CCSS Gr. 6 Find Areas by Composing and Decomposing Polygons

Jack D. Gittinger, Nov 20, 2015

[color=#c51414][i][b]Common Core Standards for Grade 6-Geometry[/b][/i][/color] [b]Solve real-world and mathematical problems involving area, surface area, and volume.[/b] 6.G.1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. 6.G.2. 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 show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. 6.G.3. Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. 6.G.4. 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 techniques in the context of solving real-world and mathematical problems. [b]Apply and extend previous understandings of numbers to the system of rational numbers.[/b] Extend coordinate axes familiar from previous grades to represent points on the plane with negative number coordinates.[list] [*]Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. [*]Find and position pairs of integers and other rational numbers on a coordinate plane. [/list]


  • 1. FINDING THE AREA OF A TRIANGLE
  • 2. Discover the formula for the Area of a Triangle
  • 3. DECONSTRUCTING THE PARALLELOGRAM
  • 4. FINDING THE AREA OF PARALLELOGRAM
  • 5. FINDING A FORMULA FOR THE AREA OF A PARALLELOGRAM
  • 6. DECONSTRUCTING THE TRAPEZOID
  • 7. FINDING THE AREA OF A TRAPEZOID
  • 8. FINDING A FORMULA FOR THE AREA OF TRAPEZOID 

1.

FINDING THE AREA OF A TRIANGLE

Instructions
[left] [b]Introduction and Background[/b][/left]Polygons can usually be deconstructed (decomposed) into smaller polygons.  Likewise, smaller polygons can be used to construct (compose) other larger polygons. [br][br]Composing and decomposing polygons can help us find the area of these polygons. The area of a rectangle, as you know from earlier [i]GeoGebra [/i]activities, can be found by counting how many unit squares it takes to completely cover the rectangle’s surface. You can do this by dragging unit squares to cover the rectangle or by using the [i]GeoGebra[/i] grid to count unit squares. [br][br]Unfortunately, not all polygons are so easy to work with when determining their area. Triangles are not so easy.  In this activity you will see how to use mathematical reasoning to determine the number of unit squares it takes to cover a triangle. [br][br][b]Step 1.  Unit Squares and Triangles[/b] [br][br]On the [i]GeoGebra[/i] workspace, you see a red triangle.  [br][br]If you try to determine how many unit squares it takes to exactly cover the triangle, you will have a problem. [br][br]Try it. Use the MOVE tool [icon]/images/ggb/toolbar/mode_move.png[/icon] to move unit squares in an attempt to exactly cover the triangle. [br]The problem is that one or more of the sides of a triangle will cut through the unit squares, so there is not an obvious way to exactly determine the area by simply counting unit squares.  You end up covering too much or too little of the triangle when using unit squares. [br][br]Move the unit square counters back off to the right side of the screen. If you try to use the grid to count unit squares within the triangle, you still have the same problem.  [br][br][b]Step 2.  Composing a New Polygon[/b] [br][br]Click on the [b]View the Clone of the Red Triangle [/b]check-box.  The new, blue triangle is [b][i]congruent [/i][/b]with the original red triangle.  [b][i]Congruent [/i][/b]means it is identical in every way – the length of the sides are the same and the angles are the same.  This cloned[br]congruent triangle does have a visible Rotator Point that you can use to rotate[br]the triangle into new positions. Use the MOVE tool [icon]/images/ggb/toolbar/mode_move.png[/icon] to move and rotate the blue triangle next to the red triangle so that you form a rectangle.  [br][br] Use the GRID or the unit square counters to determine the area of the newly created rectangle:     __________ square units[br][br]Now that you know the area of the rectangle in square units, how can you use reasoning to determine the area of the original red triangle in square units?
– Jack D. Gittinger

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