Professional Courses
Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.
Categories
Interactive Games
Fun, engaging games to boost memory, math fluency, typing speed, and English skills—perfect for learners of all ages.
Typing
Memory
Math
English Adventures
Knowledge
Enroll to start learning
You’ve not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take practice test.
Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Associativity in Addition
Unlock Audio Lesson
Today, we will talk about the associative property of addition. Can anyone tell me what happens if we change the parentheses in an addition problem, like in a + (b + c)?
I think it doesn't change the answer, like a + (b + c) equals (a + b) + c?
Exactly! That's the essence of the associative property. It means that when we add, the way we group the numbers doesn't matter.
Can you show us a quick example?
Sure! If we take the numbers 2, 3, and 4: 2 + (3 + 4) equals 2 + 7, which is 9. Now, if I do (2 + 3) + 4, that's 5 + 4, which is also 9.
So it doesn't matter how we add the numbers?
Exactly! Remember, we can use the acronym **A-Add** to remind us of 'Associative Addition.'
Got it! A-Add means I can group numbers any way I want when adding.
Great! Let's summarize: In addition, you can change how you group your numbers and still get the same total.
Associativity in Multiplication
Unlock Audio Lesson
Next, let's look at multiplication. Can anyone explain if multiplication is also associative?
Like addition, it should be, right?
Absolutely! For example, 2 × (3 × 4) equals (2 × 3) × 4.
Can we use numbers to show that?
Sure! Let's do 2 × (3 × 4). The answer is 2 × 12, which equals 24. If we compute (2 × 3) × 4 instead, we get 6 × 4, which is also 24.
So multiplication is associative too!
Exactly! Use the mnemonic **M-Multiply** to remember 'Multiplicative Multiplication.'
Awesome! I will remember M-Multiply!
Non-associativity of Subtraction and Division
Unlock Audio Lesson
Now, let's discuss subtraction and division. Are these operations associative?
I don't think so! I've heard they behave differently.
That's right! For example, in subtraction, what can we say about 5 - (3 - 2) and (5 - 3) - 2?
They don't give the same answer!
Exactly! 5 - (3 - 2) equals 4, but (5 - 3) - 2 equals 0. Therefore, subtraction is not associative.
What about division?
Good question! Just like subtraction, division is also not associative. For instance, consider 8 ÷ (4 ÷ 2) and (8 ÷ 4) ÷ 2. They lead to different results.
So subtraction and division change based on grouping?
Exactly! Remember the phrase **S-Switch** to imply that Subtraction and Division do NOT stay the same! To summarize: Addition and multiplication are associative, while subtraction and division are not.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
The section covers the associative property in various operations such as addition and multiplication, highlighting which number sets are associative under these operations. It also explains the non-associativity of subtraction and division, providing examples for clarity.
Detailed
Associativity in Mathematics
The associative property states that the grouping of numbers does not affect the result of certain operations, namely addition and multiplication. This section emphasizes the significance of this property in whole numbers, integers, and rational numbers. For addition and multiplication, rearranging parentheses in expressions yields the same result, while subtraction and division do not share this property. The section details how the associative property applies differently across the number sets, illustrated with examples and providing insight into when students can expect to apply this mathematical principle.
Youtube Videos
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Associativity of Addition in Whole Numbers
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Recall the associativity of the four operations for whole numbers through this table:
| Operation | Numbers | Remarks |
|---|---|---|
| Addition | ......... | Addition is associative |
| Subtraction | ......... | Subtraction is not associative |
| Multiplication | Is 7 × (2 × 5) = (7 × 2) × 5? | Multiplication is associative |
| Is 4 × (6 × 0) = (4 × 6) × 0? | For any three whole numbers a, b and c a × (b × c) = (a × b) × c | |
| Division | ......... | Division is not associative |
Detailed Explanation
In this chunk, we are discussing the associativity of addition for whole numbers. Associativity refers to the property where the way in which numbers are grouped in addition (or multiplication) does not change the result. For whole numbers, when adding three numbers, it doesn't matter how we group them; we will always get the same sum. For example, when evaluating (2 + 3) + 4 and 2 + (3 + 4), both will yield 9. However, subtraction is not associative, meaning grouping affects the outcome.
Examples & Analogies
Think of adding fruits. If you have 2 apples, 3 oranges, and 4 bananas, it doesn't matter whether you first count the apples with oranges, or then add bananas. The total number of fruits remains the same. Conversely, if you were to take away fruits, the order in which you take them away can change how many remain (e.g., taking away 4 fruits from 5 is different from taking away 3 then 2).
Associativity of Multiplication in Whole Numbers
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Multiplication is associative. Is 7 × (2 × 5) = (7 × 2) × 5?
Is 4 × (6 × 0) = (4 × 6) × 0? For any three whole numbers a, b and c a × (b × c) = (a × b) × c.
