How to learn math
Updated: Oct 27, 2022
Most students will learn the same math from schools, and there are not much new things about math. However, it requires both a good understanding of basic concepts and lots of practices (such as drills).
WikiHow has many articles talking about math, but each standalone article can't form a solid and comprehensive solution. Therefore, we have rewritten and integrated several articles together to create a better version. However, this is still more like a summary of math instead of teaching math.
For students with good basic math skills, the focus should be on math competitions, such as Home (mathleague.com).
Mathematics (from Ancient Greek μάθημα; máthēma: "knowledge, study, learning") is an area of knowledge that includes such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).
Math is a fundamental subject and is the foundation for many other subjects. Math is especially useful for your personal financial management. For primary school, arithmetic is the fundamental knowledge, there are also some simple algebra and geometry. For secondary school, the focus is algebra and geometry, there are also knowledge about trigonometry. For college and university, it's primarily linear algebra, calculus and analysis.
Here we quote the best way to learn math provided by wikiHow, a wiki that is building the world's largest and highest quality how-to manual. Please edit the articles and find author credits at the original wikiHow articles on How to Learn Math, How to Understand Math, How to Learn Algebra, How to Get an "A" in Geometry, How to Learn Trigonometry, How to Understand Calculus. Content on wikiHow can be shared under a Creative Commons License. We also recommend a good math website for learning math: Math Worksheets Center- Free Printable Math Worksheets and Lessons.
You can learn math both inside and outside of the classroom, and it doesn’t have to be stressful or overwhelming! Once you have a good grasp of the basics, learning the more complex stuff will feel a lot easier. This article will teach you those basics (addition, subtraction, multiplication, and division) and also give you strategies you can use in and out of the classroom to help you better learn math.
Part 1: Keys to Being a Good Math Student
Show up for class. When you miss class, you have to learn the concepts either from a classmate or from your textbook. You'll never get as good of an overview from your friends or from the text as you will from your teacher and always pay attention.
Work along with your teacher. If your teacher works problems at the front of your class, then work along with the teacher in your notebook.
Do your homework the same day as it's assigned. When you do the homework the same day, the concepts are fresh on your mind. Sometimes, finishing your homework the same day isn't possible. Just make sure that your homework is complete before you go to class.
Make an effort outside of class if you need help. Go to your teacher during his or her free period or during office hours.
Part 2: Learning Math in School - Arithmetic (Primary School)
Start with arithmetic. In most schools, students work on arithmetic during the elementary grades. Arithmetic includes the fundamentals of addition, subtraction, multiplication and division. Work on drills. Doing a lot of arithmetic problems again and again is the best way to get the fundamentals down pat. Look for software that will give you lots of different math problems to work on. Also, look for timed drills to increase your speed. Repetition is the basis of math. The concept has to be not only learned, but put to work for you to remember it!
Learning Addition and Subtraction. Use objects to count out the math visually. Practice your addition and subtraction tables. Start by adding the final digits of both numbers together. Play games with simple math. Add larger numbers. Learn about regrouping 1s into the 10s place, 10s into the 100s place, etc. Subtract with borrowing.
Learning Multiplication and Division. Understand the concept of multiplication and division. The idea behind multiplication is to shortcut adding the same things over and over again. You use multiplication when you have groups of the same size. The idea behind division is splitting equal parts into groups. Learn the properties of multiplication. For example, order doesn’t matter when you multiply. Practice your multiplication and division tables. Multiply and regroup the columns. Think of division as the opposite of multiplication. Account for remainders.
Part 3: Learning Math in School - Progress to Pre-algebra (Primary School)
This course will provide the building blocks that you'll need to solve algebra problems later on.
Learn about fractions and decimals. You'll learn to add, subtract, multiply and divide both fractions and decimals. Regarding fractions, you'll learn how to reduce fractions and interpret mixed numbers. Regarding decimals, you'll understand place value, and you'll be able to use decimals in word problems.
Study ratios, proportions and percentages. These concepts will help you to learn about making comparisons.
Solve squares and square roots. When you've mastered this topic, you'll have perfect squares of many numbers memorized. You'll also be able to work with equations containing square roots.
Introduce yourself to basic geometry. You'll learn all of the shapes as well as 3D concepts. You'll also learn concepts like area, perimeter, volume and surface area, as well as information about parallel and perpendicular lines and angles.
