# Multiplying Turtles

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```					                                                                        NYSATL Learning Experience Submission: Multiplying Turtles

Modeling Multiplication
Grade 2, May 2003, Kathy Verzoni

Learning Context

This three-day lesson provides a conceptual introduction to the concept of
multiplication for second graders. Students are taught to think about multiplication as
repeated sets. The teacher begins with off-computer, whole group instruction with
magnetic "monster baby set cards." Once students grasp ideas related to multiple
sets, the teacher introduces multiplication statements as mathematical language with
the same meaning. Students then interact with the "Multiplying Turtles" MicroWorld,
creating visuals of multiple sets in order to model multiplication statements and
determine the products of factors (with values up to 5). Experiences making sets with
Unifix cubes and relating their models to multiplication statements and creating
drawings of sets in order to illustrate given multiplication statements are alternated
wit h their use of the computer MicroWorlds in order to facilitate transfer of and assess learning across contexts.

Prior to this lesson, students have learned to (a) count
and group, (b) model, explain, and solve basic addition and
subtraction problems, and (c) relate the mathematical language
and symbolism of addition and subtraction to problem
situations and informal language. These mathematical
reasoning skills, coupled with rudimentary abilities to recognize
written English words are prerequisite for this lesson.

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NYSATL Learning Experience Submission: Multiplying Turtles

Rationale

NCTM's Curriculum and Evaluation Standards call for mathematics as problems solving, connections, communication, and
reasoning. This lesson epitomizes such an approach.
For instance, one of the problem solving standards calls for student ability to "formulate problems from everyday and
mathematical situations." One of the culminating activities in this lesson calls for students to create their own "picture stories"
to illustrate or model multiplicative situations. They draw from everyday experience in order to invent problems that require
multiplicative thinking.
Regarding mathematics as connections, the standards call for ability to recognize relationships between mathematics and
daily lives, and among different topics in mathematics and other curriculum areas. In this lesson, students relate the
multiplicative models that they interact with on the computer to concrete and visual models that they experience or create in
the classroom. These activities set the foundation for use of multiplicative understandings in real life contexts. Opportunities
for meaningful application of learned mathematics increases motivation for learning.
The ability to relate everyday language to mathematical language and symbols is one of the "Mathematics as
Communications" standards. Involving students in relating English language to mathematical expressions and concrete or visual
models is a powerful approach to helping children develop operation meaning (Van De Wall, 1998).
Finally, this lesson engages students in applying reasoning processes when analyzing mathematical situations. Students
apply inductive reasoning when learning to use mathematical expressions to generalized representations of multiplicative
situations.

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NYSATL Learning Experience Submission: Multiplying Turtles
Intended Learning Outcomes, Related NYS Standards Key Ideas, and Specific NYS Performance
Indicators

Intended Learning Outcomes                                                                                  Specific NYS Performance
Related NYS Standards
(Hereafter referred to as ILO’s)                                                                                      Indicators
The student will be able to:                    Mathematics, Science, and Technology Standard 2        Students use a variety of equipment and
Technolog

1. use mice and keyboards as input devices.     (Elementary Information Systems) Key Idea 1.           software packages to enter, process,
2. follow basic written and verbal              Information technology is used to retrieve, process,   display, and communicate information in
y

instructions when using simulation and/or   and communicate information and as a tool to enhance   different forms using text, tables, pictures,
modeling applications.                      learning. (p 8)                                        and sound.
The student will be able to:                    Mathematics, Science, and Technology Standard 1        Students use special mathematical notation
3. represent multiplicative situations          (Elementary Mathematical Analysis) Key Idea 1.         and symbolism to communicate in
(factors < 6) using (a) English phrases     Abstraction and symbolic representation are used to    mathematics and to compare and describe
(eg. four sets of triplets), (b)            communicate mathematically. (p 2)                      quantities, express relationships, and relate
mathematical expressions (4 x 3), and (c)                                                          mathematics to their immediate
concrete and visual models.                                                                        environments.
Mathematics

