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1.03 Properties of number operations

Lesson

One of the hardest things to do in mathematics is to not use the calculator. It's just so much easier than working things out in your head. But what if you don't have a calculator or simply aren't allowed one? Is it even possible to solve something like $24\times13+78\times13$24×13+78×13 in your head?

Yes. By using mental arithmetic strategies we can make questions easier by changing the way we approach them.

But we don't have the tools to solve something like this yet. Let's start with some simple mental strategies to build up our tool kit.

 

The associative property

Try calculating $29+38+12$29+38+12 in your head. Which two numbers did you add together first?
If you added the $38$38 and $12$12 together first, you were using the associative property!
The associative property lets us evaluate the operations in any order, so long as the operations are all addition or all multiplication. Mathematically, this looks like we are adding or moving parentheses to change which pair of numbers we add or multiply first.

The Associative Property for Addition

 

Careful!
This property doesn't apply to subtraction or division.
For example: $22-7-5\ne22-(7-5)$227522(75) and $24\div6\div2\ne24\div(6\div2)$24÷​6÷​224÷​(6÷​2).

 

The commutative property

Which of the following are true?

a) $4+3=3+4$4+3=3+4
b) $4-3=3-4$43=34
c) $4\times3=3\times4$4×3=3×4
d) $4\div3=3\div4$4÷​3=3÷​4

If you answered a) and c), you're correct!

These two show applications of the commutative property which lets us swap the numbers on either side of the operation. Notice that since only a) and c) are true, the commutative property can be used for addition and multiplication, but not subtraction and division.

The Commutative Property for Multiplication

The distributive property

The distributive property is useful for distributing expressions such as $3\times\left(5+2\right)$3×(5+2). The distributive property applies the operation outside the parentheses to the expression inside it in order to distribute the parentheses.

The Distributive Property for Multiplication and Addition

While this property is not very useful on its own until we do algebra (especially if we only use small numbers), it can be very useful for mental arithmetic when combined with some clever tricks.
Is there an easy way to solve $102\times13$102×13?
There is! The trick is find an "easy to multiply by" number close to $102$102. In this case we can use $100$100. So instead of trying to calculate $102\times13$102×13 we can instead calculate $\left(100+2\right)\times13$(100+2)×13. We can then use the distributive property to distribute the parentheses to get $\left(100\times13\right)+\left(2\times13\right)$(100×13)+(2×13) and from there it's just simple multiplication and addition.

The distributive property can be used to distribute the parentheses when they are being multiplied from either side or if they are being divided by from the right side.

For example:

$(24+6)\times2$(24+6)×2 $=$= $(24\times2)+(6\times2)$(24×2)+(6×2)
$2\times(24+6)$2×(24+6) $=$= $(2\times24)+(2\times6)$(2×24)+(2×6)
$(24+6)\div2$(24+6)÷​2 $=$= $(24\div2)+(6\div2)$(24÷​2)+(6÷​2)
Careful!

We cannot apply the distributive property to division from the left side.

For example: $24\div(6+2)\ne(24\div6)+(24\div2)$24÷​(6+2)(24÷​6)+(24÷​2)

We can show the distributive property visually as well through area. We can either break up the total area into two simpler areas or subtract some excess area from an approximate total.

     

Finding Areas using the Distributive Property

We can use a similar trick to make division questions like $168\div14$168÷​14 easier. What is a number close to $168$168 that is "easy to divide by $14$14"? One way we can break up $168$168 is into $140$140 and $28$28, splitting a difficult to divide number into two easy to divide numbers.

The Distributive Property for Division and Addition

Now that we're a bit more familiar with these strategies, let's see them in action.

 

Worked example

Remember the question from the start of this lesson?
Let's solve $24\times13+78\times13$24×13+78×13.

Think: It will be difficult to solve both these multiplications individually so instead we need to find an easier way to solve the problem. Notice that both expressions involve multiplication by $13$13. Does this remind you of any of the propertys? Yes, this is the result of the distributive property!

