Lesson Planning Template *
6th grade Dividing Fractions – Jeff’s class
Identify the mathematical goals of the lesson, both
short term and long term.
A major goal for this unit is that students conceptualize ways of thinking
about division with fractions. In particular, we want students to consider
how a common denominator method for dividing fractions might be
embedded within physical models of division situations.
In this lesson specifically, students will begin to explore measurement
contexts for dividing a whole number by a fraction. Our goal is for students
to develop a way of solving the story problem presented in a way that makes
sense to them.
There are some key ideas that precede this work:
- Both sharing and measuring problems can both be modeled by division
- In dealing with ‘leftovers’, we can ask either ‘what fraction of a group is
left’ or ‘what fraction of an item will be in each group’?
- A divided by B is the same as A/B
(30 m&ms shared with 4 friends, and 30 cookies with 4 per bag – What
operations would you use? How can you model these situations? How do we
describe the leftovers? )
Identify the ways in which the task can be solved.
 Which of these methods do you think your
students will use?
 What misconceptions might students encounter?
 What errors might a student make?
A turtle travels 2/3 mile in one hour. How long will it take him to travel 6
miles?
We anticipate a popular solution would be to use a linear (number line)
model numbered from 0 to 6 miles. Students will likely mark off 2/3mile at a
time, and count how many ‘2/3s’ it took to get to the end. (9 hours)
0 2/3 4/3 6/3 8/3 10/3 12/3 14/3 16/3 18/3
0
1m 2m
3m
4m
5m
6m
Adapted from The Thinking Through a Lesson Plan Protocol (TTLP). Developed by M. Smith & V. Bill, School of Education at University of Pittsburgh
Lesson Planning Template *
Another model students might use would be 6 discrete circles, since some
students are accustomed to using circles to think about fractions. These
students are likely to use a similar strategy of marking of 2/3 at a time.
Some students may convert 2/3 to approximately 66% since they have
recently done work converting fractions into equivalent forms. This method
will not be very useful here since the repeating decimal requires some
rounding off and may not make the answer as apparent.
Students may also use what they know about multiplication of fractions and
think 2/3 x (some number of hours) = 6 miles. Students may use a guess and
revise strategy to figure out what you could multiply by 2/3 to get 6.
After drawing models, some students may reflect on their work as 18 thirds
divided by 2 thirds = 9 thirds.
When asked to write a number sentence to match their work, students may
write several equations:
2/3 + 2/3 + …. =6
9 x 2/3 = 6
Launch: How will you introduce students to the
task so as not to reduce the problem solving
aspects of the task(s)?
What will you hear that lets you know students
understand the task(s)?
During this lesson, we want to be careful not to cue students to use particular
models or methods, as we are investigating what informal knowledge
students may call upon as they begin dividing by fractions.
Students will be partnered so that they can draw on each other’s ideas and
will have to articulate their thinking to a partner. Students will be paired
intentionally so that partners have different strengths to contribute.
Adapted from The Thinking Through a Lesson Plan Protocol (TTLP). Developed by M. Smith & V. Bill, School of Education at University of Pittsburgh
Lesson Planning Template *
As students work, the teacher will make choices about which students will
share their work in the group discussion. The teacher will be looking for a
Explore: As students are working independently
variety of representations to share so that students can make connections
or in small groups:
between different models.
 What questions will you ask to focus their
Some questions to ask may include:
thinking?
 What will you see or hear that lets you know how - What operation would you say you’re using to solve this problem?
students are thinking about the mathematical
- What number sentence can you write that matches your model?
ideas?
- How does this problem compare to the cookie problem we worked on
 What questions will you ask to assess students’
yesterday?
understanding of key mathematical ideas,
- What makes this a division problem? (or addition, etc depending on what
problem solving strategies, or their
students say)
representations?
 What questions will you ask to advance students’ - Where is your answer in your model? In your number sentence?
- If students use “invert and multiply”, Where is the 3/2 in your model?
understanding of the mathematical ideas?
 What questions will you ask to encourage
students to share their thinking with others or to
assess their understanding of their peers’ ideas?
We will have a student share a number line model first, being sure that
everyone understands how the model matches the story situation and where
the answer is in this model.
Which solution paths do you anticipate will come
up and which do you want to have shared during
the class discussion in order to accomplish the
goals for the lesson?
 Which will be shared first, second, etc.? Why?
 In what ways will the order of the solution paths
helps students make connections between the
strategies and mathematical ideas?
If there are other physical models (circle models), we will share those next to
discuss the similarities and differences between the models.
Then, we will discuss the number sentences. If symbolic notation seems
difficult, we will listen for kids to articulate their answer as 9 groups of 2/3 (9
x 2/3).
If someone identifies the problem as a division story, the student will to
explain how he/she saw the story that made him/her think it was division.
Adapted from The Thinking Through a Lesson Plan Protocol (TTLP). Developed by M. Smith & V. Bill, School of Education at University of Pittsburgh
Lesson Planning Template *
Ultimately, we are looking for students who can articulate thinking of the 6
as 18/3 in order to begin to develop the idea that if the pieces are the same,
we can divide the numbers as we would normally in whole number division.
What will you see or hear that lets you know that
students in the class understand the mathematical
ideas of problem solving strategies that are being
shared?
Our follow up lessons will investigate whether similar problems can also be
solved by converting both numbers to a common denominator, and how to
deal with problems whose answers don’t result in a whole number.
Adapted from The Thinking Through a Lesson Plan Protocol (TTLP). Developed by M. Smith & V. Bill, School of Education at University of Pittsburgh