4. use concrete or visual models in order to
represent multiplicative expressions        Mathematics, Science, and Technology Standard 3        Students use multiple representations
(factors < 6) and find their products.      (Elementary Mathematics, Modeling / Multiple           (simulations, manipulative materials,
5. given English descriptions and visual        Representation) Key Idea 4. Students use               pictures, and diagrams) as tools to explain
models, distinguish between additive and    mathematical modeling / multiple representation to     the operation of everyday procedures.
multiplicative situations.                  provide a means of presenting, interpreting,
communicating, and connecting mathematical             Students use physical materials, pictures,
information and relationships. (p 15)                  and diagrams to explain mathematical ideas
and processes and to demonstrate geometric
concepts.

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NYSATL Learning Experience Submission: Multiplying Turtles

Assessment Plan

Formative

One form of student assessment involves analysis of student responses during whole class discussions. Student
achievement of intended learning outcomes is monitored as students participate in discussion and translations between the
language of English, the language of mathematics, and the language of models (ILO #s 3 and 4). During class discussion,
students are asked to give English descriptions (ie. 3 sets of quads…) and equivalent multiplicative statements (…is like saying 3
x 4) when shown picture models.
In a similar manner, analysis of discussion also provides formative assessment of student ability to
find solutions to multiplication statements using modeling (ILO # 4). During discussion, students use
“monster” twin, triplet, quad, and quint cards to model multiplicative statements (ie. 2x5) and find their
products. These abilities are also assessed as the teacher watches student use of the microworlds
(correctly creating models of various multiplication statements) and their recording of product
determinations on their guiding worksheets.
Tendencies of students to add rather than multiply factors give evidence of lack of conceptual
differentiation between addition and subtraction (ILO # 5). This is a common difficulty among primary
Student achievement of technology learning outcomes (ILO #s 1 and 2) is monitored as students
work with the "Multiplying Turtles" ( Available:
http://www.mohonasen.org/staffdev/microw/multuweb2com.html ) and the “Models of Multiplication” (
Available: : http://www.mohonasen.org/staffdev/microw/modelm.html) MicroWorlds. During this time,
student understanding of sequencing, basic written English words, buttons, and keyboard and mouse use
are assessed as the teacher circulates the classroom observing and facilitating student use of the
microworlds.
Formative monitoring of student performance informs teacher pace and possible needs for re-teaching along the way.

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NYSATL Learning Experience Submission: Multiplying Turtles
Summative

Students' efforts to use unifix cubes in order to model and find the products of multiplicative expressions provide one
means for summative assessment (ILO #'s 3 and 4). Analysis of student efforts to create picture models that represent
multiplicative relationships and their equivalent mathematical statements provides a second means (see attached worksheet
entitled “Understanding Multiplication” and the “Models of Multiplication” MicroWorld (Available:
http://www.mohonasen.org/staffdev/microw/modelm.html )). Finally, student ability to distinguish between multiplicative and
additive relationships (ILO #5) is assessed with an independent activity where students interpret pictures with captions and
identify the relationship (additive or multiplicative) that the picture represents (see attached worksheet entitled “Name that
Operation”).

Through group discussion and guided practice, the teacher circulates in order to check progress. The worksheets are
designed to show student progress at modeling and finding products. When the teacher sees errors, he/she stops and works
with the student to correct misconceptions. This feedback helps students to monitor their progress. As students work with the
materials that I designed for this lesson, their experiences give me new ideas for guided practice and assessment materials. I
created the “Models of Multiplication” MicroWorld during the evening of the second day of the lesson that I presented with a
class of 2nd grade students in October. At the end of the second day, through formative monitoring of their work, it was evident
that several were not making correct translations between picture and number models. That evening, I designed the “Models of
Multiplication” MicroWorld in order to develop these abilities. We used the MicroWorld during the next day’s lesson as guided
practice. “Models of Multiplication” can also be used as an assessment tool.

