Math Responses and Discussions

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Last year I experimented with a couple different ways to encourage students to discuss mathematics.  I used a form of a number talk last year and found some success.  Students were engaged the conversations were more productive than in the past.  I also noticed that not all students participated in the conversation.  Even with manipulatives, some students participated minimally and shied away from being called on.  I found that some students dominated the discussion more than others.  This was taking place in most of my classes and I kept on reinforcing the importance of having a positive classroom climate where mistakes were honored.  I thought emphasizing the climate and providing support would help encourage participation from everyone involved. For some that worked, others not so much.

This year is a bit different.  I’m still using a form of number talks with success.  I’m still looking for ways to help improve this process.  I also introduced a more organized way to incorporate math discussion prompts with students.  I first organized students into groups using a randomizing student spreadsheet.

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Students are put into groups and a destination in the classroom.  I put a new slide on the whiteboard once everyone finds their assigned location.

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Students get into their groups and identify themselves as partner A or B.  Usually I use the spreadsheet to indicate the partners.  Partner A starts with the first prompt and I display it on the whiteboard.

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I click the timer and partner A has 40 seconds to respond to the prompt while partner B listens.  After the 40 seconds I pick a few different people in class and ask them about their thoughts about the prompt and their answer.  Partner B then gets a different prompt.

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Partner B gets to respond to the prompt while partner A listens.  I’ve toyed around with 20 – 40 seconds and have landed on 40 because it gives students an ample amount of time, but also the limit encourages them to be concise.  Students usually go through 2-3 questions each and then we have a whole class debrief session.  So far students have been receptive to this medium and I’m hoping to expand it to other classes that I teach.

Files referenced in this post:


Finding the Difference

My second grade math group started this week.  I gave a pre-test on Monday and found that students had some trouble with the word difference.  Many of the second graders saw the word difference and immediately thought subtraction. I could see why students would see this as a quick search reveals difference as being “The result of subtracting one number from another” and “How much one number differs from another.”  I think most students in my class focused on the first definition rather than the latter.  While discussing the word more than a few students brought up that they knew how to find the difference using a method.  It ended up being the standard subtraction algorithm.

On  Wednesday students were introduced to part of a 100 grid and asked to use it to find the difference between two numbers.  Some of the students started to see that difference could be interpreted as distance between.

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Students used two different colors to locate and identify the numbers.  Students then counted the space between the two numbers.  They used hops while moving to the right and then down.

30 + 3 = 33

Another student used the grid to show a different way to find the difference.

3 + 30 = 33

I showed both methods under the document camera and the class discussed how both could work.  Students were then asked to place their strategy on a number line.


Another student raised their hand and wanted to show the class something that they created.

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Next week the class will be investigating regrouping strategies.

Categorizing Numbers and Number Lines

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This week my students explored how to categorize numbers. By then end of the week students were expected identify integers and rational numbers and apply them to real-world contexts. The class reviewed what and where to place numbers on a number line and how to classify them as whole, counting, integers, rational, and/or irrational numbers.  This was an introductory lesson and the term rational and irrational were new to them.  After a brief class conversation about the differences between rational and irrational numbers the class took a deeper dive into how to identify the characteristics of each classification.  The class looked at a few true/false statements:

  • Is 1,000,000 a counting number?
  • Is 1,000,000 an integer?
  • Is every rational number in an integer?
  • Is zero is a counting number?

The class went through these types of questions and were able to respond and justify their answers.  The questions started to get more challenging as students needed to circle  multiples answers.

  • Circle all of the numbers that belong to each set.

Integers:   4.5       2/3     102     -6       8       0


This was more challenging and took some time to categorize each number to see if it fit accordingly.  Students were then asked to place numbers on vertical and horizontal number lines.  I was glad to see how well the students responded to the vertical number line as I don’t believe they get enough practice with those.

