My 7th graders are working on “percentages of” problems currently, and late last night, I saw this problem on one of Don Steward’s handouts.

There are 75 olives, 40% of which are green. I eat some of the green olives until 10% of the olives that remain are green. How many green olives did I eat?

How would you solve this?

I solved it using algebra. Then, immediately, I thought, *Fawnzie, since when do you use algebra to solve stuff like this. C’mon, do your rectangles*.

I think of 40% as 2 of 5 boxes.

So, 75 olives must split into 5 groups of 15, so there are 30 green olives.

Then, I ate some olives to end up with only 10% of the remaining olives are green.

Well, since I didn’t eat any of the 45 black olives, so these 45 must make up 90% of the olives remaining [in the 9 boxes], so 45 must split into 9 groups of 5.

Oh, look! I began with 30 green olives, I now only have 5 green ones left, so I must have eaten 25 of them.

Okay, your turn.

There are 80 olives, 75% of which are green. I eat some of the green olives until 20% of the remaining olives are green. How many green olives did I eat?

Because if I tried to show my kids the work below, or versions thereof, a few might just shit in their pants.

## 10 Comments

Since the 20 black are 80%, the 5 green left are the 20%, so you must have eaten 55.

I’d do it algebraically first too but thinking like a 7th grader this is highly amenable to guess and check. After figuring there are initially 30 olives there are are only 30 possibilities to check.

Its easiest to go by 5’s leaving only 6 values to check:

25/70, 20/65, 15/60, 10/55, 5/50 <— bingo

Okay, the boxes make sense, though I tend not to think that way, particularly because I’m not always sure that the numbers I get out in the real world will behave nicely (that is, real-world data has a nasty habit of not being integral or even rational).

Of course, I don’t often wonder about how many olives I’ve eaten because I can’t stand olives. But if this were, say, those nice Italian red peppers in a jar, which I love, I still wouldn’t worry because I’d either know how many I ate, or I probably wouldn’t be thinking in terms of percentages, particularly if I hadn’t taken note of the numbers to begin with. Plus there is only one kind of red pepper. In other words, this has a bit of a forced feel to it, but I suppose that’s not a fatal flaw, and maybe I can figure out some way to play with the situation so that it feels more plausible and less forced to me.

I also realize that grade-level tells us whether we’re going to play with irrational numbers or not in our problems. Nothing wrecks counting problems more thoroughly than irrational quantities (remind me to talk about a great tiling problem I learned from Robert and Ellen Kaplan back in the early oughts where irrational numbers throw a monkey-wrench into a perfectly nice situation).

So, yes, in the end, if you have nice numbers, the boxes certainly work and the fact that I don’t think that way in no way would stop me from teaching this approach. But I really do jump to that algebra instinctively and I have a rule that no one is allowed to shit in my classroom. :^)

Great problem!

I like teaching problem solving to my 6th graders Singapore style, using boxes like you do, Fawn. It removes the mystery, and makes the process accessible to almost all the kids.

But I get soooo much push back from kids who don’t like trying this out.

And it’s frustrating when their “better” way doesn’t work.

How can I “sell” this method more effectively?

BTW, I guffawed out loud when I read your true-but-profane comment! Thanks for the laugh!

40% of 75 taxes my Trianglemind. So I found 20% of 150 instead, which is 30. There are 30 green olives. And therefore 45 black olives.

Fawn pigged out on green olives only (WHY?!?!?!? Were the black ones from a can instead of some good Kalamata olives? Making a mental note not to go back to Fawn’s house; she buys bad olives and hogs all the good ones.)

So 45 black olives and some green ones together make what’s left, and what’s left needs to be 10 times as many as the green ones. So I need two numbers—one 10 times the other, and that also differ by 45. What about 10? No good. What about 5? SCORE!

There are 5 green olives left over. Fawn ate 20. That’s OK. I’d rather snack on a nice ripe apple this time of year anyway.

(Fawn ate 25, that is)

I didn’t know you think of yourself as Fawnzie! My grandmother used to call me Debsie-web. That’s almost as good as Fawnzie!

Awesome post, too!

Great Find! Not limited to 7th grade believe me. A good talk through High School as well.

My first intuition and the easiest way for me to solve this problem would be to solve algebraically. I’m not sure if the students (7th grade of high school students) would think the same way. The rectangles method might make more sense for them because it is visual and thus easy to explain or understand. Your explanations are very straight-forward and clear and if I were teaching 7th graders, I would first show them the rectangles method then explain how the same problem can be solved algebraically. Great problem!

Hi Fawn! Thanks for being such an inspiration. This comment isn’t related to this particular post, but I wanted to share a resource with you and I don’t have Twitter, so I figured this was the best means.

Your website, along with suggestions from colleagues, turned me on to more open problem solving activities, like Visual Patterns, Which One Doesn’t Belong, and Graphing Stories. One problem I’ve noticed working with a highly diverse, high-poverty population is that its sometimes difficult for all students to have a voice in the classroom. I recently was introduced to Peardeck (www.peardeck.com), and its really helped with student engagement. With Peardeck, I can take a Google Slides presentation and make it wholly interactive. So for example with a graphing stories activity, I can show a video and then project and send out a blank coordinate grid to my students’ devices. They can they graph the story, and I can project their responses either individually, or overlaid. This has been HUGE in getting all of my students engaged in the activities, and it has led to some super rich discussions. Looking at all the graphs overlaid and noticing similarities and differences has been a blast!

Anyways, just wanted to share a resource with you. Thanks for all you do!