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NYSATL Learning Experience Submission: Multiplying Turtles
Descriptors of Student Proficiency

Intended Learning Outcome                                                              Levels of Student Proficiency

The student will be able to:                                   Developing                         Proficient                  Distinguished
1. use mice and keyboards as input devices.            Student requires guidance         Student uses input devices     Student demonstrates fluid,
with use of the keyboard or       independently                  accurate, and efficient use
mouse                                                            of input devices
2. follow basic written and verbal instructions        Student requires guidance         Student works semi-            Student works independently
when using simulation and/or modeling applications.    interpreting directions and       independently, requiring       at every stage of use. All
sequencing inputs                 minor adult and/or peer        computer models are
guidance once directions are   generated correctly with no
3. represent multiplicative situations (factors < 6)   Student has difficulty            Student makes correct          Student makes correct
using (a) English phrases (eg. four sets of            making translations between       translations most of the       translations all of the time.
triplets), (b) mathematical expressions (4 x 3),       English phrases,                  time.                          He or she provides clear
and (c) concrete and visual models.
mathematical expressions,                                        models or English
and/or models.                                                   descriptions and generates
mathematical expressions
with fluency.
4. use concrete or visual models in order to           Student has difficulty            Student creates a correct      Student creates models and
represent multiplicative expressions (factors < 6)     generating the model to           model and finds the product    finds products with fluency
and find their products.                               illustrate the expression and     successfully most of the       and ease. He or she clearly
does not consistently             time.                          explains process to students
identify the product of the                                      in need of additional
factors.                                                         instruction and practice.
5. given verbal English descriptions, distinguish      Student has difficulty            Student correctly              Student differentiates
between additive and multiplicative situations.        differentiating between           differentiates between         correctly and with ease. He
situations. He or she guesses     situations.                    examples of each.
during discussion and
independent work.

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NYSATL Learning Experience Submission: Multiplying Turtles

Student Work

I’ve chosen the work of three students in order to illustrate three levels of student performance.

Kyle - Developing

Kyle engaged actively during class discussion and guided practice times. His responses to “picture story” problems posed during class
discussion were mixed. Some were correct, others were incorrect at first and then corrected as a result of scaffolded questioning. The
following paragraphs refer to Kyle’s work attached as Appendix A.
With his partner, Kyle completed the “Multiplying Turtles” sheets while interacting with the “Multiplying Turtles” MicroWorld. He
needed extra guidance using the microworld (ILO’s 1 and 2). As Kyle worked through from twins, to triplets, to quads, and finally to quints, we
see that he became confused at the quads level. His responses to 1, 2, and 3 sets of quads lead me to believe that he was clicking on the
triplets button in error (instead of the quads button). His understanding of the multiplicative relationship was not strong enough that he
would notice something wrong when 1 set of quads (or 1 x 4) turned up with 3 turtles. His errors continue through until “4 sets of quads”
where he generated the correct product again. He did not notice a problem, or break in pattern, with his previous responses. While working
with quints, we see that he was probably clicking on quads in error. Again, he does not notice discrepancies in pattern or a need to go back
and re-work problems (ILO 4). Kyle’s diagrams show totals, but do not show set definitions (ILO 4).
Kyle’s multiplication models show understanding at a developing level. He was able to generate a correct model for 2 sets of 3. On the
next page we see an incorrect model for 3 sets of 5. He has reverted to an additive concept and has determined that 3x5 equals 8. His
picture model does not match either the multiplicative model or his additive understanding of it. With scaffolding he was able to eventually
create the (3 sets of 5) matching picture model and complete the number model correctly (ILO’s 3, 4, and 5).
During the “Understanding Multiplication” assessment Kyle created correct picture models for both 2 sets of 4 and 3 sets of 2.
However, he was not able to generate the equivalent number models (ILO 3).

