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Level Three > Geometry and Measurement

# Houses

Achievement Objectives:

Purpose:

This unit seeks to connect learning outcomes across all five content strands, number, geometry, statistics, algebra, and measurement.  The context of houses is used to develop concepts such as drawing and modelling 3-dimensional objects, using co-ordinate systems to locate position, finding all the possibilities of events, and identifying paths through simple networks.  The unit provides an excellent vehicle for students to use a broad range of problem solving strategies.

Specific Learning Outcomes:
• draw a diagram showing the top, side, front, and back of a solid figure
• make the net for a solid figure
• use co-ordinates on a map to identify the position of houses
• construct triangles and quadrilaterals by measuring appropriate lengths, and identify the properties of the shapes
• find all the possible outcomes for a simple event
• find a path through a network
Description of mathematics:

Properties of triangles and quadrilaterals, networks, nets and diagrams of 3-dimensional solids, co-ordinates on a number plane, and combinations are all included in this unit.

Required Resource Materials:
Plasticine, clay or other modelling material
Cardboard, scissors, glue, rulers, protractors
Strips of card and split pin paper fasteners, hole punch
Coloured felt pens or pencils
Copymaster 1
Copymaster 2
Copymaster 3
Copymaster 4
Key Vocabulary:

net, model, location, angles, central point, co-ordinates, quadrilaterals, vertical, horizontal, triangular, diagram, possibilities

Activity:

#### Session 1

1. Show the students House Plan One (Copymaster 1).  Illustrate how the house designer drew the plans by setting up an object on a desktop and drawing the object as seen from the front, side, and top as you move around the object.  Provide the students with plasticine or modelling clay and ask them to make a small model of the house that is shown in these plan drawings.

2. After students have attempted this get them to check their model by sketching it on a desktop by moving around it to get the front, side, and top views.  The drawings can be checked against the original plans.  Discuss with the students what features they concentrated on in producing the model.  This might include thinking of the model in two parts - the basic cuboid (rectangular prism) and the triangular prism that forms the roofline.

3. Provide the students with House Plan Two, from Copymaster 1, and challenge them to make a model of that house.  Help scaffold the task by discussing what features of the house are different to the previous plan.  Students may note that the new house has the same roofline but the top view shows that it is L-shaped.  As with the first plan the models can be checked by using them to produce plan drawings.

4. The activity can be extended in two ways.  Firstly tell the students that architects frequently make models of their buildings for presentation purposes.  Get them to design the net (flat pattern) that could be folded up make a model of house one.  This could be produced using the plasticine model by rolling it on a sheet of paper and tracing around each face as it lies against the paper until all the faces have been traced.  Students should note that use of symmetry can make the process much easier.

5. One net of house plan one looks like this:

1. Discuss with the students how nets usually have tabs to allow them to be glued together.  The usual convention is to put a tab on every second edge as you work around the perimeter (outside) of the net but other ways are also workable.  Creating the net for House Plan Two is considerably more difficult but can be done in halves and symmetry used to create the full net.

2. Another extension of the activity is for students to make their own house model from plasticine and create front, side and top view drawings of it.  These drawings are then given to another student who must produce either the model or net for the plan.

#### Session 2

1. Ask several students in the class how far they live away from the school.  They may give their answers in distance or travel time.  Comment that they are going to investigate where the school should be to make it as central as possible.  Take two students and ask them to mark as accurately as possible where their house is on a street map of the area.  Ask the students if the school is central (in the middle) of these two houses.  This is easy to visualise by drawing a line between the two locations:

2. Finding the central location becomes more difficult when three houses are involved.  Tell the students to draw a triangle in their books and assume that each corner is the location of someone’s house.  The triangle need not be equilaterial, in fact, it is desirable that the students try several different triangles.  Challenge the students to find a method of locating the central point for the school, the point that minimises the travel of all three students.
1. Discuss the methods that students have used.  These will be focused on features of the triangle, particularly equally splitting angles and sides.  Such methods might produce solutions like:

Split the angles method.

Split the sides method that, in this case, does not appear to give a sensible solution.

Width by height method.  Draw the smallest rectangle that includes all of the corners and cut the sides in half.  The intersection of these “bisectors” gives the centre.

