GEEM - Learning Situation

E1. Geometric and Spatial Reasoning

Describe and represent shape, location, and movement by applying geometric properties and spatial relationships in order to navigate the world around them.

Learning Situation 1: Mysterious Ruins

Total Duration: approximately 90 minutes

Summary

In this learning situation, students use interlocking cubes to build a model of ruins that corresponds to a given set of views.

Learning Goals

The purpose of this learning situation is to have students:

develop their spatial sense and their ability to visualize objects in three dimensions;
build a model of a solid from different perspectives;
make connections between two- and three-dimensional representations of a solid;
develop and use problem-solving strategies.

Learning Context

Prerequisites

In previous grades, students have explored, manipulated, and classified three-dimensional objects. They have built structures with three-dimensional objects from a given model and constructed three-dimensional objects from their nets. All of these activities have helped them develop spatial sense as well as the ability to visualize any three-dimensional object in two and three dimensions, and to make connections between its two- and three-dimensional representations. In Grade 8, students continue to develop this skill, including exploring different views (front, side, and top) of a three-dimensional object.

This learning situation allows students to deepen their understanding of views in a problem-solving context and to hone their ability to visualize a three-dimensional object from its views. Students must analyze the views of the east, west, north, and south sides of the ruins of an ancient monument and build a model that corresponds to these views with as few cubes as possible. This is a fairly complex analysis and deduction task that allows them to work at the level of analysis and progress to the level of informal deduction of geometric thinking, the levels described by the Dutch van Hiele.

To be able to complete this learning situation, students must:

have developed some understanding of views as a two-dimensional representation of a three-dimensional object;
have performed various activities of drawing views of a given three-dimensional object and constructing a model from its views.

Materials

Appendices slides 6.3, 6.4, 6.6a and 6.6b
interactive whiteboard
Appendices 6.1 and 6.2 (one copy per student)
Appendix 6.3 (two copies per student and one copy per team)
Appendices 6.6a and 6.6b (one copy per team)
interlocking cubes (16 cubes per student and 100 cubes per team)
large sheets of paper (one per team)
digital camera (optional)

Mathematics Vocabulary

side view, front view, top view

Before Learning (Warm-Up)

Duration: approximately 30 minutes

Provide each student with 16 interlocking cubes and a copy of Appendices 6.1 (Model 1) and 6.3 (Grid Paper). Ask them to build a model of the structure in the picture using nine interlocking cubes.

Once the students have built their model, ask them to draw the five visible views on the graph paper, for example, the left, right, front, back and top views. Share the work that has been done with the class. Ask students to take turns drawing one of the five views on the board or the interactive whiteboard. Tell them to keep this model to compare with the next model they build.

Give each student a copy of Appendices 6.2 (Model 2) and 6.3 (Grid Paper). Have them use the seven remaining cubes to build a model that matches the structure in the picture and draw the five visible views on the grid paper. Once completed, ask students to form pairs and compare their views of models 1 and 2. Share their observations with the class.

Students should find that the left views are the same for both models, and the same is true for the right, front, and back views. However, the top views of the two models are different. Ask them to explain why it is possible to build two models that have the same side views, but require a different number of cubes. (In Model 1, the cubes are connected to each other while some are detached in Model 2, which makes it possible to eliminate cubes while respecting the views.)

This activity prepares students for the following scenario.

Scenario

Using the Appendices 6.4, 6.6a and 6.6b, present the following scenario.

Mysterious Ruins

At an archaeological site, an excavation team unearthed the ruins of a very ancient monument. One of the excavators has drawn the views of each of the four sides (east, west, north, and south) of the ruins so that the archaeologists can make a model of the ruins of them. They realize that there are several possible arrangements of the cubes which respect the views. Using interlocking cubes, build a model for one of these layouts using as few cubes as possible?

Check for student's understanding of the situation and the task at hand by asking questions such as:

What are archaeological ruins (an archaeological site, excavations, views, a model, possible layouts)?
Would anybody like to explain the task in their own words?

Active Learning (Exploration)

Duration: approximately 45 minutes

Form teams of four students. Provide each team with approximately 100 interlocking cubes and the four views of the monument ruins (Appendix 6.6a and 6.6b). Have them build a model of the ruins that respects the views while trying to use as few cubes as possible.

