Over the past couple of weeks, some colleagues and myself put together a chart outlining Mariam Small's "Big Ideas" for the Ontario Math curriculum, and matched these with the specific expectations from it as well; specifically for Grades 68. If you are unfamiliar with Small's "Big Ideas", here is a link to a PDF outlining them, with examples. Our document can be found here.
Feel free to use the chart for your own long range plans, and please comment on this post if you notice any need for changes that we should/will make looking forward!
Monday, October 26, 2015
Bansho A Mathematics Instructional Method
A colleague of mine the other day mentioned Bansho, and not having a strong math background from my teacher's college days, I had never heard of it before then. Having said that, after doing a bit of research and discussing Bansho's concept with them, I realized that more and more we are seeing this method of mathematical instruction make its way into our Ontario classrooms. I suppose then it is no surprise that the Ontario Ministry of Education has made many publications regarding this concept, one of which you can view here for more information regarding Bansho. ***UPDATE October 24th, 2015: here is also a lesson plan template/exemplar explaining how to structure a Bansho lesson.
My last blog mentioned Bump It Up (BIU) walls in the classroom, and Bansho provides the teacher as well as their students with the opportunity to create and assess student work in real time in an inclusive and cooperative educational environment. It focuses more on what Dan Meyer calls "patient problem solving" (see video below for more information on this concept), which is having students use their knowledge of math and how it works to problems in order to clarify students' understanding of how math works, and why it is meaningful.
Anyways, I wanted to post a blog about this to provide anyone who is interested with these resources to see how they can implement Bansho into their mathematics programs, and why it is a great tool for improving not only student success/understanding, but to help reshape/redefine the aging textbook/linear mathematic programs from the past. As Dan Meyer states, "Math needs a makeover", and here in Ontario we are doing this one classroom at a time!
My last blog mentioned Bump It Up (BIU) walls in the classroom, and Bansho provides the teacher as well as their students with the opportunity to create and assess student work in real time in an inclusive and cooperative educational environment. It focuses more on what Dan Meyer calls "patient problem solving" (see video below for more information on this concept), which is having students use their knowledge of math and how it works to problems in order to clarify students' understanding of how math works, and why it is meaningful.
Anyways, I wanted to post a blog about this to provide anyone who is interested with these resources to see how they can implement Bansho into their mathematics programs, and why it is a great tool for improving not only student success/understanding, but to help reshape/redefine the aging textbook/linear mathematic programs from the past. As Dan Meyer states, "Math needs a makeover", and here in Ontario we are doing this one classroom at a time!
Bump It Up Walls in the Mathematics Classroom
Ok, so Bump It Up Walls (or Interactive Performance Boards) have been around in many school boards for a few years and we are seeing them in almost every classroom. Personally, I really like the idea of what a BIU wall can provide my students, as each level clearly shows students what a particular level of achievement looks like. Besides this, students actively engage in the classroom with this wall by developing and deconstructing an assignment's success criteria and learning goals in order to gain a better understanding of what each levelled exemplar has and/or does not have. Thus, at any given point while constructing an assignment, they can look back at the wall, success criteria, rubric, etc. to help figure out where their work currently is and formulate steps of how to "bump it up". Similarly, this serves as a great tool to help students assess their peers work to provide meaningful and transparent feedback. If you'd like more information about what a BIU wall is, or what it looks like, you can check out Stephanie Kennedy's video explaining her BUI wall and/or read Raine6's blog about them.
The purpose for my blog post today is to shed light on a BIU issue that I have struggled with to date, and that is how to formulate a successful BIU wall in the Mathematics classroom. My background has typically provided me with opportunities to engage with BIU walls in Language, Art and Social Studies courses, as these disciplines encompass tasks that include formulated writing via reading comprehension, research skills, creativity, etc. Thus, providing levelled exemplars is as easy as having students assess success criteria (SC) and mark sample questions; gauging which parts of the SC are evident, and which are not. Math is a little more tricky with regard to this type of marking, as often students understand math answers as either "correct" or "incorrect", so how can we assign levels when the perception of math as a discipline is so black and white?
