Mixed Mathematics Class

This week we completed activities that would be completed in the grade 12 mixed math class. The first activity was about domain and range of a function. First four stations were set up around the class: inequalities, words, number line and list of numbers. Then each student was given 3 pieces of paper that represented one of the stations. After this, the students from each station then collaborated to match there equivalent statements that were represented in the different ways. This allowed for collaboration and for students to actively see that a range of numbers can be represented in four different ways. At the end we were given a hand out with graphs and students would state the domain and range using two different notations. This then allowed students to apply what they just demonstrated in the beginning of the activity.

The next activity we did was about investigating exponential functions. To begin the activity an example was shown using an email chain. The first person sent the email to one person the email to two people, and then those two people sent to two more people and then all of those people sent the email to two more people etc. Everyone who got an email stood at the front of the room in a line for each group of emails, It was shown that in the first group there was one person, then two, then 4 and then 8 (figure 1). This was then graphed using Desmos to show that it is an exponential function of y=2^x. This was a great way to visually show how exponential growth works as well as get the students up and moving. Then a handout was given and students were asked to use their personal devises to access Desmos and graph three groups of exponential functions. One group exponential growth, the other was exponential decay and the third was 1^x (a horizontal line of y=1). This was a great way to visually see the graphs and see how the variable a in y=a^x affects the graph. This was another great and engaging activity that incorporates technology. 

Engaging Activities

This week we completed 3 activates in class from the different presentations. The first activity was for a grade 10 applied math class and the activity dealt with measurement. In this activity we were first asked to estimate the conversions between the metric system and the imperial system. Then we were asked to measure various body parts with a partner using a meter stick. This was great for the students, especially applied level, to get them up and moving and actually physically measuring and seeing the measurements.

Next we completed an activity about triangles and using trig laws along with sine and cosine law. First we worked in small groups to complete a few simple problems using soh cah toa and cosine and sine law. After each question was complete we were given a pile of letters and once we competed all of the questions we had to unscramble a word that completed a sentence. This was a great way to motivate students to complete each question quickly because they would be excited to complete the word and phrase. Lastly we were given a real world problem that incorporated the students’ names and interests. My group’s problem was about measuring the distances of ships to a dock, to see which one was closer, only being given the distance between the boats and two angles. Since it was not a right angle triangle we used sine law to solve for the two unknown side lengths and discovered that the red ship was close to the dock than the blue ship. We then shared our solution with the class using chart paper, which can be seen in figure four. Using students interests and names in problems is a great way to get the students involved in a math class, this makes the students more excited about the problems and I think intrinsically motivates them to complete it.

The next activity was completed using graphing calculator and motion detectors. In this activity we were to recreate a circle graph using the detector. At first my group thought we simply just turn in a circle but then realized that the motion detector needs to stay fixed on one thing. We then realized that we can just move the motion detector in a circle by holding it close to you and then going around so your arm is stretched out in front of you and then back close you to you, all in a circular motion. Graphing calculators are definitely a great way to incorporate technology into the math classroom but I personally find them difficult to use, which is probably just due to my lack of experience. I think computerized graphing technologies such as desmos and/or gizmos seem to be easier to understand and are more user friendly. It is definitely important to some sort of graphing technology into the classroom, if it is available, as I think it really helps the visual learns as well as prepares students for post-secondary school. I remember getting to University and having to use graphing software in many of my math and physics classes and it was a huge learning curve since I wasn’t really exposed to a lot of it in high school.  

