<p>The purpose of this lesson is to help students increase their fluency in setting up and solving real-world problems. Students will:&nbsp;<br>- complete equations with a variety of properties, including the order of operations.&nbsp;<br>- use the order of operations to simplify expressions and solve equations.&nbsp;<br>- substitute values for variables and evaluate expressions.&nbsp;</p>

<p>- How can mathematics help to quantify, compare, depict, and model numbers?<br>- How are relationships represented mathematically?<br>- How are expressions, equations, and inequalities used to quantify, solve, model, and/or analyze mathematical problems?<br>- How can recognizing repetition or regularity assist in solving problems more efficiently?</p>

<p>- Coefficient: The numeric factor of a term with a variable.&nbsp;<br>- Distributive Property: The product of a number and a sum is equal to the sum of the individual products of addends and the number (e.g., <i>a(b + c) = ab + ac</i>).&nbsp;<br>- Expression: A variable or any combination of numbers, variables, and symbols that represent a mathematical relationship (e.g., 24 × 2 + 5 or 4<i>a</i>−9).&nbsp;<br>- Inequality: A mathematical sentence that contains an inequality symbol (&gt;, &lt;, ≤, ≥, or ≠) in which the terms on either side of the symbol are unequal.</p>

<p>- True or False? cards (M-6-6-3 _True or False Cards and KEY)&nbsp;<br>- Order of Operations Organizer (M-6-6-3_Order of Operations Organizer)&nbsp;<br>- Order Up! cards (M-6-6-3_ Order Up Cards and KEY)&nbsp;<br>- Solving Equations: True or False? worksheet (M-6-6-3_Solving Equations and KEY)<br>- Working with the Distributive Property to Solve Equations worksheet (M-6-6-3_Distributive Property and KEY)&nbsp;<br>- Number and Symbol Cards (M-6-6-3_Number and Symbol Cards)&nbsp;<br>- Number Properties Chart (M-6-6-3_Number Properties Chart)&nbsp;<br>- Order of Operations (M-6-6-3_Order of Operations The Rules Reference Sheet)&nbsp;<br>- Solving Equations and Evaluating Expressions worksheet (M-6-6-3_Solving Equations and Evaluating Expressions and KEY)&nbsp;<br>- index cards&nbsp;<br>- graham crackers&nbsp;<br>- chocolate bars&nbsp;<br>- marshmallow fluff (jar)&nbsp;<br>- spatula or spoon</p>

<p>- Use the Solving Equations and Evaluating Expressions Worksheet to assess student understanding of the order of operations.&nbsp;<br>- Teacher observation during the True or False? cards activity will assess students' ability to apply the order of operations correctly.&nbsp;</p>

<p>Scaffolding, Active Engagement, Modeling and Explicit Instruction&nbsp;<br>W: Students will simplify algebraic expressions and equations using order of operations, commutative, associative, and distributive properties to gain a deeper understanding. Begin by going over the order of operations and number properties with students. These criteria will be used to determine if given expressions and equations are true or false.&nbsp;<br>H: Engage students with the True or False card activity. After distributing out the True or False? cards, show equations or expressions to the class and ask students to show the correct side of the True or False? cards. Call on individuals to explain their thinking.&nbsp;<br>E: To learn about the order of operations, students will create instructions for a given task and have a classmate follow the directions exactly, which may mean excluding a vital step if students don't include it in their written instructions. This exercise demonstrates the significance of following the correct sequence of steps. Distribute the Order of Operations reference sheet and use it to solve questions in class.&nbsp;<br>R: To review the lesson's concepts, assign the Solving Equations and Evaluating Expressions assignment. Use the results to divide students into appropriate groups to complete further assignments based on their abilities.&nbsp;<br>E: Evaluate students' level of proficiency while they work on assignments. Provide additional teaching to those who require particular attention.&nbsp;<br>T: The lesson can be adapted to the needs of the class. The need-more-practice and advancing exercises can be used with students who require additional support. The expansion suggestions can be used with students who are prepared for a challenge. The routine ideas are designed to be used in class throughout the year to reinforce lesson concepts.&nbsp;<br>O: This lesson aims to guide students through the process of translating stories into algebraic expressions and equations, as well as simplifying expressions and determining if the equation is true or not. Students simplify the expressions by applying the order of operations principles as well as the associative, commutative, and distributive properties. Students will also practice writing several equivalent expressions.&nbsp;</p>

