Implementing Geography Into Mathematics

Integrated curriculum's are a great way to make classes fun and exciting by combining two different topics from different subjects. This is great for learners who have interests outside of math and allows them to learn about how mathematics applies to the subjects that they be more interested in. In today's blog I will be talking about an activity introduced to our class that integrates geography into the mathematics curriculum and can be used to teach many different topic areas!

For this activity students will need access to a computer or some sort of device that can access the internet. A computer would work the best though as the activity requires precise clicking.

Google maps/Google Earth has an interesting feature built in where you can place markers at various points on a satellite image and measure the distance between these markers. This is great for the measurements unit in the grade 9 locally developed, applied and academic classes and allows students to explore shapes found all around the world!

It can be used as a great way to teach perimeter to students as the measurement tool on Google Earth/Maps allows students to measure the distances from different markers. You can give students the coordinates of various famous places that replicate the shapes that need to be covered in the curriculum such as rectangles, squares and pyramids and ask students to determine various things such as perimeter and even ratios! For example, students can be asked to determine the perimeter of BMO field which is a rectangle. Once they find the perimeter of the field they can find the perimeter of any other rectangular shaped object they want. Once they have both perimeters you can ask them to use a ratio to find out how much bigger/smaller BMO field is compared to their object, which covers two topics at once.

Image retrieved from: http://travelingcanucks.com/wp-content/uploads/2011/10/Pyramids-of-Giza-Egypt-20.jpg

This activity is excellent too because it provides important geography and technology skills such as how to use coordinates to locate an object and how to use google maps to locate objects which is a very useful skill in today's technological world.

Strategies for Teaching Trigonometry

Various topics in trigonometry are covered in the mathematics curriculum throughout high school. In grade 10 students are introduced to similar triangles, right angle trigonometry and the sine and cosine law to calculate missing angles or side lengths. In grade 11 and 12 trig concepts become even more prevalent and abstract involving topics such as the ambiguous case, trigonometric identifies and the graphing and transformations of trigonometric functions.

In an earlier blog post I discussed the use of spaghetti to produce a graph of the sine function, which was one activity to make trigonometry more hands on. In this post I will discuss a few more options that can make trig a little more hands on within the classroom.

One of the ways trig can be used within the classroom is to look at Pythagorean's theorem and ride angle trigonometry. All you need is some tape, protractors, measuring sticks and square floor tiles. Students can use the tape to cover the gaps between multiple floor tiles, if the tape goes over 4 tiles then the length of that side will be 4. The tap can be used to create a right angle triangle where the length of each side is visually determined by how many squares the tape crosses. The math involved here will be with the angle and the hypotenuse side length as it goes diagonally through multiple squares, making an exact length calculation difficult. Students can measure the lengths of each side with a meter stick or calculate the length of the missing side using Pythagorean's  Theorem. The angle can also be determined using a protractor. Various groups within the classroom can create triangles of different size and record their results to share with other groups. After each group has created a triangle and calculated their missing variables a class discussion can be had about the results discussing how different size triangles effects the angles.

 

Image retrieved from: https://www.teachengineering.org/activities/view/nyu_triangles_activity1

Trigonometric identities, another trig topic seen in the senior level classes, are often infamous among students for being one of the more challenging topics. My goal within my classroom would be to eliminate that stigma against trig identities by creating a classroom culture where student's know its okay to be wrong sometimes, what's important is that you learn from your mistakes and move on! Which is why its important to tell students with trig identities that they require lots of trial and error sometimes, they don't always work out perfectly the first time which is perfectly okay. Another good way to introduce identities is by using non-trig examples first such as:

3x + 4 = 7

Hopefully these strategies can help eliminate the stigma in our classrooms surrounding trigonometry and make the topics fun and intriguing for all the different types of learners in our classrooms!

