Spring 2000 Workshop on Geometry – grades 3-5

Contents

Introduction

SCOs related to geometry and measurement, grades 3-5

Lessons on the theme of houses

Lessons on the theme of naming shapes

Lessons from 4173-2000

Introduction

There are many resources to help you teach geometry.The curriculum guide includes ideas for addressing specific curriculum outcomes.The text Interactions 4 has some good activities (I especially like the one on flags, which could be part of a larger interdisciplinary unit).The magazine Teaching Children Mathematics is also full of good ideas.You can find an index of it at nctm.org.

I have tried to assemble activities that touch on a range of outcomes.One approach would be to explore a couple of activities like the ones I propose, and then use ideas from the curriculum guide to hit specific curriculum outcomes that were missed.

SCOs for Measurement and geometry


 
Grade Three
Grade Four
Grade Five
D1- recognise and demonstrate that objects of various shapes can have the same area
D2- calculate areas of irregular shapes
D2- recognise and demonstrate that objects of the same area can have different perimeters
D1- solve simple problems involving the perimeters of polygons

E4- explore relationships between area and perimeter of squares and rectangles
D2- estimate and measure capacity in millilitres and litres
D3- measure volume, using non-standard units 
D6- Solve simple problems involving volume and capacity
D4- estimate and determine the volume of rectangular prisms, using centimetre cubes
D5- recognise that the measure of an angle indicates an amount of turn
D6- estimate and measure angles, using non-standard units
D3 determine the measures of right angles, acute angles, and obtuse angles

D7- estimate angle size in degrees
D7- read clocks
D7- use a thermometer to read temperatures
D1- estimate and measure length in metres, decimetres, and centimetres

D6- solve problems using kilometres
D8- estimate and measure in millimetres, centimetres, decimetres, metres, and kilometres
D4- demonstrate an understanding of the relationships among particular SI units
D4- estimate and measure area in non-standard units and square centimetres
D9- estimate and measure area in square centimetres
D5- Develop formulas for areas and perimeters of squares and rectangles

D8- determine which unit is appropriate in a given situation and solve problems involving length and area
D3- estimate and measure mass in grams and kilograms

D6- use appropriate units for capacity and mass

D8- continue to solve a wide variety of measurement problems
D10- solve relevant problems involving millilitres, and litres, grams and kilograms
E12- make the connection for rectangles between the arrays of squares forming them and the describing of their dimensions
D11- relate dimensions and areas of rectangles to factors and products
E6- cut and assemble net patterns for pentagonal and hexagonal prisms and pyramids
E1- draw various nets for rectangular prisms and cubes
E1- draw a variety of nets for various prisms and pyramids
E7- build skeletons of various prisms and pyramids to focus on edges and vertices
E2- construct models for various cylinders, cones, prisms, and pyramids
E2- identify, describe, and represent the various cross-sections of cubes and rectangular prisms
E3- construct shapes given isometric drawings
E3- make and interpret isometric drawings of shapes made from cubes
E11- recognise and identify various polygons, prisms, and pyramids in real world contexts 
E4- explore relationships among 3-D shapes
E8- predict the result of combining triangles and/or quadrilaterals
E5- find all possible composite figures that can be made from a given set of figures
E5- Predict and construct figures made by combining two triangles

E13- explore how figures can be dissected and transformed into other figures
E2- recognise and represent angles that are less than/more than right angles
E6- recognise, name, describe, and construct acute and obtuse triangles
E6?

E7- recognise, name, describe, and construct right, obtuse, and acute triangles
E7- recognise name, describe, and construct equilateral, isosceles, and scalene triangles
E8- make generalisations about the angle, side length, and parallel side properties of various quadrilaterals
E8- make generalisations about the diagonal properties of squares and rectangles and apply these properties
E4- recognise, name, describe, and represent kite, and some concave, convex, and regular polygons
E9- sort quadrilaterals under property headings
E5- recognise, name, describe, and represent different prisms and pyramids
E10- make generalisations about the numbers of vertices, edges, and faces of various prisms, pyramids, comes, and cylinders
E3 recognise, name, describe, and represent congruent angles and congruent polygons

E10- recognise, name, describe, and represent half and quarter turns of 2D figures
E11- predict and confirm the results of various 2D figures under slides, reflections, and quarter/half turns
E9- make generalisation bout the properties of translations and reflections and apply these properties

E10- explore rotations of one-quarter, one-half, and three-quarter turns using a variety of centres
E9- find the lines of reflective symmetry of polygons
E12-make generalisations about the reflective symmetry property of various quadrilaterals
E11- make generalisation bout the rotational symmetry properties of square an rectangles and apply them

Recognise, name and represent figures that tessellate.



