Boxes Stage 3

A thinking mathematically targeted teaching opportunity focussed on creating and representing 3D shapes using blocks

Adapted from NZ Maths – Sugar Boxes

Syllabus

Syllabus outcomes and content descriptors from Mathematics K–10 Syllabus © NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales, 2023

• MAO-WM-01
• MA3-3DS-01

You will need:

Watch

Watch Boxes Stage 3 part 1 video (3:24).

How many boxes can you make using exactly 12 or 24 blocks?

Transcript of Boxes Stage 3 part 1

(Duration: 3 minutes and 24 seconds)

[Text over a navy-blue background: Boxes (Stage 3). Adapted from NZ Maths. Small font text in the lower left-hand corner reads: NSW Mathematics Strategy Professional Learning Team (NSWMS PL team). In the lower right-hand corner is the waratah of the NSW Government logo.]

Speaker

Boxes Stage 3, Adapted from New Zealand Maths.

[Text over a white background: You will need…

· 24 cubes (blocks or sugar cubes)

· scissors

· sticky tape or blue tac

· isometric paper

· grid paper

· something to write with

· something to write on.

Images beside the text depict: 24 coloured cubes; a pair of scissors; a roll of tape; a section of centimetre grid paper; a section of isometric dot paper; a person holding a piece of paper and a cup of coloured pencils.]

Speaker

You need 24 same-sized cubes such as blocks or sugar cubes, scissors, sticky tape or blu tack, isometric paper, grid paper, something to write with and something to write on.

[Text over a blue background: Let’s investigate!]

Speaker

Let's investigate.

[Text over a white background: Watch the video to see what to do. Below the text is an image of 12 cubes, 6 of which are yellow, and 6 of which are purple.]

Speaker

Hi, mathematicians, I have a job at a toy company, I sell blocks in sets of 12 and 24. I have 12 blocks here. Let's see how we could arrange them to pack into different-sized boxes.

[The cubes are laid out on a white surface in front of the speaker. The blocks are arranged into 2 groups of 6. One group contains all the purple cubes, and the other group contains all the yellow cubes. Each group is arranged into 2 rows of 3 blocks.]

Speaker

Hmm, here is one way.

[The speaker places the 2 groups of blocks together, so that they form 2 rows of 6.]

Speaker

The box for these blocks would be six blocks long, two blocks wide and just one block high. We can also call this one layer. Here is another way.

[The speaker takes the yellow group of blocks, and places them on top of the purple group of blocks. An image shows a different view of the blocks, where the height of the 2 layers is visible.]

Speaker

The box for these blocks would be three blocks long, two blocks wide and two blocks high. There is one block on the bottom and one at the top, it's two blocks high. Which means this arrangement has two layers, six blocks on the bottom layer and six blocks on top of that.

[Text over a blue background: Over to you!]

Speaker

Over to you mathematicians. Together we looked at ways to pack 12 blocks into boxes. Pause the video here to get your materials and see how many different arrangements of 24 blocks you can make to pack into boxes.

[Text: Let’s investigate further.]

Speaker

Let's investigate further.

[3 groups of 12 blocks are arranged on a wooden surface. Each group of 12 contains 2 groups of 6 cubes of the same colour. Each group of 12 is arranged differently. The first group is 3 cubes long, 2 cubes wide, and 2 cubes high. The second group is 2 cubes long, 3 cubes wide, and 2 cubes high. The third group is 2 cubes long, 2 cubes wide, and 3 cubes high.]

Speaker

Hello mathematicians, I have started to make some different arrangements of 12 blocks to pack into boxes. Here are some of them.

[The speaker picks up the third group of blocks. She removes one of the cubes from the block, to show the blue tac that keeps them together. She puts the cube back in place, then puts the group of blocks down. She points to the first group, which has been sticky taped together.]

Speaker

I have blu tacked them together and sticky taped them together. Then I noticed something and wondered if you had noticed that too. I thought I had made different boxes here. While I was investigating, I turned them and noticed that they are the same arrangement of 12 cubes.

[She repositions the groups so that they are all arranged in the same orientation; 2 cubes long, 3 cubes wide, and 2 cubes high.]

Speaker

So have a look at your arrangements and do some investigating. Have you made some arrangements that are the same when turned?

[Text over a blue background: Over to you…

Over a grey background, the red waratah of the NSW Government logo appears amongst red, white and blue circles. Text: Copyright State of New South Wales (Department of Education), 2021.]

[End of transcript]

Instructions

• How many different sized boxes can be made with exactly 24 blocks?
• Have you made any boxes that are the same, even though they look different when turned or rotated?
• Check the boxes you have made.

Watch

Watch Boxes Stage 3 part 2 video (1:25).

Investigate how to draw a box from the top, side and front views.

Transcript of Boxes Stage 3 part 2

(Duration: 3 minutes and 17 seconds)

[Text over a blue background: Let’s draw! Text in the upper left-hand corner of the screen: NSW Department of Education. In the lower right-hand corner of the screen is the waratah of the NSW Government logo]

Speaker

Let's draw.