Structured Problem Solving

This week in class we discussed structured problem solving in math. We used the example of the growing dots problem. While completing this problem, we were told we must visualize what is happen and draw a diagram. This forced us to solve the problem in a structured way with visuals. Without these instructions many of us may have completed the problem mathematically instead of solving is using diagrams. However even though we were told to use visualizes and diagrams there were still several different approaches to the solution. Two of which can be seen in figures 1 and 2. One is the approach I took of a growing squares and since a square has 4 corners it grows by four dots every minute. The second approach was one that viewed the four corners as orbital’s that grew out by one dot each minute. Both of these approaches come to the same solution but they way the visually problem solved to reach the solution was different. As I’ve said in previous blogs, I think it is very important for teachers to realize that students can approach the same problem in many different ways, and in face teachers should encourage this creativity, as this is what leads to higher order thinking.

figure 1

figure 2

            During this class we also had Diana’s presentation on grade 10 applied math. For her activity we were to solve a problem that was different for each group and can be made based on students’ interests. My group completed a problem about choosing a banquet hall for the school athletic banquet. The solution we came up can be seen in figure 3. This activity is a creative way to show students real life situations when they would need to know how to use substitution or elimination. It was also good that we were not restricted to one method and we could choose if we wanted to use substitution or elimination. We chose substitution as that it was we were all most comfortable with, however we then saw how easy it would have been to use elimination with our two equations, and this may have been the method a grade 10 applied student would have chosen. This is an example of non structured problem solving, which is sometimes more beneficial than structured problem solving as it can lead to more creativity.

figure 3

Using Student's Work As A Mathematical Resource

This week in class we discussed materials and resources in a math classroom. This is something that may be obvious is a school that has access to things such as computers, smartboards and the finances to support a technological math classroom, but this is not always the case. Something that I think many teachers and students don't think about is using themselves as a resource. Students can be used as the primary resource in a classroom through things such as collaboration and presentations of hands on actives with each other. They have the ability to create, draw, write and research and then their work can be used as resources. If you allow students to be creative and come up with their own approach to a problem in mathematics, chances are you will have several different solutions to the same problem. As a teacher you could then use these different solutions to teach how these different approaches work and the math behind them. This allows for collaboration, creativity and higher level thinking in the classroom as well as student centered learning. Using students’ examples to teach would be a great way to use your students as a resource in the classroom, especially if you’re in a school that may not have access to many resources.

            This week I also had my presentation on the grade 10 academic mathematics course. In this presentation we (Ela and I) did an overview of the whole course looking at each major strand: Quadratic Relations, Trigonometry and Analytic Geometry. We then each did an activity that can be used in this course. I did my activity for the Quadratic Relations unit focusing on the transformations of a parabola. In this activity I use a student-centered approach while incorporating technology. In this activity 8 different groups of students would each be given 4 equations representing one a transformation. For example group one would be given 4 equations of a parabola shifted up, group 2 would be given 4 equations representing a parabola shifted down, group 3 would be given 4 equations representing a parabola shifted right etc. The eight groups would be given one of the following transformations: shift up, shift down, shift right, shift left, stretch and compression.  As a teacher you may want to strategically choose these groups to ensure that each group has 1-2 strong students depending on the class size. From here the groups would graph their functions by hand using a table of values and discover the pattern between their graphs. From here each group will present their findings. This is where you can incorporate technology. I used Desmos, which is an online graphing calculator. Using this technology you can either have each group put their graphs onto the same grid so everyone can visually see what each transformation would look like in comparison to a non-transformed parabola. This can be seen in figure one.

Figure one: Desmos Screenshot

 From here students would get into new groups where one student from each of the eight groups form a new group. Each of these new groups would then have an ‘expert’ of each transformation. They will then be given a list of parabolas that contain several transformations of the form y=a(x-d)²+h. They will then come up with a general solution for what each parameter represents and share these findings with the class through a gallery walk and/or class discussion. Desmos also has a parabola transformation software on it that can be used for this part where you adjust each parameter to see how the parabola changes. Other forms of technology can be used throughout this activity such as graphing calculators, a smartboard and gismos. This activity is a student centered lesson which uses the student’s work as resources as well as technology if it is available, but the technology is not required is just there for visual assistance. This activity also allows students to collaborate and explore.

figure two: student graphing their transformation on desmos (student: Ela)