Prime Climb Number String

Last month, I worked in Andrew Eller’s amazing 4th grade classroom in Los Angeles that was beginning an exploration of factors and multiples using the game Prime Climb. If you have never played the game, PLEASE take a minute and think about their design. What do you notice and wonder about this chart? Can you predict what 30 would look like? 21? 60?

an image of the numbers 1-20, 1 is grey, 2 is orange, 3 is green, 4 is two orange segments, 5 is blue, 6 is one orange and one green segment, and so on.
Image of the numbers 1 to 20

On the previous day, Mr. Eller led a notice and wonder on the first 20 numbers, then 100, and then the kids had time to explore the charts on their own. Some of the kids were using the colors/prime factors right away, others were not. What I realized was that these 4th graders had a lot of experience with finding factors, but only two factors. Like factor pairs. Thinking that the factors of 16 are 2, 4, 8, and 16, because 1 x 16 = 2 x 8 = 4 x 4 = 16. They didn’t have a lot of experience thinking about 16 = 2 x 2 x 2 x 2. Which would make the colored circles harder to use when playing the game.

And those circles are the key to mathematizing the game, I think. In Realistic Mathematics Education (RME), we think about how particular mathematical models are worth investing in, because they shape how kids think about mathematics. The number line, for example, is not just a visual, but a tool that offers certain affordances in HOW you think about operations (and pretty much all of mathematics). But I could write an entire post about the difficulties of representing the associative property with number lines or arrays. It gets a bit convoluted . . .

That is where I think the colored circles have a really interesting and concrete role to play as a possible model for the deep structure of numbers as factors. To think of 60 not as the pairs of numbers that multiply, but the core factors, the prime factors, that is something that is challenging to represent in a way that it becomes intuitive for kids. But that is the genius of Prime Climb. 

I wanted the number string to call attention to how the prime factors were related, so that kids could see the connections between related multiples. A number string is a beautiful way to call attention to mathematical patterns, since we do one problem at a time, and we focus on developing our collective noticing of emerging patterns.

I decided to focus on multiples of 5, since that would allow everyone to engage in thinking about the model, while feeling comfortable with the facts. We start pretty simple, with the basic- how do these colors work-problems. I noticed that there were a few kids who did not leave class the day before with that understanding- they just didn’t use the colors at all. I thought this might make it more explicit for those kids. The string design focuses on kids not just multiplying, but visualizing and justifying the prime factors of the number, so I asked, “what did you get, and what colors will the circle be? Why?” I did not show the “answer” until kids have described it and justified why it would be any particular colors, and how many segments it would have.

Before I move on, here’s what made me uncomfortable with this format. I really wanted to do a string with the circles, not just with the numbers, because I wanted the kids to start thinking WITH the factor/circles. But the best way I thought to do that was to make a powerpoint, so I could reveal the “answer.” This was an awkward part of this number string- normally I represent multiple student answers, and here there is just one? Not sure how to resolve that . . .It seemed to work well in this group, since our discussion was mostly about strategies, not answers. But I wonder if others have thoughts. Is this format TOO constrained? Am I pushing this model too hard?

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Next we get to problems with more than two factors, which led us to equivalence, and the associative property.

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With problems with more than two factors, I explored if kids rearranged the factors to multiply, like if they decided to do 2 x 3, then 6 x 5, or did 2 x 5, then multiplied by 3.  When I taught this, I didn’t represent those, but in retrospect, I would note the various equations on the side of the board.  4 x 5 was a nice place for equivalence with factors, as kids noted, “That is the SAME ONE you just gave us!”

On the second slide, I went for another equivalence example, this time with 2 x 3 x 5 and 6 x 5. There was some great conversation on the relationship between the two.

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The problems end with more equivalence, but here I want to get kids to use what they noticed about multiplication from the previous problems to make a challenging problem simpler. 

I recommend a turn and talk for the second to last problem, which is the stumper (last problem of a string, challenging, and has multiple ways to solve) here. Our discussion was around how kids dealt with multiple factors, some rearranging (using the associative property).

And then I asked what they think the last problem might be . . . 

A good final question might be, how many ways can we make 60? 

I have some ideas for other number strings that would build off the game: division could be very interesting, and perhaps repeating 2s for students to think about what patterns emerge when we multiply 2 x 2 x 2 x 2 x 2 . . . What else?

Here is the Google Slide of the string with animations.

Feet, inches and yards: Conversions on a ratio table

This post is from our friend and colleague Kathy Minas, a 4th grade teacher at PS 158, and an avid strings enthusiast.


Recently, our fourth grade team met with Math in the City co-director and staff developer Kara Imm, who has been working with our school for several years. We wanted to explore ways to introduce Common Core 4th grade standards of measurement and conversion by using strings. What follows is the string we designed together and my notes about how to lead it with kids.

Why this string? Why a ratio table?

The purpose of this string is to introduce students to conversions, using the most common units of measurement, inches and feet. We wanted to ensure that students had a familiar model that visually captured the (twelve-to-one) relationship between these units, which led us to using the ratio table. Our students have worked with the ratio table before in both our multiplication and division units of study. The model allowed them to represent and maintain the relationship between known units in order to multiply, divide, break apart, or even add groups with ease.

Performing measurement conversions on a ratio table also supported students to monitor the reasonableness of their answers. In addition, it encouraged them to keep the relationship between units in mind.  Instead of memorizing whether they needed to multiply or divide feet to get inches, they simply trusted the existing relationships on the table that we build together.  In fact, student were much more flexible about how to convert and did not rely on a memorized rule or catchy mnemonic to solve these problems.

