## 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.”

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.”

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.

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.

## 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:

## Division of fractions — The role of ratio

A 4th/5th grade teacher I am working with planned to introduce division of fractions and needed some ideas. I knew that a context would really help kids make sense of the operation so I nudged her towards the book Minilessons for Operations with Fractions, Decimals and Percents (Imm, Fosnot & Uttenbogaard, 2008). We found a good place to start and I offered to lead the first string.

After we discussed the word ratio, I set up a ratio table with # of cans of paint and portion (of wall painted).

Those of you who are fans of the middle school curriculum Connected Math Project (CMP) will especially appreciate this string. I was preparing for a visit to MS 22, a middle school in the South Bronx —  my collaborator Erica Berger, a thoughtful and dedicated teacher, asked me to design a string to introduce linear relationships and to prepare students for a potentially messy investigation in CMP.

Some would say number strings are “curriculum neutral” or “curriculum impartial.”  That is, they are not tied to or loyal to any one curriculum — the routine of number strings can be a helpful and grounding experience for all kids. The challenge, for all of us, is locating and/or designing strings that will support the lesson or investigation that follows.

## Arrays on the West Side

We recently learned that the third grade team at Manhattan School for Children (Elizabeth Frankel-Rivera, Madelene Geswaldo, Alice Hsu and Marissa Denice) started a really cool Homework Page for their classes.  In addition to reading for 20-30 minutes a night and writing two entries in their ELA homework section, third graders are also expected to think about and solve a set of math problems.

## 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.