Note: The physical and material organization of the classroom can play an important role in the success of this activity. For example, each team can be asked to gather around two desks placed side by side, without the chairs. This arrangement allows students to move around the desks more easily and to lean in close to observe the views (see photo 1 below). A variety of materials can also be made available to students, including sheets of graph paper (Appendix 6.3). Some students may choose to use graph paper as a strategy to help them arrange and align the cubes in loose groups while respecting the views (see photo 2 below).

Models of square structures on sheets of paper are reproduced with interlocking blocks

Interlocking blocks are placed on a gridded sheet.

Allow sufficient time for students to explore and discuss a variety of problem-solving strategies. Circulate among students and provide support as needed. Pay particular attention to the strategies used and the composition of the models in order to strategically select the teams that will be invited to present during the math exchange.

Possible Observations	Possible Interventions
Each team member chooses one of the four views of the ruins and builds the row of cubes that corresponds to it. Then the team builds a model by arranging the four rows to form the outer walls of a rectangular structure, not realizing that two of the walls are 11 cubes wide.	Does your model respect the four views given? How can you verify this?
One team realizes that it is possible to build a model made up of only two exterior walls (for example, the north and west views), since the back view of the north side corresponds to the front view of the south side and the back view of the west side corresponds to the front view of the east side.	Why did you choose to build a model with only two walls? How many cubes does your model include? Are you able to eliminate other cubes while maintaining the views?
A team removes several cubes from each of the exterior walls, but their model no longer respects the given views.	Why did you remove cubes from each wall? Does your model still respect the four views given?
Some teams manage to reduce the number of cubes by detaching certain cubes from the exterior walls.	How many cubes do you have in total? Are there any cubes that you don't see when looking at your model from the side? Are they all necessary? Can you remove or detach others while still respecting the given views?
A team builds a model in which the cubes are detached and strategically placed in a 9 × 9 square space.	Does your model match the given views? How many cubes did you use to build your model? Are you certain that you have used as few cubes as possible to represent these ruins?

Ask teams to record their strategies, trials and reflections on a large sheet of paper. If a digital camera is available, ask teams to take a picture of their model.

Note: In this learning situation, the minimum number of cubes to be used is 20, but this number should not be overemphasized. What is important is that students can visualize how to remove some of the cubes while respecting the given views. Photo 2 shows a possible model built with 20 cubes. Appendices 6.7a and 6.7b illustrate other examples of possible arrangements using the top view.

Consolidation of Learning

Duration: approximately 15 minutes

Ask the selected teams to present their model, explain the strategy they used to solve the problem and share their observations. Ask them to describe the spatial reasoning that they used. Possible observations could include:

When we built a rectangle using the four rows of cubes that correspond to the four views, our model did not respect the given views because two of the sides of the rectangle had 11 columns of cubes instead of 9.
Since the view from the south side is a reflection of the view from the north side, we used a single row of cubes to represent both views. The same was true for the views from the east and west sides.
It is possible to detach the cubes from the exterior walls because the ruins are not necessarily composed of cubes only connected to each other in a rectangular arrangement.

Note: For presentations, it can be quite difficult for students to move the models to the front of the class. If students have taken pictures, they can project them on the interactive whiteboard. Otherwise, students can be asked to gather around the presenting team's model.

Invite other students to respond to each presentation and to share their observations or questions. Facilitate the mathematical exchange by asking questions as needed, such as:

Who can explain to me in their own words the strategy used by this team to build their model?
Do you think their model respects the given views?
Did other teams use the same strategy? Did you use the same number of cubes?
Do you agree with what this team just said? Why?
Does anyone have a strategy for removing other cubes? Can you come demonstrate it?

Differentiated Instruction

To Facilitate the Task	To Enrich the Task
remove the restriction on using the fewest number of cubes possible; show students the possible way to remove some of the cubes while respecting the given views; hand out a copy of Appendix 6.8 (Example of a possible top view) and ask them to build their model using this view and the other four views given.	ask students to find a way to summarize, with a single view, the location of the elements that make up the ruins of the monument (see examples in Appendix 6.7a and 6.7b); ask students to make a convincing mathematical argument for the minimum number of cubes to be used, which is 20.

Follow-up at Home

Ask students to bring home their drawings of the views of models 1 and 2 (Appendix 6.1 and 6.2). Have them use cubes available at home to build the models that correspond to the two models. If necessary, allow them to bring interlocking cubes home. Ask them to show a family member the possible way to build two different structures from four given side views.

