<p>The purpose of this lesson is to assist students become proficient in determining the perimeters of rectangles and squares, translating verbal descriptions into algebraic sentences to describe perimeter in numerous ways, and solving for unknowns in one-step equations. Students will:&nbsp;<br>- use mathematical properties to check the equivalence of expressions.&nbsp;<br>- discover several ways to display the same perimeter.&nbsp;<br>- solve for an unknown in a one-step equation.&nbsp;<br>- look for numerical patterns and describe them using words or algebra.&nbsp;</p>

<p>- How can mathematics help to quantify, compare, depict, and model numbers?&nbsp;<br>- How are relationships represented mathematically?&nbsp;<br>- How are expressions, equations, and inequalities used to quantify, solve, model, and/or analyze mathematical problems?&nbsp;<br>- How can recognizing repetition or regularity assist in solving problems more efficiently?&nbsp;</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>- transparency of computer projection of the Garden Template (M-6-6-2_Garden Template)&nbsp;<br>- Blank Garden template (M-6-6-2_Blank Garden Template)&nbsp;<br>- Routine template (M-6-6-2_Routine Template)&nbsp;<br>- November and March Calendar Activity (M-6-6-2_November and March Calendar Activity and KEY)&nbsp;<br>- Quilt—There’s More Than One Way activity (M-6-6-2_Quilt Activity and KEY)&nbsp;<br>- graph paper&nbsp;<br>- chart paper&nbsp;<br>- blank large calendar for display (paper or electronic)</p>

<p>- Observation during class activities, particularly the November and March Calendar Activities, can be used to monitor understanding and progress.&nbsp;<br>- The Quilt Activity (M-6-6-2_Quilt Activity and KEY) can be used to track student understanding and highlight misconceptions.&nbsp;</p>

<p>Scaffolding, Active Engagement, Modeling<br>W: In this lesson, students will translate expressions/equations with variables, similar to the previous lesson, and, additionally, they will discover several approaches to solve those problems.&nbsp;<br>H: Encourage students to solve perimeter problems in groups using the Garden Template sheet. Inform them that there is more than one correct way to solve for the perimeter, with or without variables.&nbsp;<br>E: Encourage students to explore using the calendars in the Resources folder. Distribute calendars to students, and have them look for number patterns within the dates. Students will practice writing algebraic expressions to represent consecutive dates or one week from a specific date, such as <i>x</i>, <i>x</i> + 1, <i>x</i> - 1, or <i>x</i> + 7, etc.<br>R: Proficient students can work individually on the Quilt activity, writing as many equations as they can to find the quilt's perimeter. Encourage other students to practice and review by working on the Quilt activity with a partner or in a small group.&nbsp;<br>E: Observe students working on perimeter activities to identify if they need additional assistance and preparation on this topic.&nbsp;<br>T: Adapt the lesson to the needs of the students. Use small-group suggestions for students who need more practice, as well as expansion activities for students who are ready to be challenged. Routine activities could be used throughout the year to help students revisit the topics covered in this lesson.&nbsp;<br>O: This lesson teaches students that equations and expressions can be written in multiple ways that are equivalent.</p>

