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In this video you will learn what domain and range are. You will see how to state the domain and range using a set and an interval notation. The video covers a variety of situations and possible restrictions on the domain and range. Practice the concept by taking a short quiz: https://intomath.org/quiz/domain-and-range-of-a-function-quiz/

When you want to own a car, but your savings are not enough, you can finance your car. Financing a car means taking out a car loan and paying it back overtime. You could finance both a new vehicle or a used one. A financial institution loans you enough money to purchase your car, however, charges interest on the amount you borrowed. Usually you repay your loan by making regular payments every month or every two weeks (repaying the borrowed amount + the interest). Financing a car is a good solution for someone who is able to make regular payments for a prolonged period of time (4-5 years). If you do not think you can maintain making those payments, for various reasons, car financing is not for you. It is then better to save up and pay the total amount at once (this may mean you would have to go with a more affordable vehicle). Learn more about how to finance a car: https://intomath.org/how-to-finance-a-car/

An income tax is the portion of your earnings that you pay to support public services and Government programs. In most countries, income tax is calculated based on a progressive system – the more money you make, the higher share you pay in taxes. The tax return that you submit every year before a certain due date, specified by the Revenue Agency, usually includes the following types of income: Employment Income (when you work for someone and are an employee) Self-Employment Income (when you work for yourself and when you have contracts with others) Pension Income (if you are collecting pension because of retirement) Investment Income (if you have a property that you are renting or if you have money in the bank savings account that is gaining interest) Learn more here: https://intomath.org/what-is-income-tax/

Simplifying expressions with exponents is an important skill that is required to comfortably work with different types of functions and their equations. It is especially useful when solving polynomial and rational equations. We already looked at the concept of exponent in previous grades. However, we only operated integer exponents. In this lesson we are moving further and learning about rational exponents and their properties. A rational exponent is an exponent expressed as a fraction m/n. A power containing a rational exponent can be transformed into a radical form of an expression, involving the n-th root of a number. The n-th root of a number a is another number, that when raised to the exponent n produces a. We are also practicing how to use negative exponents in this lesson and discussing the difference between even and odd exponents. In addition, we are analyzing the base of 0 raised to a rational exponent. Simplifying expressions with exponents requires us to use a variety of exponent laws and properties. We may start with a really complex expression that, when simplified, could result in one variable or a number. More free math help and activities: intomath.org

Solving quadratic equations requires a good understanding of and proficiency with a number of concepts. A quadratic equation is an equation of the form ax2 + bx + c = 0. This is a standard form equation. A quadratic equation can also be recorded in the factored form a(x – r)(x – s) = 0, where r and s are the roots of the equation. In order to determine the solutions (roots or x-intercepts of the parabola) of the quadratic equation and when the equation is given in a standard form, we can either try to factor the equation or use the Quadratic Formula. The method of solving quadratic equations by factoring depends whether the standard form equation is given as a simple or as a complex trinomial and whether the factors could be easily identified. If the a value of the equation is a = 1 and the equation is x2 + bx + c = 0, then x1 + x2 = -b and (x1)(x2) = c However, if a is not 0 or 1, then the equation could be solved by factoring using the decomposition method (watch the video lesson to see how). Sometimes factors are not very “nice” numbers. In such cases it is difficult to factor and the Quadratic Formula should be used instead. Before setting up the Quadratic Formula it is always a good idea to check whether the equation is at all solvable. To check how many possible solutions the quadratic equation has we use the Discriminant. The Discriminant is an expression that is part of the Quadratic Formula. The note above contains all possible scenarios for when the Discriminant is positive, negative or 0. Once the quadratic equation has been solved, the solutions can be used to graph the quadratic function or a statement based on the real life problem could be made. For example, if we are talking about a kicked ball, then the x-intercepts represent the time when the ball touches the ground as it was kicked and when it landed. More free math help and activities: intomath.org

