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Assignment for EDBE 8F83 by Carrie Willick
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Mathematics Portfolio for Teaching
Carrie Willick
EDBE 8F83
Brock University
March 10, 2016
Mathematics Portfolio for Teaching 2
Percent of Empty Space
I created the Percent of Empty Space activity
during my first teaching block, as a teacher-guided
problem solving lesson. This activity requires
students to determine the amount of empty space in
a tennis ball canister containing three golf balls, by
measuring the relevant parameters of the physical
manipulative. Once they have obtained the
necessary measurements, the students can perform
the required volume and percent calculations to
arrive at a solution. Afterward, a class discussion
may be held regarding whether or not 50 percent of
the canister needs to be empty space, and how the
wastefulness of the packaging company is harmful
to the environment. Since this activity was well
received by my students, I decided that it was worth
sharing with my classmates as my Digital
Mathematics Word Problem.
The Percent of Empty Space task aligns well
with the Ontario Ministry of Education’s (2005)
mathematics curriculum expectations, for the
Number Sense and Algebra and Measurement and
Geometry strands of the grade nine academic
course. In particular, this activity relates to the
overall expectation “manipulate numerical and polynomial expressions, and solve first-degree
equations” of the Number Sense and Algebra strand, as well as the overall expectation “solve
problems involving the measurements of two-dimensional shapes and the surface areas and
volumes of three-dimensional figures” of the Measurement and Geometry strand (pp. 30-36).
The Percent of Empty Space problem also addresses the specific expectations “substitute into
and evaluate algebraic expressions involving exponents”, “solve problems requiring the
manipulation of expressions arising from applications of percent, ratio, rate, and proportion”, and
“rearrange formulas involving variables in the first degree, with and without substitution” of the
Number Sense and Algebra strand (pp. 30 - 31). Additionally, this activity relates to the specific
expectation “solve problems involving the surface areas and volumes of prisms, pyramids,
cylinders, cones, and spheres, including composite figures” of the Measurement and Geometry
strand (p. 37).
The Big Ideas embedded in the Percent of Empty Space activity include number sense,
algebra, and geometry. More specifically, students are required apply their number skills related
exponents and substitution, by using formulas to determine the volumes of the tennis ball
canister and the tennis balls. It is also necessary for the pupils to manipulate algebraic
expressions, in order to isolate and solve for unknown variables. Additionally, students are
required to utilize their understanding of volume to identify the appropriate formulas to use, as
well as the necessary parameters to measure.
Picture taken by Carrie Willick on December 13, 2015
Mathematics Portfolio for Teaching 3
The Percent of Empty Space problem allows for several formative assessment opportunities.
Specifically, a teacher may wish to use observations, anecdotal records, prompting and
questioning to gauge student understanding. The assessment information gathered from this
lesson may be used to inform future lesson planning and assessment measures.
During this activity, teachers would do well to pay attention to which students are
participating, and which ones are reluctant to become involved. Although the task is teacher-
guided, the students are required to suggest which parameters to measure, which formulas to use,
and how to complete the solution process. The educator’s role should be limited to providing
hints when students become “stuck”, guiding students in recognizing potential errors in their
solution process, facilitating the classroom discussion by posing critical thinking questions,
managing the classroom environment, and assessing student understanding. As a result, it is
necessary for a variety of students to become involved in the activity, so that everyone has an
opportunity to demonstrate their learning. Furthermore, the activity would not be very effective
if only a few students participated, because the remaining pupils would likely become bored and
off-task.
Educators would also be wise to delegate certain responsibilities to various students, in order
to ensure that a variety of pupils participate. Teachers can assign roles based on the students’
performance levels and learning styles, in order to ensure that the pupils are sufficiently
challenged but are not faced with an impossible task. For example, educators can ask several
hands-on learners to measure the circumference and height of the canister, while several other
tactile learners measure the circumference of the tennis balls. This approach not only increases
the accuracy of the obtained measurements, but also encourages more students to engage in their
own learning. The more students who partake in the activity, the more pupils the educator is able
to assess. Additionally, the students who become involved in the activity are more likely to
remember how they solved the problem, compared to the students who chose to remain passive
learners throughout the lesson.
Although this activity was designed as a teacher-guided problem-solving activity, it could
also be modified to be used as a student-guided group problem-solving activity. For instance,
educators could generate heterogeneous groups and assign group leaders. The group leader could
guide their group members through the measurement and problem-solving phase of the task,
allowing for opportunities of peer tutoring and scaffolding. Afterward, the groups could post
their solutions throughout the room and conduct a gallery walk. This would allow the groups to
see how other students may have solved the problem differently, as well as where they may have
potentially made errors in their own work. Throughout this whole process, teachers can quietly
monitor the students’ behaviour and progress, while also assessing their pupils’ understanding of
the concepts discussed in class. The benefit of this approach to the lesson is that the students are
able to choose which approach they take to solving the problem, rather than having to go along
with their classmates’ suggested solution process. As a result, this alternative problem-solving
method allows for more differentiated learning opportunities, and also acknowledges and
celebrates student differences.
