WEEK (1+2) x 4= 12!

Hey everyone!


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What a journey it has been with our math learning! Over the last 12 weeks, we have covered topics such as number sense and numeration (fractions, decimals, etc.), patterning and algebra, geometry and spatial sense, measurement, data management and probability and the types of formative assessments in math. In this blog post, I will discuss some things that really stood out to me during the journey!

Rates, Ratios, Percents and Proportions

Within our number sense and numeration strand, we covered proportional relationships. Here is an introductory video to ratios and proportions that will work well for junior-intermediate students during a Minds-On math lesson! Additionally, I created a math word problem catered to Grade 8 students, that allows them to calculate percentages as consumers making purchases. I think that students will be engaged with this problem because they can connect it to their own life experiences! Also, this problem will pave the way to financial literacy. I will discuss the specific and overall curriculum expectations along with the math processes used below: ⇩

My Digital Math Word Problem: To Buy or Not to Buy the Designer Backpack?

No attribution required.

Question:
I received an enticing offer from COACH Canada in the mail. My offer (attached photo) states that I can save 20% if I spend $300-$500 or save 30% if I spend $500 or more. My budget is $555 before tax. I then looked to see what I can buy on the online store and picked this "too cool for school" backpack that caught my eye. The price tag is a modest $1300. (Here's the link to this fancy backpack: http://ca.coach.com/en_CA/coach-gotham-backpack-in-burnished-glovetanned-leather/11106.html?dwvar_color=JIMDB)

Will my budget be enough to purchase this backpack, with my discount offer before tax?

Answer:
Since I am purchasing an item over $500, I will get 30% off. Using percents I will convert 30% to a decimal out of 1.0, so I divide 30 by 100. I move the decimal place once to the left and I am left with 0.3.
Then, I multiply my total $1300 by 0.3 to see how much 30% of $1300 is. This gave me: $390.
I then subtract that 30% or $390 from my total cost to see my final price before tax. So $1300- $390 = $910.
I am over budget. To see how much more money I need, I subtract my budget from my discounted cost. So, $910 - $555 = $355. I need even more than 150% of my budget to buy this backpack. Looks like I won't be buying this pretty backpack anytime soon!

Review of Problem:

The strand that this problem directly relates to overall is number sense and numeration, dealing more specifically with proportional relationships. This question would be part of the Grade 8 math curriculum.

Overall Expectations for this problem:
• solve problems by using proportional reasoning in a variety of meaningful contexts (p. 111)

Specific Expectations for this problem:
– solve problems involving percent that arise from real-life contexts (e.g., discount, sales tax, simple interest) (p.112)

Math Processes involved in this problem (p. 110):
Reasoning and Proving:
• develop and apply reasoning skills to make mathematical conjectures, assess conjectures and justify conclusions, and plan and construct organized mathematical arguments
Connecting:
• make connections among mathematical concepts and procedures, and relate mathematical ideas to situations or phenomena drawn from other contexts
Problem Solving:
• develop, select, apply, and compare a variety of problem-solving strategies as they pose and solve problems and conduct investigations, to help deepen their mathematical understanding

My Overall Reflections Throughout My Math Journey

Over this course, I made several connections to an inquiry question I formulated, where I asked myself, "how can I incorporate culturally diverse content during lessons throughout the curriculum?". I am sure many teachers have this question as well, since our goal is to make our classrooms as positive and inclusive as possible. I was able to find a wonderful video that contains excellent tips to guide my inquiry. Here, the teacher is focusing on an Afro-Centric math lesson. Take a look!

Concluding Thoughts

Over the last few weeks, I learned that as teachers, we must aim to engage our students in math, in order to create meaningful learning experiences. I would like to end off with a great math joke. I hope you understand it and if you do not, don't worry! You can google it! Also, you will eventually become familiar with the terms when you reach secondary school, if you are not there yet!

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Thanks for reading friends!

Love,

Teddy

WEEK 11!

Welcome back!

This week, we had a really engaging math session. This is because we focused on how to incorporate technology into the classroom! In addition, we briefly discussed the implications of assessment, evaluation and reporting in mathematics.

I would like to begin by discussing a great activity that my colleague demonstrated to us. Then, I will highlight a wonderful video that helps teachers incorporate formative assessment into their classrooms!