The co-ordinate method.  Each house location is given an ordered pair and the average of the x-co-ordinates and the average of the y-co-ordinates is found.  This gives the centroid or balance point of the triangle.
The average of the co-ordinates gives the point (7, 4.33…).In this case the co-ordinate averaging method seems to give a good indication of the centre.
1. Discuss with the students which of these methods will also work if we consider finding the centre of four or more houses.  They may conclude that the length by width and co-ordinate methods will always work, no matter the number of houses.  The split the angles and sides methods could be tried with some quadrilaterals to determine if they will work.

1. Apply both the length by width and co-ordinate methods to the task of finding the central location for a school that only included students in your class.  Use a scale map of the district.  Draw a co-ordinate plane over the map and ask each student to provide the location of their house as an ordered pair.  Note that decimal ordered pairs could be used to make the locations more accurate.  A spread sheet might be used to find the average of all the co-ordinates quickly.

 Name X co-ordinate Y co-ordinate Joe 3.5 6.4 Gill 0.9 8.7 Shirley 6.5 1.2 … … =average(B2:B?) =average(C2:C?)

1. Compare the centre as determined by both methods.  Discuss which method appears to be the most accurate.  Students may note that outlier (extreme) co-ordinates have a pronounced effect on the length by width method but minimal effect on the average co-ordinate method.  This could be modelled by introducing a hypothetical student who lives an extreme distance form the school and noting the effect using the spreadsheet and map.

#### Session 3

1. Tell the students that they are going to investigate the best way to build framing on the wall of a house.  Using cardboard strips and split pin paper fasteners get them to construct the following “walls.”

Point out the need to accurately measure the strips and location of the holes so that the walls lie flat. Which one is more stable as the walls of house need to be reasonably rigid? The students should notice that the vertical and horizontal arrangement tilts while the triangular arrangement stays fixed.

1. Ask the students why they think this occurs.  They may comment that triangles are strong while the angles of the rectangles are movable.  Discuss why the triangular wall they have made, while very strong, might not be functional in building houses.  This relates to the difficulty of measuring angles so that the walls will intersect and the difficulties of living in dwellings with non-perpendicular (right-angled) corners.  A-framed houses are a good example of the later problem.  Building a tetrahedron with polygons might be a good way to demonstrate the inefficiency of space usage (let alone trying to clean!).

2. To develop students’ ideas about the structural strength of triangles and quadrilaterals get them to carry out the following investigation.

3. Make the following lengths from cardboard strips.  The lengths show the length between the holes:

1. Ask the students to make as many different triangles as they can by fastening the cardboard strips with fasteners and moving the sides around.  They will realise that only one triangular shape is possible as the side lengths determine the angles at the corners.

2. Ask them to use another length of cardboard strip (for example 18cm between holes) and use it with the other lengths to make quadrilaterals of different shapes.  The students will see that the angles at the corners can be changed by tilting the quadrilateral.  This could be used at another time to investigate the sum of interior angles.

3. The students might try making triangles and quadrilaterals with different lengths to further validate their conjectures if need be.

4. Extend this idea by telling the students that they are allowed to add only one strip of card and fasten it to the original vertical and horizontal wall arrangement they started with.  The challenge is to stabilise the wall so it does not tilt.  This can be done by using a brace that effectively turns the quadrilaterals in the wall into triangles with rigid structural strength.

#### Session 4

1. Tell the students that they have been hired by a construction company as a consultant to look at the colour schemes for a new housing estate (Copymaster 2).  The houses all have the same design so the company is keen to use the colours to make each house appear different to the others in the estate.

2. The company has only bought three colours of paint red, blue and yellow.  They have decided that for each house the roof will be painted in one colour, the walls in a different colour, and the window sashes and door in a third colour.  They are wondering if three colours are enough.  How many houses can be painted with three colours remembering that each scheme must be unique? That is, no two houses can look the same.  Encourage the students to be systematic in planning the colour schemes.  Provide them with copies of Copymaster 2.

3. After an appropriate time of investigation bring the students together to share their ideas.  Focus on strategies they have used to systematically find all the possibilities.  Strategies might include:

Using tables

 Roof Walls Door/Windows Red Yellow Blue Red Blue Yellow Yellow Red Blue Yellow Blue Red Blue Yellow Red Blue Red Yellow

Using tree diagrams

Roof                             Walls                Door/Windows

Focusing on the structure of these models reveals that the six combinations are found by finding three possibilities for the roof colour multiplied by two possibilities for the walls (that is 3 x 2 x 1).