Detailed Explanation
This chunk focuses on the multiplication of whole numbers and highlights that multiplication is associative. This means that when multiplying three numbers together, we can regroup them without affecting the result. For instance, (3 × 2) × 4 = 6 × 4 = 24, and 3 × (2 × 4) = 3 × 8 = 24; hence, both expressions yield the same product. This property allows flexibility in calculations and simplifications.
Examples & Analogies
Consider arranging boxes in configurations. If you have 2 rows of 3 boxes and 4 layers high, whether you calculate the boxes in layers or rows first, you will always end up with the same total boxes. Just like stacking the boxes does not change the total number.
Associativity in Integers
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Associativity of the four operations for integers can be seen from this table:
| Operation | Numbers | Remarks |
|---|---|---|
| Addition | Is (–2) + [3 + (– 4)] = [(-2) + 3)] + (– 4)? | Addition is associative |
| Subtraction | Is 5 – (7 – 3) = (5 – 7) – 3? | Subtraction is not associative |
| Multiplication | Is 5 × [(-7) × (–8)] = [5 × (–7)] × (–8)? | Multiplication is associative |
| Division | Is [(–10) ÷ 2] ÷ (–5) = (–10) ÷ [2 ÷ (–5)]? | Division is not associative |
Detailed Explanation
This chunk discusses the associativity of addition and multiplication for integers, explaining that both operations are associative similar to whole numbers. For example, rearranging how we add or multiply does not change the total. However, subtraction and division are not associative since changing grouping alters the outcome. For instance, 5 - (7 - 3) gives a different result compared to (5 - 7) - 3.
Examples & Analogies
Imagine planning a trip with multiple stops. If you add the distances by regrouping the stops, the overall distance can remain unchanged (e.g., adding leg of the trip), but if you were to subtract certain distances and change the order, you'd likely end up with a different total distance remaining.
Associativity in Rational Numbers
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Refer to the addition of rational numbers:
−2 [3 (−5)] = −2 + (3 + (−5)) = (−2 + 3) + (−5) = 1 + (−5) = −4.
For any three rational numbers a, b and c, a + (b + c) = (a + b) + c.
Detailed Explanation
This section shows how rational numbers also maintain the property of associativity for addition. No matter how we group the numbers, such as in the operation above, the result remains consistent. It's essential for calculations involving fractions, as we often add more than two fractions. This property ensures that we can compute sums in a flexible manner.
Examples & Analogies
Think about sharing pizza slices. If you're sharing slices among friends, how you group the sharing does not change the number of slices eaten. Whether you combine groups of 2 and 3 first or share them together, the total number of slices each person receives remains the same.
Checking Associativity in Subtraction and Division
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Subtraction is not associative for rational numbers. Is −2 [−4 1] = −3? Check for yourself. Division is not associative for rational numbers.
Detailed Explanation
In this part, we reiterate that subtraction and division do not hold the associative property in rational numbers. For instance, when you consider (3 - 2) - 1 or 3 - (2 - 1), you get different results based on how the numbers are grouped. Similarly, for division, dividing in different orders does not yield the same results, further solidifying that these operations lack associativity.
Examples & Analogies
Imagine you have several tasks to complete based on a deadline. If you decide to subtract certain hours from your pool of time without careful consideration of the steps involved, changing the order can lead to remaining time discrepancies. For division, think of splitting a pie among friends; the way you decide to divide the pie can yield different portions to individuals based on the order in which you split it.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Associativity of Addition: The sum remains unchanged regardless of how numbers are grouped.
-
Associativity of Multiplication: The product remains unchanged regardless of how numbers are grouped.
-
Non-Associativity of Subtraction: Changing the configuration changes the result.
-
Non-Associativity of Division: Changing the configuration changes the result.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
Example 1: For addition, (2 + 3) + 4 = 9 and 2 + (3 + 4) = 9.
-
Example 2: For multiplication, (3 × 2) × 4 = 24 and 3 × (2 × 4) = 24.
-
Example 3: For subtraction, 5 - (3 - 2) = 4 but (5 - 3) - 2 = 0.
-
Example 4: For division, 8 ÷ (4 ÷ 2) = 4 but (8 ÷ 4) ÷ 2 = 1.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
-
When you add or you multiply, group them how you like, oh my!
📖 Fascinating Stories
-
Once upon a time, numbers met in a village where addition and multiplication lived happily, always getting the same result no matter how the townsfolk grouped them together. But subtraction and division, they fought over how to group, and each time they did, they forgot their previous answers.
🧠 Other Memory Gems
-
Remember the acronym 'A-M-A': Addition-Multiplication are Associative!
🎯 Super Acronyms
S-NAT
- Subtraction and Division are Not Associative.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Associative Property
Definition:
A mathematical property that states that the way numbers are grouped in an operation does not change the result, applicable in addition and multiplication.
-
Term: NonAssociative
Definition:
Refers to operations (like subtraction and division) where changing the grouping of numbers results in different outcomes.