Math and Technology Activities

This week, the activity that I participated in was so much fun! It was called the "Map Maker" and it is an interactive web-based program. The program can be accessed by clicking here. In a classroom, you would need ChromeBooks, iPads or a computer lab to do the activity. Now, to do this activity you will have a blank grid on the left (see website) and a set of blocks (that are typical landmarks in a city) on the right. The grid uses a compass with north, south, east and west. The goal is to create a city, as you input your blocks (i.e., house, school, park etc.). You can also add roads to connect all of your landmarks. You may add forestry around your city as well.

This activity is applicable to Grade 5 students. According to the Overall and Specific Ontario Math Curriculum Expectations, Grade 5 students are expected to:

Overall-
• identify and describe the location of an object, using the cardinal directions, and translate two-dimensional shapes (p. 82)

Specific- 
Location and Movement
– locate an object using the cardinal directions (i.e., north, south, east, west) and a coordinate system

This interactive game really hits the curriculum expectations because the students are using 2-D shapes and the cardinal directions with a co-ordinate system, as they are using their compass on the website. Additionally, the students are visually locating their objects and then describing their locations. The activity can also help with students' understandings of coding because they are essentially using directions to make a product (in this case, a city). I would love to use this game as an introduction to coding in a classroom too, where students can transfer their math skills to another platform. Specific math processes that are applied in this activity could be making connections, representing and selecting tools and computational strategies (p. 77).

Coding

Minecraft Hour of Code (2017). Graphic. Retrieved from https://code.org

An actual coding activity that resembles the Map Maker is the Minecraft Coding game, found here: https://code.org/minecraft. In Minecraft coding, you must move your objects on the grid and play the game according to the instructions. Thus, here you are also using cardinal directions and a co-ordinate system as you move your 2-D shapes around! I would use this application after introducing the Map Maker. Just to note, the hour of code activities on code.org are exceptionally engaging and easy to use. You can modify the coding activities to your level of preference as well!

Assessment, Evaluation and Reporting in Mathematics

As educators, we have been focusing on the types of assessments and evaluations that make sense in the classroom. We have also noted how important formative assessment is when planning instruction and specifically, allowing the necessary time for ongoing feedback. Summative assessment should only be used sparingly and for evaluation. An excellent tip that teachers can use in the classroom is a formative assessment strategy called "My Favourite No". This is a way for teachers to diagnose the understandings of each student in the class. How it works: the teacher asks the students a question, the students write it on a piece of paper and hand it back, then the teacher sorts the right and wrong answers. From the wrong pile, the teacher chooses their favourite wrong answer and shares it anonymously with the class. Then, the teacher goes through the answer and the math processes step-by-step, allowing all students to engage their thinking and benefit their understandings of the problem. Here is a video that exhibits this practice in a real Grade 8 classroom!

I hope you became inspired today and learned some new things regarding how to use technology and  formative assessments in a classroom with respect to math!

Next week, I will be wrapping up my blog posts for the course. I hope you join me then too!

L0v3,

Teddy

W33K 10

Hi friends!

What are the chances that we have met here today again? They are certain I would believe! Here's a task for you, take a moment and watch this adorable children's story. Embedded in the story is a math lesson, try to see if you can figure out what strand of math it is!

Any thoughts? If you guessed PROBABILITY, you are correct! So friends, this week we will discuss the math strands of: Data Management and Probability. 

Data Management: Stuck in the Middle

Data management is about recording, organizing and analyzing numbers. Some aspects of data management are averages/means, medians and mode. These are all tools used that help us organize and make meaning of numbers. Many people still confuse those three terms. They all have something to do with central tendencies and different types of middle points of number sets. Here are the definitions and how averages, medians and modes help us organize and interpret data.

Average/Mean

Harvey Mudd College (05 January, 2010). Example of a graph with a mean [Graphic]. Retrieved from http://bit.ly/2hyHDkQ

• HOW TO: Adding together several quantities in a SET and dividing it by the total number of quantities used.
• Average helps us understand the middle or central tendency of a quantity of numbers.

