Aim: How can we use algebraic equations to represent a visual model?

Instructional Objectives:
1. To use a strategy to determine the number of tiles needed to form the border around a square pool.

2. To articulate such strategy in words as well as using a mathematics sentence, this will later be translated into an algebraic equation.

3. To describe any noticing patterns that may help students in generating the algebraic equations.

4. To use algebraic equations to generalize the problem as the lengths of the swimming pool vary

5. To justify that algebraic equations are equivalent by substitute specific values to the variable.

6. To define equivalent equations as equations which represent the same quantity.

7. To define algebra as a generalization of arithmetic

Terms Used

border, algebra, generalization, algebraic equations, equivalent algebraic equations

Procedure

A rectangular swimming poor has a length of 10 '/2 feet and a width of 8 feet. What are the area and perimeter of this pool?

Solution: Area: 10 1/2 x 8 = 84 ; 10 x 8 = 80 and 1/2 x 4 = 4 ; 80 + 4 = 84

Perimeter: 10 1/2 + 10 1/2 + 8 + 8 = 21 + 36 = 37 (10 1/2 + 8) x 2 = 18 1/2 x 2 = 37 10 1/2 x 2 + 8 x 2 = 21 + 16 = 37 (Elicit responses from students and focus on their strategies in finding the perimeter. Leave those strategies on the board.)

To look for: If students still misunderstand perimeter as area and area as perimeter. A strategy used to find the perimeter and area of a rectangle.

Lesson

Motivation: How many of you have been to swimming pools? Did you notice that sometimes these swimming pools are surrounded by borders of tiles? What is a border? (A border is a strip around an edge or a side.)

1. Draw a rectangular pool.
2. Being a math teacher, I am always thinking how mathematics works in our day to day lives. I wonder how the construction workers know how many tiles to put around swimming pools. What do you think?
3. I have a 5 by 5 square swimming pool here surrounded by the square border tiles. These tiles measure 1 foot on each side.
4. Draw of a 5 by 5 square pool.
5. Determine the number of border tiles used to surround this 5 by 5 square pool. (Elicit responses from students without getting into their strategies.) In this problem, you will explore this question: If a square pool has sides of length s feet, how many tiles are needed to form the border? Before we can answer this question, we need to collect some data.

Instructions:

1. Make sketches on graph papers to help you figure out how many tiles are needed for the border of square pools with sides of lengths 1, 2, 3, 4, 5 and 6 feet. Record your results in a table. Describe how you figure out the number of border tiles needed in words and in a mathematical sentence.

2. Describe any patterns that you notice.

This is a partner activity. However, students will work independently first. One of you will work on square pools 1, 2 and 6 feet and the other will work on 3 and 4 feet for 3 minutes. Then you will discuss your findings with each other for about four minutes. At the end of seven minute, we will elicit strategies that you used to determine the number of border tiles needed and discuss the patterns that you come up with.

During this time, I will be collecting evidence on:
. The different ways students organize their data.
. The mathematical sentences
. The different strategies
. Any students who miscount the number of tiles.
. Any students who forget to subtract the four corners that are counted twice.

Anticipated Strategies: (Use swimming pool length 6 as an example)

. 6 + 6 + (6 + 2) + (6 + 2)
. 4 x 6 + 1 + 1 + 1 + 1
. (6 + (6 + 2)) x 2
. 4 x (6 + 2) - 4
. (6 + 1) x 4
. (6+2)^2 -6^2
* * These strategies should be on the board or on easel pad.

Anticipated Patterns:

. As the length of the square pool increases by 1 foot, the number of tiles needed for the border increases by 4.
. The shape of the border is also a square.
. The length of the border is always two feet more than the length of the swimming pool.

Pose Question: Is there an efficient way to calculate the number of border tiles needed for a square pool, no matter what the lengths of the sides of the pools are?

We will then proceed to the next two questions.

3. Without drawing the pictures, and using what you have so far, determines the number of border tiles needed when the lengths of square pools are 10 feet and 26 feet.

4. Write an algebraic equation for the number of tiles, N, needed to form a border for a square pool with sides of lengths feet.

Try to write at least one more equation for the number of tiles needed for the border. How could you convince someone that your equations for the number of tiles are equivalent?

(At this point, most of the students should have some entry points to complete parts 3 and 4. The strategies and patterns from part 1 and 2 would be good tools to assist students who are still struggling. They will have 5 minutes to complete these two parts.)

Class Discussion:

1. What strategies did you use to determine the number tiles used for 10 feet? 26 feet? (Students would probably use one of the strategies stated previously.)

2. What algebraic equations did you come up with for s feet as the length of the square pool?

Anticipated equations:
N = s + s + (s + 2) + (s + 2)
N = 4s + 1 + 1 + 1 + 1
N= 4s + 4
N = 2(s + (s + 2))
N = 4(s + 2) - 4 {counted the four corners twice)
N=(s + 2)^2 - s^2
N= 4(s+1)

(Encourage students to use the visual model to explain how they come up with the equations.)

3. How are these equations similar to the mathematical sentences that we came up before?
Instead of using a specific length of the swimming pool, we use a variable to represent the any length of the swimming pool.
Explain to students that algebra is basically a generalization of arithmetic.

4. How can you convince me that all of these equations would get me the same answer?

Anticipated Responses: substitutes with a value, set up a table for each of the equations, or graph the equations. In mathematics, we said that these algebraic equations are equivalent because they represent the same quantity. In this case, all these equations would be able to help you to determine the number of tiles needed to surround a square pool whose length is 10 feet.

Summary

Go back to our aim for today. Take about 3 three minutes to answer the question. Elicit volunteers to share their thoughts.

Extension

1. For example, the only thing you know about the rectangular swimming pool is that the length of the swimming pool is 6 feet longer than the width. Is it possible to find the number of tiles needed for the border using an algebraic equation for this swimming pool? Draw a picture to help you. Be sure to explain what the variables stand for.

2. What if you don't know anything about the swimming pool other than it is a rectangular pool? Is it now possible to determine the number of tiles needed for the border with an algebraic equation? Draw a picture to help you. Be sure to explain what the variables are.

Standards Addressed by This Unit

• New York City Math Standards: Be familiar with assorted two- and three-dimensional objects.
  Determine and understand length, area, and volume.
  Model situations geometrically to formulate and solve problems.
  Discover, describe, and generalize patterns, and represent them with variables and expressions.
  Represent relationships

Overall

This is a investigative unit on introductory algebra. It draws upon interest from the real life construction of a swimming pool

Tips

Right before or after summer vacation is the optimal season for the introduction of this lesson.