Write and construct formulae - Year 7 PDF Download

Key Points

• Understanding algebraic notation and the order of operations (BIDMAS) will help in using formulae successfully.
• A formula is a rule that links variables in a mathematical relationship. For example, the area, length, and width of a rectangle are connected by a formula. When two of the variables are given, the third can be worked out using the formula.
• Numbers are substituted into a formula to find the value of the subject.
• A formula can be rearranged to change the subject. For instance, the formula for the area of a circle (A = πr²) can be rearranged to make the radius the subject, as in r = √(A/π).

Using Formulae Given in Words

A formula provided in words offers the necessary instructions to calculate something. To apply a written formula:

• Substitute a given value for the variable.
• Follow each step in the formula instructions to determine the subject of the formula.

Examples

• Image caption: The perimeter of a regular polygon is the length of one side multiplied by the number of sides. Use this formula to find the perimeter of a regular pentagon with a side length of 12 cm.

• Image caption: Substitute the given values for the length of one side (12) and the number of sides (5). Follow the formula instruction (multiply 12 by 5), 12 x 5 is 60. The perimeter of the regular pentagon is 60 cm.

• Image caption: A plumber places an advert with a formula for the charges made. £35 per hour plus a call out charge of £60. Find the total cost of a call that takes three hours to complete.

• Image caption: A plumber places an advert with a formula for the charges made. £35 per hour plus a call out charge of £60. Find the total cost of a call that takes three hours to complete.

Plumber Charges Calculation

• Hourly Rate Calculation: The plumber charges £35 per hour. For 1 hour, the charge is 1 x £35, for 2 hours 2 x £35, and for 3 hours 3 x £35, and so on. The total cost increases as the call-out charge must be included.

• Total Cost Composition: The total cost comprises the cost of the time taken and the call-out charge. The call-out charge amounts to £60, which is added to the cost based on the number of hours the job takes.

• Calculation of Total Cost: Following each step in the formula instructions, you multiply the cost per hour (£35) by the number of hours taken (3) and then add the call-out charge (£60). For a three-hour call, the total cost amounts to £165.

Pie Cooking Time Calculation

• Cooking Time Formula: The time to cook a pie is determined by a formula based on its mass. To find the cooking time for a 1000g pie, follow the formula instructions.

Image Caption: Cooking Time Calculation

• Substitute the mass of the pie (1000) into the formula. Follow each step in the formula. Divide the mass of the pie (1000) by 500; this gives how many 500g of pie there are. Then multiply by 25. Finally, add 15 to get the total cooking time.

Image Caption: Cooking Time Calculation (Continued)

• Substitute the mass of the pie (1000) into the formula. Follow each step in the formula. Divide 1000 by 500, which equals 2 (two 500g portions of pie). Multiply this by 25, then add 15. The total cooking time is 65 minutes.

Example 1: Perimeter of a Regular Polygon

• An image of a regular pentagon is shown. The perimeter of a regular polygon is calculated by multiplying the length of one side by the number of sides.

Calculation for a Regular Pentagon

• Use the formula to find the perimeter of a regular pentagon with a side length of 12 cm.

Constructing formulae from words

• A variable or symbol is assigned for each element.
• The formula always commences with the primary variable followed by the equality symbol.
• The statement following the equality sign delineates each instruction in the written formula using appropriate algebraic notation. The variables must be explicitly defined.

Confidence in transcribing mathematical formulae from a verbal statement fundamentally hinges on a solid comprehension of algebraic notation.

Examples

• Construct an algebraic formula for speed.

Construct an algebraic formula for speed.

Physics Formulas

• Speed Calculation: The formula for calculating speed involves dividing the distance traveled (D) by the time taken (T). This can be expressed as: S = D/T.
• 
• Pie Cooking Time: The time required to cook a pie depends on its mass. Typically, you cook for 25 minutes per 500g of mass, along with an additional 15 minutes.
• 

Variables and Formulas

• Symbol Assignments: In these formulas, specific letters or symbols are assigned to represent different variables. For instance, T denotes the time taken to cook the pie, while M stands for the mass of the pie in grams. The standard format involves starting with the subject variable, followed by the equals symbol. In the case of the time formula, it initiates with T =.
• 

Paraphrased Content

Time Calculation Formula for Cooking

• When cooking a pie, each 500g portion requires 25 minutes in the oven. To determine the total time needed for a given mass (M) of pie, divide the mass by 500 (M/500), then multiply by 25. Add an extra 15 minutes to this result. The simplified formula is T = (M/500) * 25 + 15.