Grade 12 Mathematics

The content in grade 12 mathematics classes are often difficult to make hands-on and interactive due to the abstract nature of the topics. In class we presented a few activities that can be done in grade 12 data management, advanced functions and calculus and vectors that can be used.

One of the most interesting activities presented was a math escape room. If you are unfamiliar with escape rooms, escape rooms are rooms that are full of puzzles that need to be solved in order to achieve a certain goal (usually escaping the room) but within a classroom the goal can be to open a box. Escape rooms involve a large amount of collaboration and problem solving. They are also an excellent team building activity.

Escape rooms are a great activity to do in mathematics classrooms because they allow all students to collaborate and work together regardless of mathematical ability. This is because the escape room can be designed so that all types of learners can excel at the activity. For advanced functions the clues can be designed to reflect many different topics covered in the course such as trigonometry, transformations of functions or exponential growth.

Escape rooms are an excellent activity to do at the end of a unit as a review class, but could also be supplemented with a formal review session as well. The escape room is not only limited to grade 12 classes either. It can be used in any math class as long as the clues are designed accordingly to your classes mathematical ability.
Overall escape rooms are a great review activity for any mathematical topic!

Grade 11 Math Lesson Ideas

In class we had two demonstrations for teaching content in the grade 11 math curriculum. One concept was from the grade 11 U/C class and the other from the grade 11 college level. I will first talk about the grade 11 U/C activity.

Spaghetti sine is an activity that can be used in a grade 11 U/C or grade 11 university level math course to help students understand the sine function. This activity would be used during the trigonometric functions unit.

Image retrieved from: http://www.cwu.edu/math/sites/cts.cwu.edu.math/files/images/Spaghetti_Sine_lg.jpg

This activity requires students to cut pieces of raw spaghetti based on a radians/degree wheel. As students start cutting the spaghetti they should start to notice a trend once they make it all the way around the circle for cutting spaghetti pieces. This provides students with a visual representation of where the sine function originates from and can be used to explore other topics such as periods, amplitude and transformations of trigonometric functions.

The next activity was for grade 11 college mathematics and was very practically suited towards choosing an appropriate car insurance company to save you money. The goal of the activity was to figure out how much car insurance would cost with driving school, and whether or not participating in driving school will save you money on your car insurance in the long run. I found this activity very interesting because I love to save money (don't we all?!) so I thought this would be an excellent lesson during the financial unit that will grab the attention of students easily.

Streaming in High School

In class we discussed the idea of streaming in high school. Streaming is placing a student in either academic, applied or locally developed based classes. The stream chosen is often based on the students "best" learning style. Academic classes tend to be more abstract whereas applied classes tend to be more "hands-on." Locally developed classes provided the necessary skills students need to be literate in the subject area but often omit abstract concepts.

The issue I have with streaming is that it limits the students options for their future. If a student is great at mathematics they will likely be in academic math classes but if they have a learning disability for reading and writing they may struggle in English classes which would put them in applied English classes. The issue with this is that you need a grade 12 University English credit to apply to Universities and if the student is a great mathematician but not an amazing writer they unfortunately cannot attend university for anything math related due to the stream they were placed in.

Another issue with streaming is that it happens even before the students enter high school. The students grade 8 teacher will often suggest which stream certain students should be placed in when entering high school. This means the students have to decide their academic future when in just grade 8! I find that unacceptable. We should give the students the opportunity to show their potential in grade 9 and 10 and destream the courses so there isn't so much pressure on students and their parents deciding their academic future at such a young age.

For mathematics in grade 9 the two streams can be broken down as follows:

The applied stream is actually very similar to the academic stream in grade 9 with the only big difference being the omission of the analytic geometry unit. The purpose of these streams is to cater to the different learners present in the school. If we create an environment that positively fosters the learning of every student though, whether math is their strength or weakness, I believe that the streaming would not be necessary.

Strategies and Activities for Grade 9 and 10 Academic Math Classes

This week in class we looked at strategies and activities for grade 9 and 10 academic math classes. Daria started the class with a great activity called Battle Slopes which was an interesting way to reinforce the idea of slope in a grade 9 math class.