Mathematics Portfolio for Teaching 4
Buying Juice
The Buying Juice question was created by Ela Zasowski, for the Digital Math Word Problem
forum. This problem requires students to consider going to the store to buy Iced Tea, only to
realize that they have two purchase opinions: a can for $1.47 + tax or a juice box for $1.12 + tax.
The pupils are asked to determine the unit price for both choices, and to make a conclusion
regarding which item is the better buy. Since the question does not mention the volume of either
container, students are expected to read these values off of the box and can.
The Buying Juice problem aligns well with the Ontario Ministry of Education’s (2005)
curriculum expectations, for the Number Sense and Algebra strand of the grade nine applied
mathematics course. More specifically, the question addresses the overall expectation “solve
problems involving proportional reasoning” (p. 39). It also relates to the specific expectations
“make comparisons using unit rates” and “solve problems involving ratios, rates and directly
proportional relationships in various contexts” (p. 39).
The Big Idea encompassed by the Buying Juice problem is number sense. More specifically,
this question requires students to apply their number skills related to determining proportions and
unit prices, to a real-world application. The problem also calls students to make sense of the
Picture taken by Ela Zasowski on February 18, 2016
Mathematics Portfolio for Teaching 5
values they receive, by providing a sound conclusion about which container of Iced Tea to
purchase.
If used as a classroom learning activity, Ela’s word problem offers several opportunities for
formative assessment. For example, teachers may wish to present this question in the form of an
entrance or exit card, in order to gauge student learning. Additionally, educators might decide to
utilize this problem for summative assessment purposes, on a test or quiz.
I believe this problem is worth mentioning, because it deals with a real-world application of
mathematics that students are highly likely to encounter, at some point during their lifetime. I
regularly volunteer in high school mathematics classrooms, and frequently hear students
questioning why they have to learn certain concepts. In many cases, if these students are not
provided with a real-world context in which the material might be of use to them, they lack the
motivation to learn. By effectively illustrating the usefulness of knowing how to calculate unit
rates and tax, I strongly believe that Ela’s word problem will encourage applied level students to
become more engaged in their own education.
It is recommended that teachers be aware of their students’ performance level, before
assigning this particular question to their class. The Buying Juice problem entails several types of
calculations, including tax and unit rates. If students do not possess a sufficient knowledge base
to perform the necessary calculations, they may become frustrated and disengaged in the lesson.
Disengaged students may become disruptive to the learning environment, and potentially cause
classroom management concerns. As a result, it may be necessary to modify the Buying Juice
problem, so that it better meets the needs of the students. For example, a teacher may decide not
to require students to calculate the applicable tax in this question, if the majority of the class is
struggling with unit rate calculations. Educators may also increase the difficulty of the task, by
requiring students to covert their cost calculations to various monetary units, such as cents rather
than dollars. Furthermore, a teacher may decide to assign tiered versions of this question, if there
is a significant gap in student performance within the classroom. This particular method enables
educators to differentiate the activity, to meet the various skill levels within the class.
Additionally, educators might want to consider using a healthier drink option for this
question. Student health is a big concern these days, particularly with the rising number of obese
and diabetic children. Therefore, teachers would do well to promote health-wise diet habits in
their daily lessons and activities. By continually presenting healthier alternatives to various
snacks and drinks, teachers may inspire their students to lead a more health-conscious lifestyle,
and ultimately help combat child health concerns.
Mathematics Portfolio for Teaching 6
Crossing the River
For her Leading Learning Activity Presentation, Ela Zasowski crated a problem solving task
that required students to work in pairs or groups of three, to solve a personalized word problem.
Daniel Van Oosten and I were paired together to solve the following question:
Exercise created by Ela Zasowski
Our solution is as follows:
Picture taken by Carrie Willick on January 14, 2016
The Crossing the River problem aligns with the Trigonometry strand of grade ten academic
mathematics curriculum document (Ontario Ministry of Education, 2005). This question
specifically addresses the overall expectation “solve problems involving right triangles, using the
primary trigonometry ratios and the Pythagorean Theorem” (p. 51). Moreover, the Crossing the
Mathematics Portfolio for Teaching 7
River problem relates to the specific expectations “determine the measures of sides and angles in
right triangles, using the primary trigonometric ratios and the Pythagorean Theorem” and “solve
problems involving the measures of sides and angles in right triangles in real-life applications,
using the primary trigonometric ratios and the Pythagorean Theorem” (p. 51).
