Theoretical Probability

Let's tackle the Theoretical Probability conceptually using a series of visual prompts to promote spatial reasoning and deep understanding.

 

In this¬†Math Is Visual Prompt, students are given the opportunity to wrestle with the idea of probability¬†through a question in the 2018 Education Quality and Accountability Office (EQAO) Standardized Test for Grade 6 Mathematics release materials. While questions that come up on the EQAO standardized assessment for mathematics tend to look quite dry and “wordy” on the surface, they are great questions that are easy to solve for students who have experience attacking problems concretely and visually using mathematical models and representations as tools for thinking.

Let’s get started…

Spark Curiosity: What Do You Notice? What Do You Wonder?

As I do in most every 3 act math task, I use the Annie Fetter “What do you notice? What do you wonder?” to spark curiosity while watching the first portion of this video.

Here’s an image of what students/children will be looking at when you are instructed to pause the video:

 

Theoretical Probability - Marble Question From 2018 EQAO Grade 6.046 - What do you notice and wonder?

 

After students share what they notice and wonder, you can then show them the next portion of the video until you’re instructed to pause.

 

Theoretical Probability - Marble Question From 2018 EQAO Grade 6.050 - Which colour has a 0.25 chance of being selected?

 

Here, students are now able to go ahead and try solving this problem using any mathematical representation they choose. Remember, if they choose to use an abstract solution using numbers and symbols, be sure to ask them how they know that will work. If they don’t have a conceptual understanding, their method will likely be forgotten and/or cause them to get mixed up as things get more complex down the road.

Fuel Sense Making: Visualizing Theoretical Probability

When you move on, you will start to show one of many conceptual approaches to finding the theoretical probability using a combination of a set, area, and linear model.

 

Theoretical Probability - Marble Question From 2018 EQAO Grade 6.057 Setting up set, area and double number line Whole bag and 20 marbles

 

The reason this is a combination is because you can clearly see the “set” of individual marbles, however they are lined up overtop of a “rectangle” which has area. We are also labelling two variables (number of marbles and whole bag of marbles) using a linear model called a double-number line.

Wow. Definitely a lot going on here. You might need to approach this in sections at first. However, if you build this WITH students, it can be really powerful.

Now, we can partition (or “chop up”) the 20 marbles and the 1 whole bag of marbles by halving:

 

Theoretical Probability - Marble Question From 2018 EQAO Grade 6.060 halving

 

This is a great start. However, we need to see fourths. So, let’s “fourth” or “quarter” this up:

 

Theoretical Probability - Marble Question From 2018 EQAO Grade 6.063 fourthing or quartering

 

Great!

Now we can clearly see spatially how much 0.25 of the whole is, however we need to rearrange our marbles to figure out which colour represents a 0.25 chance.

Initially, some might be thinking maybe yellow or orange. Students might need to physically move them to be sure.

Theoretical Probability - Marble Question From 2018 EQAO Grade 6.066 rearranging to find the chance

 

After tinkering, you’ll see that 5 orange represents 0.25 or 25 hundredths or 1 fourth of the whole bag and this represents a probability of 0.25.

Then, students are asked what the probability of selecting a red or purple marble would be:

Theoretical Probability - Marble Question From 2018 EQAO Grade 6.068 question 2 - probability of red or purple

Pause the video and let them give it a shot.

Here’s one way students could tackle this concretely and/or visually.

First, they can partition (chop up) the entire quantity of 20 marbles and the entire bag into 20 parts:

Theoretical Probability - Marble Question From 2018 EQAO Grade 6.074 exploring number of marbles and how many twentieths

 

Thus, we can see that the chance of selecting a red or purple marble would be 8 twentieths.

Theoretical Probability - Marble Question From 2018 EQAO Grade 6.077 a solution of 8 twentieths

 

We could also “iterate” or combine parts. For every 2 twentieths we combine, we get 1 tenth:

Theoretical Probability - Marble Question From 2018 EQAO Grade 6.080 4 tenths

 

The result shows that the chance is 4 tenths.

For every tenth we combine, we get 1 fifth which results in a chance of 2 fifths:

Theoretical Probability - Marble Question From 2018 EQAO Grade 6.082 2 fifths

 

Pretty awesome what mathematics can do when we approach it from a concrete and visual standpoint first to solidify our conceptual understanding before moving on to steps, procedures, and algorithms!

 

How’d It Go?

Thanks for watching and reading!

Did you use this in your classroom or at home? How’d it go? Post in the comments!

Math IS Visual. Let’s teach it that way.

kylepearce3

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