## [3ACTS] Nana’s Paint Mixup

Nana’s back.

I messed up her chocolate milk order a few years back. This is a new ratio task I first heard from Colin Foster at the Shell Centre last winter.

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Don:

I dunno, this one made me wish the follow-up showed him just throwing the paint away and starting over. Thereâ€™s just not enough investment in materials & time to make me think past. Plus if 6 tablespoons was enough paint to do the job then 30 is just a waste of paint far in excess of throwing away 6.

@Don, what if he had used up all the white paint after putting in the 5 scoops of white? Then heâ€™d have to figure out how to do it by just adding more red.

2014 Mar 11. Great extension for Algebra students from Paul in the comments:

I used the task to set up this question in an Algebra class.

The students were very puzzled when their intuition about the solution did not match arithmetic or a demonstration with cubes.

â€œI have two cans of pink paint variations in the following ratios. Neither is perfect.

Nanaâ€™s Pink 5 Reds : 1 White
50/50 Pink 1 Red : 1 White

I think that a perfect pink will be 3 Reds : 1 White

Can I make it by mixing Nana Pink and 50/50 Pink?â€

Students expected the solution to be one cup of each. How wonderful the sound of a perplexed group of students when their arithmetic did not match their intuition.

Sense making ensued for many minutes with pictures, cups with cubes and more arithmetic.

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. #### Alex

February 20, 2014 - 1:34 pm -

This is admirably devious. Anyone who downloads the link to see the answer is proving the power of your hook.

If we can’t resist, how will the students?

2. #### Don

February 20, 2014 - 3:04 pm -

I dunno, this one made me wish the follow-up showed him just throwing the paint away and starting over. There’s just not enough investment in materials & time to make me think past. Plus if 6 tablespoons was enough paint to do the job then 30 is just a waste of paint far in excess of throwing away 6.

Which if the class brings it up organically would be a big win; then you can talk about why to choose one path or the other and it’s some good critical thinking. Otherwise it smells like pseudocontext to me.

3. #### Danny

February 20, 2014 - 5:31 pm -

I agree with Don here. Would make me ecstatic if while doing this question someone went for the easy answer and to see where the discussion led.

To cover the argument I’d just say something like this is only a sample of what we are going to do, will need much more paint than this eventually. But I would definitely discuss the option of throwing it out.

If no student brought up throwing it out on their own, I would probably try to push them toward it. Often we make a mistake and rather than starting over we try to fix the mistake and this causes bigger problems in the end. Being willing to start over can be a great asset in life.

4. #### Kevin Hall

February 21, 2014 - 10:45 am -

@Don, what if he had used up all the white paint after putting in the 5 scoops of white? Then he’d have to figure out how to do it by just adding more red.

5. #### Jill

February 21, 2014 - 1:23 pm -

Brill! I’ve just done some paint ratio problems with my students, including an ‘Oops! I mixed it up wrong’ problem. We also looked at the Pantone website…check out the colours of the year and of the day. Since the printer, Smartboard and computer screen showed three different versions of the Colour of the Year, it all led to a discussion of how we can be sure we are getting the colour that we really mean or want. The t-shirts for their up-coming school trip turned out pink instead of the expected lilac. I also suggested that they have a look at Jasper Fforde’s book, ‘Shades of Grey’, (no, not ’50 Shades of Grey’!), where a post-apocalyptic population can only see certain parts of the colour spectrum and this determines their role in society. I love it when Maths turns into a bigger lesson!

6. #### Shaun Errichiello

March 2, 2014 - 9:58 am -

This is great. I am introducing this 3act and want to connect it into my linear functions unit. I am thinking something about slope representing the ratio of each color in the mixture.

I am trying to make an animated graph (something like ditch diggers) and was hoping someone could share a general approach to using adobe software in making a line graph. Should I do all the work in After Effects? Or should I make the graph in another program like Illustrator.

7. #### Dan Meyer

March 3, 2014 - 9:13 am -

Hi Shaun, here are some basics. You use the pen tool up on the top of the screen to draw the graph. Make sure its “stroke” is set to the right color and the line thickness.

Now you have your entire graph, which isn’t super helpful. You want to animate it. So you expand the shape layer (click on the right arrow) and click on the tiny right arrow that’s revealed. You’re looking for Add Trim. Open up that property and you’ll see Start 0% and End 100%. Play with the end percentage and you’ll see the line erase and then redraw.

You want to set “keyframes” for 0% and 100%. AE will fill in the rest.

I’m happy to expand on any of this with screenshots or whatever.

8. #### Shaun Errichiello

March 10, 2014 - 4:08 pm -

I adapted this 3 act for our 8th/9th grade unit on Linear functions. The idea was to think of correlation and slope as a mixing of color. So many awesome things happened in these lessons (I will post about it someday) but I wanted to share a Geogebra applet that shows how the change in mixture changes the color of the paint. The kids loved seeing the connection between slope and color.

The first applet was designed to be a low inference visual:
http://www.geogebratube.org/student/m94597

The second is a bit more detailed:
http://www.geogebratube.org/student/m94599

I used these during the third act (after we show the video).

Also thanks for the tips on AE. I am slowly learning how to put all the pieces together.

9. #### Paul

March 11, 2014 - 2:32 pm -

I used the task to set up this question in an Algebra class.

The students were very puzzled when their intuition about the solution did not match arithmetic or a demonstration with cubes.

“I have two cans of pink paint variations in the following ratios. Neither is perfect.

Nanaâ€™s Pink 5 Reds : 1 White
50/50 Pink 1 Red : 1 White

I think that a perfect pink will be 3 Reds : 1 White

Can I make it by mixing Nana Pink and 50/50 Pink?”

Students expected the solution to be one cup of each. How wonderful the sound of a perplexed group of students when their arithmetic did not match their intuition.

Sense making ensued for many minutes with pictures, cups with cubes and more arithmetic.

10. #### Dan Meyer

March 11, 2014 - 4:56 pm -

Man. Nice extension. “One cup of each” is such an enticing answer.

11. #### Simon Biggs

April 12, 2014 - 4:07 pm -

Heya Dan,

Thanks for all your 3 Act maths. During my teaching prac I tried out your “Basketball hoop” lesson combined with Geogebra.

Wanted to let you know that I love putting your stuff up on teaching resources reddit:
http://www.reddit.com/r/teachingresources

Thank you for not just helping me, but many many other teachers :).

Cheers,
Simon