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.


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.  

Exploring Rules and Patterns


This past week my upper elementary classes started their equations, patterns, and rules units.  The units are composed of patterns, special cases, student-created rules, and solving equations.  To be honest this is one of my favorite units and involves a good amount of pattern exploration.  Through exploration, students construct their own understanding of how mathematical rules can be developed by analyzing patterns.   Many of these activities involve manipulatives or visual representations of various patterns.  I’m going to highlight three specific activities that seemed to work well this past week.

Analyzing the Perimeter

What's the Rule?
What’s the Rule?

Students were given a handful of square geometry blocks.  They were asked to find the perimeter of one block.  This was quick as students just needed to count the sides of the block.  Four!  Students then put together two blocks and found that the perimeter didn’t double, instead it was six. Students continued the patterns and discussed with their group what the rule could possible be.  Some groups used the whiteboards to write possible solutions.  Throughout this activity students struggled at first and then came to an understanding that the rule just didn’t include one operation. After the rule was discovered the students found the perimeter of 100, 200, and even 1,000 squares put together in a horizontal row.  I believe this activity also helped establish the reason for having mathematical rules.

Rule Tables

Students used four dice, a whiteboard, iPad, and dry erase marker to complete this activity. Two of the dice were operation and they had + and – on the sides.  The other two were typical six-sided 1-6 dice.   Students rolled all four dice and created a rule.  For example, if a student rolled a 6, 2, +, and – then he/she could say the rule is + 6 – 2.  Students wrote the rule on top of the whiteboard and used one of the die to roll five numbers that would be included in the in column.  Afterwards, students were asked to find the out column using the rule that was created.  A few examples are below.


The students then took a picture of their product and sent it to Showbie.  Later on that day the class discussed how to combine rules.  So instead of + 6 – 1 this rule could be + 5.  The students were then combining all of their rules.  This activity led to some productive discussions on how to simplify or expand rules.

Visual Patterns


I came across Fawn’s Visualpatterns site a couple years ago.  This is a fantastic resource that I introduced this past week.  I printed out some of the patterns and placed them in manilla file folders.  The picture of that is located near the top of this post.  The six folders were placed around the classroom.  Student groups visited each folder and determined the rule. While in the group students worked together and filled out the sheet below.

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Modified from this site.

Students took whiteboards and started to build possible rules for the pattern. Once they accomplished this they filled out the table and graphed the relationship.   I appreciate that students are asked to graph their findings.  This could lead into so many other math topics. Students only rotated through two folder stations so we’ll continue this activity next week.  By the way, the students were stoked when I showed them the visual patterns site and not because it has the answers.  A few students even said they were going to check out the other patterns on the site.  I’m looking forward to utilizing this resource a bit more next week.

How do you introduce patterns, rules, and equations?


Math and M.C. Escher


Math and M.C. Escher
Math and M.C. Escher

During the last week of school my students started to explore topography concepts. Topography usually isn’t the first thing that is thought of when someone mentions the word math. That’s why I find it so interesting.  I truly enjoy teaching this topic because it often brings out the best from my students.  I find that most upper elementary students tend to thrive when given geometric shapes and asked to explore, rotate, translate or even turn them inside out.

I generally introduce the unit with M.C. Escher.  The class learns a bit about the life of Escher and his contributions to the world of art.  Moreover, we discuss how art and math are related. This is often a deeper conversations as students start to expand on the notion that mathematics can be found throughout our world.  Topics like the golden ratio and Pi often get brought up during this time.

After learning about Escher’s life and his influencers, the class looked at his different artistic creations. Usually my students recognize at least a few different creations.  Students seem to gravitate towards his optical illusion pieces or the famous Waterfall work.  As each work of art was discussed the more students found mathematics as an integral part of Escher’s work. After reviewing the different pieces of lithograph art, the class watched a short video on how Escher’s design and math are connected.

After the video the students were asked to have a conversation about how math can be found in most art.  The words symmetry, rotations, slides, translations, reversals, surfaces, and perspective were all brought up during the discussion.  What’s nice is that the vocabulary was brought up naturally as students spoke to one another.   I was able to highlight the words and facilitate the discussion as needed.

Eventually the discussion ended and the class moved to the next activity.  I planned to have the students create their own Escher-like artwork.  The students reviewed how to have “Escher-like eyes” when creating their own pieces.  I was proud of the student responses and the imagination that came forth during this discussion.  The class then reviewed the directions to create their own Escher-like creations.

The students went through the directions and asked questions.  Once the expectations were clear I passed out a 8 inch by 8 inch square to each student.  Students created their own tessellation template.  In the future I’m probably going to cut the square dimensions in half so the patterns become more evident.

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Eventually the students used the template to create an Escher-like creation.  Students showcased their work to the class using the vocabulary mentioned above.  The students were able to bring their work home on the last day of school.  All in all, this is a lesson I’m intending on using next year and a definite #eduwin in my book.


How do you incorporate art and math?





Visual Patterns

#mtbos post three
Patterns and Algebra

My third #MTBoS post is about the visual pattern resource,  Generally, patterns (numerical and object-based) are some of the first concepts taught to introduce algebra and number characteristics.   Initially, patterns may seem simple, but they often allow opportunities to enrich and extend instruction into more challenging concepts

My third grade class has been studying patterns and rules during the past two weeks.  One of the activities that we used can be found here.   We’re using math tasks to uncover different type of numerical patterns.   I’ve had the opportunity to visit this site multiple times during the algebra unit.  One of my classes tackled the problem below last week.

Visual Patterns

The students loved that this problem was created by a sixth grade student.  I think this added to their motivation and bonus! … also had them thinking of how they could create their own problems.  The students were given whiteboards and worked with a partner to find the fourth pattern. During that time I asked questions that helped guide the students towards a solution.  With enough time, most groups were able to find a solution.  The class then had a discussion on how the solution was derived.  Then came the fun part …  the partners decided to answer “How many lego pieces are in step 43”?  Student groups then presented their answers to the class.

At some point I’d like to have my students create and submit their own patterns to