ACT Math
Articles in this topic. Browse the full ACT prep blog for every guide.
The midpoint formula on ACT Math
The midpoint is the average of the x-coordinates and the average of the y-coordinates.
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The Pythagorean theorem on ACT Math
In a right triangle, a squared plus b squared equals c squared.
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Vertical angles on ACT Math
Vertical angles are the opposite angles where two lines cross. They are equal.
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Arc length on ACT Math
Arc length is the fraction of the circumference given by the central angle.
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Area of a sector on ACT Math
Area of a sector is the fraction of the circle's area given by the central angle.
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Volume of a pyramid on ACT Math
Volume of a pyramid is one-third the base area times the height.
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Surface area of a cylinder on ACT Math
Surface area of a cylinder is two bases plus the wrapper: 2 pi r squared plus 2 pi r h.
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X and y intercepts on ACT Math
The x-intercept is where y equals 0. The y-intercept is where x equals 0.
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Change of base on ACT Math
log base b of a equals log a divided by log b.
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Log rules on ACT Math
Log of a product is a sum. Log of a quotient is a difference. Log of a power brings the exponent out front.
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The thirty-second first pass on ACT Math
On the first pass, take only the Math items you can finish in about thirty seconds.
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Mental math before Desmos on the ACT
One-step arithmetic stays in your head. Opening Desmos for 12 times 8 costs more than it saves.
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Don't rewrite every step on ACT Math
You do not get points for showing work. Stop writing when a click or a choice check answers it.
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When not to graph in Desmos
Do not graph a one-step calculation, a definition, or a pure counting item.
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Circles in Desmos on the ACT
Type the circle equation as written and click the center or a crossing.
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Inequalities in Desmos on the ACT
Type the inequality and read the shaded side.
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Piecewise functions in Desmos on the ACT
Graph each piece with its domain in braces, then click the x-value the item asked for.
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Check your work in Desmos
After you have an answer, graph the original equation or plug the value back in.
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Mean, median, and mode in Desmos
Type mean, median, or mode on a list in Desmos instead of adding by hand.
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67 seconds is an average on ACT Math
50 minutes for 45 questions is about 67 seconds each as a mean, not a timer on every item.
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The permutations formula on ACT Math
Permutations count ordered lists: nPr = n! / (n - r)!.
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Scatterplots on ACT Math
A scatterplot shows paired points. Read the trend and how tight the cloud is.
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Outliers on ACT Math
An outlier sits far from the rest of the data. It pulls the mean more than the median.
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IQR on ACT Math
IQR is Q3 minus Q1: the spread of the middle half of the data.
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Conditional probability on ACT Math
Conditional probability is given that. Restrict the sample space to the given group.
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Radical equations on ACT Math
Isolate the radical, square both sides, then check every solution in the original.
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Matrix determinant on ACT Math
For a 2 by 2 matrix, the determinant is ad minus bc.
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i squared on ACT Math
i squared equals negative one. Powers of i cycle every four.
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Amplitude and period on ACT Math
For y = A sin(Bx), amplitude is the absolute value of A and period is 2 pi over the absolute value of B.
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Reference angles on ACT Math
A reference angle is the acute angle to the x-axis. Then pick a sign from the quadrant.
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Scale factor on ACT Math
A scale factor k multiplies lengths. Area scales by k squared. Volume scales by k cubed.
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The hyperbola equation on ACT Math
A hyperbola is a difference of those squared terms. The minus is the tell.
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Area of a rectangle on ACT Math
Area of a rectangle is length times width.
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Area of a parallelogram on ACT Math
Area of a parallelogram is base times perpendicular height.
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Volume of a rectangular prism
Volume of a box is length times width times height.
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Triangle angle sum on ACT Math
The three interior angles of any triangle add to 180 degrees.
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Complementary and supplementary angles
Complementary angles add to 90. Supplementary angles add to 180.
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Corresponding and alternate interior angles
When a transversal cuts parallels, corresponding and alternate interior angles are equal.
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Radians to degrees on ACT Math
Degrees equal radians times 180 over pi.
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Cofunction identities on ACT Math
Sine of 90 degrees minus theta equals cosine of theta.
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Combined work problems on ACT Math
Add what each person does in one hour, then invert to get the time together.
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Successive percent change on ACT Math
Two percent changes in a row multiply. They do not add.
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Reread the last sentence on ACT Math
Solve for x, then reread. The item may have asked for area or 2x.
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The sixty-second skip on ACT Math
If a Math item is still a wall after about a minute, flag it and move.
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First-half sprint on ACT Math
Bank the early Math items. Those points fund the harder ones later.
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Plug the answer back in on ACT Math
When you have a value, put it back in the original sentence or equation.
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If you see x and y in Desmos
If the item shows both x and y, graph first.