After some research on the internet, and viewing countless amounts of Math BIU walls online, I'm starting to get the impression that the focus for a levelled approach in Math is on the process and representation of student answers, much like the Social Sciences/Arts BIU walls, however, there is a stronger emphasis of how the answer is represented. For example, depending on the success criteria, students are often encouraged to solve a problem however they see fit and, therefore, the levelled approach is based on what how student demonstrates how/what they know as opposed to an answer being "correct or incorrect". Here is a neat example video from Pat Johnson, showing his grade 1's Math BIU wall.
For interests sake, here is a link to some Pintrest BIU wall exemplars!
Wednesday, October 14, 2015
Baseball & Math
So I'm sitting in my living room, coming down off my high from the Blue Jays win and being the nerdy teacher that I am, I think..."How can I incorporate the world series into math lessons?"
So I look up online and of course there are hundreds of baseball themed math lessons. Obviously I wasn't the first person to think of this.
I found a few interesting activities off this website
http://illuminations.nctm.org/Lesson.aspx?id=1025
I think the easiest connection between baseball and math is statistics. You can have your students try to figure out Jose Bautista's playoff batting average by calculating the number of hits and divide it by the number of at bats. Even students that don't follow baseball can relate because it's just a different example for calculating mean.
Who else has used athletics or baseball in their math lesson planning? How effective is it for those students that are athletic and can relate to baseball a lot more easily?
So I look up online and of course there are hundreds of baseball themed math lessons. Obviously I wasn't the first person to think of this.
I found a few interesting activities off this website
http://illuminations.nctm.org/Lesson.aspx?id=1025
I think the easiest connection between baseball and math is statistics. You can have your students try to figure out Jose Bautista's playoff batting average by calculating the number of hits and divide it by the number of at bats. Even students that don't follow baseball can relate because it's just a different example for calculating mean.
Who else has used athletics or baseball in their math lesson planning? How effective is it for those students that are athletic and can relate to baseball a lot more easily?
Friday, October 9, 2015
Class Kick
I came across an app that is really interesting and can be beneficial to students and to educators. The app is called Class Kick. It's basically an online classroom. Teachers can create the classroom online through this app and can post assignments, quizzes, blog forums.
The app and website have available resources for teachers to use to create different assignments and tutorials on how to navigate the Class Kick app. I watched a video on how to create an assignment and saw how easy it was to use. You can basically copy and paste images of worksheets or draw in questions with the cursor. Another really neat aspect of this app is that you can correct students work in real time. You can view which students are having problems (they have the option of clicking a button that is a hand so the teacher will see that they have raised their hand for help)
Does anyone have any actual experience with this app? I wonder how effective it can be. It seems like it would be a great tool for the classroom, maybe as supplementary work.
http://www.classkick.com/#home
The app and website have available resources for teachers to use to create different assignments and tutorials on how to navigate the Class Kick app. I watched a video on how to create an assignment and saw how easy it was to use. You can basically copy and paste images of worksheets or draw in questions with the cursor. Another really neat aspect of this app is that you can correct students work in real time. You can view which students are having problems (they have the option of clicking a button that is a hand so the teacher will see that they have raised their hand for help)
Does anyone have any actual experience with this app? I wonder how effective it can be. It seems like it would be a great tool for the classroom, maybe as supplementary work.
http://www.classkick.com/#home
Thursday, October 8, 2015
Teaching Math using Statistics in Sports
Simple addition methods can be taught through sports
statistics. For example, in order to figure out the number of point a team has,
you must calculate the number of wins and ties in relation to how many points
are allotted for both these factors. If each win counts as two points and each
tie is worth one, these numbers can be calculated to determine the number of
points. Thus, the number of wins and ties has a direct relation to a number of
points a team has.
The steps to this problem can
be followed like so: If a team has 30 wins and 0 ties and each win is worth two
points, the number of wins would be multiplied by 2 in order to get the number
of points earned (30x2=60). Another example which can be used as a next step
would be to add in the number of ties to the equation. If a team has 30 wins
and 15 ties (ties = 1 point), how many points do they have? (30x2 + 15x1 = 75
points). In this equation, the number of wins are multiplied by two while the
number of ties are multiplied by 1. After this step is completed, both values
are added together to calculate the number of points the team has.
I believe this is an
excellent lessons to teach students how to use their multiplication and
addition skills. I also think that relating this lesson to sports will create a
higher interest in the students. While doing this lesson, a number of sports
pictures and statistical sport standings can be used to keep the lesson
interesting. I also believe this will help students who actually play sports by
giving them the tools needed to understand how to calculate statistics.
