This week, I am starting our next unit in Investigations, Prisms and Pyramids: 3D Measurement and Geometry. Every time I start a new unit, I always like to read the Teacher Notes in the back of the teacher’s book because Investigations has done such a phenomenal job explaining students’ work prior to that unit, the big mathematical ideas, student misconceptions (with dialogue boxes), and offering examples of student work for both the activities and assessments. It is an invaluable resource that often gets overlooked. After reading the teacher notes, I typically like to look at the Ten Minute Math Activities, build my Number Talks, and then dive into planning out the lesson activities. I was going to devote this entire post to thinking about this unit on Volume, however I tweeted out suggestions for a blog topic on this flight and Michael gave me this:
I feel like I just finished up such an exciting (in my mind at least 🙂 blog post about student moments with multiplication of fractions that I wanted to do something a bit different and this tweet gave me an idea. Since my students have done such amazing work with fractions, I started thinking about how I could embed that into our volume unit. So, after reading the Teacher Notes, instead of jumping into the number talks right away, I began reflecting on those great fraction moments and putting them in the context of volume.
Our first activity in the unit is building boxes, estimating how many cubes will fill the box and checking that estimate by physically filling the boxes with cubes. We then share and discuss strategies for finding the number of cubes as a class. I find every year, I have over 1/2 the class who can see the layering and arrive at an efficient strategy including length x width x height fairly early in the lesson, while others cannot visualize how to organize the cubes. They can easily find the number of cubes in a rectangular prism when the box is constructed and filled, however moving to a net or just dimensions is a very tough abstraction.
I think one of the hardest aspects of teaching is meeting every student’s needs on a daily basis. This is one of those lessons in which I feel I struggle a lot. It is beyond the numbers and context and more about the students who have the spatial reasoning to abstract volume vs those who need to always build the box to arrive at an answer. For those who need concrete models, I feel comfortable meeting their needs through using cubes and building on that concrete work to form generalizations for finding volume. However, what can I do for those who just “see it” and building those boxes is unnecessary and a tedious waste of time? They are done the lesson in like 10 minutes and are waiting to share while others build. Bigger numbers won’t challenge them, they can prove it always works, doubling/tripling volume is a class lesson after that one…..hmmmm….
This is where Michael’s tweet made me think about our fraction work. I have never done my fraction unit before volume unit before. Typically we do volume to start the year and then move into fractions. I think it is so interesting to think about those students who can reason about fraction multiplication and how they would think about a volume problem in which there are fractional dimensions. What does the picture look like? How can you have a 1/2 in x 1/2 in x 1/2 in and end up with 1/8 cubic inch? How do you end up with a smaller fraction than all of your dimensions? That is a tough concept to grasp and I THINK THAT WOULD BE SO COOL for them to think about!
Now the even harder part comes into play, planning to launch that with them. Questions I have to think about (and any/all thoughts are always welcome):
– Do I build on the problem with whole numbers (2 x 3 x 4) and ask, what would happen if that was 4 1/2 tall instead of 4? Would it increase the volume? How do you know?
– Does it make it harder or easier to manipulate a certain dimension, i.e. length, width, height?
– Do mixed numbers feel like a good starting point vs fractions less than 1?
– What will their recording look like? What are good questions to ask them when they have found the answer?
– Is there a better volume model aside from cubes (which you cannot break in fractional parts to prove your point)?
….and there are probably a million more, but the flight is about over….so much planning left to do. Not to mention the actual typing of my lesson plans and 9 million other things I should be getting done while I type this blog! I just love to write about this stuff…cannot help myself! I apologize for any typos, no time to re-read before saving and putting the laptop away!
Check out this post from Mike which incorporates decimals and volume. http://mikewiernicki.com/penny-cube/ I love this task because student curiosity leads to volume and they don’t ever need the formula. Mike taught it a couple of weeks ago to a 5th grade class and blogged about it here. http://mikewiernicki.com/2014/09/19/the-penny-cube/ He provided his students such a rich environment for learning. Great stuff that might help you out a bit.
I have done this task with students as well, and the best part is after the figure out how many pennies are in a 6inch x 6inch x 6inch cubic foot ask them “how many pennies would fit into a cubic foot?” They just want to double it because 6 inches is half of a foot. Obviously this is not right but it identifies a huge misconception and opens up the conversation about doubling all 3 dimensions.
Or you could use (shameless plug) http://gfletchy.com/got-cubes/ which incorporates the fractional reasoning you were developing in your previous units. Can’t wait ti hear about how you reflect looking back at the unit after you have taught it. What you would keep, change, etc…Thanks for sharing as always!
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