## 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.

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.

## Fraction Multiplication and Division

My fourth graders are deep into a unit on fractions.  They’ve been multiplying common fractions and tiptoeing into fraction division.  That was until Wednesday of this week.  On Wednesday students explored different ways to divide fractions.  Students used visual models to divide, but that didn’t seem to help students understanding them better.  They encountered abstract problems and used the “flip” method to find the quotient.  This still didn’t help improve much in the conceptual understanding department.  Students wanted me to show the exact process of what to do to solve fraction division problems.  I wasn’t thrilled.  It was evident that students needed more exposure and practice with fractions. So I took a step back and reviewed fraction multiplication.

The class reviewed fraction multiplication and scenarios that are needed to find products.  Students were aware of many different situations where they might need to multiply fractions.   They were able to show visual models and computation strategies to find solutions involving multiplication.  I had a few students also indicate that it’s important to simplify the product.  So the class was rolling in a  positive direction and I decided to bring the lesson back to division.  The break through moment occured when the class connected fact families to the current lesson.  Similar to addition and subtraction, multiplication and division fact families can also contain fractions.  This helped students make connections.  Students wrote out different fact families using unit fractions (1/2,1/4,1/3…).  Students then changed the fact families related to only multiplication and division.  The class was starting to wrap their heads around fraction division with a bit more ease.  I felt as though students were ready for the next activity which was related to food.

Students were placed in teams of three and given a blueberry muffin recipe.

Students reviewed the sheet and wondered where this was going.  Each group then received a sheet related to the original recipe.  Each half-sheet asked students to modify the recipe based on the serving size.

Some students were asked to make 12 muffins, while others said 36, 24, 60, 96 or 72.  I felt as though some students were relieved when they were asked to half or double the recipe.  Other groups tackled the problem with some major perseverance.  Students were asked to show their number model and explain why their answers were reasonable.  Some students wrote number models that multiplied fractions by the recipe amount.

Groups also used fraction division to show a number model.  The majority of groups connected how multiplication and division of fractions can be part of a fact family.  This was especially apparent when students started to see that 1/4 * 4/1 = 1.  I feel like this is laying groundwork for next year’s class when we start pre-algebra equations. Having a solid understanding of how to “undo” operations is a great tool to have in the math toolbox. Once students found the fractional reduction or addition they changed each ingredient accordingly.  After showing their work, students took a picture of the whiteboard and recorded their voice.  Student groups explained how they found each answer and why it was a reasonable answer.  Some student groups were amazing when communicating their reasoning.  They actually explained that the ingredients needed to be increased by a factor of 4.  Other groups were very general with their reasoning in saying that the recipe increased because they were asked to make more muffins.  I can tell this is an area that’ll need strengthening throughout the year.

Overall, this activity seemed to help reinforce skills taught earlier in the year.  The most complicated part was where to start.  Students had trouble knowing what do do with the problem at first.  Students seemed comfortable with the number model and computation components.  Explaining their reasoning needed some tweaking, but that might also be an expectation that needs to be set more in the future.

## First Few Days of School

In less than a week my school year starts. The first week is so important in helping set the tone and stage for the school year. Usually I take out my lesson plan from last year to start planning out the present school year. Some of the activities are the same from year to year and others I tend to ditch. This post/plan is by no means set in stone, but it’ll be helpful in planning as school is just around the corner. Ideally, I’d like to get to everything noted in this post, but honestly I doubt that will happen.  Flexibility is key here and this is a rough outline.

Keep in mind that I usually see four different groups of students during the first day of school. Each group stays for their math block, which is about an hour.

Day One

For the past few years I’ve always had music on as students enter the classroom for the first time. This year will be no different. Students will enter the classroom and find their own seat. The seats aren’t marked. Once everyone arrives I’ll quickly introduce myself and ask the students about their summer. I ask the students to write down one activity that they participated in this summer that they’d like to share.  Students write this down on a Post-it note. I then take all the notes and read off the activities. Each student then claims their activity and tells the class a bit more about their experience.

