## 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:

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

## Video: Division on the open number line in fourth grade

Wondering what a number string looks like in a classroom? Curious about the details of the routine? The following video features site co-founder Rachel Lambert teaching a group of 4th grade students at the Citizens of the World Charter School in Mar Vista, CA. These students have a wonderful classroom teacher, Hayley Roberts, who does number strings regularly with the students as part of a rigorous, inquiry-based mathematics curriculum. Hayley is off camera in Part 1 leading the students in a mathematics mindfulness exercise.

Students in this class had done lots of number strings with the open number line for addition and subtraction, as well as the array as a model for multiplication. On the previous day, the number string had been a multiplication doubling and halving string on the open number line. Today’s number string was designed to help students understand division on the open number line, focused on using equivalence as a strategy. Before you watch, you might want to anticipate how 4th graders might solve these problems, and how Rachel will represent strategies on the open number line.

2 x 50

4 x 100

100 ÷ 2

100 ÷ 4

200 ÷ 4

400 ÷ 8

800 ÷ 16

800/16

## 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”

## True False and Fill in the Blank Number Strings for Second Grade

A teacher recently handed me a second grade interim assessment with some true or false and fill in the blank problems:

She asked me if I could develop some number strings to help her students prepare for this assessment. It called to mind some equivalency number strings that are developed in the Trades, Jumps and Stops algebra unit (Contexts for Learning Mathematics) as well as some problems in the Thinking Mathematically book by Thomas Carpenter, Megan Franke, and Linda Levi (2003).   Continue reading “True False and Fill in the Blank Number Strings for Second Grade”

## 30 ways to make number strings more inclusive

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”

## 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?

## 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.

## Photo number strings for multiplication

Here are two photos I snapped as I walked by a 99 cent store in LA. Beautiful arrays, no?

I am thinking about how to use these kinds of images as the anchors for number strings, particularly for intervention work with older students.  Sometimes older kids need work thinking about multiplication, but in an age-appropriate way.  What kind of questions do you think of with this image?  One could most simply begin by asking what kids noticed about the image.  That would bring most of the interesting mathematics forward, I think. Beginning perhaps with how many boxes of hot chocolate do you see (nice numbers)?  And then, considering this is a 99 cent store, how much would it cost to buy all of this chocolate.  It reminds me of some work that Pamela Harris suggests in her book on Powerful Numeracy, in which she asks kids what is 99 plus any number?  A 99 cent store is a great way to think about what is 99 times any number?

## Go-to questions for teaching number strings

My four-year-old son likes to walk over to the magnetic rekenrek (math rack) I keep on the fridge, move some of the red beads to the right, and then ask me, “How many beads you see Momma?” and “How did you get that?”  My son has my routine down pat.  When I ask him or his brother questions on the math rack, I keep my questions very consistent.  I put up a particular number and ask, “How many beads” and then when they answer, “How did you get that? or “How did you know?”  or “How did you see that?”

Teaching mathematics is a very complex act- when teaching a number string, you are listening carefully to students’ responses, carefully representing their thinking, and always at the same time looking for ways to connect to the larger mathematical goals of the string.  As the mathematics educator Deborah Ball so beautifully wrote, we teach mathematics with  “ears to the ground, listening to students, eyes are focused on the mathematical horizon” (Ball, 1993, p. 376).  I found that as my mind does so many things simultaneously, it helps to keep my own words pretty simple.  When I am teaching strings without the rekenrek, thus most number strings, I tend to use these phrases a lot:

“Give me a thumb when you are ready”

“What did you get?”

“How did you get that?”

“Can anyone restate her/his strategy?”

“Does this match what you were thinking?”(about the representation I made of their work)

Much of my talk when representing strategies is my repeating the words of a student who shared a strategy, simply because it helps me remember the strategy if I restate it while I am drawing a representation of it on the board.

My aim is to get strategies up on the board, and then to make sure that at some point in the string, I ask a question that moves the discussion to the level of generalization. The best questions are closely connected to the mathematics of the string, and the strategies of the students, but I seem to often say,

“Will this strategy always work?”

“Can we name and define this strategy?”

“Are there helpful patterns in this number string?”

Often, a student will begin this generalization process for you, and you just need to follow their thinking.  They may say, I think that this strategy works because . . . , and at that point I write their words down verbatim on the side of the board, leaving space for the group to refine the generalization.

Just like my son learned his questions from me, I learned them from watching other people teach strings.  Are there other go-to questions for teaching number strings?

## Fractions as operators on money (Early fractions)

Fractions as operators,

Money

What is ½ of \$1.00?

What is 1/4 of \$1.00?

What is 2/4 of \$1.00?

What is 1/8 of \$1.00?

What is 3/8  of \$1.00?

What is 1/4 of \$2.00?

Note: I modeled this on an open (double ) number line.

This is intended as an early fraction string, best done with kids as they are beginning to think about fractions, and already have exposure to the open number line.

## Doubling with early fractions

Number String: Doubling

4 x 2

1/4 x 2

1/10 x 2

1/5 x 2

2/5 x 2

4/5 x 2

2 x 7/8

I modeled this on an open number line and by adding fractions.

## Halving with early fractions

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.