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Algebra - Algorithms and Flowcharts

Grade 8IB

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

🔑Concepts

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An algorithm is a finite, step-by-step procedure to solve a specific mathematical problem, such as solving a linear equation like ax+b=cax + b = c.

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A flowchart is a visual representation of an algorithm using standard symbols: Oval (Start/End), Parallelogram (Input/Output), Rectangle (Process/Calculation), and Diamond (Decision/Condition).

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Variables in algorithms act as containers for values. Assignment is often represented as x←5x \leftarrow 5, which means 'store the value 55 in variable xx'.

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Selection (Decision) structures allow the algorithm to branch based on a condition, such as comparing two values: x>yx > y or n(mod2)=0n \pmod{2} = 0.

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Iteration (Looping) is the process of repeating a sequence of steps. For example, to find the sum of the first nn integers, an algorithm might repeat the operation sum←sum+isum \leftarrow sum + i for i=1i = 1 to nn.

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Dry Running is the process of manually tracing the steps of an algorithm with specific input values to verify its correctness.

📐Formulae

x=c−bax = \frac{c - b}{a}

S=n(n+1)2S = \frac{n(n + 1)}{2}

Area=πr2Area = \pi r^2

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

D=b2−4acD = b^2 - 4ac

💡Examples

Problem 1:

Create an algorithm to solve for xx in the linear equation ax+b=cax + b = c, where a≠0a \neq 0.

Solution:

  1. Start. 2. Input values for aa, bb, and cc. 3. Calculate temp=c−btemp = c - b. 4. Calculate x=tempax = \frac{temp}{a}. 5. Output xx. 6. End.

Explanation:

The algorithm follows the algebraic steps of isolation: first subtract bb from both sides, then divide by aa to find xx.

Problem 2:

Design a flowchart logic to determine if a given integer NN is even or odd.

Solution:

  1. Start. 2. Input NN. 3. If N(mod2)=0N \pmod{2} = 0, then Output 'Even'. 4. Else, Output 'Odd'. 5. End.

Explanation:

The modulo operator (mod2)\pmod{2} returns the remainder when NN is divided by 22. If the remainder is 00, the number is even.

Problem 3:

Trace the values of SS and ii in an algorithm to find the sum of the first 33 natural numbers where S←0S \leftarrow 0 and ii goes from 11 to 33.

Solution:

Initial: S=0,i=1Step 1: S=0+1=1Step 2: S=1+2=3Step 3: S=3+3=6Final Sum: 6\begin{array}{r} \text{Initial: } S = 0, i = 1 \\ \text{Step 1: } S = 0 + 1 = 1 \\ \text{Step 2: } S = 1 + 2 = 3 \\ \text{Step 3: } S = 3 + 3 = 6 \\ \hline \text{Final Sum: } 6 \end{array}

Explanation:

The loop adds the current value of the counter ii to the running total SS in each iteration until ii exceeds 33.