Patterns and Functions
In this lesson, students investigate properties of perimeter, area, and volume related to various geometric two- and three-dimensions shapes. They conjecture, test, discuss, verbalize, and generalize patterns. This lesson involves using a real world problem. Exploration is used to determine a formula.
- Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge...
- Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures....
- Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between...
- Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles,...
- Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
- GLE 0606.1.3
- Develop independent reasoning to communicate mathematical ideas and derive algorithms and/or formulas.
- GLE 0606.4.3
- Develop and use formulas to determine the circumference and area of circles, and the area of trapezoids, and develop strategies to find the area of composite shapes.
- GLE 0606.4.4
- Develop and use formulas for surface area and volume of 3-dimensional figures.
- SPI 0606.4.4
- Calculate with circumferences and areas of circles.
- SPI 0606.4.5
- Determine the surface area and volume of prisms, pyramids and cylinders.
- SPI 0606.4.6
- Given the volume of a cone/pyramid, find the volume of the related cylinder/prism or vice versa.
- Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and...
- Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these...
- Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the...
- Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles,...
- Know and understand the formulas for the volumes of cones, cylinders, and spheres, and use them to solve real-world and mathematical problems.
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- Compute perimeter, area, and volume of various geometric figures.
- Compute maximum and minimum area of geometric figures, given linear dimensions restrictions.
- Fixed Area
- How many people can be seated if we have 24 unit tables?
- Or, what is the perimeter of a rectangular table whose area is 24 square units?
- Discussion. Allow the students to use the square tiles to form tables and to list all rectangles with an area of 24 square units in a chart. Stress that we are trying to find all rectangles with a given length. After the students have explored this problem, organize the data.
- Ask students to observe the patterns.
- Are they the same as the problem with fixed perimeter?
- As the length increases, the width decreases. But the perimeter decreases to a certain point and then starts to increase, but the decrease and increase are not as symmetric as in the pattern with fixed perimeter. Graphing the problem displays this pattern visually. The maximum perimeter is 50 units for the long, skinny 1 × 24 rectangle. The minimum perimeter is 20 units for the fat, "almost square" 4 × 6 rectangle.
- Ask, "Could we have a larger perimeter?" Students might suggest cutting the tables in half to form a rectangle with dimensions of 1/2 × 48. Other students will claim that if we allow rational (or real) numbers, we could go on forever.
- "What about the minimum perimeter? Could we get a smaller perimeter?" [Yes, if we try lengths between 4 and 6.] Try 5 for one side, then the other side will have to be 24/5 (the area is fixed at 24 square units). The perimeter is 19.6 units, which is less than 20 units. This process can go on until we reach a number that when squared is 24. Students can use a calculator to compute the square root of 24, or they can estimate the square root of 24 as 4.9: 4.92 = 24.01, or approximately 24.
- Call attention to the fact that this graph is not a parabola (its shape is called a hyperbola).
- Ask the students to read information from the graph: If the perimeter is 30 units, what are the dimensions of the rectangle?
- If the length is 7 units, what is the perimeter of the rectangle?
- If the length is 24 units, what is the perimeter?
- For older students, this process can be generalized as
- P = 2(l + w) and A = lw, or w = A/l. Thus P = 2(l + A/l).
- Fixed Area for Plane Figures
- For a fixed perimeter, which plane figures will have the greatest area?
- Discussion. If we extend our investigations to other figures in the plane, such as polygons and circles, students can discover that the circle is the plane figure that has the most area for a fixed perimeter.
- Square tiles
- A piece of grid paper for each student
- Tables at a Birthday Party Activity Sheet