Does the Order in Which We Do Each Operation Matter?

 In previous lessons you have worked with expressions, combinations of numbers, variables, and operation symbols,that involve either adding integers or multiplying integers. Today you will examine expressions that involve both operations. By the end of this lesson, you will be expected to be able to answer these target questions:  How can expressions with the same numbers and operations have different meanings?  Does the order in which we do each operation matter?

 3 -1. Katrina wrote the following set of instructions for Cecil: 4 · 2.5 + 1.

1. If Cecil the acrobat follows Katrina’s instructions, how far will he go? Draw a diagram of Cecil’s movements and show how far he will move.

2. Katrina drew the diagram at right. How was she thinking about Cecil’s moves? Write an expression to represent Katrina’s diagram.

3. Explain why Katrina’s diagram in part (b) will not give her the correct length for 4 · 2.5 + 1.

4. Cecil changed Katrina’s set of instructions so that the length of 1 foot came first, as shown in the diagram at right. Write an expression to represent this new diagram. How far does Cecil move here? Does this give the same length as 4 · 2.5 + 1?  3-2. Cecil’s trainers proposed each of the following movements. Which one requires the longest rope? Draw diagrams to justify your answer.

1. 8 + 2(6.48) feet

2. 2 + 8(6.48) feet

3. (8 + 2)6.48 feet

 3-3. The expression 5 + 3 · 4 + 2 can be used to represent the group of + tiles shown at right.

1. Work with your team to explain how each part of the expression connects with the groups of + tiles. How many tiles are there?

2. Each group of + tiles is represented by a different part of the expression, also called a numerical term. Numerical terms are single numbers or products of numbers. It is often useful to circle terms in an expression to keep track of separate calculations. For example, each term circled in the expression below represents a separate part of the collection of + tiles above.

Circle the terms in the expression shown below. Then explain what each term could describe about collections of + or – tiles. 4 · 5 + 1 + 3(–2) + 6

 3-4. Circle the terms and simplify each expression shown below. Simplify means to write an expression in its simplest form. In the case of numerical expressions, the simplest form is a single number.

1. 7.08 + 2.51 + (−3.84)

2. 7.8 + 2.1(−3) 3. 5 + 2 · 1.5

4. 4.35 · 2 + 5 +3 (−1) + (−10)

 3-5. GROUPING CHALLENGE

 Work with your team to add parentheses to 1 + 3 · 2 + 8 + (−4) so that the resulting expression has the smallest possible value. Then find a different way to add parentheses to the same expression so that it has the largest possible value. For each expression, draw a diagram of how Cecil would move.

 3-6. LEARNING LOG

 In your Learning Log, answer the target questions for this lesson, which are reprinted below. Be sure to include examples to support your ideas. Title this entry “Grouping Integers” and label it with today’s date.

 How can expressions with the same numbers and operations have different meanings?

 Does it matter in which order we do each operation?

 3-7. Consider the expression 7 + 3 · 4 + 2. 3-7 HW eTool (CPM). Homework Help ✎

1. What movements does this represent for Cecil walking on his tightrope? Draw a diagram to show his movements and the length of his walk.

2. How many different answers can you get by grouping differently? Add parentheses to the expression 7 + 3 · 4 + 2 to create new expressions with as many different values as possible.

 3-8. Find the distance between each pair of points if they were graphed on a number line. Represent your work using absolute value symbols. Homework Help ✎

1. −27.1 and 53.2 2. 71.54 and −28.3

3. −38.9 and −7.3

 3-9. Find the missing information from the following relationships. Homework Help ✎

1. Mark has downloaded four times as many songs on his music player as Chloe. If Mark has 440 songs, how many songs does Chloe have?

2. Cici likes to collect shoes, but she only has half the number of pairs of shoes that her friend Aubree has. If Cici has 42 pairs of shoes, how many pairs of shoes does Aubree have?

3. Tito walked three more miles than Danielle. If Danielle walked 2 miles, how far did Tito walk?

 3-10. After a pizza party, Julia has parts of five pizzas left over, as shown below. Each pizza was originally cut into 12 pieces, and the shaded areas represent the slices that were not eaten. Homework Help ✎

1. What fraction of pizza A is left?

2. If all of the pieces were put together, how many whole pizzas could Julia make? How many extra pieces would she have?

3. Julia wants to write the amount of leftover pizza as a single fraction. How could she do this?

 3-11. Copy and complete the generic rectangle below. What multiplication problem does it represent and what is the product? Homework Help ✎ 