Great Math Products!

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Multiplication Tricks

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Doubles

ThreeFingers with Numbers

Telling Time Misconceptions

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Equivalent Fractions

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Simplifying Fractions

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Clock Fractions

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Math Fact Motivation

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Bulletin Board Ideas

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Classroom Management

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Lines and Angles

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I get the cutest handwriting fonts at Fonts for Peas! kevinandamanda.com/fonts

Lesson Ideas

Need a Beginning of the Year Lesson for Symmetry?

This lesson was almost an accident, but it turned out so well!  Not knowing I was going to have to teach a particular class, I desperately went in search of the librarian’s expertise for a beginning of the year book. I couldn’t find the book I was looking for so she gladly showed me this book.


I read the students the story, calling particular attention to the stamp she receives from her grandmother, and then I had students make stamps.  In order to do this I used some bottle caps we had been collecting. Sadly, I spent too long at home hot gluing foam squares to bottle cap lids.

 

I was kind of worried that if the squares weren’t perfectly rectangular or square it would not look good when students stamped their letters, but actually kids wasn’t able to tell after the stamp was pressed down.  It really just mattered where the pressure was when the stamp was pressed down.

When I brought these to school, I had kids find the Korean letter that corresponded to one of their initials.  I used this page I found after a Google search.  Then I had students write their Korean initial once, draw a line of symmetry and flip the Korean letter over before they carved it into their stamp.  Because not every letter of the Korean alphabet corresponds to an English letter, I had students find the Korean letter that most closely corresponded.  I also gave them the option of picking a letter from their middle or last name if their first initial didn’t correspond to a Korean letter.  Students carved with their pencil into the foam.

Here are some of the results.  I let the kids take their stamps home.  NOTE:  If you don’t want stamp ink everywhere invest in some baggies for them to put their stamps into.

I did this with both first and second graders.  I was a little nervous about doing this with first graders, but they handled it like champs (one of my stamp pads was worse for the wear due to a first grader I might add).  I would do this lesson again, and it was such a rich lesson.  There were connections in the book to another country, the Korean alphabet offered a connection to another language, and the symmetry added a math connection.  The whole lesson took three class periods of 30 minutes each. Happy stamping! 🙂

Try this Fun Activity for Your Promoted Fifth Graders!



We always have a big ceremony for our 5th graders who are being promoted to 6th grade every year.  In fact it rivals many high school graduations in its attention to detail and classiness.  This year, I wanted to add little something to it, so at the prompting of my students, we made these cute profile images of their faces.  The students put quotes on them that meant something to them.  In some cases students made up their own quotes, and I let them.

I had students make up the rubric for what a good profile image would look like. Most of them agreed that a quality profile had to have no white spaces, at least two colors, and at least one quote.  They turned out SO well, and I’m so proud of them! 🙂  You will notice that some profiles look similar to others.  That is because when one student had a good idea, other students tended to copy the good ideas.

First you must know that I work next door to the art teacher (HOW convenient!).  The art teacher let me borrow these spotlights that make the perfect shadow on a large sheet of white construction paper.  I outlined the students’s shadow and had them trace their shadow in black Sharpie.  If you don’t do this first, it turns out disastrous because then students end up coloring over the pencil lines and then when they cut out their profile, their lips and nose look somewhat deformed. I had them make their colored design first, and then AT THE END they can cut out their profile.  I allowed students to be able to use oil pastels, water color, black Sharpie, and crayons to make their images.  I emphasized the fact that they must NOT use Sharpie over the top of oil pastels or crayons because the wax will ruin the Sharpie.  I had them use the oil pastels, crayons, or Sharpie first and THEN they could paint over it with water color.  The wax in the crayons and oil pastels will cause the water color to resist causing a nice effect.

Parents and students took much pride in these as they lined the hall after their “graduation ceremony”.

Do You Have a Pi Day Plan? {Giveaway Time}

Do you have any fabulous plans for Pi Day you would like to share?  I am going to spend a little time doing two things.
1.  Discover pi. I am going to have a variety of circular objects such as Pringles containers, nut canisters, and other recyclable items for students to have access to.  Students will choose at least three objects.  Then they will measure the circumference with string and divide by the diameter of each circle.  I will provide string to measure the circumference and rulers to measure the string and diameter of each circle.  Then students will divide the circumference by the diameter.  I will do this to help them discover pi by themselves.  Students should roughly get three on each circular item.
2.  Next, I will print off enough digits of pie to at least fill a page.  I will have them go on a pi scavenger hunt to find as many meaningful numbers as the students can find that have to do with their life.  For example, they can possibly find their age, birthday, house number, zip code, phone number, etc.
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Help Your Strugglers with Fraction Number Sense!

