Keeping in shape - Units of work

1.

Talk to the students about buildings and how they are constructed. Give them images of buildings to inspire them and for them to discuss. You could also ask a local builder who is a member of your community to come and talk to your class about construction.

• What material is used to construct buildings? (wood, bricks, mortar, sand, cement, etc.)
• How do builders put a house together?
• How do they build walls?

2.

Get the students to describe the brick pattern on the walls. Show them (Keeping in shape 1), or another picture of a brick wall. Talk about the features of the pattern (i.e., there are no gaps, the bricks are in rows, and each row is slightly displaced from the row below).

• Why do you want to have no gaps in a wall?
• (You might note that some walls do have gaps. Why?)

3.

Are there any other ways of using bricks to build a wall?

4.

Get the students to produce their own (2-dimensional) bricks. The bricks should be rectangles that are twice as long as they are wide. This can be done using graph paper and then tracing around one to produce many. Model this process for the class. They could also draw a number of bricks on graph paper to be photocopied before cutting up the photocopied bricks. (Keeping in shape 2) can be photocopied if you would prefer students not to make their own bricks.

5.

Ask the students to create a wall using the bricks that they have just produced. They might copy the pattern shown in (Keeping in shape 1) or arrange the bricks in any pattern they like. However, if they think up their own patterns, there are two rules that must be followed: (i) there are to be no gaps, and (ii) the pattern must be capable of being continued indefinitely (to cover a very big wall!). Draw attention back to the brick wall example shown previously and highlight each of these features.

6.

Roam and work with the students. If needed, they could work in pairs or small groups. Once they have one brick pattern, see if they can find others. Emphasise that the bricks don’t always have to be "horizontal" and that the pattern doesn’t have to be practical. It’s OK if the pattern won’t produce a very strong wall.

7.

Bring the class together, and let the students discuss the patterns that they have made. The students should be able to identify that their patterns have no gaps and that they can go on forever in all directions. Support them with their explanations as necessary.

8.

Tell the class that they have been making tessellations. A tessellation is a way of using a fixed shape to cover the whole area of a flat surface (a plane). Tessellations have two important properties: (i) they have no gaps (all of the plane is covered), and (ii) they go on forever (no matter where you go in the plane, the shapes will still be covering the part of the plane that you can see). Sometimes we call a tessellation a tiling. Record a summary of these key points on a class chart.

In this session, the students will explore tessellations by regular polygons. The key finding is that regular polygons don’t tessellate. If you don't have any solid regular polygons handy, then they can be made using (Keeping in shape 3), or displayed digitally.

1.

Ask the students to recap the work from the last lesson. They can share their ideas with a buddy, a small group, or the whole class.

• What did we do in the last lesson?
• What shapes did we use?
• What word did we learn? (tessellation, or maybe tiling)
• How many rules does a tessellation have? What are they?

Display the class chart constructed in the last session, and revise any key points that students didn’t emphasise in discussion.

2.

Today we are going to look at special types of shapes called regular polygons.

3.

Show the students a variety of relevant artworks and images that feature tessellations (this could be an opportunity to look at a local tapa cloth or kowhaiwhai panel). Ask them to identify any regular polygons. Emphasise that a regular polygon has sides that are equal and interior angles that are equal.

4.

What we want to do is see which of these shapes tessellate and which don’t. We’ll then construct this table together.

5.

Distribute the regular polygons to students and ask them to find out which shapes tessellate and which don’t. Let the students tackle the problem in any way that they like. As you go around their groups, check that they understand that there are to be no gaps and that the patterns must continue forever in all directions. Check, too, that if they can’t find a tessellation, they can explain why one doesn’t exist. It isn’t good enough to say that they have tried for 5 minutes and they can’t find one. They need to be able to justify why there have to be gaps or why the pattern can’t go on forever. It should usually be the first reason. This is fortunate, as it is easier to justify. They could record the findings from their investigations on a large sheet of paper, on which they could draw each tessellation and write a few sentences explaining why it does or does not tessellate. Encourage the students to refer back to the class chart constructed in session 1 for a reminder of the "rules" tessellations must follow.

6.

Groups who finish early could be asked to see if they could find some different tessellations using a combination of the regular polygons or using the regular polygons they have not yet investigated.

7.

Bring the class together to discuss their results. After each group has reported and justified their claims, add them to the table. A completed form of the table is given below. The angles talked about in the third column are the interior angles of the polygons. If your students are confident in measuring angles, they could measure the angles of the shapes they investigated and add this information to their charts. Otherwise, briefly model the measurement of the angles in a few shapes and explain how the total sum of the angles is found (i.e., add them together). If your students are not familiar with measuring angles and you feel this knowledge will be too much in addition to the learning in this unit, you could use the existence of gaps in the pattern of shapes to justify whether the shape does or does not tessellate.

8.

Ask the class if they have any new information or conjectures (i.e., statements that have not yet been proven to be correct or incorrect) about tilings by regular polygons that could be added to the class chart.

