Data toss – Stage 2 and 3

Data toss is a thinking mathematically context for practise resource focussed on collecting and representing data.

Adapted from Gervasoni, A. (2015). Extending Mathematical Understanding: Intervention, Ballarat Heritage Services Publishing

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
  • MA2-DATA-01
  • MA2-DATA-02
  • MAO-WM-01 
  • MA3-DATA-01 
  • MA3-DATA-02

Collect resources

You will need:

  • small object (block, rubber, peg or a small soft toy)

  • chalk or stick

  • pencil or whiteboard marker

  • paper or mini whiteboard.


Watch

Watch the Data toss part 1 video (5:33).

Hit the target and record your results.

This video was created with Wendy from Hermidale PS, Cheryl from Hornsby Heights PS and Rachel from Albury North PS.

[Text over a blue background: Data toss 1. Adapted from Number Target by Ann Gervasoni. 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.]

Speaker

Let's play data toss.

[Text on a white background: You will need…

· small object (block, rubber, peg, a small soft toy)

· chalk or stick

· pencil or whiteboard marker

· paper or mini whiteboard

Below, are three images. The first shows four small items: a 50- cent coin, a yellow button, a purple cube, and an echidna-shaped pencil eraser. The second image shows a piece of white chalk and small stick. The third image shows a small whiteboard, a whiteboard marker, a pencil and a yellow piece of paper.]

Speaker

You will need an object to toss. This can be a block, a rubber, a peg, a toy, or anything that will fit in the palm of your hand. It might be good if you do something that doesn't roll. You will also need some chalk or a stick to draw your target. We need to record our scores and we can use this by using our pencil and paper or a whiteboard and marker. We can play this game inside or outside. If you are using chalk, I suggest you play it outside on the cement. You could even use your stick to draw your target in the dirt. If you are playing inside, draw your target on a piece of paper.

[Over a blue background, text: Let’s play!

Outside, a dirt driveway. The speaker moves a ladder aside.]

Speaker

I am playing my data toss game in the dirt today. You can join me if you like. Before we begin, we need to set up that game. We need to mark the starting line.

[On the left of the frame, the speaker uses a stick to mark a line in the dirt. She places the stick over the line.]

Speaker

I'll use that stick to help us see that.

[From the line, she takes two large steps towards the right of frame.]

Speaker

Then we need to take two large steps and this is where we are going to mark our target.

[She takes a step back. She places her palm flat on the ground. She uses a stick to draw a circle around her hand. She writes “100” in the centre of the circle.]

Speaker

If you are drawing like me, what you need to do is take two steps, draw a circle around your hand and put 100 in it.

[She draws a larger circle around the original circle. Within this circle, she writes “50”.]

Speaker

Draw a larger circle around that, put 50.

[She draws a third, larger circle around the second circle. Within this circle, she writes “30”.]

Speaker

Another larger circle. Put 30 and...

[She draws a fourth, larger circle around the third circle. Within this circle she writes “20”.]

Speaker

on the outside put 20. Now, might get a bit hard to see. So I found some string. I'll just pop that on it. Then we can clearly see our circles.

[She gathers a bundle of lengths of blue string. She places the string on the ground, to mark the circles that she has drawn.]

Speaker

You can use whatever you've got at home or you might be able to see yours in the dirt. OK, our target. Nearly ready? I need a recording sheet. Could write it in the dirt, could get a bit hard.

[The speaker places a whiteboard beside the line she marked on the left of the frame.]

Speaker

So I have a whiteboard here and my pen, ready. And lastly, I need my object.

[The speaker shows the green, echidna-shaped eraser to the camera.]

Speaker

We have Eddie the Echidna, helping us out today. So we are going to toss our object 20 times and report our score. Are you all set up and ready? Let's begin.

[The speaker stands behind the starting line. She tosses the echidna at the target. It bounces a little, and lands on the line which separates the 30 and the 20 circle.]

Speaker

Oh! That one should be there at 20.

[The speaker writes on the whiteboard. She tosses the echidna again. It bounces past the target. She writes on the whiteboard.]

Speaker

Didn't land in my target, so I get zero. Zero.

[She continues to throw the echidna and record her score on the whiteboard.]

Speaker

OK, last one.

[She tosses again. It doesn’t land on the target. She writes down her score. She shows the white board to the camera. It features 4 columns of 5 rows of scores: 20, 0, 20, 0, 0, 0, 20, 20, 0, 0, 0, 0, 0, 30, 0, 20, 0, 0, 30, 0.]

Speaker

These are my scores that I scored off using Eddie the Echidna. OK, I played my game. Let's head inside now to take a look at my data.

[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 a starting line (using chalk or a stick if outside) to show where you need to stand.
  • Take 2 large steps forward then start to draw the target or use the resource sheet if inside.

  • Toss your object 20 times, aiming for the target and record each score on a piece of paper or whiteboard.


Watch

Watch the Data toss part 2 video (6:05).

Explore ways to represent and organise data toss scores.

