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.

 

One School One Book

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I’m straying a bit away from my usual math posts to highlight a literacy connection.  This week my school kicked off their One School One Book campaign.  It’s been an annual tradition to have all students in the school read one book for a period of about a month or so.  Every students gets a copy of the book and classes engage in questions and conversations about the book.  We’ve been participating in OSOB for the past few years.  In the past, we’ve used Charlotte’s Web and The World According to Humphrey for the school-wide event.  This year the school is using the Lemonade War for OSOB. One aspect that’s helped make this successful is an organized process that’s been used before and during the reading.  One of our school’s fifth grade teachers, Vicki, has helped organize a team and the process since its beginning.  Here’s the process that’s been used over the past few years:

1.)  To generate student curiosity, cutouts of certain items are placed around the school.  Sometimes the school uses an Ellison letter machine for this. For example, when our school used Charlotte’s Web, cutouts of a pink pig were placed all over the school.  They were placed on doors, windows, in the hallways and even on the ceiling.  Teachers didn’t say a thing and let students ask questions and wonder why they were placed all over the school.  This year coins were placed all over the school since we’re using the Lemonade War.  This year students had an idea that it was related to OSOB, but they weren’t exactly sure what the book was.

2.)  In the meantime, teachers read certain chapters of the book in front of a green screen.  For the past two years my school has been using Touchcast to record the readings.  The background is changed based on the chapter’s contents. Since I teach mostly math, I took the chapters involving math and graphs.

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3.)  A group of teachers participate in a skit from the book.  All students attend an assembly to see the skit. The students don’t yet know what the book will be until the skit concludes.  At the end of the skit the book is revealed.  The kids tend to get a kick out of seeing their teachers as characters in the book and it also generates additional interest in the book.

4.)  Students read the first chapter of the book the evening after the skit.  A reading schedule is sent out to all the students and the principal includes information about OSOB in their newsletter. After reading the book they can also watch the Touchcast videos.  This year questions are embedded within the videos.


Every year it seems that OSOB engages students in reading a book as a community.  The curiosity and engagement that seems to follow OSOB continues and that benefits stakeholders. The success of this has me wondering how school’s can use a similar model to promote math.

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.

We’re Back in Business

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Students are scheduled to enter my school tomorrow morning.  It’s been a whole two weeks since I’ve seen my students.  Tomorrow, routines will be reestablished, backpacks will be filled, students will be chattering about their break, and students and teachers alike will get back into school mode after a brief hiatus.  As  tomorrow approaches, I’m reflecting on what my classes have accomplished and what still is on the plate ahead of us.  I spent a good amount of time yesterday planning out the next week of instruction and it confirmed my anxiousness to know that the school year is over half-way completed.  As I look over the next few months, I’m finding curriculum pacing guides, standardized testing, school performances and field trips all impact my instruction to a certain degree.  This happens every year and it has me thinking of what time I truly have left with the kids. I’m also aware that these next few months directly impact students in meaningful ways.  For some, this will be my last year with a group of fifth graders that I’ve seen since they were in second grade. I want to ensure that I make the most of that time remaining.  That doesn’t necessarily mean speeding through the curriculum.  I’m hoping to gives students opportunities during the next few months to make connections, reflect and set goals.  As we all come back tomorrow, I want to communicate the following to my kids:


1.)  The learning experiences that you’ll encounter in the next few months are intentionally designed for you to make meaningful math connections.  Perseverance will be key in helping you create these connections.  You might find that you don’t understand a particular concept when we introduce it.  That’s okay.  Learning is a process and we’re all in this together.

2.)  Group projects, individual assignments and standardized tests are on the calendar and will be approaching in the next few months.  Keep in mind that I believe you’ll will show your potential on all of these. The scores and marks will help teachers and your parents have a better understanding of your strengths and areas that might need to be bolstered.  Also keep in mind that the scores are a number and don’t represent who you are as a person.

3.)  Let’s celebrate a milestone.  We’ve worked hard and have made significant progress since September. Each student in here has made gains and I want us to reflect a bit on our success. There’s more to accomplish, of course, but reflecting on our past growth can also encourage us to move forward with additional confidence.


I’d like to communicate this to all my classes at some point tomorrow.  I won’t necessarily read off a script, but I feel like flushing it out on here is a decent starting point.

It’s time to get back into the routine of setting my alarm clock to wake up extra early.  I’ll be joining the trove of educators heading back into their schools this week.  I’m looking forward to tomorrow.