The class then reviews the arrival / dismissal flow chart.   This is a time where I open up the floor for any questions. We then have a conversation about procedures within the classroom. This takes about 10 minutes. The class then participates in a hands-on geometry game. It’s similar to a Simon Says, but with geometry terms and movements. The students tend to enjoy this and it’s a time for them to get out of their seat and engage in a different activity related to math.

After a few rounds of the game we all find our seats again and I pass out the student consumable math journals. Students then take out their math supplies and start organizing their accordion file.  I model how the accordion file should look and place the tabs in the correct places. Students label their accordion file tabs and organize their materials. I give each student a class information sheet, curriculum guide and contact sheet. Students get all business-like and start organizing their files.

Then it’s picture time! We all line up in the front of the class and take a class picture. The picture is then usually used during Back to School Night.

Following the class picture students start filling out their hand. Students use a Sharpie and write their name on the hand and place it on the door. It remains there for the entire school year. In some sort of small way I feel like it also encourages ownership.

After all the hands have been tapped up on the door we move to the next activity, the puzzle piece community builder. This has been a staple activity for years. The puzzle starts like the picture below.

I then cut out the pieces and each student creates their own according to the directions.  Students place their name, favorite place to visit, favorite math topic, an interesting drawing or whatever you’d like them to place on the piece.  All students in the class create a puzzle piece and then the puzzle is put together once everyone finishes. Once it’s finished it hangs in the room for the year.

Students usually have around 10 minutes or so to work on the puzzle piece before they leave to their next class. Near the end of class I remind students of the dismissal flow chart as they leave.

Day Two

While students enter the classroom I’m planning on having the arrival / dismissal flow chart clearly visible. Today students will help create expectations for the classroom. This takes up a good part of the class, but I feel like it’s worth the time commitment. Once the expectations are established, students sign their name and this document is posted on a bulletin board for the year.   I’m planning on having students practice logging into their online math accounts today. This is important because the math student reference book is only online.

Students will also continue to work on their puzzle piece. Today I’m planning on introducing Estimation 180 and the student recording sheet to the class.  I haven’t yet decided on what picture to use, but I’d like to incorporate this periodically throughout the school year.  By end of the class students should (emphasis on should) have finished their puzzle piece. Today I’m also taking pictures of students as they work. I’m looking forward to using our class Twitter handle and Instagram to document our learning journey.  Students will be asked to compile Tweets in their own words that I will send out throughout the year.  This is another way to document our shared math experiences.

Day Three

Again, students will follow the flow chart that’s posted. I’ll remind students of the expectations that were created yesterday. Students will start to compile the community puzzle of the classes. Today I’ll introduce the math journal to the students. Students will write about their past experiences with math and maybe even write a short version of their math autobiography.   This is a good opportunity to talk about the learning process and how mistakes are valued in this class. I want students to be able to use the math journal as a reflection tool and a place to record their mathematical learning. While students are writing in their journal I generally play sometype type  of music in the background. Students find a comfy place in the classroom to setup their journal time. Once finished, the class will move to a math game/station discussion. Each grade level will play a math game related to their current goal. Some of the more regular games that we play are Angle Tangle, Factor Captor and Name that Number.

Day Four

Today is dedicated to the Marshmallow Challenge. Before completing the activity the class will have a discussion about the importance of being part of a community that’s supportive. We also discuss the math implications of building a tower out of food items. At the end of the time the class will measure all the towers. We then fill out a plus/delta chart indicating what worked and didn’t work. Students usually end this class by having a conversation about team work and building a classroom community of support/trust.

Day Five

Students will delve deeper into their mathematical understanding by completing different types of open-ended/response problems (similar to 1 or 2) in small groups. Students will be asked to explain their thinking and find a solution. Student groups will present their solutions to the class. Many of the open response problems have already been compiled and are found in the district-adopted curriculum. Afterwards, students will be asked to document their experience in their math journals. Students will also login into their Showbie account on their Ipads. Students will be using the iPads to turn in certain math projects throughout the year. Students will be asked to take a picture of their work, annotate their picture and turn it into their Showbie account.  This will also provide students with an avenue to share math work with others.