The Big Ideas encompassed by the Crossing the River problem are trigonometry and
geometry. More specifically, this question requires students to apply their understanding of the
trigonometric ratios to determine a missing angle measure. The problem also calls students to
employ their knowledge of the properties of right triangles, to determine the missing side length.
As a problem solving activity, Ela’s problem offers several formative assessment
opportunities. In particular, teachers would be wise to record observations of student behaviour
in an anecdotal record, in order to adjust future lesson plans and learning activities to meet the
needs of their students. Educators may also wish to conference with their students, in order to
gain further insight into their learning progress, potential misconceptions, and areas of weakness
and strength. This information will better enable teachers to determine which concepts need to be
revisited, as well as which ones are well understood by the class.
I believe this problem is worth mentioning, because it is customized to engage students in
their own learning and promote a positive learning environment. By inserting student names into
the question, Ela was able to generate a sense of excitement in the classroom, which ultimately
inspired my peers and I to be more dedicated to the activity. Additionally, Ela specifically chose
pairs and groups based on which students are able to work effectively and efficiently together.
As a result, there were few classroom management concerns and everyone was able to complete
the task in a timely manner. Overall, I believe that Ela’s Crossing the River question is an
excellent example of how differentiating instruction to meet the needs and interests of the
students, produces a more productive learning environment.
Teachers would be also wise to consider creating heterogeneous pairs or groups for this
activity. Heterogeneous grouping allows for peer tutoring and scaffolding opportunities, by
encouraging students of a wide ability level to converse about curriculum material. Peer tutoring
is often more effective than teacher remediation, because many students are afraid to ask expert
authority figures questions, for fear of appearing unintelligent. Additionally, students are better
able to communicate with one another, because they generally use the same speech patterns.
During this activity, educators would also do well to allow students to “mull” over their own
work. Teachers are always eager to help their students solve problems, but this does not allow
the pupils to develop their own problem solving skills or mathematical literacy. Therefore, it
would be best for educators to only offer assistance when it seems absolutely necessary, and
even still the help should be limited to hints or guiding questions.
One way in which Ela’s Crossing the River question may be improved, is by reducing the
amount of required reading. The problem presented to Dan and I was rather lengthy and wordy,
which delayed our problem solving process. Students who are reading below grade level may not
be able to complete the task in time, which could potentially cause their educator to make
incorrect conclusions regarding their understanding of the course content.
Mathematics Portfolio for Teaching 8
Trigonometry Word Problem
Tanya Paladino created the
Trigonometry Word Problem for the
Digital Math Word problem forum. The
question states that Tanya is standing at
some distance away from a tree, such that
the angle of elevation from her feet to the
top of the tree is 15°. Toby, Tanya’s dog,
is sitting 31.5 inches in front of Tanya,
such that the angle of elevation from his
feet to the top of the tree is 17°. Given
this information, the students are asked to
calculate the height of the tree, to the
nearest inch. It is expected that the pupils
will draw the triangles on the diagram,
along with the given information, as a
means of demonstrating their
understanding of the question.
The Trigonometry Word Problem aligns well with the Geometry and Trigonometry strand, of
the grade eleven college mathematics curriculum document (Ontario Ministry of Education,
2007). In particular, this question addresses the overall expectation “solve problems involving
trigonometry in acute triangles using the sine law and the cosine law, including problems arising
from real-world applications” (p. 73). Moreover, the problem relates to the specific expectations
“solve problems, including those that arise from real-world applications, by determining the
measures of the sides and angles of right triangles using the primary trigonometric ratios”,
“describe the conditions that guide when it is appropriate to use the sine law and cosine law, and
use these laws to calculate sides and angles in acute triangles”, and “solve problems that arise
from real-world applications involving metric and imperial measurements and that require the
use of the sine law or cosine law in acute triangles” (p. 73).
The Big Ideas embedded in the Trigonometric Word Problem are geometry and trigonometry.
More specifically, this question requires students to apply their understanding of the sum of the
interior angles of a triangle, in order to calculate the unknown values necessary for using the sine
law. The problem also calls pupils to employ their knowledge of the sine law, to determine an
unknown side length.
Tanya’s problem may be used as a formative assessment tool. For example, teachers can
assign this question to their class for practice. As students work through the solution, educators
will likely have the opportunity to conference with individual pupils regarding their strengths
and weaknesses.
Additionally, Tanya’s problem may be used as a summative assessment tool. For instance, a
teacher can utilize the question on a test or quiz, in order to evaluate student learning. The in-
Picture taken by Tanya Paladino on January 31, 2016
Mathematics Portfolio for Teaching 9
depth nature of the Trigonometry Word Problem makes it suitable for a thinking or application
question.
