Name:_________________________ Block:___________

List some places that there are rules: 1.

2.

3.

4.

5.

6.

7.

8.

Why do we have rules/laws?

2

Order of 0perations In order to complete some of the problems in the previous section correctly, you needed to know order of operations. Mathematicians worldwide follow the same order to operations. This allows everyone to compute the same answer to the same problem. Take a close look at the following problem. It illustrates why this is important. *The rugby club wants to order new jerseys. The jersey manufacturer tells the club members that they can figure out the total cost by using the equations C = 100 + 15n. C represents the total cost and n represents the number to jerseys purchases.

Pedro’s calculations: David’s calculations: C = 100 + 15 x 20 C = 100 + 15 x 20

C = 100 + 300 C = 115 x 20 C = $400 C = $2300

• Is it possible that both Pedro and David are both correct?

• If not, who is correct?

• What mistake did the other person make?

I am sure that you have all learned about order of operations (PEMDAS). Did you know there are lots of acronyms to help you remember? Here are a few found on-line.

Try these (be careful!).

1) 9 – 8 + 7 2) 21 ÷ 7 • 3 3) 9 + 9 – 6 + 5 What did you learn?

3

As you work through the following sets of problems, pay careful attention to what calculations you are doing first, second, third, etc. Level 1:

1) 5 x (4 + 6)

Step 1:________________ Step 2:________________

2) (10 + 8) ÷ 2 Step 1:________________ Step 2:________________

3) 6 + 3 – 7 Step 1:________________ Step 2:________________

4) 7 + 10 • 5 + 10

Step 1:________________ Step 2:________________ Step 3: _______________

5) 3 (6 – 1) + 4

6) 8 – 2 x 4

7) 6 + (24 ÷ 4) – 12

8) 5 x 6 ÷ 3

9) 5 – 4 + 2

10) 90 ÷ 10 – 5 x 1

4

11) 9 – 9 + 6 - 5

12) 9 – 32 ÷ 4

13) 5 + 7 – 2

14) 2 * 3 ÷ 2

15) 18 ÷ 9 x 2

16) 1 x 9 – 3 x 3

Level 2:

1) 6 + (5 – 7 x 6)

Step 1:________________ Step 2:________________ Step 3:________________

2) (2 + 52) – 5

Step 1:________________ Step 2:________________ Step 3:________________

3) (72 ÷ 9 – 1) ÷ 2

Step 1:________________ Step 2:________________ Step 3:________________

4) 22 – (9 x 4) – 6 – 9

Step 1:________________ Step 2:________________ Step 3:________________

5) (82 - 12) ÷ 9

6) (62 ÷ 3 ÷ 6)

7) (7 ÷ 7) + 9

5

8) 42 x 22 ÷ 8

9) 8 – (6 + 9) – 3

10) 32 – 9 + 82

11) (10 – 2 + 3) x 6 – 1

12) (8 + 5) x (35 ÷ 5) + 6

13) 20 ÷ (4 – (10 – 8))

14) 5 – (3 – 3) + 62 ÷ 4

15) (13 – 6) ÷ 5

Level 3: 1) {90 ÷ (6 x 4 – 9)} x 5

Step 1:________________ Step 2:________________ Step 3:________________ Step 4:________________

2) 17 – {6 – 18 ÷ 6 + 3}

Step 1:________________ Step 2:________________ Step 3:________________ Step 4:________________

6

3) {(72 ÷ 6) – 5 (11 + 3)} + 10

Step 1:________________ Step 2:________________ Step 3:________________ Step 4:________________

4) {4 + [(8 – 7) x 6 – 55 ÷ 11]} x 2

Step 1:________________ Step 2:________________ Step 3:________________ Step 4:________________

5)

�

3+ 2 × 5 − 48 ÷ 4 +1

Step 1:________________ Step 2:________________ Step 3:________________ Step 4:________________

6) 25 + {2 [3 + 8 – 7 (25 ÷ 5) + 11]}

7

7) 90 ÷ {12 + [(6 – 32) - 7] + 4}

Order of Operations with Absolute Value There are two more types of mathematical symbols that act as grouping symbols. They are absolute value and radicals (square roots). Absolute value states how far a number is from 0 on a number line. The rationale for absolute value is that there is no such thing as a negative measurement. This can be illustrated by the problem below.

