## 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”

## 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.

## 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

## 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?

## 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.