Guided Lessons
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# Order of Operations Sequencing

Challenge students to listen to instructions to create expressions with parentheses. Use this lesson on its own or as support to the lesson PEMDAS: Order Up.
This lesson can be used as a pre-lesson for the PEMDAS: Order Up! lesson plan.
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This lesson can be used as a pre-lesson for the PEMDAS: Order Up! lesson plan.
##### Academic

Students will solve expressions using the order of operations.

##### Language

Students will be able to listen to and solve mathematical expressions involving parentheses using transition words and peer supports.

(5 minutes)
• Draw an array on the board (e.g., 16 by 4) and ask students to copy the same on their whiteboards. Then, instruct students to do 3 things to the array, without any sequencing words or language (e.g., add 4, divide by 2, subtract 7). Allow them to do them in any order they choose, and restate them, mixing the order in which you say them (e.g., subtract 7, add 4, divide by 2).
• Have students compare each other's answers to see that if they did the steps in a different order from their partners, each person's answer could be different. Some of the students may not even have whole numbers any longer ("I noticed I have ____ while you have ____.").
• Explain to students that today they'll use symbols that represent the sequencing words ("first," "second," "third," etc.) they use in everyday conversation to solve math problems. Tell them that parentheses are symbols used in mathematical problems to help decide the order in which to solve the problem.
(5 minutes)
• 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.
(10 minutes)
• 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.
(10 minutes)
• 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.

Beginning

• Allow students to use their home language (L1) or their new language (L2) in all discussions. Provide bilingual reference materials to assist in their vocabulary word acquisition.
• Encourage them to use the vocabulary cards and terms in their conversations and writing. Allow them to draw pictures to support their understanding of the terms.
• Review the terms "sum," "quotient," "difference," "expression," and "product" throughout the lesson to make sure students understand they refer to the answer of specific expressions depending on the operation used. Encourage them to draw the visuals on their vocabulary cards and refer to them when saying or listening to their peers stated expressions.
• Rephrase the closing question: "Why do you need to solve the parentheses first in an expression?"

Advanced

• Pair students with mixed ability groups so they can offer explanations and provide feedback to beginning ELs when appropriate.
• Ask them to provide examples of sequencing words that are helpful when speaking about the expressions.
• On the board, write sentence frames and stems to help students during their conversations. Model using the frames before releasing them to practice in partnerships.
(5 minutes)
• Say, "First, I need to multiply 10 times 1. Then I need to add 17." Have students write the problem down on the back of their Impact of Parentheses worksheet and solve it.
• Tell students to compare their answers with partners and discuss the placement of their parentheses and the impact of the parentheses on the answer. Write previously used comparative phrases (see Introduction section) on the board and supply more for students to use in their discussions (e.g., "I placed the parentheses...").
(5 minutes)
• Ask students to turn and talk to their elbow partner about the importance of using parentheses in mathematical expressions (e.g., "It's only important to use parentheses if there is more than one operation," or "It's important to use parentheses because the answer can change," etc.).
• Have one student volunteer their answer while another student restates the volunteer's answer and add the definition for the order of operations.

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