Section 1.1: Patterns in Division Dividing by 2 All even numbers are divisible by 2. E.g., all numbers ending in 0,2,4,6 or 8. Dividing by 4 1. Are the last two digits in your number divisible by 4? 2. If so, the number is too! 3. For example: 358912 ends in 12 which is divisible by 4, thus so is 358912. Dividing by 5 1. Numbers ending in a 5 or a 0 are always divisible by 5. Dividing by 8 1. This one's not as easy, if the last 3 digits are divisible by 8, so is the entire number. 2. Example: 6008 - The last 3 digits are divisible by 8, therefore, so is 6008. Dividing by 10 1. If the number ends in a 0, it is divisible by 10. Section 1.2: More Patterns in Division Dividing by 3 1. Add up all the digits in the number. 2. Find out what the sum is. If the sum is divisible by 3, so is the number 3. For example: 12123 (1+2+1+2+3=9) 9 is divisible by 3, therefore 12123 is too! Dividing by 6 1. If the Number is divisible by 2 and 3 it is divisible by 6 also. Dividing by 9 1. Almost the same rule and dividing by 3. Add up all the digits in the number. 2. Find out what the sum is. If the sum is divisible by 9, so is the number. 3. For example: 43785 (4+3+7+8+5=27) 27 is divisible by 9, therefore 43785 is too! Page 1 of 8
Section 1.3: Algebraic Expressions Algebraic expressions contain variables such as x and n (which can represent ANY number). Note: the words "a number" in the meanings below is replaced by the variable. Examples and their meanings: x + 5 : Five more than a number OR A number plus five n - 4 : Four less than a number OR A number subtract four 4 - n : Four subtract a number (This example is often confused with the one above - BE CAREFUL) 5n : Five times a number Note: In algebra we do not use the "times" symbol x as it would be mistaken as a variable. We simply write the number in front of the variable! 6n + 2 : Six times a number, then add two OR two more than six times a number 10/n : Ten divided by a number Note: the slash symbol can be used to represent division. So can a fraction like form! In the algebraic expression: 6t - 4 6 is the numerical coefficient (this will always be the number next to the variable) t is the variable (the letter in the expression) 4 is the constant term (the number being added or subtracted in the expression) Page 2 of 8
Section 1.3: Algebraic Expressions continued. You can use an algebraic expression to solve similar problems more easily. Example: Suppose you earn $9 per hour If you worked 3 hours, you earn: 3 x $9 = $27 If you worked t hours (an unknown amount of time) you earn t x $9 = 9t (Remember: Multiplication in algebra is written without a "times" symbol and the numerical coefficient is always written in front of the variable!) We evaluate an algebraic expression by substituting in a value for the variable. Example: Evaluate 3f -2 for f = 5. Solution: We substitute the 5 for the letter f in the expression as follows: (3)(5) - 2 (Note: The order of operations tells us we must multiply before subtracting) = 15-2 =13 Page 3 of 8
Section 1.4: Relationships in Patterns A number pattern may be described by using the term number Term Number 1 2 3 4 5 6 Term 6 12 18 24 30 36 In this case each term is 6 times the term number. We can let n represent any term number. Term Number Term 6 x 1 = 6 1 2 3 4 5 6... n 6 x 2 = 12 6 x 3 = 18 6 x 4 = 24 6 x 5 = 30 6 x 6 = 36... Then the term is represented by 6 x n, or 6n (As seen in the table above) 6 x n = 6n If we compare or "relate" a variable ( n ) to an expression that contains the variable (6n), you have a relation. If we wish to determine the 15th term of this relation we substitute n = 15 in the expression 6n. 6n =6 x 15 = 90 Therefore, the 15th term of this relation is 90. The major advantage of this is we do not have to find the previous 14 numbers in the table. Page 4 of 8
Section 1.5: Patterns and Relationships in Tables We can make a table of values for a relation such as 3n + 4 is related to n. We then choose values for n (input numbers) and substitute these into the expression to get the output numbers. If n =1 then (3)(1) + 4 = 3 + 4 = 7 If n = 2 then (3)(2) + 4 = 6 + 4 = 10 Input Output n 3n + 4 1 7 2 10 3 13 4 16 You may also determine a relation given its table of values. Input Output 1 7 2 9 3 11 4 13 5 15 Let n represent any input number. When n is increased by 1, the output number increases by 2. This means the expression for the output numbers contains 2n. The multiples of 2: 2, 4, 6, 8, 10. These multiples are ALL 5 less than those in the table. So, the output is 2n + 5. Therefore this table shows how 2n + 5 relates to n. Page 5 of 8
Output Grade 7 Math Section 1.6: Graphing Relations You can use a graph to show a relation. This table and graph show how 4n -1 relates to n. Input Output n 4n - 1 1 3 2 7 3 11 4 15 Graph of 4n -1 against n 16 14 12 10 8 6 4 2 0 0 1 2 3 4 5 Input The scale on the output axis is 1 square to 2 units. The points lie on a straight line, so this represents a linear relation. Both the table and graph show that when the input number increases by 1 the output number increases by 4. Page 6 of 8
Section 1.7: Reading and Writing Equations An equation is a statement that two expressions are equal. An equation will ALWAYS have an equal sign = in it! Think 2 x 6 = 12 (The expression 2 x 6 is equal to the expression 12) In algebra: 3x + 1 is an algebraic expression. 7 is an expression 3x + 1 = 7 is an equation. In words we write: One more than triple a number is seven. How to write an equation from a statement. 1. Choose a letter for the variable. 2. Write an algebraic expression to represent the relationship. 3. Write an equals sign between the expression and the constant term. Ex. #1: Four more than a number is 17 Let x, represent the number. Four more than the number: x + 4. The equation is: x + 4 = 17 Ex. #2: A number divided by three is nine. Let h represent the number. h divided by 3: h 3 The equation is: h 3 = 9 Page 7 of 8
Section 1.8: Solving Equations Using Algebra Tiles You may use tiles to represent an expression. This unit tile represents +1 This variable tile or x-tile represents x. We can use tiles to solve an equation. For example to solve x + 2 = 8: Draw a vertical line in the center of your page. (This represents the equal sign in the equation). Arrange tiles on each side of the line to represent the expression or number on each side of the equal sign. On the left side, place tiles to represent x + 2. On the right side place tiles to represent 8. To get the unit tile by itself we remove two unit tiles (from both sides of the equation). The tiles now show that x = 6. To verify this is the correct solution, simply replace the x - tile with 6 unit tiles. As you can see both sides now have 8 unit tiles. Page 8 of 8