Pool Border Task. The Mathematics of the Task

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1 Pool Border Task How many unit square tiles does it take to build a border around a square pool? The length of the sides of the pool is a natural number of units (1, 2, 3,...) and the border is just one tile wide. Find a way to express the number of unit square tiles it will take to build the border --- without having to count individual tiles --- for a square pool of any size. MPs 1, 2, 3, 4, 5, 6, 7, 8 CCSSM Strands Expressions and Equations, Operations and Algebraic Thinking, Measurement and Data The Mathematics of the Task This task requires participants to make representations, first of the pool and the border for different square pools whose sides have natural number lengths, and second of how they found the number of unit-square tiles in a one-tile-wide border for any given pool. The mathematics involved must be precisely developed and precisely communicated if we are to fully uncover the many rich connections embedded in this task. There are many ways to generate expressions for the number of square tiles in the border of a square pool. Different strategies for finding a number of tiles can result in different algebraic expressions. A major feature of the work is determining the algebraic equivalences of the different expressions for the general statements of the number of square tiles in the border of a given pool. All of these expressions are equivalent to 4s + 4, where s represents the length of one side of the square pool. The border is made around the perimeter of the square pool. What we are finding is the area of that border. Care needs to be taken when discussing this. Referring to the "number of square tiles" in the border might pre-empt the possible confusion of thinking that the task is about the perimeter of the pool when it is about the area of the border. p. 1 / 12

2 Solutions Table 1 shows six possible strategies for finding the number of square tiles in the border of the pool. Each strategy shown is based on a square pool with a side equal to 4 units (s 4) and a border of 20 square tiles. We model the task by drawing what it looks like from above. When looking at a pool and its border from overhead, the border and pool form a square that contains the smaller square representing the pool. In the solutions below, we refer to the pool square as the "little square" and the larger square, made by the pool with the tile border, as the "big square." There are other possible ways of generating expressions for the number of tiles in the border. As new ways come up, they should be recorded on the blackboard or otherwise. Table 1. Strategies for solving the Pool Border Task. Strategy 1 Area along 4 sides of little square (the pool) plus area of the 4 corners 4(4) + 4 4s + 4 Strategy 2 Area along 2 vertical sides of big square (the pool with the tile border) plus area along 2 sides of small square 2(4 + 2) + 2(4) 2(s + 2) + 2s 2s s 4s + 4 p. 2 / 12

3 4(4 + 1) Strategy 3 Area on 4 (sides plus 1) 4(s + 1) 4s + 4 2(4 + 1) + 2(4+1) Strategy 4 Area of 2 L-shaped half borders. 2(s + 1) + 2(s+1) 2[2(s + 1)] 2[2s + 2] 4s + 4 p. 3 / 12

4 Strategy 5 Area along border on each side of the pool minus the 4 corner tiles that were counted twice. (4 + 2) + (4 + 2) + (4 + 2) + (4 + 2) (4 + 2) 4 4(s + 2) 4 4s s + 4 p. 4 / 12

5 (s + 2) 2 - s 2 Strategy 6 Difference of areas of the big square and the little square (s 2 + 4s + 2) - s 2 4s + 4 Discussion of the Pool Border Task: Where to Begin The Pool Border Task (PBT) can be introduced as a homework assignment or started in class. Either way, it is important to clarify the conditions of the Pool Border task. When working on the task, teachers may forget that the pool is a square and begin their work with a non-square rectangle. They also need to understand that in the model, the border and pool are constructed with the same size square tiles, that the border is onetile-wide, and that the border creates a larger square with the pool in the center. Some teachers will benefit from the use of graph paper to represent the different sized pools. Using different colors to represent the pool and the border is one way to start and is useful for recording work on the board or elsewhere. Keeping the color coding as part of recording the numerical steps (as we did in Table 1) helps teachers make connection between representations. If you are sharing their work using a document camera, you can have them use different colors to highlight the particular part of the drawing to which they are referring. As teachers find strategies for finding the number of tiles in the border that eliminate the need for counting, it may be helpful to point out that they are coming up with methods that can be generalized to square pools of any size. The generalizations can be expressed algebraically, as in the sample solutions (Table 1). There are two key ideas for the teachers to take from this task. First, the number of tiles in a given pool's border is constant. No matter what method is used to find the number of tiles in the border of a given sized pool, this number of tiles always is the p. 5 / 12

