• Many ancient cities were constructed inside great walls to protect and defend the city. The city of Carcassonne, France, still exists and has a double wall around it. The length of the inner wall around the city (the perimeter) measures about 1245 meters. The land inside the inner wall (the area) is approximately 105,400 square meters. In this lesson, you will work with tiles to practice measuring perimeter and area.

When measuring shapes, it can be important to look at the space the shape covers. This space is called the area. Other times, it is important to look at the length of the boundary around a shape, which is called the perimeter. In this lesson, you will be using tiles and toothpicks to measure these two attributes (area and perimeter) of various shapes. To describe the shape at right, for example, you could say, “There are 10 tiles,”“It takes 14 toothpicks to surround the shape,”“It takes four more toothpicks to surround the figure than tiles to fill it,” etc. At the end of today’s lesson, you will work with your team on a toothpick-and-tile challenge.

  • 1-5. TOOTHPICKS AND TILES
  • Cruz, Sophia, and Savanna are using toothpicks and tiles to describe the attributes of the shapes below. Cruz made a pattern and told the girls the number of tiles he used. Then Sophia and Savanna each tried to be the first to see who could call out how many toothpicks, or units of length, were on the outside.
  • Cruz made the tile pattern shown below and said, “There are six tiles.” Savanna quickly said, “There are ten toothpicks.” Copy the tile pattern on your paper and show where Savanna counted the 10toothpicks. Justify your answer with words, numbers, or pictures.
  1. Cruz put down the pattern as shown below, but he ran out of toothpicks. How would you describe this shape using toothpicks and tiles?
  1. Get a set of tilesfrom your teacher and work with your team to:
  2. Make a pattern so that there are four more toothpicks than tiles.
  3. Draw your tile pattern on your paper
  4. Label the number of toothpicks and tiles on your drawing.

Is there more than one answer?

  • 1-6. When you are working with your team to solve “Toothpicks and Tiles” in the next problem, as well as other problems in this course, it will be important to work effectively with other people. Effective math conversations are a valuable part of the learning process throughout this course. Choose a member of your team to read aloud these Collaborative Learning Expectations:

COLLABORATIVE LEARNING EXPECTATIONS

  • Working with other students allow you to:
  • Develop new ways of thinking about mathematics,
  • Learn to communicate about math, and
  • Understand ideas better by having to explain your thinking to others.
  • The following expectations will help you get the most out of working together.

T / Together, work to answer questions.
E / Explain and give reasons.
A / Ask questions and share ideas.
M / Members of your team are your first resource.
S / Smarter together than apart.
  • 1-7. TEAM CHALLENGE: TOOTHPICKS AND TILES
  • Today you and your team members will work together to participate in the “Toothpicks and Tiles” challenge. You each will have one card that shows a tile shape and there will be a fifth card to share as a team. Any extra cards should go in the center of your workspace so that everyone can see them.
  • Your task: As a team, do the following:
  • Place shape cards and all of the fact cards face up in the middle of the table so that everyone can see them.
  • Work together to match each tile shape with one fact statement so that each fact has only one shape and each shape has only one fact.
  • If you want to change the shape that matches a fact card, you must convince the person whose name is on the card of the shape you want to change. He or she is the only person who can touch that shape card.
  • Once everyone in your team is convinced that each fact is paired with one shape, call your teacher over. Be prepared to justify your choices!
  • 1-8. Does changing the number of toothpicks always change the number of tiles? Does changing the number of tiles always change the number of toothpicks? Think about these two questions as you look at the following tile shape.
  • Write a fact statement that includes information about the number of tiles and toothpicks that would describe the tile shape below.
  1. How can you add a tile to the shape in part (a) but not change the number of toothpicks? Justifyyour response.

1-9. Recall that the perimeterof a design is the length of the boundary around the outside of the design (the toothpicks). The number of squares needed to fill the design (the tiles) is called the area.

Use these words to write a fact statement with your team describing the perimeter and area in the tile pattern below.

Next, build and draw a different shape that could also be described by the fact. Label the figure with the area and perimeter.

  • Perimeter and Area
  • The perimeterof a shape is the total length of the boundary (around the shape) that encloses the interior (inside) region on a flat surface. In the game, “Toothpicks and Tiles,” the number of tile side lengths (toothpicks) is the same as the perimeter of the shape. See the examples below.
  • The areais a measure of the number of square units needed to cover a region on a flat surface. In the game, the area is equal to the number of “tiles” in the shape.
  • A rectangle is a quadrilateral (four sides) with four right angles. The opposite sides are equal in length. Two sides that come together (meet) at a right angle are referred to as the length and width, or base and height. The area (A) of any rectangle is found by the relationship A = length · width.
  • 1-10. Janelle wants to challenge you to a “Toothpick and Tiles” game (described in problem 1-8). Using exactly four tiles, solve her challenges below. Justify your answers with pictures and labels.
  • Create a tile pattern where the number of toothpicks is exactly double the number of tiles.
  • Create a tile pattern where the number of toothpicks is more than double the number of tiles.
  • 1-11. In this lesson, you looked at the number of tiles and number of toothpicks used to form shapes made of square tiles as you played the “Toothpick and Tiles” game. The math words that also describe the number of tiles and toothpicks are area and perimeter. Read the Math Notes box for this lesson to review how area and perimeter are related to tiles and toothpicks, then answer the questions below.
  • Find the area and perimeter of the figure below.
  1. Find the area and perimeter of the rectangle below.
  1. Now design your own shape with 5 square tiles.
    Record the perimeter and the area.
  • 1-12. Consider the first three figures of the pattern below.
  • On your own paper, draw what Figure4 of this pattern should look like.
  • Using words, describe what Figures 5 and 6 should look like.
  • Using words, describe how the pattern is changing.
  • 1-13. Vi is trying to figure out how a square can be divided into four equal parts. Show her at least three different ways to divide a square into four equal parts.
  • 1-14.The band students at Tolt Jr. High and Maywood Middle School have been invited to participate in the Evergreen Music Festival in Seattle. Each group has decided to have a car wash to raise money to pay for the trip. Use the graph below to answer the following questions.
  1. Which school washed more cars? How do you know?
  2. Which school has raised the most money so far? How do you know?
  3. Additional Challenge: Find how much each school is charging to wash a car. Show your work to justify your answer.