What All Great Mathematicians DoMary Lou Hessling, Jen Voorhees, and Dan Ruch

Jefferson Elementary & Mosser Elementary

Joelle wants to create gardens of various sizes on her large lot. Each garden is 3 feet wide. She wants a walkway created around each garden that is formed by 2 side-by-side square tiles that are 1 ft X 1 ft in size. The figure below shows these tiles around a 3 X 2 garden.

Your task is to develop as many different ways as you can for finding the number of tiles needed for a walkway around a 3 ft X n ft garden, where n is any positive integer. You will be required to make a presentation to Joelle showing that your different approaches give correct answers and give the same answer for any garden.

*Common Core Mathematical Practices posters downloaded from: https://www.sandi.net/Page/50909

Mathematicians use many different problem solving strategies to help them solve difficult problems. We used some of these strategies to help us solve the problem above. In addition to the four strategies we are demonstrating on this poster, the Common Core Standards include the following Mathematical Practices:

Make sense of problems and persevere in solving them

Construct viable arguments and critique the reasoning of others

Attend to precision Look for and make use of structure

We used different colored square tiles to begin modeling a solution to this problem. We decided to use green tiles to show the garden, and yellow and red tiles to show the walkway around the garden.

The first picture (a 3 ft X 1 ft garden) shows 32 tiles in its walkway.

The second picture (a 3 ft X 2 ft garden) shows 36 tiles in its walkway.

The third picture (a 3 ft X 3 ft garden) shows 40 tiles in its walkway.

We noticed that we had to add 4 new tiles for each foot that the width of the garden increased. We showed this using red tiles. We also noticed that there are always 28 tiles that stay the same. We showed this using yellow tiles.

As we were using tiles to model our problem, we noticed a pattern. We can say that the pattern has a rule of “add 4.” The rule of a pattern is a repeated calculation. A good way to show this repeated calculation is to use a chart. The width of the garden is the value that we are changing, and the number of tiles in the walkway is the value that we are trying to figure out.

Using repeated reasoning in this way allows us to extend our pattern as far as we want. No matter what the width of our garden is, we will be able to figure out how many tiles are needed to build the walkway.

Number of tilesWidth of garden (ft) in walkway

1 32 2 36 3 40 4 44 5 48 6 52

+4

+4

+4

+4

+4

We have been reasoning quantitatively. This means that we have been using numbers to represent the values in the context of this problem.

Mathematicians also learn how to reason abstractly. This means using a combination of numbers and symbols to show an equation, or rule. Our equation will help us find the number of tiles needed to build a walkway around a garden of any width.

We will use the letter “w” to stand for the width of the garden.We will use the letter “t” to stand for the number of tiles in the walkway.

Our equation is: t = 4w + 28

number of tiles needed Each time the width

increases by 1,the number of tiles

increases by 4.(This is shown by the

red tiles.)

number of tiles along each side of the

garden, which always remains constant

(This is shown by the yellow tiles.)