IntoMath Grade 10 Lessons

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 we are learning about a triangle and triangle properties. You have been working with triangles for a while now. You may already know that there are different types of triangles: equilateral, isosceles and scalene. There are right (one of the angles is 90o) and non-right angle triangles). Triangles are congruent if they completely overlap when placed on top of each other. This means that their corresponding sides and angles are equal. There is a number of important triangle properties. We label the sides using lower case letters and the angles using upper case letters and the side opposite the angle is labelled with the same letter (E-g: a and A). For example, that all interior angles inside a triangle add up to 180o. A Midpoint is a point in the middle of the side of the triangle. Midpoint Segment is a line segment, connecting two midpoints. It is parallel to the opposite side of the triangle and it half of its length. Median – a line segment, connecting one of the vertices of the triangle and a midpoint of the opposite side. Altitude – a line segment, connecting one of the vertices of the triangle and a point on the opposite side at a right angle. The point may or may not be the midpoint. More free math help and activities: intomath.org

A quadratic function is a function of the form f(x) = ax2 + bx + c, where a cannot be 0. This is called a standard form equation. There are two other forms: vertex and factored. The graph of a quadratic function is called a parabola. Parabolas may open upward or downward. They have the “U” shape. The most basic parabola has an equation f(x) = x2. In order to graph this parabola, we can create the table of values, where x is the independent input and f(x) is the output of a squared input. The vertex of a parabola is its the highest or the lowest point. When a quadratic equation is given in the vertex form, it is easy to immediately determine the vertex by looking at the values of k and h. They are the coordinates of the vertex. f(x) = a(x – h)2 + k Vertex (h,k) Sometimes it is necessary to convert the standard form equation to a vertex form equation. In order to do that, we can use a process called “completing the square”. We can also complete the square geometrically. We can also use the method of completing a square to solve certain types of quadratic equations for zeros (x-intercepts) and determine where parabola intersects with the x-axis. Quadratic functions are very common in real life. Remember the Angry Birds game and movie? In order to design the game, the creators had to have a good understanding of a parabolic trajectory. A trajectory of a kicked ball or a diver diving off of the board are also described by parabolas. In basketball a quadratic functions are used to calculate the exact position of the player in order to get the ball in the net. More free math help and activities: intomath.org

A system of linear equations is when two or more equations are considered at the same time and we are looking for the point of intersection of the equations. In Grade 10 we are only looking at the system of 2 equations. We are learning how to solve a system of linear equations in three ways: graphically, by substitution and by elimination. To solve the system of linear equations with two unknowns means to determine the values of the unknowns, that satisfy BOTH linear equations in the system or show that the solution does not exist. The System Solution = Point of Intersection of the two lines. If two lines are parallel, the system will have no solutions. When we want to solve the system graphically, we graph the equations of the lines and determine the point of intersection visually on the graph. this may not be accurate. Elimination is a method when equations are being added or subtracted. One of the variables gets eliminated. Then the new single equation is solved for one of the variables. After that, the result is used to determine the value of the other variable. When doing substitution, one of the equations is rearranged to isolate for one of the variables. Then the expression that represents the variable is substituted in the other equation. After solving the equation, the result is used to solve for the other variable. In real life, we use this concept to compare and relate quantities. For example, choosing a better cell phone plan. Or determining the exact value of both quantities at the same time (example in the accompanying note). Knowing and understanding how to solve a system of two linear equations useful and the skills developed when learning this concept can be applied to a variety of situations in mathematics. More free math help and activities: intomath.org

Coordinate geometry helps us find the distance between the points, determine the side lengths and angles of geometric shapes using a Coordinate Plane. There is a number of ways to do that and it is really fun to do when you understand why we take certain steps. In coordinate geometry you need to know some terminology. For example, you need to know that the word “origin” means a point with coordinates (0,0). Or that the word “subscript” means a small letter or number below a regular sized one (E-g.: x2). This lesson demonstrates how to determine the distance between the two points using their coordinates. It also explains how to find the midpoint on a line segment. In this lesson we are also looking at the equation of a circle. The general equation of a circle with the center not at the origin is (x – a)2 + (y – b)2 = r2 where r is the radius of a circle and a,b are not 0. When the center of a circle is at the origin, then a and b are equal to 0. In general, coordinate geometry is a fascinating branch of mathematics. It is like learning to read a mathematical map. Did you know that any pdf file on the web is an example of a coordinate geometry? The words and symbols in these files are written using the coordinate geometry. If a document contains text and images, they are placed according to a Coordinate Plane using the concepts of a distance between two points, slope of a line, as well as trigonometry. Photo editing software are based on coordinate geometry. Changing the dimensions of an image or moving it around the screen – everything is based on the concept and the position of every point has its coordinates (x,y). In a GPS system that we use to get from one place to another, the longitude and the latitude of a place are its coordinates. More free math help and activities: intomath.org