Understanding Fractions Using Food
Fractions can be a tough math unit to teach to younger grades. It
may be even more difficult when students are not as interested in the lesson
due to the difficulty of it. Therefore, as teachers, we must find other
interesting ways to engage the students with our lessons. In most cases,
students won't even realize they are learning! I believe incorporating
fractions with food such a pie or pizza (or any type of food that can be
divided into fractions) would be a great way to teach younger students.
In this type of lesson, a pizza can be
divided up into fractions. For example, if we slice a pizza into 8 parts and
give away 2 pieces, what would my fraction be? Once students understand the
basic principles of fractions they will be able to answer questions like this
one. Asking students if they like certain foods like pizza, cake, or pie at the
beginning of the lesson will get their attention. It's not even necessary to
introduce the lesson as fractions! After a couple examples the teacher could
explain what fractions are and how we can use them in this manner.
This strategy is important because it
gives the students something to relate to. The teacher may also reward the
class by brining in the foods used in the example for an endofunit task. This
will also give the students something to look forward to and keep their
participation levels up.
Money is Math
Using money in math can be a very effective way to teach addition
in the classroom. In this case, coins can also be used as a supplementary tool
in addition to students writing. Money as a symbol of numbers can be related to
addition as well as fractions. For example, using quarters can help student
remember how many quarters are in one whole or in this case a dollar. In
addition, students can learn that a dime is 1/10 (onetenth) of a whole which
also means it takes ten dimes to make one dollar. Other example are obviously
using nickles or pennies to make a dollar.
I believe that this method is also useful in terms of
teaching students how to add. Asking students to make a dollar using more than
one type of coin would be an excellent way to stimulate their minds.
Understanding math using money is not only something that is beneficial in the
classroom, but is also something that students will be doing a lot of in their
futures. This is why I believe using money is an excellent tool for math class.
Wednesday, October 7, 2015
Mathematics Question Amazes the Internet
Mathematics Question Amazes the Internet Christina Whates
In a society that is primarily dominated by social media it really isn't surprising when something goofy or phenomenal becomes viral on the internet, but the article below does surprise me. A couple months ago (see problem below) a Singapore (grade 5) math question flooded the internet with questions of how, what, why and the most common, WHAT IS THE ANSWER? I stumbled upon it myself and became stumped. I was actually flabbergasted at how difficult the problem was for me to solve. I guess my real question is does our curriculum lack the ability to build our problem solving skills? People questioned whether the problem was really too difficult or the mere fact that maybe we need to sharpen up our problem solving skills. The article states that statistics show that students from Singapore score significantly higher than any other country in mathematics, but what does that say for us Canadians? Should we be doing something different too? I'm not sure what the answers are to these questions, but I can't lie I was pretty impressed that grade 5 students can solve problems like the one below.
http://www.theatlantic.com/education/archive/2015/04/themathquestionthatwentviral/390411/
In a society that is primarily dominated by social media it really isn't surprising when something goofy or phenomenal becomes viral on the internet, but the article below does surprise me. A couple months ago (see problem below) a Singapore (grade 5) math question flooded the internet with questions of how, what, why and the most common, WHAT IS THE ANSWER? I stumbled upon it myself and became stumped. I was actually flabbergasted at how difficult the problem was for me to solve. I guess my real question is does our curriculum lack the ability to build our problem solving skills? People questioned whether the problem was really too difficult or the mere fact that maybe we need to sharpen up our problem solving skills. The article states that statistics show that students from Singapore score significantly higher than any other country in mathematics, but what does that say for us Canadians? Should we be doing something different too? I'm not sure what the answers are to these questions, but I can't lie I was pretty impressed that grade 5 students can solve problems like the one below.
Does our Mathematics Curriculum need a Facelift?
Does our Mathematics Curriculum need a Facelift?
Christina Whates
Dan Meyer proposes that we need to change our approach when it comes to mathematics in the classroom that encourages inquiry thinking. Often our lessons present mathematics in a manor that is strictly on a memorizing basis. Students answer all questions in a similar way  they find the same three pieces of information plug it into an equation and away they go. But are we really teaching students to become critical thinkers? How could incorporating real life questions that students can actually relate to change how students see mathematics? Dan Meyer has fully convinced me that if teachers create problems that allow students to question and apply real life situations, it will increase their problem solving abilities as they progress into adulthood.
