The Tile Problem

A construction worker puts 1 foot by 1 foot tiles on the floor to form shapes of different sizes that follow the pattern below.

# 2 # 4 # 5 a. After laying the tiles, the worker needs to apply some waterproof caulking around the outside of the floor pattern. The caulking is sold by the foot. Write an expression that would help you determine how many feet of caulking he would need to apply for a floor pattern of any size. Create a table and draw a graph for this relationship. b. How many tiles are needed to make each pattern?

Tile Problem: Part A

Getting Started When introducing the task, it is important to make sure that participants understand the following points:  The pattern numbers correspond to the number of tiles laid horizontally, vertically, and diagonally.  The caulking is applied around the outside of the entire pattern, not each tile.  For English language learners (ELLs): You will need to explain weatherproof caulking and how it is applied to tile. This will be important for ELLs or anyone unfamiliar with what this means. You might also ask if any students have performed such a job or can demonstrate what this would mean on the tiles of a floor or roof.  Struggling Students: Students who are slow to get started will need to draw pattern #3 and perhaps pattern #6 before trying to make a table and a graph.  Extension: Find the rule, equation, or function to represent the relationship.

Method

Pattern # Feet of Caulking 2 8

1 C = 4p is a linear relationship. Plotting the points on the graph will show this.

Content

Mathematical concepts to be made explicit include the following:

 Adding 4 to the amount of caulking as the pattern number increases by 1 shows a constant, steady increase.  The increase between the y values is adding by 4 each time, but the relationship between x and y is multiplicative. The rule is a functional relationship between x and y.  The graph is linear. The steady increase is the slope.

Tile Problem: Part B

Getting Started

The question being asked, “How many tiles,” is more definite and therefore less open to misinterpretation. However, determining the pattern and thus the solution is more difficult.  Use of grid paper is essential for this problem.  It is especially important that the table is made sequentially with pattern numbers in the x column.  Struggling Students: Encourage students to look at the pattern geometrically and not just at the numbers in the table. Seeing the pattern of the number of tiles is the first step. Expressing the pattern as a formula is more demanding.  Completing the square of the tile pattern on the grid paper then determining how that relates to the amount of tiles needed will lead to the answer—one way or another. Encourage struggling students to shade the tiles needed on the grid, and then try to express this quantity in terms of the pattern number.  Extension: Should a student be able to see the pattern either geometrically or from the table, ask the student to justify the answer in another way. For example, if the pattern was determined numerically, show how it is justified geometrically and vice versa.

Methods

There are several methods that will arrive at the general formula. Students are often able to see a pattern in the y column of the table, or other patterns within the table, but will still struggle to find a formula.

Pattern Number Number of Tiles 2 3 3 6 4 10 5 15 6 21

2 7 28 8 36

As the xs increase the y values increase by adding one more than was previously added. (First add 3, then 4, then 5, then 6 , and so on.) This is because as the pattern number increases, a new row of tiles is added that is one tile longer than the previous pattern.

The resulting y values are actually a special sequence called the “triangular numbers.”

Methods A, B, and C rely on diagrams, but actually Method B can be done using the formula for the sum of a series. See the next page for an example.

Method A:  Complete the square and shade in the tiles needed.  Draw a line through the middle.  The whole square is n2 so the diagonal line divides it into n2/2.  Now you need to add in all the ½ squares.  Counting and looking for the pattern shows that the additional squares are n/2.  The final answer is n2/2 + n/2= (n2+n)/2= n (n+1)/2.

Method B  Complete the square and shade in the tiles needed.  Now you need to subtract the extra tiles that are left unshaded.  The pattern of those tiles is 1, 3, 6, 10 . . . (as seen in the table).  The formula for these numbers is n(n+1)/2 beginning with n=1.  The tile pattern begins with n=2, so this formula becomes n(n-1)/2.  The whole square is n2 and now subtract n(n-1)/2 for the extra tiles.  The algebraic simplification is shown on the diagram page and the quantity simplifies to n(n+1)/2.

Method C  Take two tile patterns, invert one, and then turn it so it fits onto the other.  The width remains n, but the length is n+1.  Thus the number of tiles in the whole rectangle is n(n+1).  The number of tiles needed is half of this, so the formula is n(n+1)/2.

3 4 Method D  Graph the points in the table.  Note that the result is a parabola.  Use the points to make 3 equations in 3 variables: 6=9a+3b+c 3=4a+2b+c 10=16z+4b+c Eliminating a results in this possibility: -2=-4b-3c -3=-6b-5c Solving for c gives c=0, substituting for b results in b=1/2, and replacing c and b gives a=1/2. Thus obtaining y=1/2 x2+ 1/2.

Math Content

 Graphing the values from the table produces a curve, because the function is quadratic.  The number of tiles needed is the area in square units.  Emphasize the difference between Part A and Part B. The steady, constant change of Part A produces a linear relationship, whereas the changing quantity in Part B produces a non-linear relationship.