## Planning Better Learning Experiences

Spring break is now here and many schools are still bustling.  There’s not as much student laughter inside the school, but the parking lot is still busy.  A fresh batch of snow has covered the local area and vehicle tire tracks have carved their way into the teacher section of the school parking lot.  Many of the teachers inside and those at home are planning for the last few remaining months of the school year.  My plan book for each class is now starting to fill up.  Regardless of how I plan, student understanding of a particular concept doesn’t always align with my 3-inch plan book squares.  Specific curriculum and lessons can be planned to a tee, but it doesn’t guarantee an ideal learning experience for the students. This break has given me time to think of how educators plan their instruction.

Before break I was able to have a conversation with my classes about learning.  We discussed metacognition and analyzed how we learn best.  The class had a conversation about what math concepts will be introduced in April.  The conversation transitioned to what math activities are on the schedule for the months of April and May.

While discussing this I emphasized the words learning experiences instead of referring to the objectives that were posted to the board.  I find that students can easily see written objectives on the board.  Writing the objectives on the board is required, but I don’t believe many students actually internalize the meaning or they need more information to do so.  The objectives may say something specific and some benefit from reviewing them, but I want students to be able to understand that they are participating in intentional learning experiences that will give them opportunities to question, make connections, and become better math communicators.

Many of my students and parents are aware of the implications of the PARCC assessments and CCSS.  Common Core aligned material is everywhere.  Marketing and advertisers are consistently promoting the newest aligned Common Core material.  Many districts are in the process or have already purchased content that matches the CCSS and PARCC.  Regardless of what district adopted curriculum is purchased, learning experiences that meet students’ needs should be high on the priority list.  My colleagues and I are finding that there are many ways to follow the CCSS and still create engaging student learning experiences and activities.  This year I’ve modified and used different learning tasks that were created by members of my PLN.  Fawn, Dan, Julie and the MTBOS community have been generous in sharing their thoughts and resources.  These experiences don’t have to be scripted word-for-word (like the first curriculum that I was given) and many supplement the curriculum that the district provides. These student learning experiences are what will create beneficial memories that students can use going forward.  In addition, they will drive students to ask questions, make connections and develop math reasoning skills that will help them in the future.

## Moving Towards Algebra

I’ve been fortunate to have an opportunity to participate in #MTBoS over the past few weeks.  It’s been a worthwhile experience to collaborate with math teachers around the world.  I’ve been able to share/use many of the resources found through this community.  This post is associated with #MTBoS mission eight.

My upper elementary students are now starting to dabble into a few algebra concepts and will be getting a formal introduction in the next few months.  There’s algebraic concepts sprinkled through my district’s curriculum, but solving equations and inequalities isn’t formally introduced till March.  That being said, I’m always on the lookout for additional algebra resources that help gradually emphasize the topic throughout the year.  Otherwise, the unit kind of brings a sticker shock to the students that haven’t encountered writing or solving equations before.

I’ve used visual patterns and Hands on Equations in the past to prepare students for the algebra unit.  Both have been beneficial in wetting the appetite for algebra.  While searching for a few other resources I came across the msmathwiki.  If you haven’t had a chance yet, check it out and maybe contribute some of your math teaching ideas.  I was eventually directed towards @cheesemonkeysf ‘s post about the Words into Math game.  I believe the idea was created by Maria and found in her post here. Two pdfs are included for this game, one informally termed beginning and one advanced.

Both of the documents can be used to match equations and inequalities.  They’re many ways to use this activity in the classroom.  I decided to print one side on orange paper and the other on yellow.  Students cut out each rectangle.  The easiest way for my students to do this was to overlap the yellow and orange sheets and cut them at once.  Both pages line up so it wasn’t that big of an issue.  Students turned all the rectangles so the blank side faced them.

Students then took turns and were allowed to turn over one orange and yellow card.  All cards that were turned over stayed that way.  This is similar to a memory matching game except the cards all stay turned over.  Students then took turns to see if they could match any of the visible cards.  Each match resulted in one point.

As the games progressed students started to become more comfortable with using equations and inequalities.  The game was over after all the game pieces were matched.  Students then bagged up the game pieces for future use.  I shared the ideas with a colleague at another school but haven’t yet heard how it went.