Mean Median Mode – Kinesthetic Based Learning
Mean, Median, & Mode for the Kinesthetic Learner
Christina Whates
Teaching mean, median, and mode can become a routine lesson, but there are always ways to make mathematics lessons interactive for learners. I have included a short, but detailed lesson teaching mean, median, and mode. I decided to use Smarties instead of M&M's to keep my lesson nut free. Considering all the allergens present in elementary schools today peanut free is always the best option, and including a treat can always be fun! I bought a box of Smarties for a dollar, so an easy way to make this lesson more cost effective is to have students work in pairs.
Ontario Curriculum (Grade 5):

– read, interpret, and draw conclusions from primary data
Included is a definition of Mean, Median, and Mode. These are just rough examples of what can be done as a class. Together a definition of each can be created with an example, that can be posted on the class math bulletin board.
Activity:
1) Provide students with Smarties.
2) Organize the Smarties.
3) Calculate Mean, Median, Mode.
4) If time is permitted students can organize the data into a graph.
2) Organize the Smarties.
3) Calculate Mean, Median, Mode.
4) If time is permitted students can organize the data into a graph.
Sunday, October 4, 2015
Funtastic Graphing
I came across a few great examples of graphing activities to get kids involved in hands on data collecting in a fun way! Some are quite classic activities, but sometimes a helpful reminder is needed to remember that these activities exist and are easy to implement into the classroom. Here are three examples of fun and interactive graphing activities:
1. Graphing: Facebook Birthdays: This activity pulls in technology and social media into the math class! Have students graph the frequency of birthdays each month on Facebook by "friending" other students in class as their data set, and observe what patterns may occur throughout the year!
If students do not have profiles of their own, you could use this similar idea on the website Fakebook, where students could create fake profiles to collect data as an alternative.
2. Paper Airplane Graphing: Have students design and test different paper air plane models, and collect data about how far each one flies! Students will graph the different distances in order to figure out which model produces the best results. This would be great for middle school classrooms.
To make this more challenging, you could have students change variables on one plane model (adding weight, changing wing length) to make a cross curricular connection to scientific investigation, and the engineering design cycle.
3. Candy Graphing: This would be a perfect activity to perhaps put into practice in preparation for the Halloween season! Place a mixed amount of candy (or M&Ms/Smarties/jub jubes) into a jar and have students guess how many of each colour are in the jar. Students will then be able to sort, count, and graph the results. Additionally, you could use prepackaged candy bars and create a largescale graph on the classroom floor as another visual alternative.
This would be a good introductory graphing activity, but of course it would be incredibly important to consider the potential for classroom allergies, and dietary restrictions.
1. Graphing: Facebook Birthdays: This activity pulls in technology and social media into the math class! Have students graph the frequency of birthdays each month on Facebook by "friending" other students in class as their data set, and observe what patterns may occur throughout the year!
If students do not have profiles of their own, you could use this similar idea on the website Fakebook, where students could create fake profiles to collect data as an alternative.
2. Paper Airplane Graphing: Have students design and test different paper air plane models, and collect data about how far each one flies! Students will graph the different distances in order to figure out which model produces the best results. This would be great for middle school classrooms.
To make this more challenging, you could have students change variables on one plane model (adding weight, changing wing length) to make a cross curricular connection to scientific investigation, and the engineering design cycle.
3. Candy Graphing: This would be a perfect activity to perhaps put into practice in preparation for the Halloween season! Place a mixed amount of candy (or M&Ms/Smarties/jub jubes) into a jar and have students guess how many of each colour are in the jar. Students will then be able to sort, count, and graph the results. Additionally, you could use prepackaged candy bars and create a largescale graph on the classroom floor as another visual alternative.
This would be a good introductory graphing activity, but of course it would be incredibly important to consider the potential for classroom allergies, and dietary restrictions.