This video was created with Wendy from Hermidale PS, Cheryl from Hornsby Heights PS and Rachel from Albury North PS.

[The whiteboard from the previous video lays flat on a white surface. Beside it, is a marker pen and a blank sheet of paper.]

Speaker

With so many scores, it might be easier to put this in a table. I think a tally table would be a good choice, so we can see things clearly. We can draw one up.

[The speaker draws a table. In a single column, the speaker writes the word “Score”, then the numbers, “0”, “20”, “30”, “50”, “100” beneath it. The speaker then draws a large box around the text, then separates each row with a horizontal line. The column of numbers under the text “Score”, sits on the left side of the table. The speaker then draws a vertical line to the right of the column. The speaker then writes the word “Tally” at the top of the second column.]

We have our scores that we could possibly have tossed and we have our tally. Let's transfer our data from our tosses into our table.

[The speaker tallies the results from the whiteboard into the table, one by one. She crosses out each score on the whiteboard as she tallies it. The results are: zero was scored 13 times, 20 was scored 5 times, and 30 was scored 2 times. 50 and 100 were not scored.]

Now that we've transferred our data from our tosses into the table, let's have a look at our table more closely.

[The speaker moves the whiteboard out of the way and positions the tally table in the centre of the frame.]

I wonder how I can make it easier to look at the data, instead of having to count the tally all the time.

[The speaker draws a vertical line to the right of the Tally column. At the top of the third column, she writes “Frequency”. She draws a vertical line to the right of this column.]

We can do this by making another column. We call this the frequency column. It will help us tell how often we tossed that score.

[The speaker writes the frequency of each score as a decimal number: 13, 5, 2, 0, 0.]

Let's take some time to have a look at our data that we have now gathered. Which score did I toss the most? That is very easy to see on my table.

[The speaker points to the first row of tallied scores.]

13 times I tossed zero. Which score did I toss the least?

[The speaker points to the final two rows of the table, where the cells in the “Tally” column are empty.]

Good looking, I did not toss a 50 or a 100, so therefore, 50 and 100 were the scores that I tossed the least. Did you notice anything else from the data gathered? Yes, I did toss 20 more times than I tossed 30, and yes, there is a large gap between the number of times I've tossed 20 or 30, compared to how many times I tossed zero. I wonder what the total point scored would be. Let's use our table to find out that.

[At the top of the final column of the table, the speaker writes, “Points”. In the cells below, she writes the number of points gained from each score as a decimal: 0, 100, 60, 0, and 0.]

We're going to find the points for each score. Zero, I tossed 13 times, so 13 lots of zero is zero. 20, I tossed five times, so, five lots of 20, or five lots of two tens is ten tens, 100. 30, I tossed twice, two lots 30, or two lots of three tens is six tens, which is 60. 50, I tossed zero times, zero points, same with a 100.

[Beneath the table, the speaker writes, “Total Points”.]

My total points, adding all of these points together, zero plus 100, plus 60, plus zero, plus zero, so 100 plus 60, or ten tens and six tens, is 16 ten, which is renamed to 160.

[Beside “Total Points”, the speaker writes, “160”.]

My total points from this game of Data Toss is 160. As a mathematician, I take the time to reflect on my data. I wonder how I could improve it? I wonder what I could do to improve my total points? How can I increase my scores every time I toss?

[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

  • Create a table, with 4 columns and 6 rows. Across the top row include the headings, score, tally, frequency and points.

  • Record your scores as tally marks then write the total of each score in the next column.

  • Have a look at your table, which score did you toss the most? Which score did you toss the least? What else did you notice about your data?

  • Find the total of each score and record it in table.

  • Combine all of the points that you scored to find your total score.

Discuss

  • Have a look at your table, which score did you toss the most?

  • Which score did you toss the least?

  • What else did you notice about your data?

Watch

Watch the Data toss part 3 video (7:03).

Explore ways to represent and organise data toss scores.

[Text over a blue background: Data toss 3. Adapted from Number Target by Ann Gervasoni. In the lower right-hand corner of the screen is the red waratah of the NSW Government logo. Small font text in the lower left-hand side of the screen reads: NSW Mathematics Strategy Professional Learning team (NSWMS PL team).]

Speaker

Welcome back mathematicians, how did you go collating your data in the tally table? Hopefully, you managed to score higher than I did, or at least scored less zeros than I did.

Have you been thinking about your data like I have been? I've been wondering what I can do to increase my total score? So, let's have a look...

[Text on a white background: You will need…

your data from your data toss game

small object (block, rubber, peg, a small soft toy)

chalk or stick

pencil & paper or whiteboard & whiteboard maker.

Below, are four images. They depict: the speaker’s tally table from the previous video; four small objects, including the echidna-shaped eraser; a piece of chalk and a stick; and a whiteboard alongside a whiteboard marker, a yellow sheet of paper and a pen.]

Speaker

You will need your data from your data toss game, your object that you used, your chalk or stick, your pencil and paper or whiteboard and marker.

