6:12 (ratios and proportions) (week 6)

     This week we discussed ratios and proportion. To ensure my students develop a full understanding of these concepts, I plan to use the strategies we discussed in class. One of the things we did was use real life example, such as baking to understand how we need to proportion things out. Guliana used this example to demonstrate how chefs and bakers need to know their proportions and ratios because they are constantly asked to very quickly alter recipes in order to feed less or more people. This means having a complete understanding of the numbers and how they are balanced in order to make sure there is just enough ingredients. An example of that is if one adds to much salt to a batch of cookies they will not taste good. If we are doubling the batch, we have to know how much to add to each individual ingredient so it does not negatively affect the whole. The same works for ratios. Ratios are a good base for comparing things. They allow students to have an understanding of how two numbers can be contrasted against each other, or compared. This will allow the students to have a better understanding of how numbers work. The relationship of 4 blue squares :2 green squares can also be thought of as 2 green squares: 4 blue squares. This could also be expressed as a fraction, 2/6 or a percentage, 33.33%. 

6 Squares. (October 2016). Casey Made with Pages. 

   

    We also looked at percentages, because the topics are so closely related and interchangeable.With percentages, I always find the visuals a very good representation of determining and understanding percentages. I find this unit to be very important because it is extremely useful in everyday life. Even to this day, students are given marks out of 100%. This means that if they have an understanding of what a percentages is, they can take the mark of 25/40 and know that they got 62.5% on their test. Practical skills like that are always useful and can be applied in various situations. In the textbook, it does state that one of the common misconceptions is students who do not have an understanding of percents that are above 100%. Explaining this as equivalent to a mixed fraction would be a very good strategy. I think I would also try and draw a visual, like one of the grid visuals used throughout the textbook, to demonstrate what it means to have more than enough, and to show how that works as a mixed fraction. With these, pie graphs are also a great visual, for another class I made a pie graph about students who wanted to join a book club: 

    This was a silly joke, because I obviously did not actually ask 90 people to join a book club. But the percentage needed to be out of 100%, and I wanted it to be obvious that it was many more people that did not want to join, so I took the numbers of 60 people who did not want to join, 3 people who wanted to hang out, and 2 people who wanted to join and multiplied them all by 2 so that my ratio could be out of 100%. The initial ratio would have been 60: 3: 2. This pie graph is a good visual because it provides students with a representation of parts out of a whole. If they can see that the numbers represent something, they will have an easier time developing a conceptual understanding of these concepts. 
    In Chapter 16 of the textbook, a section was dedicated to Appropriate Manipulatives. From this, I found my understanding of teaching this subject was greatly increased. It listed counters, cubes, tiles, or balancing scales and paper bags. I think the activity with balancing scales would be really fun for students to do. It would provide a real life example they can understand. The algebra tiles would also be a very good manipulative for this topic. They can be used to show how the formulaic expression is equal. I think using these manipulatives to show ratio and proportion would be great. 

    To ensure my students misconceptions are addressed correctly I will listen closely as they explain things. I plan to develop resources in the classroom that ensure that each student who is misunderstanding something, or feels they do not have the right tool to address the problem, will receive extra guidance and further explanations. Being able to understand their misconceptions is the first place I would start, and as someone who will have had the same misconceptions as a math struggler, I think I will be able to see the thought process behind some of these students. I always found I applied the wrong concepts to math units. I will attempt to learn from the student and see how they thought out the problem so that I can then address it. Group work is another great resource so that the students can then help each other out. Hands on projects and real life examples are my favourite ideas to developing students complete understanding of concepts. Ensuring the students understand will come from practice and continuous learning. 

positives and negatives (5)

       This week was all about integers! We talked extensively about how hard it can be for everyone (students, teachers) to understand the concept of negative and positive numbers. They can get lost if we just show an equation with a mixture of integers and don't have a background explanation for what is going on.

     It is always hard to find real life examples that make sense to students but also teach them something they will remember. For this subject, I loved all the cool ways we learned how to keep track of the positive, negative relationships and their outcomes. One of my favourite graphics is this one: 

"Integers"" Don Steward. (2016). Retrieved From http://donsteward.blogspot.ca  

    Finding unique yet relatable ways to describe and explain things will be the most effective way to reach the students. I find that real life examples with a combination of pictures and words help me, so I will try and access my students through those means. I want to appeal to all learners while providing them with challenging material. I think the above image is a nice place to start, with that image I would then include temperature and think of all the ways I could demonstrate how integers actually visually work. One of the other ideas that was suggested is a sideways number line. I personally love number lines and how they provide a nice clear visual of what the numbers are doing, how they are moving or jumping around, so I think that could be very useful.
   One of the things discussed in the textbook was around the misconception of "two positives makes a negative" (351). Sometimes boxing students into rules makes them forget the other aspects of the question. In this situation, the students can forget that the rule applies for multiplication and division instead of addition and subtraction. This is an example of students not thinking about the real world applications and what the question actually means. 

