• Use the Solving Equations and Evaluating Expressions Worksheet to determine student comprehension regarding the order of operations.
• Teacher observation during the True or False? cards activity will measure student ability to apply the order of operations accurately.

“In today’s lesson, we are going to determine if an equation is true or false by evaluating both sides of the equation. We will be reviewing order of operations and referring to number properties to help us evaluate an expression or solve an equation. Remember from the previous lesson, equations and expressions can be written in a variety of ways and still be equivalent.”

Note: For this activity, you will be using the True or False? cards (M-6-6-3 _True or False Cards and KEY.doc). You will be presenting one card at a time to the class, so you will either need to cut them apart or create a transparency.

Give each student an index card. On one side of the card, have students write “true” in large letters; on the other side, have them write “false” in large letters. Show students the True or False Cards one at a time. For each card you show, ask students to indicate whether they think the equation is true or false by holding up the appropriate side of their index card. Call on individual students to explain why the equations are true or false. Once you have shown all the cards, say, “A good way to decide whether equations are equivalent, or equal, is to first evaluate one side of the equation and then evaluate the other side. Once you have simplified both sides, compare to see if they are equivalent.”

For this part of the lesson, prepare a chocolate graham cracker bar (or a paper replica) by breaking a graham cracker in half and putting a spoonful of marshmallow fluff on one half, followed by a piece of chocolate. Use the other half of the graham cracker to make a sandwich.

Begin the activity by explaining to students the order of operations. “The order of operations is a set of rules that you follow to solve an equation or simplify an expression; these rules help you determine which operation to do first. If you don’t follow these rules, you might not all arrive at the same answer. The order of operations is necessary when you are trying to evaluate whether two equations/expressions are equivalent. This is similar to following a set of directions.”

Say, “Today I have some ingredients to make chocolate graham cracker bars. I have graham crackers, chocolate bars, and marshmallow fluff. I am going to give you three minutes to quickly write down a set of directions for how I can make a graham cracker bar that looks just like this.” (Hold up the example you prepared earlier.) “I will follow the set of directions exactly as they are written.”

When time is up, collect the directions written by the students. Randomly pull one set of directions and follow them exactly as they are written. Do not change the order or assume anything. If the directions say to put the marshmallow fluff on the graham cracker without opening the jar, put the jar of marshmallow fluff on the graham cracker. The point of this demonstration is to show students if they do not follow steps in the correct order, they will not necessarily end up with the expected result. Model a few more examples from the sets of directions that students created, and then form a generalization about the importance of steps and order.

“There are times when one step needs to be done before the next step. If you do not do the steps in a logical order, you may not get the intended result or you may not be able to complete the task. This is true for making chocolate graham cracker bars as well as simplifying expressions and solving equations.

“Now that you have seen the importance of sequence and order, here is an expression I would like you to simplify.” Write 24 ÷ 3 + 5 × 4 on the board and give students time to simplify it. Once students have finished simplifying the expression, record their answers. Then write the correct answer of 28. Talk about any differences in students’ answers. Ask them why their answers differ. Explain that students may have done the calculations in a different order. Explain that just like the set of directions to make a graham cracker chocolate bar should have resulted in making a chocolate graham cracker bar like the example you showed them, the expression was intended for all of them to arrive at the same answer.

Give each student a copy of the Order of Operations Organizer (M-6-6-3_Order of Operations Organizer.doc) and a copy of the Order of Operations reference sheet (M-6-6-3_Order of Operations The Rules Reference Sheet.doc). “Let’s review the order of operations. The order of operations is like a set of directions that should be followed when simplifying an expression or solving an equation. Take a look at the Order of Operations Rules reference sheet you were given and notice that the sequence used in the order of operations is parentheses first, then exponents, then multiplication/division left to right, then addition/subtraction left to right. If these rules are not followed, you may not all arrive at the same answer. I have given you an Order of Operations Organizer to help keep track of the order in which you should complete the steps of simplifying and expression. Follow along while I simplify the expression.”

Model the following on the board using the Order of Operations Organizer. Explain that this is a numerical expression because it does not contain an equal sign.

“We are going to simplify the expression ten divided by two plus five times three. The organizer asks for an equation, so we will put the expression equal to itself.” Put an equal sign in the middle column of the organizer, and then rewrite the expression.

• “Look at the right side of the equation written here, and then use your rule sheet to decide which operation needs to come first.”
• “Do we have parentheses?” (no)
• “Do we have any exponents?” (no)
• “Division? Multiplication? The rule sheet says to go in order left to right. We will start by dividing 10 by 2. What is ten divided by two?” (5) “Replace the ‘ten divided by two’ with ‘5’ and then go on to the next step. Which operation did we just use?” (division) “Let’s put a division symbol in the rule column on the right side of the organizer.”
• “Which operation do we do next? What do the rules say?” (multiplication) “Correct. We will multiply five by three. What is five times three?” (15) “Since five times three is 15, we replace ‘five times three’ with 15 in the middle column. And what goes in the rule column?” (a multiplication symbol)
• “What is the next step?” (addition) “What do we add?” (15 + 5) “Twenty is the answer and there are no more operations.”

Once you have shown how to use the Order of Operations Organizer to solve an equation, ask students to use their Order of Operation Organizers to determine whether two sides of an equation are equivalent.

“What would we do if the left side and right sides of the equation were different? Could we use the same idea, working through the rules, to simplify each side? We will evaluate each side of the equation to determine if the two sides are equivalent.”

