = a³ + 3a²b + 3ab² + b³"],["Solve x² = 49","x = 7 or x = −7"],["Solve x² − 5x + 6 = 0","x = 2 or x = 3"],["Midpoint of two numbers","Their average: (a + b)/2"],["Cross-multiplication","If a/b = c/d, then ad = bc"],["Adding fractions with unlike denominators","Use a common denominator first"],["Slope between two points","(y₂ − y₁)/(x₂ − x₁)"],["Perpendicular to slope 2","Slope −1/2"],["Distance, rate, time","d = r × t"],["Percent change","(new − old) / old × 100%"],["Simple interest","I = P r t"],["Average (mean) formula","Sum of values divided by number of values"],["Order of solving with variables on both sides","Collect variable terms on one side, constants on the other, then divide"],["Check a solution","Substitute it back into the original equation"]]},{"source":"https://peabody.kids/flashcards/math-algebra-terms","id":"math-algebra-terms","title":"Algebra Terms","category":"math","subject":"Math","levels":["6-8","9-12"],"description":"The vocabulary of algebra: expressions, equations, functions, graphing and polynomials.","tags":["algebra","vocabulary"],"frontLabel":"Term","backLabel":"Meaning","cards":[["Variable","A letter that stands for a number"],["Constant","A fixed number that does not change"],["Coefficient","The number multiplying a variable","In −4y, it is −4"],["Term","A part of an expression separated by + or −"],["Like terms","Terms with the same variables raised to the same powers"],["Expression","A combination of numbers, variables and operations without an equals sign"],["Equation","A statement that two expressions are equal"],["Inequality","A statement comparing two expressions using <, >, ≤ or ≥"],["Solve","Find the value of the variable that makes the statement true"],["Evaluate","Replace variables with numbers and compute the result"],["Simplify","Combine like terms and reduce an expression"],["Factor","Write an expression as a product of simpler expressions"],["Expand","Multiply out brackets to remove them"],["Substitution","Replacing a variable with a value or another expression"],["Linear equation","An equation whose graph is a straight line; the variable has degree 1"],["Slope-intercept form","y = mx + b"],["Point-slope form","y − y₁ = m(x − x₁)"],["Standard form of a line","Ax + By = C"],["Slope","Rise over run; change in y divided by change in x"],["Rise","The vertical change between two points"],["Run","The horizontal change between two points"],["Y-intercept","Where a graph crosses the y-axis (x = 0)"],["X-intercept","Where a graph crosses the x-axis (y = 0)"],["Parallel lines (slopes)","Have equal slopes"],["Perpendicular lines (slopes)","Slopes are negative reciprocals; their product is −1"],["Slope of a horizontal line","0"],["Slope of a vertical line","Undefined"],["Function","A rule that gives each input exactly one output"],["Function notation","f(x) is read 'f of x'; the output for input x"],["Domain","All allowed input values"],["Range","All resulting output values"],["Independent variable","The input variable, usually x"],["Dependent variable","The output variable, usually y; it depends on x"],["Vertical line test","A graph is a function if no vertical line crosses it twice"],["Absolute value","The distance of a number from zero, written |x|"],["System of equations","Two or more equations solved together"],["Substitution method","Solve one equation for a variable and substitute into the other"],["Elimination method","Add or subtract equations to cancel a variable"],["Consistent system","A system with at least one solution"],["Inconsistent system","A system with no solution, such as parallel lines"],["Dependent system","A system with infinitely many solutions; the same line"],["Polynomial","An expression of terms with whole-number powers of variables"],["Degree","The highest power of the variable in a polynomial"],["Standard form of a polynomial","Terms written from the highest power to the lowest"],["Quadratic equation","An equation of the form ax² + bx + c = 0"],["Quadratic formula","x = (−b ± √(b² − 4ac)) / 2a"],["Completing the square","Rewriting a quadratic so a perfect square trinomial appears"],["Vertex form","y = a(x − h)² + k"],["Difference of squares","a² − b² = (a − b)(a + b)"],["Perfect square trinomial","a² + 2ab + b² = (a + b)²"],["Greatest common factor (GCF)","The largest factor shared by all the terms"],["Zero product property","If ab = 0, then a = 0 or b = 0"],["Root, zero or solution","An x value where the function equals 0"],["Radical","A root symbol, such as √"],["Rationalize the denominator","Rewrite a fraction so the denominator has no radical"],["Rational expression","A fraction whose numerator and denominator are polynomials"],["Inverse variation","y = k/x; as one quantity goes up, the other goes down"],["Direct variation","y = kx; the ratio y/x is constant"],["Constant of variation","The constant k in a variation equation"],["Arithmetic sequence formula","aₙ = a₁ + (n − 1)d"],["Geometric sequence formula","aₙ = a₁ · rⁿ⁻¹"],["Exponential growth","y = a(1 + r)^t with r > 0"],["Exponential decay","y = a(1 − r)^t with 0 < r < 1"],["Compound interest formula","A = P(1 + r/n)^(nt)"],["Continuous compounding formula","A = Pe^(rt)"]]},{"source":"https://peabody.kids/flashcards/math-angles-lines","id":"math-angles-lines","title":"Angles, Lines and Circles","category":"math","subject":"Math","levels":["3-5","6-8","9-12"],"description":"Kinds of angles, angle pairs, lines and the parts of a circle.","tags":["angles","lines","circles"],"frontLabel":"Term","backLabel":"Meaning","cards":[["Acute angle","Between 0° and 90°"],["Right angle","Exactly 90°"],["Obtuse angle","Between 90° and 180°"],["Straight angle","Exactly 180°"],["Reflex angle","Between 180° and 360°"],["Full rotation","360°"],["Complementary angles","Add up to 90°"],["Supplementary angles","Add up to 180°"],["Complement of 30°","60°"],["Supplement of 110°","70°"],["Vertical angles","Opposite angles made by two crossing lines; they are equal"],["Adjacent angles","Share a vertex and a side, with no overlap"],["Linear pair","Two adjacent angles that form a straight line; add up to 180°"],["Corresponding angles","Angles in matching positions at two different crossings of a transversal"],["Alternate interior angles","Angles inside the two lines on opposite sides of the transversal"],["Alternate exterior angles","Angles outside the two lines on opposite sides of the transversal"],["Same-side interior angles","Angles inside the two lines on the same side of the transversal"],["Vertex of an angle","The point where the two rays meet"],["Angle bisector","A ray that splits an angle into two equal angles"],["Protractor","A tool for measuring angles in degrees"],["Line","A straight path that goes on forever in both directions"],["Line segment","A piece of a line between two endpoints"],["Ray","A part of a line with one endpoint, going on forever one way"],["Collinear points","Points that lie on the same line"],["Coplanar","Lying in the same flat plane"],["Parallel lines","Lines in a plane that never meet"],["Perpendicular lines","Lines that meet at 90°"],["Skew lines","Lines in 3D that do not meet and are not parallel"],["Midpoint","The point exactly halfway along a segment"],["Perpendicular bisector","A line that cuts a segment in half at a right angle"],["Transversal","A line that crosses two or more lines"],["Plane","A flat surface that extends forever in every direction"],["Center of a circle","The point equally far from every point on the circle"],["Radius","A segment from the center to the circle"],["Diameter","A chord through the center; the longest chord"],["Chord","A segment joining two points on a circle"],["Secant","A line that cuts through a circle at two points"],["Tangent","A line that touches a circle at one point"],["Arc","A piece of the circle's outline"],["Minor arc","An arc smaller than a semicircle"],["Major arc","An arc larger than a semicircle"],["Semicircle","Half a circle; an arc of 180°"],["Sector","A region bounded by two radii and an arc"],["Segment of a circle","A region bounded by a chord and an arc"],["Central angle","An angle with its vertex at the center of a circle"],["Inscribed angle","An angle with its vertex on the circle"],["Concentric circles","Circles that share the same center"],["Congruent circles","Circles with equal radii"],["Degrees in a circle","360"],["Radians in a full circle","2π"],["Degrees in one radian","About 57.3°"],["Convert 180° to radians","π radians"],["Convert 90° to radians","π/2"],["Convert 60° to radians","π/3"],["Convert π/4 to degrees","45°"],["Convert π/6 to degrees","30°"]]},{"source":"https://peabody.kids/flashcards/math-calculus-concepts","id":"math-calculus-concepts","title":"Calculus Concepts and Limits","category":"math","subject":"Math","levels":["college","9-12"],"description":"Core calculus vocabulary and limit values, from continuity to series tests.","tags":["calculus","limits","series"],"frontLabel":"Term","backLabel":"Meaning","cards":[["Limit","The value a function approaches as the input approaches a number"],["One-sided limit","A limit approached only from the left or only from the right"],["Limit at infinity","The value a function approaches as x grows without bound"],["Continuous at a point","The limit equals the function value there"],["Removable discontinuity","A hole in the graph that could be filled in"],["Jump discontinuity","The left and right limits exist but differ"],["Infinite discontinuity","The function grows without bound near that point (vertical asymptote)"],["Intermediate value theorem","A continuous function takes every value between f(a) and f(b)"],["Extreme value theorem","A continuous function on a closed interval has a maximum and a minimum"],["Differentiable","Having a derivative at a point; the graph is smooth there"],["Differentiable implies","Continuous"],["Local maximum","A point higher than nearby points"],["Local minimum","A point lower than nearby points"],["Absolute (global) extremum","The highest or lowest point over the whole domain"],["Optimization","Using derivatives to find the best (largest or smallest) value"],["Linear approximation","Using the tangent line to estimate nearby values"],["Differential","dy = f'(x) dx; a small change in y"],["Antiderivative","A function whose derivative is the given function"],["Riemann sum","A sum of rectangle areas that approximates an area"],["Definite integral","A number: signed area under a curve between limits"],["Net change","The integral of a rate of change"],["Slope field","A picture of tangent slopes for a differential equation"],["Differential equation","An equation involving a function and its derivatives"],["Separable differential equation","One where the variables can be moved to opposite sides"],["Exponential growth equation","dy/dt = ky has solution y = Ce^(kt)"],["Sequence limit","The value the terms approach as n grows"],["Geometric series test","Converges if |r| < 1, with sum a/(1 − r)"],["p-series","Σ 1/nᵖ converges if p > 1 and diverges if p ≤ 1"],["Ratio test","Compare consecutive terms; convergence if the limit of |aₙ₊₁/aₙ| is less than 1"],["Alternating series test","Converges if terms alternate sign, decrease in size and go to 0"],["Taylor series","A power series that matches a function's derivatives at a point"],["Maclaurin series","A Taylor series centered at 0"],["Maclaurin series of eˣ","1 + x + x²/2! + x³/3! + ..."],["Maclaurin series of sin x","x − x³/3! + x⁵/5! − ..."],["Maclaurin series of cos x","1 − x²/2! + x⁴/4! − ..."],["Maclaurin series of 1/(1 − x)","1 + x + x² + x³ + ... for |x| < 1"],["Radius of convergence","How far from the center a power series converges"],["lim (x→0) sin x / x","1"],["lim (x→0) (1 − cos x) / x","0"],["lim (x→0) (eˣ − 1) / x","1"],["lim (x→∞) (1 + 1/x)ˣ","e"],["lim (x→∞) 1/x","0"],["lim (x→1) (x² − 1)/(x − 1)","2"],["lim (x→∞) (3x² + 1)/(x² + 5)","3"],["lim (x→0) ln(1 + x)/x","1"],["Squeeze theorem","If a function is trapped between two functions with the same limit, it has that limit"],["Vertical asymptote","A vertical line the graph approaches where values blow up"],["Horizontal asymptote","A horizontal line the graph approaches as x goes to ±∞"],["Indeterminate form","A limit form such as 0/0 or ∞/∞ that needs more work"],["Partial derivative","A derivative with respect to one variable, others held constant"],["Gradient","A vector of partial derivatives pointing in the direction of steepest increase"],["Double integral","Integral over a 2D region, finding volume under a surface"],["Multivariable calculus","Calculus of functions with more than one input"],["Differential vs. derivative","The derivative is a rate; the differential is a small change"],["Leibniz notation","dy/dx"],["Lagrange notation","f'(x)"],["Newton's notation","ẏ, a dot over the variable"],["Who developed calculus independently?","Isaac Newton and Gottfried Leibniz"]]},{"source":"https://peabody.kids/flashcards/math-coordinate-graphing","id":"math-coordinate-graphing","title":"Coordinate Plane and Graphing","category":"math","subject":"Math","levels":["3-5","6-8","9-12"],"description":"Parts of the coordinate plane, plotting points, transformations and reading graphs of functions.","tags":["coordinate plane","graphing","transformations"],"frontLabel":"Question","backLabel":"Answer","cards":[["Horizontal axis","The x-axis"],["Vertical axis","The y-axis"],["Origin","The point (0, 0)"],["Ordered pair (x, y)","First number is across, second is up or down"],["Quadrant I","x positive, y positive (top right)"],["Quadrant II","x negative, y positive (top left)"],["Quadrant III","x negative, y negative (bottom left)"],["Quadrant IV","x positive, y negative (bottom right)"],["Where is (0, 5)?","On the y-axis, 5 units above the origin"],["Where is (−3, 0)?","On the x-axis, 3 units left of the origin"],["Which quadrant is (3, −2)?","IV"],["Which quadrant is (−4, −1)?","III"],["Which quadrant is (−2, 6)?","II"],["Reflect (3, 4) over the x-axis","(3, −4)"],["Reflect (3, 4) over the y-axis","(−3, 4)"],["Reflect (3, 4) through the origin","(−3, −4)"],["Reflect (a, b) over the line y = x","(b, a)"],["Translate (2, 1) by 3 right and 2 up","(5, 3)"],["Rotate (a, b) 90° counterclockwise about the origin","(−b, a)"],["Rotate (a, b) 180° about the origin","(−a, −b)"],["Dilate (2, 3) by scale factor 2 about the origin","(4, 6)"],["Distance from (0, 0) to (3, 4)","5"],["Distance from (1, 2) to (4, 6)","5"],["Midpoint of (2, 4) and (6, 8)","(4, 6)"],["Midpoint of (−2, 3) and (4, −1)","(1, 1)"],["Slope through (0, 0) and (2, 4)","2"],["Slope through (1, 5) and (3, 1)","−2"],["Slope of y = 3x + 2","3"],["Y-intercept of y = 3x + 2","2"],["X-intercept of y = 2x − 6","3"],["Graph of y = 5","A horizontal line"],["Graph of x = −2","A vertical line"],["Graph of y = x","A line through the origin with slope 1"],["Graph of y = x²","A parabola opening up with vertex at the origin"],["Graph of y = −x²","A parabola opening down"],["Graph of y = |x|","A V shape with vertex at the origin"],["Graph of y = 1/x","A hyperbola with branches in quadrants I and III"],["Graph of y = √x","Half of a sideways parabola starting at the origin"],["Graph of y = 2ˣ","A curve that rises quickly and passes through (0, 1)"],["Graph of x² + y² = 9","A circle with center (0, 0) and radius 3"],["Shift y = f(x) up 3","y = f(x) + 3"],["Shift y = f(x) right 2","y = f(x − 2)"],["Reflect y = f(x) over the x-axis","y = −f(x)"],["Reflect y = f(x) over the y-axis","y = f(−x)"],["Stretch y = f(x) vertically by 3","y = 3f(x)"],["Vertex of y = (x − 2)² + 5","(2, 5)"],["Axis of symmetry of y = (x − 2)² + 5","x = 2"],["Parallel to y = 2x + 1","Any line with slope 2"],["Perpendicular to y = 2x + 1","Any line with slope −1/2"],["Rise over run","Slope"],["Positive slope","The line goes up from left to right"],["Negative slope","The line goes down from left to right"],["Where lines of a system meet","The solution of the system"],["Line graph","Plots points joined by lines to show change over time"],["Scale of a graph","The value each grid mark stands for"],["Interval of an axis","The step between