Source: translated from Guide d’enseignement efficace des mathématiques de la 4^e à la 6^e année, Géométrie et sens de l'espace, Fascicule 1, p. 83-91.

Learning Situation 2: Do You Tessellate?

Total Duration : approximately 100 minutes

Summary

In this learning situation, students create tessellations from a triangle using one or two different transformations.

Learning Goals

The purpose of this learning situation is to have students:

create tessellations;
visualize the transformations of a figure that allow it to tessellate;
formulate and test hypotheses;
identify certain properties of triangles.

Learning Context

Prerequisites

In previous grades, students have developed an understanding of several properties of two-dimensional shapes. They have worked with transformations, and have learned to describe and perform translations, reflections, and rotations. Students have also used the property of the sum of the angles of a triangle, and can now generate a tessellation from a pattern core using rotation.

This learning situation allows students to consolidate their understanding of the properties of triangles and tessellations, to work at the level of analysis, and to progress to the informal deduction level of geometric thinking in the van Hiele model. It also prepares students for the study of regular and semi-regular tessellations.

To be able to complete this learning situation, students must:

know the different geometric transformations and how to perform them;
know what a tessellation is;
know that a straight angle measures 180° and that a full angle measures 360°.

Materials

pattern blocks
interactive whiteboard
Appendix 6.1 (photocopied on cardboard)
Appendix 6.2 (one copy per team)
scissors (one pair per student)
glue sticks (one per team)
small sheets (several white half-sheets per team)
large sheets of paper (one per team)

Mathematics Vocabulary

rotation, translation, reflection, reflection line, centre of rotation, centre point, tessellation, vertex, degree

Before Learning (Warm-Up)

Duration: approximately 15 minutes

Review with students the definition of tessellation and ask them to create one using pattern blocks or software. Then invite a few students to show their tessellation to the class or on the interactive whiteboard. Point out that whether or not it is possible to create a tessellation with certain two-dimensional shapes depends on the properties of those shapes.

Discuss with students the importance of animatics (computer animation) in the production of some television shows, movies and video games. Explain that the effect of motion in these animations is usually created by geometric transformations of a shape or object. Thus, in order to animate the construction of a tessellation on the computer, the designer must know the properties of the shapes and objects, as well as those of transformations.

If necessary, review the four transformations described below. In particular, ensure that the students are able to perform a rotation about a centre of rotation on the side of a shape.

Present the learning situation in an interesting context with a focus on exploration and discovery.

For example, tell students, “To animate the construction of a tessellation using the computer, it is important to determine in advance the transformation that will be used to move a shape from position A to position B. I therefore suggest that you enter this wonderful world of computer animation by exploring two situations in which transformations are used to create a tessellation (see Appendix 6.2, Explorations 1 and 2). This activity will also allow you to discover an important property of triangles (see Appendix 6.2, Exploration 3).”

Active Learning (Exploration)

Duration: approximately 60 minutes

Form teams of two students. Give each team a pair of triangles, either A and B, C and D, E and F or G and H (Appendix 6.1) and ask them to cut them out. Ensure that each of the four pairs of triangles is given to at least one team. Also give each team several small sheets, one large sheet and a copy of Appendix 6.2 (Do you tessellate?).

Read with the students the process outlined in Appendix 6.2 and make sure they understand it. Explain that for each of the explorations, they must, like mathematicians, make observations, formulate conjectures, verify them and draw conclusions. Among other things, they must determine whether each of the mathematical situations presented to them is always true (this is the case for Explorations 2 and 3), never true (this is the case for the first part of Exploration 1), or sometimes true (this is the case for the activity “Spin, Spin!”). The statements in each of the explorations are intended to guide them in their work.

Circulate among the students and provide support, if necessary, to encourage students to reflect on their ideas. Pay particular attention to the conjectures and conclusions that are reached in order to strategically select the teams that will be invited to make a presentation during the mathematical exchange. See Appendix 6.3 for a summary of the key findings for each of the explorations.

