Building Castles

Allow your group about 3-4 minutes of private think time before beginning to explore each problem.

Final Product:

Each group member needs to keep track of your groups’ thinking for finding the number of blocks or tiles needed in each case. For each problem, draw a diagram on poster paper and show your work process. Each group member should be prepared to explain your group’s thinking to the class. A complete explanation will include clear connections between the strategies you used and the diagrams you make. One group member will be randomly selected to present for your group.

Task 1: The Stairs

We are building staircases out of wooden cubes. A 1-step staircase consists of one cube, and the 2-step staircase consists of three cubes stacks (see below).

1-step 2-step3-step

How many cubes will be needed to build a 4-step staircase? An 8-step staircase? A 50-step staircase? A n-step staircase?

Task 2: Skeleton Towers

A skeleton tower is made up of a stack of cubes with a triangular wing on each of the four lateral faces of the cube. The pictures represent the first three skeletons towers.

1st Tower2nd Tower3rdToweer

How many cubes would be needed to build the 4th skeleton tower? The 8th? The nth? How would you describe the pattern verbally?

Task 3: Skeleton Castles

A skeleton castle is made up of stacks of cubes that rise on four corners of a square and descend to meet at the middle of each side of the base. Skeleton castles that are 2 cubes high and 3 cubes high are shown.

How many cubes would be needed to build a skeleton castle that is 5 cubes high? 8 cubes high? n cubes high?

Task 4: The Garden

Below are three different garden borders made of square tiles.

Garden 1Garden 2Garden 3

The perimeter of the garden is determined by the number of tiles it takes to surround the garden using the pattern above.

  • What would the 100th garden border look like? How do you know?
  • Find at least 2 methods to determine the number of tiles it would take to surround any garden border. At least one of these methods should enable you to quickly determine the total number of tiles for any garden border.
  • Use these methods to find a rule that helps you determine the number of tiles needed to build the any garden border.

Extension:

Imagine the garden border was made using toothpicks instead of tiles. How many toothpicks would be needed to build any garden border?