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1.2.6 - Distributivity of multiplication over addition for rational numbers

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Understanding the Distributive Property

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Teacher
Teacher

Today, we are going to explore an important property of multiplication called the Distributive Property. Who can tell me what they think it means?

Student 1
Student 1

Is it how multiplication works with addition, like when you can multiply a number by a sum?

Teacher
Teacher

Exactly! The distributive property states that a(b + c) = ab + ac. This means if you have a number a multiplied by the sum of b and c, you can distribute 'a' to both b and c. Let's use rational numbers to illustrate this.

Student 2
Student 2

Can we see a specific example?

Teacher
Teacher

Sure! Let’s use a = -3/4, b = 2/3, and c = -5/6. If we compute (-3/4) × (2/3 + -5/6), we can distribute it as follows. It will be: (-3/4) × (2/3) + (-3/4) × (-5/6).

Student 3
Student 3

So we multiply separately and then add the results?

Teacher
Teacher

Exactly! Great understanding!

Teacher
Teacher

Let’s recap today’s lesson: The distributive property helps us simplify multiplication over addition effectively.

Application of Distributive Property

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Teacher
Teacher

Now, how do we use this property in solving problems? Can someone give an example problem?

Student 4
Student 4

What if I have to calculate -2 × (5 + 1)?

Teacher
Teacher

Great! Using the distributive property, we can rewrite it as -2 × 5 + -2 × 1. Can anyone calculate that?

Student 1
Student 1

That would be -10 + -2, so -12!

Teacher
Teacher

Correct! This shows how simplifying or expanding expressions can make the calculations easier and more efficient.

Student 2
Student 2

Does it work with subtraction too?

Teacher
Teacher

Absolutely! You can distribute over subtraction as well. For example, a(b - c) = ab - ac.

Teacher
Teacher

Let’s confirm what we learned today: Distributive property applies to both addition and subtraction.

Illustrating Zero and One

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Teacher
Teacher

Let’s talk about how the numbers zero and one interact with the distributive property? What can we say about multiplying with zero?

Student 3
Student 3

If you multiply anything by zero, the answer is zero!

Teacher
Teacher

Correct! So, if we distributed a number multiplied by zero, say a(0 + b) would equal ab + 0? What does that simplify to?

Student 4
Student 4

That simplifies just to ab, right?

Teacher
Teacher

Exactly! Now, what happens when multiplying by one?

Student 1
Student 1

It stays the same! Like one is the identity for multiplication.

Teacher
Teacher

Yes, it respects the distributive property too! So remember both zero and one play special roles.

Teacher
Teacher

Today, we learned how zero and one relate to the distributive property.

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

The section introduces the distributive property of multiplication over addition for rational numbers, establishing how it works through various examples.

Standard

In this section, the distributive property of multiplication over addition for rational numbers is discussed. Examples illustrate how multiplying a rational number by a sum can be expressed as the sum of products, and this property is crucial for simplifying expressions in algebra.

Detailed

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Number System ( Natural Number, Whole Number, Integer, Rational Number, Irrational Number, Real)
Number System ( Natural Number, Whole Number, Integer, Rational Number, Irrational Number, Real)

Audio Book

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Introduction to Distributivity

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To understand this, consider the rational numbers −3/4, 2/3, and −5/6.

Detailed Explanation

Distributivity is a property that involves how multiplication interacts with addition. This property states that when you multiply a number by a sum, it is the same as multiplying each addend separately and then adding the results. In this case, we are looking at the rational numbers -3/4, 2/3, and -5/6 to demonstrate the property.

Examples & Analogies

Think of distributivity like sharing a pizza among friends. If you have three friends and you buy a pizza that costs $10, you can either calculate the total cost by multiplying $10 by 3 (for all the pizzas) or by splitting the pizza first into 3 equal slices, counting how much each person gets, and then figuring the total cost based on the slices they took.

Applying Distributivity

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−3 × (2/4 + −5/6) = −3 × (8/24 + −20/24) = −3 × (−12/24).

Detailed Explanation

In this step, we are applying the distributive property. We take −3 and multiply it by the sum of 2/4 and -5/6. The first step is to find a common denominator, which is 24 in this case, so we convert both fractions to have this denominator. After simplifying, we multiply −3 by the resulting fraction -12/24, which simplifies to -1/2.

Examples & Analogies

Imagine you're distributing snacks among two groups of friends. You have 12 cookies (representing the total value of -12 as a negative because it’s a loss in terms of costs). Distributing them evenly among your friends demonstrates how you can break down the process—each friend gets cookies separately just like we distributed values in math.

Final Calculation

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Therefore, −3 × (2/4 + −5/6) = −3 × 2/4 + −3 × −5/6 = 1/2.

Detailed Explanation

Finally, we compute separately the products after distributing the multiplication. We calculate the parts: -3 times 2/4 gives us a product of -3/2 and for -3 times -5/6 gives us positive results. Adding these results together gives us 1/2, concluding our property demonstration.

Examples & Analogies

Returning to our pizza analogy, if you started with a negative amount (like owing cookies) and then share them evenly, your friends could potentially trade cookies back and forth, balancing each other's snack supply to finally see what’s left, just as we balance equations in math.

General Formula of Distributivity

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For all rational numbers a, b, and c, a × (b + c) = ab + ac and a × (b - c) = ab - ac.

Detailed Explanation

The distributive property can be generalized. It means you can multiply any number by a sum or difference of other numbers and achieve the same result, allowing flexibility in computation. This formula helps simplify calculations and is foundational in algebra.

Examples & Analogies

Imagine you have a job that pays you weekly based on hours worked. If you work overtime, instead of calculating the full amount each time, you can just calculate it based on the basic hours and multiply that figure by both the usual work and the overtime, thus confirming you’re using distributivity to arrive at your final paycheck.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Distributive Property: It allows multiplication to distribute over addition, making calculations easier.

  • Rational Numbers: Fractions that can be expressed as a ratio of two integers.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • Example 1: a = -3/4, b = 2/3, c = -5/6. Calculate (-3/4) × (2/3 + -5/6)

  • Example 2: Using the distributive property, calculate -2 × (5 + 1) as -2 × 5 + -2 × 1.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • If you see a sum, and multiplication is near, distribute it well and have no fear!

📖 Fascinating Stories

  • Once there was a number, let's call it a, who had two friends b and c. A wanted to share its gifts equally with b and c, so it multiplied by their total; together, they became an even better sum.

🧠 Other Memory Gems

  • Remember: D for Distributive, A for Addition, S for Subtraction, helps us keep the numbers in the right function!

🎯 Super Acronyms

DAS

  • Distributive add then solve! An easy way to recall how to apply it.

Flash Cards

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Glossary of Terms

Review the Definitions for terms.

  • Term: Distributive Property

    Definition:

    A property that states a(b + c) = ab + ac, allowing multiplication to be distributed over addition or subtraction.

  • Term: Rational Number

    Definition:

    A number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero.