## New Activity: Marcellus the Giant

In Marcellus the Giant, the new activity from my team at Desmos, students learn what it means for one image to be a “scale” replica of another. They learn how to use scale to solve for missing dimensions in a proportional relationship. They also learn how scale relationships are represented on a graph.

There are three reasons I wanted to bring this activity to your attention today.

First

Marcellus the Giant is the kind of activity that would have taken us months to build a year ago. Our new Computation Layer technology let Eli Luberoff and me build it in a couple of weeks. We’re learning how to make better activities faster!

Second

When we offer students explicit instruction, our building code recommends: “Keep expository screens short, focused, and connected to existing student thinking.”

It’s hard for print curricula to connect to existing student thinking. Those pages may have been printed miles away from the student’s thinking and years earlier. They’re static.

In our case, we ask students to pick their own scale factor.

Then we ask them to click and drag and try to create a scale giant on intuition alone. (“Ask for informal analysis before formal analysis.”)

Then we teach students about proportional relationships by referring to the difference between their scale factor and the giant they created.

You made Marcellus 3.4 times as tall as Dan but you dragged Marcellus’s mouth to be 6 times wider than Dan’s mouth. A proportional giant would have the same multiple for both.

Our hypothesis is that students will find this instruction more educational and interesting than the kind of instruction that starts explaining without any kind of reference to what the student has done or already knows.

That’s possible in a digital environment like our Activity Builder. I don’t know how we’d do this on paper.

Third

Marcellus the Giant allows us to connect math back to the world in a way that print curricula can’t.

Typically, math textbooks offers students some glimpse of the world —Â two trains traveling towards each other, for example —Â and then asks them to represent that world mathematically. The curriculum asks students to turn that mathematical representationÂ into other mathematical representations —Â for instance a table into a graph, or a graph into an equation — but it rarely lets students turn that math back into the world.

If students change their equation, the world doesn’t then change to match. If the student changes the slope of the graph, the world doesn’t change with it. It’s really, really difficult for print curriculum to offer that kind of dynamic representation.

But we can. When students change the graph, we change their giant.

There is lots of evidence that connecting representations helps students understand the representations themselves. Everyone tries to connect the mathematical representations to each other. Desmos is trying to connect those representations back to the world.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.

1. #### Kathy

October 21, 2016 - 8:38 am -

Perfect timing! I’m planning on doing life-sized Barbie next week. I’m going to do this activity the day before. I think it will really add a lot. Thanks!

2. #### Cathy

October 21, 2016 - 8:47 am -

Biscuits! This activity will be perfect in a unit we’re starting next week! You can bet our PLC meeting this very afternoon will be filled with giants!

3. #### Howard Phillips

October 21, 2016 - 8:52 am -

Don’t forget the terms “scaled up” and “scaled down”.
It would be a good idea to keep one giant fixed and the other one variable, initially at least. A model car is usually produced at a scale of 1/20 or 1/10, but the original car stays the same.
(engineering approach)

4. #### Kevin Hall

October 21, 2016 - 11:38 am -

I have a little suggestion on Screen 10. When I did the activity, I entered the correct number on Screen 7, but then forgot to “submit to teacher”. Instead, I just advanced to the next screen. That meant that when I got to screen 10, it said my Screen 7 answer was wrong, even though I knew it was right. Maybe the program could intercede between Screens 9 and 10 and say “you can’t go to screen 10 until you submit answers for screens 7, 8, and 9”?

5. #### Kevin Hall

October 21, 2016 - 11:43 am -

Would it be better to label the orange dot on the graphs as “Giant’s height”?

Typo on Screen 12: “Go back to the previous screen and try again until you’re sure you look at ANY graph” needs to say “you *can* look at any graph”.

• #### Dan Meyer

October 26, 2016 - 1:03 pm -

Would it be better to label the orange dot on the graphs as “Giant’s height”?

I’d love to find some kind of concise label for the orange point but I don’t think “Giant’s height” is accurate. The orange point affects the slope of a line that represents the proportional relationship. Like I said, not very concise.

Typo on Screen 12: “Go back to the previous screen and try again until you’re sure you look at ANY graph” needs to say “you *can* look at any graph”.

Eagle eyes!

6. #### Kevin Hall

October 21, 2016 - 4:25 pm -

Sorry for yet another comment, but one more thought. I wonder if you’re missing an opportunity to ask a question that’s quite low on the dial of mathiness: to enlarge a picture, should you take all the existing measurements and multiply them by some constant, or add some constant to them? I used to teach an activity that let students try both so they could see how wonky the picture looked when you enlarged all the sides by adding some constant.

October 21, 2016 - 7:00 pm -

Hi Dan, Eli and team Desmos,

Love the new Giant activity. It’s very fun to see the window (on screen 5) adjust according to the various numbers you can plug in. I love that students can enter values less than 1, equal to 1, and greater than 1. Entering 1000 was fun too. On that note, I second Howard Phillips’ comment about using scaling up and/or scaling down. I’m wondering if you could use your super slick Computation Layer to say something after students enter their scale. Maybe something along the lines, “we noticed your Giant is a scaled up version of Dan.”

Heads up on the typo on screen 10: “Marcellus is proportional because every one OF his dimensions…”

I like how screen 14 asks students to use academic language from the activity. I’m wondering if you could increase their exposure to the mathematical terminology (proportional, scale up, scale down, etc.) throughout the activity. For example, proportional is only used on screen 10 as far as I can tell.

I’m also wondering if some of the information on screen 10 can be put on screen 9. For example, screen 9 could include “Congrats, Marcellus’s height is X times..his eyes are X times… Now let’s go to the next screen to see what this looks like graphically.” This might place a greater focus on the equation near the bottom of screen 10 and points on the graph.

On screen 10, maybe add a teacher note or suggest students click on the 3 black points to reveal their coordinate points and better understand the proportional relationship between x and y.

I’m wondering why the card sort screen 13 does not include a completely proportional example. Would this be something worth adding? It seems there are so many opportunities toward the end of the activity where students get to play with the tools in order to see both proportional and disproportional relationships and I see the value in it. As a teacher, I would like more opportunities at the end to firm up student understanding of proportionality. For example, maybe students can click and drag a proportionally sized hat on the giant by decoding the scale first.

Overall, this is a great activity and I’d love to hear what you think.

• #### Dan Meyer

October 26, 2016 - 1:06 pm -

Lots of helpful feedback here, Andrew. No follow-up questions. I think we’ll take them all on.

8. #### Avery

October 21, 2016 - 9:44 pm -

Very cool. I vote for the teeth being triangles, though. There’s got to be some similar triangle stuff that could be “hidden” in here. Or in the sword?

9. #### Scott Immel

October 22, 2016 - 6:51 am -

Cool activity! I am starting proportions soon, so this is great! When I have an overwhelming activity (for my students who are mostly below standard) like this, I generally need to make pre-activities beforehand for this. I love the idea Kevin Hall said about adding to each side. I think I will make that activity as well. BTW, card sort is my new best friend in Desmos! We can know really assess as well as teach with Desmos!

10. #### Rachel Maloney-Hawkins

October 22, 2016 - 10:37 am -

My 7th grade kiddos did this activity on Friday to wrap up work we’ve been doing correlating graphs with events (in this case growing/shrinking proportionally) and hopefully to bridge into discussion about scale drawings. THEY LOVED IT. I was just wondering, though, if there was a way to grab measurements from the last screen where they make an accessory to determine whether or not their drawings were actually proportional? Desmos has spontaneously led to such deep mathematical discussions!! Fantastic work!!

• #### Dan Meyer

October 26, 2016 - 1:07 pm -

I was just wondering, though, if there was a way to grab measurements from the last screen where they make an accessory to determine whether or not their drawings were actually proportional?

Thanks for the feedback, Rachel. I wish that were possible. Perhaps in the future.

11. #### Kevin

October 22, 2016 - 9:17 pm -

I worry that kids won’t really pay attention to the axes labels on Screen 10. In fact, instead of stopping to consider what the graph is really showing, they may just internalize the rule that “a body part that’s above the line on the graph is too big, and one that’s below the line on the graph is too small”. That’s a correct rule, but it short-circuits the main idea of the lesson. I made a suggested Screen 9.5 to attempt to prevent that from happening. It just makes them plot Dan’s measurements as points on the x-axis, and Marcellus’ on the y-axis.

Here is that activity: https://teacher.desmos.com/activitybuilder/custom/580c41a3d2bc7c35077bb742

(Do I need to wrap links in HTML tags in your new blog format? I’m not sure)

Ideally, after they got this Screen 9.5 right, you’d have a nice animation that would then make a point a point appear in the x-y plane to for each body part, e.g., the point (Dan’s eyes, Marcellus’ eyes) would appear. Those are the points that are displayed in Screen 10, so you could make sure students understand what those points mean.

