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

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