geometry and measurement of plane figures activity...

Geometry and Measurement of Plane Figures

Activity Set 3

Trainer Guide

GEOMETRY AND MEASUREMENT OF PLANE FIGURES—AcTIvITY SET 3 Int_PGe_03_TGCopyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development

GEOMETRY AND MEASUREMENT OF PLANE FIGURES —AcTIvITY SET 3 Int_PGe_03_TG

Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development 1

GEOMETRY AND MEASUREMENT OF PLANE FIGURESACTIvITY 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-cmGrid• Transparency/Page:BreakApart• Transparency/Page:HexagonsandArea• 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

Vocabulary

• area

tiMe: 30 minutes

INTRODUCE

•Display Transparency:1-cmGrid.

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.

•Askparticipantswhattheperimeteroftheshapeis.(12)

GEOMETRY AND MEASUREMENT OF PLANE FIGURES—AcTIvITY SET 3 Trans_K6_PG_03Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development

1-cm grid

Transparency: 1-cm Grid




•Askparticipantswhattheareaoftheshapeis. (9 square units)

•Askhowtheyfoundit.(counted,ormultiplied3 3)

•Pointtooneofthesmallrainbowsquares.

•Explainthatthissquarepieceis1unitonitsside. Itmakes“1squareunitofarea.”

•Createarectangleof4 6 units.


•Askparticipantshowmanysquareunitsmakeupthearea of this shape. (24 square units)

•Askhowtheyfoundtheanswer.(counted,ormultiplied 6 4)

•Askparticipantsiftheycancreateaformulathatcanbe used to find the area of any rectangle. (A = l•w—area equals length times width)

•Createasquareof4 4 units.


•Askparticipantshowmanysquareunitsmakeupthearea of this shape. (16 square units)

•Askhowtheyfoundtheanswer.(counted,ormultiplied 4 4)

•Explainthatthisisaspecialcasewheretheperimeterand the area have the same numerical value (but not unit value). The perimeter is linear units and the area is square units.

•Laytheopaquerectangleonthegridsothatthesidesmeet a corner exactly.

•Askparticipantstoapplytheformulatotheareaofthis rectangle. (24 square units)




•Laythetwoopaquerighttrianglesbacktobackonthe 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.

•Askparticipantswhattheareaofthistriangleis. (24 square units)

•Askhowtofindthisareawhetheryougivetheanswer or they provide the answer. (The formula for the area of a triangle is A = 1

2bh.)

•Explainthattheformulaworksbecause...andshowthe following:

◆ Taketheright-handtriangleandflipitup.

◆ Move the flipped triangle to the left and down until it joins the original to create a rectangle, and label as shown:

◆ Pointoutthattherectanglecreatedbythetwotriangles is equivalent to the rectangle previously created (6 4) or 1

2 the width of the base of the triangle times its height.

◆ Explainthatanumberofformulasforareaaregroundedinthesamesortofactivity,thebreakingup of shapes to construct rectangles.




DISCUSS AND DO

•Haveparticipantsworkinpairs.

•Display Transparency:BreakApart.

•HaveparticipantstakeouttheirBreakApartpages and cut out the shape.

•Have them find the formula for the area of a parallelogrambytakingtheshapeapartandconstructing rectangles with the pieces.

•Allowparticipantstousethegridpapertohelpthemsolve 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


break apart

How can you cut this shape apart to create a rectangle?

Transparency: Break Apart




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.

•Display Transparency:HexagonsandArea.

•Askparticipantshowtheymightusewhattheyhavelearned to find the area for this hexagon.

•Giveparticipantstimetoworkoutpossibleformulasandrecordonasheetofpaperhowtheywouldbreakthis shape apart.

•Havevolunteerparticipantssharetheirsolutions. Some sample pictures are shown below:

•Haveparticipantscomeupanddrawtheirsolutionswithintheblankhexagons.

•Explainthatbecausethehexagonisregular, 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.

•Haveparticipantstakeouttheir1-cmGrid pages.

•Haveeachparticipantcreateashapefollowingthelines of the 1-cm grid that has an area of 24 square units. The shape may not be a rectangle.

•Giveparticipants2–3minutestocreatetheirshapes.





Break apart these regular hexagons in multipleways to find the area.

hexagons and area

Transparency: Hexagons and Area




•DisplaytheTransparency:1-cmGrid.

•Havevolunteerparticipantscomeupanddrawtheshapes on the transparency.

•Pointoutthatjustasdifferentshapescanhavethesame perimeters but different areas, different shapes may have the same area and different perimeters.

•Askparticipantshowtheydeterminedtheareasfortheshapes.(Mostwillhavebrokentheshapesintosmall rectangles and used multiplication and addition to find the totals.)

•Pointoutthatonceagain,theyhaveuseddeconstruction to find the areas of shapes.

•Countwithparticipantstheperimetersoftheshapesthat they have drawn on the transparency.

•Labeleachshape.

•Pointoutthatshapeswiththesameareacanhavedifferent perimeters.

•Pointouttoparticipantsthattheyhaveexploredtherelationships between shapes, perimeters, and area.

•Askparticipantstosummarizewhattheyhavelearned 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


1-cm grid

Transparency: 1-cm Grid

GEOMETRY AND MEASUREMENT OF PLANE FIGURES—AcTIvITY SET 3 Int_PGe_03_PMCopyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development

1-cm Grid


Break Apart

How can you cut this shape apart to create a rectangle?


Break apart these regular hexagons in multiple ways to find the area.

Hexagons and Area


GlossaryGeometry 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.


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.

Glossary (continued)

geometry and measurement of plane figures activity...

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