Students had about 20 minutes left and one project to complete.  I introduced students to a number line project.  I ended up going with Google Draw for this project because I don’t have enough access to iPads at the time and I was able to checkout a Chromebook cart for this particular lesson.  Students were given a prompt to use dice to create numbers and fractions to place on a number line.  They rolled and found their numbers.  Students used their Chrombooks to access

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Students make a copy of the Google Drawing and added their numbers to the number line.  It took some work to manage the tools involved in this platform.

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I explained what each icon meant and how they could use it to make the number line their own.  It wasn’t as smooth of a transition as I thought it’d be, but students persisted and were eventually able to place the numbers they created on the number line and dragged the label to each number.  Students were then expected to take their drawing, save it as an image and place it in their individual SeeSaw account.

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Not all students finished this in class and I sent it home as optional homework for students to complete.  The above example is from one student that took it home and completed it before putting it into their SeeSaw account.

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Data Landmarks and Context

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One of my classes is working on a unit related to data displays and number systems. Around a week ago the class was putting together sets of numbers to match data landmarks.  This was a challenge as students had to think differently.  The class was also asked which data landmark better represents a student’s performance.   I was meaning to write a post then, but a number of things came up and it never happened.  Fast forward a week and here we are.   

Students were given two sets of scores from two different students. 

Jack’s scores:  85, 81, 78, 100, 84, 89 

Sonja’s scores:  55, 87, 91, 92, 68, 93 

Students were asked to find the median and mean for each student.  For the most part, students were able to identify both of these landmarks.  Here comes the kicker … now students needed to determine which landmark better represents each student’s performance, mean or median?  This was a challenging prompt for a couple reasons.   

  • Students weren’t accustomed to using the word represent in this context.  Students were taking the view that the students should get the higher grade and that would be the mean or median. They explained that the student should receive the higher grade because they (the person) is a hard worker and deserves to be rewarded with the highest score. 
  • Students thought of the word represents as the typical score.  When discussing the mean earlier in the year the word typical would often come up as a synonym. 
  • Students looked at the last score as the most recent and thought that should be the final representation.  My school is heading in the direction of standards-based grading so that’s maybe why students took that approach.  I don’t know. 
  • Students looked at the lowest and highest score of each set of data and reviewed the range to help them pick the median or mean 


After struggling a bit, the class came together and we discussed a few possible solutions.  The class agreed that the question allows a lot of room for interpretation and context certainly matters.  The fruitful conversation brought about a change in perspective for some as students started to see this type of math differently than just numbers sprawled across a page.  The numbers had meaning and the context drives the answer.   

A little later in the week students were asked the following prompt: 

If you were the teacher in Jack and Sonja’s class, would you use the median or the mean to calculate students’ grades?  Explain. 

This was a bit confusing at first, but students made progress in understanding the context and how it helped determine which landmark to use.  Again, I had answers related to the teacher wanting to give the higher score to help students with confidence.  Other students used the data landmarks to find the average.  I felt like students were more comfortable using the average as they could say that they used every data point, therefore making sure all assignments counted for something.   

I’m looking forward to next week as we dive into histograms. 

Building a Classroom Community

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The 2019-20 school year officially started on Wednesday.  We’ve only been in school for three days, but it seems like a full week and then some.  Teachers had institute and professional development on Monday and Tuesday.  The first few days of school tend to be full of community building, learning new routines and organization. Everyone is nicely asked to be patient as new routines are established – such as going to lunch, specials, arrival and dismissal. Teachers also talk about the different procedures involved in all the weather and emergency related drills.  Most classrooms have and are continuing to build community and rapport in their classrooms.  They create expectations together and come to a common understanding of what they look and sound like.  This year teachers are asked to conduct morning meetings with their classes.  The idea sounds interesting and is one way in which to help reinforce the community aspect of a classroom while addressing social/emotional learning standards. This is a first in my district so the verdict is still out on how it’ll be received, but I’m optimistic and I think there’s potential in starting the day off with this structure.

