- using counters/objects
- drawing pictures
- using ten-frames
- using a 100-dot array
- using a 100 chart
- using a blank number line
- breaking numbers apart
- using operations such as adding and subtracting
- looking for patterns
- using charts or tallies
I posed this question to the students:
Luke is buying candy bars to share with his classes. They come in packages of 6. How many packages will he need to buy if there are 25 students in his class?
The students glued in their question strips and we read the question together to make sure they understood what was being asked.
I will stop here and give my opinion on an issue. I know there are a lot of math text books out there with lots of writing in them, and teachers have told me that some students who are not good readers, but are better with numbers, have little success using books that require a great deal of reading. I like using a problem-based approach to mathematics, and find that, when I am focusing the lesson primarily on a single “rich” question, then I can read the question with the students, make sure students understand what is being asked, and set the students to work. Although I am a great proponent of literacy and want students to be accomplished in that area, I do not want reading to hold a student back in my math classroom. A rich question, again in my opinion, is one with multiple possible solutions OR a single solution with multiple strategies for finding the solution (or both!). I use many of the latter, and encourage students to find as many solutions as they can. In many cases, the more ways they can solve the problem, the greater their understanding of the concept.
So, back to this particular question. The students went to work solving the problem of candy bars bought in packages of 6. The photo shows some of the strategies shared in the one class. The discussion was quite interesting. The students realized that buying four packages of candy bars would get Luke 24 bars, but most did not want him to purchase another full package. They were suggesting it would be better if he then went to a convenience store and bought only one more bar, which, in real life, is a great idea!. I asked the students, what if Luke were in a hurry and had to buy only packages, and they all agreed that he would need to buy 5 packages to have enough. There was then a discussion around the extra bars: he could save them; he could sell them; he could give them to his family. Lots of good ideas!
It is important to have those kinds of discussion around division and any remainders that come up, because in real life things are much more likely not to divide evenly than to do so. The remainder must be considered carefully. If students were doing this in a “standard” way, they might be likely to say that 25 divided by 6 is “4 remainder 1” without ever considering what the remainder of 1 would stand for. In this case, it would be a student without a candy bar! The actual answer to this question is 5, not 4 remainder 1. This is a case of the answer being forced up to the next whole number.
Remember, have your student discuss the remainders!