Grade 3 Math Lesson Study 12-8-16
Columbia and Tierra Bonita Elementary Schools
Presented by Dr. Michele Douglass
What is working and what are the struggles?
Teachers report that they are more familiar with the curriculum, and this is allowing
them more flexibility with using it as a tool. Students are coming to grade 3 with
more understanding and familiarity with simple word problems. Teachers are
looking forward to addition and subtraction units, to see how their students are able
to implement new strategies in regrouping from grade 2.
Teachers decided to make some variance to the alignment guide as December is full
of activities, so most are at Chapter 3. Students are struggling with creating word
problems with multiplication operations, and separating these from situations when
they have to add. Division is also a struggle. MD recommends setting up an anchor
chart that refers to multiplication as groups of items. We should even change our
language during the first units, and not use the word "times" for multiplication, but
rather refer to "groups" of items.
Students are not picking up time concepts, in particular elapsed time. Lack of
exposure to an analog clock is an issue. MD recommends using a number line, and
in one classroom this is being done. Ms. Hicks offered to demonstrate how this is
done. The team practiced with a problem, "David finished his soccer practice at
12:35, and his practice lasted 1 hour and 15 minutes. David took 5 minutes to walk
to the field. When did practice start, and when did David arrive to the field?"
Starting in August, using simple time problems and integrating number lines will be
a help, for example "How many minutes until recess?" MD also recommends skip
counting by 5s using a clock, and marking the classroom clock with increments of 5
minutes. This gives a double benefit of telling time and learning multiples of 5.
Elapsed time suggests addition and subtraction facts, so MD recommends that these
lessons may be best in conjunction with those units. Teachers also discussed analog
clocks available in the classrooms (from prior math sets).
Subtraction demonstration using a number line
With an example problem of 438-252, students can count using a number line by
counting back in groups that students can identify. Another method is finding the
difference between the two amounts, also on the number line, by counting up
strategy. The number line is a great strategy to help them develop the concept.
This is especially useful when students have worked on place value. Teachers
worked on different methods for showing the difference on white boards, with a
variety of results for counting up or down.
MD asked teachers to hand their number lines around the group to see how others
modeled the problem. Some counted up beginning with large increments, others
counted with smaller increments, and some worked by grouping numbers.
If students were able to hand their work around and see other solutions, how would
this help their learning? Also, when students are using blocks for addition and
subtraction, showing the number line in conjunction is very helpful. MD and
teachers worked on how blocks might be used for regrouping/ungrouping, and
discussed some instances when this might be difficult. Regrouping/ungrouping
from the left to right may help them stay organized. MD wants to be sure that
students are no longer counting by ones.
Subtraction using chips
MD emphasized that when teaching different strategies, using the same problems
over again, including word problems, helps students in their concept development.
Teachers practiced setting up chips in place value groups to solve 438-252. "There
are 438 gingerbread man cookies in the office, and 252 of them were broken. How
many were whole when they made it to the classrooms?"
MD recommends that students set chips in rows of 5 when possible so that students
recognize rows of 5 visually. Also, as students subtract, they should remove to the
bottom of the page so the teacher can see errors, or students can replace them if
needed. However, trades need to come off the page. The action of trading chips
will help students to build the concept.
MD also discussed that all regrouping can be done before any subtraction occurs.
This is starting now in second grade. It is important that you have students count
what is on the board, and make sure that the original amount (438) is still on the
board after ungrouping. Students should practice by working on problems such as
"3 hundreds, 13 tens, 8 ones = 438" to better understand how place value works.
Brain research indicates that students who have to go back and forth from
regrouping to subtraction often make mistakes, which is why regrouping all
numbers first often helps. Place value blocks are initially important as they
preserve magnitude of numbers, however students should be able to make a
transition to "non-magnitude sizes." From three digit using color chips or symbols,
we can naturally extend to four digit numbers without as much difficulty.
As students are performing the ungrouping (trading, substitution) with their coins,
teacher will document on the actual algorithm so that students can see the
relationship between the model and the problem. Be sure to show the substitution
for 13 in the tens place (write the entire number) to match the coins and show what
they are actually seeing.
MD presented a second example where more ungrouping needs to occur: 345-179.
Teachers were asked to model this with a number line, coins, place value blocks,
and at the end with the algorithm.
MD demonstrated how the adding up strategy written as an addition sequence will
show students how this will work numerically (345-179). Idea of counting up
allows students to utilize "friendly numbers." A counting down strategy will match
more closely to the coins, and will look very different to counting up strategy. With
coins, teachers ungrouped their numbers twice before subtracting.
Lesson Plan and Study - Mr. Guzman's room
MD introduced and will demonstrate Number Talks using addition with doubles,
15+15, 15+16, 15+17, and 15+18. When starting Number Talks, it is best to start
with K-2 sections to build student confidence in articulation.
Between the Number Talk and the lesson, MD will practice skip counting with the
class. The lesson will focus on addition using number lines.
In the Number Talk, MD explained hand signals to the students, and displayed
15+15. MD asked students for answers, and all students agreed on 30. She then
asked how students got their answers, and scripted as the students explained their
addition of 5s and 10s. MD then asked "What can I say about these numbers?"
Students replied that they were both odd, they were both "teens," and MD led them
to the understanding that they are doubles.
