Insights Into Algebra 1: Teaching for Learning
Variables and Patterns of Change Lesson Plan 1: Miles of Tiles – The Pool Border Problem
In this lesson students will recognize patterns and represent situations using algebraic notation and variables.
One 50-minute class period
Variables and patterns
Students will be able to:
- Identify a pattern involving the number of tiles required to form a border around a pool with length l and width w.
- Write a symbolic expression that describes the number of tiles needed to form a border around a pool.
Principles and Standards for School Mathematics, National Council of Teachers of Mathematics (NCTM), 2000:
NCTM Algebra Standard for Grades 6-8
NCTM Algebra Standard for Grades 9-12
Teachers will need the following:
- Transparencies with 1 cm grids
- Approximately 30 unit algebra tiles (click here for a set of printable algebra tiles)
Students will need the following:
- 1 large piece of poster board with a 1 cm grid
- 1 cm by 20 cm strips (click here for a set of printable 1 cm strips)
- Glue stick
- Grid paper
- Approximately 40 unit algebra tiles
Teachers Activities and Assignment
1. Begin the day’s lesson with a story, such as:
Last night, I saw the most wonderful pool. It had beautiful tiles all around it. So this morning, I asked my landlord if he would install a pool in the backyard of my apartment. At first, he thought I was crazy, but I told him I’d make him a deal. I told him that if he built my dream pool, I would install the tiles around the edges of the pool. So, he made a deal with me. He told me that he’d install a pool with an area of 36 square feet.
2. Ask the class, “If my pool has an area of 36 square feet, what are the possible dimensions of the pool?” Elicit from students all possible dimensions of the pool, using only whole numbers: 1 ft by 36 ft, 2 ft by 18 ft, 3 ft by 12 ft, 4 ft by 9 ft, and 6 ft by 6 ft.
3. Explain to students that you are on a budget, so you need their help in determining the least number of tiles that could be used around the outside edge of the pool. Using the overhead projector, display a 4 ft by 9 ft pool. Tell students that each algebra tile represents a 1 ft by 1 ft tile. Ask students to predict the number of tiles that would be needed to put a border of tiles around the entire pool.
4. On the board, record student guesses for the number of tiles needed. You may want to have the class reach a consensus regarding the number of tiles that will be necessary, or you may want them to discover this in their groups as part of the learning activities below. (For a 4 ft by 9 ft pool, the class should conclude that the border will consist of 30 tiles: the perimeter of the pool is 26 ft, and one tile is needed for each foot of perimeter; in addition, 4 tiles are needed at the corners, as shown below.)
1. Explain to the class that they will be working in groups of four to investigate the number of tiles needed for pools of various sizes. For the group exploration, provide the following directions:
- Sitting together, build pools and make borders around the pools.
- Record the number of tiles needed for each pool.
- Look for a pattern.
- Finally, come up with an algebraic expression that relates the length and width to the number of tiles needed.
You may want to have students devise their own way of working together or you may want to assign the following roles to members of the group: writer, responsible for filling in the group’s chart; cutter, responsible for the scissors; sticker, responsible for the glue; and speaker, who will present the group’s findings to the class.
2. Assign one of the various pool sizes (from the introductory activity) to each group. The students in each group are responsible for constructing a model of the pool they are assigned. In addition, the group should consider all of the various pool sizes and look for a pattern that relates the length and width to the number of tiles needed.
3. Allow students time to construct a model of the pool they have been assigned. Students should cut the 1 cm by 20 cm strips to the length needed to form a border around their pool. Students may also use the 1 cm by 20 cm strips to investigate pools of sizes other than the one they were assigned, or they can investigate using the grid paper. As students are working, circulate and use effective questions to help the groups identify the relationship between the length and width and the number of tiles.
4. After about 15-20 minutes, have each group present its findings. (Depending on the number of students in your class, this may mean that two speakers are presenting the same material, or it may mean that some sizes will not have been assigned.)