Understand some basic statistics. In pre-algebra, your introduction to statistics mostly includes visuals like graphs, scatter plots, stem-and-leaf plots and histograms.
Learn algebra basics. These will include concepts like solving simple equations containing variables, learning about properties like the distributive property, graphing simple equations and solving inequalities.
Part 4: Learning Math in School - Advance to Algebra I (Secondary School)
Solve linear equations and inequalities that contain 1-2 variables. You'll learn how to solve these problems not only on paper, but sometimes on a calculator as well.
Tackle word problems. You'll be surprised how many everyday problems that you'll face in your future involve the ability to solve algebraic word problems. For example, you'll use algebra to figure out the interest rate that you earn on your bank account or on your investments. You can also use algebra to figure out how long you'll have to travel based on the speed of your car.
Work with exponents. When you start solving equations with polynomials (expressions containing both numbers and variables), you'll have to understand how to use exponents. This may also include working with scientific notation. Once you have exponents down, you can learn to add, subtract, multiply and divide polynomial expressions.
Understand functions and graphs. In algebra, you'll really get into graphic equations. You'll learn how to calculate the slope of a line, how to put equations into point-slope form, and how to calculate the x- and y-intercepts of a line using slope-intercept form.
Figure out systems of equations. Sometimes, you're given 2 separate equations with both x and y variables, and you have to solve for x or y for both equations. Fortunately, you'll learn many tricks for solving these equation including graphing, substitution and addition.
Part 5: How to Learn Algebra (Secondary School)
Review your basic math operations. To start learning algebra, you'll need to know basic math skills such as adding, subtracting, multiplying and dividing. This primary/elementary school math is essential before you start learning algebra.
Know the order of operations. One of the trickiest things about solving an algebra equation as a beginner is knowing where to start. Luckily, there's a specific order for solving these problems: first do any math operations in parentheses, then do exponents, then multiply, then divide, then add, and finally subtract. A handy tool for remembering this order of operations is the acronym PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction).
Know how to use negative numbers. On a number line, a negative version of a number is the same distance from zero as the positive, but in the opposite direction. Adding two negative numbers together makes the number more negative (in other words, the digits will be higher, but since the number is negative, it counts as being lower). Two negative signs cancel out — subtracting a negative number is the same as adding a positive number. Multiplying or dividing two negative numbers gives a positive answer. Multiplying or dividing a positive number and a negative number gives a negative answer.
Know how to keep long problems organized. While simple algebra problems can be a snap to solve, more complicated problems can take many, many steps. To avoid errors, keep your work organized by starting a new line every time you make a step toward solving your problem. If you're dealing with a two-sided equation, try to write all the equals signs ("="s) underneath each other. This way, if you make a mistake somewhere, it'll be much easier to find and correct.
Understanding Variables. Look for symbols that aren't numbers. In algebra, you'll start to see letters and symbols appear in your math problems, rather than just numbers. These are called variables. Variables aren't as confusing as they may first seem - they're just ways of showing numbers with unknown values. Think of variables as "unknown" numbers. As mentioned above, variables are basically just numbers with unknown values. In other words, there's some number that can go in place of the variable to make the equation work. Usually, your goal in an algebra problem is to figure out what the variable is — think of it as a "mystery number" that you're trying to discover. Watch for recurring variables. If a variable appears more than once, simplify the variables. What do you do if the same variable appears more than once in the equation? Though this situation may seem tricky to solve, you can actually treat variables how you'd treat normal numbers — in other words, you can add them, subtract them, and so on as long as you only combine variables that are alike. In other words, x + x = 2x, but x + y doesn't equal 2xy.