marked numbers"],["Latitude and longitude","A coordinate system for locations on Earth"],["Battleship grid","A game that uses coordinates like B7"],["Plotting order","Start at the origin, move across x, then up or down y"],["René Descartes","Introduced the Cartesian coordinate system"],["Polar coordinates (r, θ)","Distance from the origin and angle from the positive x-axis"],["3D coordinates","Add a z-axis: (x, y, z)"],["Number of axes in a 3D grid","3"]]},{"source":"https://peabody.kids/flashcards/math-cube-roots","id":"math-cube-roots","category":"math","subject":"Math","title":"Cube Roots of Perfect Cubes","levels":["6-8","9-12"],"description":"Cube roots of the perfect cubes from 1 to 8000.","tags":["cube roots"],"frontLabel":"Root","backLabel":"Value","cards":[["∛1","1"],["∛8","2"],["∛27","3"],["∛64","4"],["∛125","5"],["∛216","6"],["∛343","7"],["∛512","8"],["∛729","9"],["∛1000","10"],["∛1331","11"],["∛1728","12"],["∛2197","13"],["∛2744","14"],["∛3375","15"],["∛4096","16"],["∛4913","17"],["∛5832","18"],["∛6859","19"],["∛8000","20"]]},{"source":"https://peabody.kids/flashcards/math-derivative-rules","id":"math-derivative-rules","title":"Derivative Rules","category":"math","subject":"Math","levels":["9-12","college"],"description":"Differentiation rules and the derivatives of the common functions.","tags":["calculus","derivatives"],"frontLabel":"Function or rule","backLabel":"Derivative","cards":[["Constant: d/dx [c]","0"],["Power rule: d/dx [xⁿ]","n xⁿ⁻¹"],["Constant multiple rule","d/dx [c f(x)] = c f'(x)"],["Sum rule","d/dx [f + g] = f' + g'"],["Difference rule","d/dx [f − g] = f' − g'"],["Product rule","d/dx [fg] = f'g + fg'"],["Quotient rule","d/dx [f/g] = (f'g − fg') / g²"],["Chain rule","d/dx [f(g(x))] = f'(g(x)) · g'(x)"],["d/dx [x]","1"],["d/dx [x²]","2x"],["d/dx [x³]","3x²"],["d/dx [√x]","1/(2√x)"],["d/dx [1/x]","−1/x²"],["d/dx [eˣ]","eˣ"],["d/dx [aˣ]","aˣ ln a"],["d/dx [ln x]","1/x"],["d/dx [log_a x]","1/(x ln a)"],["d/dx [sin x]","cos x"],["d/dx [cos x]","−sin x"],["d/dx [tan x]","sec² x"],["d/dx [cot x]","−csc² x"],["d/dx [sec x]","sec x tan x"],["d/dx [csc x]","−csc x cot x"],["d/dx [arcsin x]","1/√(1 − x²)"],["d/dx [arccos x]","−1/√(1 − x²)"],["d/dx [arctan x]","1/(1 + x²)"],["d/dx [sinh x]","cosh x"],["d/dx [cosh x]","sinh x"],["d/dx [e^(2x)]","2e^(2x)"],["d/dx [sin(3x)]","3 cos(3x)"],["d/dx [(x² + 1)³]","6x(x² + 1)²"],["d/dx [x eˣ]","eˣ(x + 1)"],["d/dx [x sin x]","sin x + x cos x"],["d/dx [ln(x²)]","2/x"],["d/dx [5x⁴ − 3x + 7]","20x³ − 3"],["Derivative as slope","The slope of the tangent line at a point"],["Limit definition of derivative","f'(x) = lim (h→0) [f(x + h) − f(x)] / h"],["Second derivative","The derivative of the derivative; tells concavity"],["Critical point","A point where f' = 0 or f' does not exist"],["First derivative test","f' changes from + to − at a local maximum, and − to + at a local minimum"],["Second derivative test","f'' > 0 gives a local minimum; f'' < 0 gives a local maximum"],["Inflection point","A point where concavity changes"],["Concave up","f'' > 0; the graph curves like a cup"],["Concave down","f'' < 0; the graph curves like a cap"],["Velocity from position","v(t) = s'(t)"],["Acceleration from position","a(t) = s''(t)"],["Mean value theorem","A smooth curve has a point where the slope equals the average slope over the interval"],["L'Hôpital's rule","For 0/0 or ∞/∞ forms, lim f/g = lim f'/g' (if that limit exists)"],["Implicit differentiation","Differentiate both sides with respect to x, using dy/dx for y terms"],["Related rates","Differentiate an equation in time to link rates of change"],["Tangent line equation","y − f(a) = f'(a)(x − a)"],["Rolle's theorem","If f(a) = f(b) and f is smooth, there is a point between where f' = 0"]]},{"source":"https://peabody.kids/flashcards/math-exponent-rules","id":"math-exponent-rules","title":"Exponent Rules","category":"math","subject":"Math","levels":["6-8","9-12"],"description":"The laws of exponents, with worked examples including zero, negative and fractional exponents.","tags":["exponents","powers","algebra"],"frontLabel":"Rule","backLabel":"Result","cards":[["Product rule","aᵐ · aⁿ = aᵐ⁺ⁿ","Same base: add exponents"],["Quotient rule","aᵐ / aⁿ = aᵐ⁻ⁿ","Same base: subtract exponents"],["Power of a power","(aᵐ)ⁿ = aᵐⁿ","Multiply exponents"],["Power of a product","(ab)ⁿ = aⁿbⁿ"],["Power of a quotient","(a/b)ⁿ = aⁿ/bⁿ"],["Zero exponent","a⁰ = 1, for a not zero"],["Negative exponent","a⁻ⁿ = 1/aⁿ"],["Fractional exponent","a^(1/n) = the nth root of a"],["Rational exponent","a^(m/n) = (nth root of a)^m"],["Exponent of 1","a¹ = a"],["2³ · 2⁴","2⁷ = 128"],["5⁶ / 5²","5⁴ = 625"],["(3²)³","3⁶ = 729"],["(2x)³","8x³"],["(x²)⁴","x⁸"],["x³ · x⁵","x⁸"],["x⁹ / x³","x⁶"],["(−2)²","4"],["−2²","−4","The exponent applies only to the 2"],["(−2)³","−8"],["10⁰","1"],["0⁰","Usually left undefined (or treated as 1 in some settings)"],["2⁻³","1/8"],["10⁻²","0.01"],["4^(1/2)","2"],["8^(1/3)","2"],["27^(2/3)","9"],["16^(3/4)","8"],["9^(−1/2)","1/3"],["Is (a + b)² equal to a² + b²?","No. (a + b)² = a² + 2ab + b²"],["Is aᵐ + aⁿ = aᵐ⁺ⁿ?","No. The product rule works for multiplying, not adding"],["Negative base with even power","The result is positive"],["Negative base with odd power","The result is negative"],["Scientific notation of 45,000","4.5 × 10⁴"],["Scientific notation of 0.0007","7 × 10⁻⁴"],["(3 × 10²)(2 × 10³)","6 × 10⁵"],["Square root as an exponent","√a = a^(1/2)"],["Cube root as an exponent","∛a = a^(1/3)"],["Reciprocal of x⁻²","x²"],["Simplify (x³y²)²","x⁶y⁴"],["Simplify x⁻²y³ with positive exponents","y³/x²"]]},{"source":"https://peabody.kids/flashcards/math-factorials","id":"math-factorials","category":"math","subject":"Math","title":"Factorials 0 to 12","levels":["9-12","college"],"description":"n factorial is the product of all whole numbers from 1 up to n.","tags":["factorial","counting"],"frontLabel":"Factorial","backLabel":"Value","cards":[["0!","1","By definition"],["1!","1"],["2!","2"],["3!","6"],["4!","24"],["5!","120"],["6!","720"],["7!","5040"],["8!","40320"],["9!","362880"],["10!","3628800"],["11!","39916800"],["12!","479001600"]]},{"source":"https://peabody.kids/flashcards/math-fraction-decimal-percent","id":"math-fraction-decimal-percent","category":"math","subject":"Math","title":"Fractions, Decimals and Percents","levels":["3-5","6-8"],"description":"Common fractions matched to their decimal and percent forms.","tags":["fractions","decimals","percent","conversion"],"frontLabel":"Convert","backLabel":"Answer","cards":[["1/2 as a decimal","0.5"],["1/2 as a percent","50%"],["1/3 as a decimal","0.333... (3 repeats)"],["1/3 as a percent","about 33.3%, exactly 33 and 1/3%"],["2/3 as a decimal","0.666... (6 repeats)"],["2/3 as a percent","about 66.7%, exactly 66 and 2/3%"],["1/4 as a decimal","0.25"],["1/4 as a percent","25%"],["3/4 as a decimal","0.75"],["3/4 as a percent","75%"],["1/5 as a decimal","0.2"],["1/5 as a percent","20%"],["2/5 as a decimal","0.4"],["2/5 as a percent","40%"],["3/5 as a decimal","0.6"],["3/5 as a percent","60%"],["4/5 as a decimal","0.8"],["4/5 as a percent","80%"],["1/6 as a decimal","0.1666... (6 repeats)"],["1/6 as a percent","about 16.7%, exactly 16 and 2/3%"],["5/6 as a decimal","0.8333... (3 repeats)"],["5/6 as a percent","about 83.3%, exactly 83 and 1/3%"],["1/8 as a decimal","0.125"],["1/8 as a percent","12.5%"],["3/8 as a decimal","0.375"],["3/8 as a percent","37.5%"],["5/8 as a decimal","0.625"],["5/8 as a percent","62.5%"],["7/8 as a decimal","0.875"],["7/8 as a percent","87.5%"],["1/10 as a decimal","0.1"],["1/10 as a percent","10%"],["3/10 as a decimal","0.3"],["3/10 as a percent","30%"],["7/10 as a decimal","0.7"],["7/10 as a percent","70%"],["9/10 as a decimal","0.9"],["9/10 as a percent","90%"],["1/20 as a decimal","0.05"],["1/20 as a percent","5%"],["1/25 as a decimal","0.04"],["1/25 as a percent","4%"],["3/20 as a decimal","0.15"],["3/20 as a percent","15%"],["1/50 as a decimal","0.02"],["1/50 as a percent","2%"],["1/100 as a decimal","0.01"],["1/100 as a percent","1%"],["1/12 as a decimal","0.08333... (3 repeats)"],["1/12 as a percent","about 8.3%, exactly 8 and 1/3%"],["2/8 as a decimal","0.25"],["2/8 as a percent","25%"],["9/20 as a decimal","0.45"],["9/20 as a percent","45%"],["11/20 as a decimal","0.55"],["11/20 as a percent","55%"],["13/20 as a decimal","0.65"],["13/20 as a percent","65%"],["17/20 as a decimal","0.85"],["17/20 as a percent","85%"],["19/20 as a decimal","0.95"],["19/20 as a percent","95%"],["3/25 as a decimal","0.12"],["3/25 as a percent","12%"],["7/20 as a decimal","0.35"],["7/20 as a percent","35%"]]},{"source":"https://peabody.kids/flashcards/math-fractions-skills","id":"math-fractions-skills","title":"Fractions: Terms and Operations","category":"math","subject":"Math","levels":["3-5","6-8"],"description":"Fraction vocabulary and how to add, subtract, multiply, divide, simplify and compare fractions.","tags":["fractions","operations"],"frontLabel":"Question","backLabel":"Answer","cards":[["What does the denominator tell you?","How many equal parts the whole is divided into"],["What does the numerator tell you?","How many of those parts are being counted"],["Proper fraction","Numerator is smaller than denominator, like 3/5"],["Improper fraction","Numerator is equal to or larger than denominator, like 9/4"],["Mixed number","A whole number plus a fraction, like 2 1/4"],["Equivalent fractions","Different fractions with the same value"],["How to make an equivalent fraction","Multiply or divide top and bottom by the same number"],["Simplify 6/8","3/4"],["Simplify 10/15","2/3"],["Simplify 12/18","2/3"],["Simplify 24/36","2/3"],["Simplify 15/45","1/3"],["Simplify 18/24","3/4"],["Simplify 20/50","2/5"],["Convert 7/4 to a mixed number","1 3/4"],["Convert 11/3 to a mixed number","3 2/3"],["Convert 2 1/2 to an improper fraction","5/2"],["Convert 3 2/5 to an improper fraction","17/5"],["Convert 1 3/4 to an improper fraction","7/4"],["1/2 + 1/4","3/4"],["1/3 + 1/6","1/2"],["2/5 + 1/10","1/2"],["3/8 + 1/4","5/8"],["1/2 + 1/3","5/6"],["3/4 + 2/3","1 5/12"],["3/4 − 1/4","1/2"],["5/6 − 1/3","1/2"],["7/8 − 1/2","3/8"],["2/3 − 1/4","5/12"],["1 − 3/5","2/5"],["2/3 × 3/4","1/2"],["1/2 × 1/2","1/4"],["3/5 × 5/6","1/2"],["2/7 × 14","4"],["1/3 of 12","4"],["3/4 of 20","15"],["2/5 of 25","10"],["1/2 ÷ 1/4","2"],["3/4 ÷ 3/8","2"],["2/3 ÷ 4","1/6"],["4 ÷ 1/2","8"],["How to divide fractions","Multiply by the reciprocal of the second fraction"],["Reciprocal of 5/7","7/5"],["Reciprocal of 4","1/4"],["Common denominator","A shared multiple of the denominators, needed for adding and subtracting"],["Least common denominator of 1/4 and 1/6","12"],["Least common denominator of 2/3 and 3/5","15"],["Which is greater, 3/4 or 2/3?","3/4"],["Which is greater, 5/8 or 3/5?","5/8"],["Which is smaller, 1/3 or 2/7?","2/7"],["Compare 4/9 and 1/2","4/9 < 1/2"],["Order 1/2, 1/3, 1/4 from least to greatest","1/4, 1/3, 1/2"],["Unit fractions get smaller as the denominator","Gets bigger"],["Benchmark fractions","0, 1/2 and 1, used for comparing and estimating"],["Is 5/9 more or less than 1/2?","More"],["Is 3/8 more or less than 1/2?","Less"],["Fraction of a day that is 6 hours","1/4"],["Fraction of an hour that is 20 minutes","1/3"],["Fraction of a dollar that is a quarter","1/4"],["Fraction as division","a/b means a ÷ b"],["1/0","Undefined; you cannot divide by zero"],["0/5","0"],["Fraction bar also means","Divide"],["Fraction of a circle in a half turn","1/2"],["Number line fraction: halfway between 0 and 1","1/2"],["Complex fraction","A fraction with a fraction in the numerator or denominator"]]},{"source":"https://peabody.kids/flashcards/math-geometry-formulas","id":"math-geometry-formulas","title":"Geometry Formulas: Area, Perimeter, Volume","category":"math","subject":"Math","levels":["6-8","9-12"],"description":"The area, perimeter, circumference, surface area and volume formulas every student needs.","tags":["formulas","area","volume","geometry"],"frontLabel":"Formula for","backLabel":"Formula","cards":[["Area of a rectangle","A = l × w"],["Perimeter of a rectangle","P = 2l + 2w"],["Area of a square","A = s²"],["Perimeter of a square","P = 4s"],["Area of a triangle","A = ½ × b × h"],["Perimeter of a triangle","Add the three sides"],["Area of a parallelogram","A = b × h"],["Area of a trapezoid","A = ½ × (b₁ + b₂) × h"],["Area of a rhombus","A = ½ × d₁ × d₂","The two diagonals"],["Area of a kite","A = ½ × d₁ × d₂"],["Area of a circle","A = πr²"],["Circumference of a circle","C = 2πr = πd"],["Diameter in terms of radius","d = 2r"],["Length of an arc","(θ/360) × 2πr, with θ in degrees"],["Area of a sector","(θ/360) × πr², with θ in degrees"],["Area of an equilateral triangle","A = (√3/4) × s²"],["Area of a regular polygon","A = ½ × apothem × perimeter"],["Heron's formula","A = √[s(s−a)(s−b)(s−c)], where s is half the perimeter"],["Area of a triangle with two sides and the included angle","A = ½ ab sin C"],["Area of an ellipse","A = πab","a and b are the semi-axes"],["Volume of a cube","V = s³"],["Surface area of a cube","SA = 6s²"],["Volume of a rectangular prism","V = l × w × h"],["Surface area of a rectangular prism","SA = 2(lw + lh + wh)"],["Volume of any prism","V = base area × height"],["Volume of a cylinder","V = πr²h"],["Lateral surface area of a cylinder","2πrh"],["Total surface area of a cylinder","SA = 2πr² + 2πrh"],["Volume of a cone","V = ⅓πr²h"],["Lateral surface area of a cone","πrl, where l is the slant height"],["Total surface area of a cone","SA = πr² + πrl"],["Volume of a pyramid","V = ⅓ × base area × height"],["Volume of a sphere","V = 4/3 πr³"],["Surface area of a sphere","SA = 4πr²"],["Volume of a hemisphere","V = 2/3 πr³"],["Slant height of a cone","l = √(r² + h²)"],["Diagonal of a rectangle","d = √(l² + w²)"],["Diagonal of a square","d = s√2"],["Space diagonal of a rectangular prism","d = √(l² + w² + h²)"],["Pythagorean theorem","a² + b² = c²"],["Distance formula","d = √[(x₂−x₁)² + (y₂−y₁)²]"],["Midpoint formula","((x₁+x₂)/2, (y₁+y₂)/2)"],["Slope formula","m = (y₂−y₁)/(x₂−x₁)"],["Sum of interior angles of an n-gon","(n − 2) × 180°"],["Each interior angle of a regular n-gon","(n − 2) × 180° / n"],["Sum of exterior angles of any polygon","360°"],["Number of diagonals of an n-gon","n(n − 3)/2"],["Equation of a circle","(x − h)² + (y − k)² = r²"],["Surface area of a triangular prism","2 × base area + perimeter of base × height"],["Volume of a torus","V = 2π²Rr²"],["Perimeter of an equilateral triangle","P = 3s"],["Area of an annulus (ring)","A = π(R² − r²)"],["Area scale factor","If lengths scale by k, area scales by k²"],["Volume scale factor","If lengths scale by k, volume scales by k³"]]},{"source":"https://peabody.kids/flashcards/math-geometry-theorems","id":"math-geometry-theorems","title":"Geometry Theorems and Rules","category":"math","subject":"Math","levels":["9-12"],"description":"Triangle, circle and polygon theorems and congruence rules.","tags":["theorems","geometry","proof"],"frontLabel":"Name","backLabel":"Statement","cards":[["Triangle angle sum theorem","The three angles of a triangle add up to 180°"],["Exterior angle theorem","An exterior angle of a triangle equals the sum of the two remote interior angles"],["Isosceles triangle theorem","Angles opposite equal sides are equal"],["Converse of the isosceles triangle theorem","Sides opposite equal angles are equal"],["Triangle inequality theorem","The sum of any two sides is greater than the third side"],["Pythagorean theorem","In a right triangle, a² + b² = c²"],["Converse of the Pythagorean theorem","If a² + b² = c², the triangle is a right triangle"],["45-45-90 triangle","Sides are in the ratio 1 : 1 : √2"],["30-60-90 triangle","Sides are in the ratio 1 : √3 : 2"],["Midsegment theorem","A segment joining midpoints of two sides is parallel to the third side and half as long"],["Triangle angle bisector theorem","An angle bisector splits the opposite side in the ratio of the adjacent sides"],["SSS congruence","Three pairs of equal sides make triangles congruent"],["SAS congruence","Two sides and the angle between