• #### Dan Meyer

October 26, 2016 - 1:08 pm -

Ideally, after they got this Screen 9.5 right, you’d have a nice animation that would then make a point a point appear in the x-y plane to for each body part, e.g., the point (Dan’s eyes, Marcellus’ eyes) would appear. Those are the points that are displayed in Screen 10, so you could make sure students understand what those points mean.

Super interesting. I like it.

• #### Kevin Hall

October 27, 2016 - 4:58 am -

Anything’s good as long as it ensures that kids are processing what the x- and y-coordinates of those points represent. I work with a population that needs to be very inclusive of ELL and special education students, so I was trying to avoid a long sentence saying, “Now graph the points for each body part, where x represents Dan’s measurement of the part, and y represents Marcellus’ measurement”.

In addition, I think there’s something beneficial about having kids make some kind of a double number-line to represent the relationship initially. I want them to see that if the original body parts are spaced like this on the number line:

x x x

then the new body parts will be spaced like this:
x x x

The points spread out, but the ratio of the distances between the points is preserved. Dragging the points to the right locations is a feeble attempt to give some tactile sense of that. After which, of course, students need to turn the second number line vertically and make an x-y graph. For a novice, it’s not easy to see that when the second number line has bigger spaces between points, it means the graph will get steeper. Will they get that understanding out of what I’ve proposed? I’m not sure, but that’s what I’m trying to drive at here.

And that the

12. #### David Petro

October 24, 2016 - 3:46 am -

I really like this. The flow is nice and the interaction is great. But I wonder about few things.

I am curious as to why you only chose to focus on the eyes, mouth and sword (IE the arms, legs, torso and head are proportional)? So for example, when I was looking at the “Not Scale” page for the first time to try to see why they were not scale, I felt like I was looking at a more difficult version of “which is not like the other” since the head, arms etc were actually proportional and it was actually some of the details that were not. That is, it wasn’t the entire person that wasn’t scale but just parts. Was that a conscious decision? I wonder if that slide needs some teacher notes to so they can anticipate that issue?

The second thing is when you get to the screens that allow you to type in the actual measurement (slides 7-9), I would love to have a “ghost” of my original guess still appear on the screen so I can see the difference between that and the correct answer once I type it in. It changes to the correct position on the screen but if you are not actually looking there at the time you click Submit then you might miss that it moves (especially if you are close)

This last thing is minor. I get why it looks odd on screen 6 if you don’t pick a scale on screen 5 and you do have it say “go back and pick a scale” but you might consider some teacher notes on screen 5 (or 6) that highlights that. This way you might get less student questions when they don’t pick a scale, go to screen 6 and then wonder why it looks like it does (because they haven’t read “go back and pick a scale”)

13. #### Dan Meyer

October 26, 2016 - 1:13 pm -

That is, it wasn’t the entire person that wasn’t scale but just parts. Was that a conscious decision? I wonder if that slide needs some teacher notes to so they can anticipate that issue?

Thanks for the feedback, David. We’ll take it all into consideration. In the quoted piece above, are you saying you wanted every body part to be disproportional?

14. #### Beth Hentges

October 26, 2016 - 4:23 pm -

Have you seen the Moris, Boris, and Doris lesson from the original Connected Math program?

Students are given (essentially) three or four tables of values and dot-to-dot instructions. Each graph becomes the face of a cat. Students are then asked to decide which ones have the same shape. What is the criteria?

The activity leads to students developing/constructing the definition to include congruent corresponding angles and proportional corresponding lengths.

When assessing students at the end of such a unit, I like to include a question in which the scale factor is given and I ask the students the typical questions about angles, lengths, areas, and volumes. However, I ALSO make sure to ask a question about the NUMBER of some item. (the number of handles on the chest, the number of windows on the front, the number of eyes on the giant). You would be surprised how many students will apply the lenght scale factor to such problems.

15. #### Dan Meyer

October 26, 2016 - 4:24 pm -

Beth:

However, I ALSO make sure to ask a question about the NUMBER of some item. (the number of handles on the chest, the number of windows on the front, the number of eyes on the giant). You would be surprised how many students will apply the length scale factor to such problems.

Awesome. I’m sure I would.

16. #### Cathy

October 26, 2016 - 5:03 pm -

Two comments from students today, regarding this activity:

1. “Desmos is too accurate!” (a student received “Oh biscuits” feedback regarding the mouth being 1.95 times larger instead of 2 times larger, and Desmos called him on it, ha!)

2. “That was so dorky! But the dorkiest things are often the funnest things!”