Write the equation (4 + 3) Ÿ• 9 ÷ 3 = 5 × 3 + 10 ÷ 2 on the board. Say, “Is this equation true or false? Is the left side equivalent to the right side? Let’s find out.”

Walk students through the left side using the Order of Operations Organizer method (Simplify what is inside parentheses first; then perform multiplication, etc.). Then walk them through simplifying the right side or have them do it on their own. “Is the equation true or false? Is the left side equivalent to the right side? We determined the left side equaled 21. Did you get 20 for the right side? The two sides are not equivalent and therefore the equation is not true. Does everyone agree?” Make sure students understand how the group arrived at 20 and 21.

“Now let’s try it again, but this time we’ll use symbols.”

Write the following equation with symbols on the board:

(∆ + 5) Ÿ □ = ∆ + (5 Ÿ □) *

*Note about the equation with the triangles and squares: This is a great way to show students that variables are just placeholders. If students seem really puzzled, eliminate one symbol or do another equation with all whole numbers and the organizer. If necessary, skip the equation with symbols altogether.

Ask students what they notice about this equation. (There are symbols instead of numbers or variables.) “Is this equation true or false?” Give students time to explore the equation. Remind them that symbols are like variables. A different symbol, like a different variable, represents a different number. Students may answer this question by substituting numbers for the symbols, or they may rely on their prior knowledge of number properties. Ask students to share how they arrived at their answer. (Possible answer: The statement is false because if you substitute 4 for the ∆ and 3 for the □, the left side of the equation would be (4 + 5) Ÿ• 3 = 27; the right side of the equation would be 4 + (5 •Ÿ 3) = 19.)

“In the equation we just worked on, you substituted numbers for the symbols. You can do the same when you are solving an equation that has variables.” Write the equation (a + 5) + b = a + (5 + b) on the board. “Is this equation true or false? Remember you are just checking to see whether or not each side of the equation is equivalent. You are not determining the value of the variables.” Give students time to explore the equation. Ask them to share how they arrived at their answer. (The statement is true because if you substitute 4 for the variable a and 3 for the variable b, then the value of the left side of the equation is (4+5)+3 =12; the value of the right side of the equation is 4+(5+3)=12. This equation is demonstrating the associative property of addition.)

Do some additional examples until students gain confidence and show some proficiency. Equations used can model the commutative and associative number properties, though these properties do not need to be directly identified. You can use the Number Properties Chart (M-6-6-3_Number Properties Chart.doc) as a resource/review if you want to layer the activity with number property identification.

As a quick assessment tool, have students complete the Solving Equations and Evaluating Expressions worksheet (M-6-6-3_Solving Equations and Evaluating Expressions and KEY.doc). Based on how well they perform on this worksheet and complete the Order of Operations Organizer, place students in flexible groups according to their levels of understanding. Activities for each group are listed below.

Proficient: Have students complete the Solving Equations: True or False? worksheet (M-6-6-3_Solving Equations and KEY.doc). Students should be able to work independently. Post answers in the room so students can receive immediate feedback.

Progressing: Have students complete the Order Up! activity (M-6-6-3_ Order Up Cards and KEY.doc). Students will cut the cards apart and match them up. They should use the Order of Operations Organizer to show the steps they used to evaluate the expressions. This way, you can see where errors may be occurring. This activity will help students refine and strengthen their ability to evaluate an expression so they can transfer this skill to solving equations and determining whether an equation is true or false. Once students have completed this activity, have them work on the Solving Equations: True or False? worksheet.

Need additional practice: These students will work in a small group with teacher guidance. Use problems with fewer steps and more prompts as guided practice. Start out with only numerical equations, and then progress to equations containing symbols and variables. Have students refer to the Order of Operations: The Rules Reference Sheet (M-6-6-3_Order of Operations The Rules Reference Sheet.doc) to remind them of the correct sequence.

Watching and listening to students interact with their classmates provides opportunities to take anecdotal notes. Completion of the Order of Operations Organizer helps determine if students have a working understanding of the order of operations necessary when evaluating numerical expressions/equations. A paper-pencil formative assessment given to students after direct instruction helps determine what level of understanding students have, and further instruction can be tailored to meet their academic needs.

Extension:

This lesson is designed to give students practice simplifying expressions and determining whether an equation is true or false. To complete such a task, having an understanding of the order of operations is necessary. Students will also gain experience using the number properties such as the commutative and associative properties.

Following are some ways to tailor this lesson to meet the needs of your students.

• Routine: Divide students into groups, and give each group a copy of the Number and Symbol Cards (M-6-6-3_Number and Symbol Cards.doc). Students should cut apart the cards. Give the class a target number and have the groups create one expression using the number and symbol cards that will equal that number. Take two groups’ expressions and write them as an equation with an equal sign to show the relationship between the two expressions.

As an alternative, hand out one number or symbol card each to between five and fifteen students, depending on how simple or complicated you want the equation to be. Be sure some students have numbers and some students have symbols. Have these designated students arrange themselves in the front of the room. Have the rest of the class solve the student-made equation.

• Expansion: Students who have shown proficiency in using the order of operations and in determining whether equations/expressions are true or false are ready for more direct instruction on the distributive property. Use Working with the Distributive Property to Solve Equations worksheet (M-6-6-3_Distributive Property and KEY.doc) to get students started.