6th Grade - Do Operations In The Correct Order

 
     
 
     
 
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6th
Algebra
Do Operations in the Correct Order
Apply algebraic order of operations and the commutative, associative, and distributive properties to evaluate expressions; and justify each step in the process.
Follow the order of operations rules to solve problems with multiple operations. Use the Commutative Property of Addition or Multiplication to evaluate expressions. This property states that the order of the numbers can be changed and not change the sum or product [Ex. 7 + 3 = 3 + 7 or 7 x 3 = 3 x 7]. Use the Associative Property to evaluate expressions with addition or multiplication operations. The Associative Property states that the way numbers are grouped does not affect the sum or product [Ex. 7 + (4+5) = (7 + 4) + 5 or 6 x (4 x 5) = (6 x 4) x 5]. Evaluate expressions using the Distributive Property of Multiplication over Addition. The Distributive Property says that it makes no difference whether you add two or more terms together first, and then multiply the results by a factor, or whether you multiply each term alone by the factor first, and then add up the results [Ex. 5 (2 + 3) = (5 x 2) + (5 x 3)]. Explain why each property was used or why each operation was performed in a particular order.
 

Sample Problems

(1)

Evaluate the expression.

25/(4 + 1) – 3

(50)

(2)

Find the missing number. Tell which property is shown.

30 + (5 + 8) = (__ + 5) + 8

(30, Associative)

(3)

Find the missing number. Tell which property is shown.

8 x 7 x 10 = 10 x ___ x 7

(8, Commutative)

(4)

Use the commutative property and solve the expression with mental math.

142 + 56 + 8

(142 + 8 + 56 = 106)

(5)

Use the Distributive Property to solve the problem.

6 (29)

(6(20)+6(9)= 120 + 54 = 174)

Learning Tips

(1)

Children often confuse the properties. However, making word family connections can often clear this up for them. For example, the word commutative is related to commute. Many children already know that commute means to move back and forth. The Commutative Properties of Addition and Multiplication mean just that. The numbers are commuting or moving from one place to another. This is possible only because the order of numbers does not change a sum or product. In other words, 2 x 3 gives us the same product as 3 x 2. For the kinesthetic learner, you may want to use a multiplication chart to show this. For a more hands on approach you can use counters (dried beans work well) to model the product or sum both ways. For example, if you’d like to use the sum 4 + 5 compared to 5 +4, you’d have your child make a pile of 4 counters next to 5 counters to show the first problem. Then, you’d have him/her make a pile of 5 next to 4. Lastly, you’d have your child count all the counters from the first problem and all the counters from the second problem. This would allow him/her to see that the sums are the same. Using this same activity for multiplication, you’d have your child put the counters in groups. If we use the expressions 2 x 3 = 3 x 2, for instance. Your child would place the counters in two groups of three for the first problem and three groups of 2 for the second.

(2)

To aid your child in remembering the Associative Property you can use the word relationship with associative and the word associate. Make sure that your child understands that the word associate can mean to join together as partners or to keep company, as a friend. Use real life examples of the people whom you or your child associate with most of the time. Next, discuss how sometimes we associate with different people (i.e. new friends, a new job, etc…). Now, bringing it back to math, help your child understand that parentheses can be used to join together numbers and operations, like in the problem (4 + 5) + 7. The 4 and the 5 are associating with one another, they’re partners. This is shown by the parentheses around them. However, since all the operations in this problem are addition, the association can change by moving the parenthesis to partner new terms, like 4 + (5 + 7). Now the 5 is associating with the 7. However, it is important to remember that this property only works with solid addition and multiplication problems. Also, remind your child that anytime he/she sees parenthesis around a problem that indicates that that part of the problem must be solved first.

(3)

When using the Distributive Property, a child will break apart a number and distribute the factor that needs to be multiplied. This will help break problems into smaller parts so that computations can be done mentally. For example, solving the problem 6 (108) can be done mentally. You simply need to split the 108 into parts and then distribute the 6. To break the 108 into parts you need to think of numbers you can add together to get 108. In doing this you want to choose numbers that will be easy to add together mentally. For instance, 100 + 8 = 108, so this is one way to break the problem up. Now the problem can be rewritten as 6 (100 + 8). Next, the six is distributed to each addend as, 6(100) + 6(8). Finally, the problem can be solved mentally because 6 x 100 = 600 and 6 x 8 = 48 and 600 + 48 = 648. No paper needed! It is important to note that this property can be used when multiplying against a sum or a difference. The Distributive Property can also be used to join numbers together to find a problem like 5 (34) – 5 (4). Looking at this problem we can see that it would be beneficial to join together the 34 – 4 because the result would end in zero and make for easy multiplication. Working backwards we’d rewrite the problem as, 5 (34 – 4). Since 34-4 is 30, we’d simply need to multiply 5 (30) as our last step. You may want to browse our links to find hands on ways to show this property using tiles.

