Learning Activities: Week 6

We had another good round of lesson plans this week in my course Teaching Mathematics I/S.  There are so many different ways that we can engage students in their learning and tailor to different student learning.  I want to specifically talk about one of the learning activities that we participated in.

One of the activities focused on Proportional Reasoning in course MEL4E: Mathematics for Work and Everyday Life.  Students were given a printout of a floor plan, a worksheet to guide them through the activity, and a large variety of different floor material (hardwood, carpet, tiles, etc.) with different prices on each.  There was also a selection of paint samples.  For this activity, students had to choose three rooms from the floor plan to redesign the floor and the walls.  Each group of students would choose the new floor and/or paint colour they wanted to use for that room, and then use the price per square foot to determine how much their redesign would cost.

There were multiple aspects of this activity that made it very effective and engaging.  One of the first things I noticed was the very large amount of physical material that students had access to.  Instead of simply giving students a list of the different materials to choose from and the prices, the teacher supplied the physical floor samples and paint samples for students to use during the activity.  This makes the activity so much more engaging because students can use the samples to help them visualize what material they want in their home.  The different prices of the material were also very realistic (for example, the hardwood flooring was more expensive than the vinyl plank), allowing students to recognize how prices differ in the real world.  We were also able to further relate this activity to real life by choosing flooring based on our own personalities.  For example, my group decided that we were going to have a dog so we decided on vinyl plank instead of hardwood so the dog would not ruin the nice (and expensive!) hardwood.

Another aspect I liked about this activity was that it could be used for students at multiple different levels of learning.  How the floor plan is designed allows for many different levels of difficulty for solving proportional reasoning problems.  Students were only required to choose three rooms to redesign; thus, students could choose rooms based on their current level of learning.  Some of the rooms were simply rectangles or squares (e.g. closet, master bathroom, living room), whereas other rooms were irregular shapes (e.g. kitchen, bedroom #2, foyer), requiring multiple calculations to solve the problem.  This encourages students to work at their own level, and therefore allowing everyone to be successful.  In addition, students could try different levels of difficulty, perhaps starting with simple rooms, and progressively trying harder rooms.

There is not much that I would change about this activity.  I believe that it is an excellent activity, tailoring to different learning styles and learning strengths.  One thing that was mentioned in class was the idea of incorporating a budget into this activity.  I believe that this would be a very good extension activity to do with students.  This would simply add another aspect of the activity, making it more engaging and challenging, as well as more realistic.

Learning Activities: Week 5

Prior to taking this course on Teaching Mathematics in the Intermediate/Senior Classroom, I thought that it would be a lot harder to incorporate fun and engaging activities within the senior grades as opposed to the younger grades.  However, some of the activities that were explored this week in class proved that even in grade 12 there are plenty of exciting things you can do with your students to keep them engaged in their learning.

Logarithm Dominoes

One of my colleagues made use of personalized dominoes to help solidify students’ understanding of logarithms.  Students played a classic game of dominoes; the only difference was that instead of a certain number of dots on each domino, there were different logarithms.  Throughout this entire game, students were able to practice their skills at solving logarithms.  Each student took turns solving the logarithms and playing one of their dominoes (if possible).  Students were told that each turn should last no longer than 45 seconds.  If a student could not play any of their dominoes, played an incorrect domino, or they took longer than 45 seconds, that student would need to pick up an extra tile.  The person to play all their tiles first wins.

I think that this is a very clever activity to do with your math class.  One thing I like about this activity is that it can be used for many different topics in almost any grade.  For example, this game could be used to help students practice understanding equivalent fractions.  In addition, I thought that this activity was a very engaging way to practice a mathematical concept.  Often, teachers simply assign textbook questions or worksheets with many practice questions on it.  Having students engage in this alternate version of dominoes allows students to practice the same skills that they would if they answered textbook questions but in a more fun and exciting way.  When students are engaged in their learning, students are more likely to better remember and understand different concepts.

