mathematics portfolio for teaching

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


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.


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


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.


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


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.


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


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


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


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.


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

http://www.edu.gov.on.ca/eng/curriculum/secondary/math910curr.pdf

http://www.edu.gov.on.ca/eng/curriculum/secondary/math1112currb.pdf

mathematics portfolio for teaching

Documents