(4)

For the visual learner, have the child draw pictures to go with each property. For the Commutative Property, he/she can draw cars with numbers on the side trading places in two steps. Make sure he/she draws and equal sign between the two steps and an + and x sign between the cars. One idea for drawing a picture to go with the Associative property would be to draw two sets 3 characters with numbered shirts on. The two sets should be separated by an equal sign.

In each set would be two characters partnered up (holding hands, in a separate room) and the other alone. Just make sure the same character isn’t alone in both drawings. In other words, show a switch of partners.

(5)

It is crucial that children memorize order of operations. They will need to use them throughout sixth grade and beyond. It is imperative that children follow these rules every time he/she solves and expression with more than one operation. Often times, the order of operations are taught using the phrase, “please excuse my dear aunt sally”, where the p stands for parentheses, e = exponents, m = multiply, d = divide, a = add, and s = subtract. The problem with this teaching strategy is that it does not teach students to move from the left side of the problem to the right using multiply and divide are interchangeably, as well as add and subtract. So, if solving a problem such as 8/4 x 2. Many students will multiply first, when he/she really needs to divide. A more appropriate way to teach would be to use a chant with hand movements to aid in memorization. Here’s an example (the words in italics are chanted aloud, the regular type is an example of the concept and bolded words are hand signals):

Grouping Symbols”

(2+4), [(3 x 4) + 5 ]

Child holds up both hands in a “c” shape to represent parentheses


Exponents

42

Child holds up two fingers with hand above head to show exponents (ex. a number squared)


Multiply or divide, divide or multiply left to right”

3x4†2 or 8/4•5

Child crosses arms in an X shape for multiply, uses the left arm to show the division slash / and points from left to right.


Add or subtract, subtract or add, left to right”

2 + 4 – 6 or 9 - 6 + 11

Child crosses arms in a + shape for add, uses right arm to show a minus sign – and points left to right.

Extra Help Problems

(1)

Find each missing number. Write the name of the property that is shown.

15 x (13 x 6) = (15 x 13) x ___

(6, Associative)

(2)

Find each missing number. Write the name of the property that is shown.

14 + 9 + 11 = 9 + ___ + 14

(11, Commutative)

(3)

Find each missing number. Write the name of the property that is shown.

75 x 55 = ___ x 75

(55, Commutative)

(4)

Find each missing number. Write the name of the property that is shown.

(___ + 11) + 8 = 17 + (11+ 8)

(17, Associative)

(5)

Find each missing number. Write the name of the properties that are shown.

(53 x 22) x (7 x 15) = (22 x ___) x (53 x 15)

(7, Associative, Commutative)

(6)

Find each missing number. Write the name of the property that is shown.

4(82) = 4(80) + 4 (___)

(2, Distributive)

(7)

Find each missing number. Write the name of the property that is shown.

5 (48) – 5 (8) = __(48 – 8)

(5, Distributive)

(8)

Use the distributive property and mental math to solve.

7 (32)

(7(30) + 7(2)= 240 + 14 = 224)

(9)

Use the commutative property and mental math to solve.

17 + 9 + 33

(17 + 33 + 9 = 59)

(10)

Use the associative property and mental math to solve.

(42 + 9) + 8

(42 + 8 = 50 + 9 = 59)

(11)

Use the distributive property and mental math to solve.

9 (43) + 9 (7)

(9(40)+9(3) + 9(7) = 450)

(12)

Use the commutative property and mental math to solve.

8 x 54 x 10

(4,320)

(13)

Use the associative property and mental math to solve.

(99 x 4) x 5

(1,980)

(14)

In the problem 47 + 19 + 3, the commutative problem could be used to perform the operations using only mental math. Which numbers should be moved to make the operations easier to compute mentally? Rewrite the problem using the commutative property and explain why you moved the numbers you moved.

(47 + 3 + 19)

(15)

When solving the expression (8 x 17) x 5, Carley used to associative and commutative properties to rewrite the problem as (8 x 5) x 17. Explain why she would do this.

(She could move 8 x 5 to mentally get 40. Next, she would be able to mentally multiply 40 x 17 to get 680.)

(16)

(6 + 18) 23

(192)

(17)

9/3 + 7 • 4

(40)

(18)

56(11) – (14 – 8 x 2)

(618)

(19)

59 + 9 – 7 x 6 + 1

(27)

(20)

175 – 5 x 52÷2

(112.50)

(21)

18 + 2 x 11 – 14

(26)

(22)

(79 + 44) x (55 ÷ 11)

(615)

(23)

(16 + 2 x 5) – 42

(10)

(24)

(15 – 9 + 11) x 4 (99 ÷ 11 x 2)

(1,224)

(25)

[(4 + 5) x (15 ÷ 5) – 7 + 125] x 72

(6,145)

 

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