Students will invariably arrive at several different expressions for finding the number of tiles, including:
- 2l + 2w + 4
- 2(l + w) + 4
- 2(l + w + 2)
- 2(l + 2) + 2(w + 2) – 4
- (l + 2)(w + 2) – lw
5.After all student groups have presented their findings, describe one of the expressions that they have not discovered, and ask them to consider whether or not this alternative method is equivalent to their expression. For instance, you might say, “I was thinking that I would add the length and the width, double that result, and then add 4.”
6.Select a student to translate your method into an algebraic expression. Be sure to discuss the order of operations.
7.Select another student to demonstrate how the equation found by their group is equivalent to the alternate expression that you suggested. (You may wish to repeat this step if several groups found different expressions. This discussion may allow for an explanation of the distributive property, the order of operations, the associative and commutative properties, and other topics.)
8.Ask again the question that was posed at the beginning of the lesson: “Which pool would require the fewest tiles?” Students should conclude that the 6 ft by 6 ft pool will only require 28 tiles, and that this is the fewest needed for any 36 sq ft pool.
Then, ask: “If I wanted a pool in which to swim laps, which would be the best one?” Students may suggest that the 1 ft by 36 ft pool is best, because it is the longest. Other students, however, will likely point out that such a pool would not be wide enough. Students may argue for the 2 ft by 18 ft and 3 ft by 12 ft pools as the best candidates. While 4 ft by 9 ft and 6 ft by 6 ft would be wide enough, they would not be long enough for swimming laps.
Explain to students that because the sizes are not ideal for the dream pool, you would like them to consider other patterns for pools. On the board or overhead, show them Design 1, which is a 1 ft by 2 ft ; Design 2, which is a 2 ft by 3 ft pool; Design 3, which is a 3 ft by 4 ft pool; and Design 4, which is a 4 ft by 5 ft pool. Ask them to use this pattern to predict what Design 5 would look like, and then use their drawing to determine the number of tiles needed for the border of the Design 5 pool. Similarly, have students determine the number of tiles needed for Design 11, as well as for Design n.
Allow students to present their findings to the class. In particular, encourage students to share their expression for the number of tiles. For Design n, the length of the pool is n + 1, and the width is n. Consequently, numerous expressions could represent the number of tiles needed for the border of Design n:
- 2(n + 1) + 2n + 4
- 4n + 6
- 2(2n + 3)
- 2(n + 1 + n + 2)
- (n + 3)(n + 2) – n(n – 1)
Have students use the expressions to confirm the number of border tiles for Design 6 and Design 11.
Ask students to express their ideas regarding what they learned about algebra and the power of algebra. Allow several students to share their thoughts.
Related Standardized Test Questions
The questions below dealing with translating words into symbols have been selected from various state and national assessments. Although the lesson above may not fully equip students to answer all such test questions successfully, students who participate in active lessons like this one will eventually develop the conceptual understanding needed to succeed on these and other standardized assessment questions.
- Taken from the Maine Educational Assessment, Mathematics, Grade 11 (2002):
Carl needed to rent a car for one week. He collected the following information.
- Hillbrook Car Rentals charges $230/week plus 20� per mile.
- Bridge Car Rentals charges $210/week plus 25� per mile.
a. Write an equation for the cost, c, to rent from Hillbrook Car Rentals if Carl drives the car x miles.
b. Write an equation for the cost, c, to rent from Bridge Car Rentals if Carl drives the car x miles.
c. For what number of miles will it cost the same to rent a car from the two companies? Show or explain how you found your answer.
Solution: For Hillbrook Rentals, c = 230 + 0.2x. For Bridge Car Rentals, c = 210 + 0.25x. The cost will be the same when the value of c is equal for both companies, which occurs when
230 + 0.2x = 210 + 0.25x, or when x = 400 miles.
- Taken from the Maryland High School Algebra Exam (2002):
Lydia has $200 in her bank account at the beginning of the year. Each month, she deposits $40 into her account. She does not withdraw any money from her account, and the account pays no interest. Which of these equations could Lydia use to find the total amount (T) in her bank account at the end of m months?