Lessons on the themes of houses

These lessons could be useful incorporated into a interdisciplinary unit on housing.Some discussion is suggested beforehand, of the need to plan houses when building them, and when remodelling in order to know the quantities of materials required.

House lesson 1

Materials

Centimetre grid paper

Information for teacher

This can be done in groups or individually.Some students may need clarification of the idea of outlining the base of the house.Looking at house plans beforehand might help to clarify the difference between a front view (an elevation) and a plan.Be sure they realise that they are measuring all the area inside the outline of the plan.Any interior walls they might add should not effect the overall area.A whole class discussion of #2 provides an opportunity for mathematical communication and reasoning.The key idea to get to is that different shapes can have the same area.

Instructions to students

1. An important thing to know about a house is the amount of living space in it.The area of the floors is used to describe the amount of living space.Draw the outline of the floor of a house with an area of 100 m2.Use centimetre grid paper so you can measure the area of your house accurately.Be imaginative about the shape of your house.

2. What shapes can you use to make a house with 100 m2 of living space? Explain.

Outcomes

4-D1, D9, 3-D4, 5-D2

House lesson 2

Materials

Centimetre grid paper, construction paper cut in 3 cm strips, adhesive tape

Information for teacher

Some students may want to draw a new plan instead of attempting to attach walls to their previous plan.The questions in 2-5 could be answered in small groups on a response sheet, or in a whole class discussion, or both. In 3 they should find the lengths were different for different shapes.The key idea to explore though these questions is that perimeter can vary while area remains the same.In terms of houses this means that two houses can have the same living space, but different lengths of wall.

Instructions to students

1. The size of the outside walls of a house influence the cost of siding, paint, and heat. Use construction paper strips to make the outside walls of your house.Tape them in place.

2. What was the total length of the construction paper strips you used?

3.Compare your answer with other people's answers.What shapes used the longest strips?What shapes used the shortest?Why?What else do you notice?Can you explain what you notice?

4. What is the most strip you would ever need to make walls for a 100 m­2 house?

5. How long is the shortest strip you could use to make walls for a 100 m­2 house?

Outcomes

4-D2, D8, 3-D1, 5-D1

House lesson 3

Materials

Centimetre grid paper, construction paper cut in 3 cm strips, adhesive tape, lima beans, miniature marshmallows, or some other roughly 1 cm objects.

Information for teacher

Counting the beans is an opportunity to review systematic methods of counting, by grouping in tens, for example.They should find that each house used about the same number of beans.This provides a foundation for understanding that the volume of a prism with a given height depends on the area of the base, not the shape of the base.

Instructions to students

1. Make sure the wall of your house are taped down securely. Fill your house with beans.

2.How many beans fit in your house?

3.Compare your answer with other people's answers. What shapes held the most beans?What shapes held the fewest? What do you notice?Can you explain what you notice?

4. Did you measure all the space in your house?Explain. 

Outcomes

4-D3, 5-D6

House lesson 4

Materials

Centimetre grid paper, construction paper cut in 3 cm strips, adhesive tape, centimetre cubes

Information for teacher

The area of the base no longer has to be 100 m2 for this activity.Question 4 provides an opportunity to evaluate their understanding of 4-D2 in lesson 2.Someone may observe that the number of cubes can be determined by repeated adding of the number in each layer.Encourage them to share such observations.

Instructions to students

1. Make a new house with a rectangular base. 

2. How many m2 of living space is there in your house?

3. How much paper did you need for the walls?

4.Is the length of paper you needed related to the living space?Why?

5. Fill your house completely with centimetre cubes.

6. How many cubes did you need?

7.Is the number of cubes you needed related to the living space?Why?

