# Archive for

## Unitizing, Pokemon Cards, and would you rather

At a recent grade level work session, we tried out John Stevens’ would you rather structure to help us design a task for first graders . His structure offers really great opportunities for kids to play around with numbers and ideas; and the best part is that they have a reason to chat about their ideas and write about the choices they make.

This school’s first graders are loving Pokemon cards, not our favorite context.  However, it is not about us and what we like. It is about what the kids are interested in and what is meaningful to them.  We were looking for a context, a topic, that would pull them into the mathematics and the conversation.  So, it works.  Figuring out which set of cards you want to buy is way, way, way more interesting than the word problems found at the end of the chapter in a textbook or the daily dose of 6 word problems on a page, with varying levels of difficulty, 2 easy, 2 medium, and 2 hard.

Your grandma gave you a gift card to Target so you can get some new cards for your Pokemon™ card collection.  When you get to Target, you find out that they sell Pokemon™ cards in 2 different packs.  PACK 1 has 9 packages of Pokemon™ cards with 10 cards in each package and 12 more cards.  PACK 2 has 100 Pokemon™ cards.  PACK 1 and PACK 2 both cost the same amount of money and you have enough money to buy 1 pack.

Would you rather buy PACK 1 that has 9 packages of Pokemon™ cards with 10 cards in a package and 12 more cards OR PACK 2 with 100 Pokemon™ cards?

Use a drawing and words to show which pack of Pokemon™ cards you would rather buy.

We were interested in developing a task focused around the standards from the CCSS domain of Number and Operations in Base Ten.

Extend the counting sequence.

1. Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Understand place value.

2. Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases

a.  10 can be thought of as a bundle of ten ones—called a “ten.”

b.  The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

c.  The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

OUR WANTS–

1.  a task that would help us learn about how students’ thinking was developing in regards to the big ideas in the domain

2.  a task that gave us data on their thinking about unitizing

3.  a taks that offers our students an opportunity to explore ideas focused around the structure and organization of numbers and the base 10 system

5.  a task that would offer students multiple ways to demonstrate what they know and would help us determine what they were ready to learn

OUR LEARNING–

The idea of unitizing is not something we had really looked at in depth, our textbooks did not focus on this at all.  So, we did some research on this important idea of unitizing.  Fosnot and Dolk offer really nice explanations and details about the idea of unitizing in their book, Young Mathematicians at Work: addition, subtraction, and number sense.  Their work gives us some insights into the transitions students need to make in their thinking, as they build their understanding of groups, the size of groups, decomposing and recomposing numbers, and unitizing.

This idea, that ten things (in a group) can also be one group, is such a big idea for children.  They have only recently figured out how to count meaningfully, attributing only one counting word to each object and now this original counting strategy seems contradictory. It’s not one word for one object, but one word for a group of 10.  How can ten simultaneously be one? (page 3)

Fosnot and Dolk go on to say,

Unitizing requires children to use number to count not only objects, but also groups and to count both simultaneously.  The whole is thus seen as a group of a number of objects.  Children have just learned to count 10 objects—one by one.  Unitizing these 10 things as 1 thing—one group—requires almost negating the original idea of number. (page 11)

In our task, we ask children to consider two amounts that are organized into different size groups.  We want to see how they compare the quantities:  do they compare the two sets in terms of groups of ones; do they create groups of ten; do they use a combination of different sized groups; or do they not use group size at all in their thinking and representation?  The second quote gave us pause, and provoked another round of discussion about possible student responses and other components and elements we might need to consider.

How does the student display his/her thinking–does he/she draw groups of ten, draw groups of one, or is it a combination?

The student identifies the pack that has the fewer number of cards, and s/he doesn’t like Pokemon cards, so s/he wants the pack with the smallest number of cards.

The student just counted the number of cards in one group.  He/She didn’t give any explanation or any indication of his/her thinking.

Is there any evidence of unitizing?  If so, how is the child working with this really important idea? If not, what do we need to do?

Does the students have another idea–one that we haven’t considered? (We are sure that this is going to happen and can’t wait to see what the kids do!)

Have we written the story context at an appropriate language level for grade 1 kids–so that they can determine which pack of cards they want to buy without too much “clarifying” from their teacher?

What might we have missed or do we need to change so ensure that the task is accessible for kids?

SIMULATED responses–

We then started creating responses that we thought kids might have; that had the elements that we considered indicated the level of understanding we wanted; that included some of the tools, manipulatives, materials that they use regularly; and that we could use in conversations with small groups or with the entire class.

