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This lesson builds up on the previous one and takes it further. You will discover a linear function and its meaning. You will also find out how to represent a linear function in 4 different ways: equation, table of values, graph and words. You will learn about two types of linear functions: one that passes through the origin when graphed and the other one that doesn’t. This lesson will explain what each letter stands for in the equation y = mx + b. When the relationship is linear, the rate of change is constant. We have already observed it in the previous lesson. The relationship between distance and time is a great example of a linear function. When graphed it produces a straight line. The slope of a line represents the rate of change of one quantity in relation to the other. It can be calculated using any two points that sit on the line. Since each point has the coordinates (ordered pair), the slope of a line is calculated by finding the difference between x and y coordinates of the points respectively. We use letter “m” to represent the slope of a line: m = (y2 – y1)/(x2 – x1) Practice exercises will help you master the skills.​ More free math help and activities: intomath.org

In this lesson you will learn how to calculate surface area and volume of the given 3D shapes. Volume is how much space the shape has inside (its capacity). Surface Area is the total outside surface of the shape. In order to calculate the volume of a square-based pyramid, we multiply the area of the base by the height of the pyramid. Sometimes, the height is unknown and needs to be determined using the height of the lateral surface and Pythagorean Theorem. The process for finding the volume of a cone is similar. In order to calculate the surface area of any 3D shape, we need to determine the area of all 2D shapes that make up the 3D figure and add them together. Thus, the surface area of a square-based pyramid would be the sum of the areas of 4 triangular faces and the area of a square base. Depending on the situation, we may need to exclude some of the faces from our calculations when determining surface area and volume (for example, the floor inside the house when determining the area of the walls to be painted, etc.) 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

In this lesson you will learn and practice a very important skill of dividing decimals using long division. For example, dividing 2.34 by 2 or 34.5 by 0.4. It is important to understand the place value well and how it affects the final answer. Sometimes it makes sense to convert a decimal into a fraction to perform division. 0.2 is the same as 2/10 84 divided by 0.2 is the same as 84 multiplied by 10/2, which results in 420. You will also learn how decimals are related to unit conversions (between systems and within each system – metric, imperial, etc). We focus on converting metric units of length and mass – cm to m, kg to g and so on. For example, 1000 grams is equal to 1 kilogram, thus 0.001 kilogram is the same as 1 gram. More free math help and activities: intomath.org

In this lesson you will learn more about decimals. You will practice how to determine the place value of the digits within the decimal and why it is important to understand the place value of each digit when converting decimals to percent. For example, does 0.1 represent 10% or 1%? Why? You would need to multiply 0.1 by 100 to find the percent equivalent of this decimal, which results in 10%. It is easy to multiply any decimal by 100, since all you have to do is just move a decimal point 2 digits to the right of the original position. More free math help and activities: intomath.org

Percent is a concept that is widely used in everyday life in a variety of situations. “30% Discount”, “Yes, definitely, 100%” – are just some of the expressions that we encounter frequently. It is important to understand what the above means and how it can be useful. A percent is a number out of a 100. The term “percent” is derived from the Latin per centum, meaning “by the hundred”. In Ancient Rome, before the decimal system was invented, computations were often made in fractions which were multiples of 1/100 (or 0.01 if recorded as a decimal). Thus, computation with these fractions was the same as computing percentages. More free math help and activities here: intomath.org

In this lesson you will learn and practice how to multiply any number by a double-digit number vertically. For example, how to multiply 142 by 32. You will also learn how to divide any number by a double-digit number using long division. For example, how to divide 345 by 21. It is important to understand why place value of each digit within each number is important and why it is crucial to follow a certain form when performing operations in order to get the right answer. In this lesson you will also discover division with a remainder. More free math help and activities: intomath.org

In this lesson you will learn about the concept of a fraction and comparing fractions with unlike denominators. A fraction is a part of a whole. A fraction has two components. The number on the top of the line is called the numerator. It tells how many equal parts of the whole are given. The number at the bottom of the fraction line (bar) is called the denominator. The denominator represents the total number of parts that make up a whole. You will practice how to compare fractions with different denominators and why, in order to do that, you need to bring them to the Lowest Common Denominator (LCD). To find the LCD you would need to find a number that is evenly divisible by the denominators of the fractions you are comparing. But remember, when the denominator changes, the numerator changes with it. More free math help and activities here: intomath.org

Numbers can be broken down into smaller numbers or themselves and 1. In this lesson we are learning about prime factors, how to determine them and why we need them. Prime factors are numbers that cannot be split further into other whole numbers, other than themselves and 1. Factors in general are numbers that the given number can be divided by. Or whole numbers that are multiplied together to produce another number. The original numbers are factors of the product. If a x b = c then a and b are factors of c. For example, the factors of 8 are 1, 8, 2, 4. In this lesson we are also discussing the concept of the Greatest Common Factor for the given two numbers. The GCF is the largest number the two given numbers could be divided by. The Least Common Multiple is the smallest number that could be divided by the given two numbers. More free math help and activities here: intomath.org