Median

• HOW TO: Take the entire quantity of numbers you are using, call this your SET of numbers. Sort them consecutively from the least to the greatest. Find the number in the middle of the list. If the total quantity is even, take the 2 numbers in the middle, add them and divide them by 2, to find their average and thus your median of the SET. 
• Medians help us with finding out the central number of a SET of numbers. 

Mode

• The mode is the most frequently occurring number in a SET. Take your data set and record how many times each individual number occurs. The most frequent is your mode. There may be several modes or no mode at all. 

Not so hard after all right? Anything can be accomplished with a little bit of practice! ⬇

Practicing the Means, Medians and Modes

Would you like to practice and fine-tune your skills? The Khan Academy is a renowned resource with hundreds of instructional videos, lessons and practice opportunities. Here is a practice page for mean, median and mode!

It is Certain That We Will Talk About: Probability

No attribution required. 

In her book "Making Math Meaningful to Canadian Students, K-8", Marian Small discusses that, "probability is the study of measures of likelihood for various events or situations. How likely it is to rain tomorrow, how likely it is that a contestant will spin a particular number on a game show, or how likely it is that a particular candidate will win an election are all examples of probability situations" (p. 628). Small (2016) also states that using pictorial models like probability lines are helpful because students can see, describe and compare likelihoods of events. A probability line has specific terminology and should be out of a whole, like 100 or 1, as in the image. It thus starts at 0 or impossible, continues to 50 or even chance, and stops at 100 or certain. These are all tools to help students visualize and understand the likelihood and chances of events!

Activities

I would like to share a super fun activity that I participated in this week. My colleague made a great worksheet that could be used as a Grade 7 activity. The Ontario Math Curriculum states that the 3rd overall expectation for Grade 7 students is, "compare experimental probabilities with the theoretical probability of an outcome involving two independent events". This activity demonstrated just that! It used 3 bags filled with a set amount of blue and red lollipops (i.e., bag 1 has 5 red, 5 blue, bag 2 has 8 red, 2 blue etc.). Our colleague drew these lollipops for 10 independent trials and we were to predict which bag she was using, based on the event probability. It is always so engaging when there are visuals and manipulatives to consolidate mathematical concepts. Using these resources helps teachers differentiate to visual and kinaesthetic learners and is a fun mind-break and observational time for the logical-mathematical learners!

Thanks for participating in blog today friends! There is a very likely chance I will be posting next week, so stay tuned!

Teddy

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W33K 3^2

H3ll0 fr13nd5!

Have you been finding yourself in any awesome math scenarios since we last me? I hope so, because I sure have! This week in math we looked at measurement. I will be focussing on a few of the great activities that my colleagues introduced to our class this week because I really think that they are noteworthy and helpful for teaching and student learning.

There are so many units of measurement out there and students should develop skills to understand when different units make sense, depending on what they are measuring. In this blog, we will be using the metric system because it is the most used measurement system across the world! Only THREE out of 195 countries use the Imperial system of measurement, can you guess which ones?

1. United States of America
2. Myanmar, formerly known as Burma
3. Liberia

Pretty interesting, right?!

Now, let's take a look at an engaging video used to introduce the idea of non-standard units of measurement. ⬇

Standardized and Non-standardized Units of Measurement

Non-standardized units of measurement are important because they get students using their mathematical thinking processes for measurement strands. Marian Small (2016) states in her book "Making Math Meaningful to Canadian Students, K-8" that, "ideas about area [and measurement] can be introduced through the same three stages as other measurement[:] concepts-definition/comparison, using nonstandard units, and standard units". Measurement is basically a number of pre-determined units, that can extend as much as the object they are placed against. Moreover, non-standard units are used to ease students into eventually applying standard unit measurements, specifically with metric units. The base unit of the metric system is meters, litres and grams depending on the item you are measuring such as, volume, weight and distance.

As you saw in the engaging video, using items like a potato and a football to build a non-standard unit and measure objects with it is fun for students and gets them thinking about what measurement really is plus how to do it! Once students get this basic understanding, they can begin to use standard units of measurement from the metric system. Standard units for distance in meters that students need to understand by Grade 6 are Kilo, Hecto, Deca, Base, Deci, Centi, Milli. You then add the standard unit name, for example, a kilometer.

Using Starbursts Candy for Measurement

No attribution required.