them equal makes triangles congruent"],["ASA congruence","Two angles and the side between them equal makes triangles congruent"],["AAS congruence","Two angles and a non-included side equal makes triangles congruent"],["HL congruence","Right triangles with equal hypotenuse and one equal leg are congruent"],["CPCTC","Corresponding parts of congruent triangles are congruent"],["AA similarity","Two triangles with two equal angles are similar"],["SAS similarity","Two proportional sides with equal included angle make triangles similar"],["SSS similarity","Three proportional sides make triangles similar"],["Vertical angles theorem","Vertical angles are equal"],["Linear pair postulate","Two angles that form a straight line add up to 180°"],["Corresponding angles postulate","Parallel lines cut by a transversal make equal corresponding angles"],["Alternate interior angles theorem","Parallel lines cut by a transversal make equal alternate interior angles"],["Same-side interior angles theorem","Parallel lines cut by a transversal make supplementary same-side interior angles"],["Perpendicular bisector theorem","A point on the perpendicular bisector is equally far from the segment's endpoints"],["Angle bisector theorem (points)","A point on an angle bisector is equally far from the angle's sides"],["Centroid","The point where a triangle's three medians meet"],["Circumcenter","The point where perpendicular bisectors meet; the center of the circumscribed circle"],["Incenter","The point where angle bisectors meet; the center of the inscribed circle"],["Orthocenter","The point where a triangle's altitudes meet"],["Centroid ratio","The centroid divides each median in a 2 : 1 ratio, longer part toward the vertex"],["Inscribed angle theorem","An inscribed angle is half its intercepted arc"],["Central angle theorem","A central angle equals its intercepted arc"],["Thales' theorem","An angle inscribed in a semicircle is a right angle"],["Tangent-radius theorem","A tangent is perpendicular to the radius at the point of tangency"],["Two tangents theorem","Tangent segments from one outside point to a circle are equal"],["Chord-chord theorem","If two chords cross, the products of their pieces are equal"],["Opposite angles in a cyclic quadrilateral","They add up to 180°"],["Parallelogram theorem: sides","Opposite sides are equal"],["Parallelogram theorem: angles","Opposite angles are equal; consecutive angles add up to 180°"],["Parallelogram theorem: diagonals","Diagonals bisect each other"],["Rectangle diagonals","Diagonals of a rectangle are equal"],["Rhombus diagonals","Diagonals are perpendicular and bisect the angles"],["Kite diagonals","Diagonals are perpendicular"],["Isosceles trapezoid base angles","Base angles of an isosceles trapezoid are equal"],["Side-splitter theorem","A line parallel to one side of a triangle divides the other two sides proportionally"],["Geometric mean (altitude) theorem","The altitude to the hypotenuse of a right triangle is the geometric mean of the two segments it makes"],["Law of sines","a/sin A = b/sin B = c/sin C"],["Law of cosines","c² = a² + b² − 2ab cos C"],["Euler line","The centroid, circumcenter and orthocenter of a triangle lie on one line"],["Ceva's theorem","Cevians through a point in a triangle satisfy (AF/FB)(BD/DC)(CE/EA) = 1"],["Pick's theorem","Area of a lattice polygon = i + b/2 − 1, with i inside points and b boundary points"],["Euler's polyhedron formula","V − E + F = 2"]]},{"source":"https://peabody.kids/flashcards/math-integral-rules","id":"math-integral-rules","title":"Integral Rules and Antiderivatives","category":"math","subject":"Math","levels":["college","9-12"],"description":"Antiderivatives of common functions, integration techniques and the fundamental theorem of calculus.","tags":["calculus","integrals"],"frontLabel":"Integral","backLabel":"Result","cards":[["∫ 0 dx","C"],["∫ k dx","kx + C"],["∫ xⁿ dx (n ≠ −1)","xⁿ⁺¹/(n + 1) + C"],["∫ 1/x dx","ln|x| + C"],["∫ eˣ dx","eˣ + C"],["∫ aˣ dx","aˣ/ln a + C"],["∫ sin x dx","−cos x + C"],["∫ cos x dx","sin x + C"],["∫ sec² x dx","tan x + C"],["∫ csc² x dx","−cot x + C"],["∫ sec x tan x dx","sec x + C"],["∫ csc x cot x dx","−csc x + C"],["∫ tan x dx","−ln|cos x| + C"],["∫ cot x dx","ln|sin x| + C"],["∫ sec x dx","ln|sec x + tan x| + C"],["∫ ln x dx","x ln x − x + C"],["∫ 1/(1 + x²) dx","arctan x + C"],["∫ 1/√(1 − x²) dx","arcsin x + C"],["∫ x² dx","x³/3 + C"],["∫ x³ dx","x⁴/4 + C"],["∫ √x dx","(2/3) x^(3/2) + C"],["∫ 1/x² dx","−1/x + C"],["∫ e^(2x) dx","e^(2x)/2 + C"],["∫ cos(3x) dx","sin(3x)/3 + C"],["∫ 2x dx","x² + C"],["∫ (3x² + 2) dx","x³ + 2x + C"],["Constant of integration","+ C, since many functions share the same derivative"],["Linearity of integrals","∫ (af + bg) dx = a∫f dx + b∫g dx"],["Fundamental theorem of calculus, part 1","d/dx ∫ₐˣ f(t) dt = f(x)"],["Fundamental theorem of calculus, part 2","∫ₐᵇ f(x) dx = F(b) − F(a), where F' = f"],["Definite integral","The signed area between a curve and the x-axis over an interval"],["Indefinite integral","The family of all antiderivatives of a function"],["Integration by parts","∫ u dv = uv − ∫ v du"],["Substitution (u-substitution)","Let u = g(x), then replace g'(x) dx with du"],["Partial fractions","Split a rational function into simpler fractions before integrating"],["Riemann sum","A sum of rectangle areas that approximates a definite integral"],["Trapezoidal rule","Approximating area using trapezoids instead of rectangles"],["Average value of a function","(1/(b − a)) ∫ₐᵇ f(x) dx"],["Area between curves","∫ (top − bottom) dx"],["Disk method","V = π ∫ [R(x)]² dx"],["Washer method","V = π ∫ ([R(x)]² − [r(x)]²) dx"],["Shell method","V = 2π ∫ (radius)(height) dx"],["Arc length","L = ∫ √(1 + [f'(x)]²) dx"],["∫₀¹ x dx","1/2"],["∫₀¹ x² dx","1/3"],["∫₀^π sin x dx","2"],["∫₁^e (1/x) dx","1"],["∫₀^1 eˣ dx","e − 1"],["Improper integral","An integral with an infinite limit or an unbounded function, defined with a limit"],["Position from velocity","s(t) = ∫ v(t) dt"],["Velocity from acceleration","v(t) = ∫ a(t) dt"],["Even function integral over [−a, a]","2 ∫₀ᵃ f(x) dx"],["Odd function integral over [−a, a]","0"],["Swapping limits of integration","∫ₐᵇ f = −∫ᵦᵃ f"]]},{"source":"https://peabody.kids/flashcards/math-linear-algebra","id":"math-linear-algebra","title":"Linear Algebra Terms","category":"math","subject":"Math","levels":["college","9-12"],"description":"Vectors, matrices and the core ideas of linear algebra.","tags":["linear algebra","matrices","vectors"],"frontLabel":"Term","backLabel":"Meaning","cards":[["Scalar","A single number, as opposed to a vector"],["Vector","An ordered list of numbers; has magnitude and direction"],["Magnitude (norm) of a vector","Its length, √(x² + y² + ...)"],["Unit vector","A vector of length 1"],["Zero vector","The vector whose entries are all 0"],["Vector addition","Add matching entries"],["Scalar multiplication","Multiply every entry of a vector or matrix by the same number"],["Dot product","Sum of products of matching entries; a · b = |a||b| cos θ"],["Perpendicular vectors","Their dot product is 0 (orthogonal)"],["Cross product","A vector perpendicular to two 3D vectors; length |a||b| sin θ"],["Linear combination","A sum of vectors each multiplied by a scalar"],["Span","The set of all linear combinations of some vectors"],["Linear independence","No vector in the set is a linear combination of the others"],["Linear dependence","At least one vector can be built from the others"],["Basis","A linearly independent set of vectors that spans a space"],["Dimension","The number of vectors in a basis"],["Standard basis of R²","(1, 0) and (0, 1)"],["Vector space","A set where vectors can be added and scaled by numbers following set rules"],["Subspace","A vector space inside a larger one"],["Matrix","A rectangular array of numbers in rows and columns"],["Order (size) of a matrix","Rows × columns, such as 2 × 3"],["Square matrix","A matrix with the same number of rows and columns"],["Identity matrix","A square matrix with 1s on the main diagonal and 0s elsewhere; AI = A"],["Zero matrix","A matrix with all entries 0"],["Diagonal matrix","A square matrix with zeros off the main diagonal"],["Symmetric matrix","A matrix equal to its transpose"],["Transpose","Swap rows and columns; written Aᵀ"],["Matrix addition","Add matching entries; matrices need the same size"],["Matrix multiplication rule","An m × n matrix times an n × p matrix gives an m × p matrix"],["Is matrix multiplication commutative?","No. AB is usually not equal to BA"],["Inverse matrix","A⁻¹ such that A A⁻¹ = I"],["Invertible (nonsingular) matrix","A square matrix with an inverse; its determinant is not 0"],["Singular matrix","A matrix with no inverse; determinant 0"],["Determinant of a 2×2 matrix [a b; c d]","ad − bc"],["Determinant meaning","Scale factor of area or volume under the matrix's transformation"],["Trace of a matrix","The sum of the diagonal entries"],["Rank","The number of linearly independent rows (or columns)"],["Null space","All vectors x with Ax = 0"],["Column space","The span of the columns of a matrix"],["Row space","The span of the rows of a matrix"],["Linear transformation","A function T where T(u + v) = T(u) + T(v) and T(cu) = cT(u)"],["System of linear equations as a matrix","Ax = b"],["Augmented matrix","The coefficient matrix with the constants column attached"],["Gaussian elimination","Using row operations to turn a matrix into row-echelon form"],["Row operations","Swap rows, multiply a row by a nonzero number, add a multiple of one row to another"],["Row-echelon form","A staircase form with leading entries moving to the right"],["Reduced row-echelon form","Row-echelon form with leading 1s and zeros above and below them"],["Pivot","A leading nonzero entry in a row used in elimination"],["Eigenvalue","A number λ such that Av = λv for some nonzero vector v"],["Eigenvector","A nonzero vector that only gets scaled by a matrix"],["Characteristic polynomial","det(A − λI); its roots are the eigenvalues"],["Orthogonal matrix","A square matrix whose columns are perpendicular unit vectors; Qᵀ = Q⁻¹"],["Projection","The shadow of one vector onto another"],["Rotation matrix in 2D","[cos θ, −sin θ; sin θ, cos θ]"],["Cramer's rule","Solve a system using ratios of determinants"],["Orthonormal basis","A basis of perpendicular unit vectors"],["Gram-Schmidt process","A method to turn a basis into an orthonormal one"],["Diagonalization","Writing A = PDP⁻¹ with D diagonal"],["Singular value decomposition (SVD)","Factoring a matrix into rotations and stretches: A = UΣVᵀ"]]},{"source":"https://peabody.kids/flashcards/math-log-rules","id":"math-log-rules","title":"Logarithm Rules and Values","category":"math","subject":"Math","levels":["9-12","college"],"description":"What a logarithm means, the laws of logarithms and common log values.","tags":["logarithms","exponentials","algebra"],"frontLabel":"Rule","backLabel":"Result","cards":[["Definition of a logarithm","log_b(x) = y means bʸ = x"],["Product rule","log_b(xy) = log_b x + log_b y"],["Quotient rule","log_b(x/y) = log_b x − log_b y"],["Power rule","log_b(xⁿ) = n log_b x"],["Change of base","log_b x = log x / log b (any base)"],["Log of 1","log_b 1 = 0"],["Log of the base","log_b b = 1"],["Inverse property 1","b^(log_b x) = x"],["Inverse property 2","log_b(bˣ) = x"],["Common logarithm","Base 10, written log x"],["Natural logarithm","Base e, written ln x"],["Binary logarithm","Base 2, written log₂ x"],["Domain of log x","x > 0"],["log of a negative number","Not defined among the real numbers"],["log 100","2"],["log 1,000","3"],["log 0.1","−1"],["log 0.01","−2"],["log 10","1"],["log₂ 8","3"],["log₂ 64","6"],["log₂ 1024","10"],["log₃ 81","4"],["log₅ 125","3"],["log₄ 2","1/2"],["log₂ (1/8)","−3"],["ln e","1"],["ln 1","0"],["ln(e²)","2"],["e^(ln 5)","5"],["Expand log(ab/c)","log a + log b − log c"],["Condense 2 log x + log y","log(x²y)"],["Solve 2ˣ = 16","x = 4"],["Solve 3ˣ = 5","x = log 5 / log 3 (about 1.465)"],["Solve log₂ x = 5","x = 32"],["Is log(x + y) = log x + log y?","No. The product rule needs log(xy)"],["Is log(x)/log(y) the same as log(x/y)?","No. They are different quantities"],["Graph of y = log x: vertical asymptote","x = 0"],["Graph of y = log x: x-intercept","x = 1"],["Value of e (to 5 decimals)","2.71828"],["Half-life formula idea","Amount = start × (1/2)^(t / half-life)"],["Decibel scale uses","A logarithm of a power ratio (10 log₁₀)"],["Richter scale uses","A base-10 logarithm of shaking amplitude"],["pH formula","pH = −log₁₀[H⁺]"]]},{"source":"https://peabody.kids/flashcards/math-mathematicians","id":"math-mathematicians","title":"Famous Mathematicians","category":"math","subject":"Math","levels":["6-8","9-12","adult"],"description":"Mathematicians through history and what each is known for.","tags":["history of math","mathematicians"],"frontLabel":"Mathematician","backLabel":"Known for","cards":[["Euclid","Wrote the Elements, the foundation of geometry (about 300 BCE)"],["Pythagoras","Greek thinker linked to the Pythagorean theorem (6th century BCE)"],["Archimedes","Greek scholar who found area and volume of the sphere and estimated pi"],["Eratosthenes","Greek scholar who estimated Earth's size and made a sieve for primes"],["Thales","Early Greek geometer; the inscribed angle in a semicircle theorem"],["Hypatia","Mathematician and teacher in Alexandria, about 400 CE"],["Diophantus","Greek writer on equations, often called a father of algebra"],["Aryabhata","Indian mathematician and astronomer (born 476 CE); worked on pi and sines"],["Brahmagupta","Indian mathematician who gave rules for zero and negative numbers (628 CE)"],["Al-Khwarizmi","Persian scholar whose book on equations gave us the word algebra; 'algorithm' comes from his name"],["Omar Khayyam","Persian poet and mathematician who studied cubic equations"],["Fibonacci (Leonardo of Pisa)","Brought Hindu-Arabic numerals to Europe; the Fibonacci sequence"],["Zu Chongzhi","Chinese mathematician who found pi to about 7 digits (5th century)"],["Liu Hui","Chinese mathematician who estimated pi with polygons (3rd century)"],["Madhava","Indian mathematician who found infinite series for sine, cosine and pi (14th century)"],["Ramanujan","Indian mathematician known for remarkable formulas and number theory (1887-1920)"],["René Descartes","Created the coordinate plane linking algebra and geometry"],["Pierre de Fermat","Number theory; Fermat's Last Theorem"],["Blaise Pascal","Pascal's triangle and early probability"],["Isaac Newton","Co-inventor of calculus; laws of motion and gravity"],["Gottfried Leibniz","Co-inventor of calculus; the integral and derivative notation we use"],["Leonhard Euler","Prolific Swiss mathematician; e, i, graph theory, Euler's formula"],["Carl Friedrich Gauss","German 'prince of mathematicians'; number theory, the bell curve, 1 to 100 sum story"],["Joseph-Louis Lagrange","Mechanics and number theory"],["Pierre-Simon Laplace","Probability and celestial mechanics"],["Sophie Germain","French mathematician who worked on primes and elasticity"],["Évariste Galois","Showed which polynomial equations can be solved by formulas; founded group theory"],["Niels Henrik Abel","Norwegian mathematician; proved the quintic has no general formula"],["Bernhard Riemann","German mathematician; Riemann hypothesis, geometry of curved spaces"],["Georg Cantor","Founded set theory; showed that some infinities are bigger than others"],["David Hilbert","German mathematician who posed 23 famous problems in 1900"],["Henri Poincaré","French mathematician; topology and chaos"],["Emmy Noether","German mathematician; abstract algebra and Noether's theorem"],["Ada Lovelace","Wrote the first published computer program (1843) for Babbage's engine"],["Charles Babbage","Designed early mechanical computers"],["Florence Nightingale","Nurse who used statistical charts to improve health care"],["Mary Somerville","Scottish science writer and mathematician (1780-1872)"],["Maria Gaetana Agnesi","Italian mathematician who wrote an early calculus textbook"],["Kurt Gödel","Showed math systems have