1. The company has built 120 houses in the estate so obviously three colours are not sufficient to make each colour scheme unique.  Ask the students to predict how many colours will be needed to produce 120 unique colour schemes.  Intuitively it seems that a large number of colours will be required.

2. Suggest to the students that they progress systematically in finding out the number of combinations when each new colour is added.  This should involve “adding on” to the tables, diagrams, or other models that they used in the simpler example with three colours.

3. The students may notice that the addition of a fourth colour, say green, results in a four fold increase in the number of possibilities.  Extending the table illustrates this clearly:

 Roof Walls Door/Windows Red Yellow Blue Green Red Blue Yellow Green Red Green Blue Yellow Yellow Red Blue Green Yellow Blue Red Green Yellow Green Red Blue Blue Yellow Red Green Blue Red Yellow Green Blue Green Red Yellow Green Red Yellow Blue Green Yellow Red Blue Green Blue Red Yellow

Similarly drawing the tree diagram will reveal four possibilities for the roof colour multiplied by three possibilities for the wall colour multiplied by two possibilities for the door/windows colour (4 x 3 x 2 = 24).

1. Students will have to look for some generalisation in determining how many possibilities occur with five colours.  Creating all of these possibilities would be a painstaking process.  Ask the students to discuss in groups what might happen when the fifth colour is used.  Look for reasoning like:

Roof                             Walls                            Door/Windows

5 colours          x          4 colours left     x          3 colours left    gives 60 possibilities.

1. This would mean that a sixth colour would create 6 x 5 x 4 = 120 possible colour schemes which would be enough to make each of the 120 houses unique.

2. Students might explore other scenarios that are either simpler or more complex.  For example:
What happens to the number of possibilities if it is decided that the roof, windows, and doors are painted the same colour?
What happens to the number of possibilities if the windows and door are painted in different colours?

3. Students might also be asked to solve problems that are similar in structure.  For example:
You have these digit cards in a pack: 0, 1, 2, 4.  You deal out three cards.  4 then 2 then 0 come up.
What other card combinations might have come up?
How many different combinations are there?

#### Session 5

1. Discuss with the students why people might be interested in very old houses.  Tell them that touring large, old houses is very popular with tourists.  As the houses are very precious it is important to design tours that do not involve returning frequently to the same room or doorway.  Tourists don’t like repeating themselves either.

2. Use an overhead projector to show the plan of the rooms in “”Comfy Cottage” (Copymaster 3).  The gaps indicate doorways and the start and finish indicate where the tour is to begin and end.  Tell the students to mark a tour through Comfy Cottage.  Note that there are several possible tours.  One possible tour is given on Copymaster 3.

3. Tell the students that they are going to be given several historic homes to design tours for.  They will need to decide which rooms to start and end the tour on and check to see whether the tour will go through each doorway only once.  Provide the students with copies of Copymaster 4 and encourage them to work in pairs in solving the tour problems.

4. After an appropriate time bring the class together to share solutions.  Concentrate firstly on how the students identified which rooms to begin and end the tour at.  Look for common features of these rooms.  Students might notice that these rooms always have either one or three doorways.  Discuss what they notice about the rooms that are passed through.  Focus on the number of doorways in these rooms.  Students may notice that they have either two or four doorways.

5. Look at both scenarios, an odd or even number of doorways.  For a room with an even number of doorways a tourist can either start inside or outside it.

If a tourist starts inside the room they must return to it.
If a tourist starts outside the room they must end up leaving it.

For a room with an odd number of doorways the opposite scenario is true.

If a tourist starts inside the room they can return to it many times but always end up leaving it.
If a tourist starts outside the room they can pass through it several times but will always end up returning to it.

1. In this analysis it is easy to see that any house cannot have more than two odd doorway rooms if the tour is to work.  This means that the tour must start in one odd room and end in the other.  Along the way, any even-doored rooms are entered then left.

2. Challenge the students to try this theory out by creating two house plans where a tour is possible and two plans where a tour is not.  Students can share these plans to help confirm that a tour is only possible given no more than two odd-doored rooms.

AttachmentSize
HousesCopymaster1.pdf42.92 KB
HousesCopymaster2.pdf50.34 KB
HousesCopymaster3.pdf51.43 KB
HousesCopymaster4.pdf44.41 KB

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