Sad but true…Most kids start out struggling with fractions.  In real life, we don’t count by fractions and fractions are smaller than our normal counting numbers.  Sometimes students get the “top number” and “bottom number” confused.  The computer makes fractions with a slash, and teachers tell students to write fractions with a straight line and not a slanted line.  There is so much to stumble over as a student!

Could making fractions a part of your daily routine actually help students have a better conceptual understanding?!  But of course, darling (with godfather accent)!  I mean, after all, when teaching kids to count, we count over and over again EVERY DAY in kindergarten.  Students count by 2’s, by 5’s, by 10’s etc. and that is how we teach them to develop number sense.  We somehow lose this idea when it comes to fractions.  What if we actually gave the same tenacity to counting with fractions?

I am going to show you the tool to use to be able to support your students through scaffolded understanding of counting with fractions.  Behold!  Fill in Fraction Number Charts! 😉

 

Students have the opportunity to count by ½, ⅓ , ¼, 1/5, 1/6, ⅛, 1/10, 1/12, and 1/100. There are a variety of number charts included so that students can start out finding patterns when counting by a unit fraction.  Then there are three levels of charts when counting by each fraction.  Each chart level gets increasingly more difficult as it scaffolds learning.  This could also provide differentiated practice for your learners.  When students become comfortable counting by unit fractions, they can then try the three levels of simplified charts if the unit fractions can be simplified.  Then after daily practice, ta-daaaaaa, better fraction understanding!

Oh, my gosh!  What a great idea for morning work!  Great way to start the day!

I’ll be back in a few days to show you a special fraction freebie I have in store for you!

Try These Fun Christmas Stations!

I wanted my students to enjoy their last class with me before the Christmas break, so I planned several stations for them to work on.  The students really enjoyed the engagement of every station.

Station 1:  Students picked out a pine cone and were instructed to find seeds inside the pinecone.  I showed them a picture of what pine cone seeds looked like.  I gave them magnifying glasses so they could look closely and deeply into the pine cones.  I instructed them to draw a picture of their pine cone and to draw a picture of the seed.  In preparation for the next station, I instructed them to find two more pine cones that were really “pretty” and “tree shaped”.  I really wanted them to find two that were opened up really well so they could see the difference in station 2.

 

Station 2:  I had students place two nice flared out pinecones in water.  Students made predictions before they placed one pinecone in cold water and one pinecone in warm water.  I have no sink in my room, so I had a gallon jug set up for the cold water and a crock pot set up with warm water for them to dip from with a measuring cup.  Make sure to have paper towels on hand for any water that is spilled.  Now, students can watch the pinecones for a while, but it will take about 30-45 minutes before students will see any changes.  I make students look and notice what happened.  I don’t tell them.  Now the last group that does this station probably won’t notice what happens.  They won’t have enough time in an hour long class period to notice the pinecones change.

(So what does happen?  Pinecones close up when wet.  Why does this happen?  Pinecones don’t want to release their seeds when weather is wet because they want to allow their seeds to fly as far away as possible during drier weather so that they won’t compete with the parent tree.)

After students have noticed the pinecones closing up, make sure you leave time at the end of the stations to ask the students why they think the pine cones are closing up in the water.

In case you are wondering, yes the pinecones will open back up when dried out.

 

Station 3:  What happens when you place a peppermint under water?  After students have watched, then what happens when you place M & M’s under water?  This experiment wasn’t that impressive in my opinion, but the kids loved it!  The experiment works best in a white bowl.  The plates were way too messy when kids tried to throw out the water.  Water, of course, spilled everywhere.

Students predicted first what happened with the peppermint.  Most guessed it would turn white, but few guessed it would streak the way it did.  You must keep the dish still and not shake the desk to get the full effect of this.  

 

After students watch the peppermint for about five minutes, have them predict what would happen when they place out M & M’s around the bowl.  I bought the small ones so there would be more color.  If students place them around the edge of the bowl, it makes a dramatic rainbow like this.  Again keep the bowl as still as possible.

 

Station 4:  Light up a Christmas light with alligator wires and batteries.  This station by far had the biggest WOW factor for the kids.  I didn’t give kids many instructions with this station.  I wanted them to do experimentation to figure this out.  I made the kids predict which type of batteries would work to light up the battery.  They lit up a small LED bulb.  (this wasn’t actually a Christmas light bulb).  To get these bulbs  I bought a $1 LED flashlight at War-Mart and banged the flashlight with a hammer until I broke it enough to get the lightbulbs out.  It’s cheaper to buy bulbs that way :0.  To light up the bulb students need at least three batteries placed end to end with at least two alligator clip wires–one each touched to the positive and negative ends of the battery.  I used D batteries because I had a bunch in a science kit, but this will work with smaller batteries.  I hovered around this station because I knew students might struggle here.  I made it a point to talk about how scientists struggle until they figure things out.  We talk about how Edison didn’t give up the first time his improvements of the light bulb didn’t work and they shouldn’t either.  Now in reality, it would have been a lot cooler if I actually did have Christmas light bulbs, but I already had LED bulbs on hand so we used them.