Possible Observations	Possible Interventions
Some students believe that it is possible to create a tessellation using only translation.	Did you identify all the angles using the numbers 1, 2 and 3? How did you manage to fill this space (referring to the free space between two adjacent triangles)? If ceramic tiles were shaped like your triangle, could you cover a floor using only translation? How would you do it?
Some students confuse the rotation in which the centre is located on one of the vertices of the triangle with the rotation in which the centre is located on the midpoint of one of the sides of the triangle.	Why did you place the triangle this way? What transformation have you made? Where is the centre of rotation?
Students do not have angles 1, 2 and 3 at a point where three triangles meet.	Can you show me each of the steps in creating your tessellation? Is it possible to cover the entire plane by continuing in this way?
Some students leave spaces between the triangles.	What is the definition of a tessellation? What would happen if you used your tessellation as a model for a floor covering with ceramic tiles?

Once the task is completed, allow sufficient time for students to prepare for the mathematical exchange. Clarify that they must provide clear, fair and convincing mathematical arguments to justify their conclusions.

Consolidation of Learning

Duration: approximately 25 minutes

Invite the selected teams to take turns presenting their conjectures and conclusions. Encourage the use of precise vocabulary and causal terms in communication. Examples of conclusions include:

It is impossible to make a tessellation using only translation because you don't fill all the spaces.
Since angles 1, 2, and 3 are found at a point where three triangles meet and form a straight line, then the sum of the measures of the angles of a triangle is equal to 180°, the measure of a straight angle.

After each presentation, encourage other students to respond and ask questions. If necessary, ask questions such as:

Who can explain in his or her own words the conclusion just presented?
Did any other teams come to the same conclusion?
From what we have just presented, can we draw a conclusion?
Is this conclusion true for all triangles?

Note: Students' development of conclusions about whether or not they can create tessellations in this learning situation reflects a progression to the informal deduction level of geometric thinking. Students cannot yet prove the validity of these conclusions, but they can deduce it informally from the results obtained in a limited number of cases.

Differentiated Instruction

To Facilitate the Task

To Enrich the Task

give the team the C and D triangles which are easier to use;

Note: It is easier to create a tessellation with these triangles because triangle C is an equilateral triangle and triangle D is a right-angle isosceles triangle.

determine the midpoint of each side before handing the triangles to the team.

Note: By indicating the midpoint on each of the sides, students will have an easier time performing the rotation from the center of rotation that is on one of these points.

ask students to do the same activity using only the reflection from a reflection line that is one side of the triangle;

Note: This is sometimes possible, for example with the E triangle.

have students determine the measure of each of the angles in the triangles and determine what kind of triangle it is according to its side or angle measures;
distribute sheets of graph paper (Appendix 6.3). Some students may choose to use graph paper as a strategy to help them arrange and align the cubes in loose groups while respecting the views (see photo 2 below).
ask students to solve the problem using dynamic geometry software or coding (for example, Scratch, mBlock).

Follow-Up at Home

Time for Change

At home, students look for a tessellation (for example, floor covering, wall covering). Based on the classroom activity, students create and colour a tessellation that they would like to see in place of the existing tessellation.

Source: translated from Guide d’enseignement efficace des mathématiques de la 4^e à la 6^e année, Géométrie et sens de l'espace, Fascicule 2, p. 83-89.

Activity: What about Quadrilaterals?

Challenge students to discover the property of the sum of the measures of the interior angles of a quadrilateral. If necessary, suggest that they follow the approach used in the learning situation Do You Tessellate? and create a tessellation from a quadrilateral of their choice using only rotations starting from a center of rotation located on the midpoint of one of the sides of the quadrilateral.

Note: Students should first identify the four interior angles of the initial quadrilateral, for example, using the numbers 1, 2, 3, and 4, and then do the same with the interior angles of all the other quadrilaterals in the tessellation. They will then be able to see that at each point where four quadrilaterals meet, there is a full angle formed by the four interior angles of the quadrilateral (angles 1, 2, 3 and 4). Since a full angle measures 360°, students can conclude that the sum of the measures of the interior angles of a quadrilateral is equal to 360°.

Source: translated from Guide d’enseignement efficace des mathématiques de la 4^e à la 6^e année, Géométrie et sens de l'espace, Fascicule 2, p. 90.

Overall Expectation	Specific Expectation
E1 Describe and represent shape, location and movement by applying geometric properties and spatial relationships to navigate the world around them.	E1.2 Make objects and models using appropriate scales, given their top, front, and side views or their perspective views.

Overall Expectation	Specific Expectation
E1 Describe and represent shape, location and movement by applying geometric properties and spatial relationships in order to navigate the world around them.	E1.1 Identify geometric properties of tessellating shapes and identify the transformations that occur in the tessellations.