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.

 

 

 

A count around for fractions

This February, I led a Number Strings Writers’ Retreat, as part of my role as staff developer at Math in the City.  One participant was 4th grade teacher Kathy Minas, a former colleague from PS 158.

Kathy wanted to design strings and other routines to help her students move past rote strategies (e.g., stacking) when subtracting fractions, but also to support them to think flexibly about the relationships involved in situations that involved fraction subtraction.

At the retreat, we began thinking about two central questions:

  • How could we use a visual component to help children during a fractions count around?
  • What context would support the students to reason about the quantities involved?

We chose the context of brownies as this is often something children can visualize. We used actual pieces of paper to represent the pans and pieces of brownies with the intention that kids would hold and move these pieces as the count around progressed. We hoped that this “manipulative in hand” would make the experience concrete, helpful and memorable.

But before she led the count around with kids, she and I acted it out with together. This helped us anticipate:

  • what kids in her class would experience
  • what part of the conversation she would record
  • what strategies her fourth graders would have
  • how she might support anyone who struggled or who needed a challenge

Below you will find Kathy’s notes from the count around.  We hope it’s helpful for you and your kids, too.

— Nicole Shield, Staff Developer, Math in the City

 

Materials:

  • Five wholes cut into fourths
  • An empty number line set up to 5 wholes
  • White board or document camera with paper for recording jumps on number line, equations and kids’ strategies

Start with 5 wholes cut into fourths displayed on the rug with the class sitting in a circle around it.

Introduce the context:

I want to tell you a story about my friend Sonya and I brought some materials for us to use to help us to visualize this situation. On Saturday, my friend Sonya baked 5 trays of brownies for her family, which included her husband and three kids. She cut each tray into fourths. Using the model here, can you tell how many fourths she had?  Turn and tell your partner.

Record what kids say — 5 pans is 20 pieces OR 5 = 20/4

After dinner on Saturday, Sonya brought out the trays of brownies. She ate ¼ of a tray. How many trays of brownies were left? How do you know? (4 ¾)

Record — 5 -¼ = 4 ¾

Then her husband Mike ate ¼ of a tray. So now how many trays of brownies were left? How do you know? Is there another way to think about this portion?

Record what kids say — 4 ¾ – ¼ = 18/4 or 9/2 or 4 ½

But be sure to push their thinking:

I thought we were talking about fourths. Where are the nine halves here? Who can show us in the model?

Mathematicians, are you claiming that that 4 ¾ – ¼ = 18/4 = 9/2 = 4 ½. Talk to your partner about whether or not you agree with this statement, and if you do, how would you convince those of us who are not yet convinced?

Count 2
4th graders preparing to convince each other. Photo taken with parent consent.

Bring the class back so that a student (or two) can try to use their model or other reasoning to convince others of this equivalence.

Well, Sonya’s family loves those brownies, so now each of Sonya’s three boys eats ¼ of a tray. Can you picture this, using our model?

You may want to invite a student act this out, using the shared model in the center of the circle.

So, what do we know now? What problem or problems did we just solve? Turn and talk.

4 ½ – ¼ = 4 ¼

4 ¼ – ¼ = 4

4 – ¼ = 3 ¾

OR maybe even….

4 ½ – 3/4 = 3 ¾

So, how many trays of brownies were left after dessert on Saturday?

Well, there is more to the story.  On Sunday, Sonya brought out the remaining trays of brownies. First, she ate ½ of a tray of brownies. So, now how many trays of brownies are left? (3 ¼ trays)

Record — 3 ¾ – 1/2 = 3 1/4

OR

3 ¾ – 1/4 = 3 1/2

3 1/2 – 1/4 = 3 1/4

So, now, using our model let’s make some predictions. Sonya just ate 1/2 of a tray for herself, right? I’m wondering: Are there enough trays of brownies left for the rest of her family to also each eat 1/2 of a tray? What do you think?  Will there be enough? Turn and talk.