Learning to Solve Equations by "Canceling". Try to get the variable by itself in algebra equations. Solving an equation in algebra usually means finding out what the variable is. Algebra equations are usually set up with numbers and/or variables on both sides, like this: x + 2 = 9 × 4. To figure out what the variable is, you need to get it by itself on one side of the equals sign. Whatever is left on the other side of the equals sign is your answer. In the example (x + 2 = 9 × 4), to get x by itself on the left side of the equation, we need to get rid of the "+ 2". To do this, we'll simply subtract 2 from that side, leaving us with x = 9 × 4. However, to keep both sides of the equation equal, we also need to subtract 2 from the other side. This leaves us with x = 9 × 4 - 2. Following the order of operations, we multiply first, then subtract, giving us an answer of x = 36 - 2 = 34. Cancel addition with subtraction (and vice versa). As we just saw above, getting x by itself on one side of the equals sign usually means getting rid of the numbers next to it. To do this, we perform the "opposite" operation on both sides of the equation. For instance, in the equation x + 3 = 0, since we see a "+ 3" next to our x, we'll put a "- 3" on both sides. The "+ 3" and "- 3", leaving x by itself and "-3" on the other side of the equals sign, like this: x = -3. Cancel multiplication with division (and vice versa). Multiplication and division are a little harder to work with than addition and subtraction, but they have the same "opposite" relationship. If you see a "× 3" on one side, you'll cancel it by dividing both sides by 3, and so on. Cancel exponents by taking the root (and vice versa). Exponents are a fairly advanced pre-algebra topic — if you don't know how to do them, see our basic exponent article for more information. The "opposite" of an exponent is the root that has the same number as it. For example, the opposite of the 2 exponent is a square root (√), the opposite of the 3 exponent is the cube root (3√), and so on.
Sharpening Your Algebra Skills. Use pictures to make problems clearer. If you're having a hard time visualizing an algebra problem, try using diagrams or pictures to illustrate your equation. You can even try using a group of physical objects (like blocks or coins) instead if you have some handy. Use "common sense checks" (especially for word problems). When converting a word problem into algebra, try to check your formula by plugging in simple values for your variable. Does your equation make sense when x=0? When x=1? When x = -1? It's easy to make simple mistakes by writing down p=6d when you mean p=d/6, but these are easily caught if you do a quick sanity check on your work before going further. For example, let's say we're told that a football field is 30 yards (27.4 m) longer than it is wide. We use the equation l = w + 30 to represent this. We can test whether this equation makes sense by plugging in simple values for w. For instance, if the field is w = 10 yards (9.1 m) wide, it will be 10 + 30 = 40 yards (36.6 m) long. If it's 30 yards (27.4 m) wide, it will be 30 + 30 = 60 yards (54.9 m) long, and so on. This makes sense — we'd expect the field to get longer as it gets wider, so this equation is reasonable. Be aware that answers won't always be integers in algebra. Answers in algebra and other advanced forms of math aren't always round, easy numbers. They can often be decimals, fractions, or irrational numbers. A calculator can help you find these complicated answers, but keep in mind that your teacher may require you to give your answer in its exact form, not in an unwieldy decimal.
Try expanding your skill. When you're confident with basic algebra, try factoring. One of the trickiest algebra skills of all is factoring — a sort of shortcut for getting complex equations into simple forms. Factoring is a semi-advanced algebra topic, so consider consulting the article linked above if you're having trouble mastering it. Below are just a few quick tips for factoring equations: Equations with the form ax + ba factor to a(x + b). Example: 2x + 4 = 2(x + 2). Equations with the form ax2 + bx factor to cx((a/c)x + (b/c)) where c is the biggest number that divides into a and b evenly. Example: 3y2 + 12y = 3y(y + 4). Equations with the form x2 + bx + c factor to (x + y)(x + z) where y × z = c and yx + zx = bx. Example: x2 + 4x + 3 = (x + 3)(x + 1).
Practice, practice, practice! Progress in algebra (and any other kind of math) requires lots of hard work and repetition. Don't worry — by paying attention in class, doing all of your assignments, and seeking out help from your teacher or other students when you need it, algebra will begin to become second nature.
Ask your teacher to help you understand tricky algebra topics. If you're having a hard time getting the hang of algebra, don't worry — you don't have to learn it on your own. Your teacher is the first person you should turn to with questions. After class, politely ask your teacher for help. Good teachers will usually be willing to re-explain the day's topic at an after-school appointment and may even be able to give you extra practice materials.
