## 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?

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.

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.

## 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”

## Beyond Skip Counting

This post comes to us from strings enthusiast and middle school teacher Marcelle Good, who works at School of the Future (Brooklyn, NY) and is a Math for America (MfA) Master Teacher.

————————————————————————–

My 6th graders love to skip count. If you were to ask them whether, for example, 6 was a factor of 96, many would be willing to skip count (by sixes) all the way to 96. Even if they don’t find this tedious and inefficient, I do. So I’ve been working this week on generating some faster strategies for finding factors of numbers — also known as divisibility rules. Continue reading “Beyond Skip Counting”

## Developing Algebra

This string was developed by our colleague Bill Jacob, University of California, Santa Barbara, who is, among other things, an algebraist.  The post is written by our colleague Monica Mendoza, University of California, Santa Barbara who leads a summer algebra institute for teachers as part of her work at The Center for Mathematical Inquiry with Bill. Continue reading “Developing Algebra”

## New percentage number strings

We recently saw the problem above on a New York CCSS-M 6th grade benchmark test. A few teachers I was working with argued about what the question required.  Some teachers thought this called for converting a percent into a decimal and multiplying.  Another teacher reasoned persuasively, “I don’t want kids to do any calculations here.  None. Without a calculator they would be tedious and time consuming. And I also don’t want them setting up some equation like .12x = 3.84.  Unless they are really fluent in algebra and division of decimals, and there is a lot that can go wrong there.”

## “Stack of Bills” string

Here’s a string I designed with a team of fifth grade teachers who were looking for creative ways to encourage a multiplicative understanding of place value.  When students are reasoning additively about number, they might see 321 as equivalent to (3 x 100) + (2 x 10) + (1 x 1).  And this is very common in classrooms, and is often confused as thinking multiplicatively because it does involve some multiplication. But in essence you are splitting the number into place value parts and adding those together. When students are able to reason multiplicatively about number it looks more like this: 321 = 32.1 x 10 or 3210 ÷ 10.

## 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

## 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.