Right triangle trigonometry is a branch of mathematics that deals with angles and sides in a right triangle. A right triangle is a triangle with one right angle. The other two angles add up to 90o. Together, the sum of all interior angles in a right triangle is 180o. The side relationships in a right triangle are based on the Pythagorean Theorem. If the triangle is right, then the following will be true for the three sides of the triangle: c2 = a2 + b2 where a and b are shorter sides and c is the longest side. When the lengths of any two sides are known, the third side can always be found using the Pythagorean Theorem. When two sides are given and a measure of an angle other than the right angle needs to be found. Or only one side and an angle other than the right angle are given – trigonometric ratios should be used. Trigonometric ratios are ratios between any two sides of a right triangle that can then be used to determine the measure of an angle between those two sides. sin(x) = opposite/hypotenuse cos(x) = adjacent/hypotenuse tan(x) = opposite/adjacent In this lesson we are discussing how to use these ratios to find angles and side lengths in right triangles. We are also looking at some special angles and their trigonometric ratios. We are discovering sines and cosines of common angles. These angles are frequently used and their trigonometric ratios should be memorized just like the multiplication tables. Right triangle trigonometry is widely used on engineering and construction. For example, when homes are designed and built, it is crucial that architects and builders accurately determine and construct the angles of the roof. There are tools for that, however, it is always good to understand the concept and be able to double check. More free math help and activities: intomath.org

In this lesson you will learn about linear relations and what they mean. One of the most common examples of a linear relation is the relationship between distance and time when a distance changes over time at a constant rate. A linear relation is represented by a straight-line – a relationship between a variable and a constant. Linear relations can be expressed in 4 different ways: a graph, a table of values, a word description and an equation. In this lesson we will compare two objects moving and discuss the relationship between distance, time and speed while comparing the graphs representing the motion patterns of the two objects. You will learn why the equation D = vt (or D = st) is a direct proportion and how knowing the time and speed one can determine the distance traveled. Watch the video lesson, complete the note and additional practice in order to improve your understanding of the concept. More free math games and activities: intomath.org

In this lesson we are discovering the distributive property of multiplication, as well as common factoring. According to the distributive property, a(b + c) = ab + bc. When multiplying the sum or difference of two terms by another term, it is possible to find the product of each pair of terms first and then add/subtract them. This is how we expand brackets and simplify algebraic expressions. ​Distributive property is used to simplify expressions, solve equations and word problems. It is a part of virtually every math course in high school and the skill is transferable to other concepts. ​The opposite (inverse) operation to distribution is common factoring. This is when we are given the expanded form of an algebraic expression and need to determine the Greatest Common Factor, that all the other terms of the expression are divisible by – the term that goes outside of the brackets. ​For a more detailed explanation of these processes with examples watch the video lesson above, then check out the accompanying note and complete additional math practice on this topic. More free math help and activities: intomath.org

Exponents are very important and powerful. Read this blog post based on an ancient Indian legend about the creator of chess. This lesson teaches about exponents. An exponent shows how many times the number needs to be multiplied by itself. For example, 5³ = 5 × 5 × 5 = 125. In the order of operations, BEDMAS, exponents come right after operations in brackets. Sometimes exponents are used with variables. Then, exponent laws are used in order to simplify expressions with variables. This lesson explains how to use general exponent laws and provides insight into operations with exponents. Negative exponents make the base turn into its reciprocal. Any number (except 0) or an algebraic term raised to the exponent of 0 is equal to 1. When raising a fraction to an exponent it is necessary to raise both the numerator and the denominator to that exponent separately. More free math help and activities: intomath.org

In this lesson you will learn about a coordinate plane (Cartesian Plane). You will learn how to plot a point and record its coordinates – an ordered pair (x, y). You will discover how to determine the relationship between two quantities. One of those quantities is dependent on another like distance and time or like the circumference of a circle and its diameter. You will also learn how to graph the relationship. You will learn about the rate of change (slope of a line) and how to calculate it using the coordinates of the two points. When the relationship is linear, the rate of change is constant. You will see how the slope of a line (linear relationship) depends on the change in the x and y coordinates. Additional practice will help you master the skills. More free math help and activities: intomath.org