If I were to use this in my classroom, one of the things I might eliminate is the timer.  Although I understand that this is so students do not take forever on their turn, I believe that it discourages students who may not be as fast at solving problems.  Students should be told that they should take a “reasonable amount of time” on their turn (encouraging them not to take too long), however, I would not restrict them to a specific amount of time.  Another thing I might do if I were to use this activity in my classroom is create the groups myself.  I believe that creating the groups for this activity could help students learn and practice better.  There are a couple different ways that I would like to try splitting up the groups.  The first way would be to split students up into groups with similar abilities (i.e. students with high skills in one group, students with lower skills in another, etc.).  Splitting the groups up this way would make the game fair because everyone would be on the same level.  In addition, students could help each other throughout the game.  The other way that I would maybe split up the groups would be with a range of abilities in each group (i.e. low, average, and high level students in each group).  I believe that splitting the groups up in this way would allow students to help each other more efficiently.  Higher level students could practice their skills and understanding by assisting the lower level students; whereas lower level students would receive this assistance and therefore better their own understanding.  How I would split up the class would be completely dependent on my specific class and the students within it.

I hope to find ways to use this activity within my classroom, as it is a fun and engaging way for students to solidify their learning. 

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Learning Activities: Week 4

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“When are we ever going to use this in life?”: One of the most common questions students like to throw at their math teachers.  It is very important to not just teach students math concepts, but to show them how it is applicable in their everyday life.  Show them that math is everywhere.  When students are able to see how a mathematic concept can be applied to the real world, it not only helps them better understand the concept, but it also makes math more interesting and fun.

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ice-bucket-challenge-clipart.html

One of the lessons that we had this week in Teaching Mathematics I/S was very application based.  With the use of three application problems, students were introduced to the concept of exponential functions.  Each student received a handout that had sections for each of the three stations.  The sheet had empty tables of values and graphs on it to lead students through the activities, as well as extension and reflection questions.  At each of the three stations, there was an application problem and some type of manipulative to help students with the problem.  An example of one of the stations was the “Ice Bucket Challenge” station.  Students were given counters to help them visualize how many people were nominated to do the ice bucket challenge each day (assuming that each person nominates three people after they complete the challenge).  Students could fill out the table of values through the help of the counters, and then create a graph representing the day number versus the number of nominations on that day.  Students could try to find a pattern because they were then asked how many people were nominated on the eleventh day (that would be a lot of counters!).

I really enjoyed this activity and will use it in my future classroom.  There are multiple different aspects of this lesson that I believe were very beneficial.  First of all, like mentioned in the introduction of this blog post, it is very important to relate mathematics to the real world.  All three of the stations in this activity involved some sort of application problem for students to solve.  Including application problems allows students to better understand the material, rather than just memorize the information and erase it from their brain the second it is no longer needed.  Relating to something that students understand and enjoy, such as the ice bucket challenge, increases students conceptual understanding as opposed to just their procedural understanding.  Another thing I think was very beneficial about this lesson was the use of manipulatives.  Two of the three stations had counters for students to visualize the exponential growth of the application.  This helps students who are visual learners.  The third station involved how many sections are created when folding a piece of paper multiple times.  This station had a stack of paper for students to use while solving the problem.  Again, this is beneficial for visual and tactile learners to help with their understanding.  A final thing I liked about this activity was the kinesthetic aspect of it.  Each station was set up at a different table in the room and students had to move to each of the stations in order to solve the problem.  I think this was an excellent idea, as opposed to simply giving students the tools to solve all three problems at their own desk.  Allowing students to get up and move around during the lesson not only makes the activity more fun and exciting, it also can be very beneficial for kinesthetic learners.

