Probability and Tree Diagrams

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My fifth grade students are in the midst of a unit on probability.  This is one of my favorite units to teach for a couple different reasons.  One is that it follows a massive pre-algebra unit and it’s so different than what students have been working on for the past few months.  I feel like it’s time students see a different strand of mathematics. Another reason, is that students have to think logically about probability and it’s something that impacts their daily life.  Also, students haven’t had a lot of time to discuss probability in math class.

Near the beginning of the week students started to explore the different terms related to probability.  They completed a random selection activity the week prior and students are starting to have a better understanding of the terms.  Around mid-week students investigated tree diagrams and their usefulness in determining actual probability.  One of the highlights on Tuesday was a maze activity.  Students were given a scenario where they needed to find the probability that students would win or exit the maze without running into a dead end.  They used number cards 1-4 to accomplish this. It looked similar to the image below.

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Students first estimated the probability that they’d win and then created a tree diagram to find the actual results.  They tested out the game by playing six times with a partner.  The class was asked what they found and if their estimations were in the ballpark.  For the most part they weren’t, which was good news because the class used a tree diagram to find the actual probability.

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Students were then asked to use the maze as a fundraising activity. The next question is below.

If 100 students entered the maze, how many would end up being the winner?  Let’s say that the winner receives $25. How much profit would be made If students were charged $5 to enter the maze?  

This was a turning point in the lesson because students started to become even more vested in what was happening.  I gave them about 3-5 minutes to work independently and then they shared their findings with their table group.  Most groups were right on target and were able to explain their math reasoning.

On Thursday, students were asked to use their probability skills with spinners and tree diagrams.  I found an amazing resources in this book that spurred me to recreate a diagram that my students could use. I gave a copy of the diagram to each student.


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I placed five minutes on a timer and gave students that time to work independently to read the prompt and start to find a solution.  Students wrote on the sheet and attempted to put together a cohesive tree diagram that made sense to them.  I had a few students that thought it was impossible  After the five minutes were up, students were asked to share their strategy with partners.  The answers were interesting and all over the place.  Some students were confused with the spinners as they had to convert them to fractions.  Other students had issues with the actual directions.  I helped answer questions and students presented their ideas on the solution.  This entire activity took 30+ minutes to discuss.  Students finished up their ideas on the paper and turned it in.  I’m reviewing the results right now and can tell that I need to follow-up with the class.  The majority of students did very well, although simple mistakes seem to be evident in quite a few.  The class will be discussing this on Tuesday.  

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Random Numbers and Sheets

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My fifth grade classes started their data analysis and probability unit this week.  On Monday the class had a conversation about the terms we use when discussing data.  The words, likelihood, probability, experiment and chances were all discussed.  After reviewing the terms we dove into the first lesson of the unit.

One of the first activities I generally use asks students to draw a card (between 1-5) 20 times.  The data is supposed to be collected and then shared.  The class then looks at the predicted probability compared to the actual results.

I decided to change the lesson a bit by incorporating a technology component and possibly save some time in the process. The class also just finished a pre-algebra unit and I thought the formulas used in a spreadsheet could reinforce some of the learning.  I’ve had success with using Excel with my fifth grade class so I decided to use that medium for this lesson.  Also, my students now all have Google Drive passwords so they’re all able to login with a Chromebook.

Earlier in the day I put together a Google Sheet with a tab for every student in the class.  I shared it with all my students during our math block. Students retrieved a Chromebook, logged in and found the shared document.  I modeled the formula within Sheets and the students followed along.

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Students were able to randomly select the digits between 1-5.  Students observed their data and how it changed.  We had a classroom discussion on how the sample that they created was based only on 20 trials.  They were then able to observe their personal total.

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After reviewing their total, they could view the tab called data set.  This showcased the data of the entire class.  The total, over 300, was much closer to the predicted results.

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After students compared the two they filled out a writing prompt asking them. to compare their individual results to the class. What were the similarities or differences?  How does a larger data set impact reliability?  Students wrote down their responses.  I’m in the midst of grading those right now.

The activity was great, but also had some issues.  Getting everybody to stick to their individual tab took some work.  Some students were caught viewing other students’ tabs.  Also, the data sets kept changing when someone clicked certain cells.  This was tedious near the beginning.  Regardless, once those two kinks were taken care of it was smooth sailing. I ended up freezing some of the cells so students couldn’t change them.

At some point the class will revisit the spreadsheet to discuss tree diagrams.  Click the image to copy and use the spreadsheet in your own class.

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I changed the names to S1, S2 … so you can change them as needed.