Next, students were asked the sum of 15+16. Students responded with 21, 36 and
31. MD asked "Who can justify their answer?" MD scripted as student reported
5+6=11, 10+10=20, and then 11+20=31. "Does anyone have a completely different
way?" One student mentioned counting on his fingers 16 times from 15, and MD
discussed that although this method would work it would not be efficient. She used
an example of larger numbers for this. MD then asked the class if 16 is 15 plus one
more? "So, is 15+16 the same as 15+15 (the double) plus one more?"
Students went to 17+15, and responded with three answers. In justifying their
answers, MD asked students to describe numbers with correct place value ("not
1+1, but 10+10"). She also asked students to repeat the justification from other
students, and to describe if there was another way to solve the problem. She also
asked if anyone used doubles to solve this problem, and then asked students to
explain how this would work.
As students returned to their desks, students asked them to count by 3s in a whisper,
then reviewed this with them when they arrived at their desks.
In the addition lesson, MD drew a number line on the board, and asked students to
show 37+40 on the number line, beginning with 37. She modeled and checked first
to be sure that students had their number line set up. Students were asked ways to
count to 40, and MD displayed 4 sets of ten on the number line. Students then
counted with MD to reach 77. All students were asked to model the problem on
their number lines, and she checked around the room before proceeding.
Students were then asked to use the number line to solve 37+46. MD and the
students added 40 by tens once again (previous problem was left on the board to
help scaffold students), but she stated that she did not know what 77+6 was, and
asked students what she might do. She remarked that she does know how to count
by tens. Students were guided from 77+3=80, then 80+3 more = 83.
MD then presented 48+70, and asked students to create a number line. Students
created the line, and were able to create seven groups of 10. In counting, students
encountered issues between 98 to 108, and 108 to 118.
Students were asked to show their own counting strategy for 48+38, and display it
on their number lines. Students shared out and MD drew various answers. In one
case, students counted up by 10s, then counted the final 8 by 2s. The last group
counted up by 10s four times, and then counted back (subtracted) by 2 to arrive at
the answer.
In reviewing the lesson, teachers commented that students were able to look at other
students and be more certain that their answers were correct, which provided a
comfort level. Some students were tempted to number their lines by ones, but then
adjusted later. When adding up, some students would stop at the number they were
adding with, not by the specific amount. Some were confused about using the
number line to count up.
When adding on the number line, MD states that students could practice by looking
at the problem in both directions. When initial problem was 48+38, starting with
48, students could start with 38 and add up and see if they land on the same answer.
It was helpful that MD was leading them to add to friendly numbers. It may be that
these students did not learn to add to friendly numbers in earlier years, but moving
forward teachers will try it. The Number Talk Provided a good segue to the
addition on number lines. As a result of the demonstration, teachers are considering
using Number Talks during Power Hour time. When a student had a wrong answer
in Number Talks, MD scripted her answer and she was able to realize (without
being corrected) that her answer was wrong. It is important to script students’
discussions in Number Talks, and guide them to the correct answer without
correcting them directly.
MD reiterated that students need to skip count from various numbers by 10s, and be
sure to go beyond 100. When skip counting - students should start by counting up
and down, and add a few numbers each day. Then, students can start at various
places and count up, and at teacher's signal count back down, then up again. MD
added that place value needs to be discussed, as it is a key issue in upper grades.
The group discussed chapter 4, and place value cards that are part of Math
Expressions sets.
Number Talks review
MD distributed the Number Talks review, and guided teachers to the 5 Productive
Talk Moves. This includes how to get multiple students involved, to be sure that
they restate and are listening, and can continue to explain where a student left off.
Wait time is also significant during Number Talks.
Fluency program review
Materials were designed by John Woodward based on studies of memory capacity.
The premise came from working in special education. When we work in small sets
of facts, and then complete cumulative reviews of that set, we will master fluency.
He also looked at what was easy for us to learn, and what is hard. Once students
have mastered 0, 1, 2, 5, and 10 in multiplication, there are not as many facts left
for students to master.
MD displayed Facts Overview. This provides all of the pretests and templates to
begin the program. Grade 3 would likely start with multiplication first. The pretest
is 2 minutes, and students need to pass at 90% proficiency (36 out of 40). MD
would recommend that students take both pretests before the holiday break, to
assess where students are at.
In the Multiplication file, work is presented with new facts mixed in to known facts,
and are presented in a sequence for mastery. Mix is 50% of new facts, and 50% of
known facts. This is based on research that students who can master 70% will
continue to be motivated to achieve, and if they are unsuccessful in the endeavor
they will remain at a low mastery level.
Practice pages and recommended strategies are included. This will lead to extended
facts (for example, 2 x 6 will extend to 2 x 60). In the fluency program, problems
are written vertically. MD recommends that students also see the problems
horizontally. 6 x 2 = ______, or 6 x _____ = 12. EngageNY.org has modules that
set problems up in this manner, and there are challenges in Math Expressions. Some
students get stuck with these - MD recommends to go back to drawing and models
as needed. Also, do not attack math facts in order, but in related math facts (for
example 2, 4, 8 as they are all doubles). A way to introduce doubles is by showing
ten frames, asking students to double those numbers, and then introduce doubling
the digits.
Homework
MD described a system for homework from math researchers:
 2 problems - what I just did in class
 4 problems - cumulative review of what they have done before
 2 problems - thought provoking, where student needs to articulate and write
Another homework suggestion is to take questions from DOK 1 level to DOK 2 or
3 by providing answers to a subset of math questions, and then ask students to
justify why the answer is correct or incorrect.