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The only thing I would slightly alter about this activity is some of the wording on the worksheet.  For example, when I first started at Zombie Apocalypse station, I thought that the numbers we were suppose to fill out were the total number of zombies, as opposed to the number of zombies that were infected that day.  Although this is specified on the main sheet at the station, the wording in the table of values on our individual worksheets had me slightly confused at first, until it was clarified by the teacher.  However, this was a very small thing that, once clarified, did not at all impact the great outcome of the activity.  I would definitely use this lesson activity to introduce exponential functions to my future math class.

Learning Activities: Week 3

Different students learn in different ways.  Some students learn best through writing out notes; others learn best through getting up on their feet; some learn best through using tactile objects; other students learn best through simply listening.  There are so many different types of learners that need to be accommodated for in the classroom.  This is why it is important to provide a large variety of instructional activities that are beneficial to many different types of learners.  The learning activities done this week in my course on Teaching Mathematics at the Intermediate/Senior Level were great examples of how to incorporate different types of learning into an instructional activity.

Introduction to Rates of Change Activity

One of the activities this week focused on the math concept rates of change.  Each group of students received a CBR (calculator-based ranger) which can be used to collect data on motion (including distance and speed).  For the first part of the activity, students were given six different distance-time graphs.  The groups had to try to create each graph using the CBR.  Then students were given blank graphs with specific descriptions (example: “walking away from an object slowly”), and students were required to draw what the line would look like.  After this group activity, the class came back together as a whole to do some specific examples on calculating the rate of change of a line.

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I really enjoyed this activity and would definitely use it in my future classroom.  When learning rate of change, distance-time graphs are often used.  In addition, these graphs are often related to real world scenarios (example: “describe a scenario that would produce this distance-time graph”).  However, students rarely get to create distance-time graphs themselves.  What I like about this activity is that students are actually able to get up on their feet and explore with and create distance-time graphs.  By bringing CBRs into the classroom for an activity like this, students get to see their own ideas play out in real life.  It also allows students to discover their own learning.  For example, a student may have thought the graph of someone walking slowly towards the sensor would look one way; however, after testing it using the CBR, perhaps the student discovers that he/she forgot that the slope should be negative.  This activity is a good example of tailoring to different styles of learning.  Clearly from what I previously mentioned, this activity is excellent for kinesthetic/tactile learners.  It can also help students visualize the different types of lines created when comparing distance and time.  I also believe that it is good for students who learn best through discovery.  Students are able to make mistakes and try again.  This activity helps visual learners because students will likely think of this activity when trying to solve future problems.  In addition, I believe it was very beneficial that this activity started with the more application based part of the lesson and then went into the next more mathematical part (i.e. mathematically calculating the slope of a line).  I believe that this made the activity much more engaging.  I look forward to hopefully trying this activity with my future math class.

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If I were to use this activity, I would likely also add a part of the activity where students can create their own distance-time graphs.  For example, students could draw a random graph, predict how they will need to move in order to create that graph, and then test it out using the CBR.

Learning Activities: Week 2

Having an engaging classroom is very beneficial for student learning.  I believe that this is especially important in a mathematics classroom.  Math often has a bad reputation for being boring, hard, or a subject that only people with “math brains” can succeed in.  I strongly believe that math can be fun and engaging for all students, and that every single student is capable of success.  Incorporating interesting lessons within a math classroom can help keep students engaged and improve their desire to learn.

Speed Dating and Equation Making

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One of the math activities that we did this week in my course on Teaching Mathematics at the Intermediate/Senior Level was called Speed Dating and Equation Making.  For this activity, students sat in two circles: an inner circle and an outer circle.  Students would face each other so that each student on the inner circle was paired with a student on the outer circle.  Every student received a graph that had hearts on it.  In addition, each student on the outer circle obtained a card describing a specific slope, and each student on the inner circle obtained a card describing a specific y-intercept.  Each pair had to combine their information to create a linear equation in slope y-intercept form and then graph the line on the grid; if the line went through a heart, those two students were a match!  The inner circle then rotated, changing up the partners, to test for more matches.  This continued until students got back to their original partner.