6 same. Second, all methods of finding the border's area, when expressed algebraically, should yield expressions equivalent to 4s + 4. To reinforce these two ideas, you might want to assign a table, like the one on the next page, for the teachers to complete after summarizing what they have found (See Table 2. Organizing Work on the Pool Border Task.). Many will use the formula 4s + 4; others will use their favorite method. After they complete the table, you can have them look at the number of tiles needed for the border and ask them what they notice. (For example: All the numbers are multiples of 4. Why?) Table 2. Organizing work on the Pool Border Task. Square Pool Size s Number of tiles needed for the border Your strategy for finding the number of tiles needed for the border 4 x 4 10 x x x 25 A key task that you as facilitator has is helping the teachers make a connection between the representation created and the numerical expressions used to find the number of tiles in the border. 1 Here is where the use of a square pool of size 4 comes in handy. One of the first expressions given for the solution is We deliberately chose 4 as the side of the pool to highlight this point. This could lead to a discussion of the idea that which examples are selected can lead to confusions or opportunities to highlight important ideas. p. 6 / 12

7 Asking what each of the 4s represents and how it is connected to the representation results in a comment like the first four 4s are the number of tiles along the side of the pool (Notice we did not say the length of the side of the pool the perimeter/area confusion.) and the last four represents the four corners. This often leads to the expression 4(4) + 4. Again there are too many 4s, each representing something different. The first 4 represents the number of sides of the square pool; the second 4 represents the number of tiles along each of these sides; the third 4 represents the corners of the border. This can lead to an algebraic expression 4s + 4. Similar analyses of the other numerical expressions for finding the number of tiles in the border of the pool makes this connection between the representation and the numerical and/or algebraic expressions. This is the heart of the Mathematical Practices of the Common Core State standards for Mathematics. Using the Pool Border Task with K 12 students The Pool Border Task can be used at all grade levels with different goals. (See the Ferrini-Mundy, Lappan, & Phillips article for specific suggestions.) In the earliest grades, it presents the opportunity for students to make representations using blocks or graph paper and reinforces ideas such as square and border and understanding what the task is about. From grade 3 on, students, in addition to what primary-grade students learn, can find numerical answers to how many tiles are in the border. Once students begin articulating the solution as four sides plus four, they are getting ready for the introduction of algebraic representations. By grade 4 they can write different algebraic expressions and tie the problem to geometric ideas such as those found in Table 1. For middle- school and high- school students, this task becomes a vehicle for undirected or open-ended problem solving. When sharing their solutions to the Pool Border Task, students can share their different approaches. This sharing broadens all students ways of thinking about doing mathematics while reinforcing ideas about equivalent algebraic expressions. At all grade levels, color coding of the pools and borders with parallel color coding of the numerical expressions facilitates making connections between representations. Which Mathematical Practices are highlighted in solving the Pool Border Task? The Pool Border Task is an extremely rich exploration that can provide teachers with opportunities to use each of the eight mathematical practices. To claim that one mathematical practice is less significant than another would require that an essential ingredient of the process be ignored. Therefore, all eight mathematical practices are addressed below. p. 7 / 12

