Bernadine Cotton
Marita Lindbeck
Latisha Montoya
A 3rd
Brownie Bits
Grade Lesson Study on the Distributive
Property
Section 1: Introduction
The Common Core State Standards in the elementary grades have made a shift in the
focus and attention in the numbers and operations domains. Never before has there been such a
targeted focus on a deep and rich understanding of number concepts. As we have gone though
one year in LIFT, we have begun to understand the importance a solid foundation in number
sense can provide for our students as they embark on their mathematical journey. Young
children with a strong number sense understand that breaking apart numbers when you add can
make finding the sum easier. For our lesson study our team explored the idea of how children
can use the same strategy to break apart numbers when trying to find the product in a
multiplication problem.
Our third grade team met with our school support team leader (Lisa Virag), two former
LIFT Cohort members (Jana Ward and Felicity Valle), and a mathematician (Ted Stanford).
After a long discussion that included exploring a variety of number concepts and lessons, we
decided to focus on CCSS 3.OA.B.5 (apply properties of operations to multiply and divide –
distributive property). A lesson using the distributive property tied into our overarching question
and was an important topic that we felt our third graders needed to explore. After working with
landmark numbers, arrays and area during the fall semester, we felt the students were ready to
use what they had previously learned in a lesson that focused on dividing rectangular shapes into
smaller arrays to find the area.
Bernadine Cotton
Marita Lindbeck
Latisha Montoya
Our lesson was called Brownie Bits. The lessons were done In Latisha’s class in the
morning and Marita’s class in the afternoon. Both classes contain special education students and
have a wide range of academic learners. The objective of the lesson was to find the area of large
rectangular brownie pans. Each pan could hold 1 inch X 1 inch brownie bits. The children were
to work in groups to figure out how many brownie bits could fit into each pan. Instead of tiling
the large pans and counting the pieces, we wanted our students to use what they had learned
about landmark numbers, arrays and area to divide each pan into smaller pieces to figure out the
area. For example, one pan had a length of 17 inches and a width of 8 inches. The ultimate goal
was for the students to divide the pan into two pieces represented in an equation similar to this
one: (10 X 8) + (7 X 8). Solving multiplication problems that contain two-digit by one-digit
multiplicands using models such as arrays is an important skill for third grade students to master.
Fourth grade students are required to multiply a whole number of up to four digits by a one digitnumber and multiply two two-digit numbers using strategies based on properties of
multiplication. They can use equations, rectangular arrays and area models to represent their
work. Fifth grade students should be able to use these same concepts in regards to multi-digit
whole numbers and decimals to hundredths. By developing a basic understanding of the
distributive property in third grade, we hope our students will be more proficient in the work
they do in fourth and fifth grade.
Section 2: Mathematics Learning
The first semester of third grade saw the students working with a variety of area models. In
all three third grade classes, the children had worked on finding area of rectangular and irregular
shapes. Latisha’s students had done some work with the distributive property in lessons prior to
our lesson study. Most of the students had a solid understanding of the formula for finding area.
Bernadine Cotton
Marita Lindbeck
Latisha Montoya
The children have also worked with dividing numbers into landmark numbers to make addition
easier. Both of these skills were important to building the understanding of the distributive
property (Latisha’s class) or introduction of the distributive property (Marita’s class).
The lesson began with a scoot activity to activate background knowledge and introduce
vocabulary. After a brief discussion of the vocabulary words, the children were divided into
groups and given their supplies (one inch tiles, yard sticks, post-it notes and their poster). They
worked as a team on one brownie pan to find the area. There was a wide range of strategies as
the students approached this problem. Some of the groups used what they knew about area and
measured the length and width to find the area of their pan. However, because the numbers were
not friendly and familiar, some struggled with finding the area of a two-digit by one-digit
number. Some of these teams were able to break the numbers part using the distributive property
to find the area. Other groups started with tiling the pan. There were a variety of problems that
arose from this strategy, but a few of them were able to find the area, despite not using the
distributive property.
As the journey of the brownie bites launched and progressed to the Explore and Summarize,
student misconceptions were evident throughout the experience. During the launch activity it
was evident that students possessed a vast vocabulary. However, their ability to apply the
vocabulary displayed during their discussions was limited during the lesson. For example, the
students had a general understanding of the meaning of distributive property. However, when
distributive property was paraphrased into what was assumed was student friendly language it
caused confusion and frustration for some of the students. In order to determine mastery,
students were given an exit ticket with an array. The directions read: Decompose this array into
smaller pieces. Show multiplication equations for the decomposed shape to find the total area.