<p><strong>"In today's lesson, we'll see if an equation is true or false by evaluating both sides of it. We will go over the order of operations and refer to number properties to help us evaluate an expression or solve an equation. Remember from the last lesson that equations and expressions can be written in a variety of ways while still being equivalent."</strong>&nbsp;<br><br>Note: This activity will require the True or False? cards (M-6-6-3_True or False Cards with KEY). You will be presenting one card at a time to the class, so either cut them apart or make a transparency.<br><br>Give each student an index card. Students should write "true" in large letters on one side of the card, and on the other side, have them write "false" in large letters. Show students one True or False Card at a time. For each card you display, ask students to indicate whether they think the equation is true or false by holding up the relevant side of their index card. Ask individual students to explain why the equations are true or false. After you've presented all of the cards, say, <strong>"A excellent method to determine whether equations are equivalent, or equal, is to evaluate one side of the equation first, and then evaluate the other side. After simplifying both sides, compare to see if they are equivalent."</strong>&nbsp;<br><br>For this section of the lesson, make a chocolate graham cracker bar (or a paper version) by breaking a graham cracker in half and spreading a tablespoon of marshmallow fluff on one half, followed by a piece of chocolate. Make a sandwich using the second half of the graham cracker.<br><br>Begin the activity by explaining to students the order of operations: <strong>"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 do not follow these rules, you may not get the same answer. When determining if two equations or expressions are equal, you must consider the order of operations. This is similar to following a set of directions."&nbsp;</strong><br><br>Say,<strong> "Today I have some ingredients for chocolate graham cracker bars. I've got graham crackers, chocolate bars, and marshmallow fluff. I'm going to give you three minutes to rapidly jot down a set of directions for making a graham cracker bar that looks just like this."</strong> (Present the example you prepared previously.) <strong>"I will follow the set of directions exactly as they are written."</strong><br><br>When the timer goes off, collect the student-written directions. Choose one set of directions at random and follow it exactly as written. Don't change the order or make any assumptions. If the directions say to put the marshmallow fluff on the graham cracker without opening the jar, place the jar of marshmallow fluff on the graham cracker. The goal of this demonstration is to demonstrate students that if they do not follow the steps in the exact order, they will not always get the expected outcome. Model a few more instances from the sets of directions that students developed, and then draw conclusions about the importance of steps and order.<br><br><strong>"Sometimes, a particular step must be completed before moving to the next. If you do not follow the steps in a logical order, you may not achieve the intended result or may be unable to complete the task. This applies to creating chocolate graham cracker bars, as well as simplifying expressions and solving equations.&nbsp;</strong><br><br><strong>"Now that you have seen the importance of sequence and order, here is an expression I would like you to simplify."</strong> Write 24 ÷ 3 + 5 × 4 on the board and let students to simplify it. Once students have finished simplifying the expression, record their answers. Then write the correct answer, 28. Discuss any differences in the students' answers. Ask them why their answers differ. Explain that students may have completed the math in a different order. Explain that, just as the set of directions to make a graham cracker chocolate bar should have resulted in a chocolate graham cracker bar, as in the example you gave them, the expression was designed for everyone to get the same result.&nbsp;<br><br>Give each student a copy of the Order of Operations Organizer (M-6-6-3_Order of Operations Organizer) as well as the Order of Operations Reference Sheet (M-6-6-3_Order of Operations The Rules Reference Sheet). <strong>"Let's go over the order of operations. The order of operations is similar to a set of directions that should be followed while simplifying an expression or solving an equation. Take a look at the Order of Operations Rules reference sheet you were given and observe that the order of operations is parentheses first, then exponents, then multiplication/division left to right, and finally addition/subtraction left to right. If these guidelines are not followed, you may not get the same answer. I've provided you with an Order of Operations Organizer to help you keep track of the order in which the simplification and expression steps should be completed. Follow along as I simplify the expression."