## 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.

## 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”

## Feet, inches and yards: Conversions on a ratio table

This post is from our friend and colleague Kathy Minas, a 4th grade teacher at PS 158, and an avid strings enthusiast.

Recently, our fourth grade team met with Math in the City co-director and staff developer Kara Imm, who has been working with our school for several years. We wanted to explore ways to introduce Common Core 4th grade standards of measurement and conversion by using strings. What follows is the string we designed together and my notes about how to lead it with kids.

Why this string? Why a ratio table?

The purpose of this string is to introduce students to conversions, using the most common units of measurement, inches and feet. We wanted to ensure that students had a familiar model that visually captured the (twelve-to-one) relationship between these units, which led us to using the ratio table. Our students have worked with the ratio table before in both our multiplication and division units of study. The model allowed them to represent and maintain the relationship between known units in order to multiply, divide, break apart, or even add groups with ease.

Performing measurement conversions on a ratio table also supported students to monitor the reasonableness of their answers. In addition, it encouraged them to keep the relationship between units in mind.  Instead of memorizing whether they needed to multiply or divide feet to get inches, they simply trusted the existing relationships on the table that we build together.  In fact, student were much more flexible about how to convert and did not rely on a memorized rule or catchy mnemonic to solve these problems.

We build ratio table together with the class, instead of revealing the entire completed ratio table all at once. We add values to the ratio table — one at a time, increasingly more complex — and ask students to determine the corresponding number of inches or feet. Some values are added to the ratio table as students explain their process.

The string:

Begin by naming a true statement to ground the conversation:

Mathematicians, we know that there are 12 inches in one foot.

Draw a ratio table and label the columns, number of inches and number of feet. If this piece of social knowledge is not known by your students, having a 12-inch ruler to see and touch is also useful at this moment.

Since we know that there are 12 inches in one foot, how many inches are there in 3 feet? How do you know?

Student responses may include:

• I know that there are 12 inches in one foot, so there are 36 inches in 3 feet because you multiply the number of feet by three, which means you have to multiply the number of inches by three.

Note: In order to move away from additive reasoning on the ratio table  also known as repeated addition or “chunking” I purposely do not record the unit rate of  12 inches in 1 foot on the table until the students bring it up as part of their reasoning. I’m nudging students from additive to multiplicative reasoning on the ratio table. You may wish to ask students to visualize three feet, “What does this look like? What does this make you think of?”

A fourth grader (and sports enthusiast) in my class introduced the unit of a yard immediately, so I added it as a third column to the ratio table and made a post it note about his unique contribution. I left the yards column blank for most of the string, but then we returned to it later to reason about this third unit.

Mathematicians, how many feet are there in 60 inches? How do you know?

Student responses might sound like:

• I know that there are 12 inches in 1 foot, 60 inches is 5 times greater than 12 inches, so 60 inches are equivalent to 5 feet.
• I know that 36 inches is equivalent to 3 feet, and 60 inches is 24 more inches than 36. 24 inches is equivalent to 2 feet, so I need to add 2 feet to 3 feet and that’s 5 feet.

So, how many feet are there in 72 inches? How do you know?

Students will likely reason:

• I know that 72 inches is 12 more inches than 60 inches, which means I need to add one more foot to 5 feet, which is 6 feet.
• I know that 72 inches is equal to 36 inches times 2, if I double the 36 inches, I have to double the 3 feet.

Follow up question, “How many yards are there in 72 inches or in 6 feet? How do you know?”

Okay, how many inches are there in 9 feet? Can you picture this?  What are you thinking?

I heard students say:

• I know that 9 feet is three times 3 feet, so there are 108 inches in 9 feet because I have to triple 36 inches.
• I know that 6 feet is 3 feet away from 9 feet, so I need to add 36 more inches to 72 inches, which is 108.

Follow up question, “How many yards are there in 72 inches or in 6 feet? How do you know?”

Once we were done with the inches and feet component of the string, we tackled the yards. We returned to the relationship between feet and yards. I asked students to consider,

If we know that there are 3 feet in 1 yard and we only have 1 foot, what part of a whole yard do we have?

If we have 2 feet, do we have a whole yard yet? What part of a whole yard do we have?

Later I said,

Using all you know about the relationship between feet and yards and all you know about fractions, if we have 5 feet, how many yards do we have?

My students tackled this problem with ease:

• I know that there is 1/3 yard in 1 foot, so there are 5/3 yards in 5 feet.
• So, there is 1 yard in 3 feet and 2/3 yard in 2 feet, which means that in 5 feet there is 1 2/3 yards. I just put them together to make 5.

As big ideas or important strategies come up, my colleagues and I have begun annotating the strings poster so that kids can both see and hear these ideas.  I listen carefully for students to make these contributions we write and display them so that the ideas are shared and accessible to all kids, even if they are still emergent.  Sometimes, when students are ready, I nudge kids towards a generalization, which helps us move beyond the specific string and into other related quantities and relationships. Examples of this practice of annotation are below:

## 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”

## 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”