We have already learnt about Natural numbers in our grade 5 lessons. Now we are moving on to working with much larger numbers. Sometimes we need to use really large numbers when measuring distances or calculating weight of a big object. What if we wanted to calculate the distance around the Earth? One million (1000,000) is a very large number. In this lesson you will learn and practice how to add and subtract numbers up to 1000,000, as well as how to multiply and divide them using a variety of strategies and methods (breaking the number down by place value of its digits, vertical multiplication, long division). For example, you will see how breaking the large numbers down by place value of the digits in each number helps to quickly add and subtract the two numbers. In 234,560 + 345,678, add 200,000 and 300,000, then 30,000 + 40,000 and so on.. More free math help and activities here: intomath.rog

In this lesson you will learn about decimals and their properties. A decimal number can be defined as a number whose whole and part are separated by a decimal point. The digits following the decimal point show a value less than one. Decimals are based on the powers of 10. As we move from left to right, the place value of the subsequent digit is divided by 10, meaning the place value becomes tenths, hundredths and thousandths. One tenth is 1/10. In decimal form, it is 0.1. This concept is frequently used in everyday life and can represent many different quantities, relationships and situations. More free math help and activities here: intomath.org

We have looked at the concept of a simple fraction in one of the previous lessons. Fractions can be proper (when the value of a number in the numerator is lower than that in the denominator) or improper (when the value of a number in the numerator is greater than or equal to that in the denominator). An improper fraction is always 1 or greater than 1. Improper fractions can be converted into mixed numbers (also called mixed fractions) by isolating the whole in the improper fraction. A mixed number is a combination of a whole number and a proper fraction. Rewriting an improper fraction as a mixed number can be helpful. It helps us identify more easily how many whole components there are. Mixed numbers can be represented visually as several wholes and parts of something. For example, in the short animation below we use pizzas to demonstrate the concept. It is important to understand how to connect the visual representation of mixed numbers and their arithmetic representation on paper (tablet). We can add and subtract mixed numbers by first adding and subtracting their whole parts and then their fractional parts. If the sum of the fractions is an improper fraction, then we change it to a mixed number. When the denominators of the fractions are different, we need to find equivalent fractions with a common denominator before adding or subtracting. More free math help and activities here: intomath.org

In this Grade 5 lesson you will learn about simple fractions and their properties. A fraction is a part of a whole. It has a numerator and a denominator. The numerator above the fraction line (bar) is used to count the parts out of the total. The number below the fraction line (bar) is called the denominator. It is used to name the total number of parts (fifth, tenth, thousandth, etc). The denominator can never be 0, because we cannot divide by 0. We can compare fractions, order them and perform various operations. You will learn and practice how to add and subtract fractions with the same denominators, as well as how to compare them. For example, you will discover that when the denominators are the same, the largest fraction is the one with the most parts (only comparing the numerators). More free math help and activities here: intomath.org

In this lesson you will discover that 1 dollar equals 100 cents and that cents can be expressed as part of a dollar using fractions or decimals ($0.05 = 5 cents). You will learn and practice how to add and subtract dollars and cents, as well as how to multiply and divide them. For example, if one item costs $2 and 50 cents and the other item costs $3 and 14 cents, what is the total cost of both items? We would add dollars and cents separately and get: $2 + $3 = $5; 50 cents + 14 cents = 64 cents. The total cost is $5 and 64 cents. The lesson teaches how to group the digits based on their place value in order to do operations with money faster and without a calculator. It also discusses sales tax and how to calculate it when making a purchase in order to find the grand total. A sales tax is what a customer pays on top of the actual price of the item. In some countries it is included in the price right away, in others it need to be calculated and added at the check out. More free math help and activities here: intomath.org

Division is the opposite operation to multiplication. When we are trying to distribute something evenly or with a remainder among a group of people or split something into smaller components – we are dividing. We cannot divide by 0, since no splitting or sharing is taking place. We use the ÷ symbol, or sometimes the / symbol to demonstrate division. The symbol ÷, called obelus or obel (from the Greek οβελοσ), was introduced by the Swiss mathematician Johann Henrich Rhan in 1659. Knowing the multiplication tables can help with division. More free math help and activities here: intomath.org

100,000 (one hundred thousand) is the natural number following 99,999 and preceding 100,001. In this math lesson you will learn about numbers up to 100,000 and how to do operations with these numbers. You will discover their properties (for example, that any number added to 0 is that number). In addition, you will learn about the importance of the place value of each digit within the number. Moreover, you will also learn how to pronounce and write these numbers in words correctly. In this lesson, we demonstrate how to add and subtract numbers up to 100,000, as well as how to multiply and divide them vertically. The process involves “borrowing” a digit or “carrying” a digit. Our place-value system is called the “decimal” system, because it’s based on 10 fingers and when we move to the next ten, we add another zero, thus creating a new place value for a new digit. More free math help and activities here: intomath.org

We use numbers everyday to count, to measure, to order and estimate and so on. Natural numbers are positive numbers that we use to count and order. 0, 3, 6, 10, 204, 1000… In mathematical terminology, numbers used for counting are called “cardinal numbers” and numbers used for ordering are called “ordinal numbers”. Mathematicians use N to represent a set of all natural numbers. The set of natural numbers is an infinite set. This infinity is called countable infinity. Properties of natural numbers: 1. 0 is a natural number (but some mathematicians do not include it in the list of natural numbers) 2. Every natural number has a number that comes after it and is also a natural number 3. 0 does not come after any natural number 4. If the number that comes after x is equal to the number that comes after y, then x = y More free math help here: intomath.org