Count 4
Kathy’s 4th graders during a turn and talk. Photo taken with parent consent.

Mathematicians, what do we think? How do we take away ½ of a tray from 3 ¼ trays of brownies?

Invite your students to use the model to act this situation out.  You might start by asking a student to just model what happens when Mike, Sonya’s husband eat his 1/2 a tray.

3 ¼ – ½ = 2 ¾ or 11/4

Together with your students, model the removal of ½ of a tray of brownies three times, one for each of the boys.

2 ¾ – ½ = 2 ¼

2 ¼ – ½ = 1 ¾

1 ¾ – ½ = 1 ¼

So, how many trays of brownies are left? How many fourths is that? (1 ¼ trays or  5/4 trays).

Here’s our last prediction. On Monday night after dinner the 5 members of Sonya’s family want to share the remaining trays of brownies equally. Is this possible? If so, how much of a tray of brownies would each person eat? Turn and talk.

Have a student or two act this out using the physical model, while you record on a number line, making note of the equations that correspond to each action in the model.

1 ¼ – ¼ = 1

1 – ¼ = ¾

¾ – ¼ = ½

½ – ¼ = ¼

¼ – ¼ = 0

So, now I’m thinking about this question: Over the course of these three days, how many total trays of brownies did each person in Sonya’s family eat? How do you know? (1 tray)

Two big ideas that emerged:

  1. When subtracting fractions, mathematicians may find it helpful to rename whole numbers into fractions with equivalent denominators.
  2. When subtracting mixed numbers, we may need to break apart wholes in order to make it work.
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Kathy’s notes (recorded on paper under the document camera) at the end of the count around

 

Cars, coffee, and climbing stairs: Inviting students into the story

Here’s another important contribution from our friend and colleague Marcelle Good — a 6th grade teacher at School of the Future (Brooklyn) and a Math for America Master Teacher.

In this post she illustrates the role of number strings in helping students to reason quantitatively. This idea  — codified as one of eight Standards for Mathematical Practice — means that students of all ages can “make sense of quantities and their relationships
in problem situations…the ability to contextualize.”

It also suggests that students have developed the “habit of creating a coherent representation of the problem at hand….considering the units involved, attending to the meaning of quantities, not just how to compute them.” In other words, giving students the chance to situate numbers and other values in a story and using those stories to make sense of the mathematics.


 

To be honest, I was slow to come to the idea of loving context in number strings — the numbers were so beautiful on their own!  An even bigger issue was that the context never seemed to be taken up by students during independent work or when the problems got more complex.

Recently with my students, though, I came to really appreciate the power of story as a referent for kids. When my entire class tried to convince me that since 100 cars would have 400 tires, 99 cars would have 399, I knew I had a problem.

With just a bare number ratio table, my kids would not have found their way out of this misconception. But, with the context in mind, one of my 7th graders explained to the class, “This is how you think about it: You have 100 cars for some reason. Some jerk comes and steals one of your cars. He doesn’t drive off with one tire — he drives away with four tires, so you have 396 left.”

Marcelle Photo 1

Before another example, some background about me and my students. At my school, School of the Future (located in East New York) I am really struggling to help students access grade-level content. When they enter my school in 6th grade, most of my students are 3 to 4 years below grade level. To address the challenge, our approach has been to take the long view: we are not too concerned with getting them to do 6th grade math in 6th grade. Instead, our goal is to get them to do 8th grade math by 8th grade.

That wasn’t our initial approach. Originally, we tried to teach them grade level content, and scaffold the work by re-teaching or reviewing topics like double-digit multiplication or generating equivalent fractions. What we found, as a school community,  was that this approach was not working, and just not enough. Students were entering 7th grade with a partial understanding of 6th grade, and still lacking a weak foundation.

As a result, I spend a lot of time figuring out exactly what my students know.

Do they have a concept of how the number ten works in addition and why it’s so powerful?

What ideas so they have about multiplication?

Do they have a concept of area?