Bernadine Cotton
Marita Lindbeck
Latisha Montoya
We assumed the terminology was basic and students would be able to complete the ticket with
ease. After evaluating the struggles the children had, we should have worded the exit ticket:
Use the distributive property to find the area. One student became emotionally frustrated and
said, “If you wanted me to use distributive property, why didn’t you just say that.”
Another misconception, was noted while students were attempting to tile their brownie
pan. As one of the students began to draw lines, another student pointed
out the lines were not straight and would therefore not give the accurate
number. The students disagreed and continued with the drawing shown in
the example to the left. Using the incorrect measurements did not allow
the correct dimensions to be established.
Students were knowledgeable with the concept of area. The diverse ranges of learners were all
very familiar with area and were able to use multiple entry
points to solve the brownie dilemma. Some used the
manipulative square inch tiles to tile the pan. Some tiled a
side of the pan and listed multiples, some used distributive
property, and some used rulers to determine the length and
width. When using the length and the width formula some
students attempted to use the standard algorithm to solve
the problem. Many misconceptions were evident as they attempted to solve the equation as
illustrated in the example.
When attempting to complete the algorithm, students multiplied the one’s place with
ease; 9 x 3 equals 27 and carried the 2 to the ten’s place. Then they added one instead of
Bernadine Cotton
Marita Lindbeck
Latisha Montoya
multiplying the nine completely across. The students came up with a product of 37 for 13 x 9.
When students were asked if this answer made sense, they all reasoned that their answer was
probably not correct and began to explore other options.
Decomposing numbers was a vocabulary word students grasped well during the initial
activity. When applying the word, students experienced difficulties as shown in the example
below.
The dimensions of the rectangle 5 x 18 were noted
accurately. The students then began to decompose the
number. First they did 5 x 10 = 50. Instead of multiplying 8
by 5, they multiplied 50 x 8 = 90. They noted that 90 was
their answer.
The next misconception observed was when
students were decomposing numbers. Students were confident in decomposing numbers.
However, some groups were not able to identify the purpose of using landmark numbers to make
the multiplication equations more accessible. Instead of
decomposing the 12 into landmark numbers and using a 10
and 2 they instead decomposed the 8 into two 4s. This was
determined to be an initial step in using the distributive
property.
In addition to continuous observation, students had the
opportunity to present their work to classmates during a mini presentation. As students were able
Bernadine Cotton
Marita Lindbeck
Latisha Montoya
to articulate their thinking to classmates during this time their level of understanding was
evident. In addition to presentations, the exit ticket (shown below) sufficed as a clear
achievement level of students individually. We discussed the importance of using landmark
numbers with students in order to effectively
master using the distributive property.
Section 3: Instructional Strategies
The entire third grade lesson study, “How Many Bits?” was designed in such a way as to
foster active learning and engage students cooperatively. The lesson initially began with a
“scoot” to activate prior knowledge and pre-assess student understanding of the essential terms
being used in the lesson. Students were put into five groups. There were approximately four
students in each group. Each group was given a distinct color of marker and the groups then
circulated around the classroom to different essential terms, brainstorming about each term’s
meaning. The students displayed their schema and became actively engaged for the lesson.
The lesson was introduced as a meaningful dialogue with the students. A variation of the
question, “Which pan should I use to make the most brownies so that more people can be
served?” piqued student interest and helped the problem seem relevant and personal to the
students. Additionally, once the problem was presented to the students, each group was able to
select the “brownie pan” they preferred to measure. Manipulatives were provided during the
Bernadine Cotton
Marita Lindbeck
Latisha Montoya
exploration as a means for participants to self-select the manner with which to measure their
“brownie pan.” Students were given 1x1 square tiles and yardsticks to work with. Additionally,
they had access to various other familiar math tools around the classrooms (such as hundreds
charts and multiplication charts).
Groups worked cooperatively to determine the area of their brownie pan. Communication
between students and teachers was facilitated mostly through the use of probing and questioning.
When students had questions, they were directed back to the information they had produced
during the launch of the lesson or they were questioned in such a way as to hopefully maintain
the cognitive demand of the lesson. Although it was frustrating to see groups of students
struggling so close to breakthrough and yet not quite reaching that point of breakthrough, much
of the critical learning took place during this time of disequilibrium.