[A piece of isometric grid paper lays on a wooden surface. Above the piece of paper, a person lays a rectangular prism made from 12 coloured cubes that have been stuck together with blu tack. The top layer consists of 6 yellow cubes. The bottom layer consists of 6 purple cubes. The person rotates the block to show its other sides.

Using a pencils, the person points to one of the cubes on the corner of the prism. They point the pencil along the three edges that meet at the corner of the block. They draw these three edges on the isometric grid. They then point to the edges on the top face of that cube, and then draw them on the isometric grid. They then point to the edges on the side face of the cube, and then draw them on the isometric grid. They then point to the edges on the front face of the cube, and then draw them on the isometric grid. They then point to the edges of the other cubes of the top layer of the rectangle and draw each of the visible edges on the isometric grid.

The person then points to the purple cubes on the bottom layer of the shape’s front face. They point to the visible edges of the cubes along the front and left-side face, and then draw them on the isometric grid.

Once completed, the shape on the isometric grid appears like the rectangular prism, viewed from the upper left-hand corner. The person holds the prism so that it appears like its representation on the isometric grid.

Text over a blue background: Over to you!]

Speaker

Over to you. Choose your favourite arrangement of blocks, draw it on isometric paper.

[Text: What’s (some of) the mathematics?]

Speaker

What's, some of, the mathematics?

[Text:

· Different rectangular prisms can be made using the same number of cubes. For example, all of these prisms are made using 12 blocks but they all look different. This shows us that 3D objects can look different but be made using the same number of blocks.

Below are three images, each depicting a rectangular prism made from 12 blocks. They first, is the prism that has been used in this video. The second is a long prism, in which all of the cubes have been placed in a single-file line. The third is a flat, wide prism in which the blocks have been arranged in a single layer, which is 3 cubes long and 4 cubes wide.]

Speaker

There are a number of important mathematical ideas this task helped me see. For example, we can see that different rectangular prisms can be made using the same number of cubes, for example, all of these prisms are made using 12 blocks, but they all look different. This shows us that 3D objects can look different but be made using the same number of blocks.

[Text:

· The same rectangular prism can look different depending on what angle you are looking at it from (your ‘point of view’, or ‘perspective’). For example, this is a drawing of the top, side and front view of the same object. The top and front views look the same but the side view is different.

Images below show the yellow and purple rectangular prism that has been used in this video from three different angles: the top view, side view and front view. The different views have been drawn on one centimetre grid paper.]

Speaker

The same rectangular prism can look different depending on what angle you are looking at it from, your point of view, or perspective. For example, this is a drawing of the top side and front view of the same object. The top and front views look the same, but the side view is different.

[Text:

· We can make nets by tracing around 3D objects.

Images below show the construction of the box net. The sides of the rectangular prism are outlined. The box net is cut out, and the sides are being folded. The box net is folded around the rectangular prism.]

Speaker

We can make nets by tracing around 3D objects.

[Text:

· Mathematicians can use different tools to draw representations of 3D objects. For example…

Images below show the rectangular prism drawn on grid paper and isometric paper.

Text: …we can use grid paper and isometric paper.]

Speaker

Mathematicians can use different tools to draw representations of 3D objects. For example, we can use grid paper and isometric paper.

[Over a grey background, the red waratah of the NSW Government logo appears amongst red, white and blue circles. Text: Copyright State of New South Wales (Department of Education), 2021.]

[End of transcript]

Instructions

Choose your favourite box and draw the top, side and front views.

Watch

Watch Boxes Stage 3 part 3 video (2:42).

Investigate a way to create a net from a box.

Transcript of Boxes Stage 3 part 3

(Duration: 2 minutes and 42 seconds)

[Text over a blue background: Let’s draw! Text in the upper left-hand corner of the screen: NSW Department of Education. In the lower right-hand corner of the screen is the waratah of the NSW Government logo]

Speaker

Let's draw.

[A blank sheet of white paper lays on a wooden surface. A person handles a rectangular prism, made from 12 coloured cubes that have been stuck together with blu tack. The top layer consists of 6 yellow cubes, and the bottom layer consists of 6 purple cubes. The person lays the prism on the paper. On the top-facing side of the prism, the word “TOP” has been written. The person picks up the prism and rotates it, showing its other sides to the camera.

They place the prism on the sheet of paper. Using a pink highlighter pen, they trace along the outside of the rectangle. They then rotate the prism, onto its other faces, and outline each one with the highlighter, to create a box net.

First, they rotate the shape onto its left side, a 2 by 2 square, and trace an outline so that it shares a common side with the previous outline. They rotate the prism onto its right side and draw an outline on the other side of the original outline. They rotate the shape back into its original position, then rotate it down onto its front-facing side and draw an outline, below the original one. They rotate the shape toward themselves two more times, each time drawing an outline.

All sides have been outlined. The person removes the prism from the screen. The box net appears in the shape of a capital letter “T”.]

