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.

Biking on the fractional number line . . .

The following post comes from Rafael Quintanilla, a Mathematics Instructional Coach for the Los Angeles Unified School District and a long-time teacher of number strings.

About eight years ago, I was teaching fifth grade and I wanted students to begin to see relationships between halves, fourths and eighths, be able to decompose fractions into unit fractions, and have an understanding of scaling when multiplying.  So I decided to write and try out some strings that focused on benchmark fractions as helpers and then go from there.  I used a number line as a model because that is what we tended to use as a fraction model because the students were able to connect it to real life applications.  I decided to use the context of a bicycle ride because students can relate to to riding bicycles and because they can picture a straight bicycle ride (I let them know I ride my bicycle on the riverbed next to my house), hence the number line.

Students are really engaged in this string because as the string develops, I tell a story of why we stopped at certain places. Some of the “look-fors” in the string is that students might  just give you numbers instead of saying the numbers represent miles on a bicycle ride.  Be careful because students will see the numeric patterns that exist but not make connections that these numbers represent fractional distances on the ride.

Multiplying Fractions by a Whole Number

Model: Number Line

Context: On weekends, my friend and I enjoy riding our bicycles. Our goal is 24 miles. But sometimes, we stop for breaks.

Draw number and teacher labels 0 miles on left side and 24 miles on right side

String:

1/2 of 24 (“tell the story, we stopped half way through the bike ride to get ice cream”)

(take one or two strategies, helper problem)

 1/4 of 24 (“Oh, I forgot, we also stopped one fourth of the way to drink some water”)

(take one or two strategies, helper problem)

3/4 of 24

(this is where the string begins to develop, allow more time, ask students to think-pair-share and then model their thinking whole class)

1/8 of 24 

(take one or two strategies, helper problem)

3/8 of 24

(this is where the string continues to develop, allow more time, ask students to think-pair-share and then model their thinking whole class)

7/8 of 24 

(this is where the string continues to develop, allow more time, ask students to think-pair-share and then model their thinking whole class)

5/8 of 24

(usually comes up during other problems or else you can use it as your last problem)

Two notes:

  1. Keep in mind that the goal of the string is to multiply fractions by a whole number. Therefore, the teacher puts the tick mark on the number and labels the fraction.  The students are computing the number of miles and developing strategies.
  2. The goal of the lessons is to allow students to discuss the relationship between the fraction and the whole. The goal is not to determine where the fraction is located on a line. That is why the teacher marks the location. Also do not worry about rushing into writing the expression or equation.

 

 

 

30 ways to make number strings more inclusive

Image of teacher-made classroom sign which says Alisha's strategy for multiplication, break up the numbers, 8 times 12, 8 times 10 equals 80, 8 times 2 equals 16, 80 plus 16 equals 96, then shows an array model of that equation.

As a teacher in inclusive settings, number strings were a critical part of my daily mathematical work.  Routines are highly effective in inclusive classrooms, particularly routines like number strings which externalize complex cognitive processes.  By that delicious turn of phrase, I mean that a number string is not the kind of routine that teaches low-level thinking like memorization.  Instead, it allows kids to participate in the strategic thinking of other kids, giving them access to complex processes that too often only go on in individual minds. Continue reading “30 ways to make number strings more inclusive”

A Number String for Angle Measures — Before We Kick the Bucket

This post was co-created by Jesse Burkett, Ranona Bowers, and Adrian Sperduto — teachers in the Hazelwood School District and their colleague Cheryl Montgomery of the Parkway School District (both in St. Louis, MO). They were recently selected as participants in the Mathematicians in Residence program — a three year, three district project involving almost 100 teachers and eventually a summer math academy for approximately 200 students. Jesse, Ranona, Cheryl and Adrian just completed a two-week professional development academy, focused on numeracy routines in grades K – 5.  Jesse Burkett, a 4th grade teacher at Brown Elementary School, led the writing for his colleagues. Continue reading “A Number String for Angle Measures — Before We Kick the Bucket”

What’s in a name?

This post is by reader and strings enthusiast Laura Bofferding, Assistant Professor, Purdue University

I was first introduced to number strings by Jennifer DiBrienza (one of the teachers highlighted in Young Mathematicians at Work: Constructing Number Sense, Addition and Subtraction by Fosnot and Dolk) when we both worked as teaching assistants for an elementary mathematics methods course. Captivated by their complexity and ability to hook students of all ages, I began to use number strings myself with teacher candidates. Now, as an assistant professor at Purdue University, I routinely explore them with my undergraduate mathematics methods students and require them to try a number string in their practicum classrooms. As I have looked for resources and talked with colleagues about this practice over the past few years, I’ve noticed an increased focus on mathematics instructional routines that people refer to as math talks, number talks, number strings, math strings, cluster problems, and problem strings. Some of these things are not like the others…

Continue reading “What’s in a name?”

Division – Whole number by fraction

During the spring of the 2012-2013 school year Kara Imm and I were working with my 5th graders to help them visualize what it meant to divide a whole number by a rational number. The students were very quick to invert the fraction and multiply. They loved saying, “Flip and multiply.” Mind you, I had never uttered those words in the classroom. However, this class was very used to string work and representing their work through models.

Continue reading “Division – Whole number by fraction”

The Power of Strings

I’m embarrassed to say this, I have a vivid memory of around my fourth year of teaching 5th grade and my second year of using number strings during a staff development meeting with a very patient and kind staff developer, “I’m not sure I understand why I am teaching strings. I do the string, the kids do it, we discuss it, there’s a chart up and then poof it disappears and I see no transfer.” It took me a while to understand the purpose and powerfulness of strings. Here are some of my initial questions about strings and my responses to those thoughts after much practice with strings. Continue reading “The Power of Strings”

Closed to Open Array

After looking over beginning-of-the-year assessments, a new 4th grade teacher was concerned that half of her students were still unsure of the open array model. Some were simply still not convinced of the empty boxes! As her coach, we planned a number string together that would engage the students through questioning, get to know the students even more through open discussions (since it’s still September), and to help each student “hook in” and trust the open array. I modeled this string, hoping to model the questioning but to also model for the teacher who is coming across the open array for the very first time.

Continue reading “Closed to Open Array”

Making Thinking Visible

I recently worked with a 6th grade teacher, Miss T, as she led a string for the very first time.  Twenty-eight middle school students quietly re-arranged desks and chairs and situated themselves at the front of her room — itself, no small feat — as she prepared to facilitate a messy multiplication string:

17 x 10 =

17 x 2 =

17 x 12 =

17 x 20 =

17 x 19 =

17 x 21 =

Continue reading “Making Thinking Visible”

A Dilemma with Models

A 5th grade teaching team I work with recently raised the issue of how to model the problem 1/2 x 3/10 on an array. They saw their students use a variety of models and one teacher got “bothered” by how some of the models felt “imprecise” or “not to scale.” We had a conversation together about this issue.  To prepare myself for the conversation I did a little sketching of possible models that kids might use.  What do you think?

Fraction Models_Multiplication