One of my colleagues presented an excellent measurement activity this week. The activity was designed for students in Grade 4 and it made use of the best manipulative, Starbursts!
This activity was taken from the book "Making Math Meaningful to Canadian Students, K-8" by Marian Small, and it gets students ready to measure area. The activity states to:

"Ask students to do the following: cut five pictures out of a magazine, then [ask] which picture covers the biggest? How do you know? [Then] put your five pictures in order from smallest area to largest area". 

Nonstandard units of measurement are applied here because students are using unconventional units, Starbursts, to figure out what shapes are bigger and smaller than each other. As you can see, Starbursts are made into small rectangles and this works well for conceptualizing measurements. We also had a piece of graph paper to use if we needed to outline the shapes and count the squares to make the comparisons. Students could also use estimation with this activity and build those skills as well.

This activity touches on the Ontario Curriculum for Grade 4 students and it states:

Overall Expectations

• estimate, measure, and record length, perimeter, area, mass, capacity, volume, and elapsed time, using a variety of strategies

Specific Expectations

• estimate, measure using concrete materials, and record volume, and relate volume to the space taken up by an object

Thus, the Starbursts activity is a great way to get students engaged and introduced to the concepts of area, perimeter and measurement. Moreover, the spatial relationships between objects and their measurements.

King Henry Died By Drinking Chocolate Milk

Stacy Pearson (2013, March 26). Measurement [Image]. Retrieved from http://bit.ly/2hib7Ub

It appears that back in the middle ages, one of the Kings of England really love his chocolate milk and it was also, sadly his last drink. According to this acronym, that chocolate milk was probably poisoned by someone that wanted take over King Henry's throne! Whatever actually happened and why we may or may never know. This acronym, however, is a great way for students to remember their standard units.

This fun chart was introduced as an additional activity today for  principles in measurement. It is used to demonstrate how exactly students can learn to convert between standard metric units. This activity had the engaging acronym "King Henry Died By/Unexpectedly Drinking Chocolate Milk (note: By and Unexpectedly are used interchangeably as the base unit)" displayed as an anchor-chart for students to answer conversion questions along with more challenging, thinking and application questions. Students can really use the simplicity and cognitive benefits of acronyms and this is a perfect way to introduce this skill into the math curriculum!

Next Week

I hope that you, my friends, enjoyed the fun activities that you can do or have your students do in the junior and intermediate classrooms to help foster measurement skills! These activities were fun and quite yummy and I believe students will really enjoy trying them out.

Join me next week for more math adventures!

Teddy

W33K [64/8] =

Hi there!

This week in math was super interesting. We talked about geometry and spatial sense! In her book "Making Math Meaningful to Canadian Students K-8, 3rd Edition", Mariam Small describes this strand as, "one, but not the only, aspect of mathematics where visualization is important". She also discusses how teachers and students enjoy this particular strand because there are many opportunities for engaging hands-on activities that, in my opinion, can benefit both visual and kinesthetics learning styles. Essentially, Small (2016) states that geometry is the "study of shapes and spatial relationships". I will explore more of these notions below.

No attribution required.

To understand how students learn geometry in school, we discussed several levels that they begin with during their primary level all the way to adult-hood and post-secondary school, potentially. Theory is a very important foundation in all math strands, especially geometry. Thus the van Hieles Theoretical Model was created, stemming from extensive research. I will discuss the levels below↓. Then I will discuss a great activity that we did in class to drive these points into context and application for students in the classroom!


The van Hieles Theoretical Model of Geometrical Thinking for Students

There are 5 sequential levels, starting with level 0 which begins at the primary level and progresses.

0. Visualization - This is the first phase for students when learning geometry. Here, they view different shapes and they have to recognize the differences and classify the shapes. Orientation and symmetry. Examples would be students can arranging basic shape blocks and using Tangrams.

1. Analysis - Here, students try to understand the different shapes. They must know the properties and attributes of shapes that define the shapes. Students study the shapes and analyze. Properties are now more important than the simple appearance of the shape. Examples students can do is sort shapes into groups like which triangles are obtuse and which triangles are symmetric. Students can also use Geoboards to make shapes themselves.  

2. Informal deduction/Abstraction - Students now understand shapes and compare them to each other, they understand and recognize relationships. For example, questions you may ask students are 'how are squares connected to parallelograms?' and 'how are shapes different from one another and how are they similar?' 

