Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Introduction to Expanding Brackets
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Today, we're going to talk about expanding single brackets. Does anyone know what expanding brackets means?
I think itโs when we take something like (x + 2) and change it to a different form?
Exactly! When we expand brackets, we use the distributive property. This means we multiply the term outside the bracket by each term inside. For example, in 3(x + 5), how do we expand it?
We multiply 3 by x and then 3 by 5!
That's correct! So, what do we get?
It becomes 3x + 15!
Good job! Remember, we follow the rule: a(b + c) = ab + ac. Let's explore more examples together.
Real-life Applications
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Now, letโs talk about why we need to expand brackets. Can someone think of a situation where this might be necessary?
Maybe something like calculating total prices in a shop?
Exactly! If you see a deal that says 3 items for `x` dollars, you might express the total cost as 3(x). Expanding helps us figure out the total easily!
So when we expand, it helps us see the total in a clearer way?
Right! By expanding, we can simplify expressions that resemble equations in real life.
Solving Problems Together
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Let's do some practice problems together! First, can someone expand `-2(y - 4)`?
I think itโs -2y + 8?
Yes! Great job! Now for another example, try to expand `5a(2a + 3b)`.
I can do that! Itโs 10a squared + 15ab!
Excellent work! Remember, practice makes perfect.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
In this section, students learn the process of expanding single brackets. By applying the distributive property, they can simplify expressions involving brackets effectively. The section covers various examples and practice problems to reinforce the concept.
Detailed
Expanding Single Brackets
Expanding single brackets is a crucial algebraic skill that involves applying the distributive property of multiplication. The distributive property states that when you multiply a term by a sum inside brackets, you add together the products of the term and each of the individual terms in the brackets. This section provides examples and systematic steps to expand expressions effectively.
The Rule for Expanding Brackets:
For a general case, if you have an expression like a(b + c), the expansion would be:
$$ a(b + c) = ab + ac $$
Key Examples:
- Positive Term Outside: Expanding
3(x + 5)gives: - Step 1: Multiply 3 by x ->
3 * x = 3x - Step 2: Multiply 3 by 5 ->
3 * 5 = 15 -
Result:
3x + 15 -
Negative Term Outside: Expanding
-2(y - 4)involves: - Step 1: Multiply -2 by y ->
-2 * y = -2y - Step 2: Multiply -2 by -4 ->
-2 * -4 = +8 -
Result:
-2y + 8 -
Variable and Number Outside: Expanding
5a(2a + 3b), we see: - Step 1: Multiply ->
5a * 2a = 10a^2 - Step 2: Next,
5a * 3b = 15ab - Result:
10a^2 + 15ab
Practice Problems:
The section includes several practice problems designed to reinforce the skill of expanding single brackets, providing students with the opportunity to apply what they've learned and gain confidence in their abilities.
Audio Book
Dive deep into the subject with an immersive audiobook experience.
What is Expanding Single Brackets?
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Expanding brackets involves multiplying the term outside the bracket by every term inside the bracket. This is based on the distributive property of multiplication.
Rule: a(b + c) = ab + ac
Detailed Explanation
Expanding single brackets means taking a number or variable that's outside a pair of parentheses and multiplying it by every term inside the parentheses.
The essential rule here is the distributive property, which says that when you multiply a number (a) by a sum of two other numbers (b + c), itโs the same as multiplying the number by each of the two numbers separately and then adding the results together.
For example, if you have 2(x + 3), you would do:
1. Multiply 2 by x, which equals 2x.
2. Multiply 2 by 3, which equals 6.
3. Combine these results to get 2x + 6.
Examples & Analogies
Imagine you have a box of chocolates, where each chocolate costs $2, and you have 3 friends, each with their own chocolate. You can think of this as 2(x + 3) where 'x' represents the number of chocolates. When you multiply 2 by each person's chocolates, you're finding the total cost you need to pay for all the chocolates.
Example 1 - Positive Term Outside
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Expand: 3(x + 5)
- Step 1: Multiply 3 by x. 3 * x = 3x
- Step 2: Multiply 3 by 5. 3 * 5 = 15
- Step 3: Combine the results.