Kody – Proficient

Kody engaged actively during class discussion and guided practice times. His responses to “picture story” problems posed during class
discussion were usually correct. He corrected incorrect responses quickly with minimal scaffolding. The following paragraphs refer to Kody’s
work attached as Appendix B.
With his partner, Kody completed the “Multiplying Turtles” sheets while interacting with the “Multiplying Turtles”
MicroWorld. He led his partner in use of the microworld and interacted with the microworld with minimal adult guidance (ILO’s 1 and 2). His

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NYSATL Learning Experience Submission: Multiplying Turtles
responses to most problems are correct. Kody’s diagrams show totals and set definitions (ILO 4). However, he does have some difficulty with
“switching” in his picture models on the “Multiplying Turtles” sheets. For example, he showed 3 sets of 5 when he should have showed 5 sets
of 3.
Kody’s multiplication models show understanding at a proficient level. He generated correct picture models for all of the number
models. The first one that he attempted was 3 sets of 5. Kody’s first printout shows only 2 sets of 5. When his mistake was pointed out, he
easily corrected his picture model and multiplication equation (ILO’s 3, 4, and 5).
During the “Understanding Multiplication” assessment Kody created correct picture models for both 2 sets of 4 and 3 sets of 2.
However, he generated the equivalent number models, but, in error, used the “plus sign” instead of the “times” sign. (ILO 3).

Andrew – Distinguished
Andrew was from a high ability mathematics class. He engaged actively during class discussion and guided practice times. His
responses to “picture story” problems posed during class discussion were always correct. The following paragraphs refer to Andrew’s work
attached as Appendix C.
With his partner, Andrew completed the “Multiplying Turtles” sheets while interacting with the “Multiplying Turtles”
MicroWorld. He interacted with the microworld with no adult guidance (ILO’s 1 and 2). His responses to all problems are correct. Andrew’s
diagrams show totals and set definitions (ILO 4).
Andrew also created multiplication models with ease. He made no errors (ILO’s 3, 4, and 5). Andrew completed two additional
assessments that Kyle and Kody did not complete. In “Name that Operation,” Andrew consistently distinguished between addition and
multiplication models correctly. During the problem posing assignment he authored an original story problem that required multiplication (16 x
19), correctly solved the problem, and converted cents to dollars and cents.

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NYSATL Learning Experience Submission: Multiplying Turtles

Procedure

Day 1
Begin the class with a fun discussion of some hypothetical monster moms and have students tell about types
of babies that monster moms might have. Elaborate the scenario explaining that monster moms have monster
babies and that the monster moms that we will be talking about today often have babies in sets. Use the monster
baby cards to talk about sets of twins, triplets, quads, and quints. Begin using the cards to pose questions such as….
This monster mom had 3 sets of quads. How many monster babies did she have? Gradually pull scaffolding away
during the discussion to the point where students are creating the scenarios, modeling with the cards, and giving
the total number of monster babies. The final step of the introduction is to introduce the concept of multiplication,
explaining that they were already multiplying, and giving the syntax and symbols for multiplicative expressions. (~15
minutes) During this time, mentally note conceptual problems that students have with multiplication. Keep this time
short and fun. Don't worry about students that don't quite "get it" yet. The lesson is designed to keep addressing
the same ideas in multiple contexts.
Next, introduce the "Multiplying Turtles" MicroWorld. Demonstrate use of the microworld, showing
students sequences necessary to model the situations represented in the problems on the worksheets (see
attached). Once you have demonstrated modeling of a few problems and have provided students with instructions
for completing worksheets, allow them about 15 - 20 minutes to work independently. Circulate while they work. A
common problem that students have is that they forget to press "NEW" between models. Remind them and watch
for incorrect answers on the sheets. When "NEW" is not pressed between models, the previous sets of turtles remain on the screen.
Students will have another chance to work with the "Multiplying Turtles" MicroWorld during the Day 2 lesson. They will not have worked
through all of the problems at the close of this part of the lesson.
As closing, re-group students for whole-class modeling practice with Unifix cubes. Give verbal scenarios (eg. 3 sets of 4 or 3 sets of
quads) and have students (1) create the models using Unifix cubes, (2) translate to mathematical expressions (eg. 3 x 4), and (3) find the
products. (~15 minutes)