I often loop with my students for multiple grades so I tend to stay away from using the whole-class community builders year after year.  In my mind it keeps our activities fresh, although there have been time where I use similar projects, but in those cases the final products are different.  I’ll be highlighting two activities that I used this week in this posts.  Both activities are designed for teachers and students to get to know each other better.  Both were found on Sarah’s fantastic blog post.

How well do you know _____?

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Hints aren’t given and students are asked to fill in every blank – even if they don’t have an answer.  I give the students around 5-10 minutes to come up with a solution.  Sometimes students will work in groups. The class then reviews the answers as I display the sheet under the document camera.  The students’ answers are telling and a few downright made me laugh. Those shared experience can be brought out as the school year progresses.  The template can be found here.

Getting to know you quiz

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After I collect the responses from the first activity I move on to this one.  I tell the students that it’s now their turn.  They’re asked to create a quiz that I’ll fill out, except this one is multiple choice.  I give them a few sample question ideas and they’re off to the races.  Some students even use the “none of the above” or “all of the above” for a letter.  Students have around 15 minutes to complete this and it gets turned in.  Here’s an example from two years ago.   I answer the questions that evening and return the sheets the next day.  Students get a “teacher marker” and grade my work.  From a teacher’s perspective, I’m hoping I get at least one right on each page.  Sometimes I get more and sometimes less.  I’ve had my share of humble pie in these instances, but it’s worth it.  I might even ask a sibling to even the odds : ) .  Anyway, students get a kick out of “grading” the work and then the class discusses how we’re all learning and getting to know each other.  This will help our community grow and how we react to perceived mistakes is important.   Feel free to use this doc to make it your own.

I’m hoping that these activities are a good start to helping build a worthwhile community of learners.  Being able to reference the student responses throughout the year is another shared experience that benefits all involved.  I’m looking forward to next week as we dive into more content, but I’m also keeping in mind that community is key.


Supply Lists

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Like many teachers, I’m reporting back to school soon.  My classroom has changed and I’ve been told that all of my furniture and supplies have been moved to a new location.  At some point next week I’ll enter the school, grab a new set of keys and find an unassembled room full of boxes.  It’s overwhelming at first.  I’ll probably spend the first few hours just unpacking and setting up shop.

I’ll be bringing my own supplies like I usually do at this time of the school year.  I try to be somewhat minimal with what I purchase each year as I tend to move classrooms almost every year.  Also, my district provides a sum of money for classroom purchases and this is where I spend the bulk of that funding. For the past few years or so I tend to get a staple items that are standard in my room.  This post will highlight what I tend to purchase before school and the reasoning behind it.  It’s also a good reminder for me to look back at and reference (think next August). The supplies work for me and I’m not affiliated in any way with the companies involved with the products.  This post actually stemmed from the whole #clearthelist initiative that seems to making the rounds on Twitter.  As I look over those lists I wonder what’s truly needed and what’s considered a priority purchase.

To give context context, I teach multiples sections of math to different grade levels.  I have around 100 kids pass through my door each day along with four different grade levels.

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Student math journals

I tend to purchase one of these notebooks for each student.  I try to color code them so that a certain grade level is a specific color.  That doesn’t always work as it depends on the supply they have at Big Lots or Walmart.  These notebooks are primarily used for student math journals.  Students write about their math experience, take notes and reflect on assessments within the journal.  Since I loop with the students they keep this journal for three years.  At the end of fifth grade they take the journal home.

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Student folders

Each student also gets a folder.  I usually get the folders and notebooks at the same place around the end of July.  The folders are color coded based on the grade level.  I usually purchase the cheap ones that don’t include special hole punches or brads.  Students grab these folders when they enter the classroom and their morning routine work is located inside.  Students take and return these folders from the same place when they arrive and leave the classroom.  The folders barely last one whole year.

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Popsicle sticks for groups

These popsicle sticks/tongue depressors are winners in my book.  I get a large back from Hobby Lobby, but I’m sure you can also find them at other places. I organize them by color and assign each group a grade level.  I use white address labels to print out the names on my roster and attach them to the sticks, folders and math journals.  These sticks are used to group students or randomly pick a student for a response.