Making Math Notes Interactive
Interactive notebooks have become increasingly popular, and I have noticed them widely appearing on education websites. These are great resources to help students create notes that are engaging and help to highlight the major concepts for a particular topic. As you will see below, I found the following images of an interactive notebook on another blog: Interactive Notebook Entry: Graphing Using SlopeIntercept Form
Personally, I am a big fan of this concept, and could see myself using these widely in my classrooms to incorporate principles of metacognition into the math classroom; it can be challenging for students to learn how to effectively take notes, and turn a blank notebook page into great study notes for the future. If you would like to learn more about math interactive notebooks, and see a few more examples of their implementation, visit the following website: Everybody is a Genius: Interactive Notebooks
Example: Interactive Notebook for Graphing Using SlopeIntercept Form
Personally, I am a big fan of this concept, and could see myself using these widely in my classrooms to incorporate principles of metacognition into the math classroom; it can be challenging for students to learn how to effectively take notes, and turn a blank notebook page into great study notes for the future. If you would like to learn more about math interactive notebooks, and see a few more examples of their implementation, visit the following website: Everybody is a Genius: Interactive Notebooks
Example: Interactive Notebook for Graphing Using SlopeIntercept Form
Slope Intercept Foldable (y=mx+b) and Brief Notes over Rearranging Equations 
Outside view of y=mx+b Foldable 
Inside View of y=mx+b foldable. The x and y flaps define y as the dependent variable and x as the independent variable. 
Frayer Model over YIntercept 
Notes over graphing using slopeintercept form. This page is kinda blah. 
Reminders of which way to rise/run to get a positive or negative slope. Some of my students found this really confusing. Need to fix in the future! 
View of Left and Right Page in INB. 
Battleship: Cartesian Edition
Recreating an inclass version of the classic game Battleship is a great way to get student engaged and practicing how to plot points on the Cartesian Plane! The website Education World has a detailed activity plan to show teachers how to implement this strategy into classrooms.
Simply put, students plot points on their grids to represent their ships of different sizes before facing off against one another to see if they can locate and sink each others ships.
This activity can be modified for different age groups to show plotting points in one quadrant, two quadrants, or all four quadrants depending on the grade level and needs of the teacher. I can also see the potential for a larger scale version of this activity using painter's tape and tiles on the classroom floor as the coordinate system guide. This activity is well explained in the clip below by teacher/youtuber Mr. Dunaway of the Mr. Dunaway's Math Videos channel:
Simply put, students plot points on their grids to represent their ships of different sizes before facing off against one another to see if they can locate and sink each others ships.
This activity can be modified for different age groups to show plotting points in one quadrant, two quadrants, or all four quadrants depending on the grade level and needs of the teacher. I can also see the potential for a larger scale version of this activity using painter's tape and tiles on the classroom floor as the coordinate system guide. This activity is well explained in the clip below by teacher/youtuber Mr. Dunaway of the Mr. Dunaway's Math Videos channel:
HandsOn Visualization: Orange Peel Geometry
Visualizing how formulas came to be is no easy task, especially for students who struggle in math. It is important to accompany algebraic models with visual ones, in order to enhance understanding for all students. One area where this can implemented is explaining surface area formulas; incorporating nets is a great example of showing a visual (and even kinesthetic) representation of these concepts. One great example I have come across to visualize uses an orange peel to help show the relationship between area of circles, and surface areas of spheres.
Basically students measure the diameter of the orange, and draw four circles with the same diameter. Students then peel and flatten the orange peel and try to fit together the peel into as many circles as needed. Students will observe that the surface area of the orange is equal to the same amount as the area of the four circles that have the same diameter. This will help students to see how the two formulas are connected to each other! Plus they have the chance to get kinesthetic during class.
You can find this activity on the Math Solutions Webpage and searching for the activity "Orange You Glad...?" in the search bar. Happy teaching!
Basically students measure the diameter of the orange, and draw four circles with the same diameter. Students then peel and flatten the orange peel and try to fit together the peel into as many circles as needed. Students will observe that the surface area of the orange is equal to the same amount as the area of the four circles that have the same diameter. This will help students to see how the two formulas are connected to each other! Plus they have the chance to get kinesthetic during class.
You can find this activity on the Math Solutions Webpage and searching for the activity "Orange You Glad...?" in the search bar. Happy teaching!
Tuesday, September 8, 2015
Thursday, August 6, 2015
Cool Classroom Activity
I was searching the internet for interesting activities that I could implement into my own classroom some day and I came across this that I thought I would share.
http://www.coolmath.com/SurvivorAlgebra
It is an elaborate activity and the creator has shared it for us to use in our own classrooms.