Exploring Intermediate Topics. Learn to graph x/y equations. Graphs can be valuable tools in algebra because they allow you to display ideas that you'd usually need numbers for in easy-to-understand pictures. Usually, in beginning algebra, graphing problems are restricted to equations with two variables (usually x and y) and are done on a simple 2-D graph with an x axis and a y axis. With these equations, all you need to do is plug in a value for x, then solve for y (or do the reverse) to get two numbers that correspond to a point on the graph. Learn to solve inequalities. What do you do when your equation doesn't use an equals sign? Nothing much different than what you'd normally do, it turns out. For inequalities, which use signs like > ("greater than") and < ("less than"), just solve as normal. You'll be left with an answer that's either less than or greater than your variable. Tackle quadratic equations. One algebra topic that many beginners struggle with is solving quadratic equations. Quadratics are equations with the form ax2 + bx + c = 0, where a, b, and c are numbers (except that a can't be 0.) These equations are solved with the formula x = [-b +/- √(b2 - 4ac)]/2a . Be careful — the +/- sign means you need to find the answers for adding and subtracting, so you can have two answers for these types of problems. Experiment with systems of equations. Solving more than one equation at once may sound super-tricky, but when you're working with simple algebra equations, it's not actually that hard. Often, algebra teachers use a graphing approach for solving these problems. When you're working with a system of two equations, the solutions are the points on a graph that the lines for both equations cross at.
Part 6: How to Learn Geometry. (Secondary School)
In geometry, you'll learn about the properties of lines, segments, angles and shapes. You'll memorize a number of theorems and corollaries that will help you to understand the rules of geometry. You'll learn how to calculate the area of a circle, how to use the Pythagorean theorem and how to figure out relationships between angles and sides of special triangles. You'll see a lot of geometry on future standardized tests like the SAT, the ACT and the GRE.
Learning the Concepts of Geometry. Know Euclid’s five postulates of geometry. Geometry is founded upon the basis of five postulates put together by the ancient mathematician, Euclid. Knowing and understanding these five statements will help you understand many of the concepts in geometry. 1: A straight line segment can be drawn joining any two points. 2: Any straight line segment can be continued in either direction indefinitely in a straight line. 3. A circle can be drawn around any line segment with one end of the line segment serving as the center point and the length of the line segment serving as the radius of the circle. 4. All right angles are congruent (equal). 5. Given a single line and a single point, only one line can be drawn directly through the point that will be parallel to the first line.
Recognize the symbols used in geometry problems. When you first start learning geometry, the various symbols can seem overwhelming. Learning what each of them means and being able to immediately recognize them will make things easier. Here are some of the most common geometry symbols you will come across: A small triangle refers to the properties of a triangle. A small angle shape refers to the properties of an angle. Letters with a line over them refer to the properties of a line segment. Letters with a line over them with arrows at each end refer to the properties of a line. One horizontal line with a vertical line in the middle means that two lines are perpendicular to each other. Two vertical lines mean two lines are parallel to each other. An equal sign with a squiggly line on top means that two shapes are congruent. A squiggly line means that two shapes are similar. Three dots forming a triangle means "therefore".
Understand the properties of lines. A line is straight and extends infinitely in both directions. Lines are drawn with an arrow at the end to indicate that they continue on. A line segment has a beginning and an end point. Another form of a line is called a ray: it only extends infinitely in one direction. Lines can be parallel, perpendicular, or intersecting. When two lines are parallel they never intersect with each other. Perpendicular lines are two lines that form a 90° angle. Intersecting lines are two lines that cross each other. Intersecting lines can be perpendicular, but can never be parallel.
Know the different types of angles. There are three different types of angles: obtuse, acute, and right. An obtuse angle is one that measures greater than 90°, an acute angle is one that measures less than 90°, and a right angle is one that measures exactly 90°. Being able to identify angles is an important part of geometry. A 90° angle is also a perpendicular angle: the lines make a perfect corner.
Understand the Pythagorean Theorem. The Pythagorean Theorem states that a^2 + b^2 = c^2. It is the formula that allows you to calculate the length of the side of a right triangle if you know the lengths of the other two sides. A right triangle is a triangle with one 90° angle. In the theorem, a and b are the opposite and adjacent (straight) sides of the triangle, while c is the hypotenuse (angled line) of the triangle.
Be able to identify the types of triangles. There are three different types of triangles: scalene, isosceles, and equilateral. A scalene triangle has no congruent (identical) sides and no congruent angles. An isosceles triangle has, at least, two congruent sides and two congruent angles. An equilateral triangle has three identical sides and three identical angles. Knowing these types of triangles helps you identify properties and postulates associated with them. Remember, that an equilateral triangle is technically also an isosceles triangle, because it does have two congruent sides. All equilateral triangles are isosceles, but not all isosceles triangles are equilateral. Triangles can also be classified by their angles: acute, right, and obtuse. Acute triangles have angles that are all less than 90°; right triangles have one 90° angle; obtuse triangles have one angle that is greater than 90°.
