Mathematics - Term 3 - Week 6

Problem solving in mathematics involves identifying a problem, devising a plan, carrying out the plan, and reviewing the solution. It is essential to approach problems systematically. Visual models, such as diagrams and charts, can help clarify complex ideas and make abstract concepts more tangible. For instance, when solving word problems, drawing a picture or using a table can help organize information and identify relationships.

To find the area of a triangle, we use the formula: Area = 1/2 × base × height. Let's say we have a triangle with a base of 10 cm and a height of 5 cm. Substituting these values into the formula gives us: Area = 1/2 × 10 × 5 = 25 cm². This example illustrates how to apply a formula and shows the importance of clearly stating each step in the solution process.

Consider the following problem: A farmer has 120 meters of fencing to create a rectangular garden. What dimensions will maximize the area of the garden? Start by letting the length be 'l' meters and the width be 'w' meters. The perimeter is given by the equation 2l + 2w = 120. Rearranging gives us w = 60 - l. The area A can be expressed as A = l(60 - l). Now, we can find the maximum area by completing the square or using calculus. Work through this problem step by step, discussing the importance of each stage in the solution.

1. Calculate the circumference of a circle with a radius of 7 cm. Use the formula C = 2πr. 2. A train travels 150 km in 2 hours. What is its average speed? 3. If a rectangle has a length of 8 cm and a width of 3 cm, what is its perimeter? 4. Solve for x in the equation 3x + 5 = 20. Show all working clearly.

• 40 cm²
• 30 cm²
• 50 cm²
• 20 cm²

Answer: 40 cm²

Area = length × width = 10 cm × 4 cm = 40 cm².

• A = πr²
• A = 2πr
• A = r²
• A = 2r

Answer: A = πr²

The area of a circle is calculated using the formula A = πr², where r is the radius.

Answer: 8

Subtracting 7 from both sides gives x = 15 - 7 = 8.

• 20 cm
• 25 cm
• 15 cm
• 10 cm

Answer: 20 cm

Perimeter of a square = 4 × side length = 4 × 5 cm = 20 cm.

Answer: 7

Adding 3 to both sides gives 2y = 14, then dividing by 2 gives y = 7.

• 100 km/h
• 90 km/h
• 80 km/h
• 70 km/h

Answer: 100 km/h

Average speed = total distance / total time = 300 km / 3 h = 100 km/h.

Answer: 25

3² = 9 and 4² = 16, so 9 + 16 = 25.

• y = mx + b
• y = x²
• y = sin(x)
• y = e^x

Answer: y = mx + b

The equation y = mx + b is the slope-intercept form of a linear equation.