Speaker

Let's check together.

[The person on screen places each of the prism’s faces into their corresponding outlined areas.]

Speaker

Cut out the net.

[Using a pair of scissors, the person on screen cuts out the net.]

Speaker

Fold the net.

[The person folds a crease along the edges of the net. They fold the net into a rectangular prism. They place the block of cubes into the middle of the net, and fold around it. They remove the block of cubes from the net.]

Speaker

Sticky tape the net together.

[They tape the outside edges of the net. Once finished, they place the net on the table in front of them, and then place the block of cubes beside it. They rotate both shapes onto their side faces.

Text over a blue background: Over to you!]

Speaker

Over to you. Select your paper arrangement of blocks. Draw and make the box net for those blocks.

[Over a grey background, the red waratah of the NSW Government logo appears amongst red, white and blue circles. Text: Copyright State of New South Wales (Department of Education), 2021.]

[End of transcript]

Instructions

Draw and make the net of your favourite box.

Watch

Watch Boxes Stage 3 part 4 video (3:17).

Investigate using isometric dot paper to draw a box.

Transcript of Boxes Stage 3 part 4

(Duration: 3 minutes and 17 seconds)

[Text over a blue background: Let’s draw! Text in the upper left-hand corner of the screen: NSW Department of Education. In the lower right-hand corner of the screen is the waratah of the NSW Government logo]

Speaker

Let's draw.

[A piece of isometric grid paper lays on a wooden surface. Above the piece of paper, a person lays a rectangular prism made from 12 coloured cubes that have been stuck together with blu tack. The top layer consists of 6 yellow cubes. The bottom layer consists of 6 purple cubes. The person rotates the block to show its other sides.

Using a pencils, the person points to one of the cubes on the corner of the prism. They point the pencil along the three edges that meet at the corner of the block. They draw these three edges on the isometric grid. They then point to the edges on the top face of that cube, and then draw them on the isometric grid. They then point to the edges on the side face of the cube, and then draw them on the isometric grid. They then point to the edges on the front face of the cube, and then draw them on the isometric grid. They then point to the edges of the other cubes of the top layer of the rectangle and draw each of the visible edges on the isometric grid.

The person then points to the purple cubes on the bottom layer of the shape’s front face. They point to the visible edges of the cubes along the front and left-side face, and then draw them on the isometric grid.

Once completed, the shape on the isometric grid appears like the rectangular prism, viewed from the upper left-hand corner. The person holds the prism so that it appears like its representation on the isometric grid.

Text over a blue background: Over to you!]

Speaker

Over to you. Choose your favourite arrangement of blocks, draw it on isometric paper.

[Text: What’s (some of) the mathematics?]

Speaker

What's, some of, the mathematics?

[Text:

· Different rectangular prisms can be made using the same number of cubes. For example, all of these prisms are made using 12 blocks but they all look different. This shows us that 3D objects can look different but be made using the same number of blocks.

Below are three images, each depicting a rectangular prism made from 12 blocks. They first, is the prism that has been used in this video. The second is a long prism, in which all of the cubes have been placed in a single-file line. The third is a flat, wide prism in which the blocks have been arranged in a single layer, which is 3 cubes long and 4 cubes wide.]

Speaker

There are a number of important mathematical ideas this task helped me see. For example, we can see that different rectangular prisms can be made using the same number of cubes, for example, all of these prisms are made using 12 blocks, but they all look different. This shows us that 3D objects can look different but be made using the same number of blocks.

[Text:

· The same rectangular prism can look different depending on what angle you are looking at it from (your ‘point of view’, or ‘perspective’). For example, this is a drawing of the top, side and front view of the same object. The top and front views look the same but the side view is different.

Images below show the yellow and purple rectangular prism that has been used in this video from three different angles: the top view, side view and front view. The different views have been drawn on one centimetre grid paper.]

Speaker

The same rectangular prism can look different depending on what angle you are looking at it from, your point of view, or perspective. For example, this is a drawing of the top side and front view of the same object. The top and front views look the same, but the side view is different.

[Text:

· We can make nets by tracing around 3D objects.

Images below show the construction of the box net. The sides of the rectangular prism are outlined. The box net is cut out, and the sides are being folded. The box net is folded around the rectangular prism.]

Speaker

We can make nets by tracing around 3D objects.

[Text:

· Mathematicians can use different tools to draw representations of 3D objects. For example…

Images below show the rectangular prism drawn on grid paper and isometric paper.

Text: …we can use grid paper and isometric paper.]

Speaker

Mathematicians can use different tools to draw representations of 3D objects. For example, we can use grid paper and isometric paper.

[Over a grey background, the red waratah of the NSW Government logo appears amongst red, white and blue circles. Text: Copyright State of New South Wales (Department of Education), 2021.]

[End of transcript]

Instructions

Draw one of your boxes on isometric paper.

Discuss/ reflect

Have a go at making and drawing a 3D object using 18 blocks.

• How many different arrangements of 18 blocks can be made?

• How will you know when you have made them all?