As the class becomes more familiar with algebra, it’s my hope that students are better able to connect past concepts to algebra topics later in the school year. This was an #eduwin for my class as we continue to explore algebra.

## Student Content Creators – InstaCollage

My school’s second grade measurement unit began last week.  By the end of the unit students are expected to be able to measure objects in metric and U.S. Customary units of length.  The students are now starting to measure items in the classroom to the nearest centimeter.  One of my colleagues in a school nearby mentioned that their classroom was having a challenging time measuring different objects.  Students were performing consistent errors, such as measuring using the wrong side (cm vs. inch), not starting the measurement at zero, measuring with the ruler at an angle, not lining up the ruler and object, using the ruler as a helicopter propeller (okay … maybe not the last one).  Anyway, students were getting all types of different measurements and my colleague was getting a bit flustered over the issue at hand.  The teacher continued to teach the concept over the next few days and then decided to assign a brief formative checkpoint to assess student understanding.

We ended up discussing the possibility of using Showbie and InstaCollage app (free version) for this project.

Students were given two minutes to find an object in the classroom that was less than 30 centimeters in length and bring it back to their desk.  They then opened up the Instacollage app and took a picture of the item with a ruler.  The text feature was used to label the measurement.

The ruler needed to be lined up correctly to measure the object to the nearest centimeter.  Students were asked to add their name and the measurement to the photo.  Once the students edited their projects they saved the project and imported it into Showbie.

Once everyone was finished, I reviewed the different projects to assess understanding.  Some students were asked to redo the project.  Most were able to immediately identify the error, correct it and resubmit the project within a few minutes.

I’m planning on showing the students their projects during the next class session.  Not only was this an opportunity to assess learning, but it will also be available in a digital format for retrieval.  I’m looking forward to sharing this with other colleagues.

Student content creators is a blog post series.  Click on the links that follow to find additional posts related to how Educreations, InstacollageHaiku Deck, Playback and Prezi can be used in the classroom.

## Student Content Creators – Playback

One of my goals this year is for students to take more ownership of their learning.  To do this, this year I’ve been focusing on student digital content creation.  I believe that students at any age can show their learning in a variety of ways.  How that learning is measured and the accountability involved can be contentious, as states and districts measure student learning through standardized assessment programs.

I believe my math students need to be able to demonstrate their learning through a variety of modalities.  One way in which my students are showcasing their learning this year is through digital projects.  In the past students have created Educreations and HaikuDeck projects. These projects gave my students opportunities to use a tool that they weren’t familiar with, understand the digital content creation process, express themselves, highlight the learning that’s happening in class through a presentation, and reflect on the learning process.  Most students would prefer to use a technology tool to demonstrate their learning, as opposed to a standardized test/worksheet.

While searching for additional free student content creation apps I came across a lesser known app called Playback.  The developer is actually located in Christchurch, New Zealand.  My students took a quick field trip via GoogleMaps to find out where Christchurch was located.  I took a few screenshots of the app and they are below.

Playback is a free app that allows students/teachers to create a screencast with a video of themselves demonstrating some type of lesson.  Students can use a stylus and draw on the screen by hand or text, as a streaming video can be recorded at the same time.

There are a few limitations thought.  The app can record presentations up to one minute in length (for the free version).  The 60 second limit might make a few teachers cringe as it’s not a ton of time to teach a lesson.  I didn’t mind too much as it helped the students become more concise while explaining their math procedures and calculations.  The videos can then be exported to many different apps.  I tend to have my students export their video to Showbie.

I believe this app could be used for many different purposes.  Students in my class were asked to teach a lesson related to a specific content goal.  Students were given the opportunity to choose one objective and teach it in any way they found necessary using a rubric.  The rubric is still a work in progress as I’m fine-tuning some of the criteria.  I was impressed with their ability and creativity during the content creation process.  The class reviewed all the presentation last week in preparation for the upcoming test.  One of my younger students told me that everyone is an “expert” at some concept in the unit.  How true.   I develop a larger smile when I hear comments like that.