</strong>&nbsp;<br><br>Model the following on the board using the Order of Operations Organizer. Explain that this is a numerical expression because it does not include an equal sign.<br><br><strong>"We will simplify the expression ten divided by two plus five times three. The organizer requests an equation, so we shall put the expression equal to itself."</strong> Place an equal sign in the middle column of the organizer and then rewrite the expression.&nbsp;<br><br><strong>"Look at the right side of the equation written here, and then use your rule sheet to decide which operation needs to come first."&nbsp;</strong><br><strong>"Do we have parentheses?"</strong> (<i>no</i>)&nbsp;<br><strong>"Do we have any exponents?"</strong> (<i>no</i>)<br><strong>"Division? Multiplication? According to the rule sheet, you should proceed from left to right. We'll start by dividing 10 by 2. What is ten divided by two?"</strong> (<i>5</i>) <strong>"Replace the 'ten divided by two' with '5' and go to the next step. What operation did we just use?"</strong> (<i>division</i>) <strong>"Let's put a division symbol in the rule column on the right side of the organizer."</strong><br><strong>"Which operation do we do next? What do the rules state?"</strong> (<i>multiplication</i>) <strong>"Yes. We'll multiply 5 by 3. What is five times three?"</strong> (<i>15</i>) <strong>"Because five times three is 15, we replace 'five times three' with 15 in the middle column. And what goes into the rule column?"</strong> (<i>a multiplication symbol</i>)&nbsp;<br><strong>"What is the next step?"</strong> (<i>addition</i>) <strong>"What do we add?"</strong> (<i>15 + 5</i>) “<strong>Twenty is the answer and there are no more operations.”</strong></p><figure class="image"><img style="aspect-ratio:586/228;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_32.png" width="586" height="228"></figure><p>After you've demonstrated how to use the Order of Operations Organizer to solve an equation, have students use their Order of Operations Organizers to determine whether two sides of an equation are equivalent.&nbsp;<br><br><strong>"What would happen if the left and right sides of the equation were different? Could we apply the same idea, working through the rules, to simplify each side? We will evaluate both sides of the equation to see if they are equivalent."</strong><br><br>Write the equation (4 + 3) Ÿ• 9 ÷ 3 = 5 × 3 + 10 ÷ 2 on the board. Ask, <strong>"Is this equation true or false? Is the left side equal to the right? Let's find out."</strong>&nbsp;<br><br>Walk students through the left side using the Order of Operations Organizer approach (first simplify what is behind parentheses, then perform multiplication, and so on). Then take them through simplifying the right side, or let them do it on their own. <strong>"Is the equation true or false? Is the left side equal to the right? We determined that the left side equaled 21. Did you receive 20 for the right side? The two sides are not equivalent, therefore the equation is not true. Does everybody agree?"</strong> Ensure that students understand how the group arrived at 20 and 21.<br><br><strong>"Now let's try it again, but this time we'll use symbols."</strong>&nbsp;<br><br>Write the following equation using symbols on the board:<br><br>(∆ + 5) Ÿ □ = ∆ + (5 Ÿ □) *<br><br>*Note regarding the equation with the triangles and squares: This is an excellent approach to demonstrate to students that variables are simply placeholders. If students are seem really puzzled, remove one symbol or do another equation using all whole numbers and the organizer. If required, skip the equation using symbols altogether.&nbsp;<br><br>Ask students what they notice about this equation. (<i>There are symbols rather than numbers or variables.</i>) <strong>"Is this equation true or false?"</strong> Give students time to explore the equation. Remind them that symbols are just like variables. A different symbol, like a different variable, represents a different number. Students can solve this question by substituting numbers for the symbols or using prior knowledge of number properties. Ask students to explain how they arrived at their answer. (<i>Possible answer: The statement is false because if you replace 4 for the ∆ and 3 for the □, the left side of the equation would be (4 + 5) Ÿ• 3 = 27; the right side would be 4 + (5 •Ÿ 3) = 19.</i>)&nbsp;<br><br><strong>"In the equation we just solved, you replaced numbers for the symbols. You can do the same thing when solving an equation that has variables."