true statements that cannot be proven (1931)"],["Alan Turing","Founded theoretical computer science; broke wartime codes"],["John von Neumann","Game theory and computer architecture"],["Claude Shannon","Founded information theory"],["Katherine Johnson","NASA mathematician who calculated orbits and flight paths"],["Dorothy Vaughan","NASA mathematician and computer programmer"],["Mary Jackson","NASA mathematician and the agency's first Black female engineer"],["Grace Hopper","Computer scientist and Navy admiral; early programming languages"],["Maryam Mirzakhani","Iranian mathematician; first woman to win the Fields Medal (2014)"],["Terence Tao","Australian-American mathematician; Fields Medal in 2006"],["Andrew Wiles","Proved Fermat's Last Theorem (1994-95)"],["Paul Erdős","Hungarian mathematician who wrote with hundreds of collaborators"],["Benoit Mandelbrot","Fractal geometry"],["John Napier","Scottish inventor of logarithms (1614)"],["Robert Recorde","Introduced the equals sign in 1557"],["William Oughtred","Invented the slide rule; used × for multiplication"],["George Boole","Boolean algebra, the basis of computer logic"],["Carl Jacobi","German mathematician; elliptic functions and determinants"],["Augustin-Louis Cauchy","French mathematician who made calculus rigorous"],["Johannes Kepler","Astronomer and mathematician; laws of planetary motion"],["Galileo Galilei","Italian scientist who used math to describe motion"],["Thomas Bayes","English statistician; Bayes' theorem"],["Karl Pearson","Founded modern statistics; the correlation coefficient"],["G. H. Hardy","English mathematician who recognized Ramanujan's genius"],["Hipparchus","Greek astronomer who began trigonometry tables"]]},{"source":"https://peabody.kids/flashcards/math-metric-units","id":"math-metric-units","title":"Metric System: Units and Prefixes","category":"math","subject":"Math","levels":["3-5","6-8","9-12"],"description":"Metric prefixes, conversions between metric units and benchmarks for estimating.","tags":["metric","measurement","conversion"],"frontLabel":"Question","backLabel":"Answer","cards":[["Kilo- means","1,000 times the base unit"],["Hecto- means","100 times the base unit"],["Deka- means","10 times the base unit"],["Deci- means","One tenth of the base unit"],["Centi- means","One hundredth of the base unit"],["Milli- means","One thousandth of the base unit"],["Micro- means","One millionth of the base unit"],["Nano- means","One billionth of the base unit"],["Mega- means","One million times the base unit"],["Giga- means","One billion times the base unit"],["Tera- means","One trillion times the base unit"],["Base unit of length in the metric system","Meter (m)"],["Base unit of mass in the metric system","Kilogram (kg)","Gram is the small everyday unit"],["Base unit of volume often used for liquids","Liter (L)"],["Millimeters in 1 centimeter","10"],["Centimeters in 1 meter","100"],["Millimeters in 1 meter","1,000"],["Meters in 1 kilometer","1,000"],["Centimeters in 1 kilometer","100,000"],["Grams in 1 kilogram","1,000"],["Milligrams in 1 gram","1,000"],["Kilograms in 1 metric ton","1,000"],["Milliliters in 1 liter","1,000"],["Liters in 1 kiloliter","1,000"],["Cubic centimeters in 1 milliliter","1"],["Cubic centimeters in 1 liter","1,000"],["Mass of 1 liter of water","About 1 kilogram"],["Mass of 1 milliliter of water","About 1 gram"],["Convert 3 m to cm","300 cm"],["Convert 250 cm to m","2.5 m"],["Convert 4.5 km to m","4,500 m"],["Convert 7,000 g to kg","7 kg"],["Convert 2.5 L to mL","2,500 mL"],["Convert 600 mm to cm","60 cm"],["Convert 0.8 kg to g","800 g"],["Convert 35 mg to g","0.035 g"],["Which is larger: 1 cm or 1 mm?","1 cm (10 mm)"],["Which is longer: 1 km or 1,200 m?","1,200 m"],["About how thick is a dime?","About 1 millimeter"],["About how wide is a fingernail?","About 1 centimeter"],["About how tall is a door?","About 2 meters"],["About how long is a 15-minute walk?","About 1 kilometer"],["About how much does a paper clip weigh?","About 1 gram"],["About how much does a big textbook weigh?","About 1 kilogram"],["About how much does a spoonful hold?","About 5 milliliters (a teaspoon)"],["About how much does a soda bottle hold?","About 2 liters (large) or 0.5 liter (small)"],["Symbol for centimeter","cm"],["Symbol for kilogram","kg"],["Symbol for milliliter","mL"],["Symbol for micrometer","μm"],["Which direction when going from a large unit to a small unit?","Multiply (there are more small units)"],["Hectare","A metric unit of land area, 10,000 square meters"],["Square meters in 1 hectare","10,000"],["Celsius freezing point of water","0 degrees C"],["Celsius boiling point of water (sea level)","100 degrees C"],["Normal human body temperature in Celsius","About 37 degrees C"],["Room temperature in Celsius","About 20 to 22 degrees C"],["Kelvin value of 0 degrees C","273.15 K"],["Absolute zero in Celsius","−273.15 degrees C"],["Mass of 1 cubic meter of water","About 1,000 kilograms (1 metric ton)"]]},{"source":"https://peabody.kids/flashcards/math-money","id":"math-money","title":"Money Math","category":"math","subject":"Math","levels":["k-2","3-5","6-8"],"description":"US coins and bills, making change, and simple money math like tax, tips and interest.","tags":["money","coins","percent"],"frontLabel":"Question","backLabel":"Answer","cards":[["Value of a penny","1 cent"],["Value of a nickel","5 cents"],["Value of a dime","10 cents"],["Value of a quarter","25 cents"],["Value of a half dollar","50 cents"],["Value of a dollar coin","100 cents, or $1"],["Cents in a dollar","100"],["Pennies in a nickel","5"],["Nickels in a dime","2"],["Value of 2 dimes and a nickel","25 cents, the same as a quarter"],["Quarters in a dollar","4"],["Dimes in a dollar","10"],["Nickels in a dollar","20"],["Pennies in a dollar","100"],["Quarters in $5","20"],["Total of 2 quarters and 1 dime","60 cents"],["Total of 3 dimes and 2 nickels","40 cents"],["Total of 1 quarter, 1 dime, 1 nickel","40 cents"],["Total of 4 dimes and 3 pennies","43 cents"],["Fewest coins for 41 cents","1 quarter, 1 dime, 1 nickel, 1 penny"],["Fewest coins for 30 cents","1 quarter and 1 nickel"],["Fewest coins for 65 cents","2 quarters, 1 dime, 1 nickel"],["Common bills in US money","$1, $2, $5, $10, $20, $50, $100"],["Change from $5 for a $3.50 purchase","$1.50"],["Change from $10 for a $6.25 purchase","$3.75"],["Change from $20 for a $13.80 purchase","$6.20"],["Change from a dollar for 35 cents","65 cents"],["Counting up to make change","Start at the price and count up to the amount paid"],["Write 7 dollars and 5 cents with a dollar sign","$7.05"],["Write 275 cents in dollars","$2.75"],["Write $0.08 in cents","8 cents"],["Cost of 3 items at $1.25 each","$3.75"],["Cost of 4 items at $2.50 each","$10.00"],["Unit price","The cost for one unit, such as one ounce or one item"],["Better buy: 6 for $3 or 10 for $4.50","10 for $4.50 (45 cents each vs 50 cents each)"],["Sales tax","A percent of the price added at the register"],["Total with 10% tax on $20","$22"],["Total with 5% tax on $40","$42"],["Tip of 20% on $30","$6"],["Tip of 15% on $40","$6"],["Quick 10% of a price","Move the decimal one place left"],["20% off a $50 item","Pay $40"],["25% off a $60 item","Pay $45"],["50% off a $18 item","Pay $9"],["Discount","An amount taken off the price"],["Profit","Money earned minus money spent"],["Revenue","Money coming in from sales"],["Budget","A plan for how to spend and save money"],["Income","Money you receive"],["Expense","Money you spend"],["Savings","Money set aside for later"],["Interest","Money paid for using money, or earned on savings"],["Simple interest on $100 at 5% for 2 years","$10"],["Compound interest","Interest earned on both the original amount and past interest"],["Principal","The starting amount of a loan or deposit"],["Allowance saved: $4 a week for 6 weeks","$24"],["Need vs. want","A need is required to live; a want is nice to have"],["Wage","Pay for work, often per hour"],["Pay for 8 hours at $12 per hour","$96"],["Salary","Fixed pay per year, not by the hour"],["Exchange rate","How much of one currency equals another"],["Currency","The money used in a country"],["Check vs. cash","A check is a written order for the bank to pay; cash is physical money"]]},{"source":"https://peabody.kids/flashcards/math-number-theory","id":"math-number-theory","title":"Number Theory and Divisibility","category":"math","subject":"Math","levels":["3-5","6-8","9-12"],"description":"Divisibility rules, factors, multiples, GCF, LCM, primes and modular ideas.","tags":["divisibility","factors","number theory"],"frontLabel":"Question","backLabel":"Answer","cards":[["Divisible by 2 if","The last digit is even (0, 2, 4, 6, 8)"],["Divisible by 3 if","The sum of the digits is divisible by 3"],["Divisible by 4 if","The last two digits form a number divisible by 4"],["Divisible by 5 if","The last digit is 0 or 5"],["Divisible by 6 if","It is divisible by both 2 and 3"],["Divisible by 8 if","The last three digits form a number divisible by 8"],["Divisible by 9 if","The sum of the digits is divisible by 9"],["Divisible by 10 if","The last digit is 0"],["Divisible by 11 if","The alternating sum of the digits is divisible by 11"],["Is 2,346 divisible by 3?","Yes, digit sum 15"],["Is 4,125 divisible by 9?","No, digit sum 12"],["Is 1,716 divisible by 4?","Yes, 16 is divisible by 4"],["Is 121 divisible by 11?","Yes: 1 − 2 + 1 = 0"],["Factors of 12","1, 2, 3, 4, 6, 12"],["Factors of 24","1, 2, 3, 4, 6, 8, 12, 24"],["Factors of 36","1, 2, 3, 4, 6, 9, 12, 18, 36"],["Factors of 48","1, 2, 3, 4, 6, 8, 12, 16, 24, 48"],["Factors of 60","1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60"],["Factors of 100","1, 2, 4, 5, 10, 20, 25, 50, 100"],["Prime factorization of 12","2² × 3"],["Prime factorization of 36","2² × 3²"],["Prime factorization of 60","2² × 3 × 5"],["Prime factorization of 100","2² × 5²"],["Prime factorization of 72","2³ × 3²"],["Prime factorization of 84","2² × 3 × 7"],["Prime factorization of 90","2 × 3² × 5"],["Prime factorization of 120","2³ × 3 × 5"],["Prime factorization of 144","2⁴ × 3²"],["GCF of 12 and 18","6"],["GCF of 24 and 36","12"],["GCF of 15 and 25","5"],["GCF of 48 and 72","24"],["GCF of 17 and 31","1 (both prime)"],["LCM of 4 and 6","12"],["LCM of 6 and 8","24"],["LCM of 3 and 5","15"],["LCM of 9 and 12","36"],["LCM of 10 and 15","30"],["Relationship of GCF and LCM","GCF(a, b) × LCM(a, b) = a × b"],["Coprime (relatively prime)","Two numbers whose only common factor is 1"],["Euclidean algorithm","Find the GCF by repeatedly replacing the larger number with the remainder"],["Sieve of Eratosthenes","A method that crosses out multiples to find primes"],["Fundamental theorem of arithmetic","Every integer above 1 has exactly one prime factorization"],["Largest prime factor idea","Test divisors only up to the square root"],["Number of factors of a prime","2"],["Is there a largest prime?","No. Euclid proved there are infinitely many primes"],["Is 97 prime?","Yes"],["Is 101 prime?","Yes"],["Is 111 prime?","No: 3 × 37"],["Is 143 prime?","No: 11 × 13"],["Is 221 prime?","No: 13 × 17"],["Remainder of 17 ÷ 5","2"],["17 mod 5","2"],["Modular arithmetic","Arithmetic where numbers wrap around after reaching a set value"],["Clock arithmetic example","5 hours after 10 o'clock is 3 o'clock (15 mod 12)"],["Even + even","Even"],["Odd + odd","Even"],["Odd + even","Odd"],["Odd × odd","Odd"],["Even × any integer","Even"],["Sum of the first n odd numbers","n²"],["Sum of the first n whole numbers","n(n + 1)/2"],["Sum of 1 to 100","5,050"],["Goldbach's conjecture","Every even number above 2 is the sum of two primes (unproven)"],["Fermat's Last Theorem","No positive integers satisfy aⁿ + bⁿ = cⁿ for n above 2; proved in 1994-95"],["Pythagorean triple","Three whole numbers a, b, c with a² + b² = c²"],["Common Pythagorean triples","3-4-5, 5-12-13, 8-15-17, 7-24-25"]]},{"source":"https://peabody.kids/flashcards/math-number-types","id":"math-number-types","title":"Types of Numbers","category":"math","subject":"Math","levels":["3-5","6-8","9-12"],"description":"Natural, whole, integer, rational, irrational, real and special number families.","tags":["number systems","number sense"],"frontLabel":"Term","backLabel":"Meaning","cards":[["Natural (counting) numbers","1, 2, 3, 4, ..."],["Whole numbers","0, 1, 2, 3, ..."],["Integers","..., −3, −2, −1, 0, 1, 2, 3, ..."],["Rational numbers","Numbers that can be written as a fraction of two integers"],["Irrational numbers","Numbers that cannot be written as a fraction"],["Real numbers","All rational and irrational numbers"],["Imaginary numbers","Multiples of i, where i² = −1"],["Complex numbers","Numbers of the form a + bi"],["Even numbers","Whole numbers divisible by 2"],["Odd numbers","Whole numbers not divisible by 2"],["Prime numbers","Numbers above 1 with exactly two factors"],["Composite numbers","Numbers above 1 that are not prime"],["Square numbers","1, 4, 9, 16, 25, ..."],["Cube numbers","1, 8, 27, 64, 125, ..."],["Triangular numbers","1, 3, 6, 10, 15, 21, ..."],["Fibonacci numbers","1, 1, 2, 3, 5, 8, 13, 21, ..."],["Perfect numbers","Numbers equal to the sum of their proper factors: 6, 28, 496"],["Is 6 perfect?","Yes: 1 + 2 + 3 = 6"],["Is 28 perfect?","Yes: 1 + 2 + 4 + 7 + 14 = 28"],["Palindromic number","A number that reads the same backward and forward","121"],["Twin primes","Primes that differ by 2","11 and 13"],["Mersenne prime","A prime of the form 2ⁿ − 1","7, 31, 127"],["Pi is rational or irrational?","Irrational"],["√2 is rational or irrational?","Irrational"],["√9 is rational or irrational?","Rational (it equals 3)"],["0.5 is rational or irrational?","Rational"],["0.333... is rational or irrational?","Rational (1/3)"],["Is e rational or irrational?","Irrational"],["Is zero positive or negative?","Neither"],["Is zero even?","Yes"],["Is 1 prime?","No"],["Is 0 an integer?","Yes"],["Is −4 a whole number?","No"],["Is −4 an integer?","Yes"],["Is 3/4 an integer?","No, it is rational"],["Is every integer a rational number?","Yes (n = n/1)"],["Is every rational number an integer?","No"],["Is every real number rational?","No"],["Ordinal numbers","First, second, third: tell position"],["Cardinal numbers","One, two, three: tell how many"],["Nominal numbers","Numbers used as names or labels, like a jersey number"],["Opposite of 7","−7"],["Reciprocal of 7","1/7"],["Absolute value of −7","7"],["Negative of a negative","A positive: −(−7) = 7"],["Googol","10^100"],["Googolplex","10 to the power of a googol"],["Infinity","Not a number; the idea of going on without end"],["Golden ratio","About 1.618, an irrational number: (1 + √5)/2"]]},{"source":"https://peabody.kids/flashcards/math-order-of-operations","id":"math-order-of-operations","title":"Order of Operations (PEMDAS)","category":"math","subject":"Math","levels":["3-5","6-8"],"description":"The order to do operations in, with practice expressions.","tags":["order of operations","PEMDAS"],"frontLabel":"Question","backLabel":"Answer","cards":[["What does PEMDAS stand for?","Parentheses, Exponents, Multiplication and Division, Addition and Subtraction","Please Excuse My Dear Aunt Sally"],["What does BODMAS stand for?","Brackets, Orders, Division and Multiplication, Addition and Subtraction"],["Which comes first, parentheses or exponents?","Parentheses"],["Which is done first, multiplication or division?","Whichever comes first, going left to right"],["Which is done first, addition or subtraction?","Whichever comes first, going left to right"],["Is multiplication always done before division?","No. They have equal rank; work left to right"],["2 + 3 × 4","14"],["(2 + 3) × 4","20"],["10 − 2 × 3","4"],["(10 − 2) × 3","24"],["8 ÷ 2 × 4","16"],["8 ÷ (2 × 4)","1"],["12 − 4 + 2","10"],["12 − (4 + 2)","6"],["3² + 4","13"],["3 × 2²","12"],["(3 × 2)²","36"],["5 + 2³","13"],["20 ÷ 5 + 3","7"],["20 ÷ (5 + 3)","2.5"],["6 + 4 ÷ 2","8"],["(6 + 4) ÷ 2","5"],["2 × (3 + 4) − 5","9"],["18 ÷ 3 × 2","12"],["7 − 3 × 2 + 1","2"],["4 + 6 × (5 − 3)","16"],["100 − 5² ","75"],["3 + 2 × 5²","53"],["(8 − 2)² ÷ 4","9"],["2 + 3 × 4 − 5","9"],["9 − 3 ÷ 3 + 1","9"],["5 × 4 ÷ 2 + 1","11"],["−3²","−9","Square first, then negate"],["(−3)²","9"],["6 ÷ 2(1 + 2)","9 by strict left-to-right; the notation is ambiguous, so write it clearly"],["Where do roots fit in PEMDAS?","With exponents"],["Where does the fraction bar fit?","It groups the top and bottom like parentheses"],["Evaluate (4 + 2) / (5 − 2)","2"],["Evaluate 2 + |3 − 5|","4","Absolute value bars group like parentheses"],["√16 + 3 × 2","10"],["Evaluate 5 + 3(2 + 1)²","32"]]},{"source":"https://peabody.kids/flashcards/math-perfect-cubes","id":"math-perfect-cubes","category":"math","subject":"Math","title":"Perfect Cubes 1 to 20","levels":["6-8","9-12"],"description":"The cube of every whole number from 1 to 20.","tags":["cubes","number sense"],"frontLabel":"Cube","backLabel":"Value","cards":[["1³","1"],["2³","8"],["3³","27"],["4³","64"],["5³","125"],["6³","216"],["7³","343"],["8³","512"],["9³","729"],["10³","1000"],["11³","1331"],["12³","1728"],["13³","2197"],["14³","2744"],["15³","3375"],["16³","4096"],["17³","4913"],["18³","5832"],["19³","6859"],["20³","8000"]]},{"source":"https://peabody.kids/flashcards/math-perfect-squares","id":"math-perfect-squares","category":"math","subject":"Math","title":"Perfect Squares 1 to 40","levels":["3-5","6-8","9-12"],"description":"The square of every whole number from 1 to 40.","tags":["squares","number sense"],"frontLabel":"Square","backLabel":"Value","cards":[["1²","1"],["2²","4"],["3²","9"],["4²","16"],["5²","25"],["6²","36"],["7²","49"],["8²","64"],["9²","81"],["10²","100"],["11²","121"],["12²","144"],["13²","169"],["14²","196"],["15²","225"],["16²","256"],["17²","289"],["18²","324"],["19²","361"],["20²","400"],["21²","441"],["22²","484"],["23²","529"],["24²","576"],["25²","625"],["26²","676"],["27²","729"],["28²","784"],["29²","841"],["30²","900"],["31²","961"],["32²","1024"],["33²","1089"],["34²","1156"],["35²","1225"],["36²","1296"],["37²","1369"],["38²","1444"],["39²","1521"],["40²","1600"]]},{"source":"https://peabody.kids/flashcards/math-place-value","id":"math-place-value","title":"Place Value and Big Numbers","category":"math","subject":"Math","levels":["k-2","3-5","6-8"],"description":"Names of place values from ones to trillions and from tenths to millionths, and big number names.","tags":["place value","big numbers"],"frontLabel":"Question","backLabel":"Answer","cards":[["How many ones in a ten?","10"],["How many tens in a hundred?","10"],["How many hundreds in a thousand?","10"],["How many thousands in a million?","1,000"],["How many millions in a billion?","1,000"],["How many billions in a trillion?","1,000"],["Which place is the first to the left of the ones?","Tens"],["Place values from the right in 5-digit numbers","Ones, tens, hundreds, thousands, ten thousands"],["What is the value of the 4 in 4,567?","4,000"],["What is the value of the 5 in 4,567?","500"],["What is the value of the 6 in 4,567?","60"],["What is the value of the 7 in 4,567?","7"],["Write 3,000 + 400 + 20 + 1","3,421"],["Write 5 hundreds + 0 tens + 8 ones","508"],["Expanded form of 7,306","7,000 + 300 + 6"],["Expanded form of 24,018","20,000 + 4,000 + 10 + 8"],["Which place is the first to the right of the decimal point?","Tenths"],["Second place after the decimal point","Hundredths"],["Third place after the decimal point","Thousandths"],["Fourth place after the decimal point","Ten-thousandths"],["Fifth place after the decimal point","Hundred-thousandths"],["Sixth place after the decimal point","Millionths"],["What is the value of the 3 in 0.37?","3 tenths, or 0.3"],["What is the value of the 7 in 0.37?","7 hundredths, or 0.07"],["What is the value of the 2 in 5.012?","2 thousandths, or 0.002"],["How many hundredths in a tenth?","10"],["Name for 1,000,000","One million"],["Name for 1,000,000,000","One billion"],["Name for 1,000,000,000,000","One trillion"],["Name for 10^15","One quadrillion"],["Name for 10^18","One quintillion"],["How many zeros in one million?","6"],["How many zeros in one billion?","9"],["How many zeros in one trillion?","12"],["What is a googol?","1 followed by 100 zeros (10^100)"],["Round 3,468 to the nearest hundred","3,500"],["Round 7,249 to the nearest thousand","7,000"],["Round 85 to the nearest ten","90","Round half up"],["Round 3.14159 to the nearest hundredth","3.14"],["Which is greater, 0.5 or 0.45?","0.5","0.50 versus 0.45"],["What does multiplying by 10 do to place value?","Moves every digit one place to the left"],["What does dividing by 10 do to place value?","Moves every digit one place to the right"],["What does multiplying by 100 do to a decimal point?","Moves it two places to the right"],["Standard form of seven million, four hundred thousand","7,400,000"],["How is 10,000 read?","Ten thousand"],["How is 100,000 read?","One hundred thousand"],["Digit in the ten-thousands place of 82,915","8"],["Digit in the millions place of 6,248,000","6"]]},{"source":"https://peabody.kids/flashcards/math-powers","id":"math-powers","category":"math","subject":"Math","title":"Powers of 2, 3, 5, 6, 7 and 10","levels":["6-8","9-12","adult"],"description":"Common powers worth knowing by heart, including every power of 2 up to 2 to the 20th.","tags":["powers","exponents"],"frontLabel":"Power","backLabel":"Value","cards":[["2⁰","1"],["2¹","2"],["2²","4"],["2³","8"],["2⁴","16"],["2⁵","32"],["2⁶","64"],["2⁷","128"],["2⁸","256"],["2⁹","512"],["2¹⁰","1024"],["2¹¹","2048"],["2¹²","4096"],["2¹³","8192"],["2¹⁴","16384"],["2¹⁵","32768"],["2¹⁶","65536"],["2¹⁷","131072"],["2¹⁸","262144"],["2¹⁹","524288"],["2²⁰","1048576"],["3²","9"],["3³","27"],["3⁴","81"],["3⁵","243"],["3⁶","729"],["3⁷","2187"],["3⁸","6561"],["3⁹","19683"],["3¹⁰","59049"],["5²","25"],["5³","125"],["5⁴","625"],["5⁵","3125"],["5⁶","15625"],["5⁷","78125"],["5⁸","390625"],["6²","36"],["6³","216"],["6⁴","1296"],["6⁵","7776"],["6⁶","46656"],["7²","49"],["7³","343"],["7⁴","2401"],["7⁵","16807"],["7⁶","117649"],["10²","100"],["10³","1000"],["10⁴","10000"],["10⁵","100000"],["10⁶","1000000"]]},{"source":"https://peabody.kids/flashcards/math-prime-numbers","id":"math-prime-numbers","category":"math","subject":"Math","title":"The First 60 Prime Numbers","levels":["3-5","6-8","9-12"],"description":"Prime numbers in order, by position, plus a few prime facts.","tags":["primes","number theory"],"frontLabel":"Question","backLabel":"Answer","cards":[["What is the 1st prime number?","2"],["What is the 2nd prime number?","3"],["What is the 3rd prime number?","5"],["What is the 4th prime number?","7"],["What is the 5th prime number?","11"],["What is the 6th prime number?","13"],["What is the 7th prime number?","17"],["What is the 8th prime number?","19"],["What is the 9th prime number?","23"],["What is the 10th prime number?","29"],["What is the 11th prime number?","31"],["What is the 12th prime number?","37"],["What is the 13th prime number?","41"],["What is the 14th prime number?","43"],["What is the 15th prime number?","47"],["What is the 16th prime number?","53"],["What is the 17th prime number?","59"],["What is the 18th prime number?","61"],["What is the 19th prime number?","67"],["What is the 20th prime number?","71"],["What is the 21st prime number?","73"],["What is the 22nd prime number?","79"],["What is the 23rd prime number?","83"],["What is the 24th prime number?","89"],["What is the 25th prime number?","97"],["What is the 26th prime number?","101"],["What is the 27th prime number?","103"],["What is the 28th prime number?","107"],["What is the 29th prime number?","109"],["What is the 30th prime number?","113"],["What is the 31st prime number?","127"],["What is the 32nd prime number?","131"],["What is the 33rd prime number?","137"],["What is the 34th prime number?","139"],["What is the 35th prime number?","149"],["What is the 36th prime number?","151"],["What is the 37th prime number?","157"],["What is the 38th prime number?","163"],["What is the 39th prime number?","167"],["What is the 40th prime number?","173"],["What is the 41st prime number?","179"],["What is the 42nd prime number?","181"],["What is the 43rd prime number?","191"],["What is the 44th prime number?","193"],["What is the 45th prime number?","197"],["What is the 46th prime number?","199"],["What is the 47th prime number?","211"],["What is the 48th prime number?","223"],["What is the 49th prime number?","227"],["What is the 50th prime number?","229"],["What is the 51st prime number?","233"],["What is the 52nd prime number?","239"],["What is the 53rd prime number?","241"],["What is the 54th prime number?","251"],["What is the 55th prime number?","257"],["What is the 56th prime number?","263"],["What is the 57th prime number?","269"],["What is the 58th prime number?","271"],["What is the 59th prime number?","277"],["What is the 60th prime number?","281"],["What is the only even prime number?","2"],["Is 1 a prime number?","No. A prime has exactly two factors, and 1 has only one."],["Is 91 prime?","No: 91 = 7 × 13","It looks prime but is not"],["Is 51 prime?","No: 51 = 3 × 17"],["Is 57 prime?","No: 57 = 3 × 19"],["Is 87 prime?","No: 87 = 3 × 29"]]},{"source":"https://peabody.kids/flashcards/math-probability-terms","id":"math-probability-terms","title":"Probability and Counting","category":"math","subject":"Math","levels":["6-8","9-12"],"description":"Probability vocabulary, rules and counting formulas with simple examples.","tags":["probability","counting","combinatorics"],"frontLabel":"Term","backLabel":"Meaning","cards":[["Probability","A number from 0 to 1 telling how likely an event is"],["Impossible event","Probability 0"],["Certain event","Probability 1"],["Outcome","One possible result of an experiment"],["Event","A set of outcomes"],["Sample space","The set of all possible outcomes"],["Experiment (trial)","An action with an uncertain result, like rolling a die"],["Theoretical probability","Favorable outcomes divided by all equally likely outcomes"],["Experimental probability","Number of times it happened divided by number of trials"],["Complement of an event","All outcomes where the event does not happen; P(not A) = 1 − P(A)"],["Independent events","One event does not change the probability of the other"],["Dependent events","One event changes the probability of the other"],["Mutually exclusive events","Events that cannot happen at the same time"],["Multiplication rule (independent)","P(A and B) = P(A) × P(B)"],["Addition rule","P(A or B) = P(A) + P(B) − P(A and B)"],["Conditional probability","P(A | B) = P(A and B) / P(B)"],["Bayes' theorem","P(A | B) = P(B | A) P(A) / P(B)"],["Law of large numbers","With many trials, experimental probability gets close to theoretical probability"],["Expected value","The long-run average: sum of each value times its probability"],["Random variable","A variable whose value comes from a random process"],["Probability of heads on a fair coin","1/2"],["Probability of a 4 on a fair die","1/6"],["Probability of an even number on a die","1/2"],["Probability of a heart from a standard deck","13/52 = 1/4"],["Probability of an ace from a standard deck","4/52 = 1/13"],["Cards in a standard deck","52"],["Probability of two heads in two fair flips","1/4"],["Probability of a sum of 7 with two dice","6/36 = 1/6"],["Number of outcomes when rolling two dice","36"],["Number of outcomes in 3 coin flips","8"],["Fundamental counting principle","Multiply the number of choices at each step"],["Outfits from 3 shirts and 4 pants","12"],["Factorial n!","n × (n − 1) × ... × 1"],["Permutation","An ordered arrangement"],["Permutation formula","P(n, r) = n! / (n − r)!"],["Combination","A selection where order does not matter"],["Combination formula","C(n, r) = n! / (r!(n − r)!)"],["Ways to arrange 5 books on a shelf","5! = 120"],["P(5, 2)","20"],["C(5, 2)","10"],["C(6, 3)","20"],["C(10, 2)","45"],["C(n, 0) and C(n, n)","Both equal 1"],["Pascal's triangle","A triangle where each number is the sum of the two above it; rows give combinations"],["Binomial distribution","The distribution of the number of successes in a fixed number of independent yes/no trials"],["With replacement","Each item is put back before the next draw"],["Without replacement","Items are not put back; later probabilities change"],["Tree diagram","A branching picture that lists outcomes of a multi-step event"],["Venn diagram","Overlapping circles showing how sets share members"],["Odds","Ratio of favorable outcomes to unfavorable outcomes"],["Fair game","A game where each player has the same expected value"],["Simulation","Using random numbers or objects to imitate a real chance process"]]},{"source":"https://peabody.kids/flashcards/math-roman-numerals","id":"math-roman-numerals","category":"math","subject":"Math","title":"Roman Numerals","levels":["3-5","6-8","9-12","adult"],"description":"Read Roman numerals and write them, from I to MMMCMXCIX.","tags":["roman numerals"],"frontLabel":"Number","backLabel":"Roman numeral","cards":[["What do the seven letters I V X L C D M stand for?","1, 5, 10, 50, 100, 500, 1000"],["1","I"],["2","II"],["3","III"],["4","IV"],["5","V"],["6","VI"],["7","VII"],["8","VIII"],["9","IX"],["10","X"],["11","XI"],["12","XII"],["13","XIII"],["14","XIV"],["15","XV"],["16","XVI"],["17","XVII"],["18","XVIII"],["19","XIX"],["20","XX"],["21","XXI"],["22","XXII"],["23","XXIII"],["24","XXIV"],["25","XXV"],["26","XXVI"],["27","XXVII"],["28","XXVIII"],["29","XXIX"],["30","XXX"],["35","XXXV"],["40","XL"],["44","XLIV"],["49","XLIX"],["50","L"],["60","LX"],["70","LXX"],["80","LXXX"],["90","XC"],["99","XCIX"],["100","C"],["105","CV"],["144","CXLIV"],["200","CC"],["400","CD"],["444","CDXLIV"],["500","D"],["900","CM"],["1000","M"],["1492","MCDXCII"],["1776","MDCCLXXVI"],["1999","MCMXCIX"],["2000","MM"],["2024","MMXXIV"],["2026","MMXXVI"],["3000","MMM"],["3999","MMMCMXCIX"],["Which Roman numerals can be repeated up to three times?","I, X, C and M"]]},{"source":"https://peabody.kids/flashcards/math-sequences-series","id":"math-sequences-series","title":"Sequences, Series and Patterns","category":"math","subject":"Math","levels":["6-8","9-12"],"description":"Arithmetic and geometric sequences, formulas for sums, and famous number patterns.","tags":["sequences","series","patterns"],"frontLabel":"Question","backLabel":"Answer","cards":[["Sequence","An ordered list of numbers following a rule"],["Term of a sequence","One number in the list"],["Arithmetic sequence","Each term is found by adding the same number"],["Common difference","The number added each time in an arithmetic sequence"],["Geometric sequence","Each term is found by multiplying by the same number"],["Common ratio","The number multiplied each time in a geometric sequence"],["Explicit formula, arithmetic","aₙ = a₁ + (n − 1)d"],["Explicit formula, geometric","aₙ = a₁ · rⁿ⁻¹"],["Recursive formula","Gives each term from the term before it"],["Sum of an arithmetic series","Sₙ = n(a₁ + aₙ)/2"],["Sum of a finite geometric series","Sₙ = a₁(1 − rⁿ)/(1 − r)"],["Sum of an infinite geometric series","S = a₁/(1 − r), when |r| < 1"],["Next term: 2, 4, 6, 8, ...","10 (add 2)"],["Next term: 5, 10, 15, 20, ...","25 (add 5)"],["Next term: 3, 6, 12, 24, ...","48 (multiply by 2)"],["Next term: 1, 4, 9, 16, ...","25 (squares)"],["Next term: 1, 3, 6, 10, ...","15 (triangular numbers, add 5)"],["Next term: 100, 90, 80, ...","70 (subtract 10)"],["Next term: 81, 27, 9, 3, ...","1 (divide by 3)"],["Next term: 1, 1, 2, 3, 5, 8, ...","13 (Fibonacci)"],["Next term: 2, 6, 18, 54, ...","162 (multiply by 3)"],["Next term: 1, 2, 4, 7, 11, ...","16 (differences grow by 1)"],["Next term: 1, 8, 27, 64, ...","125 (cubes)"],["Next term: 2, 3, 5, 7, 11, ...","13 (primes)"],["10th term of 3, 7, 11, ...","39"],["20th term of 5, 8, 11, ...","62"],["8th term of 2, 4, 8, ...","256"],["Common difference of 7, 12, 17, ...","5"],["Common ratio of 5, 15, 45, ...","3"],["Common ratio of 16, 8, 4, ...","1/2"],["Sum of 1 + 2 + ... + 10","55"],["Sum of 1 + 2 + ... + 100","5,050"],["Sum of the first 5 odd numbers","25"],["Sum of the first 10 even numbers","110"],["Sum of 1 + 1/2 + 1/4 + 1/8 + ... forever","2"],["Fibonacci rule","Each term is the sum