Order of Operations Sequencing

Explicit Instruction/Teacher modeling

• Use chart paper to outline and define order of operations (i.e., PEMDAS), and model how to follow the correct steps using the array you drew from the introduction section (e.g., "First, divide the array into 2, then add 4 more dots to one group, and lastly subtract seven from that same group.").
• Ask students to turn and talk to each other about the steps you followed. Ask a volunteer to share some of the sequencing phases you used to help clarify the steps you followed. Write the phrases on the board ("First... Then... Next... Lastly...").
• List the the following expressions on the chart and ask students to share how they are similar or different (e.g., one has parentheses while the other does not):
  • 45 ÷ 5 – 3
  • (45 ÷ 5) – 3
• Model how to follow the order to solve the expression, making sure to refer to the order of operations and use sequencing terms. Emphasize that the parentheses mean they need to solve within the parentheses first.

Guided Practice

• Write a new expression that contains two different operations, and provide a scenario to accompany it while you write it on the board. For example, "Each book cost $4 and $3 to ship ($4 + $3). I want 7 books total at ($4 + $3) per book. So, I need to add ($4 + $3) and then multiply by $7." Solve the problem aloud, saying the order you followed.
• Have students rephrase the steps for solving the full expression with partners ($4 + $3) x 7 (e.g., "You should add ($4 + $3) and then multiply the sum by seven.").
• Review the vocabulary terms for expression solutions involving different operations (i.e., "sum," "difference," "product," "quotient").
• Write some expressions on the board and have students write on their whiteboards what the answer is called. For example, the answer five for the expression 20 ÷ 4 is called the quotient. After practicing a few times with different operations, transition to describing more expressions.
• Model stating the steps needed to solve expressions involving parentheses. For example, "For the expression, 2 x (4 + 5), I need to multiply by two the sum of four and five."
• Write some of the following sentence frames on the board:
  • "I need to ____ (add, subtract, multiply, divide) the____(product, sum, difference, quotient) of ____(number) and ____ (number)."
• Use the same phrasing a few times for different expressions (e.g., "I need to add three to the product seven times two"). Then ask students to turn and talk to a partner to rephrase your expression. Note: make sure to have some expressions that have the second operation after the parentheses to show that you can say the expression the same way, regardless of the position of the second operation.

Group work time

• Ask students to independently complete Part 1 and 2 from the Impact of Parentheses worksheet. Make sure to read the directions and have students rephrase the directions to show understanding. Note that it has some problems where parentheses affect the answer and some that do not.
• Tell students we're using simple math here so they can easily solve the problem and see the effect of the parentheses on their answer.
• Have partners turn and work with their elbow partner to practice telling the equations in Part 1 or Part 2 on their worksheet, while the partner listens and writes the equation on their whiteboard. For example, for question one, a student might say, "I need to subtract 34 from the product of 8 and 2. My answer is 18." Have the partner write down the equation 34 – (8 x 2) = 18.
• Write the following question on the board and ask students to discuss their answers in partners: "Why, when thinking about 7+ (2 x 8) does the answer not change when it's written like (2 x 8) + 7?" (e.g., "The rule for order of operations tells me I need solve the expression in the parentheses before I add the 7 to the product.") Encourage students to use their whiteboards to sketch out the answers during their discussions with their partners.