3. Deduction - This phase requires higher order thinking. Here, students can apply theories and use logical processes to understand concepts and definitions. 

4. Rigor - Abstract thinking is involved and can extend to higher level thinking, for example it can be used and applied in University courses. Here, students must be able to use geometric reasoning to solve complex problems and issues that arise in the geometry strand.

Please review this great video for more clarification on this theory:

Activities that Help Students Understand Geometry 

I found one activity in particular that my colleague presented to be very useful in presenting students. The activity was titled "Navigating Geo City" and it was composed of a cartesian plane grid with some coloured shapes distributed on it. The goal of the activity was to answer questions based on the shapes' locations and then practice doing reflections. The activity was very well presented because it was so colourful and easy to follow. This made the worksheet visually appealing and very engaging!

Ties to the Ontario Math Curriculum

We can tie this into the Ontario Math Curriculum. For the geometry and spatial sense strand, the overall expectation in Grade 4 states, "identify and describe the location of an object, using a grid map, and reflect two-dimensional shapes". Grade 4 would be a good grade to implement the second level of van Hieles' theory of abstraction, where students recognize relationships between shapes and compare them to each other. 

Other Resources to Teach Geometry

Before we end, I would like to share this video we watched in class. Using virtual story telling is a great way to get students engaged in the material, especially while setting up context and background for a math lesson during the Minds On activity. This video would be great for primary learners in stages 0 and 1 of van Hieles' theory. 

That's it friends, enjoy exploring these resources and have a great week!

Teddy

WEEK [square root of 49]

Dear Math Friends,

I am excited to share that last week, we started one of my favourite units in math class: patterning and algebra! In her book "Making Math Meaningful to Canadian Students K-8, 3rd Edition", Marian Small states that, "algebraic reasoning is the process students use when they generalize numerical situations using equations, variables, and when they study how quantities are related". Additionally, she states that algebraic reasoning can help you focus on the specifics of a problem while looking more generally at patterns and structure.

In this post I will be exploring these concepts and relating them to a unit that I really like in the Grade 9 math curriculum!

Patterning and Algebra in the Ontario Curriculum 

No attribution required.

What might one experience when doing patterning and algebra activities you ask? Here, the Ontario Curriculum breaks down patterning and states, "one of the central themes in mathematics is the study of patterns and relationships. This study requires students to recognize, describe, and generalize patterns and to build mathematical models to simulate the behaviour of real-world phenomena that exhibit observable patterns".

In the junior and intermediate grades, patterning requires a lot of visualizations strategies because students must see how a pattern may continue before necessarily computing an equation for it. Examples of  how students may approach patterning is by using shapes that make a pattern and also use numbers. Specifically, the curriculum states that students need to engage their representational learning strategies, "...students represent mathematical ideas and relationships and model situations using concrete materials, pictures, diagrams, graphs, tables, numbers, words, and symbols". Oftentimes, students struggle with the particular strand because they cannot visualize the connections between math and the shapes, symbols and numbers. Luckily, many students are diagnosed as visual learners and teachers can help easily scaffold their class into understanding patterning.

Another part of the strand is linked with the term "equality". This is my favourite part of the strand! The concept of equality is described as undestanding how equations come from and what makes an equation equal. See that pattern? The root word equal is part of all those words. To put this into math context, students may manipulate equations based on the fundamental principal that the left side and right side of the equal sign should always be equal to each other! With this basic algebra fact, students can solve the most complicated equations!

No attribution required.

Sample Problems

An excellent Ontario Curriculum website is edugains.ca. From one of their documents titled "Continuum and Connections", I found a great sample problem on patterning and algebra for Grade 9 math. This particular example ties into linear equations, where numbers will increase or decrease on a graph at the same rate, creating a pattern. You can find the first differences between points on the graphs and see if you get a pattern! Check out the Grades 9 and 10 math curriculum for more information.

I personally love linear functions as I have tutored this for Grades 8, 9 and 10. Students are usually introduced to linear functions in Grade 8 because once they enter Grade 9, they explore this unit in detail. In Grade 10, students expand on linear functions and start exploring how to solve algebraic equations with substitution and elimination methods. It is really fun, I believe, because you re using your patterning and algebra skills to the max, in addition to other skills like data management!