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Rewrite in terms of x in Desmos
A letter other than x becomes a slider. Rename it before you graph.
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LCM, GCF, and mod in Desmos
Desmos has lcm, gcd, and mod. Type the function instead of listing factors.
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Percent calculations in Desmos
Type percents as decimals or with the percent sign.
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Find a vertex in Desmos
Graph the quadratic and click the turning point.
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Find x-intercepts in Desmos
Graph the function and click where it meets the x-axis.
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Set the window in Desmos
Set the axis ranges so the useful part of the graph is on screen.
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Exponentials in Desmos on the ACT
Graph the exponential and click the point the item asked for.
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Logs in Desmos on the ACT
Type log(x) for common log and ln(x) for natural log.
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The discriminant on ACT Math
The discriminant is b squared minus 4ac. Its sign counts the real roots.
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Absolute value inequalities on ACT Math
Absolute value less than k is a strip. Greater than k is two rays.
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Distance equals rate times time
If you have any two of distance, rate, and time, you have the third.
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Arithmetic series sum on ACT Math
The sum is n times the average of the first and last terms.
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The combinations formula on ACT Math
Combinations count unordered groups: nCr = n! / (r!(n − r)!).
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Mutually exclusive events on ACT Math
Mutually exclusive events cannot both happen. Add their probabilities.
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Histograms on ACT Math
A histogram shows counts in bins. The height is how many values fell in that interval.
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Two-way tables on ACT Math
Read the row, the column, and whether the question wants a cell, a total, or a probability.
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The triangle inequality on ACT Math
Any two sides of a triangle must add to more than the third.
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The exterior-angle theorem on ACT Math
An exterior angle equals the sum of the two remote interior angles.
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Volume of a sphere on ACT Math
Volume of a sphere is four-thirds pi r cubed.
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Volume of a cone on ACT Math
Volume of a cone is one-third pi r squared times height.
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Area of a triangle on ACT Math
Area of a triangle is one-half base times height. The height must be perpendicular to the base.
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Area of a trapezoid on ACT Math
Area of a trapezoid is the average of the two parallel sides times the height.
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Circle circumference and area
Area is pi r squared. Circumference is 2 pi r, or pi times the diameter.
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The equation of a circle on ACT Math
A circle with center (h, k) and radius r is (x minus h) squared plus (y minus k) squared equals r squared.
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Volume of a cylinder on ACT Math
Volume of a cylinder is pi r squared h. That is the area of the circular base times the height.
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The distance formula on ACT Math
Distance between two points is the square root of the sum of the squared coordinate gaps.
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Arithmetic sequences on ACT Math
An arithmetic sequence adds the same number each time. The nth term is a1 plus (n minus 1)d.
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Geometric sequences on ACT Math
A geometric sequence multiplies by the same number each time. The nth term is a1 times r to the (n minus 1).
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30-60-90 triangles on ACT Math
In a 30-60-90 triangle the sides are x, x root 3, and 2x.
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45-45-90 triangles on ACT Math
In a 45-45-90 triangle the legs are equal and the hypotenuse is a leg times root 2.
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Find intersections in Desmos
Graph both sides, then click the intersection. That point is the solution.
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Domain restrictions in Desmos
To graph a piece of a function in Desmos, type the equation, then restrict x with curly braces.
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Plot a point in Desmos
To plot a point in Desmos, type (x, y) on its own line, like (3, -2).
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Sliders in Desmos on the ACT
If Desmos offers a slider for a letter you typed, you can drag a parameter and watch the graph move.
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Absolute value graphs in Desmos
Graph y = abs(f(x)) in Desmos. The V shape and the vertex are on the screen.
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FOIL on ACT Math
FOIL multiplies two binomials: first, outer, inner, last. Then combine like terms.
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Average speed on ACT Math
Average speed is total distance over total time. Do not average the two speeds.
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Independent events on ACT Math
If two events do not affect each other, multiply their probabilities.
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Vertex form on ACT Math
Vertex form is y = a(x minus h) squared plus k. The vertex is (h, k).
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The ellipse equation on ACT Math
An ellipse centered at (h, k) is (x minus h) squared over a squared plus (y minus k) squared over b squared equals 1.
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Space diagonal on ACT Math
The space diagonal of a rectangular prism is the square root of l squared plus w squared plus h squared.
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Law of cosines on ACT Math
The law of cosines is c squared equals a squared plus b squared minus 2ab cos C. Use it when the triangle is not right.
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Reciprocal trig identities on the ACT
Cosecant is 1 over sine. Secant is 1 over cosine. Cotangent is 1 over tangent.
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Trig values to memorize for the ACT
Memorize sine, cosine, and tangent at 0, 30, 45, 60, and 90 degrees.
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Inverse functions on ACT Math
Inverse functions undo each other. If f(a) equals b, then f inverse of b equals a.
