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.  

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.

 

Fraction Blocks and Strategies – Part 2

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Last week my second grade crew explored fraction blocks.  They cut out and used the blocks to compare fractional pieces.  Students enjoyed the trial-and-error component and they started to visualize fractions in a different way.

I decided to use a similar activity with my third graders. Instead of labeling the bars, I decided to leave off the label. This initially confused the students as they expected to see the label. Students moved beyond the confusion when they were given the value of one of the blocks.  They then used that value to compare all the other blocks. Students were asked to cut out the blocks and start comparing them.  I didn’t give them any directions beyond that.  After about 4-5 minutes I placed the sheet below on the overhead projector.

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We completed the sheet as a class.  I used the document camera and students compared the pieces on their own desk. It took multiple attempts and a number line, but eventually the class was able to finish the sheet.  Students were then off on their own to find the whole or part of certain blocks.  Students used many different strategies since they couldn’t rely on the label.  screen-shot-2017-01-28-at-7-53-24-am

While the students were working I went to the different tables and observed the strategies. Almost all the students compared the shapes to one another to find one whole.  Other students created a number line and placed where they thought each shape would be located.  I had a few students take out a ruler and measure the blocks.

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I collected the sheets once everyone was finished, marked them up with feedback, and returned them the next day.  I used the NY/M model for this assignment. Every student in the class needed to make some type of correction.  After a brief review, I gave the students back their sheets and they made corrections.  There were few perfect scores after the second attempt, but everyone improved – an #eduwin in my book.

Download the file for this activity here.

Next week we’ll be learning about equivalent fractions and how to find common denominators.

Fraction Blocks and Strategies

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My second and third graders started a unit on fractions last week.  Students are used to identifying typical pie fraction pieces.  Generally, I find students are introduced to fractions using this type of visual representation.  Students then count the amount of pieces and place that number as the numerator.  I find moving towards mixed-numbers has some students changing their strategy as they can’t just count the pieces, but they have to recognize that a certain amount of equal parts are one whole.  Based on their pre-assessment results, it seems as though my second grade and some of my third grade students are at this point.

Using a number line has helped.  Placing the fractions on the line has brought a better understanding of the placement of fractions in relation to a whole number.  Currently, students can identify certain benchmark fractions on a number line.  We’re working on bolstering this skill and connecting it to fraction computation in the near future.  Before that happens I want to ensure that they have a decent understanding of mixed numbers and where they fall on a number line.

On Thursday and Friday I introduced students to a fraction block activity.  Students were given a sheet with fraction parts.  Each block was split into a certain amount of equal square parts.

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Each student was given an envelope to put their pieces in once they were finished with the activity.  Students cut out each block and were asked to put them in order from least to greatest value.  Students were able to complete the task.

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We then had a conversation about quarters, halves and wholes.  I then gave each student the card below.

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Students placed the A block near the top of their desk and started comparing the different blocks.  The class completed the first question together.

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I then gave students time to work on the rest of the problems.  Students were then given time to use trial-and-error to find which blocks worked for each problem.  I went around to the different table groups and asked students questions about their strategies. Students ended up matching the squares with other shapes to determine what was a quarter, half, almost a half, and what happens when you combine shapes.  After about 10 minutes the class reviewed the sheet and found that some problems could be answered with multiple solutions.  Students put the sheets in their envelopes since we ran out of time.

The next day students completed some more challenging half-sheets involving their blocks.

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Students struggled a bit with this as they had to look at A as half instead of one whole.  This changed the value of all of the other blocks.  I allowed students to work in groups for about five minutes and then independently for another five.  This gave them an opportunity to gain another perspective and a different strategy.  Afterwards, I reviewed the possible solutions with the class.

Next week I’m taking this activity one step further and using the blocks without markings.  I’m borrowing this idea from Graham’s post on defacing manipulatives.

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Students will complete similar half-sheets, but without the evident markings. I’m looking forward to seeing how students’ strategies change and the math conversations that follow next week.   Click here to download the activity that I used.

 

Reasonable Solutions

 

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I believe teaching multiple grade levels within the same day has value. Being able to observe how students think about numbers and the strategies that they use over time gives teachers a different perspective.  It also shows some of the linear progression of math skills and strategies. I found this especially evident as I read through Kathy Richardson’s book during July. I currently serve as a math teacher for students in grades 2-5.  I get to see how students progress over time and what tends to trip them up.  I also see the problems that emerge when students start to rely on tricks and formulas before having a deep understanding of a particular concept.  One thing that I also continue to observe is that students sometimes struggle to be reasonable with their estimates. Part of that may be due to an over-reliance on algorithms and the other part may relate to exposure. Students aren’t given (or take the) time to reflect and ask themselves whether the answer truly makes sense or not.  This tells me that students are relying on a prescribed process or algorithm and reasonableness comes second.