Simplified Time Calculation Formula

• By simplifying the previous formula, dividing the mass by 500 and then multiplying by 25 reduces to dividing the mass by 20. This simplification arises from recognizing that 25/500 is the same as 1/20 (both 500 and 25 divided by 25). The formula can be expressed as T = M/20 + 15.

Example: Speed Calculation

• Speed is calculated as the distance traveled divided by the time taken. To construct an algebraic formula for speed, you divide the distance by the time.

Question

Substitution using Formulae

A formula is a rule expressed in words or mathematical symbols. It involves substituting values into the formula to solve problems.

Definition of Formula

• A formula is a mathematical rule or principle represented using symbols or words.
• It helps in solving mathematical problems by substituting values into the given rule.
• Formulas are essential tools in algebra and other branches of mathematics.

Understanding Substitution

• Substitution in algebra is the act of replacing a variable with a specific number.
• It simplifies complex expressions and equations by assigning concrete values to variables.
• Example: In the formula 2x + 3, if x = 4, then after substitution, the expression becomes 2(4) + 3 = 11.

Application of Formulas

• Formulas are used in various fields like physics, chemistry, and engineering to solve problems.
• They enable us to make calculations, predictions, and understand relationships between different quantities.
• Example: The formula for calculating the area of a circle is A = πr², where r is the radius.

To use an algebraic formula effectively, it's crucial to understand the concept of substitution and how it simplifies mathematical expressions.

Variable Substitution

• When you substitute a given value for a variable, you are essentially replacing the variable with a known quantity.

Working with Formulas

• Follow each step outlined in a formula to correctly solve for the desired quantity.
• It's crucial to maintain the correct order of operations when dealing with formulas that involve multiple steps.

Examples

• Image caption: Formulas are sets of rules expressed using mathematical symbols.
• Image caption: Utilize a formula to determine the area of a rectangle given its dimensions.
• Image caption: By substituting the provided values into the formula, you can find the area of a rectangle.

Image caption: Formulas are rules written using mathematical symbols.

Image caption: The length of the rectangle is 15 cm. The width of the rectangle is 3 cm. Substitute the given values to calculate the area.

Image caption: The length of the rectangle is 15 cm. The width of the rectangle is 3 cm. Follow the formula to determine the area.

Geometry and Measurement Concepts

• Understanding Area Calculation:

  Calculate the area of a rectangle by multiplying its length by its width. For instance, if the length is 15 and the width is 3, the area would be 45 cm².

• Volume Calculation for a Cuboid:

  To determine the volume of a cuboid, multiply its length, width, and height. For a cuboid with dimensions 10 cm, 4 cm, and 7 cm respectively, the volume would be 280 cm³.

Area Calculation Example

Cuboid Volume Calculation Example

Speed Calculation Example

Physics Concepts Overview

Formulae are rules expressed using mathematical symbols for various physical quantities and relationships.

• Area Calculation:
  • A = l * w
• Volume Calculation:
  • V = l * w * h
• Speed Calculation:
  • S = D / T

Question

Explain the following physics concepts in detail with practical examples:

• Area Calculation (A = l * w):
• When calculating the area of a rectangle, you multiply its length by its width. For instance, if a rectangle has a length of 4 units and a width of 3 units, the area would be 12 square units (4 * 3 = 12).
• Volume Calculation (V = l * w * h):
• Calculating the volume of a rectangular prism involves multiplying its length by its width and then by its height. If the dimensions are 2 units by 3 units by 4 units, the volume would be 24 cubic units (2 * 3 * 4 = 24).
• Speed Calculation (S = D / T):
• Speed is determined by dividing the distance traveled by the time taken. For example, if an object covers 100 meters in 20 seconds, its speed would be 5 meters per second (100 / 20 = 5).

Formulae are essential tools in physics, providing a concise and precise way to represent relationships between different physical quantities.

Practise writing and constructing formulae

Quiz

Practise writing and constructing formulae with this quiz. You may need a pen and paper to help you with your answers.

Real-life maths

Formulae are used often in real life, for example when calculating the amount of paint to decorate a room, the size of a radiator to heat a room or the amount of ribbon needed to add to a wrapped gift.

A decorator buys the amount of paint needed to complete a job. A litre of emulsion paint will cover approximately 12 square metres (12 m²).

To work out how many litres of paint are required, the formula is:

𝑃 =

𝑃 = number of litres of paint needed

ℎ = height of walls

𝑤 = total width of all walls

c = number of coats of paint required

For a room that will need three coats of paint with a wall height of 2.08 metres and a total wall width of 14 metres, the calculation will be

The paint required is 9.08 litres.

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