of the two before it"],["Fibonacci numbers up to 144","1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144"],["Ratio of consecutive Fibonacci numbers","Approaches the golden ratio, about 1.618"],["Triangular numbers formula","n(n + 1)/2"],["Square numbers formula","n²"],["Pentagonal number idea","n(3n − 1)/2: 1, 5, 12, 22, ..."],["Pascal's triangle rows","1; 1 1; 1 2 1; 1 3 3 1; 1 4 6 4 1"],["Row sums of Pascal's triangle","Powers of 2"],["Sigma notation","Σ means add up the terms for the given range"],["Σ from k=1 to 4 of k","10"],["Σ from k=1 to 3 of k²","14"],["Σ from k=1 to 5 of 2","10"],["Pattern","Something that repeats or changes by a rule"],["Growth pattern","A pattern in which a number or shape grows by a rule each step"],["Limit of a sequence","The value the terms get close to"],["Converge","The terms or sum settle toward a number"],["Diverge","The terms or sum do not settle toward a number"],["Harmonic series","1 + 1/2 + 1/3 + 1/4 + ... ; it grows without bound"],["Arithmetic mean of two numbers","Their average"],["Geometric mean of two numbers","The square root of their product"],["Geometric mean of 4 and 9","6"],["Compound interest as a sequence","A geometric sequence, multiplying by (1 + r) each period"]]},{"source":"https://peabody.kids/flashcards/math-sets-logic","id":"math-sets-logic","title":"Sets and Logic","category":"math","subject":"Math","levels":["6-8","9-12","college"],"description":"Set vocabulary, Venn diagrams and basic logic statements.","tags":["sets","logic","venn diagrams"],"frontLabel":"Term","backLabel":"Meaning","cards":[["Set","A collection of distinct objects"],["Element (member)","An object that is in a set"],["Empty set","The set with no elements, written ∅ or { }"],["Subset","A set whose elements are all in another set"],["Proper subset","A subset that is not the whole set"],["Union (A ∪ B)","Elements in A or B or both"],["Intersection (A ∩ B)","Elements in both A and B"],["Complement (A′)","Elements in the universal set that are not in A"],["Universal set","The set of everything under consideration"],["Disjoint sets","Sets with no elements in common"],["Cardinality","The number of elements in a set, written |A|"],["Roster form","Listing the elements: {1, 2, 3}"],["Set-builder notation","{x | x > 2} means all x such that x is greater than 2"],["Venn diagram","Overlapping circles showing how sets relate"],["Cardinality of a union","|A ∪ B| = |A| + |B| − |A ∩ B|"],["If A = {1,2,3} and B = {3,4}, A ∪ B","{1, 2, 3, 4}"],["If A = {1,2,3} and B = {3,4}, A ∩ B","{3}"],["Number of subsets of a set with n elements","2ⁿ"],["Number of subsets of {a, b, c}","8"],["Is ∅ a subset of every set?","Yes"],["Natural numbers symbol","ℕ"],["Integers symbol","ℤ"],["Rational numbers symbol","ℚ"],["Real numbers symbol","ℝ"],["Statement (proposition)","A sentence that is either true or false"],["Negation","The statement with the opposite truth value (not P)"],["Conjunction","P and Q; true only when both are true"],["Disjunction","P or Q; true when at least one is true"],["Conditional","If P then Q; false only when P is true and Q is false"],["Hypothesis","The 'if' part of a conditional"],["Conclusion","The 'then' part of a conditional"],["Converse","Swap the parts: if Q then P"],["Inverse","Negate both parts: if not P then not Q"],["Contrapositive","Swap and negate: if not Q then not P; same truth as the original"],["Biconditional","P if and only if Q; true when both have the same truth value"],["Truth table","A table listing the truth value of a statement for every case"],["Tautology","A statement that is always true"],["Contradiction","A statement that is always false"],["De Morgan's law 1","not (P and Q) is the same as (not P) or (not Q)"],["De Morgan's law 2","not (P or Q) is the same as (not P) and (not Q)"],["Counterexample","An example showing a statement is false"],["Direct proof","Proving a statement by a chain of logical steps from facts"],["Proof by contradiction","Assume the opposite and show it leads to something impossible"],["Proof by induction","Show it for a first case, then show each case leads to the next"],["Inductive reasoning","Reaching a general rule from patterns"],["Deductive reasoning","Reaching a conclusion that must follow from given facts"],["Axiom","A basic statement accepted without proof"],["Lemma","A small proven result used to prove a bigger one"],["Corollary","A result that follows easily from a theorem"],["Q.E.D.","Latin 'quod erat demonstrandum': which was to be shown"],["Necessary condition","Something that must be true for another thing to be true"],["Sufficient condition","Something that, if true, guarantees another thing"],["Quantifier ∀","For all"],["Quantifier ∃","There exists"],["Boolean value","True or false, often written 1 or 0"],["AND gate output","1 only if both inputs are 1"],["OR gate output","1 if at least one input is 1"],["NOT gate output","The opposite of the input"],["Binary number system","Base 2, using only digits 0 and 1"],["Binary 101 in decimal","5"],["Decimal 10 in binary","1010"],["Binary 1111 in decimal","15"]]},{"source":"https://peabody.kids/flashcards/math-square-roots","id":"math-square-roots","category":"math","subject":"Math","title":"Square Roots of Perfect Squares","levels":["6-8","9-12"],"description":"Square roots of the perfect squares from 1 to 1600.","tags":["square roots","number sense"],"frontLabel":"Root","backLabel":"Value","cards":[["√1","1"],["√4","2"],["√9","3"],["√16","4"],["√25","5"],["√36","6"],["√49","7"],["√64","8"],["√81","9"],["√100","10"],["√121","11"],["√144","12"],["√169","13"],["√196","14"],["√225","15"],["√256","16"],["√289","17"],["√324","18"],["√361","19"],["√400","20"],["√441","21"],["√484","22"],["√529","23"],["√576","24"],["√625","25"],["√676","26"],["√729","27"],["√784","28"],["√841","29"],["√900","30"],["√961","31"],["√1024","32"],["√1089","33"],["√1156","34"],["√1225","35"],["√1296","36"],["√1369","37"],["√1444","38"],["√1521","39"],["√1600","40"]]},{"source":"https://peabody.kids/flashcards/math-statistics-terms","id":"math-statistics-terms","title":"Statistics Terms","category":"math","subject":"Math","levels":["6-8","9-12","college"],"description":"Descriptive and inferential statistics vocabulary, from mean and median to confidence intervals.","tags":["statistics","data"],"frontLabel":"Term","backLabel":"Meaning","cards":[["Data","Facts or measurements that have been collected"],["Population","The whole group you want to learn about"],["Sample","A smaller group taken from the population"],["Random sample","A sample where every member of the population has an equal chance of being chosen"],["Bias","A systematic error that tilts results in one direction"],["Survey","A way of collecting data by asking people questions"],["Census","A survey of every member of the population"],["Variable","A characteristic that can take different values"],["Categorical data","Data sorted into groups or names","Eye color"],["Numerical (quantitative) data","Data that are numbers you can measure or count"],["Discrete data","Data that can only take separate values, like counts"],["Continuous data","Data that can take any value in a range, like height"],["Mean","Sum of all values divided by how many values"],["Median","Middle value of ordered data"],["Mode","Most frequently occurring value"],["Range","Maximum minus minimum"],["Outlier","A value far away from the rest of the data"],["Quartile","One of three values that split ordered data into four equal parts"],["Q1 (first quartile)","The median of the lower half of the data"],["Q3 (third quartile)","The median of the upper half of the data"],["Interquartile range (IQR)","Q3 − Q1"],["Five-number summary","Minimum, Q1, median, Q3, maximum"],["Outlier rule (1.5 × IQR)","Values below Q1 − 1.5·IQR or above Q3 + 1.5·IQR are flagged as outliers"],["Percentile","The value below which a given percent of data fall"],["Variance","The average squared distance from the mean"],["Standard deviation","The square root of variance; typical distance from the mean"],["Z-score","How many standard deviations a value is from the mean: (x − mean)/s"],["Frequency","How many times a value occurs"],["Relative frequency","Frequency divided by the total count"],["Cumulative frequency","A running total of frequencies"],["Histogram","A bar graph for numerical data grouped into intervals"],["Box plot (box-and-whisker)","A graph of the five-number summary"],["Dot plot","A graph with a dot above a number line for each data value"],["Stem-and-leaf plot","A table that splits each value into a stem and a leaf"],["Scatter plot","A graph of ordered pairs showing how two variables relate"],["Bar graph","A graph using separate bars to compare categories"],["Pie chart","A circle graph that shows parts of a whole"],["Line graph","A graph that connects points to show change over time"],["Symmetric distribution","The left and right sides mirror each other"],["Skewed right","A longer tail on the right; mean is usually greater than median"],["Skewed left","A longer tail on the left; mean is usually less than median"],["Normal distribution","A symmetric bell-shaped distribution"],["Empirical rule","About 68%, 95% and 99.7% of normal data fall within 1, 2 and 3 standard deviations of the mean"],["Correlation","A measure of the strength and direction of a linear relationship"],["Correlation coefficient (r)","A number from −1 to 1 measuring linear correlation"],["Positive correlation","As one variable increases, so does the other"],["Negative correlation","As one variable increases, the other decreases"],["Correlation vs. causation","Two things moving together does not prove one causes the other"],["Line of best fit","A line drawn to match the trend in a scatter plot"],["Regression","A method for fitting a model to describe how one variable depends on another"],["Residual","Actual value minus predicted value"],["Extrapolation","Predicting beyond the range of the data"],["Interpolation","Predicting between known data points"],["Hypothesis test","A procedure for deciding whether data support a claim"],["Null hypothesis","The default claim that nothing special is going on"],["P-value","The probability of results at least as extreme as observed if the null hypothesis is true"],["Confidence interval","A range of values likely to contain the true population value"],["Margin of error","The amount added and subtracted from an estimate to form an interval"],["Sampling distribution","The distribution of a statistic over many samples"],["Central limit theorem","Sample means tend toward a normal shape as sample size grows"],["Control group","The group in an experiment that does not get the treatment"],["Random assignment","Using chance to place subjects in experiment groups"],["Observational study","A study that observes without assigning treatments"],["Experiment","A study where researchers assign treatments"]]},{"source":"https://peabody.kids/flashcards/math-symbols","id":"math-symbols","title":"Math Symbols and Their Names","category":"math","subject":"Math","levels":["3-5","6-8","9-12","college"],"description":"Common math symbols and what each one means.","tags":["symbols","notation"],"frontLabel":"Symbol","backLabel":"Name and meaning","cards":[["+","Plus: add"],["−","Minus: subtract"],["×","Times: multiply"],["÷","Divided by: divide"],["=","Equals: is the same value as"],["≠","Is not equal to"],["<","Is less than"],[">","Is greater than"],["≤","Is less than or equal to"],["≥","Is greater than or equal to"],["≈","Is approximately equal to"],["≡","Is identical to or congruent (mod) to"],["±","Plus or minus"],["∓","Minus or plus"],["√","Square root (radical)"],["∛","Cube root"],["π","Pi: about 3.14159, circumference over diameter"],["∞","Infinity: unbounded, without end"],["%","Percent: out of 100"],["‰","Per mille: out of 1,000"],["°","Degree: a unit of angle or temperature"],["′","Prime: minutes, feet, or a related variable"],["″","Double prime: seconds, inches"],["|x|","Absolute value of x"],["!","Factorial"],["Σ","Sigma: sum of a series"],["Π (capital pi)","Product of a series"],["Δ","Delta: change in"],["∂","Partial derivative"],["∫","Integral"],["∮","Contour or closed-loop integral"],["lim","Limit"],["θ","Theta: commonly an angle"],["λ","Lambda: wavelength, eigenvalue"],["μ","Mu: mean of a population"],["σ","Sigma (lowercase): standard deviation"],["ε (epsilon)","A very small positive number in limits"],["φ","Phi: the golden ratio (about 1.618) or an angle"],["e","Euler's number, about 2.71828"],["i","Imaginary unit, where i² = −1"],["ℝ","The set of real numbers"],["ℚ","The set of rational numbers"],["ℤ","The set of integers"],["ℕ","The set of natural numbers"],["ℂ","The set of complex numbers"],["∈","Is an element of"],["∉","Is not an element of"],["⊂","Is a subset of"],["∪","Union: in either set"],["∩","Intersection: in both sets"],["∅","Empty set: no elements"],["∀","For all"],["∃","There exists"],["∴","Therefore"],["∵","Because"],["⇒","Implies"],["⇔","If and only if"],["∥","Is parallel to"],["⊥","Is perpendicular to"],["∠","Angle"],["△","Triangle"],["≅","Is congruent to"],["∼","Is similar to"],["∝","Is proportional to"],["f(x)","Function f of x"],["f⁻¹","Inverse function of f"],["n!","n factorial: product of 1 to n"],["a:b","Ratio of a to b"],["⌊x⌋","Floor: greatest integer not above x"],["⌈x⌉","Ceiling: least integer not below x"],["mod","Modulo: the remainder after division"],["·","Dot: multiplication"],["→ (arrow)","Approaches, or maps to"]]},{"source":"https://peabody.kids/flashcards/math-time","id":"math-time","title":"Time and the Calendar","category":"math","subject":"Math","levels":["k-2","3-5","6-8"],"description":"Time units, telling time, elapsed time and the calendar.","tags":["time","clocks","calendar"],"frontLabel":"Question","backLabel":"Answer","cards":[["Seconds in a minute","60"],["Minutes in an hour","60"],["Seconds in an hour","3,600"],["Hours in a day","24"],["Minutes in a day","1,440"],["Seconds in a day","86,400"],["Days in a week","7"],["Days in a year","365"],["Days in a leap year","366"],["Weeks in a year","52 (plus 1 or 2 days)"],["Months in a year","12"],["Years in a decade","10"],["Years in a century","100"],["Years in a millennium","1,000"],["Months with 31 days","January, March, May, July, August, October, December"],["Months with 30 days","April, June, September, November"],["Days in February","28, or 29 in a leap year"],["Knuckle trick for month lengths","Knuckles are 31-day months, valleys between are shorter months"],["Leap year rule","Every 4 years, except century years not divisible by 400"],["Is 2000 a leap year?","Yes"],["Is 1900 a leap year?","No"],["Is 2028 a leap year?","Yes"],["Is 2026 a leap year?","No"],["Order of the days of the week","Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday"],["Order of the months","January through December"],["Seasons in order (northern hemisphere)","Winter, spring, summer, fall"],["Quarter past 3","3:15"],["Half past 7","7:30"],["Quarter to 9","8:45"],["Ten minutes to 6","5:50"],["Meaning of a.m.","From midnight to just before noon"],["Meaning of p.m.","From noon to just before midnight"],["12:00 a.m. is","Midnight"],["12:00 p.m. is","Noon"],["3:00 p.m. in 24-hour time","15:00"],["18:30 in 12-hour time","6:30 p.m."],["00:00 in 12-hour time","12:00 a.m. (midnight)"],["Elapsed time from 2:15 to 4:45","2 hours 30 minutes"],["Elapsed time from 9:40 a.m. to 11:10 a.m.","1 hour 30 minutes"],["Time 45 minutes after 3:30","4:15"],["Time 2 hours before 1:00 p.m.","11:00 a.m."],["Minutes in 2 and a half hours","150"],["Hours in 3 days","72"],["Hours in a week","168"],["Minutes in a quarter hour","15"],["Convert 90 minutes to hours","1.5 hours"],["Convert 2 hours to seconds","7,200"],["Convert 3 weeks to days","21"],["Convert 18 months to years","1.5 years"],["Fortnight","Two weeks (14 days)"],["Biweekly","Every two weeks"],["Semester","Half of a school year"],["Which hand moves faster, hour or minute?","Minute hand"],["How far does the minute hand move in 5 minutes?","From one number to the next (30 degrees)"],["Degrees the