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


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 “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?”

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.”

Continue reading “New percentage number strings”

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)”

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”

Multiplying fractions: Why context matters

Our fifth grade team was trying to encourage students to use a visual model to represent their thinking when they multiplied fractions. So many students were so fast — multiplying the numerators, then multiplying the denominators — but had no context and demonstrated very little number sense. Did their answer, the product, make any sense? What would happen to the size of the first fraction as it was multiplied by the second fraction? Was the product bigger or smaller than the fractions? Should it be? We saw students who were simply carrying out some steps without thinking about what multiplying fractions really means.

Continue reading “Multiplying fractions: Why context matters”

New strings for Measurement + Unit Conversion

Our 5th grade team was trying to address several challenges with implementing Common Core 5th grade standards of multiplying and dividing decimals and measurement conversions. Together with our Math in the City coach, Kara Imm, we created these strings.

The purpose of these strings is multi-faceted – we wanted these strings to do a lot, but seem easy and attainable to the students. First, we wanted to use a mathematical model that had the purpose of showing the relationship between multiplication and division, hence a ratio table. We did not want children to see a number, a unit of measure, then wrack their brains trying to decide whether or not to multiply or divide. The model would build habit, confidence, and reasonableness of answer.

Reasonableness of answer was a huge reinforcement of these strings. We felt if students could visualize a cm (say your pinky finger nail) then visualize a meter (arms length) that would push them away from getting stuck on what the conversion rule and help simplify and clarify what operation to preform.

Second, we needed to design the numbers to build off of their prior string work with multiplication and division – so you will see doubling and halving, numbers associated with money. Building upon this foundation of whole number multiplication and division would (hopefully) help us to alleviate some fears of operations with decimals. The ratio table also helped reinforce how to treat the decimal point when working between multiplication and division.

— Nicole Shields, Mariel Simon, Sybil Esenyan, Lori Krellenstein and Allie Minicone (5th grade teaching team, PS 158, Manhattan)

Tips for leading the string: In each case we started the conversation with kids by establishing the unit rate (first line of each ratio table). Then once we had some agreement there, we said, “So if it’s true that 100 centimeters is equivalent to 1 meter, what about 2 meters? Suppose I only was measuring 80 centimeters, now what?” We added values to the ratio table and slowly built the table, using the values here and children’s thinking to guide us. That is, we did not reveal the entire ratio table all at once. It emerged as the conversation did. In some cases we left the last line empty so that kids could make up a new true statement or we could pose a challenging one at the end.

Ultimately, we kept seeing conversion tasks for 5th graders and wanted kids to be able to solve problems like:

  • How many 200 mL paper cups can be filled from a 2 liter jug of lemonade?
  • I make 2.5kg of popcorn and eat 750g of it while watching a movie. How much popcorn is left?

We knew that having the ratio table as a tool to think with would be helpful.








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

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.