- Result: 3x + 15
Detailed Explanation
In this example, we take the number 3 that is outside the bracket and multiply it with both terms inside:
1. Start with the expression 3(x + 5).
2. First, multiply 3 by x to get 3x.
3. Next, multiply 3 by 5 to get 15.
4. Then, combine these two parts (3x and 15) to arrive at the final result of 3x + 15.
Examples & Analogies
Think of having three bags, each containing 5 apples. If you want to know how many apples you have in total, instead of counting them individually, you can say 3 bags times (x + 5) apples, which allows you to quickly calculate a total of 15 apples added to any number of apples represented by 'x' from each bag.
Example 2 - Negative Term Outside
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Expand: -2(y - 4)
- Step 1: Multiply -2 by y. -2 * y = -2y
- Step 2: Multiply -2 by -4. (Remember: negative times negative is positive) -2 * -4 = +8
- Step 3: Combine the results.
- Result: -2y + 8
Detailed Explanation
Here, we are looking at multiplying a negative number with the terms inside the parentheses:
1. Start with -2(y - 4).
2. First, multiply -2 by y to get -2y.
3. Then, multiply -2 by -4. When two negatives are multiplied, the result is a positive, so -2 * -4 gives you +8.
4. Finally, combine these results to get -2y + 8.
Examples & Analogies
Imagine you owe your friend money every time they give you a gift. If your friend gives you 4 gifts, the -2 signifies you owe them $2 per gift. Hence, you can express this as -2(y - 4), allowing you to quickly calculate the total amount you owe when you subtract the value of each negative transaction.
Example 3 - Variable and Number Outside
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Expand: 5a(2a + 3b)
- Step 1: Multiply 5a by 2a. 5 * 2 = 10; a * a = a squared, so 5a * 2a = 10a squared
- Step 2: Multiply 5a by 3b. 5 * 3 = 15; a * b = ab, so 5a * 3b = 15ab
- Step 3: Combine the results.
- Result: 10a squared + 15ab
Detailed Explanation
In this example, we go a bit deeper because we are dealing with variables as well:
1. Start with 5a(2a + 3b).
2. First, multiply 5a by 2a, which results in 10aยฒ (since multiplying variables means adding their exponents).
3. Next, multiply 5a by 3b which gives you 15ab.
4. Finally, bring both results together to get 10aยฒ + 15ab.
Examples & Analogies
If you are making smoothie packs, and each pack contains a mixture of 2 apples and 3 bananas, represented by 5a, you can multiply the ingredients together to determine how many pieces of fruit you have per smoothie and simplify your shopping list.
Practice Problems
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
- Expand: 4(w + 7)
- Expand: -5(2k - 3)
- Expand: 7x(x + 2y)
- Expand and simplify: 3(m + 4) + 2(m - 1)
Detailed Explanation
These practice problems allow students to apply what they've learned about expanding brackets:
1. For each problem, follow the steps of multiplying the outside term by each term inside the brackets, just as weโve practiced in the examples.
2. Make sure to combine like terms where necessary to simplify your final expression.
Examples & Analogies
These practice problems simulate real-world scenarios where you might need to combine amounts. For example, if you're planning for a party and need to know how many snacks to buy based on the number of guests, performing these expansions helps calculate the right quantity.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Distributive Property: The method of multiplying a term across terms within brackets.
-
Expansion: The process of removing brackets by distributing the term outside.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
Expanding
3(x + 5)gives3x + 15. -
Expanding
-2(y - 4)results in-2y + 8. -
Expanding
5a(2a + 3b)leads to10a^2 + 15ab.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
๐ต Rhymes Time
-
Brackets we must expand, multiply through as we planned!
๐ Fascinating Stories
-
Imagine a chef distributing ingredients evenly for multiple servings; that's how we expand!
๐ฏ Super Acronyms
DISTRIBUTE - Distribute each ingredient, Multiply outside, and then Sum it up!
E.B.E โ Expand Brackets Everywhere!
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Distributive Property
Definition:
A property that states a(b + c) = ab + ac, allowing us to multiply a single term across terms inside a bracket.
-
Term: Expand
Definition:
To express an expression in a simpler form without brackets.
-
Term: Bracket
Definition:
A symbol used to group parts of an expression, typically appearing as ( ) in algebra.