"Printable Number Line- Vertical" (2016). Retrieved from https://www.printablepaper.net/preview/Numberline-Vertical

    Another one of the activities I really enjoyed was a game that was a cross between Jeopardy and Battleship. We had math equations that we had to quickly figure out that featured both positive and negative numbers, and then had a sheet with the answers that acted as our Battleship board. It was a fun way to get students to really quickly do several math problems. I found it was very helpful at reminding me what adding and subtracting positives and negatives entails. I definitely would consider thinking of a similar game to play with my students.

     We also talked about exponents this week. I do not mind exponents, because it seems to be a very straightforward topic to me. It is a set of rules that are to be followed, and because of that we have to ensure that students remember what the exponent represents. 

"Exponents". Sixth Grade Math Blog. (2016). Retrieved From http://hemsmath6.weebly.com/exponents.html

    

    

       I think like most topics, this is one that just needs to be broken down into smaller steps for students to understand. It is easy to do once we have the base of remembering the rules and knowing the terminology. For a topic like this, I think it is really important to spend time on the base work such as fully understanding the terminology used to describe these numbers. From there, I would try and explain why we simplify by making it an exponent, like the above example demonstrates. If students first have the basis of when and why to use exponents, they will then know what to do and how to do it to solve the problem. 

   

    One of the other topics we discussed was manipulatives. I know how hard it can be to decide when to introduce manipulatives into a topic. When assisting grade three's in learning multiplication, I had chosen to introduce manipulatives far too early in the conversation, and found my students did not have a good enough base to work with the manipulatives on. It was a hard lesson to learn but I definitely noticed right away that they did not have that foundation, and then the manipulatives because useless instead of being a good tool to use. I think teaching the lesson with demonstrations and visuals is always a good thing, but the students need to have that clear outlined lesson first, where they are taught how to do it, before giving them the tools to explore how it works on their own. The manipulatives are an amazing tool for kinaesthetic and visual learners, but I think mathematic lessons need to be fully explained and detailed out to students before they get to that point. 

   Overall, another great week with various advances in mathematics. I am thrilled to be learning more ways to teach and hope to continue having fun and inventive examples to present to the students. 

My Math Forum Question

Due to the upcoming Thanksgiving vacation, I am planning a road trip to Kitchener- Waterloo, Ontario to visit friends and family. My starting point is Fort Erie, Ontario. 

Delicious lasagna that is awaiting my arrival

1. Please draw the route that I should take on this map 

B) Now that we have the route planned out, we need to figure out how long it is going to take me to get there. The trip is 165 kilometres one-way. I have to be there in time for dinner at 5:30pm. What time should I leave home at if I am driving 100km/hour constantly throughout the trip?

C) What if I decided to take the back roads, and would have to alter my speed and travel at 80 kilometres for the trip? What time would I have to leave home at then? 

D) The next step is to figure out how much money I will need for gas. I have a small car that holds 76 litres of gas in the tank at one time. I plan to fill the gas tank to full before I leave, gas is priced at $0.94 per litre. How much will it cost me to fill up my tank?

Fractions and Decimals! (week 4)

I will ensure that my students have a conceptual understanding of fractions and decimals by displaying to them the basic concept of both terms before elaborating on those concepts. I think that most confusion can stem from not having an understanding of why those numbers are in the place value they are, as well as what it means to switch fraction to a decimal and vice versa.

The most basic part of understanding fractions is realizing what the numbers mean and what they actually represent. I will work to show my students that through interactive conversations and examples. The textbook provided some great activities involving games and creativity that I would consider using. The pizza game we played in class, where we all started with a number and that number had a portion of the pizza such as 3/4ths of a pizza, and that had to be compared to everyone else’s number even though they had different denominators, showed us the importance of having the same denominator to compare fractions. I would definitely want to try to do something like that to start off my students, to make sure that they fully grasped the concept of fractions. 

As far as decimals, I find the place value charts to be the most effective for me so far, so I would first try that with my students. Placing the decimal into a chart and demonstrating the movement of the number while having the title of “tenth” or “ones” columns was very helpful to my understanding. I think that creating a place value chart would be the first place to start with students. I have done this with the grade three students I work with, and they use a thing called a “place value house” and act as though the columns are neighbours and when they have to borrow it is the neighbour knocking on the door asking to borrow some eggs. This was a good technique because it reminded the kids of the values in the columns. 

I also really enjoyed Nicolina's example of having to match the fraction to the decimal. It was very similar to this online game, that I found when I was looking up fractions and decimals.  The matching game of having a mixture of fractions and decimals and having to match the equivalent fraction to its decimal was a quick, fun way to practice conversion skills.

Fraction Pizza. Clipart Kids. (2016). Accessed through : http://www.clipartkid.com/pizza-fraction-cliparts/

In my placement, I have yet to see the students work on fractions. They were working on scatter plot graphs when I went in last time, and it was really interesting to see them work through the problems. I started with some questions that were actually kind of hard to answer, luckily after talking it through, my memory was jogged to remember how to create the algebraic expression and keep the y axis the same while changing the steepness of the slope. I cannot wait to find out some more techniques that I can use in the classroom. I have tried to implement a few so far, such as encouraging students to work through the problem in pairs. I have found that this math course is really helping me in my placement since they do math the first two periods I am there.