Patterning and Algebraic Thinking, Week 7

Hello Everyone,

Katic, K (2015) ©. Patterning Worksheet.

This past week was the week that I was looking forward to since the beginning of the term...that's right presentation week! Every week throughout this course, a select few students are responsible for presenting a 10-minute lesson based off of one of the chapters in the Small textbook. It's great as it allows everyone to teach the class about a topic they have become an expert in, and it's so interesting to see what kind of creative solutions my peers come up with when presenting math. The section I was responsible for was the patterning strand for Grades 5s and 6s; it was actually a lot of fun seeing as this week was Halloween and I was able to create an engaging, fun presentation and worksheet using Halloween graphics!

Within the textbook, it gave quite a good background on patterning and algebra. Small discussed the importance of teaching students about patterns and pattern rules before students can be able to expand and create their own. As stated in the textbook, "A pattern rule is an unambiguous, unarguable description of the pattern; every element, including the first element in the pattern, must be described" (Small, 2013: 610). This is a critical element to understand with patterning as this is the foundation upon which students build and expand their mathematical knowledge off of. As such, this directly relates to pages 77 and 87 within the Ontario curriculum, where problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, and representing and communicating are all key processes and expectations that students are expected to utilize.

Teaching Treasures (2014). Triangles Puzzle 1. 

After doing extensive research about this topic to ensure my lesson went off without a hitch, I began to realize that, given more time during my presentation, I would have discussed tools and useful games that teachers can use to engage their kids. While it is admittedly difficult to make algebra fun, I found this great online resource that can aid kids with thinking algebraically. It's called Triangles Puzzle 1 located on the gamequarium website. This game meets the patterning and algebra expectations for Grade 6s (specifically on p. 95 under the variables, expressions and equations specific expectations). It is not as action packed as some of the other math games I have discovered, but it certainly contributes to algebraic understanding. Very simply, students are expected to find a pattern illustrated in the top two triangles (it is the same pattern rule), and then apply it to the bottom triangle. The answer is inputted into an answer box at the bottom, answer correctly and another puzzle will be generated.

Teaching Treasures (2014). Triangles Puzzle 2.

Expanding off of this game is Triangles Puzzle 2 located on the same website! It meets the exact same curriculum expectations as the Triangles Puzzle 1, however instead of trying to compete the bottom right triangle within the larger triangle, in this game students are expected to answer the middle portion of the triangle. In this game, it's great that there's a help button available for students if they really are stuck with trying to solve the pattern and determine the pattern rule. I believe that both of these puzzles are great resources for students to access in order to get them thinking about patterns and algebra.  By utilizing all of the above resources, I am hoping that I will become an effective math teacher one day who will be able to show my students that in fact, yes, math can be fun! I hope these expectations and games I have discussed above will help all of those who read my post. Feel free to post any questions or comments below! Have a safe and happy Halloween.

References 
Ontario. (2005). The Ontario Curriculum, Grades 1-8. (6th Ed). Toronto: Ontario, Ministry of Education. 
Small, M. (2013). Chapter 22 In Making Math Meaningful to Canadian Students, K-8. (2nd Ed.) Nelson Education. Toronto: Canada. 
Teaching Treasures (2014). Triangles Puzzle 1 and Triangles Puzzle 2. Retrieved from http://www.teachingtreasures.com.au/maths/yr7-puzzle1.htm 

Ratio, Rate and Percent, Week 6

Hi Everyone,

This week was dedicated to exploring and discussing ratio, rate and percent. As you will begin to see, many weekly topics covered in this mathematics course have real-world application. This one is no different. This topic can be extremely fun once it is fully understood!

Within the Small textbook, there are many useful activities for teaching ratio, rate, and percent to students. This chapter emphasizes the importance of reading symbols meaningfully when teaching this topic to your students. A mix of grid and non grid examples should be utilized so students get a sense of how percents, ratios and rates are all associated with various fractions of amounts (Small, 2013: 314).  It is also important to keep in mind that there are not always the best approach to solving certain solutions; it is important to give students the freedom to set up the solution strategy in whatever way makes the most sense to them.

That being stated, in the Junior and Intermediate level of math, students are expected to be able to build off of their preexisting knowledge, create and expand. The ratio, rate and percent topic falls within the number sense and numeration sections of the Curriculum, where students will be expected to represent, compare, and order numbers including integers by the time they are in Grade 7. They will also need to demonstrate an understanding of proportional relationships using percent, ratio and rate, expanding off of the preexisting knowledge of math in the Junior math level.

Every week, I try to continue building math game knowledge as a way to increase student's level of engagement. This week I discovered two really fun math games on the Math Playground Website. The first game is Math at the Mall. In this game, you select a player and have to walk around a virtual mall to complete math tasks. Upon successful completion of the math tasks at each store in the mall, the player receives a gold coin. The object of the game is to receive a total of four gold coins. While this game is not a high-paced game that involves competing with other players, I find this to work to its advantage. Students are able to take all the time they need to successfully and correctly complete the math problems posed, and when they are finished they are rewarded with gold coins. This game incorporates ratio and percent and successfully supports Grades 5 and 6 levels of the Math Curriculum. 

The second game is Ratio Stadium. In this game, it gives you a rate which you need to convert to the correct fraction in order to win against the other players. This game is great and supports Grades 6 and 7 levels of the Math Curriculum, specifically in regards to ratios and proportional relationships.

I hope these expectations and games I have listed in this post help all of those who read it! If you have any questions or comments feel free to post them below.

References 
Ontario. (2005). The Ontario Curriculum, Grades 1-8. (6th Ed). Toronto: Ontario, Ministry of Education. 
Small, M. (2013). Chapter 13: Ratio, Rate, and Percent. In Making Math Meaningful to Canadian Students, K-8. (2nd Ed.) Nelson Education. Toronto: Canada. 