<p><strong>"In today's lesson, we'll continue working on creating expressions and equations with variables. We'll look at different problems and the many ways we can solve those problems."</strong>&nbsp;<br><br>Divide students into small groups and show the Garden Template (M-6-6-2_Garden Template) on the wall or screen. <strong>"Look at the garden. It is shaped like a rectangle, measuring 9 feet long and 6 feet wide. You are asked to find the perimeter. What does perimeter mean?"</strong> (<i>The perimeter is the distance surrounding a shape.</i>) <strong>"In your group, discuss the various methods for determining the garden's perimeter. Write the equation that you used. Remember, there could be more than one method to do this."</strong> Once students have done, ask them to discuss their answers. Record the many ways in which students addressed this problem on the overhead or board. (<i>Possible answers: (9 × 2) + (6 × 2) = 30; (9 + 6) × 2 = 30; 9 + 9 + 6 + 6 = 30.</i>) <strong>"Even though these equations look different, they all produce the same result. In our next lesson, we will look at the various number properties of addition and multiplication that allow so many different representations to be equivalent."</strong><br><br>Display the Blank Garden template (M-6-6-2_Blank Garden Template) on a wall or screen, or distribute a copy to each group. Give each group a chart paper on which to write their ideas. <strong>"This flower garden is located in the courtyard of a city building. A local art museum has been hired to design a mosaic tile border around the flower garden. With the border, the flower garden will be w feet wide and l feet long. I'm not providing you a certain length or width. With your group members, brainstorm several expressions using variables that can be utilized to calculate the perimeter of the flower garden with the border."</strong><br><br>Give students enough time to come up with as many solutions as possible to the situation. Have students create matching expressions that "show" what they did to solve for the perimeter. Remind students that there are a variety of ways to demonstrate multiplication and division. While students are working, examine their discussion and interaction, the strategies they take to solve the problem, and how they arrange their thoughts. After groups have finished, have students place their group's chart on the wall.<br><br>Encourage students to walk from chart to chart and make generalizations about the similarities between the charts. Discuss the observations students make. Ask students to share various expressions they think could be used to solve the flower garden problem.<br><br><i>Possible answers:</i><br><br><i>l + l + w + w &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 2(l + w)</i><br><i>2l + 2w &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;w + w + l + l</i><br><i>l + l + 2w &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 2w + 2l</i><br><i>2l + w + w &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 2(w + l)</i><br><i>w + l + w + l</i><br><br><i>Since you know that the perimeter is the sum of the four sides, you can add each side independently.&nbsp;</i><br><i>Since opposite sides of a rectangle are the same length, you can add two lengths and two widths.&nbsp;</i><br><i>You can also add one length and one width, then double them.&nbsp;</i><br><strong>"All the expressions are equivalent. Sometimes we group the numbers differently, and sometimes we can rearrange the order of the numbers to reach the same result."</strong><br><br>The following activity will entail using a calendar. Give each student a copy of the November calendar, which is the first page of the November and March Calendar Activity (M-6-6-2_November and March Calendar Activity and KEY). Explain to students that a calendar contains many different patterns. Ask students to consider the first question on the sheet, record their first observations, and then explore more patterns with classmates surrounding them. When students have done, invite them to share their observations; write the patterns they suggest on the board. Ask students to describe and demonstrate how their patterns work. Place a transparency or enlarged calendar in the front of the classroom for students to use as a model.&nbsp;<br><br>Use the remainder of the November calendar sheet as guided practice. The purpose is for students to identify patterns and demonstrate how equations/expressions may be expressed in many ways while still yielding the same solution. Instruct proficient students to create an equation for the sum of four numbers in a row and/or three numbers on a diagonal (from the November calendar). They should replace one of the numbers with a variable, and then write the remaining two or three numbers as expressions with that variable. Encourage students to practice writing these equations in various ways. Provide additional practice in small groups to students who may require further supervision.&nbsp;<br><br>Bring the class back together and distribute copies of the March calendar, which is on the third page of the resource (M-6-6-2_November and March Calendar Activity and KEY). Ask students if what they did on the November calendar will apply to the March calendar, despite the fact that March has 31 days. (<i>Yes, the numbers are still consecutive.</i>) Remind students that variables can represent the value of a number. The value can vary; sometimes the value relates directly to the condition set up in a problem. In this case, the variable reflects a starting number on the calendar.<br><br>Work through the March calendar sheet as a class. Ensure that students understand how the variables and expressions representing the numbers are formed. Refer back to the patterns described at the start of this lesson.&nbsp;<br><br>Give each student the remaining pages of the November and March Calendar Activity. Students can explore the "magic" part of the March calendar sheet by filling out the "Guess my Number! It's <i>Magic</i>!" section. The goal of this lesson is not to master how to replace values into variable expressions, but simple exposure to the procedure can help some students build their foundation of algebraic thinking. This part of the activity can be completed through guided practice and/or small-group work. The primary purpose is for students to understand how numbers can be expressed as variables.&nbsp;<br><br>Students who are proficient in writing equivalent expressions should do the Quilt—There's More Than One Way activity (M-6-6-2_Quilt Activity and KEY). Encourage students to use a variety of expressions to determine the perimeter of the square quilt. Remind students that changing the order and grouping of numbers are different ways of expressing expressions that are equivalent.<br>Small-group instruction should be provided for those who require more guidance. (See the prompts in Small Group, which is part of the Extension section.) When students begin to show more fluency and confidence, have them work on the Quilt—There's More Than One Way exercise alone or with a partner.&nbsp;<br><br>Throughout the class, keep track of which students are progressing toward mastery and which require additional assistance with the concepts. By looking at the Quilt—There's More Than One Way exercise results, you can establish whether students have developed fluency in writing equivalent expressions. The strategy a student takes to find multiple expressions that are equivalent can determine whether or not the student has identified patterns and relationships between the numbers and/or variables. Being able to use mathematical terms to compare a rectangle to a square, as well as recognize the pattern of grouping and ordering, are indicators that a student is proficient.&nbsp;<br><br><strong>Extension:</strong>&nbsp;<br><br>This lesson is intended to demonstrate to students that equations and expressions can be written in a variety ways that are equivalent. Fluency in writing these equations and expressions will help to provide the foundation of algebraic thinking. Students who are ready can solve for the value of the variables, however this is not the expected learning outcome.&nbsp;<br><br>The following are some suggestions for tailoring this activity to the needs of your class.<br><br><strong>Routine:</strong> Use the Routine Template (M-6-6-2_Routine Template) to teach students the irregular shapes, with the lengths of the sides expressed as variables. Give students two minutes to come up with as many different equivalent expressions or equations as possible to represent the perimeters of the shapes. Remind students that changing the grouping and ordering of the numbers will result in different equations and expressions that are equivalent.<br><br><strong>Small Group:</strong> Create a similar garden problem, but scale it down so that students can use graph paper. Encourage students to draw a rectangle and shade in a border. Keep a record of the dimensions. Give students verbal instructions to guide their thinking. Have students record equations that meet the criteria. Students can then replace variables for the respective length and width to generate several equations using variables. Prompts to use include:&nbsp;<br><strong>"How many sides does the shape have?"&nbsp;</strong><br><br><strong>"What is the length of each side?"&nbsp;</strong><br><br><strong>"To find the distance around the shape, what must you do?"&nbsp;</strong><br><br><strong>"If you put string around the shape, how long would it be?"</strong>&nbsp;<br><br><strong>Expansion:</strong> Ask students to develop a swimming pool problem similar to the flower garden problem at the beginning of this lesson. Students can demonstrate several expressions with variables that can be used to calculate the number of bricks or tiles required to form a border.<br>Students can identify other patterns on a calendar and write equivalent expressions/equations. For example, write a column of various equations for the sum of three numbers, with the variable <i>n</i> representing the middle number. Write some additional equations for the sum of the numbers that make the letter "t" on a calendar, with the variable <i>b</i> representing the middle number.</p>