Do they count only by ones, or do they have some strategies?

Once I know where they are, I work to address their needs in class by meeting them where they’re at. My students who are the least ready for 6th grade material are also assigned to a numeracy class, in addition to their regular 6th grade math class. And in this class, I give myself permission to teach what would be considered K-5 mathematics.

Now, as I prepare to lead number strings where I know the math is more challenging for students, my first question is always, “What story can you tell about what happened?”

Today, the story was that I had to buy coffee for a meeting (this part was true), and that I had bought 4 cups of fancy coffee for $7. The next part of the string was: Then, something happened and I had bought 8 cups. Turn and talk to your partner to tell them a story about what might have happened.

Students said things like, “Then you got back to school and 4 new teachers showed up to the meeting so you had to go back.” We are on the 5th floor of my building, so the students were feeling pretty bad for me.

They groaned when I put a 10 on the chart under the cups of coffee, possibly a sign that they were invested in the context with me. “What story can you tell now?” I asked them. KellyAnn said, “The principal called you and said we had visitors and that we needed two more cups.” Andrew said, “Now we have to figure out what two cups cost, I’m not sure about that.”

Marcelle Photo 2

A few takeaways from this lesson:

  • Students who struggled with the math were able to get started on the problem. I’d much rather have students dealing with, “How much for two cups?” then, “What should I do?”
  • This string had a tone of, “What’s going to happen next?” When they found out that I needed 10 cups of coffee now, we started thinking about all the stairs I had to climb (and this became fodder for a new string).
  • The decision to draw pictures of the situation felt authentic and not a tool to use with students who needed “remedial” math education — because we were all imagining it together. There was no point where I felt like I needed to offer a picture, we were just right there in it together.
  • The pictures were a key to transfer. Students who struggled to reason on their own during independent and partner work could be prompted to draw a picture and suddenly they were able to reason through problems.

Marcelle Photo 3

One student’s notebook who struggled a lot with proportional reasoning, but then was able to draw pictures to work through the story.

When we think about context in number strings, this question, “What happened next?” gives students an entry point to get started, and often, this translates so quickly to a picture. My students now know they have been invited into a world where maybe we can have a garage with 100 cars, or I can spend an entire day just on coffee runs, and we can wonder about how many stairs we have climbed. After many such invitations, they’re willing to go there with me to think about the math. The numbers are beautiful on their own, but that’s because they tell us a story.

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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

Continue reading “Video: Division on the open number line in fourth grade”

A “juicy” dilemma

This number string is inspired by an investigation (Comparing and Scaling, Investigation 1.2) from the Connected Math Project curriculum for 7th graders.  It can be used as a way to extend and solidify ideas that will develop as kids investigate, or it could be used to launch the task. In the investigation, students are asked to compare the following orange juice recipes, according to their “oranginess.” Continue reading “A “juicy” dilemma”

So, what’s the story?

This number string attempts to do something a little different it begins with the model (instead of the problem), and uses the model as an anchor to make sense of the context.

This means inviting students to “tell the story” to explain what is happening in the model. Then the story (and the model) expand, while the ratio stays the same. It is modified from ACE problems from the unit Comparing and Scaling (Connected Math Project, grade 7), and is intended to serve as a template for how existing curriculum can be used to design new flexible, interesting numeracy routines for students. Continue reading “So, what’s the story?”

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”

Chocolate Arrays

Lately, I’ve been seeing arrays everywhere I go: at the grocery store, at the pharmacy, at the farmers’ market. And, of course, at Costco. The big, bad bulk retailer is bursting with interesting items arranged in perfect columns and rows.

Naturally, I made a beeline for the chocolate.

What follows is a quick-image string for exploring the associative property, the patterns that occur when multiplying, and the relationship between columns and rows. It supports the development of some key strategies for multiplying: doubling and halving to maintain equivalence, doubling a dimension to double to product, and using partial products to solve.

Continue reading “Chocolate Arrays”

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”