Day 2
Begin Day 2 with a review of the prior day's learning. Use the “monster baby” set cards again. Have students explain what they
learned, giving examples of translations between English, mathematical symbols, and models. (~10 minutes)

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NYSATL Learning Experience Submission: Multiplying Turtles
Return students' "Multiplying Turtles" problem sets and re-introduce the "Multiplying Turtles" MicroWorld. Remind students of
sequences for use of the microworld and take some time to ensure that each student knows where he or she has left off in the problem set.
Direct students to complete their problem sets using the microworld to model their problems. Assign students to computers and allow ~15
minutes for completion.
Re-group the class and introduce use of the “Multiplication Models” MicroWorld. Demonstrate the use of buttons to get around in the
microworld and then show students how to create the picture model and complete the number model for 2 sets of 3. In order to draw circles
around the sets, students will need to know how to use the pencil tool (pencil tool not available on Web version of microworld). Begin another
session of hands-on computer time. Allow ~20 minutes. As students finish number models, they may print them using the print button.
Re-group students for closing, allowing time for students to show and discuss their printouts.

Day 3
Prepare students for independent assessment time. Distribute the “Understanding Multiplication” sheets (attached as Appendix D).
These sheets contain an English language expression at the top of each page (3 sets of 2, 2 sets of 4). Direct students to draw a model of
the expression by creating sets of anything they would like (hearts for example). At the bottom of the page, have students write the
equivalent mathematical expression and show the product (eg. 3 x 2 = 6). (~10 minutes) Students' finished work from this activity will provide
formal evidence of their progress with ILO #'s 3 and 4.
In classes where most students do very well with initial assessments, assess student abilities to differentiate between additive and
multiplicative situations (ILO #5) using “Name that Operation.” Distribute “Name that Operation” handouts and direct students to identify
each of the 5 situations as multiplicative or additive. Lastly, conduct a problem posing activity in order to assess achievement of ILO’s 3, 4,
and 5. Distribute blank sheets of paper and direct students to create their own multiplicative story problems. Each problem’s solution should
contain both a picture model and the symbolic representation of the factors and product.

Resources

“Monster Baby” set cards (samples available via email: kverzoni@mohonasen.org

PC or Macintosh computers (one per student is optimal)

The “Multiplying Turtles” MicroWorld, available free for PC users, request via email to kverzoni@mohonasen.org
Or online at: http://www.mohonasen.org/staffdev/microw/multuweb2com.html

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NYSATL Learning Experience Submission: Multiplying Turtles
The “Modeling Multiplication” MicroWorld, available free for PC users, request via email to kverzoni@mohonasen.org
Or online at: http://www.mohonasen.org/staffdev/microw/modelm.html

OPTIONAL: MicroWorlds Logo software by LCSI (see website: http://www.microworlds.com/index.html)

Handout worksheets: Multiplying Turtles (4 pages), Understanding Multiplication (2 pages), Name that Operation (1 page) available online at
http://www.mohonasen.org/staffdev/primaryti/timicroworlds.htm , pencils

Unifix cubes or other manipulatives suitable for modeling sets

Instructional/Environmental Modifications

This lesson facilitates learning at different levels for students of various abilities. Students with high quantitative reasoning abilities
will gain a strong conceptual understanding of multiplication as a result of the lesson. They will recognize situations where multiplication can
be used and will be able to differentiate between multiplication and addition. They will develop mental math strategies to find products of
factors less than 6 on their own, in order to avoid having to create visual models and count.
Students with low quantitative reasoning abilities may not fully grasp the concept of multiplication. They will likely have difficulty
differentiating between multiplication and addition. However, they will gain understanding of sets and counting skills and they will gain
practice with using words, models, and symbols in order to represent mathematical situations.
Average ability students will gain some understanding of multiplication. They will also gain experience with the idea of multiple
representations – that there are many ways to communicate the same phenomenon or idea. This is a concept important to all subject areas.
For academically diverse classes, help from a teacher assistant is beneficial. Teachers and assistants should interact with students as
they raise questions, scaffolding student thinking to more advanced levels.