Outcomes

4-D9, D4, D8 3-D1, D4, 5-D5, D8, E4

House lesson 5

Materials

Construction paper, scissors, rulers

Information for teacher

Some children might find this difficult.Group work is a good idea for this.Suggest that they look closely at the rectangular houses they made before.They may even want to cut up a house to see how it would look flat.Encourage experimentation. Students how finish early could be invited to assist others, or to try to draw patterns for more complex houses.

Instructions to students

1. In building it is often important to plan how to cut materials in advance.Draw cutting lines and folding line on construction paper, so that you can cut and fold one shape into a house with a rectangular floor.

2.Describe the shapes that you used in your drawing.

3.What is the living space of the house?

4.What is the length of its outside wall?

5.How many centimetre cubes would fill it?

Outcomes

4-E1, D4, D8, D9, E2, 5-E1 

Lesson on the theme of naming shapes

Materials

Blank information tables, pipe-cleaners cut to 5 cm lengths and straws cut to various lengths (colour coded) or geoboards, teacher answer sheet.

Information for teacher

This lesson is a systematic exploration of the names of shapes, in two parts: triangles, and quadrilaterals.After handing out the triangle tables, ask if there are questions about the meanings of "Obtuse" "Acute" and "Right."Encourage a systematic exploration.Questions 3 and 4 might be better for a whole class discussion.You may need more than one copy of the quadrilaterals table.There are a number of quadrilaterals that do not have standard shapes.Encourage creative naming as well as naming that indicates something about the shape. 

Part 1: Instructions to students

1.We are going to investigate how many different kinds of three sided shapes there are, and find names for them all.Fill in the Triangle Table. Start by putting zero (0) in as many columns as you can, and then try increasing numbers.Be systematic so you don't miss any.Use the straws and pipe-cleaners or geoboards to build your triangles.This will help you draw them and check to make sure they have the correct numbers for each column.If you don't know the name of a triangle you find, ask the teacher.Maybe you can stump her!If so, invent your own name for that triangle.

2.Now that the table is complete, what do you notice?

3.Why are there no triangles with two obtuse angles?

4.Why are there more kinds of isosceles triangles than equilateral triangles?

Part 2: Instructions to students

1.We are going to investigate how many different kinds of four sided shapes there are, and find names for them all.Fill in the Quadrilateral Table. Start by putting zero (0) in as many columns as you can, and then try increasing numbers.Be systematic so you don't miss any.Use the straws and pipe-cleaners or geoboards to build your quadrilaterals.This will help you draw them and check to make sure they have the correct numbers for each column.If you don't know the name of a quadrilateral you find, ask the teacher.Maybe you can stump her!If so, invent your own name for that quadrilateral.

2.Now that the table is complete, what do you notice?

3.Why are there no quadrilaterals with four obtuse angles?

4.Why are there no quadrilaterals with four acute angles?

Outcomes

Part 1:4-D5, E6, E7, 5-E7 

Part 2:4-D5, E8, E9 

Triangle Table
 
Number of sides the same length
Number of right angles
Number of obtuse angles
Number of acute angles
Sketch
Name



Triangle Table (teacher's answers)
 
Number of sides the same length
Number of right angles
Number of obtuse angles
Number of acute angles
Sketch
Name

0

0
0
3
Acute scalene triangle

0

0
1
2
Obtuse scalene triangle

0

1
0
2
Right scalene triangle

2

0
0
3
Acute isosceles triangle

2

0
1
2
Obtuse isosceles triangle

2

1
0
2
Isosceles right triangle

3

0
0
3
Equilateral triangle


Quadrilateral Table 

Number of sides the same length
Do the same length sides touch?
Number of right angles
Number of obtuse angles
Number of acute angles
Number of parallel sides
Sketch
Name


Quadrilateral Table (Teacher's answers) 

