Activity Set 3. Trainer Guide

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1 Geometry and Measurement of Plane Figures Activity Set 3 Trainer Guide GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set 3 Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development

2 GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set #3 Arranging for Areas In this activity, participants are exposed to the concept of area. Then, they derive the formulas for the areas of common shapes. Materials Transparency/Page: 1-cm Grid Transparency/Page: Break Apart Transparency/Page: Hexagons and Area overhead rainbow cm cube manipulatives 1 opaque rectangle that is 6 4 cm 2 opaque right triangles with a height of 6 cm and a base of 2 cm scissors for each participant 1-cm grid Vocabulary area Time: 30 minutes Introduce Display Transparency: 1-cm Grid. GEOMETRY AND MEASUREMENT OF PLANE FIGURES AcTIvITY SET 3 Trans_K6_PG_03 Copyright 2002 by the McGraw-Hill Companies McGraw-Hill Professional Development Transparency: 1-cm Grid Teaching Tip: Use all the same color manipulative pieces for each shape. Use the overhead rainbow squares to create a square that is 3 3 pieces. Ask participants what the perimeter of the shape is. (12) GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set 3 Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development 1

3 GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set #3 Ask participants what the area of the shape is. (9 square units) Ask how they found it. (counted, or multiplied 3 3) Point to one of the small rainbow squares. Explain that this square piece is 1 unit on its side. It makes 1 square unit of area. Create a rectangle of 4 6 units. Ask participants what the perimeter of the shape is. (20) Ask participants how many square units make up the area of this shape. (24 square units) Ask how they found the answer. (counted, or multiplied 6 4) Ask participants if they can create a formula that can be used to find the area of any rectangle. (A = l w area equals length times width) Create a square of 4 4 units. Ask participants what the perimeter of the shape is. (16) Ask participants how many square units make up the area of this shape. (16 square units) Ask how they found the answer. (counted, or multiplied 4 4) Explain that this is a special case where the perimeter and the area have the same numerical value (but not unit value). The perimeter is linear units and the area is square units. Lay the opaque rectangle on the grid so that the sides meet a corner exactly. Ask participants to apply the formula to the area of this rectangle. (24 square units) GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set 3 Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development 2

4 GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set #3 Lay the two opaque right triangles back to back on the grid to create a single triangle as shown below. The bottom-left corner of the shape must be exactly at the corner of one of the grid squares. Ask participants what the area of this triangle is. (24 square units) Ask how to find this area whether you give the answer or they provide the answer. (The formula for the area of a triangle is A = 1 2 bh.) Explain that the formula works because... and show the following: Take the right-hand triangle and flip it up. Move the flipped triangle to the left and down until it joins the original to create a rectangle, and label as shown: Point out that the rectangle created by the two triangles is equivalent to the rectangle previously created (6 4) or 1 2 the width of the base of the triangle times its height. Explain that a number of formulas for area are grounded in the same sort of activity, the breaking up of shapes to construct rectangles. GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set 3 Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development 3

5 GEOMETRY AND MEASUREMENT OF PLANE FIGURES AcTIvITY SET 3 Copyright 2002 by the McGraw-Hill Companies McGraw-Hill Professional Development Trans_K6_PG_03 GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set #3 Discuss and Do Have participants work in pairs. break apart How can you cut this shape apart to create a rectangle? Transparency: Break Apart Display Transparency: Break Apart. Have participants take out their Break Apart pages and cut out the shape. Have them find the formula for the area of a parallelogram by taking the shape apart and constructing rectangles with the pieces. Allow participants to use the grid paper to help them solve the problem. Give participants 4 5 minutes to complete the activity. Conclude Have the group come together and share their answers for the formula and ways to find it. Parallelogram A = bh GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set 3 Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development 4

6 GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set #3 Teaching Tip: The key to the deconstruction of the parallelogram is that a line is drawn or cut that is perpendicular to both bases forms a right angle with both bases. The line can be drawn from the corner of one base, as is shown in the previous example, but it also can be drawn from any place where the bases are aligned with one another. Note: If a parallelogram is regular (all sides the same length), cutting from corners to their opposites will create four right triangles that can also be combined to form rectangles. This method works only with regular parallelograms, including rhombuses and squares. It is a special case. GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set 3 Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development 5

7 GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set #3 hexagons and area Break apart these regular hexagons in multiple ways to find the area. Display Transparency: Hexagons and Area. Ask participants how they might use what they have learned to find the area for this hexagon. Give participants time to work out possible formulas and record on a sheet of paper how they would break this shape apart. Have volunteer participants share their solutions. Some sample pictures are shown below: GEOMETRY AND MEASUREMENT OF PLANE FIGURES AcTIvITY SET 3 Copyright 2002 by the McGraw-Hill Companies McGraw-Hill Professional Development Trans_K6_PG_03 Transparency: Hexagons and Area Have participants come up and draw their solutions within the blank hexagons. Explain that because the hexagon is regular, we can multiply 6 times the area of the individual triangles or we can find the area of the 2 isosceles triangles and add it to the rectangle. Have participants take out their 1-cm Grid pages. Have each participant create a shape following the lines of the 1-cm grid that has an area of 24 square units. The shape may not be a rectangle. Give participants 2 3 minutes to create their shapes. GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set 3 Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development 6

8 GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set #3 Display the Transparency: 1-cm Grid. Have volunteer participants come up and draw the shapes on the transparency. 1-cm grid Point out that just as different shapes can have the same perimeters but different areas, different shapes may have the same area and different perimeters. Ask participants how they determined the areas for the shapes. (Most will have broken the shapes into small rectangles and used multiplication and addition to find the totals.) Point out that once again, they have used deconstruction to find the areas of shapes. Count with participants the perimeters of the shapes that they have drawn on the transparency. GEOMETRY AND MEASUREMENT OF PLANE FIGURES AcTIvITY SET 3 Trans_K6_PG_03 Copyright 2002 by the McGraw-Hill Companies McGraw-Hill Professional Development Transparency: 1-cm Grid Label each shape. Point out that shapes with the same area can have different perimeters. Point out to participants that they have explored the relationships between shapes, perimeters, and area. Ask participants to summarize what they have learned about those relationships. Sample answers include: Different shapes can have the same perimeters or the same areas. Shapes with the same area can have different perimeters. Shapes with the same perimeters can have different areas. End of Arranging for Areas GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set 3 Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development 7

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10 1-cm Grid GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set 3 Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development Int_PGe_03_PM

11 Break Apart How can you cut this shape apart to create a rectangle? GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set 3 Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development Int_PGe_03_PM

12 Hexagons and Area Break apart these regular hexagons in multiple ways to find the area. GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set 3 Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development Int_PGe_03_PM

13 Glossary Geometry and Measurement of Plane Figures angle Geometric figure made of 2 rays or 2 line segments that share the same endpoint, called a vertex. area The number of square units in a region. congruent Having the same shape, size, and/or measure. degree A unit for measuring angles. irregular polygon A polygon in which not all the sides are congruent and/or not all the angles have the same measure. line A set of points forming a straight path in 2 directions that are opposite each other. perimeter The distance around the outside of a shape or figure. plane A flat surface that extends forever in all directions. point A location in space. polygon A closed shape made up of a minimum of 3 line segments. quadrilateral A polygon with 4 sides. rectangle A quadrilateral with 4 right angles. GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set 3 Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development Int_PGe_03_PM

14 Glossary (continued) regular polygon A polygon in which all the sides are congruent and all the angles have the same measure. triangle A polygon with 3 sides. GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set 3 Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development Int_PGe_03_PM

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