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NYSATL Learning Experience Submission: Multiplying Turtles

Time Required

Planning
Allow time to gain comfort with the MicroWorlds Logo programming environment and the “Multiplying Turtles” and “Modeling
Multiplication” MicroWorlds (~ 30 minutes). Three sets of handouts must be copied. Depending upon the computer use skills of your students,
you may wish to load the “Multiplying Turtles” and “Modeling Multiplication” MicroWorlds before the lesson begins.

Implementation
Three class periods, approximately one hour per class

Assessment
Although assessments are administered during implementation time, allow time (~ 30 minutes) to analyze student work, looking for
evidence of learning over the three-day experience and categorizing levels of proficiency.

Reflection

This lesson introduces second graders to the concept of multiplication. Emphasis is on conceptual understanding, not on memorization
of multiplication facts. Students interact with models of multiplication in various contexts and use multiple representations in order to
communicate multiplicative relationships.

Since I used this lesson during the month of October, I had never attempted an introduction to multiplication at such a young age
before. My observations of students throughout the lesson gave me insight to their reasoning abilities and processes. For several, the
concept of “sets” was new. Some have trouble identifying quantities over 10 and are not yet proficient with addition. For these students, gain
in counting skills, understanding of sets, and set making were the notable achievements. A number of students were classified as AIS for
reading. In many cases lack of reading ability impeded progress regarding understanding of multiplication. For average students, this lesson is
best placed during the winter of second grade. High ability math students however, develop conceptual understanding of multiplication from
this set of experiences as early as spring of first grade.

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NYSATL Learning Experience Submission: Multiplying Turtles

I have led this lesson three times. The first was during May of last year with a non-tracked first grade class. The second and third
were during October and February with average and high tracked second grade classes, respectively. Each time, the classroom teachers
participated and provided suggestions for modifications. As a result, the lesson evolved over time. The lesson and accompanying microworlds
will continue to evolve as I participate in regional peer review and use it with new groups of learners.

Regardless of subject matter, one of our ultimate goals as teachers is to develop meaningful understandings, reasoning, and problem
solving skills. What types of classrooms promote this kind of learning? Here are some characteristics (adapted from Stein, M. K. & Smith,
M.S. (1998). Mathematical Tasks as a framework for reflection. Mathematics Teaching in the Middle School. 3, 274) that one might look for.
- Care is taken to ensure that learning experiences build on students’ prior knowledge and that understandings build through
connection making between prior conceptual knowledge or procedural skill and new learning.
- The teacher scaffolds student thinking and reasoning without giving too much away. Extended problem solving experiences are
structured enough to prevent floundering, yet are not overly structured. Overly structured learning experiences reduce true
problems to mere exercises.
- Learning activities are orchestrated in such a way that students are able to monitor their progress throughout extended problem
solving tasks (through use of rubrics, for example).
- The teacher frequently models his or her thinking processes and cognitive strategies when faced with problems. In turn, the
teacher expects his or her students to explain their reasoning strategies and justify their solutions or decisions.
- Exploration is appropriately timed. Task focus tends to diminish as length of time allowed increases beyond sufficiency. Too little
time communicates the message that the end-result procedure, concept, or relationship is the priority and that experiential
aspects of the inquiry process are secondary.

Achieving that “press” for reasoning and connection making in the classroom is a continual struggle. Through reflective study and action
research we grow in our abilities to plan and conduct learning episodes that engage students in thinking conceptually, reasoning, making
personal sense of things, and making connections. The present lesson epitomizes such an approach. I hope to continually refine the lesson and
the microworlds as teachers use the resources and provide feedback regarding student progress and stumbling blocks.

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