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.
Screen Shot 2016-03-19 at 5.45.00 PM
Kathy’s notes (recorded on paper under the document camera) at the end of the count around