Take a look and let me know what you think.
Could this be something that you see yourself implementing in your own classroom?
http://www.coolmath.com/SurvivorAlgebra
It is an elaborate activity and the creator has shared it for us to use in our own classrooms.
Take a look and let me know what you think.
Could this be something that you see yourself implementing in your own classroom?
Wednesday, August 5, 2015
We NEED math
I am a firm believer of the idea that mathematics is the most important school subject. Simply put, I feel that studying mathematics makes you an overall smarter person, even in other study areas. Studying mathematics develops critical thinking and problem solving skills that can be applied to other areas of life and other school subjects. As a part of everyday life we run into problems that need to be solved and although it might not seem like it, mathematics help us solve these everyday problems. When a person reaches a certain level of mathematical understanding they begin to apply these understanding to real life subconsciously, without realizing that mathematics is to thank for their heightened ability to come up with solutions and think in a way that people who do not prioritize math are able to do.
Here is a link to a website that posted an article about why mathematics is so important: http://www.mathworksheetscenter.com/mathtips/mathissoimportant.html
This article gives examples of what I was previously talking about, math makes you a generally smarter person. It provides scenarios in which math is necessary to get by and makes it easier to understand how a person with skills in mathematics would be able to get by these scenarios in a more efficient way than a person who lacked these same mathematical skills.
Here is a link to a website that posted an article about why mathematics is so important: http://www.mathworksheetscenter.com/mathtips/mathissoimportant.html
This article gives examples of what I was previously talking about, math makes you a generally smarter person. It provides scenarios in which math is necessary to get by and makes it easier to understand how a person with skills in mathematics would be able to get by these scenarios in a more efficient way than a person who lacked these same mathematical skills.
Thursday, July 30, 2015
Perfect Squares and Patterns
Perfect Squares and Patterns
If you have been following my last blog, you may find this new post helpful for you in solving questions of Pythagorean Theorem.
Firstly, I am going to talk about perfect squares.
The following are some examples and nonexamples of perfect squares from the online resource:
http://www.mathwarehouse.com/arithmetic/numbers/whatisaperfectsquare.php.
Examples of perfect squares
 9
 9 is a perfect square because it can be expressed as 3 * 3 (the product of two equal integers)
 16
 16 is a perfect square because it can be expressed as 4 * 4 (the product of two equal integers)
 25
 25 is a perfect square because it can be expressed as 5 * 5 (the product of two equal integers)
Non examples of perfect squares
 8
 8 is a not perfect square because you cannot express it as the product of two equal integers
 5
 5 is a not perfect square because it cannot be expressed as the product of two equal integers
 7
 7 is a not perfect square because you cannot express it as the product of two equal
 integers
Hopefully, these above examples has helped you understand more about perfect squares.
Secondly, I would like to introduce some easier and faster ways for students to calculate their questions as I did in my last post.
Usually, students would just know the square of one digit number which is from 1^{2} up to 9^{2}. However, it would be even better if students can memorize up to 19^{2}. They could either memorize them just by memorizing or they could memorize them using a pattern.
1) Perfect Squares  Pattern 1
#’s Squares Difference between squares Difference increases by
11^{2} = 121 [+23] +2
12^{2 }= 144 [+25] +2
13^{2 }= 169 [+27] +2
14^{2 }= 196 [+29] +2
15^{2 }= 225 [+31] +2
16^{2 }= 256 [+33] +2
17^{2 }= 289 [+35] +2
18^{2 }= 324 [+37] +2
19^{2 }= 361
Pattern: the difference between 11^{2} and 12^{2} is +23, and then the difference increases by +2 as the number increases by 1.
2) Perfect Squares  Pattern 2
#’s Squares Difference between squares Difference increases by
152 = 225 +400 +200
252 = 625 +600 +200
352 = 1225 +800 +200
452 = 2025 +1000 +200
552 = 3025 +1200 +200
652 = 4225 +1400 +200
752 = 5625 +1600 +200
852 = 7225 +1800 +200
952 = 9025
Pattern: the difference between 15^{2} and 25^{2} is +400, and then the difference increases by +200 as the number increases by 10.
By knowing and memorizing these patterns, students can calculate squares faster.
Hope this post helps you in calculating squares as well.
