Flowers
Team members:
Cheri Kinkhorst
Katherine McMurry
Danette O’Keefe
Florence Thisquen
Grade Level:
Third Grade (can be adapted for Second & Fourth)
Title & Topic of Lesson:
Students will know the meaning of factors.
Multiplication and division can be represented as arrays.
There is a connection between multiplication and division.
Where does this fit in:
Collaborative, mid-module assessment in the first unit before the
individual mid-module assessment.
This was chosen to fit into the unit we’re teaching before October.
Engage NY: https://www.engageny.org/resource/grade-3mathematics-module-1
Prior knowledge:
Students need to be able to skip count [Second Grade]
Arrays (Rows and Columns) [Second Grade]
Students need to have had practiced working in groups using
positive talk together.
Standards Addressed:
3.OA.1 (Products of Whole Numbers)
3.OA.2 (Interpret Whole Number Quotients)
3.OA.5 (Apply Properties of Operations as Strategies to Multiply &
Divide)
3.OA.6 (Division as an Unknown–factor Problem)
Future Use:
Area of rectangle
Apply visual understanding of arrays to other products and their
related factors.
Greatest Common Factor problems
Student Behaviors:
Collaboration (using talking frames for discussions)
Explain their thinking (Read/Draw/Write)
Using resources available as needed
Misunderstandings:
Mixing up rows and columns
Equal Groups (meaning)
Appropriate visual representation of arrays
Students will be able to: See the connection between an array and number sentence.
See the connection between repeated addition and multiplication.
See the connection between multiplication and division.
Represent multiplication and division factors in an array model.
Model with mathematics correctly (MP4).
Make use of structure (MP7).
Support for ELLs:
Vocabulary (rows, columns, factors, arrays, garden, plant)
Manipulatives
Collaboration (Collaborative Discussion Frames: I agree with ___
because ___. I also got ___ as my answer, but I found it by ___. I
disagree with ___ because ___. I partly agree with ___, but I found
___ instead.)
Evaluation:
Hear the vocabulary of rows and columns used…I chose 3 rows and
4 flowers in each row because 3 x 4 = 12.
Make correct visual representations using arrays.
Hear ___, ___ times and not ___ + ___ + ___...moving away from
repeated addition and onto multiplication.
Every student is accountable for participating.
Evidence:
Monitoring for collaboration and problems 1 & 2
Status poster for problem 3
Anticipating
Drawings (Question 1)
Monitoring Questions/Teacher
Response
• What number sentence would
match your drawing?
8 + 8 + 8 (Question 1)
8 + 3 (Question 1)
• What does 8 represent? What does
3 represent?
• Explain what your number sentence
means.
35 divided by ___ = 7 (Question 2)
7 + 7 + 7 + 7 + 7 (Question 2)
Unequal groups drawn (Question 2)
• Why did you group them that way?
• Reread the question to verify what it
is asking you to do.
• Are you confused about any part of
the question?
• Are there the same number of
flowers in each row?
60 or another far from correct
response (Question 3)
• Explain your solution to me.
• Where in the question does it say
something to make this a logical
answer?
Possible Student Responses
Sequence of Sharing Desired
Question 1:
• Drawing Only
• First, to make the link between
drawings and number sentences
•8+8+8
• Second, to make the connection
between repeated addition and
multiplication
•3x8
• Third, the previous solutions should
lead to this and the students should
be able to articulate what each
number represents
Question 2:
•5+5+5+5+5+5+5
• First, to make the connection
between repeated addition and
multiplication
• 5 x ___ = 35
• Second, to point out the unknown
factor represented
•7+7+7+7+7
• ___ x 5 = 35, 7 x 5 = 35
• 35 ÷ 5 = ___
Question 3:
• Drawings
• Multiple Factors Listed
• Third, to connect the problem to
unknown factors in division
Connecting
Questions to Promote Student Thinking
and Connections
1. 8 + 8 + 8 = 24
• Can anyone think of another
number sentence to represent the
same relationship? Are there other
number sentences to show this
relationship?
• What’s the relationship between the
number sentences: ___ x ___ and ___
+ ___ + ___ ?
