In the Making Number Talks Matter chapter on Division, one of the strategies described was “Make a Tower.”
This strategy immediately reminded me of the 4th grade lesson in Investigations called just that, Multiple Towers. To make the timing even better, I am planning and teaching that lesson with a 4th grade teacher this week. The goal of this lesson is not necessarily eliciting a division strategy, but there is so much possibility for so many things in this lesson. Because of that, however, I am struggling with how we can leave it open enough for the many ideas to emerge, but focused enough to have a purpose in our planning.
The structure of the lesson, as it is the book, looks roughly like this:
Students count around the classroom by 3’s to 72 and it is recorded on the board. They then count by 30’s to 720 and it is recorded on the other side of the board, so both lists are visible. Asking students what they notice and what relationships they see. We record these noticings and leave this because the investigation of multiplying by a multiple of 10 continues to emerge throughout the unit.
Next, as a class, we introduce multiple towers through a context of stacking boxes of 30 oranges. Using post-its on the wall, students place the post-its one above the other, labeling the amount of oranges as the tower grows until it reaches the shoulder height of one student. Â We then ask if there is anything in the tower that would help us figure out how many oranges are there without counting each one? They estimate how many to the top of the student’s head, ask how many multiples are there, and record equations for it.
Choosing from the list below, students work with a partner to create their own multiple towers as tall as one of them on a strip of paper:
After creating their tower, they answer questions in their activity book about the number of multiples, how they could find the 20th multiple without counting, and writing equations for some pieces of their work. They will continue to reference these towers in the following lessons in which they multiply 2-digit numbers and discuss various strategies and representations for multiplication.
I love this lesson, but as always, I like to play around with different ideas when I am lesson planning and these are some things I cannot wait to chat with Malorie, the 4th grade teacher, about on Monday:
- Instead of a number talk do we open with the count around the classroom to save time for the multiple towers?
- How do we record the multiples? Will different recordings draw out different noticings?
- Do we start with estimation of how many cases of oranges will it be before we just start the tower? Could that draw out their use of multiplication combinations they know?
- Do narrow the list of starting numbers to choose from so there are some really great relationships that emerge, like one factor that is 1/2 of another so the towers have a relationship? or a factor that is 1/3? or do we let them choose their own off of that list within a range?
- Do we put the towers up, walk around and do a notice/wonder before they jump into the activity book pages?
- Do we want to end with a journal prompt? The answer is obviously yes, but what do we want it to be?
- How will this leave her for the next day’s lesson?
Feel free to chime in with thoughts/suggestions and I will post the plan we decide upon after our planning on Monday!
~Kristin
MathMinds 
This sounds fun too! I don’t teach fourth so take from this what you will. I think it would be good to start with either a number talk or estimation task that would support their thinking in the lesson. This could be especially helpful for those that may struggle. Then, if they do struggle in the lesson you could ask questions like, “How could what we did in number talks help you with this task? You made an estimation earlier. How did you arrive at ___? How could you prove that your estimation is correct or close?” I look forward to reading what you and Mallory decide!
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I really love that idea Jamie and completely planned on doing that today, however due to an assembly today the schedule was a mess and the class was cut short and we didn’t get to meet beforehand. Luckily, Malorie had planned and read this post too so we were able to adapt but not add in a number talk or estimation. Working on follow up post now…
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I love your idea of a. narrowing down the list of starting numbers and b. having the list numbers share a fractional relationship to other starting numbers. It can be hard to see relationships between numbers and any amount of scaffolding will support all student learning. If you chose to do this let me know how it turns out. Good Luck!
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Just a thought, but you could have different groups count up in different multiples (e.g. 20s, 30s, 40s, etc.) as this gives those slower on the uptake more references (for example the picture in the book is 40s, which might be a big step if they’ve only seen counting in 30s). This also gives the option in the next lesson of linking, with string, common numbers between stacks, e.g. 20s and 30s link at 60, 120, etc.
Count around could come towards the end – at the beginning, if they don’t get it immediately, there’s going to be some intimidating pauses as someone tries to count from 450 to 480 on their fingers while everyone stares at them.
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I completely agree about the pausing on the count around, so we did a choral count where everyone said the answer in unison and we recorded it on the board for them. The volume definitely went down on the 39 to 42 jumps over a ten, but it definitely eliminated putting students on the spot! Thanks so much for commenting!
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There is a lot for the learner to engage with here. It looks something like the “chunking” method that was prescribed in the UK some years ago and which I had great concerns over. What feels slightly different is how the learner needs to recognise, in the example provided, how 13 X 4 = 52 is connected to 531. However if the original calculation had been, say, 513 divide by 13 the method might be less easy to use. This would require the learner to have to fiddle around either with finding the difference between 390 and 513 or go backwards from 520 to 513… I love the rhetorical question in the sixth bullet point about journal writing.
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Yes, I really like this building into the the partial products and partial quotients in a pattern and repeated reasoning sense! There is definitely so much here to work with! The journal we didn’t get to today because of a crazy schedule change…blogging about the follow up now!
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