Battle Slopes had the same goal as the game Battleship, you had to find the location of the other persons markers to win the game. The twist with battle slopes though was that you could only find the other persons markers using equations such as x+1 and graphing said equation on the grid. If the graph of your equation goes through one of your opponents markers you can deduce the location of their "ship."

This game was very fun and had a good combination of strategy and fairness involved. Some strategies would involve doing things such as starting off with a small slope ie. y=x then translating that line either up or down ie. y=x+1 or y = x - 1. The fairness is a result of the randomness with the markers placed by your opponent as both students playing the game will have an equal opportunity to find the location of the markers based solely on the luck of your equation line hitting one of the markers. 

  

Image retrieved from: https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbm3uzi3RoJ9EVo9b4W_RWbmoYykoVyshOjxcCoQt64iMvljmsPn2dqHOhtck4Cg9GWb9ZITsV5qU1ASEGYV9DX5edKGVKBolXI_ikopcznVknY-Y1EuTOJ-qd8vnIIXMggUP-xp6xil2v/s1600/DSCN0747.JPG

Overall I believe this activity would be a good reinforcing tool for practicing slope. This activity could also be used if a lesson finishes early and students need work to do or if students finish a task early. It could even be used later in a future unit as a way for students to review for the upcoming exam.

There are other resources and technologies that can be used to help reinforce other topics in grade 9 and 10 academic classes such as:

Desmos
Manipulatives
Graphing Calculators

It should be noted though that technology should only be used during a lesson if it helps reinforce the learning of the new material, not just as a gimmick tool. 

Expanding Your Curriculum Knowledge Beyond High School

Secondary school teachers are often well versed in understanding the components of the grade 9-12 mathematics curriculum's and how to develop lessons that address each of the curriculum requirements. One component that is often overlooked by high school teachers, especially those teaching grade 9 mathematics classes, is delving into the elementary mathematics curriculum. This gives teachers an understanding of where our students are coming from and an idea of the knowledge that they have likely already obtained.

It is important for teachers to understand the knowledge basis from which our students are coming from. This allows us to develop effective lessons that will scaffold on the previous knowledge that the students have already obtained in elementary level mathematics.

During assessment it could also be noted that students may be struggling with a certain concept. This could be due to the lack of prerequisite knowledge in a certain topic from elementary which is hindering students progress in learning new material. Although not part of the high school curriculum it is your role as a teacher to ensure each student is achieving their individual successes, which may require you giving a brief lesson on elementary level topics to refresh student's minds on certain topics.

In our class this week we were given curriculum expectations for grades 7-9 and tried to develop continuum's that would link concept ideas across each grade. This proved to be a challenging task as not all of the topics made direct correlations, especially when going from grade 8 to 9. This showed how big the gap was between content in grade 9 and and elementary, which is why it is important that teachers teaching that grade should become familiar with the elementary level content.

Continnum chart linking curriculum expectations from grades 7-9.

There is also a variety of resources available on Edugains that link curriculum expectations across the grades.

First Teaching Block

For my first teaching block I was teaching the light and geometric optics unit of a grade 10 academic science class. During the practicum I learned many new skills and strategies that could be applied to any type of classroom. In this post I will talk about some of the experiences I had while on my teaching block as well as some of the challenges and difficulties that I had to overcome.

This practicum was the first time I had taught in a classroom setting. I had some tutoring and coaching experience but had never run a classroom before. One of the first challenges I encountered while teaching was pacing. Due to being in a university style setting for the past 4 years, my pacing was a little bit too fast at the beginning of block but as I gained more experience the pacing became much better and was definitely one of my biggest improvements.

