Crossing a rectangle

Crossing a rectangle by youcubed

​Crossing a rectangle is a thinking mathematically targeted teaching resource focussed on investigating and justifying the longest and shortest pathways through a grid.

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, 2021

Outcomes

  • MAO-WM-01
  • MA1-GM-01
  • MA1-GM-02

Collect resources

You will need:

  • 5x8 grid

  • a marker


Watch

Watch Crossing a rectangle – part 1 video (4:07)

Use a 5x8 grid to create the longest and shortest pathways.

[Text over a blue background: Crossing a rectangle. By youcubed. Small font text at the bottom of the screen reads – NSW Mathematics Strategy Professional Learning team (NSWMS PL team). In the bottom right-hand corner of the screen is the red waratah of the NSW government.

Text over a white background – You will need…

  • A marker or colourful pencil
  • 5x8 grid printable

Beside the text are two images, each containing 6 5x8 grids over a plain white background. In the image on the top, the grids are arranged in a horizontal orientation. In the bottom image, the grids are arranged in a vertical orientation.]

Speaker

Before we begin our challenge today, you will need both of the grid printables, one showing grids in a horizontal orientation and the other in a vertical orientation, and a colour marker or pencil.

[Text over a blue background – Let’s play.]

Speaker

Let's play.

[A white sheet of paper on a blue background. On the paper, are two 5x8 grids. The grid on the left is oriented horizontally, while the grid on the right is oriented vertically.]

Speaker

Hello there mathematicians, today we'll be exploring a task from Youcubed called Crossing a Rectangle. Firstly, let's have a look at these rectangles here, what do you notice? Yes, some of you may have noticed that they have been partitioned with vertical and horizontal lines to create a grid.

[The speaker points to grid on the left. She traces her finger over the grid’s horizontal axis, which is 8 squares long, then along its vertical axis, which is 5 squares high.]

Speaker

Some of you may have also noticed that this one here has eight squares going across horizontally and five squares going up or vertically.

[The speaker points to the grid on the right. She traces her finger along the grid’s horizontal axis, which is 5 squares long, then along its vertical axis, which is 8 squares high.]

Speaker

And then some of you may have noticed that this one has five squares going across, or horizontally, and eight squares going up vertically. Some of you may have noticed that they are the same in dimensions but different in their orientation. Some of you may have also noticed, they are the same in the amount of squares that they have.

[The speaker points to the two grids; first the horizontally oriented grid, then, the vertically oriented one.]

Speaker

This one has eight by five, eight fives, which we know is 40 and so does this one, five eights, which we also know is 40. Now today, we have a challenge. We have to try and find the longest pathway we can create along the lines of a grid, that starts at the top and ends at the bottom without touching either of the sides.

[The speaker points to the left and right sides of the horizontally oriented grid. Using a green marker pen, she draws a line on the grid on the left. She starts on the line one square across from the left side. She then draws down 2 squares. She then draws to the left, one square, and touches the left side. She then draws down for one square. She draws an X beside the line, and then circles it.]

Speaker

So that means that if I tried something like this... that wouldn't fit into the rules of our challenge, because that went out on to the side.

[The speaker continues the line she has drawn. From the left side, she draws across to the right 4 squares. She then draws down one square. She then draws all the way across to the right edge. The line touches the right edge and finishes one square up from the bottom. She draws an X beside the point where the line ends, and then circles it.]

Speaker

Also, if I drew a line that's stopped on this side instead of the bottom, that also wouldn't fit into the rules of the challenge.

[The speaker draws a line on the vertically oriented grid. She starts one square across from the left edge, then draws down one square. She then draws to the right 2 squares. She then draws down 6 squares. She then draws across one square, then down one square, finishing on the bottom of the grid, one square across from the right edge. She draws a tick at the start of the line, then circles it. Then she draws another tick at the bottom of the line, and circles it.]

Speaker

Something like this, however, would fit into the rules because it started at the top, it ended at the bottom, and it didn't touch either of the sides.

[The sheet of paper is replaced by two sheets of paper, each with a vertically oriented grid. Each grid has a line drawn on it, and the number of moves marked at the bottom. Beside each grid is a Lego person. The person on the left is green, while the person the right is blue. Their outfits match the colours of the lines on their grids. The green line on the grid in the left has made 11 moves. It starts at the top, one square over form the left edge, then goes down 2 squares, then right one square, then down one square, then right one square, then down one square, then right one square, then down four squares, until it reaches the bottom, one square to the left of the right edge.

