“In today’s lesson, we are going to continue working on writing expressions and equations using variables. We will look at different problems and the variety of ways we can solve those problems.”
Divide students into small groups and project the Garden Template (M-6-6-2_Garden Template.doc) on a wall or screen. “Take a look at the garden. It is shaped like a rectangle with a length of 9 feet and a width of 6 feet. You are asked to find the perimeter. What does perimeter mean?” (Perimeter is the distance around a shape.) “In your group, talk about the different ways you can find the perimeter of the garden. Write the equation you used. Remember, there may be more than one way to do this.” Once students have finished, have them share their answers. Record the different ways students solved this problem on the overhead or board. (Possible responses: (9 × 2) + (6 × 2) = 30; (9 + 6) × 2= 30; 9 + 9 + 6 + 6=30). “Even though these equations look different, they all give the same answer. In our next lesson, we will look at the different number properties of addition and multiplication that make so many different representations equivalent.”
Project the Blank Garden template (M-6-6-2_Blank Garden Template.doc) on a wall or screen or hand out a copy to each group. Give each group chart paper on which to record their ideas. “This flower garden is located in the courtyard of a city building. A local art museum has been hired to create a mosaic tile border around the flower garden. With the border, the width of the flower garden will be w feet, and the length of the flower garden will be l feet. I am not giving you a specific length and width. With the members of your group, brainstorm different expressions using variables that can be used to find the perimeter of the flower garden with the border.”
Give students time to generate as many ways as possible to solve the problem. Have students generate corresponding expressions that “show” what they did to solve for the perimeter. Remind students that there are many different ways to show multiplication and division. While students are working, monitor student dialogue and interaction, observe strategies students use to solve the problem and note how they organize their thinking. Once groups are finished, have students post their group’s chart on the wall.
Have students walk from chart to chart and note generalizations about the similarities among the charts. Discuss the observations students make. Ask students to share which expressions they think can actually be used to solve the flower garden problem.
Possible answers:
- l + l + w + w
- 2l + 2w
- l + l + 2w
- 2l + w + w
- w + 1 + w + 1
|
- 2(l + w)
- w + w + 1 + 1
- 2w + 2l
- 2(w + l)
|
- Since you know perimeter is the sum of the four sides, you can add up each side separately.
- Since you know in a rectangle opposite sides are the same length, you can add two lengths and two widths.
- You also can add one length and one width, and then double that.
“All of the expressions are equivalent. Sometimes we group the numbers differently, and sometimes we can change the order of the numbers to get the same answer.”
The next activity will involve using a calendar. Distribute to 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.doc). Explain to students that there are many different patterns on a calendar. Ask students to think about the first question on the sheet, record their immediate observations, and then discuss further patterns with classmates around them. When students are finished, ask them to share their observations; record the patterns students suggest on the board. Ask students to explain and show how their patterns work. Display a transparency or enlarged calendar in the front of the classroom for students to use as a model.
Use the rest of the November calendar sheet as guided practice. The goal is for students to locate patterns and show how equations/expressions can be written in varied ways and yet still have the same answer. Ask proficient students to write an equation for the sum of four numbers in a row and/or three numbers on a diagonal (from the November calendar). They should substitute one of the numbers with a variable and then write the other two or three numbers as expressions using that variable. Have students practice writing these equations in varied ways. For those students who may need more guidance, provide additional practice in small groups.
Bring the class back together and give each student a copy of the March calendar, which is the third page of the resource (M-6-6-2_November and March Calendar Activity and KEY.doc). Ask students if what they did on the November calendar would be the same for the March calendar, even though March has 31 days. (Yes, the numbers are still consecutive.) Remind students that variables can be used to represent the value of a number. The value can vary; sometimes the value is directly related to the condition set up in a problem. In this case, the variable represents a starting number on the calendar.
Work through the March calendar sheet together as a class. Be sure students understand how the variables and expressions representing the numbers are being created. Relate back to the patterns identified at the beginning of this lesson.
Hand out to each student the remaining pages of the November and March Calendar Activity. Have students explore the “magic” part of the March calendar sheet by completing the “Guess my Number! It’s Magic!” section. Mastery of understanding how to substitute values into variable expressions is not the target of learning for this lesson but, for some students, simple exposure to the process can begin to strengthen the foundation of algebraic thinking. This part of the activity can be done as guided practice and/or small-group work. The real goal is for students to see how numbers can be written as variables.
Those students who demonstrate proficiency in the skill of writing equivalent expressions should complete the Quilt—There’s More Than One Way activity (M-6-6-2_Quilt Activity and KEY.doc). Encourage students to use many different expressions to find the perimeter of the square quilt. Remind students that changing the order and changing the grouping of numbers are different ways of writing expressions that are equivalent.
For those students who may need a little more guidance, provide small-group instruction. (See prompts in Small Group, part of the Extension section.) Once students start demonstrating more fluency and confidence, have them work on the Quilt—There’s More Than One Way activity independently or with a partner.
Throughout the lesson, observe which students are heading toward mastery and which students need more help with the concepts. By looking at the Quilt—There’s More Than One Way activity responses, you can determine if students have acquired fluency for writing expressions that are equivalent. The way a student approaches finding multiple expressions that are equivalent can determine if the student has identified patterns and relationships among the numbers and/or variables. Being able to compare a rectangle to a square using mathematical terms and recognizing the pattern of grouping and ordering are indicators that a student is proficient.
Extension:
This lesson is designed to show students that equations and expressions can be written in many ways that are equivalent. Fluency in writing these equations and expressions will help to build the foundation of algebraic thinking. Those students who are ready can solve for the value of the variables, but that is not an expected target of learning.
Following are some ways to tailor this activity to meet the needs of your class.
- Routine: Using the Routine Template (M-6-6-2_Routine Template.doc), show students the irregular shapes with the lengths of the sides represented as variables. Give students two minutes, and see how many varied equivalent expressions or equations they can create to represent the perimeter of the shapes. Remind students that changing the grouping and ordering of the numbers will create varied equations and expressions that are equivalent.
- Small Group: Do a similar garden problem and scale it down so that students can use graph paper. Have students draw a rectangle and shade in a border. Record the dimensions. Give students verbal prompts to help guide their thinking. Have students record equations that fit the criteria. Students can then substitute variables for the respective length and width to create multiple equations using variables. Prompts to use include
“How many sides does the shape have?”
“What is the length of each side?”
“To find the distance around the shape, what must you do?”
“If you put string around the shape, how long would it be?
- Expansion: Have students create a swimming pool problem similar to the flower garden problem at the start of this lesson. Students can show multiple expressions containing variables that could be used to solve for the number of bricks or tiles needed to create a border.
Students can identify other patterns on a calendar and write equivalent expressions/equations. For example, write various equations for the sum of three numbers in a column where the middle number is represented by the variable n. Write some additional equations for the sum of the numbers that make the letter “t” on a calendar, where the middle number is represented by the variable b.