2. 35 ÷ ___ = 5
• What’s the relationship between this
sentence and ___ x ___ or ___ + ___ +
___ ?
3. Arrays Drawn
• What’s the relationship between
rows and columns?
• Can you take this picture and
represent it as a number sentence?
Are there other number
sentences/arrays to show this
relationship?
Flowers Brainstorming Sheet
Problem 3
Problem 2
Problem 1
Possible Solutions
Group Members:
Possible
Misconceptions
Teacher
Questions
Flowers Monitoring Sheet
Strategy
Problem 1
Drawing
Repeated
Addition
Multiplication
Other Solutions
Problem 2
Drawing
Repeated
Addition
Division
Missing Factor
Other Solutions
Problem 3
Drawing
Number
Sentence
6x2
3x4
12 x 1
Other Solutions
Group Members:
Who and What
Sharing Order
Flowers
Student Task
Core Idea 2
Number
Operations
Core Idea 3
Patterns &
Functions
Use knowledge of multiplication to form flowerbeds with
equal rows. Use the inverse of multiplication to find the
number of rows to plant. Use mathematics or diagrams to
show their thinking.
• Understand multiplication as repeated addition, an area
model, an array, and an operation on scale.
Understand patterns and use mathematical models to
represent and to understand qualitative and quantitative
relationships.
• Illustrate general principles and properties of operations,
such as commutativity, using specific numbers.
Third Grade - 2002
pg. 13
Third Grade - 2002
pg. 14
Third Grade - 2002
pg. 15
Third Grade - 2002
pg. 16
Looking at student work on Flowers While this problem tests a student's knowledge about multiplication
facts and problem solving, it also gives teachers an opportunity to see
how students are progressing in the continuum of multiplication.
Student A understands that the problems are based around number
and uses multiplication and division to solve part 1 and 2. The student
does not need to verify these facts by using the table. In part 3 the
student knows the facts and offers alternative solutions. The student
simplified the drawing to show the alternative arrays for part 3.
Student A
Third Grade - 2002
pg. 17
Student B also has a score of 8. This student uses addition,
multiplication, and drawing to solve for the total answer in part one.
The student uses division and drawing to solve part two.
Third Grade - 2002
pg. 18
Student B
Student C also has a score of 8. This student understands the idea that
multiplication is about equal groupings and how arrays work. The
student also knows that he can simplify the flowers to numbers. The
student is still not completely comfortable with all the multiplication
facts and is still transitioning between counting on and multiplication.
Third Grade - 2002
pg. 19
Student C
Student D can use number facts to help solve parts 1 and 2. The
student has difficulty with two missing constraints in part 3. The
student knows that 3 x 4 = 12, but is unsure what that represents in the
context of the problem. The student then does more arithmetic to get
an answer for flowers that makes no sense. The student knows
number facts, but is less clear about the multiplication representation.
Third Grade - 2002
pg. 20
Student D
Third Grade - 2002
pg. 21
Student D, Page 2
Student E can successfully solve parts 2 and 3, but struggles with the
incomplete diagram in part 1. The student adds the existing flowers in
the diagram to get a total of 8, rather than thinking about the context of
the problem and making equal rows. This is a common error.
Third Grade - 2002
pg. 22
Student E
Student F shows work of low scoring students. This student knows a
lot of number facts, but can't use them in a practical application. The
student chooses an addition example to put in the answer blanks.
Then the student does not illustrate that fact in the picture below. The
student sees the picture as an art project, rather than a mathematical
model.
Third Grade - 2002
pg. 23
Student F
Teacher Notes
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________________________
Third Grade - 2002
pg. 24
Grade 3 - Flowers
Flowers
Mean: 5.36, S.D.: 2.69
6000
100%
90%
Frequency
5000
80%
70%
4000
60%
3000
50%
40%
2000
30%
20%
1000
10%
0
0
1
2
3
4
5
6
7
8
905
802
1160
1181
802
1660
1249
1242
5266
%<=
6.3%
12.0%
20.1%
28.4%
34.0%
45.6%
54.4%
63.1%
100.0%
%>=
100.0%
93.7%
88.0%
79.9%
71.6%
66.0%
54.4%
45.6%
36.9%
Frequency
0%
Score
The maximum score available on this task is 8 points.
