Nets of of rectangular prisms

Lesson 5

Lesson outcomes

By the end of this lesson, students will be able to

visualise and recognise the nets of cubes and rectangular prisms
recognise the net of a rectangular prism is a composite figure of three pairs of congruent rectangles

Content indicators

visualise and name a right prism, given its net
visualise and sketch the nets of right prisms
solve a variety of practical problems involving the surface areas of rectangular and triangular prisms

WM indicators

recognise whether a diagram represents a net of a right prism (Reasoning)
recognise the positioning of the congruent faces of a rectangular prism (Reasoning, Understanding)
sketch and construct the different configurations of the net of a rectangular prism (Understanding)

Prior knowledge and understanding

Students are assumed to be able to:

recognise the 'length', 'width' and 'height' as the 'dimensions' of a rectangular prism
identify, visualise and draw right prisms
recognise that the net of a rectangular prism is a composite figure comprised of rectangles

Lesson content

Lesson length: 60 mins

Materials/Resources

Net of open cube printout (Introduction)
2 cm grid paper
Scissors
GeoGebra applet link:https://www.geogebra.org/m/fmbmkpj7

10:00 Introduction

Collect homework task (Courtyard Design) as students settle into class (AFL/AoL)

Check if completed
Assess if students have a well-rounded understanding of the dissection and areas of composite figures involving triangles, special quadrilaterals and parts of a circle.

(7 min) Introducing nets (LIT)

Hold up an open, cubic cardboard box. Ask students to identify the (AFL: Diagnostic assessment)

Name of the prism (Cube)
Number of faces (5)
- Shapes of each face, and are they the same? (Squares, yes)
Edges
- Length, width and height

Ask students How was this box made? Is it moulded into its 3D shape from the beginning?

Students should realise that the box is fashioned from a flat, 2D piece of cardboard first called a net.

1. Unmarked net of open cube

Provide students with the following figure, the net of box/open cube. Ask

Is this a regular or basic shape? Is it an irregular shape?
What does this remind us of? (Composite figures)

Ask students to dissect the figure such that:

The number of dissected pieces is equal to the number of faces of the box.
The shape of dissected pieces correspond to the shape of the faces.

Ask students to hold up their dissection and check that they've dissected the figure as follows. (AFL: diagnostic assessment).

Reaffirm that nets of prisms, including cubes and rectangular prisms, are a common example of composite figures.

Diagrams were produced using GeoGeobra Classic 6.

2. Dissected net of open cube

10:07 Body activity 1

(25 mins) Pentominoes and the nets of open cubes

(2 mins) Have students count the number of squares in the dissection previously. State that this is an example of a Pentomino, which is a 2D shape made of 5 side-by-side squares of equal size. Show one more example and one non-example of Pentominoes on the board for students to familiarise themselves with them. Ask:

Are these both Pentominoes? Why/Why not?
- One on the right has 6 squares

Left: The 'T' Pentomino, a possible configuration for a Pentomino.Right: Non-example of a PentominoImage adapted from https://en.wikipedia.org/wiki/Pentomino

(18 mins) Provide students with grid paper. In groups of 3 or 4, students work together to find as many different Pentomino configurations by drawing them onto the grid paper and cutting them out. Explain that rotating or reflecting a configuration of a does not count as a separate Pentomino.

Set of all 12 possible Pentomino configurations (denoted by colour) Image retrieved from https://en.wikipedia.org/wiki/Pentomino

Once groups have found as many configurations as they can, they explore which configurations are the nets of an open cube. They do this by folding their Pentominos in order to produce a 3D, open cube.

As students are working, roam around the classroom and ask questions such as (AFL):

What makes this a Pentomino?
- How many squares are there?
- Are all the sides joined together?
How are you changing the Pentomino so that it makes another configuration?
- Why did you think of that?
If you rotate/flip this, does it look like any of your other Pentominos?
What can you tell me about the area of all the Pentominoes? (Ext.)
What might prevent a configuration from folding into an open cube? (overlaps, square in a position that doesn't allow it to fold, etc.)

Students either take a photo of their found Pentominoes, grouped by "Forms net of open cube vs Does not form net of open cube", or draw all possible configurations in their workbooks.

(5 mins) Ask students for their number of Pentominos found. Go through the 12 different configurations and students check if the are missing any/found extra. If so, they should check to see which they have missed/which they doubled up on.

Discuss which configurations were also the nets of open cubes and what features of a Pentomino might prevent it from folding into an open cube. (AFL)

The Pentominoes that yield open cubes are outlined in red.

10:32 Body activity 2

(20 mins) Visualisation of nets of rectangular prisms using digital models (ICT)

(5 mins) Discuss how a cube is just a special type of rectangular prism. As with the introduction, hold up a tissue box and ask students to identify:

Name of the prism (Rectangular prism)
Number of faces (6)
- Shapes of each face, and are they the same? (Rectangles, no)

Ask students to describe the main differences between this and the open cube.

Closed 3D shape, has 6 faces, not 5
Not all faces are the same shapes/sizes
- Opposite faces are the same shape/size (congruent)

(15 mins) Visualising rectangular prisms and congruent faces

Individually, students use the GeoGebra application (link) and with have the dimensions and settings of the rectangular prism set as shown. Students identify the 'base', 'top', front', 'back' and 'sides' of the prism by manipulating the viewing angle of the rectangular prism.

By dragging the Options slider, students can visualise the position of the three pairs of congruent faces.

Return the Options slider to its initial position. Students then move the Create Net slider to visualise the 3D object 'opening up' to create a 2D net and they may alter the viewing angle of the net.

Students observe how the faces have moved to created the net. They identify pairs of congruent/same sides and their relative position to each other in the net. Students redraw the net on grid paper, conserving the dimensions of edges.

With their sketches, students identify and label the

number of faces of the rectangular prism
the 'base', 'top', front', 'back' and 'sides' of the prism
dimension of each rectangular face
pairs of congruent faces

They then paste these in their workbooks. Students repeat this process for two other rectangular prisms with dimensions of their own choosing.

As students are working, roam around the classroom to ensure students are on task. Ask questions such as:

How does the shape of the prism change as you move the length/width/height sliders?
What happens to the sides as you move the "Create Net" slider?
- This has transformed from a 3-dimensional object to a ....
Which sides are the same/congruent? How can you tell?

10:52 Conclusion

(4 mins) Class review. Ask students:

What did we cover today?
How is a net of a prism a composite shape? What do the dissected pieces correspond to on a 3D cube/rectangular prism?
Are there multiple ways of forming the nets of a cube?
What is distinctive about the sizes of the faces of a rectangular prism?

(4 mins) Homework and pack up

Find at least three other configurations for the net of a closed cube. Draw these in your workbook.
- You may use your results from the Pentomino activity to help you. Remember, rotations and reflections of already formed nets do not count as a separate net!

Evaluation

Did students have the relevant prior knowledge?
- Did they recall the features of rectangular prisms and cubes?
- Could they then recognise that a net was another form of a composite figure?
Were my instructions clear?
Were all students engaged in the activities?
- Were all students participating during group work?
- Were students all on-task during the ICT portion of the lesson?
Can students effectively visualise the nets of a rectangular prism and cube?
Were my questions clear/encouraged students to think about the concept?
What do I need to modify in my upcoming lessons to support the needs of all students?

Additional resources

The Pentomino activity or the homework may be modified/extended to cover Hexominoes: https://sites.google.com/view/areasurfacearea-s51/additional-resources#h.p_OD0XyZ5mCjbZ

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