In my opinion, this question is worth mentioning because it incorporates a variety of
trigonometry and geometry concepts, such as the sum of the angles in a triangle, the
trigonometric ratios, and the sine law. Often times, students are taught these concepts in isolation
and are not provided much practice with applying them in one question. As a result, the
Trigonometric Word Problem encourages pupils to think about when to use various geometric
and trigonometric relationships to solve problems, as well as how various concepts are
interconnected.
The problem also emphasizes that there are usually multiple valid ways of solving a question.
It is necessary for students to understand this reality, so that they do not always feel obligated to
solve questions according to the method employed by their teacher or peers. For example, Tanya
and I used slightly different computational strategies to solve this problem, and yet still arrived at
the same solution.
If assigning this problem as a formative assessment measure, educators would do well to
allow students to “mull” over their own work. “Mulling” is an important part of becoming a
good problem solver, and teachers should be wary of hindering the development of their pupils’
mathematical thinking skills, by offering inopportune assistance. Therefore, educators would do
well to only offer guidance when it appears to be absolutely necessary, and to limit their help to
hints or guiding questions. Sometimes, all a student requires is a good prompt, in order to
recognize how to proceed with the problem solving process.
Solution by Tanya Paladino Solution by Carrie Willick
Mathematics Portfolio for Teaching 10
Solving Oblique Triangles
Solving Oblique Triangles is a problem solving activity designed by Emily Clemits, for her
Leading Learning Activity Presentation. For this task, Emily instructed my peers and I to
construct a triangle with all three sides, two sides and one angle, or two sides and two angles
labelled. We were also asked to label one of the unknown angles or sides as “x”. After we had
created our diagrams, my peers and I were required to solve another classmate’s triangle for “x”.
Michelle Strauss and I solved each other’s triangles, and came up with the following solutions:
Solutions by Michelle Strauss and Carrie Willick
The Solving Oblique Triangles learning activity aligns well with the Trigonometric Functions
strand, of the grade twelve Mathematics for College Technology curriculum expectations
(Ontario Ministry of Education, 2007). In particular, this question addresses the overall
expectation “determine the values of the trigonometric ratios for angles less than 360°, and solve
problems using the primary trigonometric ratios, the sine law, and the cosine law” (p. 130).
Additionally, the problem relates to the specific expectation “solve problems involving oblique
triangles, including those that arise from real-world applications, using the sine law and the
cosine law” (p. 130).
The Big Idea embedded in the Solving Oblique Triangles problem solving task is
trigonometry. This activity specifically requires students to apply their knowledge of when to use
the sine or cosine law, in order to create solvable triangles. The same knowledge-base is
necessary for pupils to be able to solve a classmate’s problem.
This particular problem solving task may be used as a formative assessment activity. For
example, teachers are able to observe their students’ competency with the sine and cosine law, as
they work through the Solving Oblique Triangles task. Additionally, educators may wish to
Mathematics Portfolio for Teaching 11
conference with individual students, as a means of better gauging their learning progress. The
formative assessment information gathered from this task, may be used to modify future lessons
to meet the needs of the students.
In my opinion, Emily’s Solving Oblique Triangles learning activity is worth mentioning,
because it allows students to construct their own mathematics problems, involving the sine and
cosine law. Most commonly, students are asked to solve teacher-generated questions. Although
this instructional approach is sufficient in terms of familiarizing students with problem solving
techniques, it does not allow pupils much opportunity to engage with the theory behind certain
concepts. For example, the sine and cosine laws can only be applied if certain conditions are met.
While students might know these conditions from solving problems, it is unlikely that they will
fully understand the limitations until they try to construct a solvable problem. Therefore, I
believe there is merit in offering students the chance to construct their own practice problems.
During this activity, it is necessary for teachers to constantly circulate the classroom. By
moving among the various groups of students, educators are more likely to hear student
misconceptions relating to the sine and cosine law. Circulating the classroom, could also
potentially provide teachers with an opportunity to gather useful formative assessment
information regarding student strengths and weaknesses, through observations and student-
teacher conferencing.
Educators would also do well to assign heterogeneous pairs for this activity, in order to ensure
that students of differing ability levels work together. Heterogeneous grouping is beneficial in
the sense that it offers more peer tutoring and scaffolding opportunities, which are likely to
improve student performance. Judiciously deciding student pairs for this activity also enables the
teacher to ensure that only students who work effectively together, are matched. This is
necessary in order to ensure a positive classroom learning environment, with minimal classroom
management concerns.
Mathematics Portfolio for Teaching 12
References
Ontario Ministry of Education. (2005). Mathematics curriculum document: Grades 9 and 10.
Retrieved from http://www.edu.gov.on.ca/eng/curriculum/secondary/math910curr.pdf
Ontario Ministry of Education. (2007). Mathematics curriculum document: Grades 11 and 12.
Retrieved from http://www.edu.gov.on.ca/eng/curriculum/secondary/math1112currb.pdf
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