A few ago, I was invited to teach a lesson on division (Grade 3). First, I read Bean Thirteen aloud – once just for fun. About Bean Thirteen, from the author:
Ralph warns Flora not to pick that thirteenth bean. Everyone knows it’s unlucky. Now that they’re stuck with it, how can they make it disappear? If they each eat half the beans, there’s still one left over. And if they invite a friend over, they each eat four beans, but there’s still one left over! And four friends could each eat three beans, but there’s still one left over! How will they escape the curse of Bean Thirteen?
(A funny story about beans, that may secretly be about . . . math!)
Next, we revisited several of the pages. I asked students to write an equation to match the picture. I modelled this using magnetic “bean counters.” For the page above, students suggested 2 × 6 + 1 = 13; I introduced 13 ÷ 2 = 6 R 1. We discussed and recorded the meaning of this:
In pairs, students then chose their own number of beans (counters) and built different division as sharing stories for this number. They recorded (.doc) their stories using pictures, numbers, and words:
I called on students to share their stories with the class. They observed that some numbers gave remainders more so than others; Bean Thirteen can also be used to explore even/odd and prime/composite numbers.
This lesson served as the students’ introduction to division. I wrestled with the decision to introduce remainders at this time. An alternative problem – one consistent with both the prescribed learning outcomes and recommended learning resources – might be to start with 18 beans – a “nice” dividend – and share equally among 2, 3, 6, and 9 bugs – “nice” divisors. Note 15 ÷ 3 = 4 R 3 (and 15 ÷ 2 = 6 R 3) above. This mistake would not have happened had I not introduced remainders. I wonder if including remainders makes it more difficult for students to understand division and relate division to multiplication.
Then again, children will have already experienced remainders in everyday contexts.
Chris Hunter 
Great lesson! I need that book! I wouldn’t sweat the uneven thing too much. Sometimes I think it’s harder when kids are used to problems where everything works out evenly, and then it suddenly doesn’t. CGI shows young kids can handle uneven fair shares. The notation may throw them, though. One thought would be to let them name the extras however they want, at least at first. TERC has very good lessons called “What do you do with the extras?” where there are different contexts, so the remainder needs different treatments (round up, give away, make a fractional share, etc.) I think it’s more useful than the R. notation, because kids have to think about what that remainder really means in the story. If you’re dividing 16 by 3, it makes a difference if it’s 16 kids going into 3 vans; 16 cookies shared by 3 people; 16 balloons shared by 3 people, etc. Unless we’re going to leave a kid behind (“Sorry, you can’t go to the zoo! Blame remainders!) or only eat whole cookies, 5 R. 1 is of limited usefulness, you know? The beans are like balloons, so it made sense here. I would still keep the R. at bay, though.
Thanks, Tracy!
I’m not familiar with TERC (b/c Canada) but it sounds very similar to a lesson I’ve seen my friend and colleague Marc Garneau teach. Three problems, each solved by 18 ÷ 4: (1) 18 cm of string cut into 4 equal pieces, (2) 18 kids being driven to the zoo (no kidding!) in cars that hold 4 passengers (not including the driver), and (3) 18 donuts shared by 4 people. Interesting to me is that the most common answers for (1) and (3) are 4.5 and 4½ (or 4 and 2/4 b/c ¼ of a cruller and ¼ of an apple fritter is not the same as ½ of either). Missing from these three problems is the context where the remainder is given away, such as your balloon context. (I had Marc’s lesson in mind when I read your “beans are like balloons” comment which had me thinking, “I dunno… you can have a third of a bean.”)
In my curriculum document, this – interpreting remainders to solve problems – is a Grade 5 learning outcome. Remainders are introduced in Grade 4; it is not intended that they be expressed as fractions or decimals. So, in Grade 3, the focus is on solving division problems without remainders, which is how I think I’d follow up this lesson. This seems like a reasonable progression across the grades.
“Sometimes I think it’s harder when kids are used to problems where everything works out evenly, and then it suddenly doesn’t” was my argument for including remainders. That, and I know my own kids can handle it. You’re right: keeping the R notation at bay, or having students name the extras (in conversation, they were calling them “leftovers”) would be a better way to go.
You may be interested in One Hundred Hungry Ants by Elinor Pinczes. Rather than single file, 100 ants march in 2, 4, 5, and 10 rows. (Arrays!) A Remainder of One by the same author introduces remainders: 25 beetles in 2, 3, 4, and, finally, 5 rows.