I absolutely loved this activity.  I would, without a doubt, use it in my future mathematics classrooms.  I found it fun, engaging, and a very unique way to explore slope y-intercept form of a line.  One part I like about this activity is that it is repetitive.  Depending on the size of the class, students have to create many linear equations and graph them.  The repetition of this process helps students internalize their learning and remember it for the future.  I also think this activity would be useful in teaching students about parallel lines.  Since I was in the outer circle, I received a card with a slope on it.  All of my lines were parallel to each other.  This could help students realize that parallel lines have the same slope.  In addition, people in the inner circle had all their lines going through the same point (i.e. the y-intercept).  Another part of this activity that I enjoy is that it has the potential to be used for many different grades.  For example, this activity could be used for grade ten quadratic equations, possibly giving students the vertex of the parabola and a root.

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One thing I might add to this activity is having students prove (or double check) that they are a match.  This would allow students to make another connection between the graph and their equation.  Students would be required to plug in the point of the heart into their equation and show that the point does, in fact, lie on the line.  This could also be done if the line is very close to a heart and students are unsure if it is a match or not.  By substituting in the point, students can say for certain whether they are a match or not.  This game could also be made more difficult by giving students a slope and any point (not the y-intercept), or by giving students two points.  This would allow students to practice determining equations when given different information about the line.

I definitely plan on using this activity in my future mathematics classrooms.  It is a very unique and engaging activity that I believe students would really enjoy.  It definitely shows students that math can be fun!

Learning Activities: Week 1

Hi everyone! Welcome back to my mathematics blog! I will be continuing my reflections as a make my way through my course on Teaching Mathematics at the Intermediate/Senior Level.

Over the next few weeks, my fellow teacher candidates and I will be leading a mathematics learning activity.  I will be reflecting on some of these activities.

Steve the Stick Figure

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The learning activity from week one that I would like to discuss is the Steve the Stick Figure activity. In this activity, students follow a path around the room using transformations. At each station, students are given two coordinates that represent where Steve the Stick Figure's head and bottom are on the grid. There is a problem involving transformations at each station that students need to solve in order to determine where they need to go next.  Students follow the path around the room until they have visited all the stations, at which point the students have to send Steve home (i.e. a specific point on the grid) using any transformations they choose. Students then can create a story for Steve's adventure based on where he went on their grid.

I would definitely use this activity in my classroom for multiple reasons.  First of all, I believe that it is beneficial for a large variety of different learners.  Students who are tactile/kinesthetic learners are benefitted through walking around the classroom, as opposed to sitting at their desks.  Students are given the opportunity to choose whether they want to work alone or with others.  This helps different types of learners because some students work more effectively by themselves, whereas other students like discussing and problem-solving with a partner.  Another aspect of this activity that tailors to different types of student learning is the fact that students are able to use both the graphs and the coordinates.  Actually plotting the stick figure on the Cartesian grid, as opposed to just determining the coordinates, will likely help students who are more visual learners.  Students are able to use the graphs to understand the different transformations visually.  In addition, students are able to use the coordinates to assist them in understanding the different transformations.  A final way that I believe this activity incorporated different types of learning is the final part of the activity.  Students are asked to make a story that describes Steve's journey throughout the day.  This is beneficial for students who are good at creative writing.  In addition, it increases the students' interest during the activity because students can create any story they want. It also allows students to review the different transformations that took place.  I strongly believe that this learning activity was an excellent example of differentiation and tailoring to the many different types of learners that will be in a mathematics classroom.

https://www.tes.com/lessons/l-IGpdZk72gmSg
/transformations

The only aspect of this learning activity that I may have done differently is to possibly have some directions in writing.  When the presenter was first explaining the activity, I was slightly confused as to what we were supposed to do.  However, once we actually started the activity, it began to make sense.  If I were to use this activity in my future classroom, I would make some written instructions that I could hand out to the students.  However, I would still go over the instructions as a class for clarity.  I believe having a written form of the instructions would make the activity clearer in regards to what is expected of the students.  This would help students who understand better from written instruction rather than verbal instruction.  This would also give students something to refer back to.