8 MP1 Make sense of problems and persevere in solving them. A critical part of making sense of the Pool Border Task is understanding the use of the term "any" in the statement of the task. Without knowing what "any" means, it is impossible to understand the conclusion the task asks for, i.e., a generalized conclusion. Another crucial part of making sense of the task is extracting the conditions of the task from the task statement: that the pool is square, the tiles are unit squares, the border is one-tile-wide, and the pool's dimensions are natural numbers. Persevering through the Pool Border Task is important because it opens opportunities for the other Practices. Here, perseverance means going further than numeric expressions for specific pool sizes and pursuing the task demand of considering "any" size pool. Whether teachers can persevere depends on how well they have made sense of the task. A question such as Did anyone else use that method with a pool of a different size? may help teachers make sense of the task as well as help them push their thinking further. MP2 Reason abstractly and quantitatively. A major component of work on this task is creating visual representations of pools and borders for specific sizes and in general and then using them to find the number of square tiles in the border. The Pool Border Task can be a nice example of decontextualizing and contextualizing if the teachers are pushed to generalize specific solutions to "any" size pool, explain the connection between their expressions and the representation of the pool, establish equivalences between different solutions to the task, and explain how the equivalences can be established symbolically and visually. When teachers explain how their expressions relate to a visual representation, they are contextualizing the expressions. When teachers first conjecture a general expression, they may be working with patterns within numeric expressions. This is an example of decontextualization, as conjecturing has to do with algebraic expressions that can stand separate from the pool context. However, verifying the conjecture might involve connecting it back to a representation. This is an example of contextualization. Similarly, establishing equivalences symbolically is an example of decontextualizing whereas explaining equivalence in terms of the representation is an example of contextualization. For the Pool Border Task to demonstrate contextualization and decontextualization, it is important that teachers persevere through the problem (MP1) and that teachers are asked to make connections between different solutions both symbolically and visually. MP3 Construct viable arguments and critique the reasoning of others. Teachers construct arguments based on diagrams/pictures of the Pool Border Task. They analyze situations (pools of different sizes) by breaking them into cases. They make conjectures and represent aspects of the conjectures. They build a logical progression of statements to explore the truth of their conjectures. They also are able to compare the effectiveness of two plausible arguments by listening to others explanations of how they developed their formula/rule. p. 8 / 12

9 For the Pool Border Task to exemplify MP3, it is important to help teachers go beyond specific cases and substantiate their reasoning. It can be helpful to ask teachers to describe which part of the representation a particular piece of their expression refers to or to encourage teachers to do so with each others' representations. Having teachers write either numerical or algebraic expressions for each step of their arguments shows others the constructions used. This also forces teachers to make their thinking explicit. It also can be helpful to describe some incorrect expressions and ask teachers to explain why the incorrect expression may have been conceived and what is incorrect about it. MP4 Model with mathematics. This context of this task, finding the number of tiles for the pool s border, allows individuals to apply algebraic and geometric reasoning to a practical situation. In using models, such as diagrams, tables, or colored tiles, teachers explore the possible solutions that fit within a given set of conditions (e.g., the pool must be square). To do this analysis, teachers need to identify quantities such as the measure of each side of the pool, the number of tiles in the border, and how these are related. Testing these relationships may involve creating and interpreting diagrams, two-column tables, graphs, and, especially, variable expressions. By writing algebraic expressions to describe the pool s border s area, participants exemplify mathematical modeling. MP5 Use appropriate tools strategically. Strategic use of tools (such as paper, graph paper, pencil/pens, or square tiles or cubes) arises in using representations of the pool to generate or verify expressions. For instance, shading or coloring may be used to clarify how pieces of expressions are related to pieces of the pool border as well as how different solutions express equivalent expressions. MP6 Attend to precision. The Pool Border Task requires participants to attend to conditions and definitions in making sense of the task, working on the task, and communicating solutions to the task. First, they need to remember that the pools are square pools. Second, when communicating their solutions, they need to be precise about which aspect of the border is represented by a given number or variable. Though the pool border is built around the perimeter of the pool, they are finding the area of the border or the number of square tiles in the border. Describing the tile border as a "perimeter" of the pool is imprecise language that is both incorrect and confusing, and worth remarking upon with teachers. If this comes up, it may be helpful to point out the phrase "number of square tiles" and why this is a clear and precise way of talking about what the task is asking. Lastly, when calculating or counting the number of square tiles in the border, teachers need to keep track of any double counting of tiles as well as making sure that all tiles have been accounted for. Any loss of p. 9 / 12

