Algebra: Equations & Patterns Math 6

Expressions & Equations Equations, Patterns, & Graphs Expressions & Equations

Expressions & Equations Numbers and operations (+, -, x, ÷ ,) grouped together that show the value of something 2 × 3 3 + 5 – 7 x 2 ÷ 10 It is one side of an equation

Expressions & Equations A mathematical statement that tells that two sides are equal. It will include an equals sign "=“. 2 + 5 = 7 1 + 6 = 2 + 5 Solve the following 3 + 4 =

Expressions & Equations An equations says: “The value of the expression on the left (2 + 5) is equal to the value of the expression on the right (7)” An equation is a statement "this equals that“ Think of it like a scale

Expressions & Equations

Expressions & Equations Variable: A variable is a symbol for a number we don't know yet. It is usually a letter like x or y. Variables on their own (without a number next to them) actually have a hidden 1 x is really means 1(x) ex. How many students will be absent tomorrow?

Expressions & Equations Algebraic Equation: A mathematical statement that tells that two sides are equal. It will include a variable. The variable must be “solved” is also like a scale x + 2 = 6 4x − 7 = 5

Expressions & Equations

Expressions & Equations Find the missing number to make the equations true: 8 x  = 112 show “HOW” you solved it

Expressions & Equations What strategy did you use? 112 ÷ 8 = 14 This method is called: Inverse Operations An operation that undoes another operation (the opposite)

Expressions & Equations Examples 4 + 7 = 11 / 11 – 7 = 4 (11 – 4 = 7) 6 x 3 = 18 / 18 ÷ 3 = 6 (18 ÷ 6 = 3)

Expressions & Equations You Try 216 = 48 +  216 – 48 = 168

Expressions & Equations Algebra Method (solve for x) x ÷ 6 = 144 The variable x is not alone on one side . It has a ÷ 6 beside it. To find out what the value of x is we need to “isolate the variable”

Expressions & Equations Algebra Method (solve for x) Step 1: Perform the inverse operation to both sides, which means to multiply 6 from both sides. x ÷ 6 = 144 x 6 x 6

Expressions & Equations Algebra Method (solve for x) Step 2: Solve x ÷ 6 = 144 x 6 x 6 x = 864

Expressions & Equations Algebra Method (solve for x) Step 3: Check x = 864 864 ÷ 6 = 144

Expressions & Equations You Try: x + 2 = 3 - 2 -2 x = 1 1 + 2 = 3

Expressions & Equations You Try: x - 2 = 3 + 2 = +2 x = 5 5 - 2 = 3

Expressions & Equations You Try: 2x = 8 ÷ 2 = ÷ 2 x = 4 2 (4) = 8

Expressions & Equations You Try: x ÷ 2 = 8 *2 = *2 x = 16 16 ÷ 2 = 8

Expressions & Equations Translating Phrases: Addition: increased by, more than, sum of Subtraction: decreased by, difference, less than Multiplication: multiplied by, product of Division: per, quotient of, groups of

Patterns, Graphs, & Equations

Patterns Function Machine: Input Output Patterns Function Machine: Shows the relationship between input & output values The input column is a variable number (changes) The middle line represents a mathematical “rule” The output is the result of the “rule”. It is dependent on the variable number

Patterns Function Machine Input Output 1 2 3 4 Patterns Function Machine The input column is a variable number (changes) What is the pattern? Start at 1 add 1 each time

Patterns Function Machine Input Output 1 2 3 4 Patterns Function Machine The middle line represents a mathematical “rule” In this case the rule is “+8”

Patterns Function Machine Input Output 1 9 2 10 3 11 4 12 Patterns Function Machine The output is the result of the “rule”. It is dependent on the variable number What is the pattern? Start at 9 add 1 each time

Patterns You Try: The rule is multiply by 5 Input Output 1 2 3 4

Patterns 1 5 2 10 3 15 4 20 You Try: The rule is multiply by 5 Input Output 1 5 2 10 3 15 4 20

Input Output 1 9 2 10 3 11 4 12 Patterns We can use the input pattern and the output pattern to solve the “rule” between the 2 columns. Write the input pattern: start at 1 add 1 Write the output pattern: start at 9 add 1

Patterns The rule can have more than one operation. Input Output 2 10 4 14 6 18 8 22 Patterns The rule can have more than one operation. This rule is multiply by 2, then add 6. Algebra expression: 2x +6

Patterns You Try: Write the input pattern: Write the output pattern: Write the algebra expression:

Patterns 2 4 6 8 10 You Try: multiply by six, then add one Input Output 2 4 6 8 10

Patterns You Try: multiply by six, then add one Input Output 2 13 4 25 6 37 8 49 10 61

Patterns You Try: Write the input pattern: start at 2 add 2 each time Write the output pattern: start at 13 add 12 each time Write the algebra expression: 6x + 1

Patterns You Try: Write the input pattern: Write the output pattern: Write the algebra expression:

Patterns 30 60 90 120 150 You Try: divide by 3 then subtract 2 Input Output 30 60 90 120 150

Patterns You Try: divide by 3 then subtract 2 Input Output 30 8 60 18 90 28 120 38 150 48

Patterns 𝑥 3 −2 You Try: Write the input pattern: start at 30 add 30 each time Write the output pattern: start at 8 add 10 each time Write the algebra expression: 𝑥 3 −2

Input Output 1 2 5 3 9 4 13 17 Patterns We can use the input pattern and the output pattern to solve the “rule” Write the input pattern: start at 1 add 1 each time Write the output pattern: start at 1 add 4 each time

Input Output 1 2 5 3 9 4 13 17 Patterns Since the output pattern increase by 4, that is a clue about the operation. (repeated addition) This suggests that the input numbers are multiplied by 4. 1 x 4 ≠ 1 2 x 4 ≠ 5

Patterns Think…If you have 4 how do you turn it into a 1? Input Output 1 2 5 3 9 4 13 17 Patterns Think…If you have 4 how do you turn it into a 1? (divide by 1)? Think…I you have 8 how do I turn it into a 5?

Patterns Multiply the input by 4. Then subtract 3. Output 1 2 5 3 9 4 13 17 Patterns Multiply the input by 4. Then subtract 3. algebra expression: 4x – 3 = Test for 5 4 * 5 = 20 - 3 = 17

Patterns Step 1: Find the output pattern.

Patterns Step 2: Multiply the INPUT numbers by the gap.

Patterns Step 3: What must you do to each number in the second column?

Patterns Step 4: Write a rule for the T-table: Multiply by 3 then add 2 Write an expression 3x + 2

Patterns You Try: Write the input pattern: Write the output pattern: Write the algebra expression:

Patterns Input Output 1 7 2 14 3 21 4 28

Patterns You Try: Write the input pattern: start at 1 add 1 each time Write the output pattern: start at 7 add 7 each time Write the algebra expression: 7x

Patterns You Try: Write the input pattern: Write the output pattern: Write the algebra expression:

Patterns Input Output 1 9 2 14 3 19 4 24

Patterns You Try: Write the input pattern: start at 1 add 1 each time Write the output pattern: start at 9 add 5 each time Write the algebra expression: 5x + 4

Patterns Use a pattern to solve a problem Dara works at a fishing camp in the Yukon. Dara earns $25 a day, plus $8 for each fishing net she repairs. On Saturday, Dara repaired nine nets. How much money did she earn?

Patterns Strategy: What is the variable? Dara earns $25 /day. If she doesn’t fix any nets she still gets $25. She gets $8 per every net she fixes. The variable is “how many nets she fixes” 8n + 25 =

Patterns Input Output 1 33 2 41 3 49 4 57