Know the difference between similar and congruent shapes. Similar shapes are those that have identical corresponding angles and corresponding sides that are proportionally smaller or larger than each other. In other words, the polygon will have the same angles, but different side lengths. Congruent shapes are identical; they are the same shape and size. Corresponding angles are identical angles in two shapes. In a right triangle, the 90-degree angles in both triangles are corresponding. The shapes do not have to be the same size for their angles to be corresponding.
Learn about complementary and supplementary angles. Complementary angles are those angles which add together to make 90 degrees and supplementary angles add to 180 degrees. Remember that vertical angles are always congruent; similarly, alternate interior and alternate exterior angles are also always congruent. Right angles are 90 degrees, while straight angles are 180. Vertical angles are the two angles formed by two intersecting lines that are directly opposite each other. Alternate interior angles are formed when two lines intersect a third line. They are on opposite sides of the line they both intersect, but on the inside of each individual line. Alternate exterior angles are also formed when two lines intersect a third line; they are on opposite sides of the line they both intersect, but on the outside of each individual line.
Remember SOHCAHTOA. SOHCAHTOA is a mnemonic device used to remember the formulas for sine, cosine, and tangent in a right triangle. When you want to find the sine, cosine, or tangent of an angle, you use the following formulas: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, and Tangent = Opposite/Adjacent. For example: Find the sine, cosine, and tangent of the 39° angle of a right triangle with side AB = 3, BC = 5 and AC = 4. sin(39°) = opposite/hypotenuse = 3/5 = 0.6; cos(39°) = adjacent/hypotenuse = 4/5 = 0.8; tan(39°) = opposite/adjacent = 3/4 = 0.75
Writing a 2-Column Proof. Draw a diagram after reading the problem. Sometimes the problem will be provided without an image and you will have to diagram it yourself to visualize the proof. Once you have a rough sketch that matches the givens in a problem, you might need to re-draw the diagram so that you can read everything clearly and the angles are approximately correct. Make some observations about your diagram. Label right angles and equal lengths. If lines are parallel to each other, mark that down as well. If the problem does not explicitly state two lines are equal, can you prove that they are? Make sure you can prove all of your assumptions. Work the proof backwards. When you are proving something in geometry, you are given some statements about the shapes and angles, then asked to prove why these statements are true. Sometimes the easiest way to do this is to start with the end of the problem. Make a 2-column grid labeled with statements and reasons. In order to make a solid proof, you have to make a statement and then give the geometric reason that proves the truth of that statement. Underneath the statement column, you will write a statement such as angle ABC = angle DEF. Under the reason, you will write the proof for this. If it is given, simply write given, otherwise, write the theorem that proves it. Determine which theorems apply to your proof. There are many individual theorems in geometry that can be used for your proof. There are many properties of triangles, intersecting and parallel lines, and circles that are the basis for these theorems. Determine what geometric shapes you are working with and find the ones that apply to your proof. Reference previous proofs to see if there are similarities. There are too many theorems to list, but here are a few of the most important ones for triangles: CPCTC: corresponding parts of the congruent triangle are congruent; SSS: side-side-side: if three sides of one triangle are congruent to three sides of a second triangle, then the triangles are congruent; SAS: side-angle-side: if two triangles have a congruent side-angle-side, then the two triangles are congruent; ASA: angle-side-angle: if two triangles have a congruent angle-side-angle, then the two triangles are congruent; AAA: angle-angle-angle: triangles with congruent angles are similar, but not necessarily congruent. Make sure your steps flow in a logical fashion. Write down a quick sketch of your proof outline. Write down the reasons for each step. Add the given statements where they belong, not just all at once in the beginning. Re-order the steps if necessary. The more proofs you do, the easier it will be to order the steps properly. Write down the conclusion as the last line. The final step should complete your proof, but it still needs a reason to justify it. When you have finished the proof, look it over and make sure there are no gaps in your reasoning. Once you have determined that the proof is sound, write QED at the bottom right corner to signify it is complete.