minute hand turns in an hour","360"],["Time zone","A region that shares the same standard time"],["Number of time zones in the contiguous US","Four: Eastern, Central, Mountain, Pacific"],["Greenwich Mean Time (GMT)","Time at the 0 degree longitude line through Greenwich, England"],["Calendar year starts and ends","January 1 to December 31"],["BCE and CE","Before Common Era and Common Era, used to number years"],["Digital vs. analog clock","Digital shows numbers; analog uses hands on a dial"]]},{"source":"https://peabody.kids/flashcards/math-trig-identities","id":"math-trig-identities","title":"Trig Definitions and Identities","category":"math","subject":"Math","levels":["9-12","college"],"description":"Right-triangle ratios, reciprocal, Pythagorean, angle-sum, double-angle and half-angle identities.","tags":["trigonometry","identities"],"frontLabel":"Name","backLabel":"Identity","cards":[["SOH-CAH-TOA: sine","sin = opposite / hypotenuse"],["SOH-CAH-TOA: cosine","cos = adjacent / hypotenuse"],["SOH-CAH-TOA: tangent","tan = opposite / adjacent"],["Cosecant","csc θ = 1 / sin θ"],["Secant","sec θ = 1 / cos θ"],["Cotangent","cot θ = 1 / tan θ = cos θ / sin θ"],["Tangent in terms of sine and cosine","tan θ = sin θ / cos θ"],["Pythagorean identity","sin² θ + cos² θ = 1"],["Pythagorean identity with tangent","1 + tan² θ = sec² θ"],["Pythagorean identity with cotangent","1 + cot² θ = csc² θ"],["Even-odd: sin(−θ)","−sin θ (sine is odd)"],["Even-odd: cos(−θ)","cos θ (cosine is even)"],["Even-odd: tan(−θ)","−tan θ (tangent is odd)"],["Cofunction: sin(90° − θ)","cos θ"],["Cofunction: cos(90° − θ)","sin θ"],["Cofunction: tan(90° − θ)","cot θ"],["Supplementary: sin(180° − θ)","sin θ"],["Supplementary: cos(180° − θ)","−cos θ"],["Shift: sin(θ + 180°)","−sin θ"],["Shift: cos(θ + 180°)","−cos θ"],["Shift: tan(θ + 180°)","tan θ"],["Sum formula for sine","sin(A + B) = sin A cos B + cos A sin B"],["Difference formula for sine","sin(A − B) = sin A cos B − cos A sin B"],["Sum formula for cosine","cos(A + B) = cos A cos B − sin A sin B"],["Difference formula for cosine","cos(A − B) = cos A cos B + sin A sin B"],["Sum formula for tangent","tan(A + B) = (tan A + tan B) / (1 − tan A tan B)"],["Difference formula for tangent","tan(A − B) = (tan A − tan B) / (1 + tan A tan B)"],["Double-angle: sine","sin 2θ = 2 sin θ cos θ"],["Double-angle: cosine","cos 2θ = cos² θ − sin² θ"],["Double-angle: cosine (other form 1)","cos 2θ = 1 − 2 sin² θ"],["Double-angle: cosine (other form 2)","cos 2θ = 2 cos² θ − 1"],["Double-angle: tangent","tan 2θ = 2 tan θ / (1 − tan² θ)"],["Power reducing: sin² θ","(1 − cos 2θ) / 2"],["Power reducing: cos² θ","(1 + cos 2θ) / 2"],["Half-angle: sin(θ/2)","±√((1 − cos θ) / 2)"],["Half-angle: cos(θ/2)","±√((1 + cos θ) / 2)"],["Product to sum: sin A cos B","½[sin(A + B) + sin(A − B)]"],["Product to sum: cos A cos B","½[cos(A − B) + cos(A + B)]"],["Law of sines","a / sin A = b / sin B = c / sin C"],["Law of cosines","c² = a² + b² − 2ab cos C"],["Area of a triangle with trig","½ ab sin C"],["Arc length formula (radians)","s = rθ"],["Sector area formula (radians)","A = ½ r² θ"],["Degrees to radians","Multiply by π/180"],["Radians to degrees","Multiply by 180/π"],["Amplitude of y = A sin(Bx)","|A|"],["Period of y = A sin(Bx)","2π / |B|"],["Phase shift of y = sin(x − C)","C units to the right"],["Inverse sine range","−π/2 to π/2"],["Inverse cosine range","0 to π"],["Inverse tangent range","−π/2 to π/2 (not including the ends)"],["Euler's formula","e^(iθ) = cos θ + i sin θ"],["Euler's identity","e^(iπ) + 1 = 0"],["De Moivre's theorem","(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ"]]},{"source":"https://peabody.kids/flashcards/math-trig-values","id":"math-trig-values","title":"Trig Values at Special Angles","category":"math","subject":"Math","levels":["9-12","college"],"description":"Sine, cosine and tangent at the common unit-circle angles in degrees and radians.","tags":["trigonometry","unit circle"],"frontLabel":"Value","backLabel":"Answer","cards":[["sin 0°","0"],["sin 30°","1/2"],["sin 45°","√2/2"],["sin 60°","√3/2"],["sin 90°","1"],["sin 120°","√3/2"],["sin 135°","√2/2"],["sin 150°","1/2"],["sin 180°","0"],["sin 210°","−1/2"],["sin 225°","−√2/2"],["sin 240°","−√3/2"],["sin 270°","−1"],["sin 300°","−√3/2"],["sin 315°","−√2/2"],["sin 330°","−1/2"],["sin 360°","0"],["cos 0°","1"],["cos 30°","√3/2"],["cos 45°","√2/2"],["cos 60°","1/2"],["cos 90°","0"],["cos 120°","−1/2"],["cos 135°","−√2/2"],["cos 150°","−√3/2"],["cos 180°","−1"],["cos 210°","−√3/2"],["cos 225°","−√2/2"],["cos 240°","−1/2"],["cos 270°","0"],["cos 300°","1/2"],["cos 315°","√2/2"],["cos 330°","√3/2"],["cos 360°","1"],["tan 0°","0"],["tan 30°","√3/3 (1/√3)"],["tan 45°","1"],["tan 60°","√3"],["tan 90°","Undefined"],["tan 120°","−√3"],["tan 135°","−1"],["tan 150°","−√3/3"],["tan 180°","0"],["tan 225°","1"],["tan 270°","Undefined"],["sin(π/6)","1/2"],["sin(π/4)","√2/2"],["sin(π/3)","√3/2"],["sin(π/2)","1"],["cos(π/6)","√3/2"],["cos(π/4)","√2/2"],["cos(π/3)","1/2"],["cos(π)","−1"],["sin(π)","0"],["cos(2π/3)","−1/2"],["sin(5π/6)","1/2"],["sin(3π/2)","−1"],["sin(7π/6)","−1/2"],["cos(5π/4)","−√2/2"],["cos(7π/4)","√2/2"],["tan(π/4)","1"],["tan(π/3)","√3"],["tan(π/6)","√3/3"],["csc 30°","2"],["sec 60°","2"],["cot 45°","1"],["sec 0°","1"],["Where is sine positive?","Quadrants I and II"],["Where is cosine positive?","Quadrants I and IV"],["Where is tangent positive?","Quadrants I and III"],["Memory rule for signs (All Students Take Calculus)","All positive in I, Sine in II, Tangent in III, Cosine in IV"],["The unit circle point at 90°","(0, 1)"],["The unit circle point at 180°","(−1, 0)"],["The unit circle point at 45°","(√2/2, √2/2)"],["Range of sine and cosine","−1 to 1"],["Period of sine and cosine","360° or 2π"],["Period of tangent","180° or π"]]},{"source":"https://peabody.kids/flashcards/math-us-units","id":"math-us-units","title":"US Customary Units and Conversions","category":"math","subject":"Math","levels":["3-5","6-8","adult"],"description":"Inches, feet, pounds, gallons and how they convert, plus metric-to-US benchmarks.","tags":["measurement","customary","conversion"],"frontLabel":"Question","backLabel":"Answer","cards":[["Inches in 1 foot","12"],["Feet in 1 yard","3"],["Inches in 1 yard","36"],["Feet in 1 mile","5,280"],["Yards in 1 mile","1,760"],["Ounces in 1 pound","16"],["Pounds in 1 ton","2,000"],["Teaspoons in 1 tablespoon","3"],["Tablespoons in 1 fluid ounce","2"],["Tablespoons in 1 cup","16"],["Fluid ounces in 1 cup","8"],["Cups in 1 pint","2"],["Pints in 1 quart","2"],["Cups in 1 quart","4"],["Quarts in 1 gallon","4"],["Pints in 1 gallon","8"],["Cups in 1 gallon","16"],["Fluid ounces in 1 gallon","128"],["Square feet in 1 square yard","9"],["Square inches in 1 square foot","144"],["Square feet in 1 acre","43,560"],["Cubic feet in 1 cubic yard","27"],["Cubic inches in 1 cubic foot","1,728"],["Convert 5 ft to inches","60 inches"],["Convert 2 yd to feet","6 feet"],["Convert 48 inches to feet","4 feet"],["Convert 3 lb to oz","48 oz"],["Convert 32 oz to lb","2 lb"],["Convert 3 gal to qt","12 qt"],["Convert 6 cups to pints","3 pints"],["Convert 10 qt to gal","2.5 gal"],["Convert 1 mile to yards","1,760 yards"],["Convert 4,000 lb to tons","2 tons"],["Inches in 1 centimeter (about)","0.39 inches"],["Centimeters in 1 inch","2.54 exactly"],["Meters in 1 yard","0.9144 exactly"],["Kilometers in 1 mile (about)","1.61 kilometers"],["Miles in 1 kilometer (about)","0.62 miles"],["Pounds in 1 kilogram (about)","2.2 pounds"],["Grams in 1 ounce (about)","28.35 grams"],["Liters in 1 gallon (US, about)","3.785 liters"],["Liters in 1 quart (about)","0.95 liters"],["Feet in 1 meter (about)","3.28 feet"],["Fahrenheit freezing point of water","32 degrees F"],["Fahrenheit boiling point of water (sea level)","212 degrees F"],["Normal body temperature in Fahrenheit","About 98.6 degrees F"],["Formula: Fahrenheit to Celsius","C = (F − 32) × 5/9"],["Formula: Celsius to Fahrenheit","F = C × 9/5 + 32"],["Convert 212 degrees F to Celsius","100 degrees C"],["Convert 0 degrees C to Fahrenheit","32 degrees F"],["Convert 20 degrees C to Fahrenheit","68 degrees F"],["Convert 100 degrees F to Celsius","About 37.8 degrees C"],["At what temperature are Celsius and Fahrenheit equal?","−40 degrees"],["Which is longer, a foot or a meter?","A meter (about 3.28 feet)"],["Which is heavier, a pound or a kilogram?","A kilogram"],["Which is bigger, a quart or a liter?","A liter is slightly bigger"],["Which is bigger, a mile or a kilometer?","A mile"],["What is a baker's dozen?","13"],["What is a dozen?","12"],["What is a gross?","144 (12 dozen)"],["What is a score?","20"],["What is a ream of paper?","500 sheets"],["What is a nautical mile?","About 1.15 miles (1,852 meters)"],["What is a knot?","A speed of one nautical mile per hour"],["What is a fathom?","6 feet, used to measure water depth"],["What is a furlong?","One eighth of a mile (220 yards)"],["What is a hand (horse height)?","4 inches"]]},{"source":"https://peabody.kids/flashcards/math-vocab-3-5","id":"math-vocab-3-5","title":"Math Words for Grades 3-5","category":"math","subject":"Math","levels":["3-5"],"description":"Key math vocabulary for multiplication, fractions, decimals, geometry and data.","tags":["vocabulary","elementary"],"frontLabel":"Word","backLabel":"Meaning","cards":[["Factor","A number that multiplies with another to make a product","3 and 4 are factors of 12"],["Product","The answer to a multiplication problem"],["Quotient","The answer to a division problem"],["Dividend","The number being divided","In 12 ÷ 3, it is 12"],["Divisor","The number you divide by","In 12 ÷ 3, it is 3"],["Remainder","What is left over after dividing"],["Multiple","What you get when you multiply a number by 1, 2, 3, ...","Multiples of 5: 5, 10, 15"],["Prime number","A number greater than 1 with exactly two factors, 1 and itself"],["Composite number","A number with more than two factors"],["Array","Objects arranged in equal rows and columns"],["Equation","A math sentence with an equals sign"],["Expression","Numbers and operation signs with no equals sign","3 × 4 + 2"],["Variable","A letter that stands for an unknown number"],["Numerator","The top number in a fraction; how many parts we have"],["Denominator","The bottom number in a fraction; how many equal parts in the whole"],["Equivalent fractions","Fractions that name the same amount","1/2 and 2/4"],["Mixed number","A whole number and a fraction together","2 1/3"],["Improper fraction","A fraction whose numerator is greater than or equal to its denominator","7/4"],["Unit fraction","A fraction with 1 on top","1/5"],["Simplest form","A fraction that cannot be made smaller; the top and bottom share no factor but 1"],["Decimal","A number written with a decimal point to show parts of a whole","0.75"],["Decimal point","The dot that separates ones from tenths"],["Tenths","The first place right of the decimal point"],["Hundredths","The second place right of the decimal point"],["Round","Replace a number with a nearby number that is easier to use"],["Estimate","Find an answer that is close but not exact"],["Regroup","Trade between place values, such as 10 ones for 1 ten"],["Place value","The value of a digit based on where it sits in a number"],["Perimeter","The distance around a shape"],["Area","The amount of flat space inside a shape, in square units"],["Volume","The amount of space inside a solid, in cubic units"],["Capacity","How much a container can hold"],["Angle","Formed where two rays meet at a point"],["Right angle","An angle of exactly 90 degrees, like a square corner"],["Acute angle","An angle less than 90 degrees"],["Obtuse angle","An angle greater than 90 degrees and less than 180 degrees"],["Straight angle","An angle of exactly 180 degrees"],["Parallel lines","Lines that stay the same distance apart and never meet"],["Perpendicular lines","Lines that meet at a right angle"],["Intersecting lines","Lines that cross each other"],["Line segment","Part of a line with two endpoints"],["Ray","Part of a line with one endpoint that goes on forever in one direction"],["Point","An exact location with no size"],["Polygon","A closed flat shape made of straight sides"],["Quadrilateral","Any polygon with 4 sides"],["Pentagon","A polygon with 5 sides"],["Hexagon","A polygon with 6 sides"],["Octagon","A polygon with 8 sides"],["Symmetry","When a shape can be folded so both halves match exactly"],["Line of symmetry","The fold line that makes two matching halves"],["Congruent","Exactly the same size and shape"],["Face","A flat surface of a solid shape"],["Edge","Where two faces of a solid meet"],["Vertex","A corner point of a shape or solid; plural is vertices"],["Elapsed time","The amount of time that passes from start to end"],["Data","Facts or numbers collected to learn something"],["Bar graph","A graph that uses bars to compare amounts"],["Line plot","A number line with marks above it to show how often values occur"],["Pictograph","A graph that uses pictures to show data"],["Survey","Asking a group of people questions to collect data"],["Mean","The average: add all values and divide by how many there are"],["Median","The middle value when the data are in order"],["Mode","The value that appears most often"],["Range","The highest value minus the lowest value"],["Commutative property","You can swap the order of numbers when adding or multiplying","3 × 4 = 4 × 3"],["Associative property","You can regroup numbers when adding or multiplying and get the same answer"],["Distributive property","Multiplying a sum equals multiplying each part and adding","3 × (4 + 2) = 3 × 4 + 3 × 2"],["Identity property","Adding 0 or multiplying by 1 leaves a number unchanged"],["Inverse operations","Operations that undo each other","Addition and subtraction"],["Ordered pair","Two numbers in parentheses that name a point on a grid","(3, 5)"],["Coordinate plane","A flat grid made by a horizontal and a vertical number line"],["Digit","One of the symbols 0 to 9"],["Compatible numbers","Numbers that are easy to compute with in your head"],["Unit","A fixed amount used to measure, like an inch or a gram"],["Weight","How heavy something is"],["Mass","The amount of matter in an object"],["Fraction","A number that names equal parts of a whole"]]},{"source":"https://peabody.kids/flashcards/math-vocab-6-8","id":"math-vocab-6-8","title":"Math Words for Grades 6-8","category":"math","subject":"Math","levels":["6-8"],"description":"Middle school math vocabulary: ratios, integers, expressions, equations, geometry and statistics.","tags":["vocabulary","middle school"],"frontLabel":"Word","backLabel":"Meaning","cards":[["Ratio","A comparison of two quantities by division","3 to 5, or 3:5"],["Rate","A ratio that compares two different kinds of units","60 miles per hour"],["Unit rate","A rate with 1 as the second quantity","$2 per pound"],["Proportion","An equation that says two ratios are equal"],["Percent","A ratio out of 100"],["Integer","A whole number or its opposite: ..., −2, −1, 0, 1, 2, ..."],["Rational number","A number that can be written as a fraction of two integers"],["Irrational number","A number that cannot be written as a fraction; its decimal never ends or repeats","π, √2"],["Real numbers","All rational and irrational numbers together"],["Absolute value","A number's distance from zero on the number line"],["Opposite numbers","Two numbers the same distance from zero on opposite sides","5 and −5"],["Negative number","A number less than zero"],["Greatest common factor (GCF)","The largest factor two numbers share"],["Least common multiple (LCM)","The smallest multiple two numbers share"],["Prime factorization","Writing a number as a product of primes","12 = 2 × 2 × 3"],["Exponent","A small raised number that tells how many times to use the base as a factor"],["Base","The number that is multiplied by itself in a power"],["Power","A number written with a base and an exponent"],["Square root","A number that, multiplied by itself, gives the original number"],["Scientific notation","Writing a number as a number from 1 to 10 times a