They like 100.  It is their favorite number, so that is why they picked the pack of 100.  Lots of first graders love 100.  They think it is a pretty awesome number, and it seems like a lot of cards, so they pick Pack 2.

Different models can be used to demonstrate the elements of understanding that the teachers identified:   use of groups of tens or groups of ones or a combination; some symbolic notation that describes how the quantity was determined; symbols and words and drawings match; some math reasoning; just a representation of the amount of the pack of cards that were selected; and some language to explain thinking.

The drawings are our adult representations.  They helped us clarify what elements we will use to determine different levels of understanding.  Most of the exemplars are incomplete, but we felt that we would not get a “perfect” answer first time out.  From there, we will want to talk about next steps for instruction in small groups and full group, resources, etc.

FINAL THOUGHTS–

Try it with kids.  Then, get together and talk about what great ideas the kids had, what worked, what didn’t,  and go from there.

## Fraction Circles

About two weeks ago, I was fortunate to spend the day talking and learning with and from a group of fourth grade teachers.  We spent the day working on our understanding of the content contained in the CCSS Domain of Numbers and Operations–Fractions.  Our conversation focused on the shifts we are going to need to make as we implement the common core standards for mathematical content and mathematical practice.  We talked about pedagogy, pedagogical content, reasoning, communication, and a whole lot more.  We definitely will be doing some stretching in our professional practice.

During the day, we spent some time using fraction circles.  Fraction circles are a tool that can be used to foster conversation and to promote opportunities for students to reason.  They can be a vehicle to support the precise use of language.  In addition, the fraction circles can also be used to explore both grade 3 and grade 4 standards.  The more we worked with the fraction circles and explored the content of the domain of Numbers and Operations–Fractions, we determined that they can be utilized to create opportunities for students to build their foundational knowledge.  We wanted to ensure that their experiences include working with fractions as numbers (3.NF.1 and 2), unit fractions, the idea of partitioning, and number lines.  In addition, we want students to work with equivalence and comparison (3.NF.3).  These essential understandings are the foundation for the fourth grade standards, particularly, 4.NF.3 and 4.NF.4.

So, we worked on how the fraction circles could help us accomplish some of the goals we had outlined. We created several sets of number strings to offer our students so that they could explore ideas and concepts that are essential and enduring.

JUST A HINT:  Included with the number strings are photos of the set of fraction circles we used.  We learned that different manufacturers use different colors for fractional pieces.  So, if you use some of the number strings, which we hope you do, check to see if the fraction circles you have match what we have.

Number String #1

1.  If the black circle = 1, build 2/3.  What color pieces did you use?  How many 1/3 sized pieces are there?

2.  If the black circle = 1, build 5/6.  What color pieces did you use?  How many 1/6 sized pieces make up 5/6?  How could you represent 5/6 using addition?  Using multiplication?

3.  If the black circle = 1, build 7/12.  What color pieces did you use?  How many 1/12 sized pieces make up 7/12?  How could you represent 7/12 using addition?  Using multiplication?

4.  If the black circle = 1, what value do three brown pieces have?  How many 1/8 sized pieces make up 3/8?  What is the value of one brown piece?  Each brown piece?  How can you represent 3/8 using addition?

Number String #2

1,  Take out 4 red pieces. If the black circle = 1, how many red pieces does it take to make the whole?  What is the value of one red piece?  What is the value of each red piece?  What is the value of the 4 red pieces?

2.  Take out 1 green piece.  If the black circle = 1, how many green pieces does it take to make the whole?  What is the value of one green piece?

3.  If the black circle = 1, how many 1/6 pieces are needed to make 1/2?  What color is a 1/6 sized piece?  What color is a 1/2 sized piece?  How many 1/6 sized pieces does it take to make 2 black circles?  How many 1/6 sized pieces does it take to make 3/2?

4.  If the black circle = 1, how many 1/6 sized pieces does it take to make 1/3?

5.  If the black circle = 1, how many 1/12 sized pieces does it take to make 1/2?

Number String #3

1.  Build 2/2 with the fraction circles, if one black circle = 1.

2.  Build 3/2 with the fraction circles, if one black circle = 1.

3.  Build 7/2 with the fraction circles, if one black circles = 1.

4.  Build 8/4 with the fraction circles, if one black circles = 1.

5.  Build 10/4 with the fraction circles, if one black circles = 1.

6.  Take out a number line. Plot each of the numbers from questions 1-5 on the number line.

We know that these are just an initial set of number strings that we will use to provide students with opportunities to explore and investigate ideas about fractions.  But, we’re feeling pretty good about setting up some new learning situations for our students and the professional conversations we had as we designed them.  We look forward to talking about the impact on student learning at our next conversation.