As the first groups reached a solution to the area of their “brownie pan,” they were provided
with a lesson extension of finding the area of half of the pan (“to put nuts on.”). Once every
group had reached a solution, each group was given a space to share aloud about the strategies
they had used, problems they had encountered, and the solutions they had reached with the rest
of the class. Students from other groups were then encouraged to ask questions.
Two primary changes were made from the morning lesson study presentation to the one in the
afternoon. First, it was determined that when we presented the concept of an array as a series of
dots rather than squares, it may have confused some of the students or detracted from the concept
of square units. So the array was changed from the figure on the left (below) to the figure on the
right (below).
Bernadine Cotton
Marita Lindbeck
Latisha Montoya
The second change included providing more specific instructions pertaining to the use of
yardsticks as a tool for the lesson. In the first classroom, students were not specifically told to
“measure” with the yardstick, although we assumed that they would use the yardsticks for that
very purpose. However, in the morning class, students primarily used the yardsticks as a
straightedge rather than a tool for measurement. In the second classroom, students were
specifically encouraged to consider using the yardsticks to measure with, which assisted a couple
of the groups in focusing more upon the use of the distributive property rather than tiling the area
of the brownie pans.
Section 4: Lesson Study Process (Individual Reflection)
My overall involvement as a LIFT participant and, most recently, as a member of the
Lesson study has helped me not to take myself nor my instruction so seriously. I am learning to
take a step back and observe the process of math instruction from a more objective perspective
for the benefit of my students.
Planning a lesson is a very….intimate…process. It is kind of like baring your soul because it
conveys your understanding about a concept in addition to your personal biases with regard to
teaching that concept. Therefore, working with a team of people with whom I am not
accustomed to working felt risky at times. However, it forced me past the “safe zone” and into a
place of disequilibrium that caused me to grow.
Bernadine Cotton
Marita Lindbeck
Latisha Montoya
I found it incredibly interesting to compare and contrast two classroom environments. I am a
better math teacher now because of what I witnessed in my colleague’s classroom. I am clearly
aware that my passion for math instruction is not an isolated event, and that my student success
can be improved through my interactions with other teachers.
A professional development goal for next year is to implement a vertically aligned lesson
study at my school. It would be based upon an identified deficit that may show up in our
NMSBA test results. If there is a significant area in which our school shows a mathematical
deficit, a vertically aligned lesson study could identify misconceptions in student understanding
and instruction which would help us to more effectively address the deficit. – Marita Lindbeck
The Lesson Study process was a valuable learning experience for me. Working with a very
strong group of teachers that included two teachers from the first LIFT was a beneficial
experience. Everyone had great insight and having the ability to bounce ideas off of one another
was very powerful. For many years collaborating with my peers was a challenging experience
for me. I had tremendous ownership of my students and felt sharing and collaborating was time I
could spend preparing for my own class. I have morphed in my thinking the last few years and
the work I have done in LIFT is reinforcing a positive change that has occurred in my teaching.
My students are the beneficiaries of this important personal discovery.
There were many challenging parts to the Lesson Study process. Finding time to meet in a
busy day was not an easy task, and the amount of meeting, planning and preparing time was a
lot. I do appreciate the fact that we didn’t have a lot of extra math homework in LIFT during this
process and that subs were provided for the actual Lesson Study day. Working with a variety of
Bernadine Cotton
Marita Lindbeck
Latisha Montoya
personalities on the team could have been tricky, but I appreciate the fact that even though we
had disagreements, everyone was very professional in the work we did together.
When I think of the work that needs to be done at my school, I am excited to take small pieces
of the Lesson Study to my building next year. I am going to address this further in my
cornerstone project. This process will be two-fold, one personal experience with a vertical
lesson including a second and a fourth grade teacher and the other with a school wide Lesson
Study plan Kathryn Million and I are working on for next year. – Bernadine Cotton
Completing this lesson study enabled me to determine the importance of mathematical
vocabulary. Vocabulary is often relayed to students and they are able to cite definitions.
However, when asked to apply the vocabulary when not used word for word their knowledge is
limited. After observing this happen I developed an understanding that students need to be more
active participants in the vocabulary while practicing them in a variety of ways and also
paraphrasing them.
As a third grade team we researched the common core standards content Measurement
and Data. We focused on the distributive property and the depth of the Learning Targets:

Students are to model with mathematics and use appropriate tools to find area using the
distributive property

Students using distributive property to decompose a more challenging multiplication
problem into two easier ones - Latisha Montoya