</strong> Put the equation (<i>a</i> + 5) + <i>b</i> = <i>a</i> + (5 + <i>b</i>) on the board. <strong>"Is the equation true or false? Remember, you're just checking to verify if each side of the equation is equal. You are not determining the values of the variables."</strong> Give students time to investigate the equation. Ask them to explain how they arrived at their answer. (<i>The statement is true because if you substitute 4 for the variable a and 3 for the variable b, the left side of the equation is (4+5)+3 =12, and the right side is 4+(5+3)=12. This equation demonstrates the associative property of addition.</i>)<br><br>Do some more examples until students feel confident and proficient. Equations can be used to express the commutative and associative number properties, although these properties do not have to be directly identified. If you want to add number property identification to the activity, use the Number Properties Chart (M-6-6-3_Number Properties Chart) as a resource/review.&nbsp;<br><br>As a short evaluation tool, students should complete the Solving Equations and Evaluating Expressions worksheet (M-6-6-3_Solving Equations and Evaluating Expressions and KEY). Place students in flexible groups based on their level of understanding and performance on this worksheet and the Order of Operations Organizer. The activities for each category are detailed below.<br><br>Proficient: Did students finish the Solving Equations: True or False? worksheet (M-6-6-3_Solving Equations and KEY). Students should be able work independently. Post answers in the classroom so students can get rapid feedback.&nbsp;<br><br>Progressing: Have students complete the Order Up! task (M-6-6-3_Order Up Cards and KEY). Students will cut and match the cards. They should use the Order of Operations Organizer to demonstrate the steps they took to evaluate the expressions. This allows you to identify potential errors. This task will help students improve and increase their ability to evaluate expressions, allowing them to use this skill to solve equations and evaluate if an equation is true or false. After students have finished this task, assign them to work on the Solving Equations: True or False? worksheet.<br><br>Need additional practice: These students will work in small groups, guided by teacher. Use problems with fewer steps and more suggestions for guided practice. Begin with only numerical equations, then advance to equations that include symbols and variables. Ask students to refer to the Order of Operations: The Rules Reference Sheet (M-6-6-3_Order of Operations The Rules Reference Sheet) serves to remind them of the correct sequence.<br><br>Observing and listening to students engage with their classmates provides possibilities for taking anecdotal notes. Completing the Order of Operations Organizer determines whether students have a working comprehension of the order of operations required when evaluating numerical expressions or equations. A paper-pencil formative evaluation given to students following direct instruction helps evaluate what level of comprehension they have, and further instruction can be modified to meet their academic needs.<br><br><strong>Extension:</strong><br><br>This lesson will allow students to practice simplifying expressions and assessing if an equation is true or false. To complete such a task, understanding the order of operations is required. Students will also gain experience using number properties such as commutative and associative properties.<br><br>The following are some methods to modify this lesson to your students' requirements.<br><br><strong>Routine:</strong> Divide students into groups and distribute copies of the Number and Symbol Cards (M-6-6-3_Number and Symbol Cards). Students should cut the cards apart. Give the class a target number and instruct the groups to develop one expression using the number and symbol cards that equals that number. Take the expressions of two groups and write them as an equation with an equal sign to demonstrate the relationship between the two expressions.&nbsp;<br>As an alternative, provide one number or symbol card to each of five to fifteen students, depending on how easy or complex the equation should be. Make sure that some students have numbers and some have symbols. Arrange these designated students in the front of the room. Have the rest of the class solve the student-created equation.<br><br><strong>Expansion:</strong> Students who show proficiency in using the order of operations and deciding whether equations/expressions are true or false are prepared for more direct instruction on the distributive property. Use Working with the Distributive Property worksheet (M-6-6-3_Distributive Property and KEY) to get students started.</p>