Do: If we can use the distributive property to distribute parentheses, we can also use it to bring them back. This is called 'factoring' and will give us:

$24\times13+78\times13=(24+78)\times13$24×13+78×13=(24+78)×13

But why did we do this? We did this because $24$24 and $78$78 are not easy numbers to multiply by, so instead let's find some nicer numbers to work with. We can first add $24$24 and $78$78 together:

$(102)\times13$(102)×13

How can we break up $102$102 into some easy to multiply numbers?
Let's break up $102$102 into $100$100 and $2$2 since they are very easy to multiply by. This will give us:

$(100+2)\times13$(100+2)×13

Now we can use the distributive property to distribute the parentheses into easy multiplication:

$(100\times13)+(2\times13)$(100×13)+(2×13)

Then we can evaluate the multiplication:

$(1300)+(26)$(1300)+(26)

And finally we can evaluate the addition to get the final solution:

$1326$1326

Reflect: Every step involves writing the expression in a way that is easier to solve mentally by using one of the mental arithmetic strategies. It is worth noting that a lot of the steps in the work do not need to be written down and can be done in your head. With enough practice we can sometimes skip all the way from the second step to the solution without writing down anything; it's not called "mental arithmetic" for nothing!
 

As we can see, mental arithmetic strategies can make difficult questions much easier by applying a few clever techniques. While it is true that this leads to more steps of working than what a calculator would need, the advantage of these techniques is that there are no difficult calculations and we can use them wherever we are, whenever we want, even if there isn't a calculator in sight.

Practice questions

Question 1

Consider $27+49+13$27+49+13.

  1. Which pair of numbers will be easiest to add together first?

    $27+13$27+13

    A

    $49+13$49+13

    B

    $27+49$27+49

    C

    $27+13$27+13

    A

    $49+13$49+13

    B

    $27+49$27+49

    C
  2. Fill in the boxes to complete the work.

    $27+49+13$27+49+13 $=$= $27$27$+$+$\editable{}$$+$+$\editable{}$   Apply the commutative property
      $=$= $\editable{}$$+$+$\editable{}$   Evaluate the first addition
      $=$= $\editable{}$    

Question 2

Consider $52-24-12$522412.

  1. Which of the following expressions is equal to $52-24-12$522412?

    $\left(52-24\right)\times12$(5224)×12

    A

    $52-12-24$521224

    B

    $52-\left(24-12\right)$52(2412)

    C

    $52\times24-52\times12$52×2452×12

    D

    $\left(52-24\right)\times12$(5224)×12

    A

    $52-12-24$521224

    B

    $52-\left(24-12\right)$52(2412)

    C

    $52\times24-52\times12$52×2452×12

    D
  2. Which of the arithmetic rules can we apply to $52-24-12$522412 to transform it into $52-12-24$521224?

    Associative Law

    A

    Reordering

    B

    Commutative Law

    C

    Distributive Law

    D

    Associative Law

    A

    Reordering

    B

    Commutative Law

    C

    Distributive Law

    D

Question 3

Consider $4\times13\times5$4×13×5.

  1. Which pair of numbers will be easiest to multiply together first?

    $4\times5$4×5

    A

    $4\times13$4×13

    B

    $13\times5$13×5

    C

    $4\times5$4×5

    A

    $4\times13$4×13

    B

    $13\times5$13×5

    C
  2. Fill in the boxes to complete the work.

    $4\times13\times5$4×13×5 $=$= $4$4$\times$×$\editable{}$$\times$×$\editable{}$   Apply the commutative property
      $=$= $\editable{}$$\times$×$\editable{}$   Evaluate the first multiplication
      $=$= $\editable{}$    

Question 4

Consider $42\div3$42÷​3.

  1. Fill in the box to complete the sentence.

    $42\div3$42÷​3 is the same as $($($30$30$+$+$\editable{}$$)\div$)÷​$3$3.

  2. Fill in the box to complete the sentence.

    $\left(30+12\right)\div3$(30+12)÷​3 is the same as $30\div3$30÷​3$+$+$\editable{}$$\div$÷​$3$3.

  3. Which arithmetic rule explains the equality between $42\div3$42÷​3 and $30\div3+12\div3$30÷​3+12÷​3?

    Commutative Property

    A

    Associative Property

    B

    Distributive Property

    C

    Commutative Property

    A

    Associative Property

    B

    Distributive Property

    C
  4. Fill in the boxes to complete the work.

    $42\div3$42÷​3 $=$= $\left(30+12\right)\div3$(30+12)÷​3   Separate into two manageable components
      $=$= $30\div3+12\div3$30÷​3+12÷​3   Distribute the parentheses using the distributive property
      $=$= $\editable{}$$+$+$\editable{}$   Evaluate the division
      $=$= $\editable{}$   Evaluate the addition
             

     

Outcomes

6.NS.B.4

Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor.

6.EE.A.3

Apply the properties of operations to generate equivalent expressions.

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