In an effort to move towards reasoning, I’ve been using Estimation 180 on a daily basis.  I feel that the class is become better at estimating and their justification has improved.  Making sense with number puzzles also seem to be helping students create reasonable estimates and solutions.  Basically, students are given a story that has blanks.

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Students are then are given a number bank. Sometimes too many numbers are in the bank.

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Then students have to justify why they picked each answer.  This can be completed in verbal or written form.

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Usually I have students explain their reasoning with a partner.  The class has completed a number of these types of making sense with numbers puzzles.  I can say that students are now looking more closely at the magnitude of the actual numbers before estimating or finalizing an answer. That’s progress and I’m confident that students are more willing to use that strategy along their math journey in the future.

Random Sampling

Before break my students tackled the challenging topic of random sampling.  I feel like it’s challenging because some students tend to view their opinion as one that applies to other people around them.  It can be a tough concept for students to wrap their heads around. When I introduce this topic students have many questions.  Usually they follow along the lines of …

  • why can’t you ask everyone?
  • who determines if the random sampling is accurate?
  • how many people do you need to ask?
  • is their always bias involved in random sampling?

Some of these questions are more challenging than others.  Some I don’t even approach and let students make their own determination.  In the past, I had students create questions and ask a random sampling of students.  Students would then create charts and indicate whether they truly sampled the students fairly.  For the most part the activity hit the objective, although the sampling available at my school was minimal.  Students were able to ask questions about our school and students within.  Issues came up because of the lack of age groups and diversity.

Last Monday I participated in #msmathchat.  The conversation surrounded the topic of teaching about data and statistics.  Elizabeth sent out the Tweet below.

I saved the Tweet for later as my students are in the midst of their data unit.  I looked at it later that evening and thought I could immediately use it with my kids.  I put together a template that students could use as they progressed through the site.

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The next day students started at the skate park activity and used three random sampling techniques.  Afterwards, students were able to see the how their actual results compared to the entire population.  Students then moved on to complete the rest of the scenarios.  For the most part students started to change the way they asked the questions to get a better estimate.  This was a better activity than what I’ve used in the past.  The students responses to the last question brought a better insight to how students perceive random sampling.  I believe they’re making headway.  I’m hoping that the class can reflect back on this activity after break and they can take the benefits of that experience moving forward.

Two Different Camps

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My school has six days of school left before break.  Between now and then I’ll be giving a unit assessment to my fifth grade crew.  We’ve been studying angle relationships for the past few weeks.  To be honest, it’s been a great unit but it’s also been challenging.    There’s been a good amount of struggle in this unit. It’s the good type of struggle.  Right now I feel like students are in one of two camps.

One camp is focused on the measurement and precision component.  When given a question about angles they want to take out a protractor and start measuring.  They want to be precise and get an exact answer.  I’d say that some in this camp perceive this type of geometry as a measurement skill, rather than a looking at it as a problem associated with angle relationships.

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Where’s my calculator?

The other camp is all about looking at the angles and the relationships that exist.  They’re at the point of not even bothering to use their protractor.  They also look at the lines, rays and line segments that make up the construction of a shape.

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A quadrilateral is 360 degrees and a triangle is 180 so …

Getting both of these camps on the same page has been an interesting adventure.  Both have positive aspirations and have been showing a tremendous amount of effort. I believe it’s important for students to use mathematical tools to solve problems, but that’s not what this unit is about.  For so many years students have been asked to be specific and precise when calculating and finding math solutions.  This is still the case, but students are now asked to use their understanding of angles and shapes to come to conclusions.

We had a classroom discussion last week about this very issue.  I asked students to put away their protractors and calculators.  They were asked to identify specific shapes and describe the characteristics of them in detail.  The class then explored the different polygons on the Illuminations site.  Click on the image to visit the actual site.

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Students were allowed time to play and create connections.  The focus of the exploration was targeted towards sum of the angles in polygons.  The students in the first camp started to put their protractors away while the students in camp two looked at how the angle measurements changed when the triangle was stretched.  Looking back, this was such an important period of time.  Afterwards, students were given time to review angle relationships without using a measurement tool.  They were using their prior knowledge of shapes and relationships solve problems.  This was a bit of shift.  So, I decided to build upon the first task and added a reasoning component.

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Getting camp one and two on the same page

I’ll be grading the task above tonight.  Including an “explain your reasoning” component added a bit for vigor to the task.  Based on the class conversations I heard today I’m thinking that students looked at precision as well as angle relationships while tackling the problem.  After grading them at some point tonight, I’ll review the results with the kids tomorrow.