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Like many teacher, I tend to grade using some type of Flair pen (long live the Flair!).  The problem is that it’s not erasable and I haven’t found one that works for me.  When I make a typo or error with the Flair I dig out this.  They cover the mistake and I’m able to write over it without issue.  I gave up the liquid white-out a long time ago as I can’t wait for it to dry – my issue.

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Notes and sometimes grading

I’m a fan of the Sharpie pen.  I use these for grading or for taking notes.  Love that it doesn’t bleed through my notebook.  Was able to get this on Amazon for a decent deal about a week ago.  A 12 pack will last me all school year and then some.  I ration these and keep them under lock and key.

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Did you forget a pencil?

Last year I ran out of pencils around March.  This year I’m trying to be a bit proactive and ordered 150 pre-sharped pencils from Amazon.  I almost went with golf pencils, but decided to go with a regular size. I have a pencil jar/bin that’s for anyone that needs one.  I try to keep it stocked up as much as possible, but usually after a few months I see pencils that are on life support as you can’t even jam them into a pencil sharpener.  I’m going to put our a few each week and then resupply them the following week.  Like that they’re pre-sharped as students don’t have to line-up to sharpen them and lose out on time in class.

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Not really a staple, but a good find nonetheless

I found this stool at Harbor Freight about a week ago.  This will be perfect for my small group table.  It has wheels which is a major bonus in my view.  Last year I remember kneeling down or bending over to work with my early elementary students.  This might be another flexible way to get to my students during small group time.

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STEM project

My third grade students have been creating paper roller coasters for the last five years.  I keep on going back and getting another copy because of how much teamwork, math and creativity it showcases.  The booklet comes with different colored yardstick.  Students basically follow the instructions and create different parts of the roller coaster track.  Students will need to fold, score, cut and tape the tracks together to create the final project.  It takes my students almost the entire year to create the entire coaster.  Students work on this after they finish independent work or during small snippets of scheduled time throughout the year.

Every classroom is different and each teacher has their own supplies that they can’t live without.  This post highlight just a few staples that I tend to get before the year starts.


White Fragility and Perspectives

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This summer I’ve had the chance to read a few books on my list. Now with just a few weeks left before school starts I’m left with an opportunity to reflect on what I’ve read and am looking forward to applying what I gleaned from the pages.  Two books in particular have peeked my interest related to white fragility and awareness:  White Fragility by Robin DiAngelo and This is Not a Test by José Luis Vilson.  My local library was able to find a copy of each through their interlibrary loan system.

To start I’ll talk about White Fragility.  This book was eye-opening and powerful (also has an amazing discussion guide for educators).  Kudos to Sonja for giving the recommendation. To be honest, I’ve had my own fair share of “equity” type of pd trainings and whenever the discussion of race is brought up the room gets very quiet. I hear statements like “I treat all students the same” or “I don’t see color.”  The author discusses these specific phrases among others and why people get defensive. White privilege exists. White people might squirm in their seats and censor their language carefully during discussions that creep towards equity and racism.  The book discusses why white people feel uncomfortable while discussing racism.  The historical perspective detained was thought-provoking and brought me an awareness piece that I haven’t encountered before.  One particular quote that stood out is below.

“I repeat:  stopping our racist patterns must be more important than convincing others that we don’t have them.” White Fragility p. 129

Many equity trainings often start with “getting to know you” activities.  These are generally designed so that participants feel comfortable discussing racism and their perspective.  An enormous time is set aside for this.  The author suggests that this isn’t because participants don’t feel physically safe, but it’s more because they want to convince others in the room that they certainly aren’t racist.  Being willing to discuss the important issue of systematic racism means being able to accept our own white fragility and the role that plays in often hindering these important conversations from moving forward.  Also need to keep in mind white consensus and how quickly people are at defending each other when being asked to reflect on actions.