Student content creators is a blog post series.  Click on the links that follow to find additional posts related to how EducreationsInstacollageHaiku Deck, Playback and Prezi can be used in the classroom.

## Math Games in the Classroom

This post relates to #MTBoS assignment four.  For this mission I decided to listen to one of the Global Math Department‘s webinars.  I came across GMD about a year ago and look back occasionally at the webinars that I miss.  While reviewing I found the math games webinar back in January of last year, so that’s the one I picked for this mission.  Plus, I’ve always enjoyed using math games (1,2,3) to review and believe that I can always improve in this area of my practice.

Math games have always been a part of my own teaching practice, but I want to learn how to use them more effectively.  I’m fortunate to have a curriculum that highlights the use of math games in/out of the classroom.  I use math games with my classes approximately once per week and primarily use them during math stations. Most of the math games that I use deal with dice, cards, and/or some type of online component.  For me, the reason for using the games goes back to the concept of learning and engagement.  I believe engagement can be heightened with the appropriate use of a math game.  Math games also allow opportunities to develop skills related to critical thinking and problem solving.  Also, guided math has played a role in how I use math games in the classroom.  With a push for guided math at the elementary level, students that are not immediately with an instructor need to be able to engaged in mathematical thinking, self-govern themselves, and use their time wisely.  Math games at a particular math station provide an opportunity to do just that.

Understanding what makes a good math game is important.  Ensuring that the students are engaged is key.  Students that drift their attention in and out of the game can cause issues; especially if the teacher isn’t directly at that particular math station.  As I watched the webinar, I began to see affirmation and areas where I need to start thinking more critically about how math games are used.

A few takeaways/questions from this webinar include:

• Does the math actually interrupt the game/fun?
• Is the math action the same as the game action?
• Time limits can encourage math anxiety
• Games can be used to introduce concepts, not just for review
• Games can encourage math exploration
• Inferencing, prediction, critical thinking and logic reasoning can all be part of the game
• Rote mathematics doesn’t have to be the emphasis of game
• Math games can reinforce gamification thinking
• Keep in mind the game design process

How do you use math games in the classroom?

## Visual Patterns

My third #MTBoS post is about the visual pattern resource, visualpatterns.org.  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.

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 visualpatterns.org.

## Geometry Explorations

My fourth grade class is now studying geometry.  Geometry at the elementary level allows opportunities for students to get out of their seats and learn while using their compass and protractor.  Last week my students dusted off their protractor and compass in preparation for the geometry unit.  Throughout the years I’ve added different geometry explorations to this unit.  Some of the activities in this post have been modified from the curriculum and others I’ve created or borrowed from some amazing teachers.  I’m going to highlight four specific geometry explorations that I find valuable.

1.

Students are given different types of polygons and asked to find interior angle measurements.   I tend to group the students and have them work collaboratively to find a solution.  Students can use any method to find a solution.  I find that some groups use a protractor, while others find the measurement of the triangle and use it to find the interior angles of other regular polygons. Near the end of the session the class creates an anchor chart that shows similarities/differences between the polygon shapes and their sum of measures.

2.

I pass out a notecard/piece of paper to each student.  Students are asked to make an arc on each corner of the sheet.  The arcs don’t have to be the same size.  The arcs are cut out and put together to form a circle.  Essentially, students use a rectangle and turn the rectangle into a circle and both have the same interior angle measurements. Students are then asked what conclusions can be made by completing this activity.

3.

I generally use this activity before teaching about adjacent and vertical angles.  Students are asked to draw and label two intersecting lines.  This should be review, but most students haven’t been using angles in math class for about eight months.  Once the angles have been created, students measure each angle.  Students are then asked what they notice about the measures of the angles?  Do they notice any similarities?  This is a great opportunity to fill out an anchor chart indicating what angles are close in measurement.

4.

After all the above activities take place I give students a quick formative assessment.  It looks like this:

Students are asked to find and explain the reasoning for the measurement of angle Z.

Overall, these exploration activities allow opportunities for students to engage in math in unique ways.  Math manipulative and explorations often open doors that ignite interest in many students.