Welcome to the World of Averages!
Hello super mathematicians! Ever wondered how your teacher figures out your overall grade from many different tests? Or how we can talk about the 'typical' weather for a month? The secret is a cool maths tool called the average!
In these notes, we're going to learn what an average is, how to find it, and how to use it backwards and forwards to solve tricky problems. You've got this!
What is an Average?
Think of an average as a way to find a single, 'middle' or 'typical' number that can represent a whole group of numbers. It's like finding the 'fair share'.
The "Fair Share" Idea
Imagine you and three friends go picking apples.
- You pick 4 apples.
- Sam picks 6 apples.
- Lily picks 2 apples.
- Tom picks 4 apples.
Everyone has a different amount! If you wanted to share them all equally, you would first put them all into one big basket (that's finding the total) and then divide them evenly among the four of you. The number of apples each person gets is the average.
Looking at Averages with Charts
Sometimes, we get information from charts. Let's say this bar chart shows the number of books read by four children in a month.
Imagine four bars on a chart with heights of 4, 6, 2, and 4.
The average is like leveling out all the bars so they are the same height. To do this, we'd take some from the tallest bar (Sam's 6 books) and give them to the shortest bar (Lily's 2 books) until everyone's bar is the same height. That final, even height is the average!
Key Takeaway
An average is a single number that represents the 'fair share' or 'middle value' of a group of numbers.
How to Find the Average
Finding the average is like following a simple, two-step recipe. All you need to know is how to add and how to divide! The main formula is:
\(\text{Average} = \frac{\text{Total of all data}}{\text{Number of items}}\)
A Simple Trick to Remember
To find the average, you just need to remember two words: Add, then Divide!
Step-by-Step Guide
Follow these simple steps, and you'll be an averages expert in no time!
Step 1: ADD them up!
Add all the numbers in your group together. This gives you the total.
Step 2: COUNT how many!
Count how many numbers you just added. This is the number of items.
Step 3: DIVIDE!
Divide the total (from Step 1) by the number of items (from Step 2). The answer is your average!
Worked Example 1: Quiz Scores
Maria scored 7, 9, and 8 in three different maths quizzes. What is her average quiz score?
Let's follow the steps:
- ADD: Find the total score.
7 + 9 + 8 = 24 - COUNT: Count the number of quizzes.
There are 3 quizzes. - DIVIDE: Divide the total score by the number of quizzes.
24 ÷ 3 = 8
Answer: Maria's average quiz score is 8.
Worked Example 2: Collecting Shells
Five friends collected shells at the beach. They found 10, 15, 8, 12, and 10 shells. What is the average number of shells they found?
Let's use our steps again:
- ADD: Find the total number of shells.
10 + 15 + 8 + 12 + 10 = 55 - COUNT: Count the number of friends.
There are 5 friends. - DIVIDE: Divide the total shells by the number of friends.
55 ÷ 5 = 11
Answer: The average number of shells found is 11.
The Average Triangle: Working Backwards
At Primary 6, questions often ask you to work backwards! If you know two of the quantities, you can always find the third using the Average Triangle:
- \(\text{Average} = \text{Total} \div \text{Number of items}\)
- \(\text{Total} = \text{Average} \times \text{Number of items}\)
- \(\text{Number of items} = \text{Total} \div \text{Average}\)
Worked Example 3: Finding the Total
The average mass of 4 parcels is 15 kg. What is the total mass of all 4 parcels?
- Identify the values: Average = 15 kg, Number of items = 4.
- Multiply: Total = \(15 \times 4 = 60\text{ kg}\).
Answer: The total mass of the parcels is 60 kg.
Worked Example 4: Finding an Unknown Value (Change in Average)
Ali took 3 tests and got an average score of 70. After taking a 4th test, his new average score became 75. What did Ali score on his 4th test?
- Find the first total (3 tests): \(3 \times 70 = 210\).
- Find the new total (4 tests): \(4 \times 75 = 300\).
- Find the difference: \(300 - 210 = 90\).
Answer: Ali scored 90 on his 4th test.
Solving Average Problems
Now that you know how to find both the average and the total, let's look at another word problem.
Quick Review Box
- What is an average? A 'fair share' or 'typical' value.
- To find Average: Total ÷ Number of Items.
- To find Total: Average × Number of Items.
Word Problem: Daily Exercise
Ben wants to know his average time spent playing outside during the week. He played for 30 minutes on Monday, 25 minutes on Tuesday, 40 minutes on Wednesday, 25 minutes on Thursday, and 50 minutes on Friday. What was his average daily playtime?
1. Find the numbers: 30, 25, 40, 25, 50.
2. ADD them up: 30 + 25 + 40 + 25 + 50 = 170 minutes.
3. COUNT the days: There are 5 days.
4. DIVIDE: 170 ÷ 5 = 34 minutes.
Answer: Ben's average playtime per day was 34 minutes.
Common Mistakes to Avoid!
- Forgetting to divide: A common mistake is to add all the numbers up and then stop! Remember, the average should be a number that is somewhere in the middle of your original numbers, not the big total.
- Using the wrong count: When solving multi-step problems, make sure you multiply or divide by the correct number of items (for example, whether an item was added or removed!).
Did you know?
The word "average" has been around for hundreds of years! It comes from an old word used by sailors that meant sharing the cost of any damage to a ship's cargo fairly among all the merchants. So, the idea of a 'fair share' has always been part of what an average means!
Key Takeaway
You are now an averages expert! Remember the connection between Total, Average, and Number of Items. Keep practising both finding the average and working backwards to find the total, and it will become second nature. Well done!