The second book that I read was called This is Not a Test by José Luis Vilson.  I enjoyed reading about his experience growing up in NYC and how he went from being a computer science major to moving into teaching.  I enjoyed how José discusses his life and how he landed on being a teacher.  The end of the second part and the third resonated.  I appreciate how the author illustrates how teachers can have a voice in the current system that often perpetuates the status quo. José also addresses power structures and the different dynamics involved with standardized testing and teacher shaming.  I thought this quote was interesting:

“Educators have to get involved in planning curriculum and pedagogy.  We also have to believe in ourselves as powerful change agents or else we perpetuate the same power structures we say we’re against.” This is Not a Test p. 108

This hit home.  Teachers need to get involved in the curriculum and instruction development and rollout.  It’s challenging for me and it’s most likely due to my own white fragility, but educators need to be vocal and address systematic policies or pedagogy that might not be as inclusive as needed.  Same goes with resources.  Say no to historical simulations involving slavery, the holocaust and a myriad of other painful scenarios.  Be culturally aware that situations involving escaping from ______ might not be a good options.  Having the students recreate objects of hate such at the confederate flag shouldn’t exist.  Speak up.  If we’re silent than that inaction continues to kick the can down the road.  This should also apply to specialized advanced programs in schools.  Take a peek at the demographics and look at how they compare to the overall population. One concept that the book emphasizes is that intent isn’t enough. – the idea of intent vs. impact.  People need to take the initiative while also understanding that teachers are in a position that isn’t always recognized as a place where decisions are made.  Teachers have a voice and using it is important.

Both of these books have given me an opportunity to think about my perspective and privilege.  I’m kicking myself because I should’ve read these earlier.  Realizing how you’re raised, perception of what’s defined as racism, the history behind white supremacy, systematic pieces related to education, availability to access, status, and why it’s so hard to discuss racism all play a role in how you can make an impact moving forward. Both books have reinforced the idea that intent isn’t enough. I’m becoming more aware of my thoughts on white privilege and am still learning.  As I look at my school calendar I’m looking forward to seeing and possibly sharing how these two books have impacted my perspective with others.



Building Awareness and Knowledge

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During the past couple weeks I’ve created a brief routine of reading in the mornings. Sitting out on the deck, reading a book and slowly drinking my coffee has been time well spent.  I’m taking advantage now since this won’t be happening once school starts in about a month. One of the books is related to culturally responsive teaching and the other I’m just starting to dig into is about white fragility.  Both books are somewhat similar and I’ve been able to spend a decent amount of time reflecting on systematic education practices.

One particular takeaway in chapter four of CRT was related to building awareness and knowledge before making judgements. The author makes the case that teachers should widen their interpretation aperture when interacting with students. Aperture refers to the amount of light that is allowed in and out while taking pictures. Aperture is used instead of lens in this context. Widening that interpretation aperture takes time and a process is involved. The paraphrased process below is from the Mindful Reflection Protocol by Dray and Wisneski.

1.) The author discusses replaying student and teacher interactions in your mind.  That replay involves describing what’s seen. It can be challenging to replay conflicting interactions, but keep in mind that the replay is neutral – it’s stating the facts.

2.) Make assumptions and attempt to interpret the behaviors.  Teachers have to make so many decisions everyday and generally the interpretation falls into two categories:  intentional or non-intentional – positive or negative.  Behavior interpretation has the potential to be a sticky situation as it depends on the aperture of the teacher.

3.) The author suggests to offer alternative explanations.  What would a child behave in ____ way.  Do cultural norms or beliefs play a role in why the behavior happened?  How are directives given at home?  After reviewing the assumptions a couple times it’s time to check the explanation.  Explain the observation with other teachers.

4.) Hear from their perspective and check your explanation.  It might be helpful to go outside of your team to discuss this to receive alternative perspectives.  Here’s where it takes an extra effort to research and build more of an awareness of cross-cultural knowledge.  Trainings and PD can play a role with this.

5.) Make a plan of how to address similar behaviors and continue to review when you might be overgeneralizing situations.