In this session, the students try to extend their results about regular polygons to more general polygons. The key findings are that regular polygons don’t tessellate and all quadrilaterals tessellate.

1.

Get the class to recall what happened in the last session. The key points are (a) what is a tessellation; (b) tessellations were found for only three types of regular polygons; and (c) which types? The key points are (a) what is a tessellation; (b) tessellations were found for only three types of regular polygons; and (c) which types? Display the class chart and highlight any information that students did not identify.

2.

Ask:

• Which shapes do we now know will tessellate?" (The three regular polygons and the rectangle from the brick patterns.)
• We know that two kinds of 4-sided figures will tessellate. What other types of quadrilaterals will tessellate?

3.

Provide the students with various quadrilaterals (see Keeping in shape 4) and let them experiment. A table can again be constructed like the one in Session 1.

4.

Give the students the opportunity to experiment for themselves with other quadrilaterals. (For instance, the students who finish quickly and can justify what they have done should go on to invent quadrilateral tilings of their own.) Ask them to record their findings on the chart they constructed in the previous session. Remind them that there are two things that can be varied: the lengths of the sides and the sizes of the interior angles. Suggest that they vary the interior angles first. If your students are not familiar with measuring angles, ask them to look at the lengths of the sides and whether any gaps are formed when they try to tessellate the new shapes.

5.

Let the students report back on what they find. What new information did they discover? What conjecture do they have about tessellations by quadrilaterals? Can they justify that conjecture?

Teaching Notes:

The students will discover that any quadrilateral will tessellate. Because the interior angles of any quadrilateral add up to 360°, we need to put four figures together at a point so that each one of the four (possibly) different angles is used. In this pattern, sides of equal length also have to fit together. The pattern below shows how to do this.

We have shown that we can fit four of these polygons together at a vertex without gaps. How can we be sure that we can continue the pattern indefinitely? You can see that the pattern in the figure is made up of a wiggly strip of quadrilaterals. In this strip, one quadrilateral is placed one way, and then it’s placed another. These wiggly strips can be put side by side forever. What you see in the drawing is what you would see anywhere on the plane.

Notice that it doesn’t matter whether the quadrilateral is convex (no interior angles bigger than 180°), as above, or concave, as below.

If all quadrilaterals tessellate, is the same true for all triangles? Repeat the last session, but this time use triangles. Provide students with a range of triangles to experiment with (e.g., scalene, equilateral, isosceles, right-angle, obtuse, and acute). Briefly introduce the names of the triangles you include, and if appropriate, make reference to the types of angles they demonstrate.

Teaching Notes:

It is easier to establish whether or not triangles tessellate. Since the interior angles of a triangle add up to 180°, we need to make sure that each angle is represented at a point. In the diagram below, the angles form a continuous, straight line. This means that we can put the triangles together to make a row. We can fit two such rows together. In fact, we can fit as many rows together as we like until the entire plane is covered.

Review the facts:

• All triangles tessellate, and all quadrilaterals tessellate. What do we know about regular polygons? (They don’t tessellate.)
• Does that mean that no pentagon tessellates?

Ask the students to experiment using the pentagons from (Keeping in shape 5) and by inventing their own pentagonal shapes.

• What results can you find?

If the students can’t find any pentagonal tessellations, you might remind them how they constructed the triangular and quadrilateral tessellations. Rows were very important. So it might be an idea to try to use rows in some way to try to form pentagonal tilings.

The situation with pentagons is more complicated than with either triangles or quadrilaterals. Some pentagons do tessellate, and some do not. From the evidence of the regular pentagon, it is unlikely that all the interior angles of a pentagon could cluster around a similar point in a similar way. That means that there have to be at least two types of points where pentagons come together. So we have to distribute the sizes of the interior angles so that this can happen. This enables us to get tilings like the one below.

1.

Discuss with the class the results of the last few days. Refer back to the charts constructed and record some summary statements.

• What have you discovered?
• What is a tessellation?
• What polygons do you know that tessellate?
• What polygons definitely do not tessellate?

2.

Now we know that all triangles tessellate, all quadrilaterals tessellate, and regular hexagons tessellate.

• What questions do you think we ought to ask now? (Do you think that all hexagons tessellate? What do you think? Why?)

3.

Experiment for yourselves. Make your own hexagonal shapes and come up with a conjecture.

4.

After they have tried this for a while, get them to report back on their conclusions.

5.

Revisit their conjecture about regular tessellations.

• How could you prove this?

Teaching Notes:

There are other hexagons apart from regular hexagons that do tessellate, such as the concave hexagon shown below.

If we can arrange the angles of a hexagon in such a way that no combination of them will add up to 360° (or 180°), then it won’t be possible for that hexagon to tessellate.

The conjecture that only regular polygons that tessellate are the equilateral triangle, the square, and the hexagon is proved in Fitness, Level 4.