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There are three basic number properties that help to form the backbone of algebra. One of those properties is the associative property.

This lesson will define the associative property and give some examples of how it works.

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I Call Shotgun

How many times have you been out with friends and heard, or yelled, that statement: ‘I call shotgun!’? The unquestioned right to sit in the front seat and have room to stretch your legs and de facto control of the radio. At the next stop, someone else may call out for their turn in the front seat.I bet the thought never crossed your mind that sitting in different seats in the vehicle did not change the makeup of the people that were with you.

They are all the same, no matter where they are sitting. Bob is still Bob wherever he is riding, and Jody is still herself, even if she’s no longer in control of what music you are listening to.

Who’s in Charge of the Math Radio?

So, who’s in charge of the math radio? There is a mathematical rule governing that very question! OK, it’s not presented in those terms, but it means the same. And the answer is the same as when you are out with your friends.

But first, let’s do a bit of background.In mathematics, there are three basic principles for how equations work. They form the backbone of all higher math. These properties are:; The commutative property; The associative property; The distributive propertyThey all govern different aspects of how you can manipulate and solve mathematical equations correctly.

The Associative Property

The associative property is the focus for this lesson. It states that terms in an addition or multiplication problem can be grouped in different ways, and the answer remains the same.

In other words, it doesn’t matter which terms are in the back seat and which are in the front – the makeup of the equation is the same, just like it was in the car in the introduction.Let’s look a little bit at how that works. If you have an addition problem such as (3 + 6) + 13, you can also write it as 3 + (6 + 13), and, when you solve the problem, the answer will be the same either way: 22. Remember that the parenthesis signify the portion of the problem that should be completed first.So, the associative property states that it doesn’t matter which portion of the problem you do first, the answer will be the same. Again, this only works with addition and multiplication problems, and not if they are mixed.

With mixed operations, you need to always follow the order of operations, which is: multiplication and division then addition and subtraction.However, if your problem contains only addition or only multiplication, you can group them in any way and still get the same answer.Look at this next example: (2 * 5) * 7 is the same as 2 * (5 * 7). It doesn’t matter if you multiply the 2 and 5 first or the 5 and 7 first, the answer is still 70.Let’s try it both ways:(2 * 5) * 7 = 10 * 7, or 70.

2 * (5 * 7) = 2 * 35, which is also 70.The associative property can work with subtraction, but only if you convert your subtraction problem to an addition problem. If you remember, subtraction is the opposite of addition. Because of this, you can turn any subtraction problem into an addition problem.

This means that 6 – 3 is equal to 6 + (-3).When you turn your subtraction problem into an addition problem, you can use the associative property to rearrange the groupings.

Why is This Important?

With simple problems such as these, you might be scratching your head, wondering why mathematicians go to all this trouble; why is the associative property so important?While these examples might seem simplistic, the associative property can be very useful when working with more complicated problems. When mathematicians or scientists or engineers are working with a complex equation, it can help them with the solution if they are sure that they can regroup the terms without negatively affecting the outcome of the problem.

Lesson Summary

The associative property states that in addition and multiplication problems, the grouping of the terms does not matter to the final outcome of the problem. You can group the terms in any order and still obtain the correct answer.

The property is also true for subtraction if you convert your subtraction problem to an addition problem and are very careful to keep the negative with the correct number.

Learning Outcomes

After you’ve completed this lesson, you should be able to:

  • Understand what the associative property states and when you can use it
  • Remember the order of operations
  • Use the associative property with subtraction problems
  • Determine when the associative property can be helpful

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