Let’s say you are on the corner of 10th street and Winnetka and go 4 blocks north to the movie theatre. The total distance you have traveled is 4 blocks. However, once you reach the movie theatre you realize that you left your money at home, so you run 4 blocks south. If walking North represents a +4 and South a -4 then your total distance would be +4 + -4 = 0. This doesn’t make sense. So, distance is the positive distance a number is from 0. In other words, you have walked +4 + |-4| = 8

Absolute Value can be modeled on a number line. Use the number line below to show the absolute value of each given number.

a) |-7| = b) |9| = c) |-3| =

8

Use what you know about order of operations and absolute value to calculate the value of each below. d) | -7 + 2| = e) |-7| + |2| = f) |-4| x |3| = Square roots, also referred to as radicals, act as another type of grouping symbol. You have to complete the operations under the square root symbol before you go on the rest of the problem. So, what is a square root anyway? a)

�

16 = b)

�

81 = c)

�

100 = Here are some simple problems to try out. d) 6 +

�

5 +11 = e)

�

1+ 8 − 6 − 2 = g)

�

8 • 12x3 = Practice Problems: 1. |2 – 7| = 2. -8 + |-7| = 3. 1 – |-1| =

4.

�

−4| −4 |

= 5. |0 – 6| = 6. |2| – |5| =

7.

�

−−126

=

9

Order of Operations and Substitution The word substitution is used in many different situations. Describe a situation where this word is used and tell what is means. In math, the word substitution means to take out the variable and replace it with a number that is equal to the variable. How is this like your example and definition above? Practice: Re-write the problem substituting the number for the given variable. Then simplify the expression using order of operations. Level 1: Evaluate the following problems for t = 7.

1. t + 3

2. 18 – t

3.

�

21t

4. 3t – 5

5. t2 + 1

6. 2t – 4

7. 9t ÷ 3

8.

�

t 27

9. 5t + 6

10.

�

t 2 − 76

11. 4t + 5t

12. 6t ÷ 3

10

Level 2: Evaluate the following problems for a = 4, b = -2, c = 5, and d = 10.

1. 4a + 3c

2. 3bc – d

3.

�

acd

4. d – 2a

5. a2 – b

6. abd

7. 5c – ad

8. cd + bc

9.

�

6ba

10. 9a + b

11. 5 + 3bc

12. d2 + d + 1

Level 3: Substitute and evaluate the problems below.

1. The volume of a rectangular pyramid is determined by using the following formula:

�

V = lwh3

. Find the volume of the pyramid if l = 6 in, w = 6 in, and h = 11 in.

11

2. Calculate the approximate volume of a cone with a radius of 30 inches and a height of 60

inches using the formula

�

V = 13πr 2h (remember

�

π = 3.14 )

3. Lumber is measured by the following formula: Number of board feet =

�

LWT12

. Calculate

the number of board feet if L = 14 feet, W = 8 feet, and T = 6 feet.

4. The perimeter of a square is figured by the formula P = 4s. Calculate the perimeter if s = 6 m.

5. What is the approximate circumference of a circle with a diameter of 8 cm? C = πd and π = 3.14.

6. Find the area of the trapezoid.

A =

�

12h(a+b)

a = 23 inches b = 11 inches h = 18 inches

7. Calculate the approximate volume of a sphere with a radius of 6 cm. π = 3.14

�

V = 43πr 3

8. The formula for converting from degrees Fahrenheit to degrees Celsius is

�

C = 5(F − 32)9

. If it

is 680 F outside, how many degrees Celsius is it?

12

MCA Practice: Complete each problem below.

1. What is the value of when x = -4 and y = 5? a. -27 b. -9 c. 9 d. 27

2. Which of the following shows the associative property of addition? a. 5(2 + 3) = 10 + 15 b. 5 x 1 = 1 x 5 c. 5 + (4 + 3) = 5 + (3 + 4) d. 5 + (4 + 3) = (5 + 4)+ 3

3. If x = 4 and y = 3, then xy – 2x = a. 4 b. 6 c. 19 d. 40

4. The formula below can be used to calculate S, the sum of all integers from 1 through n, where n is any positive integer.

�

S = n(n+1)2

What is the value of S when n = 50?

a. 1250 b. 1275 c. 2500 d. 2550

5. Evaluate the expression

�

8 + 2 ba

when a = 2 and b = 4.

a. 9 b. 8 c. 7 d. 6

6. Evaluate the expression

�

3x + 5y when x = 5 and y = 2.

�

−3−2x − y