6th Grade Math Lesson Study 9-7-16
Columbia, Eastside, Enterprise, and Tierra Bonita Elementary Schools
Presented by Dr. Michele Douglass
What is working and what are the struggles?
Teachers explained that the different division methods are difficult. Students are
confused by the myriad of different methods, and parents are confused with new
strategies. MD stated that it is all right to have more than one way to solve
problems. Also, students shouldn't take home homework that they have not
mastered, and should be able to show their parents new strategies they have learned
to reinforce their own learning. MD showed that the toolbox method requires
students to have less command of multiplication facts. The standard algorithm
overloads students on multiplication facts and they shut down and disregard place
value. The toolbox helps with estimating and getting to the answer by other
methods. Students should estimate division problems because they will use
calculators later.
Strategies to teach doubling are necessary to build number sense - have students
double a single digit number, then decade numbers, and then other numbers.
Example of doubling 17 – first double 10 (20), then double 7 (14), and add together.
Other useful questions may include, “What do I add to this to make this the next
multiple of ten? What do you add to it to get to 100?”
Division using place value mats
Before presenting the tool box method, use place value mats to show division by
one digit. An example was shown of putting 6,123 goats into 5 pens:
Teachers need to help students to make connections between methods. In the above
photo, the place value method was translated to the algorithm, however teachers
should also write the algorithm with associated place value words.
The model of math teaching we have used in the past followed three stages - show,
practice, and assess. Now we look at a more circular model - concepts,
connections, procedures, and assessments.
MD suggests using a story problem as a hook or stimulus during instruction as
much as possible, at least daily.
Decimal division
By using units of measurements when describing the divisor, students learn through
context when working on division with decimals.
MD demonstrated moving away from the decimal version by writing the division
problem as a fraction, and then multiplying by a power of 10 fraction over itself (in
the example below, 100/100). This is where fraction sense comes back.
Ratios and proportions
Teachers and students should start with drawings because you are comparing two
things and it leads to making a table. Double number lines can be used where the
two lines are counting in two different ways; use skip counting to correlate the two
amounts to figure out missing parts. This should help students to move from
repeated addition to multiplication. Tape diagrams allows for discussion of
fractions, comparisons, percents, and seeing the whole (see below).
MD recommends using the same numbers and scenarios when introducing a new
model, so that students can see how they compare.
When presenting ratios and proportion situations, consider models to use, pictures,
and graphs. Unit rate comparisons should be shown visually, including graphing
pairs, in order to extend knowledge and make more connections. Tables and graphs
should always be labeled purposefully to ensure understanding of the scenario. MD
suggests practice during the first week of drawing pictures, identifying ratios in
stories, and labeling graphs. During the second week, students should learn to use
those ratios to make comparisons.
Once ratios and proportions are practiced, students should extend to percent
applications.
Lesson study – division strategies
"Who can tell me how you read this problem?" Some students replied with two
divided by 316, so MD clarified this for them through a story problem: “Your
principal has $316, and he is giving it to two students. How much will each student
receive?”
Students were asked to discuss answer without writing anything (estimate) for one
minute, and then shared their answers. Students were asked to set up 316 with
colored chips on white boards on a place value chart.
MD reviewed the problem, and modeled how to make it work. Questions for
students, "How do we show what one student will get versus the other? How will
they share the money?"
As students divided the chips, MD asked guiding questions, including “What do
you do with the leftover $100 bill? Why can’t you divide it into 50 dollar bills or
20 dollar bills?” Similar questioning occurred through the process, focused on how
to regroup amounts and how to account for leftover amounts.
The lesson concluded with students adding back regrouped amounts to ensure that
the original amount was accounted for.
After the conclusion of the meeting, teachers were provided with the Math
Framework document for review (online). Final remarks included suggestions to
add more equity sticks for students who need more attention (not just one per
student), and a reminder to include daily word problems for stimulus.