In this lesson we are discussing two topics: square root of a number and percent. A square root of a number is a number that has been squared to get the number under the square root. For example, the √4 is 2, because 2 x 2 = 4. Watch the video lesson to see the examples and learn why we cannot take a square root of a negative number. We have already discussed the concept of percent in our previous lessons. However, since it is a very commonly used concept, we will go over it again in more detail. In this lesson you will learn the general formula for finding the percent of a number. You will see how to use proportions in order to calculate percent. Together, we will go over some real life problems with percentages. More free math help and activities: intomath.org

In this lesson you will learn how to solve one step and multi step linear equations. For example, x – 3 = 10 or 2h + 1 = – 9. The lesson demonstrates how to solve linear equations with brackets. For example, 2(x – 1) = 16. Such equations are solved using inverse operations and reverse BEDMAS. The first step is always expanding the expression with brackets using the distributive multiplication property. You will also discover the Pythagorean Theorem. You will learn when and why to use it, how to set it up and solve the equation. The challenge of using a Pythagorean Theorem is that it involves squares (multiplying the number by itself) and it is important to follow the BEDMAS and equation solving algorithm carefully in order to avoid mistakes. For example, the quantities should always be squared first, then added. More free math help and activities: intomath.org

In this lesson you will learn about scientific notation and its significance when working with very large and very small numbers. Scientific notation is widely used in Physics, Astronomy and Chemistry. For example, it could be used to express the volume of water in a large lake. Instead of writing all the zeros we multiply the number by a certain power of 10. You will also learn how to calculate area and circumference of a circle. You will discover the importance of the Pi number. Here are the first 100,000 digits of Pi. The lesson teaches you how to determine the volume of a cylinder (a 3D shape with circular bases and a rectangular lateral part). You will also learn what calculating the volume means and what units should be used. More free math help and activities: intomath.org

In this lesson you will learn about proportions, their importance, how to set them up and how to solve them. A proportion is an equivalent relationship of quantities. It is a name we give to a statement that two ratios are equal. It can be represented in two ways: two equal fractions or as a : b = c : d. We can determine whether a proportion is true or false. We can also find an unknown in a proportion or solve real life problems using proportions. We often use proportions to convert to or from percentages. For example, what is 10% of 45? We can set up a proportion: 45 corresponds to 100%, then what corresponds to 10%? 10 : 100 = n : 45 n = (45 x 10)/100 n = 4.5 There are many examples in math in which the exact answer is required. For example, if a doctor prescribed to a patient a drug requiring a specific measure, the doctor would have to prescribe the exact amount; if the doctor simply estimated the amount, a patient could have severe side effects or it could even lead to a fatal outcome. Thus, understanding proportions could save lives! Proportions can be solved using visual models. However, it is important to understand and learn how to quickly set them up and solve algebraically. More free math help and activities: intomath.org

In this lesson you will learn about multiplication properties and their significance. For example, that a × b = b × a or that a(b + c) = ab + bc. You will learn how distributive property is used to simplify algebraic expressions. This property is also often used to solve equations. For example, when solving an equation 2(3x – 1) = 10 the first step would be to expand the expression with brackets. It would become 6x – 2 = 10. Then the equation could be solved using inverse operations and reverse BEDMAS. The lesson covers simplifying algebraic expressions by collecting like terms. Algebraic expressions are symbols or combinations of symbols used in algebra, containing one or more numbers, variables, and arithmetic operations. Like terms are terms that have the same variables with the same exponents or no variables – constants (4n and 5n, 6 and 89). Additional practice will help you master the skills. More free math help and activities: intomath.org