The cut score for a level 3 response is 5 points.
Most students (88%) could either find the total flowers planted in question 1 and
explain how they found their answers or give 2 factors that multiply to 12 for question
three. A little more than half the students (66%) could answer questions 1 and 3, find
the total when they know the number of rows and number of flowers in a row or find
the number of rows and flowers in a row when they know the total. Only 36% of the
students could combine information about the total number of plants and number of
rows to find the number of flowers in a row in question 2.
Third Grade - 2002
pg. 25
Flowers
Points
0/1
2
3/4
5
8
Understandings
Most students could do all
of question one or give the
right multipliers for
question 3.
Students with this score
could generally do all of 1
and part of question 3.
Misconceptions
Most students in this category
showed no understanding of the
problem. Students did not
understand equal rows.
Answers often had no relation to
numbers in the task with
answers like 2002 or 137.
Students were inconsistent in
their approach to the problem.
Students may have had
difficulty with the format and
only put one correct multiplier
and the number 12. Then they
made an appropriate picture.
Students could find the total Students had difficulty with
flowers when given the
inverse operations. In question 2
number of plants per row
they did not easily recognize
and the number of rows.
that 5 x 7 = 35, so the number of
Given the total number of
plants per row should be 7.
plants students could find
the number of plants to put
in each row and the total
number of rows.
In general students did very In question 3, About 50% of the
well with this problem.
students used 2x6 as their
30% of all the students
combination, and 40% used 3x4.
showed math facts in
However those who used 3x4
addition to their drawings.
were more likely to show their
About 8 % of the students
math or use modeling. Teachers
knew that could use
predicted that students would
modeling instead of
misinterpret question 2 and
drawing out all the
multiply 5 x 35. This happened
necessary flowers.
only 5% of the time. In question
3, less than 2% of the students
made rows of 12. Less than 5%
of the students tried to draw all
the flowers. Most of those did a
clear enough drawing to show
correctly the section to get credit
for question 3.
Third Grade - 2002
pg. 26
Based on teacher observations, this is what third graders seemed to know and to be
able to do:
• Draw arrays and count to find the number of flowers in a row
• Use multiplication facts to find total number of flowers and to find
combinations of rows and flowers to make twelve.
Areas of difficulty for third graders:
• Using inverse operations to work from the product to the factor
pairs
• Using mathematics to show their thinking, writing number
sentences instead of drawing pictures
• Using some form of modeling to simplify making the array
• Filling in partial diagrams
Teacher Notes
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
Questions for Reflection on Flowers:
How many of your students:
Made drawings
only
•
•
•
•
Used number
sentences
Simplified the
flowers
Offered multiple
solutions for 3
Do you students seem comfortable with the idea of equal groups
for multiplication? Do they seem to understand the idea of an
array?
Do your students have a good understanding of operation or like
the final students above are they unclear about what the results of
their calculations represent?
How many of your students had difficulty with the incomplete
diagrams in parts 1 and 2? Could they use math facts to go around
this obstacle?
Are students' experiences pushing them toward efficient, accurate
diagramming and away from drawing detailed pictures of items?
What opportunities have students had with mathematical
modeling? How do you help transition them from art to using a
representation, like a tally mark or the circle used by the first
student?
Third Grade - 2002
pg. 27
•
•
•
Did students try to do perspective drawings versus abstractions
when filling the blank chart on question 3?
Do you think your students were focusing more on the
mathematics and factors than the quality of the flowers? How can
you help students make this transition?
What are the classroom expectations around knowing multiple
ways to solve a problem?
Teacher Notes
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
Instructional Implications:
Students need more work with inverse operations. When learning to
multiply, working with number fact families and using those facts in
problems should be a natural part of the learning experience. Making these
connections should not wait until division is formally taught. Students need
to be encouraged to start making models, such as tallies or x's to represent
symbols. They need to focus more on the mathematics, than on the actual
physical representation. Many students should have been able to solve
problems one and two with just number sentences or problems. Students
need to see calculations as a valid way of showing work. Students should
have opportunities to work with problems with multiple solutions, such as
question 3.
Teacher Notes
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
____________________________________________________________________
Third Grade - 2002
pg. 28