Tips for leading the string:

We build ratio table together with the class, instead of revealing the entire completed ratio table all at once. We add values to the ratio table — one at a time, increasingly more complex — and ask students to determine the corresponding number of inches or feet. Some values are added to the ratio table as students explain their process.

The string:

Begin by naming a true statement to ground the conversation:

Mathematicians, we know that there are 12 inches in one foot.

Draw a ratio table and label the columns, number of inches and number of feet. If this piece of social knowledge is not known by your students, having a 12-inch ruler to see and touch is also useful at this moment.

Since we know that there are 12 inches in one foot, how many inches are there in 3 feet? How do you know?

Student responses may include:

  • I know that there are 12 inches in one foot, so there are 36 inches in 3 feet because you multiply the number of feet by three, which means you have to multiply the number of inches by three.

Note: In order to move away from additive reasoning on the ratio table  also known as repeated addition or “chunking” I purposely do not record the unit rate of  12 inches in 1 foot on the table until the students bring it up as part of their reasoning. I’m nudging students from additive to multiplicative reasoning on the ratio table. You may wish to ask students to visualize three feet, “What does this look like? What does this make you think of?”

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A fourth grader (and sports enthusiast) in my class introduced the unit of a yard immediately, so I added it as a third column to the ratio table and made a post it note 20151218_160434about his unique contribution. I left the yards column blank for most of the string, but then we returned to it later to reason about this third unit.

Mathematicians, how many feet are there in 60 inches? How do you know?

Student responses might sound like:

  • I know that there are 12 inches in 1 foot, 60 inches is 5 times greater than 12 inches, so 60 inches are equivalent to 5 feet.
  • I know that 36 inches is equivalent to 3 feet, and 60 inches is 24 more inches than 36. 24 inches is equivalent to 2 feet, so I need to add 2 feet to 3 feet and that’s 5 feet.

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So, how many feet are there in 72 inches? How do you know?

Students will likely reason:

  • I know that 72 inches is 12 more inches than 60 inches, which means I need to add one more foot to 5 feet, which is 6 feet.
  • I know that 72 inches is equal to 36 inches times 2, if I double the 36 inches, I have to double the 3 feet.

Follow up question, “How many yards are there in 72 inches or in 6 feet? How do you know?”  

Okay, how many inches are there in 9 feet? Can you picture this?  What are you thinking?  

I heard students say:

  • I know that 9 feet is three times 3 feet, so there are 108 inches in 9 feet because I have to triple 36 inches.
  • I know that 6 feet is 3 feet away from 9 feet, so I need to add 36 more inches to 72 inches, which is 108.

Follow up question, “How many yards are there in 72 inches or in 6 feet? How do you know?”  

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Once we were done with the inches and feet component of the string, we tackled the yards. We returned to the relationship between feet and yards. I asked students to consider,

If we know that there are 3 feet in 1 yard and we only have 1 foot, what part of a whole yard do we have?

Then I asked,

If we have 2 feet, do we have a whole yard yet? What part of a whole yard do we have?

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Later I said,

Using all you know about the relationship between feet and yards and all you know about fractions, if we have 5 feet, how many yards do we have?

My students tackled this problem with ease:

  • I know that there is 1/3 yard in 1 foot, so there are 5/3 yards in 5 feet.
  • So, there is 1 yard in 3 feet and 2/3 yard in 2 feet, which means that in 5 feet there is 1 2/3 yards. I just put them together to make 5.

 

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Poster from Kathy’s class, end of string

As big ideas or important strategies come up, my colleagues and I have begun annotating the strings poster so that kids can both see and hear these ideas.  I listen carefully for students to make these contributions we write and display them so that the ideas are shared and accessible to all kids, even if they are still emergent.  Sometimes, when students are ready, I nudge kids towards a generalization, which helps us move beyond the specific string and into other related quantities and relationships. Examples of this practice of annotation are below:

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Chocolate Arrays

Lately, I’ve been seeing arrays everywhere I go: at the grocery store, at the pharmacy, at the farmers’ market. And, of course, at Costco. The big, bad bulk retailer is bursting with interesting items arranged in perfect columns and rows.

Naturally, I made a beeline for the chocolate.

What follows is a quick-image string for exploring the associative property, the patterns that occur when multiplying, and the relationship between columns and rows. It supports the development of some key strategies for multiplying: doubling and halving to maintain equivalence, doubling a dimension to double to product, and using partial products to solve.

Continue reading “Chocolate Arrays”

The delight of disequilibrium

Disequilibrium is Piaget’s term to describe when what a learner already knows comes into conflict with new information. Learners must work through the confusion to reconstruct new knowledge. How does the process feel to a learner?  How as a teacher can we respond during a number string when students demonstrate disequilibrium?

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“Stack of Bills” string

Here’s a string I designed with a team of fifth grade teachers who were looking for creative ways to encourage a multiplicative understanding of place value.  When students are reasoning additively about number, they might see 321 as equivalent to (3 x 100) + (2 x 10) + (1 x 1).  And this is very common in classrooms, and is often confused as thinking multiplicatively because it does involve some multiplication. But in essence you are splitting the number into place value parts and adding those together. When students are able to reason multiplicatively about number it looks more like this: 321 = 32.1 x 10 or 3210 ÷ 10.

Continue reading ““Stack of Bills” string”