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

Text over a white background: Remember our game of data toss? Three images below show moments from the two previous videos: The speaker stands behind the starting line, facing the target drawn in the dirt; they speaker’s scores are written on a whiteboard; and the scores are tallied in a tally table. The points total, 160, is noted below the table.]

Speaker

Remember our game of data toss and how my total points were very low? Well, I have been thinking. How can I adjust my target, so that I can increase my chance of gaining a high score? Let's take a look at my original target.

[Text: Changing the target. An image below shows a bird’s eye-view of the speaker original target. The scores of each circle are written in red text over the image. The scores in each circle, moving from the centre circle outward, are: 100, 50, 30, and 20.]

Speaker

This was the target I used, what do you notice about this target? Yes, I can also see circles around each other, we can see that the smallest circle in the centre is worth the most points and the outer circle is worth the least points. Hmm, I'm wondering what I could do to this target to increase my chances of a higher score whilst keeping the same score as at 20, 30, 50 and 100.

[Another image appears. It shows the same target. The scores in each circle are inverted and re-written in green text, meaning that the scores for each circle, from the centre moving outward, are: 20, 30, 50, and 100.]

Speaker

When looking at my data and knowing that I did toss some twenties and thirties. I'm thinking that I could change the position of the target's numbers, by having 100 as the score for the outside circle and 20 as the centre score, I should see an increase in my score using this target.

What else could I do?

[Another image appears. It shows another target, outlined in the dirt. The size of the target areas is increased and are marked with larger circles. The score for each circle is written in red text. The scores for each circle, starting from the centre and moving outward, are: 100, 50 and 30.]

Speaker

Yes, I could just make my target larger, this might really help me when I toss Eddie the Echidna , I sometimes was very close to the target but didn't land inside the circles, maybe it could look like this, your data might be telling you a different story to mine. So you might like to think about different ways of adjusting your target to increase your total score.

[The two additional images disappear from the screen, leaving just the image of the original target.]

Speaker

I also had some other ideas about how I could create a target that would assist me in getting the maximum score every time. I could use what I know about chance and design a target where all of the scores were equally likely. So my circle could look something like this.

[Another image appears. It shows a large circle that has been quartered into four even segments. The lines that divide the segments form a cross shape in the circle. Each of the segments are marked with a score. From the top left, moving clockwise, the scores are: 100, 50, 30, 20.

Another image appears. It shows the same quartered target, but the scores have been inverted. Form top left, moving clockwise, they are: 20, 30, 50, 100.]

Speaker

This means that I need to petition my target into quarters of four equal parts, as I have four scores, giving me an equal chance to toss each score. I'm also thinking about how I could change the orientation of my target to help maximise my chances of scoring higher than what I did in my first game.

[Another image appears. It shows the same quartered target, but it has been tilted so that the cross that divides the segments is shaped like an X. The scores have been re-written. From the top segment, moving clockwise, they are: 100, 20, 50, 30.]

Speaker

What if I position my target this way will it help me?

We have the 100 over on the left, this could be helpful as Eddie the Echidna seem to keep bouncing in that direction or, I could put the 100 at the front so I can do a shorter toss or right at the back as I know Eddie the Echidna was very bouncy.

Do you think any of these targets might help me improve my score?

Ah, yes, I think you're on to something because my data showed that I scored thirties and twenties and thinking about my tossing and where the Eddie the Echidna landed, I'm also thinking that the second option would be best for me, as 100 and the 50 are at the front of the target. I still could throw zeros though, by not reaching the target, but by having 100 and the 50 closer, I should hopefully beat my last score. Let's try it out.

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

Speaker

Over to you mathematicians to begin creating a new target for your new data toss game. Remember to think about how best to change your target to suit you and your original data. You may need to change your target a few times before it works for you. Don't forget to collect your data with your new target to compare it to your original data toss results.

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

Text over a white background: We can interpret data to help us adjust targets or gameboards to increase our chances of winning or improving our scores. Four images below show: the speaker’s original tally table; the original target, but with the scores inverted and written in green; a target with larger, circular target areas; and a target with quartered target areas, and with the scores 50 and 100 in the segments closest to the starting line.]

Speaker

So mathematicians, what's some of the mathematics?

We can interpret data to help us adjust targets or gameboards to increase our chance of winning or improving our scores. Your target and the adjustments or changes you made will probably look quite different to mine because we have used data to make decisions about what we would adjust and why. Maybe your new target will have a larger inner circle and really small outer circles, or maybe your data suggests that you need to place 50 and 100 towards the back of your target because you tossed too far. Have fun investigating, I am off to set my new target, helps me to get a high score and hopefully a lot less zeros.

[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

  • Create a new target for your own Data toss game.

    • Remember to think about how best to change your target to suit you and your data from your first game.

    • You might think about changing aspects of your target such as:

      • the size and shape of each section of the game board

      • the location of the scores in the game board

      • the numbers in your game board.

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