Pythagorean Theorem
Pythagorean Theorem
A right triangle has a 90^{。}angle. The longest side of the triangle, c is opposite to the right angle and it is called the hypotenuse. The other two sides, a and b, are called legs.
c^{2 }= a^{2 }+ b^{2 }
The Pythagorean Theorem is the most famous and most important result in geometry. It allows us to compute the unknown length in a rightangled triangle, if given the other two sides.
Example: Calculate the length of the unknown side.
^{ } = 9 + 16 = 25
c = √25 = 5 m
Example: Calculate the length of the unknown side.
100 = 64 + b^{2}
10064 = 64 + b^{2}64
36 = b^{2}
B =√36 = 6 cm
Practice:
Calculate the length of the unknown side, to one decimal place.
Calculate the length of the unknown side, to one decimal place.
After introducing Pythagorean Theorem, I really want to talk about one convenient way to calculate the length of the three sides in a right angle triangle.
If you have done a lot of calculations on Pythagorean Theorem, you would have noticed that there is a ratio for the three sides that is always true.
The ratio is a : b : c = 3 : 4 : 5, with 5 being the c (hypotenuse) side.
For example:
The length of side a (adjacent) and b (opposite) are 6cm and 8cm. Find the hypotenuse.
Then you can find out the hypotenuse by:
a : b : c = 3 : 4: 5
=> a : b : c = 6 : 8 : 10
Since 3 : 4 = 3 * 2 : 4 * 2 = 6 : 8
Therefore 5 * 2 = 10 cm.
a : b : c = 6 : 8 : 10
Ans: Hypotenuse is 10 cm.
This way of calculation is NOT applicable to Isosceles Right Angle Triangle.
Calculator Concerns
I have had many people, students and parents alike, ask me the point of learning how to do mental math when we have calculators. I mean, it seems like students as young as elementary school students have personal devices these days, so they always have a calculator on them. I can't argue that.
However, we cannot forget that calculators, like other pieces of technology, are tools to help us. They do not replace our brains. We still have to tell the calculator what we want to do, and it will give us an answer we may or may not reject (this is where number sense and mental math comes in!).
Calculators help us become more efficient, but here are some challenges around calculators that we must keep in mind:
1. A calculator is a piece of technology, and it must be explicitly taught.
Now that electronics are part of our everyday lives, it’s easy to forget that we don’t just learn how to do things by osmosis. Back when I was in elementary school, we had computer classes, where we learned to type through software such as All The Right Type. Students these days are expected to “just know how” and guess what? They type with two index fingers at about ten words per minute. Some skills just need to be taught and practised, and using a calculator is one of those skills. Human error and syntax are two problems that can be minimized with practice.
Now that electronics are part of our everyday lives, it’s easy to forget that we don’t just learn how to do things by osmosis. Back when I was in elementary school, we had computer classes, where we learned to type through software such as All The Right Type. Students these days are expected to “just know how” and guess what? They type with two index fingers at about ten words per minute. Some skills just need to be taught and practised, and using a calculator is one of those skills. Human error and syntax are two problems that can be minimized with practice.
2. We need to use the same calculator all the time.
Remember the previous point about how using a calculator is a skill that must be practised? If a student uses one calculator at school and another calculator at home, how ready will that student be for a test at school, when they’ve studied for it at home?
Remember the previous point about how using a calculator is a skill that must be practised? If a student uses one calculator at school and another calculator at home, how ready will that student be for a test at school, when they’ve studied for it at home?
The most basic of calculators have numbers, the four operations, and an equal sign. These calculators do not have brackets, so imagine what happens when a middle school student tries to type in an order of operations question without the brackets. The answer would be completely off, unless of course they were taught the appropriate way to use a calculator (i.e., only typing in the part of the equation that is being solved). I suppose this particular problem – human error – wouldn’t exist if #1 was done well.
Another type of calculator is one where syntax matters. Syntax is the order of the input of your numbers (e.g., square root button first and then number, or number first and then square root button?). Calculators made today don’t really have this problem. However, older calculators as well as those on devices (like the iPhone calculator) do care about syntax. I have personally witnessed students using their device calculators (probably for the first time in their lives) on a test, and seeing the frustration on their faces because this supposed tool to help them actually messed them up. Again, using a calculator is a skill that needs to be developed and refined.
