Thank you to all that participated in the Pre-Conference at COABE for the Adult Numeracy Network (ANN). Attendees were given a call to action to continue the discussion about what you learned, experienced, or took back to the classroom here. Let's continue to learn from each other.
A few of us from the Adult Numeracy Network would like to share the activities we explored in our annual meeting during the pre-conference at COABE in Phoenix. Our goal for the day was to use a set of activities that help us improve our math content knowledge as teachers, so that we would be better prepared to help our students. We tried to organize our sessions so that they started with concrete, fundamental concepts and moved towards more abstract understandings. Over the course of the day, five of us (Brooke Istas, Sarah Lonberg-Lew, Stephanie Stewart-Reese, Heidi Schuler-Jones and I) co-presented three sessions. We would like to share some of what we did that day in the hopes that we can continue interesting conversations and think about how to bring some of this to our classrooms.
Brooke started us off with the following image. She asked a couple questions to get us started:
1. Write the number above each representation.
2. What do you see?
3. Use colors to show patterns.
So, I would ask the same questions here. Maybe take a few minutes to look over the visualizations, print it out if possible, use some markers to find patterns, and then share what you found here.
I'm so excited to see this here since I missed the morning session of the pre-conference. My schedule is full today so, sadly, I can't grab my colored pencils and get to work right now, but I'll be joining in this conversation as soon as I can. One thing I did notice, before I've put pencil to paper is that prime numbers seem to be in a circle of circles.
Question: after I've worked with it, can I share it in this discussion as an image. I remember having trouble with that in the past.
Unfortunately, there isn't an easy way to share photos through LINCS, but I can provide a work-around for this conversation. If you email me (and this goes for anyone here) a photo of your work, I'll upload it to the CollectEdNY post about number visuals, then send you a link you can use to insert an image in your post here on LINCS. It's a bit of a long way around, but that's how I shared the original image with you.
Email me separately if you're interested: email@example.com
I finally got around to finishing up coloring and I noticed that if you multiply a prime number by two you get an even number, so I wondered what would happen if you multiplied a prime number by an odd number and my findings thus far, indicate you'd always come up with an odd number. In addition, I'm thinking that five is a special prime number because it is a factor of 10, which is the base number for our number system. Lots of other takeaways, in terms of patterns, but those are my main takeaways reflected in my coloring. https://twitter.com/PatriciaHelmuth/status/995304894646116352 Looking forward to a numeracy adventure in my classroom with this activity.
If the prime number you're multiplying by is two, you'll get an even number.
Other discoveries that can be helpful are ... okay, if 8 is a factor, then all the prime factors of 8 are factors of that number.
Our Transitions math (pre-pre-algebra) includes teaching reducing fractions by breaking into prime factors because enough students make discoveries like that...
So this was pretty cool. I started out just counting circles and quickly realized that the number of circles in each figure was counting up. At that point I thought "why do the shapes change that way as circles are added?" I then started to look for groups, so nine was three groups of three and twelve was three groups of four. Then I noticed that some of the figures weren't broken into groups and I thought to myself "why would some of these just be circles" and I looked closer. It turns out that all of the circles-of-circles are prime numbers (except two and three don't look like circles, and four could be a circle even though it isn't prime). The rest looked like some form of prime factorization, but visual.
I was then distracted for a little bit...
When I came back I thought I would look at some of the numbers which had more than two prime factors to check for something. Was there an order to which factor was the primary shape that got repeated? In other words, would it be two groups of two groups of three to make twelve, or would it be three groups of two groups of two?
Were the creators of these figures consistent in how they ordered these levels of groupings? In other words, could I predict exactly how they would organize a number like 42 (2*3*7 or 7*3*2 or 3*2*7)? I'm going to stop there and see what others think about those questions.
I notice that there are 3 groups of 6 for 18... but 5 groups of 4 for 20. That tells me they don't 'always do the smaller number of groups w/ the bigger number in the group.' I'll bet there's a system, though.
We spent some time on this question during the pre-conference as well, and didn't get to an answer. We did notice that there seems to be a desire to show groups of 3. We noticed that 6, 12, 18, 24 are arranged in groups of 3, though 30 is not (at least at the highest level). There are groups of 3 in the visualization of 30, but the total shape isn't 3-pointed.
Here's an extension question we considered: Draw a number visualization for 36.
If anyone wants to share what they drew, email me and I'll give you a link to post the image: firstname.lastname@example.org
So I think that the smallest factors are used at the "smallest" level. Every even number has groups of two collected into other groups. The largest factor determines the biggest grouping. That's why the overall shapes are three-armed when 3 is the largest prime factor, five-armed when five is the largest prime factor, seven-armed when seven is the largest prime factor, and so on.
Therefore, the following image is how I think 36 should be displayed. It is groups of two, grouped into twos, grouped into threes, grouped into threes (2*2*3*3).
I'm torn. I feel like I want to see that 6/dowsing rod shape. Here's what I drew (and colored). https://twitter.com/mtrushkowsky/status/992421506499129345
I'm fascinated by your way of seeing these images. So, you're saying a number like 30 would be organized at smallest level in groups of 2, which are then organized into groups of 3, which are then organized into groups of 5. That corresponds to what I see in the image.
And 42 (which is 2 x 3 x 7) would be drawn something like this?
That's exactly how I saw them after studying and looking for a pattern for a while.
I take back what I said! I am completely convinced that you have unlocked the underlying structure to these.