A. T = 40m
B. T = 240m
C. T = 200m + 40
D. T = 40m + 200 (correct answer)
- Taken from the New York Regents High School Examination (January 2003):
The equation P = 2L + 2W is equivalent to
A. (correct answer) B. C. D.
- Taken from the Mississippi Algebra I Test, Version 2 (2003):
Jan took a taxi to visit the Petrified Forest northwest of Jackson, Mississippi. The taxi fare for the trip was $14.80, based on a fixed charge of $2.00 plus a charge of $0.80 for each mile. Which of these could be used to determine m, the number of miles that Jan rode in the taxi?
A. 14.80 = 2 + 0.80m (correct answer)
B. 14.80 = 2m + 0.80
C. 14.80m = 2 + 0.80m
D. 14.80m = 2m + 0.80
- Taken from the Colorado State Assessment, Grade 5 (2002):
Which problem can be solved using the number sentence shown below?
6 � 2 = __
A. There are 6 children in a swimming pool. Two more children are getting in. How many children are in the pool?
B. There are 6 children eating lunch. Each child ate 2 slices of cheese. How many slices of cheese were eaten? (correct answer)
C. There are 6 children playing basketball. Two left to get a drink of water. How many children are left playing basketball?
D. There are 6 children walking to the library. They are walking in groups of 2. How many groups of 2 are there?
Student Work: Pool Border Problem
I was pleased with the students’ work. I saw that students understood how to combine like terms and represent the same expression in more than one way. After reviewing the students’ work, I was able to determine that they were on task and that they understood the lesson. This particular student did a good job of showing all the steps she took to solve the problem. She made an arithmetic error in the last step. This will be a good opportunity to talk with students about the importance of checking their work.
Workshop 1 Variables and Patterns of Change
In Part I, Janel Green introduces a swimming pool problem as a context to help her students understand and make connections between words and symbols as used in algebraic situations. In Part II, Jenny Novak's students work with manipulatives and algebra to develop an understanding of the equivalence transformations used to solve linear equations.
Workshop 2 Linear Functions and Inequalities
In Part I, Tom Reardon uses a phone bill to help his students deepen their understanding of linear functions and how to apply them. In Part II, Janel Green's hot dog vending scheme is a vehicle to help her students learn how to solve linear equations and inequalities using three methods: tables, graphs, and algebra.
Workshop 3 Systems of Equations and Inequalities
In Part I, Jenny Novak's students compare the speed at which they write with their right hands with the speed at which they write with their left hands. This activity enables them to explore the different types of solutions possible in systems of linear equations, and the meaning of the solutions. In Part II, Patricia Valdez's students model a real-world business situation using systems of linear inequalities.
Workshop 5 Properties
In Part I, Tom Reardon's students come to understand the process of factoring quadratic expressions by using algebra tiles, graphing, and symbolic manipulation. In Part II, Sarah Wallick's students conduct coin-tossing and die-rolling experiments and use the data to write basic recursive equations and compare them to explicit equations.
Workshop 6 Exponential Functions
In Part I, Orlando Pajon uses a population growth simulation to introduce students to exponential growth and develop the conceptual understanding underlying the principles of exponential functions. In Part II, a scenario from Alice in Wonderland helps Mike Melville's students develop a definition of a negative exponent and understand the reasoning behind the division property of exponents with like bases.
Workshop 7 Direct and Inverse Variation
In Part I, Peggy Lynn's students simulate oil spills on land and investigate the relationship between the volume and the area of the spill to develop an understanding of direct variation. In Part II, they develop the concept of inverse variation by examining the relationship of the depth and surface area of a constant volume of water that is transferred to cylinders of different sizes.
Workshop 8 Mathematical Modeling
This workshop presents two capstone lessons that demonstrate mathematical modeling activities in Algebra 1. In both lessons, the students first build a physical model and use it to collect data and then generate a mathematical model of the situation they've explored. In Part I, Sarah Wallick's students use a pulley system to explore the effects of one rotating object on another and develop the concept of transmission factor. In Part II, Orlando Pajon's students conduct a series of experiments, determine the pattern by which each set of data changes over time, and model each set of data with a linear function or an exponential function.