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.

 

 

 

Number string structure and design

How are number strings designed? Typically, people tend to describe number strings as having the following structure

Entry problem

Helper problems

Challenge problem (or clunker)

 

This post from Math Coach on Demand (which also has a bunch of addition and subtraction number strings) describes the structure like this:

Screen Shot 2016-03-29 at 5.27.20 PM

Again, the concept of helper problems. But is there just one “formula” for a number string? Continue reading “Number string structure and design”

Complications with representing constant difference on an open number line

Representing student thinking during a number string is complex. Certain strategies are particularly challenging to represent. For addition and subtraction, representing constant difference and compensation can both be challenging, for different reasons. I will tackle compensation in another post. For today, let’s look at what makes constant difference tricky to represent. Continue reading “Complications with representing constant difference on an open number line”

Fractions as Operators (Dot Arrays)

Here’s a collection of strings written by teacher participants at the Summer Institute at Math in the City (City College, NY).

When students share their strategies, you might ask, “How do you know?  How are you seeing it on the array?”  Then circle or shade what they saw.  Remember to open it up to other ways of seeing, “Did anyone think of it differently? Oh great. Ronald, what did you see?”  Then the second student’s strategy or envisioning is shown on a different array.  I like to print several copies of the array and have them ready to go up.  Otherwise, it takes too long to draw the dots each time. Continue reading “Fractions as Operators (Dot Arrays)”

The “King of Strings” teaches us that strings are maatwerk

I recently reached out to Willem Uittenbogaard. Willem was one of the original collaborators between Math in the City (founded by Cathy Fosnot) and the Freudenthal Institute in the Netherlands.  He spent two years in New York City — working with teachers to develop the idea that realistic contexts in mathematics problems help children to build on their understanding of the world. He also taught many New York City teachers how to lead number strings. I was one of those teachers. I was lucky enough to be spend two weeks of the summer of 1999 with Willem, as he challenged me to solve mathematics mentally through number strings. Willem went on to co-author all of the Minilessons Resource books for the Contexts for Learning Mathematics series.

Continue reading “The “King of Strings” teaches us that strings are maatwerk”

Subtraction string – where is the answer?

My name is Jennifer DiBrienza.  I taught elementary school in New York City public schools for 9 years and began using number strings then.  When I moved to California, I completed my PhD in elementary mathematics education and now I teach at Stanford University and consult with school districts and education companies.

Last week I worked in a 2nd/3rd grade classroom. The students had started the school year with data collection, graphing and sorting, so they had done very little computation at about 5 weeks into the school year. The classroom teacher and I decided we’d build a subtraction number string to introduce the second graders to strings and to revisit them with the third graders.

21-19

22-19

30-20

31-19

35-19

75-50

75-48

Continue reading “Subtraction string – where is the answer?”

How to extend a string

We are often asked about how we can “stretch” strings beyond their place as a short mental math activity in the classroom.  How can they be used as part of a formative assessment of individual kids? What might come after a string that isn’t necessarily another string? What can kids do at home that builds on the thinking and reasoning that we are developing by doing strings?

Continue reading “How to extend a string”

Halving with early fractions

IMG_2821

Minilesson: Halving

(Rachel Lambert)

What is ½ of 1?

What is ½ of ½?

What is ½ of 1/4?

What is ½ of 1/8?

What is ½ of 2/3?

Notes: I modeled this on the open number line.

This is intended as an early number string when kids are beginning work on fractions.  Doubling and halving are great places to start to develop rational number sense.