Overall, this was a very unique activity that I would definitely use in my classroom.

Lesson Play: The Importance of Anticipating Student Response

Many teachers do not recognize the importance of anticipating students’ reactions to instruction and questioning.  However, this is a very important part of planning a lesson.  Teachers will often think about the questions they would like to ask, as well as the ideal answer they want, but will neglect to think about the responses that students may actually have.  This is vital for lesson planning because teachers should be aware of what students may struggle with and how to bring students to the desired answer without just telling the students.  This week in my course on Teaching Mathematics at the Intermediate/Senior level, we were required to make a Lesson Play.  What this entails is making a script that would align with your lesson plan and trying to anticipate the types of responses students would have.  This can assist us in determining how we would respond to certain responses or difficulties students may have.  The following is the script co-created with Laura Gravina that aligns with our lesson plan on linear relations and points of intersection.

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Teacher: Now that everyone has presented their information to the class, we are going to discuss as a group.  So would you still choose your payment method, now that you have done some mathematical research? And why? Billy.

Billy: I originally chose option 1 but now I think I would choose option 3.

Teacher: What made you change your mind?

Billy: Well I was thinking that the first one had the most money that I could get.  But after making the graphs I realized that I could make more money using option 3.

Teacher: What aspect of the graph led you to the conclusion that option 3 was the best?

Billy: Well once we found the point of intersection I realized that as long as I sold 9 hats I would make more than $16. And I could do that no problem at a Jays game!

Teacher:  Okay great.  Did someone have a different answer? Sarah.

Sarah: I chose option one too but I decided to stick with it because I wouldn’t have to sell any hats in order to get paid. No matter what I’d get paid $16.

Teacher: That is true. Ok so what did you notice about the steepness of the lines? Jessica.

Jessica: They all had different slopes.

Teacher: How does this relate to how much money you would earn?

Jessica: Uh I don’t know.

Teacher:  That’s okay. Can anyone help Jessica out? Bryan

Bryan: If the slope was greater, then if I sold more hats I could make more money.

Teacher: Right! So because the slope is steeper for option 3, for example, if you sell tons of hats you can make more money.  The amount of money you make increases more quickly with a steeper slope. Therefore, how hard you work will affect how much you make.  How is this different for option 1? Joseph.

Joseph: Option one doesn’t have a slope.

Teacher: So the slope is not increasing or decreasing but that doesn’t mean there is no slope.  What would the slope be in this case? Rachel.

Rachel: Zero

Teacher: Great.  So the slope of option 1 is zero.  Therefore, in option one, the number of hats you sell doesn’t change the amount of money you make.  You will always get $16.  Okay so you all discussed in your groups what the meaning of the points of intersection are.  What do these mean for our real-world problem? Trish.

Trish: It’s where the two lines overlap.

Teacher: Okay good, so that is what it means mathematically.  But we want to know the real world meaning.  What does the lines overlapping mean in terms of money?

Trish: That when you sell that many hats you make the same amount of money no matter which payment method you choose.

Teacher: So I could choose any one of the three payment methods?

Trish: No, I think it is only the two lines that are intersecting.

Teacher: Awesome! So the intersection point of two lines means that if you sell x amount of hats, you will make the same amount of money for those two payment methods.  Great job class! So now we are going to move on to our final activity.

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As evident in this script, Laura and I tried to anticipate the types of responses students would have, the specific mistakes students might make, as well as how we would respond and get students to the correct answer.  Through creating this script, I am even more convinced of the importance of anticipating student responses.  Since we know the desired answers to the questions, it can be very difficult to guess what mistakes students might make.  However, this part of lesson planning is vital for being prepared and organized in your classroom.  It is important to keep in mind what you want the final takeaway to be, while still making sure you lead students to the correct answer rather than tell them the answer.  This will likely contribute to students better understanding the content.