10 precision here makes it difficult or impossible to make connections between solutions. MP7 Look for and make use of structure. The structure of the border of square tiles around the square pool is inherent in any solution to the task. Asking teachers to explain each others' solutions is a way to engage with looking for and making use of structure. To explain how another solution was developed requires discerning relationships between the structure of the border and terms inside the expression. For instance, 2(s+2) + 2s contains the terms "2(s+2)" and "2s", and each of these are represented in the border. "2(s+2)" could refer to the left and right sides of the border with the corners, and multiplication by "2" makes sense because there are two of these congruent portions of the border. As another example, 4s + 4 contains two instances of 4. The first instance is as a coefficient of s. The variable s is defined as the length of one side of the pool. There are 4 sides, along which to lay any number of tiles. The number of tiles laid corresponds to the length of one side of the pool. The remaining 4 is a constant. The reason it is a constant is that the number of corners of the pool is the same no matter what the size of the pool is. However, the number of border tiles that are along the edge of the pool should change when the pool changes size. Thus one way of engaging teachers in MP7 using the Pool Border Task is to use the geometry of the border to make sense of the meaning of algebraic expressions. This may include the meaning of coefficients, constants, variables, addition, multiplication, or other operations. Similarly, teachers can engage in this kind of analysis of the structure of numeric expressions: 2(4 + 2) + 2(4) can be related to the pool border in much the same way that 2(s+2) + 2s can be. In developing the notion of a variable, asking for explanations of relationships between numeric expressions and the representation of the pool border may help. Doing this is also important because it supports MP8 MP8 Look for and express regularity in repeated reasoning. Whether or not teachers engage with MP8 when working on the Pool Border Task depends on how well they have made sense of the task statement. Understanding how "any" is used in the statement impacts understanding that the task is asking for generalizations. MP8 is then critical for pushing from specific cases to the general case: to arrive at 2(s + 2) + 2s from cases involves recognizing that analogous reasoning was used for [2(3+2) + 2(3)], [2(4+2) + 2(4)], [2(5+2) + 2(5)], etc. Thus, algebra becomes a language for expressing the regularity we have noticed in pool-border patterns. Similarly, understanding why 4s + 4 is a generalization of cases involves recognizing that [4(3) + 4], [4(4) + 4], etc., counts the border tiles along the little square and the corner tiles. To use the task to work on MP8 thus requires that MP1 and MP7 be in place. p. 10 / 12

11 What grade level standards can this task address? The Pool Border Task is an extremely rich exploration that provides students with opportunities to engage with mathematical content at various grade levels and strands. For example, students work with concepts of geometry as they use drawings and diagrams to represent the layout of the pool and border. They use algebraic expressions and notation to complete the task and represent the number of tiles needed for a pool of different sizes. They recognize, compare and create patterns as they create multiple pool borders for different sized pools. Some of the mathematical standards from the Common Core State Standards that are represented in this task are quoted below: 6.EE 2. Expressions and Equations 2. Write, read, and evaluate expressions in which letters stand for numbers. 3. Apply the properties of operations to generate equivalent expressions 5.OA. Operations and Algebraic Thinking 1. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. 2. Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation add 8 and 7, then multiply by 2 as 2 (8 + 7). Recognize that 3 ( ) is three times as large as , without having to calculate the indicated sum or product 3.MD. Measurement and Data 5. Recognize area as an attribute of plane figures and understand concepts of area measurement. a. A square with side length 1 unit, called a unit square, is said to have one square unit of area, and can be used to measure area. b. A plane figure that can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. 6. Measure areas by counting unit squares (square cm, square m, square in, square ft., and improvised units). 7. Relate area to the operations of multiplication and addition. a. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths. 2 6.EE means Grade 6 Expressions and Equations p. 11 / 12

12 b. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning. c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a b and a c. Use area models to represent the distributive property in mathematical reasoning. d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems. Extensions of the Pool Border Task 1. Is it possible to find a formula for finding the number of 1 x 1 square tiles needed for a one square tile-wide border of a rectangular pool that is not a square? If so, what is that formula for a pool with a length of l and a width of w? 2. We found that 4s+4 is the number of tiles needed to build the border. What is the relationship between 4s+4 and the perimeter of the border? What is the relationship between 4s+4 and the perimeter of the pool? How would you explain this relationship using a diagram of the pool and its border? 3. Suppose s is the length of one side of a square made by the border (the "big square") rather than the length of the side of the pool (the "little square"). If s is defined that way, how do you express the number of tiles in the border? Resources Ferrini-Mundy, Joan, Lappan, Glenda, & Phillips, Elizabeth (1997). Experiences with patterning. Teaching Children Mathematics, 3, This article is also available in: Moses, Barbara (Ed.). (1999). Algebraic Thinking Grades K-12: Readings from NCTM s School-based Journals and Other Publications (pp ). Reston, VA: National Council of Teachers of Mathematics. p. 12 / 12