Gallery Problems Exercise (Groups)

Gallery Problems Exercise (Groups)

Finding the Missing Base

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Find the Missing Base

A trapezoid’s height is 67.533.

  1. Use this information and the formula for the area of a trapezoid to calculate the length of the other base. Show your work.

  2. Sketch the trapezoid and label the base lengths and the height.

Utah Units

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Utah Units

The state of Utah has a rather geometrical shape, as shown in the image.

The shape is close to a rectangle, with a small rectangle cut out of the upper right corner.

  1. Use the scale to estimate the total area of Utah in square miles.
  2. Use the area of Utah as a unit of measure to estimate the area of the continental United States. That is, estimate the area of the continental United States in “Utahs.”
  3. Use your answers to problems 1 and 2 to estimate the area of the continental United States in square miles.

Growing Rectangles

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Growing Rectangles

Look at this series of growing rectangles on the right.

  1. Calculate the areas of rectangles B, C, D, and E.

  2. Find the coordinates of the points b, c, d, and e .

  3. What is the distance between points (2,212) and (13,212)?

  4. What is the distance from each of the points b, c, d, and e to the x-axis?

  5. If you continue the series and draw rectangles F and G, what are the coordinates of corresponding points f and g?

  6. What are the areas of rectangles F and G?

The Volume of Solids

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The Volume of Solids

  1. The following solids are all built out of cubes. Find the volume of each solid, and describe your solution strategy.
  2. Find the total surface area of the last building.
  3. Make your own building using the Cube Builder, and ask a classmate to find the volume of your building.

INTERCTIVE: Building Viewer

INTERACTIVE: Cube Builder

From 3-D to 2-D and Back

Work Time

From 3-D to 2-D and Back

While traveling in the Netherlands, Denzel bought a liter of milk. This image shows what the carton looked like.

This type of carton is constructed to allow a combination of pushing and pulling to make a convenient pour spout.

Denzel cut and unfolded the carton as shown.

He flattened the carton to get the net shown below.

  1. The base of the carton is a square with side lengths of 7 centimeters. How high must the carton be to have a volume of 1 liter? (1 liter = 1 cubic decimeter)
  2. The net for the carton is almost a perfect square with dimensions 29 centimeters by 29 centimeters. The triangles at the top and bottom of the net are what make it possible to transform the square into a milk carton just by folding. Explain why the heights of the triangles that form the top of the carton are taller than the heights of the triangles that form the bottom of the carton.
  3. The four isosceles triangles at the bottom of the net each have a base length of 7 centimeters and a height of 3.5 centimeters. Draw four isosceles triangles with these dimensions. Cut out the triangles and show that they form a square.
  4. Explain why four isosceles triangles each with a base length of 7 centimeters and a height of 4 centimeters do not form a square.
  5. Draw a precise net of a half-liter carton, and build it.

The Geometry of Gardening

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Geometry of Gardening

Many gardens have geometrical designs, which often include squares and rectangles, as shown in the image. Gardeners often use grid paper to sketch their designs. Sketch a garden on grid paper. The length of each square on the grid represents 10 meters.

  1. The coordinates of the corners of the garden are (10,5)(20,5)(10,15)(20,15). Plot these points, connecting them in order. What figure is the garden?
  2. A house with a square base will be built inside the garden area. Three corners of the house have coordinates (0,5)(0,10)(5,5). What are the coordinates of the fourth corner? What is the area of the house’s base?
  3. Sand will be used to cover the ground under the house. The layer of sand will be 0.5 meter high. What volume of sand is needed?
  4. A fence will be built around the house 20 meters away from the house. How long is the fence?
  5. Outside the fence, grass will be planted. How many square meters is the grassy area?
  6. Inside the grassy area, a rectangular swimming pool measuring 10 meters by 25 meters will be built. Draw the swimming pool in the grassy area any place you like. What are the coordinates of the corners of this swimming pool?

Net of a Number Cube

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Net of a Number Cube

A number cube is a cube with the numbers 1 through 6 labeled on its faces. The numbers on opposite faces add up to 7.

  • Draw a net of a number cube.

Dividing Parallelograms

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Dividing Parallelograms

Emma says that she can divide any parallelogram into four triangles with equal areas using this method:

  • On parallelogram ABCD, I draw diagonal AC.

  • Next, I mark the midpoint of line segment BC, which is point E. I draw line segment AE.

  • Finally, I mark the midpoint of line segment CD, which is point F. I draw line segment AF.

  • For any parallelogram, these steps result in four triangles with equal areas.

Is Emma right?

State your decision and justify it mathematically—that is, if you think Emma’s method works for any parallelogram, show why it does.

If you think it does not, show why not.

Area of Triangles

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Area of Triangles

  1. Find the area of triangle A. Describe step by step the method you used to find its area.

  2. Find the area of triangle B. Describe step by step the method you used to find its area.

Placing a Rug

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Placing a Rug

The image on the right shows a floor plan of Mrs. Thompson’s living room. She wants to put a square rug with an area of 25 square feet exactly in the middle of the floor.

  1. Shade the floor plan to show where the rug will go.
  2. Find the number of square feet that will not be covered by the rug.