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Exponential growth and decay on ACT Math
Exponential growth and decay look like y = a(1 plus or minus r) to the t, or y = a b to the t.
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SOH-CAH-TOA on ACT Math
Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent.
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Completing the square on ACT Math
Rewrite x squared plus bx as a perfect square minus a constant so you can read a vertex or a circle center.
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The quadratic formula on ACT Math
For ax squared plus bx plus c equals 0, x is negative b plus or minus the square root of b squared minus 4ac, all over 2a.
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Difference of squares on ACT Math
a squared minus b squared factors as (a minus b)(a plus b). Two squares, a minus, no extra middle term.
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Slope-intercept form on ACT Math
y equals mx plus b. The m is the slope. The b is where the line hits the y-axis.
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Point-slope form on ACT Math
y minus y1 equals m times (x minus x1). You need one point and the slope.
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Pythagorean triples on ACT Math
Memorize 3-4-5, 5-12-13, and 8-15-17, plus their multiples, so you can skip squaring from scratch.
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sin squared plus cos squared on the ACT
sin squared theta plus cos squared theta equals 1, and tangent is sine over cosine.
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Expected value on ACT Math
Expected value is each outcome times its probability, added together. It is a weighted average of possible results.
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Simple interest on ACT Math
I equals P r t. Interest equals principal times rate times time, with the rate as a decimal and time in years.
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Evaluate a function at a point in Desmos
To evaluate f(3), type the function, then type the function name at 3, or drop 3 into a table.
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Graph y equals a constant in Desmos
Graph y equals k as a horizontal line and click where it meets your curve. Those x-values are the solutions.
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Trig graphs in Desmos on the ACT
Set degrees or radians before you graph sine or cosine. Then read amplitude, intercepts, or a value off the curve.
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Statistics functions in Desmos
Put the data in a table, then ask Desmos for mean, median, or a regression. Do not retype a long average by hand.
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Box plots on ACT Math
A box plot shows the median, the quartiles, and the spread. The line inside the box is the median, not the mean.
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Scatterplots and correlation on ACT Math
A scatterplot plots pairs of numbers. Correlation describes the direction. Correlation is not causation.
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Compound interest on ACT Math
Compound interest adds interest to the balance, then charges interest on that new total. Simple interest does not.
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Law of sines on ACT Math
Each side sits over the sine of the angle opposite it. Use it when the triangle is not a right triangle.
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Surface area on ACT Math
Surface area is the total area of the faces. Volume is the space inside. Do not swap the formulas.
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Consecutive integers on ACT Math
Consecutive integers are n, n plus 1, n plus 2. Consecutive evens or odds go up by 2.
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Remainders on ACT Math
A remainder is what is left after division into whole groups. Keep it smaller than the divisor.
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Order of operations on ACT Math
Parentheses, then exponents, then multiply and divide left to right, then add and subtract left to right.
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Proportions on ACT Math
A proportion is two equal ratios. Cross-multiply to solve. a over b equals c over d means ad equals bc.
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Direct and inverse variation on ACT Math
Direct variation is y equals kx. Inverse variation is y equals k over x. Find k from one pair, then use it on the next.
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Isosceles and equilateral triangles
Isosceles has two equal sides and two equal base angles. Equilateral has three equal sides and three 60 degree angles.
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Slope on ACT Math
Slope is the change in y divided by the change in x. Parallel lines share it. Perpendicular slopes multiply to -1.
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Midpoint and distance on ACT Math
The midpoint averages the coordinates. Distance is the straight-line length between the two points.
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Pythagorean theorem and special right triangles
On a right triangle, a squared plus b squared equals c squared. The special triangles have fixed side ratios.
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Parallel and perpendicular lines on ACT Math
Parallel lines have the same slope. Perpendicular slopes flip the fraction and change the sign.
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Percent change on ACT Math
Percent change divides the difference by the original amount. That is not the same as percent of.
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Weighted averages on ACT Math
Multiply each value by how many times it counts, then divide by the total count.
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When not to use Desmos on the ACT
Open Desmos for graphs and systems. Skip it for arithmetic you can do in your head.
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Systems of equations in Desmos
Type each equation and read the intersection. Those coordinates are the solution.
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Quadratics in Desmos on the ACT
Graph the quadratic and read the roots and the vertex from the curve.
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Regression in Desmos on the ACT
Put the points in a table, add a regression, and read the line of best fit.
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Radians vs degrees in Desmos
Check the wrench before any trig graph. The wrong angle mode draws a confident wrong curve.
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What Desmos features are off on the ACT
The test build graphs, tables, and regresses. It turns off notes, images, and saved folders.
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Graph a circle in Desmos
Type the circle equation and read the center and radius from the graph.