Being organized was extremely important during the first block. I learned many strategies that I would use in a future classroom to help keep everything organized within the classroom. One of the key organizational strategies that my teacher used was a binder that was always within the classroom that contained extra notes/worksheets. It was the students responsibility if they missed a lesson or multiple lessons to pick-up the notes/worksheets they are missing from the binder which was organized by date of the notes and divided by each class as well. This made it much easier for putting student's who had missed days back on track.

Having an academic class, classroom management and behaviour was not a huge issue for me. Some students would be a little rowdy at times but I found that proximity during lessons and changing pace often would quiet them down.

Although I wasn't teaching a mathematics course there was still some math involved in the optics unit. The mirror equation shown above was one of the equations we would use in the class to find the distance of an image produced in a mirror. It was interesting to see how all of the students had different math capabilities within the class. I broke down each question step by step and included the class when solving problems to help cater to all of the different leveled math learners within the class. We also did a lab which had a component that applied the equation which the students found useful!

Image retrieved from: http://previews.123rf.com/images/alexraths/alexraths1209/alexraths120900001/15134543-Teacher-with-a-group-of-high-school-students-in-classroom-Stock-Photo.jpg

Overall I found my first block to be very successful. I learned many useful skills that I could use in any future classroom and gained the valuable experience of teaching in a real classroom. Can't wait until my next practicum!

Online Session Reflections

Online Session 1

In the first part of the online session we looked at Five Practices for facilitating mathematical discussions around cognitively demanding tasks within the classroom. As a beginning teacher I found that some of these tasks may be a bit challenging for me such as anticipating likely student responses. Because we are all different learners there are students in my class that will likely think of different ways to solve mathematical problems I present that I did not think about. This is perfectly acceptable but as a beginning teacher I may find it more difficult to anticipate all of the student responses to questions due to my lack of experience as a teacher. With more experience though and getting to know my students better it will make it easier to anticipate individual student's responses to questions.

I find the Five Practices beneficial for the classroom because they provide a student-centered approach to learning that creates a more dynamic and engaging classroom. For example, one of the practices is selecting students to present their mathematical responses. This can be beneficial because it shows student's how their peers solve mathematical problems and in an ideal classroom where there is no judging other students you can also occasionally select students with a common error in their formulation of their solution to the problem which can generate discussion as well as enlighten other students who had made the same mistake.

The second part of the first online module challenged us to think outside the box by solving problems using different approaches that students may use. This exercise was useful because it challenged me to use different approaches to solving a problem then what I would normally be used to which is good preparation for when I will have to anticipate student's responses to math problems while teaching.

Online Session 2

Formative assessment is important tool used to determine the level of understanding students have on certain topics/concepts/ideas. There are different ways that you can approach using formative assessment but there are five key universal steps that can be used to help achieve effective formative assessment. One of these steps is sharing learning goals with the students. This allows the students to have an idea of what they should learn at the end of a lesson so that they can asked themselves "did I achieve the learning goal today?" Another very important step is providing feedback that moves learning forward. Feedback can be provided in written, oral or through a demonstration and should force students to engage cognitively about their own work.

Written feedback to a student should provide a question to the student rather than just a "correct" or "incorrect." You should provide the student with a reasoning for why their answer was good or incorrect that allows them to think about other options or methods. For example in a substitution/elimination problem that a student solves using substitution only you may write down as feedback "great answer! can you think of another method that you can use to solve the problem?"

The teacher should also guide student's using feedback rather than providing them with the correct answer right away. This allows students to think back to the problem rather than just seeing the correct answer and moving on.


Another important thing with feedback that I believe is very important is providing enough time for students to read and think about their feedback. You can provide students with feedback on an assignment or test and tell them to read the feedback and they may or may not do so. Providing time in class to look at the feedback makes it more likely that student's will look at their feedback and cognitively engage in it.

Overall I believe feedback is important tool for all students whether they are struggling or excelling at a topic it can be beneficial for both those types of students if used effectively and this module was a great way of thinking about feedback and how it can be used in a math classroom effectively.