The blue line, on the right grid, has made 23 moves. It starts one grid to the right of the left edge, then goes down one square, then right one square, then down one square, then right one square, then up one square, then right one square, then down 2 squares, then left one square, then down one square, then right one square, then down one square, then left 2 squares, then up 2 squares, then left one square, then down 4 squares, then right one square, then down one square. It finishes at the bottom, 2 squares to the right of the left edge.

The speaker points to the green Lego person on the left, Monster Man. She traces her finger along the green line, counting all the gridded lines it travels along.]

Speaker

Let's have a look at two possible solutions, Monster Man created a path using 11 moves, one, two, three, four, five, six, seven, eight, nine, ten, 11.

[She points to the blue Lego person on the right, Policeman. She traces her finger along the blue line, counting all the gridded lines it travels along..]

Speaker

Policeman created a path using 23 moves. One, two, three, four, five, oh, he did something different there to Monster Man. Let's keep counting though. Six, seven, eight, nine, ten, 11, 12, 13, 14, 15... did you see that? He went back up. 16, 17, 18, 19, 20, 21, 22, 23! Wow!

[ Text over a white background: Our challenge.

  • What is the longest path you can create?
  • What is the shortest path you can create?]

Speaker

So our challenge is, what is the longest path we can create that starts at the top and ends at the bottom without touching either of the sides? Then, once you've explored that challenge, see if you can create the shortest path.

[6 vertically oriented 5x8 grids appear below the text.]

Speaker

We'll explore the path on the vertically orientated grids first. Have fun exploring mathematicians.

[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

  • Using the 5x8 (5 eights) grid standing tall, find the longest pathway you can, travelling along the lines of the grid from the top to the bottom of the grid.

  • Can you find the shortest pathway?

  • Your pathway can’t go along the side borders of the grid and you can't go outside of the grid either!


Watch

Watch Crossing a rectangle – part 2 video (4:07)

Does changing the orientation of a rectangle change the outcome?

[Text on a white background: What about when we change the orientation? Below are two images. The image on the right shows two 5x8 grids, one oriented vertically, and another oriented horizontally. The second image shows 6 horizontally oriented 5x8 grids on a white background.]

Speaker

So now we're wondering whether changing the orientation matters. Will we be able to make a longer path? Or will it mean we can make a shorter one? Over to you to explore on the horizontally orientated grid paper. Does changing the orientation matter?

[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

  • Turn your grid paper so it lays flat, so you can see an 8x5 grid (8 fives).
  • Turn the grid paper so you can see an 8x5 grid (8 fives)
  • Can you make a shorter pathway?


Watch

Watch Crossing a rectangle – part 3 video (0:29)

What's (some of) the mathematics in Crossing a rectangle?

[Text on a blue background: What’s (some of) the mathematics? Small font 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

So, what's some of the mathematics that we've looked at today?

[Text over a white background: What’s (some of) the mathematics?

  • We noticed that you can make paths shorter or longer by changing the direction of your path. Making the path straight, created shorter paths, while making a path with lots of changes in direction, made longer paths.

The image below depicts two horizontally oriented grids, each with a Lego person beside them. The blue line of the grid on the left has made 11 moves, while the green line of the grid on the right has made 23 moves.

Text:

  • We also found changing the orientation of the grid from vertical to horizontal affected the longest and shortest possible paths we could create.

The image below shows 4 5x8 grids. The two grids on top have straight, short, vertical lines, while the two grids on the bottom have long, snaking lines. The two grids on the left are vertically oriented and have lines drawn in red pen. The two grids on the right are horizontally oriented and have lines drawn in blue pen. The line on the grid in the top left has made 8 moves. The line on the grid in the top right has made 5 moves. The line on the grid in the bottom left has made 38 moves. The line on the grid in the bottom right has made 39 moves.]

Speaker

We noticed that you can make paths shorter or longer by changing the direction of your path. Making the path straight created shorter paths and making a path with lots of changes in direction made longer paths. We also found that changing the orientation of the grid from vertical to horizontal affected the longest and shortest possible paths we could create.

[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]


Investigate further

  • Create different pathways that have exactly 12 steps.

    • How many pathways can you create that are exactly 12 steps?

  • Create different pathways that have exactly 15 steps.

    • How many pathways can you create that are exactly 15 steps?

  • Create different pathways that have exactly 30 steps.

    • How many pathways can you create that are exactly 30 steps?

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