Number of sides the same length
Do the same length sides touch?
Number of right angles
Number of obtuse angles
Number of acute angles
Number of parallel side pairs
Sketch
Name
0
No
0
1
3
0
0
No
0
2
2
0
0
No
0
2
2
1
Trapezoid
2
No
0
2
2
1
Isosceles Trapezoid
0
No
0
3
1
0
0
No
1
1
2
0
0
No
1
2
1
0
0
No
2
1
1
0
0
No
2
1
1
1
Trapezoid
2
No
0
1
3
0
2
No
0
2
2
0
2
No
1
1
2
0
2
No
1
2
1
0
2 pairs
No
4
0
0
2
Rectangle
2
Yes
0
1
3
0
2
Yes
0
2
2
0
2
Yes
1
1
2
0
2
Yes
1
2
1
0
2 pairs
Yes
2
1
1
0
Kite
3
Yes
0
1
3
0
3
Yes
0
2
2
0
3
Yes
1
1
2
0
3
Yes
1
2
1
0
4
Yes
0
2
2
2
Rhombus
4
Yes
2
1
1
2
Right rhombus
4
Yes
4
0
0
2
Square
2 pairs
No
0
2
2
2
Parallel-ogram

Lessons from 4173-2000

qGeoboard area

qArt symmetry 

qshape classification

qLinking cube drawings

Geoboard area

Show square of area 1.

Pose problem: Make a triangle and find its area.Discuss responses

Show square of area 2.

Pose problem: Can you make a square of area 3, 4, 5, 6, 7, etc?

Show a shape containing a peg. Count Inside pegs, Edge pegs.

Pose Problem: Find a relationship between Inside pegs, Edge pegs, and area.

[Materials: Geoboards and elastics]



Art symmetry

Look for symmetry in the letters of the alphabet. [Need letter tiles]

Review (ask them to define): Rotation (turn) symmetry, Reflection (flip) symmetry.

Demo using the Mira to find a line of reflection symmetry (a mirror line) in a plane pattern.[Need tiles and Escher patterns]

Demo turning a transparency over a paper copy to find a point of reflection.

Pose Problem: Find the lines of reflection symmetry and points of rotational symmetry in the tile pattern or Escher work provided

Answers (listed in David's perceived order of difficulty):

Tile 39: lines through centers of diamonds, half turn points in centers of diamonds

E117 ½ turn on crab legs, reflections though centers of crabs.

A 11: Reflections through centers of figures, ¼ turn at corner

E6: Lines through centers of figures, half turn on angels toes, quarter turn on wingtips.

E4: Reflections through centers of figures, 1/3 turn on sleeve

A14: Reflections in lines at 60 degree angles, 1/6 turns at intersections

A6: Reflections through diamonds, QT in centers of Squares, HTs where reflection lines cross.

E91: Reflections through centers of bugs, translation up, (glide reflection)

Tile 141: reflections through centers of squares and trees, translation up 

E3: HT on third leg, HT on mid backs, HT on mouth

E8: 1/3 turns on lizards ears, knees and toes

A10: QT on centers of 8 pt stars and centers of crosses, HT on crossing points between QT points. 

A7: QT on centers of 8 pt stars and centers of crosses, HT on crossing points between QT points.

A3½ turn on 6 pointed star, ¼ turn on 8 pointed star, ¼ turn on 4 pointed star.

E941/3 turn on fish noses, ½ turn on fins, 1/6 turn on back 

A5: Hard!, 1/6 turns, 1/3 turns



Shape classification, 

1.Distribute plane shapes.

2.Pose task: Classify

3.Discuss classifications, use of properties: Number of sides, length of sides, size of angles, parallelism

4.Pose problem:How many obtuse angles can you have in a quadrilateral?

1.Distribute Solids

2.Pose task: Classify

3.Discuss classifications, use of properties: Number of faces, Shape of base, parallelism, "pointiness"

[Materials: Set of plane shapes, solid shapes]

Linking cube drawings

Show face views and isometric drawing of cube sculpture (See below)

Pose problem: Build this object

Pose problem: Build objects of your own, and draw pictures to challenge others.

[Materials: Isometric dot paper, graph paper, linking cubes]

Cube sculptures (Educ 4173 Geometry activity)

Below are "isometric" and "face" views of two cube sculptures.Both fit in a box the same size as a 3x3x3 block of cubes.

Try to construct them.Then construct a cube sculpture of your own and draw the views of it to challenge your neighbour.Do you need all the views?

Sculpture 1
 
Left-Front isometric view:
FrontRightBackLeftTopBottom

In the face views the shading represent how far the blocks are from you:



Sculpture 2
 
Left-Front isometric view
Front-Right isometric view
FrontRight BackLeftTopBottom