Check out this example and see if you can follow it. ↡

It is also explained in more detail below:
http://www.edugains.ca/resources/LearningMaterials/ContinuumConnection/PatternsAndAlgebraicModelling.pdf.

Here's an additional video resource on linear function patterns.

I hope you explored these resources and learned something new about patterning and algebra, friends!


See you next week!

Teddy

WEEK 61X *insert Drake's voice*

Hey friends!

Long time no see! I was on a small, but well deserved break because it was my reading week! Over the past few weeks, I've been thinking and reflecting on my experienced with math. I have been thinking about what I've been learning so far and how I've been using math in my daily life.

Quick! 
Solve this part 1 problem to proceed.. how many minutes have I been away, since my last blog post?

I have been away since September 29th, 2017. Today is September 27th, 2017.
How many weeks is that? It has been exactly 4 weeks in total.

So let's break that down. How many days are in a week? Yes, 7 days. So 7 x 4 = 28.

I have been away for 28 days! Now, how many hours in a day? 24! Keep going...you are so close... how many minutes in a hour? 60!

So.. let's work our way back up. 60 minutes in an hour multiplied by 24 hours= 1440 minutes in one day! Wow! Now,  1440 minutes multiplied by 28 days...

= 40, 320 minutes!

Now, those are a lot of minutes! I hope that you make every minute meaningful in your life, because as you see, time flies! Now that I've got your attention, I want to talk about what we focused on this week in math. This week, we discussed ratios and proportions. I've had quite extensive work with tutoring math and I've noticed that my students between Grades 4 and 10 really struggle with the concept. They usually need a lot of coaching and repetition to really cement ratio rules in math. I've also noticed that when students receive an application question, where they need to make connections, they can't organize their ratios properly. In this case, teachers might use the aid of manipulatives to try and cement the mathematical concepts and processes involved.

Manipulatives Activity for Ratios and Links with our Course

No attribution required.

The Ontario Curriculum for mathematics states that, "using manipulatives to construct representations helps students to: see patterns and relationships; make connections between the concrete and the abstract; test, revise, and confirm their reasoning; remember how they solved a problem; communicate their reasoning to others". Tying that into our class textbook, a quote from Small (2016) states that, "it is believed that manipulatives help students by providing • models the student can refer to (i.e., visualize) even when the manipulatives are no longer present • a reason for students to work cooperatively to solve problems • a reason for students to discuss mathematical ideas and verbalize their thinking • a level of autonomy since students could work with the materials without teacher guidance". It is clear that manipulatives play an important role in helping students visualize and make math thinking meaningful.

This week, my colleague had our class group do an Oreos activity to show how ratios work. She gave us each a plate and we had to divide the Oreo cookies with each other based on the ratio she called out. We were able to see how the Oreos were distributed and we had a chance to visually understand the magnitude of the numbers as they were made into ratios. The best part was eating them at the end!

Links With The Curriculum
Last question before you go, part 2! How much time did I spend studying if I spent my time 3 times as much time studying than relaxing? So in terms of ratio, that would look like, 3:1. Follow along... 

3:1 = 40, 320 minutes                        

Total 4 = 40, 320 minutes
40, 320 minutes/4= 10, 080 minutes
10, 080 minutes multiplied by 3= 30, 240 minutes
10, 080 minutes multiplied by 1= 10, 080 minutes

So, I spent 30, 240 minutes studying and 10, 080 minutes relaxing! I prefer a flipped ratio for this, but what can you do as a student, right?

This problem would probably be introduced to a Grade 6 classroom as a proportion strand through number sense and numeration. The Ontario Curriculum states that, "by the end of Grade 6, students will: – represent ratios found in real-life contexts, using concrete materials, drawings,and standard fractional notation (Sample problem: In a classroom of 28 students, 12 are female.What is the ratio of male students to female students?)". You can see how the sample problem is very similar to ours!

Make your minutes count and your days meaningful, friends! Have a SP00KTACULAR HALLOWEEN!

Nathan Shields. (2013, September 26). Halloween Math Pumpkin [Photo]. Retrieved from https://www.flickr.com/photos/maaorg/9955531593

Teddy