Is that what you think 60 would look like? - https://twitter.com/mtrushkowsky/status/997495498427822081
Apologies for the delay! Your 60 looks right. Two circles together, which are in groups of two, and those groups are in groups of three, of which there are five. 2x2x3x5
I think if we only focus on groupings that are prime factors, the pattern is consistent. For example, there are not groups of 4, instead there are two groups of two. My brain wants to see 4, but I think the system is more consistent when only prime groups are used. What are your thoughts?
I'm wondering when are spokes used, and when are clumps used? When four is a factor, there are clummps of four... but twos tend to be arranged as spokes...
I like how you described the difference there. So when two is only a factor once, there are spokes, but when two is a factor twice there are clumps of two twos. When a third two is a factor, it becomes spokes made of clumps (check out 24 for the only example, but I bet 40 would be five spokes each made up of a pair of clumps).
I noticed that each of the rows has seven numbers in it, which means that each row ends with a multiple of seven. All these multiples are arranged around the seven circle. (So, seven groups.) I also noticed that all the multiples of four have groups of four in the square shape (rather than having four groups). I thought I saw a pattern with the threes that altenated between groups of three (for odd numbers) and three groups (for even numbers), but it broke down.
There does seem to be preference for making visible as many factors as possible... Has anyone used this with students yet? What did they see?
I am also curious about how students deal with this. I might try it out, but won't be able to for a few weeks. What other sorts of things have teachers seen?
Dear Heroes of Adult Numeracy!
I've done some work with this visual before, but this time I tried to only focus on the patterns I saw in the shapes and then I color accordingly. It was cool to do that and then go back and think about what I know about the numbers and why different color combinations may have shown up.
Here's how it came out: https://twitter.com/mtrushkowsky/status/992417293727657984
I also wanted to share another visual representation of numbers using circles that I love to look at from Math Pickle - https://www.youtube.com/watch?v=-SHZ3pdjaMw
yours in productive struggle,
P.S. A big thank you to all the Adult Numeracy Network presenters from the ANN COABE pre-conference session who are keeping the conversations and explorations going!
Thank you for the wonderful noticings and wonderings related to the Jo Boaler number visuals. I wanted to continue sharing some of the activities we explored in the ANN pre-conference at COABE. The 2nd activity we did was related to number tiles. It went something like this...
Take a look at these tiles (print and cut your own, if you like). Spread them out and look at them with a partner. What do you notice?
After you've shared a few things you notice, try counting to 30 using these tiles.
We'll skip 1 for now.
Is there a 2 tile? Yes.
Is there a 3 tile? Yes.
Is there a 4 tile? No. Using multiplication, is there a way to get 4? Yes. 2 x 2.
Is there a 5 tile? Yes.
Is there a 6 tile? No. Using multiplication, I can make 6 with 2 x 3.
Can you continue counting up to 30 in this way? If you have the tile, use it. If not, see if you can build the number with other tiles and multiplication. You can use this worksheet if you like.
How high can you count? Are there any numbers you can't make? Is there more than one way to make any numbers?
This CollectEdNY page has the tiles and some teaching suggestions. I'll check back soon with some suggestions.
Still sharing stuff from the ANN COABE preconference...
After exploring how to find the factors of a number using factor tiles, Sarah Lonberg-Lew presented a problem. Sarah's version led a group of volunteers through a process of standing and sitting: <Last Mathematician Standing>. You might also consider a similar situation with this video of dealing playing cards: <Face Up, Face Down>
What questions could you explore here?
Hey there, math gurus!
I wanted to pick up where Eric left off with the ANN COABE Pre-conference conversation by sharing some of what we did in Session 3 following Sarah Lonberg-Lew’s “Last Mathematician Standing” activity.
Brooke Istas, Stephanie Stewart-Reese, and I started the afternoon with a partnered activity by Dr. Jo Boaler called *Cover the Field. This activity builds on prior knowledge about arrays (or area models) and applies understanding of composing and decomposing numbers to focus on binomials with numbers.
You’ll need 4 six-sided dice, a piece of grid paper per person for your “field”, a sheet of blank paper per person to record your dimensions, and something with which to write. If you don’t have 4 dice, you can use this random number generator.
Here’s how to get started:
2. Use values shown on the four dice to come up with the length and width of the rectangle you want to make to cover the field, i.e. your entire sheet of grid paper. You must choose how to make two pairs of dice add (or subtract) together to become each side of the rectangle.
- For example, if you roll 6, 3, 2, and 1, you might choose to add 2 + 6 to get a side of 8 and subtract 3 - 1 to get a side of 2. Your rectangle would have dimensions of 8 x 2.
- Think about the rectangles you can make and which would be the most useful in covering the field. Explore what would happen with the same numbers arranged in a different way with all adding, all subtracting, or a mixture of adding and subtracting.
3. Once you decide on your rectangle, draw it on your field in any place you want. You may not overlap with any existing rectangle. You may not break up your rectangle into smaller pieces. Record your equation on a piece of paper to keep track of your work. Using the example above, you would record (2 + 6) x (3 – 1) = 8 x 2 = 16.
4. Play continues until you roll the dice and cannot make a rectangle that fits on your field. The winner is the person (or team) who covers the most area of the field.
Who wants to play? Try it with a partner or on your own and post your results in this forum. Once you’ve had a chance to explore the activity a bit, share the kinds of questions you might ask students or fellow math practitioners playing this game.
Heidi*Note: The entire lesson plan for this activity (Big Idea 7) can be found in Mindset Mathematics: Visualizing and Investigating Big Ideas, Grade 4.
I like this a lot because it seems to me to have solid connectiosn with different topics and concepts and visual-concrete and symbols.