An equation is an equality of two quantities and it always contains an equal sign. To solve an equation means to determine the value of the unknown(s) that makes the equation true, so that when the unknown is being substituted with the value in the equation, the left side is equal the right side. An expression does not have an equal sign, but it may contain variables that represent a certain quantity or relationship. To simplify an expression means to complete operations within the expression that would bring it to its simplest form. In order to do that collecting like terms ins required. Like terms are terms that have different coefficients but identical variables with the same exponent. You are going to learn how to solve a linear equation by completing the reverse BEDMAS and performing opposite operations. For example, an equation 2x + 3 = 9 could be solved by first subtracting 3 from both sides of the equation and then dividing both sides of the equation by 2. Thus, x = 3. It is important to make sure that the same operation is done to both sides of the equation to keep the equation balanced. Collecting like terms is often necessary on one or both sides of the equation. Doing the “left side, right side check” is an important component of solving equations, as it would help you determine whether the solution is correct. This lesson will teach you how to do it. More free math help and activities: intomath.org

In this lesson you will learn about multiplying and dividing integers (positive and negative). You will learn the general rules that apply to every problem. For example, when multiplying or dividing two negative integers, the product or a quotient is positive. When one of the integers is negative and the other one is positive, the result is always negative. The absolute value of each number does not affect the sign of the answer like it does with addition and subtraction of positive and negative integers. You will discover how to multiply a negative number by 0 and what happens when you try to divide by 0. You will see how understanding operations with integers can be helpful in real life. More free math help and activities: intomath.org

In this lesson you will learn about adding and subtracting integers. Integers are positive and negative whole numbers and when first learning about integers it is useful to visualize how one integer is related to the other. This is when we use a number line. A number line is a line that has the origin, 0, to the right of which there are positive numbers and to the left of which there are negative numbers. We can move along the number line and determine the position of any integer. We have previously used it to compare integers. You will discover the concept of the absolute value and why it is important when performing operations with integers. It is also useful when solving certain types of equations and later, when analyzing absolute value functions and their graphs. On a number line, the distance between the two numbers is given as an absolute value. For example, the distance from -10 to +10 is 20. However, if we add -10 and +10 the answer is 0 – they are the exact opposites. If we were to subtract +10 from -10, it would be recorded as -10 – (+10) and would equal -20 ( because on a number line a positive direction has changed to a negative direction). More free math help and activities: intomath.org

Multiplying and dividing fractions is not difficult. Knowing how to multiply and divide fractions will help you succeed when other, more complex mathematical concepts, such as simplifying algebraic expressions, solving equations and so on. In order to multiply and divide fractions it is not required to find the GCF, unlike when adding and subtracting them. Most students find the process of multiplying fractions the easiest of all operations with fractions. All you have to do is multiply numerators and denominators respectively (the product of the numerators over the product of the denominators), recording the final answer as a reduced simplest fraction. In order to reduce a fraction, divide both the numerator and the denominator by their GCF (greatest common factor). The process of dividing the two fractions is a bit more complex. But as long as you understand why things are done a certain way everything is really simple. ​In order to divide one fraction by another fraction, find the reciprocal of the second fraction and then change the operation to multiplication. To get the reciprocal of a fraction, just turn it upside down. Switch the numerator and denominator. More free math help and activities: intomath.org

Adding and subtracting fractions involves representing them as a part of the same whole. Thus, their denominators must be the same. In order to bring them to the same denominator, we need to change all of the existing denominators while at the same time changing the numerator such that the fractions remain equivalent to the original ones. If two fractions with different denominators are being added or subtracted, first bring them to the Lowest Common Denominator by determining their Least Common Multiple. For example, for denominators 2 and 5, the Lowest Common Denominator is 10 – the lowest number divisible by both given denominators. For mixed numbers, first convert the mixed numbers into improper fractions and then follow the steps above. In order to convert a mixed number to an improper fraction multiply the whole portion of the mixed number by the denominator of the fractional portion and then add the numerator – the result becomes the new numerator over the unchanged denominator. The numerator will end up being a greater number than the denominator. In general, becoming proficient in adding and subtracting fractions will help you with many other more complex concepts, such as solving equations with fractions, simplifying algebraic expressions and solving rational equations. More math help and activities: intomath.org