Integers, Week 5

Hi Everyone!

This week was dedicated to exploring and discussing integers. As with many topics discussed so far in this course, this topic is interesting because it has real-world application. As such, this topic is one that I find easy to understand and communicate to others.

Integer Contexts are hugely important to discuss with students to help them understand the larger idea. Some useful contexts Small (2013) addresses are temperatures, floors above or below a main level, being below or above sea or ground level, golf scores below or above par, and being in debt or not. For myself, I have always been able to understand integers through the context of temperature, however all of the above examples are all useful.

Katic, K. (2015). Orbit Integers Screenshot.

Within the Math Curriculum, integers become a focus in grades 7 and 8 within the Number Sense and Numeration strands. In grade 7, students are expected to represent, compare and order numbers including integers, demonstrate an understanding of adding and subtracting fractions and integers, and demonstrate an understanding of proportional relationships (Ontario Curriculum, p.99). In grade 8, students are expected to represent, compare, and order equivalent representations of numbers including positive exponents, solve problems involving whole numbers, decimals, fractions and integers, and solve problems by using proportional reasoning in a variety of contexts (Ontario Curriculum, p. 111). Understandably, each increasing grade gains a more complex understanding of mathematical processes, and integers are no exception, however, there will be students who genuinely need extra help in mathematics. This is why I believe that a variety of tools need to be used in order to enhance student learning, such as math games.

Katic, K. (2015). Spider Match Screenshot.

I have recently discovered two really fun games that students can access in order to further increase their knowledge of integers. These two games are specifically related to the grade 7 curriculum expectations, however students in grade 8 can also benefit from these games, and students in grade 6 who are demonstrating a thorough mathematical understanding. The first game above, the Orbit Integers Game, is a high paced, energetic game that brings out the competitive side in its players. Students compete against each other, or the computer, to answer the most mathematical questions correctly to try to finish first. This is a good beginner game for students as it only involves the addition operation.

The second game is the Spider Match Integer Game. Like the Orbit Integers game, it involves the competitive side of students, where they can play against each other or the computer. They must try to answer as many pairs as possible to equate to the number in the middle. The spiders eat the flies, where the flies represent either a positive or negative number. This game is one I would consider to be for students who have a more thorough understanding of integers, as opposed to the Orbit Integers game.

My next steps as an emerging educator will be to try to increase my teaching tool box, where I can access a variety of resources for my students to gain a more holistic understanding of the subjects I am teaching them.

References 
Ontario. (2005). The Ontario Curriculum, Grades 1-8. (6th Ed). Toronto: Ontario, Ministry of Education. 
Small, M. (2013). Chapter 14: Integers. In Making Math Meaningful to Canadian Students, K-8. (2nd Ed.) Nelson Education. Toronto: Canada. 

Fractions and Decimals, Week 4

Hi Everyone,

This week was dedicated to exploring and discussing fractions and decimals. This particular topic I find extremely interesting because it has real-world application. This is not to say that other areas of math do not, however, I personally find myself using fractions and decimals very frequently throughout the day, and because I find it so applicable, it is easy for me to understand and communicate to others.

Small, M. (2013). 277

Within the math curriculum, it states that students must have a deep and thorough understanding of number sense and numeration, which is the category that fractions and decimals fall into. Understandably, with each increasing grade the complexity of the problems increase, and the textbook that we utilize in class is a great example of how this occurs. Additionally, the textbook demonstrates many useful student assessment examples to try to demonstrate where the student is in understanding the mathematical processes and what needs to be improved upon. To the right I have included an example straight out of the textbook within Chapter 11: Fractions. We are to determine where the student understands the question and the areas that need improvement. It is great that this student is trying to use visual representations to demonstrate their knowledge of fractions, however, the future focus for this student needs to be understanding equivalent fractions, and one strategy that can be utilized is by using manipulatives. Through the use of manipulatives, this student will be able to get a more thorough understanding of equivalent fractions, and the next step would be to try to use mental, conceptual math for understanding equivalent fractions. If this student had manipulatives available, they may more clearly understand that 6/9 is equivalent to 2/3 (see below for manipulative examples)

Katic, K©. 2015. Math Manipulatives, Polygons.

Katic, K©. 2015. Math Manipulatives, Circles.

Through this discussion, I have discovered that these manipulatives will be extremely useful within my own educational activities. I find that through personal experiences and conversations I have had with a variety of individuals (as well as textbook examples) that visual representations of mathematical problems make the question and solution more tangible. For example, a student once told me that by having manipulatives readily available in class, it makes math more fun because they understand the problem better! Isn't that what we want as educators, for math to be more fun for students through active engagement? When I was taking the two photos to the side, I forgot how much fun and useful manipulatives are to try to figure out fraction and decimal problems. This methodology for solving problems is going to be readily available to my students, along with helpful instruction and applicable, real world math questions. They allow students to explore their understanding of math deeper, which will be extremely beneficial to them in their adult lives. As an educator, that is the ultimate goal, for my students to be able to take their math skills they learn in the classroom with them into their adult lives.

My next steps will be to further discover resources that my students can access in order to gain a more holistic, deeper understanding of math, as well as gaining insightful feedback from experienced teachers and what they believe are successful strategies for teaching math effectively.

References 
Ontario. (2005). The Ontario Curriculum, Grades 1-8. (6th Ed). Toronto: Ontario, Ministry of Education. 
Small, M. (2013). Chapter 11: Fractions and Chapter 12: Decimals. In Making Math Meaningful to Canadian Students, K-8. (2nd Ed.) Nelson Education. Toronto: Canada.