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Pattern and Function - Introduction to Variables and Simple Equations

Grade 5IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A variable is a letter or symbol that represents an unknown number. In the expression x+5x + 5, xx is the variable. Using variables allows us to describe relationships that work for any number, creating a generalized pattern.

Visual representation of a variable x added to a constant value of 3.
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An Equation is like a balanced scale. The equals sign == tells us that the value on the left side is exactly the same as the value on the right side. To solve an equation, we must perform the same operation on both sides to keep it balanced.

A balance scale showing x + 2 on one side and 10 on the other, representing an equation.
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A Function Machine takes an input, applies a mathematical rule (like multiplicationmultiplication or additionaddition), and produces an output. If the rule is ×3\times 3, an input of 44 yields an output of 1212.

Flowchart of a function machine showing input, rule, and output.
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Solving equations using Inverse Operations: To isolate a variable, we do the opposite of what is being done to it. The inverse of addition is subtraction, and the inverse of multiplication is division.

📐Formulae

General Equation Form: Expression 1=Expression 2Expression \ 1 = Expression \ 2

Addition Inverse: If x+a=bx + a = b, then x=b−ax = b - a

Subtraction Inverse: If x−a=bx - a = b, then x=b+ax = b + a

Multiplication Inverse: If atimesx=ba \\times x = b, then x=bdivax = b \\div a

Division Inverse: If xdiva=bx \\div a = b, then x=btimesax = b \\times a

Function Rule: Output=(Inputtimesm)pmcOutput = (Input \\times m) \\pm c

💡Examples

Problem 1:

Solve the equation for nn: n+18=45n + 18 = 45

Solution:

  1. Identify the operation being applied to the variable nn: Addition of 1818.
  2. Identify the inverse operation: Subtraction of 1818.
  3. Apply the inverse operation to both sides of the equation to keep it balanced: n+18−18=45−18n + 18 - 18 = 45 - 18
  4. Simplify both sides: n=27n = 27

Explanation:

To isolate nn, we undo the addition of 1818 by subtracting 1818 from both sides of the equation. This leaves nn alone on the left side.

Problem 2:

A pattern starts at 55 and follows the rule Output=(Inputtimes2)+3Output = (Input \\times 2) + 3. If the input is 44, what is the output?

Solution:

  1. Write down the given rule: Output=(Inputtimes2)+3Output = (Input \\times 2) + 3.
  2. Substitute the input value 44 into the rule: Output=(4times2)+3Output = (4 \\times 2) + 3.
  3. Perform the multiplication inside the parentheses first: 4times2=84 \\times 2 = 8.
  4. Add 33 to the result: 8+3=118 + 3 = 11.
  5. The final output is 1111.

Explanation:

This problem uses a function rule. By replacing the word 'Input' with the number 44 and following the order of operations, we calculate the resulting output.

Problem 3:

Find the value of yy in the equation: y−12=25y - 12 = 25. Show the steps using the balance method.

Diagram showing the addition of 12 to both sides of the equation to balance it.

Solution:

y−12=25y - 12 = 25 y−12+12=25+12y - 12 + 12 = 25 + 12 y=37y = 37

Explanation:

To find yy, we need to undo the subtraction of 1212. We use the inverse operation, which is addition. By adding 1212 to both sides of the equation, the −12-12 and +12+12 on the left side cancel out, leaving yy alone. On the right side, 25+12=3725 + 12 = 37.

Problem 4:

A pattern is formed using squares. The rule for the number of squares is S=2n+1S = 2n + 1, where nn is the figure number. How many squares are in Figure 4?

Visual pattern showing Figure 1 with 3 squares and Figure 2 with 5 squares.

Solution:

S=(2×4)+1S = (2 \times 4) + 1 S=8+1S = 8 + 1 S=9S = 9

Explanation:

Substitute the figure number n=4n = 4 into the given function rule. First, multiply the figure number by 22 to get 88, then add 11 to find the total number of squares, which is 99.