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Shade inequalities in Desmos
Type the inequality and read which side is shaded, and whether the boundary is solid.
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Fraction answers in Desmos
If the choices are fractions, show the Desmos result as a fraction before you compare.
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Zoom when a Desmos graph is missing
An empty grid usually means the curve is off the window. Zoom out before you decide there is no graph.
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Use Desmos to check, not to start
Solve first when you know the method. Use the graph to confirm the number.
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Desmos tables on the ACT
A table evaluates a function at the x-values you care about, instead of plugging each one in by hand.
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Practice Desmos the way the ACT locks it
Rehearse in a blank window. Notes and saved graphs from a personal account will not be on the test.
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Four answer choices on ACT Math
Enhanced ACT Math has four choices. One elimination is a real lift, and a blank is the only wasted shot.
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Factors, multiples, LCM, and GCF
Factors divide the number. Multiples stretch it. The GCF is the largest shared factor. The LCM is the smallest shared multiple.
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Odd, even, positive, and negative
Odd and even follow rules under addition. The sign of a product depends on how many negatives you multiply.
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Decimals and place value on ACT Math
Tenths and hundredths are not the same place. Line up the decimal before you add.
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The counting principle on ACT Math
If both stages happen, multiply the ways. If you are choosing one path or the other, add.
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Permutations vs combinations
Order matters for a lineup. Order does not matter for a committee. Divide only in the second case.
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Arc length and sector area
Multiply the fraction of the circle by the circumference for an arc, or by the area for a sector.
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Inscribed vs central angles
A central angle equals its arc. An inscribed angle is half that arc.
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Venn diagrams on ACT Math
Add the two groups and subtract the overlap. The people in both circles are not a third group.
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Mixture problems on ACT Math
Each amount times its strength, added, equals the total times the final strength.
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Unit conversions on ACT Math
Convert everything into one unit before you calculate. Minutes and hours in the same product will not balance.
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Function composition on ACT Math
f(g(x)) means do g first, then put that result into f. The reverse is a different value.
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Piecewise functions on ACT Math
Use the formula whose interval actually contains the input. Check the endpoint when the inequality is strict.
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Backsolving on ACT Math
Backsolving means plugging the four answer choices into the question until one works. Start in the middle. It is not plug-in-numbers.
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How many Math questions can you miss for a 30?
On a typical enhanced Math section (45 items), a 30 usually allows a small handful of misses. Use that form's conversion table, not a fixed number.
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The most common ACT Math mistakes (and how to avoid them)
Most lost Math points come from careless errors, not missing knowledge. Here are the mistakes that cost students the most and how to stop making them.
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How to plug in numbers and backsolve on ACT Math
When the algebra feels heavy, plug in your own numbers or test the answer choices. Here's how these two strategies work and when to use each.
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How to improve your ACT Math score
Focus on high-frequency topics, memorize the formulas the ACT doesn't give you, and drill timed sets with good review.
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What math topics are on the ACT?
ACT Math covers pre-algebra, algebra, geometry, and some trig, with algebra and geometry making up most of the section.
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What formulas should you memorize for ACT Math?
The ACT gives no formula sheet, so you'll want area, volume, slope, distance, the quadratic formula, and basic trig ratios memorized.
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How to get a 36 on ACT Math
Close every content gap, tackle the hardest end-of-section problems, and cut careless errors to reach a perfect Math score.
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What are the hardest ACT Math topics?
Advanced functions, trig, multi-step word problems, and rare topics like matrices and logs trip up a lot of students. Here's how to tackle them.
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How much geometry is on the ACT?
Geometry is about a third of ACT Math, covering plane and coordinate geometry. Here's what's tested and which formulas to memorize.
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How much algebra is on the ACT?
Algebra is the biggest part of ACT Math, over half the section. Here's the pre-algebra through intermediate algebra breakdown.
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- How to solve linear equations on the ACT
- How to solve systems of equations on the ACT
- How to solve quadratic equations on the ACT
- How to factor quadratics on the ACT
- Exponent rules you need for ACT Math
- Inequalities on ACT Math
- Polynomials on ACT Math
- Absolute value on ACT Math
- Rational expressions on ACT Math
- Radical Expressions on ACT Math
What calculator should you use on the ACT?
The best ACT calculator is one you know well and that's on the approved list, often the TI-84. Check the policy and avoid banned CAS models.
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ACT Math functions explained
Functions are about 17 to 20 percent of ACT Math. Here is what the category covers, why students lose points, and the four skills to learn first.
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Statistics and probability on ACT Math
This area is about 12 to 15 percent of ACT Math and the math is light. Here is what it covers and the three skills worth locking down first.
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ACT Math practice tests: what to use and what to avoid
Official 45-question sections, topic-based review loops, and quality checks—Math-specific practice guidance.
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