For the purposes of proving that all calculators need to be practised, I will describe one more type of calculator that is usually free from human error and syntax problems, but we should still be cautious about. These are the amazing handwritinginput calculator apps on devices. An example of this is the MyScript Calculator (available on both iOS and Android). These calculators recognize handwriting input and immediately solves it on screen, so it minimizes the chances of human error, since you can see exactly what is in the equation. They also don’t care about syntax, because you can easily add or delete numbers and operations with the stroke of a finger. What is a problem, however, is that handwriting input isn’t perfect at the moment.
In my opinion, all calculators have potential for error (i.e., human error, syntax error, or other). The only way we can minimize these errors is to make sure that we as teachers reinforce the fact that students need to always be using the same calculator in class, at home, and on assessments.
Prioritizing "RealLife" Mathematics
As you can probably infer from the photo above, I went to the shopping mall the other day and walked into a store with lots of savings. I overheard the following conversation between a middleschoolaged boy and (presumably) his mother:
Boy: I like this shirt.
Mom: How much is it?
Boy: Um... $37.99 with 30% discount.
Mom: So how much is it?
Boy: I don't know, this chart doesn't have it.
Mom: Use the calculator on your phone to figure it out.
Boy: How do I do that?
To be fair, I'm not exactly sure how old this boy is, but it did make me rethink our Ontario math curriculum. Correct me if I'm wrong, but the learning of percentages (e.g., how they relate to fractions and decimals) begins in Grade 6, but it's not until Grade 8 that students actually learn to calculate percentage discounts and sales tax. In my opinion, it's far too late.
According to CtrlF, the word "real" appears in the mathematics curriculum 50 times (e.g., real world, reallife). In the section under "Principles Underlying the Ontario Mathematics Curriculum" (page 3), it explicitly states that it will challenge students to make connections between mathematics and the real world. If that's the case, shouldn't we prioritize "reallife" mathematics in the curriculum? Perhaps introduce concepts like sales tax much earlier in the curriculum, rather than in Grade 8? I realize that comes with some complications in the sequential learning of mathematics, but there's really nothing more relevant than calculating sales tax, discounts, and gratuity in real life.
Wednesday, July 29, 2015
Making Math Fun Is Possible
For decades there has been a common trend of many students not liking Math for various reasons. Some students don’t like it because they don’t understand concepts and find it difficult while other students don’t enjoy math because they find it repetitive and boring.
As a new teacher, I want to change this trend by not only changing instruction so that students can understand various concepts being taught but to also change instruction so that students LOVE math and have fun while learning.
From my past experience, I have found that most classes love competition. Therefore incorporating games into math instruction where students can compete against each other is one way to get them excited about math. These math games can be as simple as using math activities from IXL Learning https://ca.ixl.com/math/grade7 where there are multiple exercises that follow the curriculum for all grades. When using these activities you could split your class into groups and have students compete against each other to solve the questions.
I also like incorporating math into other subjects that students enjoysometimes that might not even realize they are using their math skills as well! For example, including geometry into an art project where students have to create a figure with a certain area and perimeter. Or incorporating math into physical education where students have to complete an ‘Amazing Race’ using their physical abilities as well as their mental skills to complete certain tasks. There are many ways you can incorporate math into other subjects, making fun and crosscurricular activities.
These are few of many ideas that I would like to apply to my teachings someday as a classroom teacher, follow this blog for more to come!
Mathemagic
Videos are a great way to get students interested in Math content that is being presented to them.
TED Talks is a great resource for videos that students can learn from and get them excited about Math.
The video posted below is wonderful for talking about squaring large numbers. In this video we have ‘Mathemagic’ where a gentleman can square large numbers in secondsit is a fun and interesting video! At the end of the video it shows his process of thinking to solve these questions.
There are many more videos on TED Talks that can be applied to a math lesson.
Math Trick
At the end of a lesson, unit, or week I like to incorporate
something fun at the end of class to make students happy and get their minds off
of the topic at hand; I feel this is useful because it gives students a mental
break. A cool way to do that and look like a math genius at the same time is by
following this video. You can use this trick any time and it will get the whole
class engaged and they will never figure out how you are doing it! It will be a
very fun component added to class whenever you feel it would be useful. Then,
when you decide to teach your class this trick, you will have a class full of
math geniuses.
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