In order to display positive and negative numbers we often use a number line. A number line is a line that contains all negative and positive integers, including zero (the origin). One of the real life examples of a number line is a thermometer. A thermometer has a scale that usually goes from -50 to +50 degrees Celsius. 0 degrees is the middle between the coldest and the hottest temperatures. When we add and subtract positive and negative integers, we can always use a number line to calculate the absolute distance between the two digits – in mathematics this distance is called “absolute value”. A number line is an excellent tool that helps understand operations with integers visually. For example, what is -8 + 9? We would locate the -8 on the number line and move 9 digits in the positive direction (from left to right). We would land on 1, which would be the answer to the problem. A number line can also be used to compare positive and negative integers. More math help and activities: intomath.org

We are often required to calculate the area and perimeter of objects or spaces in 2 dimensions (flat surface like a piece of paper or a field). In real life, this skill is helpful when designing living spaces or calculating the amount of fencing required to enclose a certain piece of land, etc. Perimeter is the length and width around the object – the sum of all outer sides. Area – the 2 dimensional space surrounded by the sides of the shape. The two shapes we are looking at in this lesson are rectangle and square. A square is also a rectangle, but a regular rectangle. “Regular” means all sides and angles are equal. We develop formulas for both area and perimeter and analyze different situations where those formulas can be applied. It is important to not only memorize the formulas or substitute numbers into them. It is important to understand where the formulas are coming from, how they were derived. Making connections and understanding relationships is one of the key components of being successful in math. More free math help and activities: intomath.org

In this lesson you will learn what a mean is. You will discover how the mean is related to the average. For example, given a set of numbers: 2, 5, 7, 7, 8 the mean is calculated by first finding the sum of all numbers and dividing the sum by how many numbers there are in a set: (2 + 5 + 7 + 7 + 8)/ 5. The mean is often used in research, academics and in sports. You often see these graphs used by media to show the relationships between various quantities ( for example, during the elections, bar or pie graphs are used to show the ratings of the candidates. You will learn how to use bar graphs when representing the relationships between quantities that are not dependent on each other. More free math help and activities: intomath.org

In this lesson you will learn about different types of algebraic expressions: monomials, binomials, trinomials and polynomials. You will see how two monomials could be multiplied or added/subtracted. For example 2m × 3m = 6m² or 4m + 12m = 16m. In math, a polynomial is an expression consisting of coefficients and variables. These coefficients and variables are called “terms”. Terms in a polynomial are separated by operations of addition, subtraction, multiplication. Variables have only positive integer exponents. This lesson also explains how algebraic expressions with brackets can be simplified by distributing a monomial in front of the bracket by every term inside the bracket. For example, an expression 3(2s -3) could be expanded to 6s – 9. This may be followed by collecting like terms (monomials within polynomials that only differ from each other by their coefficient) if there are more than one bracket. More free math help and activities: intomath.org

In this lesson you will learn what a linear equation is and what it means to “solve an equation”. In order to solve an equation, we need to complete inverse operations in the reverse BEDMAS order. The solution or root is the value that we get when we solve an equation. This lesson explains the steps of solving linear equations with brackets. From simplifying the expressions with brackets on both sides of the equal sign; identifying, rearranging and collecting like terms to isolating for the unknown variable by dividing by its coefficient. In this lesson you will see a concrete example of how to solve an equation that contains brackets, does not initially look like a linear equation, but simplifies to it. Practice exercises will help you master the skills required for solving linear equations with brackets and multi-step equations. More free math help and activities: intomath.org