power of 10","3.2 × 10^4"],["Term","A number, variable, or product in an expression"],["Coefficient","The number multiplied by a variable","In 7x, the coefficient is 7"],["Constant","A term that does not change; a plain number"],["Like terms","Terms with the same variable part","3x and 5x"],["Inequality","A statement that two values are not equal","x > 4"],["Solution","A value that makes an equation or inequality true"],["Slope","How steep a line is; rise over run"],["Y-intercept","The point where a graph crosses the y-axis"],["Function","A rule that gives exactly one output for each input"],["Input","The value you put into a function"],["Output","The value that comes out of a function"],["Linear equation","An equation whose graph is a straight line"],["Quadrant","One of the four regions of a coordinate plane"],["Origin","The point (0, 0) where the axes cross"],["Pythagorean theorem","In a right triangle, a² + b² = c²"],["Hypotenuse","The longest side of a right triangle, opposite the right angle"],["Legs","The two shorter sides of a right triangle"],["Circumference","The distance around a circle"],["Radius","The distance from the center of a circle to its edge"],["Diameter","The distance across a circle through its center; twice the radius"],["Pi (π)","The ratio of a circle's circumference to its diameter; about 3.14159"],["Surface area","The total area of all the faces of a solid"],["Prism","A solid with two identical parallel bases and rectangular sides"],["Pyramid","A solid with one base and triangular faces that meet at a point"],["Net","A flat pattern that folds up into a solid"],["Similar figures","Shapes with the same angles and proportional sides"],["Scale factor","The number you multiply lengths by to enlarge or shrink a figure"],["Transformation","A move that changes a figure's position or size"],["Translation","Slide a figure without turning or flipping it"],["Reflection","Flip a figure over a line"],["Rotation","Turn a figure around a point"],["Dilation","Enlarge or shrink a figure from a center point"],["Complementary angles","Two angles that add up to 90 degrees"],["Supplementary angles","Two angles that add up to 180 degrees"],["Vertical angles","Opposite angles formed by two crossing lines; they are equal"],["Adjacent angles","Two angles that share a side and a vertex"],["Transversal","A line that crosses two or more other lines"],["Mean","The average of a set of numbers"],["Median","The middle value of ordered data"],["Mode","The most frequent value"],["Range","Greatest value minus least value"],["Outlier","A value far from the rest of the data"],["Quartile","One of the three values that split ordered data into four equal parts"],["Interquartile range (IQR)","The third quartile minus the first quartile"],["Box plot","A graph that shows the median, quartiles and extremes of data"],["Histogram","A bar graph with no gaps that shows how data are spread across intervals"],["Probability","A number from 0 to 1 that tells how likely an event is"],["Sample space","The set of all possible outcomes"],["Compound event","An event made of two or more simple events"],["Variable","A letter that stands for a number that can change"],["Evaluate","Find the value of an expression by putting in numbers"],["Simplify","Rewrite an expression in its simplest form"],["Inverse operation","An operation that undoes another"],["Reciprocal","The number that multiplies with another to make 1","The reciprocal of 3/4 is 4/3"],["Terminating decimal","A decimal that ends","0.25"],["Repeating decimal","A decimal with a digit or group of digits that repeats forever","0.333..."],["Markup","An amount added to a cost to set the selling price"],["Discount","An amount subtracted from a price"],["Simple interest","Interest earned only on the starting amount: I = Prt"]]},{"source":"https://peabody.kids/flashcards/math-vocab-9-12","id":"math-vocab-9-12","title":"Math Words for High School","category":"math","subject":"Math","levels":["9-12"],"description":"High school vocabulary from algebra, geometry, trigonometry and precalculus.","tags":["vocabulary","high school"],"frontLabel":"Word","backLabel":"Meaning","cards":[["Domain","The set of all possible inputs of a function"],["Range (of a function)","The set of all possible outputs of a function"],["Relation","A set of ordered pairs"],["Inverse function","A function that undoes another function","f⁻¹(x)"],["Composition of functions","Using the output of one function as the input of another","f(g(x))"],["Polynomial","A sum of terms, each a number times a variable to a whole-number power"],["Degree (of a polynomial)","The highest exponent of the variable"],["Leading coefficient","The coefficient of the term with the highest power"],["Monomial","A polynomial with one term"],["Binomial","A polynomial with two terms"],["Trinomial","A polynomial with three terms"],["Root (zero)","An input that makes a function equal zero"],["Quadratic","A polynomial of degree 2","ax² + bx + c"],["Parabola","The U-shaped graph of a quadratic function"],["Vertex (of a parabola)","The highest or lowest point of the parabola"],["Axis of symmetry","The line that splits a parabola into mirror halves"],["Discriminant","b² − 4ac; tells how many real roots a quadratic has"],["Imaginary unit (i)","The number whose square is −1"],["Complex number","A number of the form a + bi"],["Conjugate","The binomial with the opposite sign in the middle","a + bi and a − bi"],["Asymptote","A line that a graph gets closer and closer to"],["Exponential function","A function where the variable is in the exponent","y = 2^x"],["Logarithm","The exponent you need on a base to get a number"],["Natural logarithm","A logarithm with base e, written ln"],["Euler's number (e)","An irrational constant, about 2.71828"],["Sequence","A list of numbers in order"],["Arithmetic sequence","A sequence with a constant difference between terms"],["Geometric sequence","A sequence with a constant ratio between terms"],["Series","The sum of the terms of a sequence"],["Sigma (Σ)","The symbol meaning add up a list of terms"],["Matrix","A rectangular array of numbers in rows and columns"],["System of equations","Two or more equations with the same variables"],["Linear programming","Finding the best value of a quantity under linear limits"],["Theorem","A statement that has been proven"],["Postulate (axiom)","A statement accepted as true without proof"],["Proof","A logical argument that shows a statement is true"],["Conjecture","A guess based on a pattern that has not been proven"],["Counterexample","An example that shows a statement is false"],["Converse","A conditional statement with its parts swapped","If Q, then P"],["Contrapositive","The converse of the inverse; it always has the same truth as the original statement"],["Inductive reasoning","Drawing a general rule from patterns"],["Deductive reasoning","Drawing a conclusion from facts that are known to be true"],["Congruent triangles","Triangles with all matching sides and angles equal"],["Locus","The set of all points that follow a given rule"],["Inscribed","Drawn inside another figure and touching it"],["Circumscribed","Drawn around another figure and touching it"],["Tangent (to a circle)","A line that touches a circle at exactly one point"],["Secant","A line that cuts through a circle at two points"],["Chord","A segment with both endpoints on a circle"],["Arc","A part of a circle's edge"],["Sector","A pie-slice region of a circle"],["Radian","An angle measure; the angle whose arc equals the radius"],["Sine","In a right triangle, opposite over hypotenuse"],["Cosine","In a right triangle, adjacent over hypotenuse"],["Tangent (trigonometry)","In a right triangle, opposite over adjacent"],["Unit circle","A circle of radius 1 centered at the origin"],["Amplitude","Half the distance between the highest and lowest values of a wave"],["Period","The length of one full cycle of a repeating graph"],["Limit","The value a function gets close to as the input approaches a number"],["Continuous","Having a graph with no breaks, jumps or holes"],["Derivative","The rate at which a function changes; the slope of its tangent"],["Integral","The accumulated total under a curve; area under a graph"],["Vector","A quantity with both size and direction"],["Scalar","A quantity with size only"],["Permutation","An arrangement of items in which order matters"],["Combination","A selection of items in which order does not matter"],["Standard deviation","A measure of how spread out data are around the mean"],["Normal distribution","A bell-shaped, symmetric distribution"],["Correlation","A measure of how two variables move together"],["Regression line","The line of best fit through a scatter plot"],["Residual","The difference between an actual value and the predicted value"],["Conic section","A curve from slicing a cone: circle, ellipse, parabola, hyperbola"],["Ellipse","An oval shape; the set of points whose distances to two foci add to a constant"],["Hyperbola","A curve with two branches; the set of points whose distances to two foci differ by a constant"],["Polar coordinates","Describing a point by distance from the origin and an angle"],["Rational function","A function that is a ratio of two polynomials"],["Extraneous solution","A solution that comes from the working but does not fit the original equation"]]},{"source":"https://peabody.kids/flashcards/math-vocab-k-2","id":"math-vocab-k-2","title":"Math Words for Grades K-2","category":"math","subject":"Math","levels":["k-2"],"description":"The first math words kids meet: counting, adding, shapes, comparing and measuring.","tags":["vocabulary","early math"],"frontLabel":"Word","backLabel":"Meaning","cards":[["Add","Put numbers together to find how many in all"],["Plus sign (+)","The sign that means add"],["Subtract","Take away to find how many are left"],["Minus sign (−)","The sign that means subtract"],["Equals (=)","Means the same amount as"],["Sum","The answer when you add","In 3 + 4 = 7, the sum is 7"],["Difference","The answer when you subtract","In 9 − 4 = 5, the difference is 5"],["Number","A word or symbol that tells how many"],["Digit","One of the ten symbols 0 to 9 used to write numbers"],["Zero","The number that means none"],["Even number","A number you can split into two equal groups with none left over","2, 4, 6, 8, 10"],["Odd number","A number that leaves one left over when you make pairs","1, 3, 5, 7, 9"],["Greater than (>)","Bigger than","The open side faces the bigger number"],["Less than (<)","Smaller than"],["Compare","Look at two things to see which is more, less, or the same"],["More","A larger amount"],["Fewer","A smaller number of things you can count"],["Same","Not different; equal"],["Order","Put numbers or things in a line by size or by rule"],["Count","Say numbers in order to find how many"],["Skip count","Count by jumping over numbers","2, 4, 6, 8"],["Number line","A line with numbers placed in order, used to count and add"],["Ones","The place on the right in a number","In 36, the 6 is in the ones place"],["Tens","The place next to the ones; each ten is a group of 10","In 36, the 3 is in the tens place"],["Hundreds","The place that counts groups of 100"],["Half","One of two equal parts"],["Fourth (quarter)","One of four equal parts"],["Whole","All of something, with nothing missing"],["Equal parts","Parts that are exactly the same size"],["Circle","A round shape with no corners"],["Triangle","A shape with 3 sides and 3 corners"],["Square","A shape with 4 equal sides and 4 square corners"],["Rectangle","A shape with 4 sides and 4 square corners"],["Side","A straight edge of a flat shape"],["Corner (vertex)","The point where two sides meet"],["Cube","A solid shape like a dice; 6 square faces"],["Sphere","A solid shape shaped like a ball"],["Cone","A solid shape like an ice cream cone"],["Cylinder","A solid shape like a can"],["Pattern","Something that repeats in the same way","red, blue, red, blue"],["Measure","Find out how long, tall, heavy or full something is"],["Length","How long something is"],["Height","How tall something is"],["Longer","Having more length"],["Shorter","Having less length"],["Heavier","Weighing more"],["Lighter","Weighing less"],["Inch","A small unit for length; about the width of a thumb"],["Foot","A unit of length equal to 12 inches"],["Penny","A coin worth 1 cent"],["Nickel","A coin worth 5 cents"],["Dime","A coin worth 10 cents"],["Quarter (coin)","A coin worth 25 cents"],["Hour hand","The short hand on a clock"],["Minute hand","The long hand on a clock"],["Graph","A picture that shows information so it is easy to compare"],["Tally mark","A mark used to keep count; four lines and a slash make five"],["Before","Comes earlier in order","4 comes before 5"],["After","Comes later in order","6 comes after 5"],["Between","In the middle of two things","5 is between 4 and 6"],["Double","Two times a number; add it to itself","Double 4 is 8"],["Number sentence","A math idea written with numbers and signs","2 + 3 = 5"],["Fact family","Add and subtract facts that use the same three numbers","2, 3, 5"],["Group","A set of things that belong together"],["Row","Things lined up side to side"],["Column","Things lined up top to bottom"]]},{"source":"https://peabody.kids/flashcards/math-word-problem-keywords","id":"math-word-problem-keywords","title":"Word Problem Keywords","category":"math","subject":"Math","levels":["k-2","3-5","6-8"],"description":"Phrases in word problems and the operation or symbol they usually suggest. Always read the whole problem, since keywords are clues and not rules.","tags":["word problems","operations"],"frontLabel":"Phrase","backLabel":"Usually means","cards":[["In all","Add"],["Altogether","Add"],["Total","Add"],["Sum","Add"],["Combined","Add"],["Plus","Add"],["More than","Add (but 'A is 3 more than B' means A = B + 3)"],["Increased by","Add"],["Gained","Add"],["Both","Add the two amounts"],["How many are left","Subtract"],["Remain","Subtract"],["Difference","Subtract"],["Fewer than","Subtract"],["Less than","Subtract (order is flipped: '5 less than 9' is 9 − 5)"],["Decreased by","Subtract"],["Take away","Subtract"],["How many more","Subtract to compare two amounts"],["Lost","Subtract"],["Gave away","Subtract"],["Minus","Subtract"],["Times","Multiply"],["Product","Multiply"],["Of (with a fraction or percent)","Multiply","1/2 of 10 is 1/2 × 10"],["Each (with a total needed)","Multiply"],["Every","Multiply or divide, depending on the question"],["Twice","Multiply by 2"],["Double","Multiply by 2"],["Triple","Multiply by 3"],["Per","Multiply or divide; it marks a rate","Miles per hour"],["At this rate","Multiply a rate by an amount"],["Groups of","Multiply"],["Rows of","Multiply"],["Quotient","Divide"],["Divided equally","Divide"],["Shared equally","Divide"],["Split among","Divide"],["Half of","Divide by 2"],["Average","Add the values, then divide by how many"],["Per person","Divide the total by the number of people"],["How many in each group","Divide"],["How many groups","Divide"],["Ratio of","Compare using division or a fraction"],["Percent of","Convert to a decimal and multiply"],["Is (as an equals sign)","="],["What is","The unknown value, a variable"],["At most","≤"],["At least","≥"],["No more than","≤"],["No less than","≥"],["More than (inequality)",">"],["Fewer than (inequality)","<"],["Is the same as","="],["Consecutive numbers","Numbers in order with no gaps, such as n, n + 1, n + 2"],["Consecutive even numbers","n, n + 2, n + 4"],["Perimeter","Add the lengths of all the sides"],["Area","Multiply (for a rectangle, length times width)"],["Step 1 of solving a word problem","Read it and decide what the question asks"],["Step 2 of solving a word problem","Pick out the numbers and units you need"],["Step 3 of solving a word problem","Choose the operation and write an equation"],["Step 4 of solving a word problem","Solve it, then check that the answer makes sense and has the right units"],["Extra information","A number in a problem you do not need"],["Multi-step problem","A problem needing two or more operations"]]}]