14 Lesson 1.4 ~ Order of Operations with Grouping Symbols

ordEr oF opEraTions wiTh grouping symbols

lesson 1.4

You have learned the order of operations for numerical expressions including powers, multiplication, division, addition and subtraction. Grouping symbols group parts of an expression. Grouping symbols must be completed before any of the other steps are completed. The two types of grouping symbols you will learn to work with are parentheses and fraction bars.

Find the value of each expression.

a. 5 × (3 + 4) − 8 b. 30 + (1 + 3)2 ÷ 2

a. Find the value inside the parentheses. 5 × (3 + 4) − 8 = 5 × 7 − 8 Multiply. 5 × 7 − 8 = 35 − 8 Subtract. 35 − 8 = 27 b. Find the value inside the parentheses. 30 + (1 + 3)2 ÷ 2 = 30 + 42 ÷ 2 Find the value of the power. 30 + 42 ÷ 2 = 30 + 16 ÷ 2 Divide. 30 + 16 ÷ 2 = 30 + 8 Add. 30 + 8 = 38

ExamplE 1

solutions

Lesson 1.4 ~ Order of Operations with Grouping Symbols 15

Find the value of each expression.

a. 8 − 2 _____ 1 + 1 b. 6² _____ 3 + 6 − 2

a. Find the value of the numerator and the denominator. 8 − 2 _____ 1 + 1 = 6 __ 2 Divide the numerator by the denominator. 6 ÷ 2 = 3 8 − 2 _____ 1 + 1 = 3 b. Find the value of the numerator and the denominator. 6² _____ 3 + 6 − 2 = 36 ___ 9 − 2

Divide. 36 ___ 9 − 2 = 4 − 2

Subtract. 4 − 2 = 2 6² _____ 3 + 6 − 2 = 2

Use the order of operations to find answers to these facts.

step 1: The highest temperature (F°) ever recorded in the (6 + 4)² + 34 United States was in Death Valley, CA in 1913. What was the record temperature? step 2: The Missouri River is the longest river in the United States. 2 × 1000 + 8 × 100 − 2 × 130 What is the length of the river in total miles? step 3: As of 2010, how many active volcanoes are there in Hawaii? 26 + 24 _______ 3 + 7 − 3 step 4: How many states were part of the United States in the 3 × (7 − 4) + 7 year 1800? step 5: Florida has how many electoral votes for president? (5 + 1)² − 9² ____ 4 + 5

ExamplE 2

solutions

ExplorE! FacT puzzlE

16 Lesson 1.4 ~ Order of Operations with Grouping Symbols

ExErcisEs

1. List the complete order of operations in words or pictures.

2. Give an example of each of the two grouping symbols used in this lesson.

3. Explain why it is necessary to have an order of operations for numerical expressions.

Find the value of each expression.

4. 8 + 12 ______ 5 − 1 5. 7 × (1 + 3) 6. (2 + 3)² − 12 7. 21 ÷ 7 + 3 × (6 + 2) 8. 2 × 20 ÷ 5 + 3 9. 35 − 3 ______

2³

10. 10³ + 2 × 100 11. 5 × (4 − 1)² − 10 12. 3 × 5 + 3 + 7 _____ 2 13. 3 + 4 _____ 5 − 4 + 8² __ 2 14. 9 ÷ 3 + 4 × 2 − 11 15. 3³ + 2 × (2 + 4) 16. 8 + 2³ × 4 × 3 17. 5² __ 2 + 2.5 18. (10 − 3)² + 1² Insert one set of parentheses in each numerical expression so that it equals the stated amount.

19. 2 × 4 + 6 = 20 20. 4 + 2 + 8 ÷ 2 = 9 21. 15 + 1 × 4 + 5 × 2 = 74

22. Abram makes $10 per hour at his job at the local gas station. He worked 4 hours on Monday, 5 hours on Tuesday and 3 hours on Saturday. a. How much did Abram earn? b. Explain two ways you could find the answer in part a.

23. Three friends order a pizza for $14, a bottle of soda for $3 and a large piece of cake to share for $4. The friends want to evenly split the costs. How much does each friend owe? Use mathematics to justify your answer.

rEviEw

write each power in expanded form. Find its value.

24. 7² 25. 3³ 26. 12²

27. 2⁴ 28. 10³ 29. 1⁵

Lesson 1.4 ~ Order of Operations with Grouping Symbols 17

tic-tAc-toe ~ num Be r tr ick Try this number trick: Pick a number. Add 5. Double the result. Subtract 4. Divide the result by 2. Subtract the number you started with. Your result is 3, right?

Try a different number in the trick above. Write out your computations in the order you do them. Do you get 3 again? Look at your written operations. Can you tell why this number trick works?

Josephine created the number trick below. Shown are two of her friends' computations. Copy the table and use words to describe each step of the number trick.

Try to create your own number trick with at least four steps. Show that your trick works by creating a table, like the one above, that shows at least three different starting numbers and a pattern for the solution on each.

Description Maria’s Computations August’s ComputationsChoose a number 7 1

10 420 824 1212 65 5

tic-tAc-toe ~ hA l f-l if e The half-life of an element is the time it takes for half of the element to turn into another element. For example, Carbon-10 has a half-life of 20 seconds. If you began with 100 grams of Carbon-10, there would only be 50 grams of Carbon-10 left after 20 seconds. After 40 seconds, there would be 25 grams. Carbon-10 decays so fast that it cannot be found in nature. Choose one element listed below. Examine the amount of the element remaining after 10 half-lives. Make a chart, table or graph to illustrate how much of the original element is remaining at different times.

element half-life starting AmountPromethium 17.7 years 1000 gramsRadon 3.8 days 200 gramsFrancium 22 minutes 80 grams