Next week I’ll be reading about how to recognize common triggers and look at building learning partnership.  I’m also going to to be diving into chapter two of White Fragility.



Reflections on 2018-19

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It has been about a month since school let out and I’ve been enjoying the summer so far.  I’ve been reading, working on the lawn, painting and took a vacation.  During the last month I decided to focus my time on things not related to school work.  This balance of time tends to give me a better perspective when I do come back to working on items related to school.  Now I’m starting to see school supplies (already!) in stores and am looking back at how the school year went last year.  Every year I attempt to gather information about my students and how they perceived the school year as a whole.  I give a survey and use that information moving forward for the next year.  I decided to wait a bit over the summer to look over the results.

Back in June I gave a survey to all of my students in 3-5th grade.  The survey was related to instruction models and preferences.  This year I intentionally varied my models throughout the year and didn’t stick with one particular tool for activities.  I started off the survey with a brief question about their favorite math topic this year.

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Before giving the survey I went into detail about each topic.  The purple is measurement and I’m not sure why it didn’t show up with my advanced table Gform add-on. The next question was related to why they felt this was their favorite topic. Here are a few responses:

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Next time I’m going to put a minimum character limit to extract more information.

The next section, which was the largest, was related to instruction models/activities. Students rated them (1-5) 1 being the least effective for learning and 5 being the most.  A brief explanation of the items is in each caption.

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Teacher gives prompt and students discuss as a group and share out responses to the class

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Used for retrieval practice – students used this individually and in groups of two

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Similar to Kahoot, but more self-paced – used to reinforce concepts

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Similar to Number Talks – prompt is displayed and class discusses concept

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Used  for notation on slides and showing student’s work to the entire class

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Used K-5 math for whole class or independent exploration of math concepts

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Students are digitally randomized into groups and then they work on a particular task with that partner.  Students report out their responses afterwards.

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I display work (sometimes student work) on the document camera and the class discusses strategies

I had 59 students take the survey, but I have 65 + students in 3-5th grade.  Some of them were out for other activities during the time the survey took place.  Something to consider … some of these activities were used more frequently so students had a larger sample size.  Overall though, it seems students enjoyed most of the tools/activities for learning about mathematics.  I think it should be mentioned that there’s a difference between a tool and strategy and I might be blending the lines a bit in this post.


The tools in the box were used independently or with a partner.  They also required some type of technology (iPad, Chromebook), while the other four didn’t.  I think having a blend between the tools/strategies is helpful and students aren’t dependent on using one medium to show their learning.  I’m looking forward to diving more into this data as the summer progresses.


Scale Factor – Part Two

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During the last week of school one of my classes explored dilatations.  It was a rather short lesson since there were only a couple days of school left.  After some review, I pulled out a project from last year and thought might be applicable since it addressed the same standard for that particular day. I looked it over and made a few changes so this year it would run smoother.  Here’s what changed:

  • I had the students create an exact 4cm by 6cm grid using rulers.  This was different than my initial project.  I made sure to check each grid before students moved on to the next step.  I’m not a fan of having a simple mistake or unclear directions derail an entire project (which it did for some last year) – so I decided to check each students initial grid.

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  • I also created a random piece to the amount of dilation this time around.  This picture is from last year’s post.update.pngLast year students already knew the grid to use and basically used a “paint by number” approach to fill in each square.  Although that was fun, it didn’t really hit the objectives as much as I’d like.  I had students roll a die to determine the dilation this time.  This gave four different options for students.

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  • I put together a criteria for success component where students could check-off items when completed.  I set up the different dimension papers on one of the tables so students could easily grab them depending on their dilation.  I also added a short debrief piece near the end of the project where students discussed how they increased the size of the image.

These changes helped improve this particular project and I believe it created a better learning experience for the students.   There are times where I completely scrap a project and other times I make tweaks in order to make it better.  I opted for the second option this time around.

* Next year I’m planning on updating the project to include dilations that involve reducing the size of an image.