I got alligator clips like these here.

ChristmasScience Here is an editable document I used when students visited science stations.  This is nothing fancy, but you are welcome to use it for your classroom and make it your own.

Cheap and Quick STEM Lesson Your Kids will Love #2

This STEM lesson was more fun than I had anticipated.  We picked up ice cubes with a piece of string.  There are few materials needed.  The materials are cheap and it doesn’t take long to carry out the lesson in the classroom.

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You will need:

  • an ice chest or freezer
  • ice cubes (the kind you freeze in ice trays) make about 5 per student to make sure you have enough
  • string (like the kind you fly kites with)
  • styrofoam bowls
  • table salt (one container should be enough for a class of about 20)
  • small cups or containers to distribute the salt
  • paper towels
  • water

First, we watched a video about how salting roads helps salt trucks melt ice and snow on roadways here:

Then I explained to the students that we were going to make a string stick to an ice cube and that they would be able to pick it up.  I had the students predict how long they thought that it would take to attach the string to the ice.  I realized mid lesson that the kids thought I wanted them to literally tie a string around the ice cube.  I had to clarify that we were not lassoing the ice cube, but that the salt would make it stick if they were patient (insert lesson about perseverance and patience here, wink wink).

These are the student directions in order.

  • In your bowl of water put 1 ice cube (I passed these out when I was ready for them to begin).
  • Put a pinch of salt on top of the ice cube
  • Lay the string on top of the ice cube
  • One team mate needs to watch time (I had this on smart board). The other team mate needs to watch the ice cube and pull on the string when the predicted time is up.
  • If your string doesn’t stick, make a new prediction and try again. Switch partners.

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I had a sheet in which the students predicted, and I also took some of their predictions and wrote them on the board.  Some students predicted up to 30 minutes.  Eventually, I reeled this in and said it wouldn’t take 30 minutes.  In honesty it takes somewhere between 30 seconds and a minute.  Also, I reminded students that they would only need a pinch or so of salt on the ice cube.  Some students think that the more salt they use the better the string will stick and end up using WAY TOO MUCH!  The important part of this experiment is the waiting.  I told students that if they added salt that they would need to add it a pinch at a time.

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For those students who are successful, I challenge them to make more than one ice cube stick and predict how long it will take for more than once ice cube.  I walk around with a ziplock full of ice cubes during this time.  When students get more than one ice cube to stick it becomes a contest about who can get the most ice cubes to stick at one time.  The most I had students able to stick was 5 ice cubes.  Not many kids were able to do this.

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After this experiment, it is fun to discuss more of the science behind why this activity works.  Because I did this during a 30 minute slot at a summer camp, we didn’t really have time to talk much.

I hope you can enjoy this experiment with your students, too!

 

The After-School Book Club that Worked For Me

I am teaming up with my dear blogger friend Pat McFadyen, a talented 5th grade teacher,  from Growing Grade by Grade to host a post she has written about book clubs.  We met at a recent TPT conference and have kept in touch ever since. I hope you enjoy!

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“Book clubs can be a valuable part of a school’s literacy program.  They can:

•Improve student reading fluency
•Present new vocabulary to students
•Introduce new ideas and concepts
•Teach students to consider others’ perspectives
•Teach students to present their own ideas
•Teach students to analyze books
•Encourage turn-taking

However, book clubs can only do all of the things if they work. Not all book clubs work. Here’s what  happened to mine…

About fifteen years ago, I was asked to coach an after-school book club for our 4th and 5th grade students. At the time, I had a self-contained 5th grade class, meaning I taught all subjects, so I felt confident enough with my ELA skills that I thought I could do it. I’d always wanted to have a book club, so I was really excited as I scheduled and organized.

Unsurprisingly, there was no money for books. I had to get really creative digging through old stashes of books in classrooms and the library to have enough copies for everyone to read the same title of a book they hadn’t already read. At this point, I don’t even remember the book we started with, but it doesn’t matter. The results would have been the same.

Within a couple of weeks, the group was experiencing problems. Faster readers were already finished with the first book and anxiously ready to discuss it and move on to the next one. Slower readers were struggling. Less serious club members were using the time as a social club. Over-scheduled kids missed meetings. I was moving away from “friendly book club coach and mentor” to “teacher who has to fuss to get kids to do their work”.  No one was really happy.

We stumbled along like this for a while. I was ready to throw in the towel and tell my administrator that the club wasn’t working and I wanted to close it out. I was dejectedly web surfing one night (and keep in mind this was long before really good browsers – or even Facebook! – so results could be sketchy), when I found a teacher chat group. One wonderful teacher mentioned that she had had good results with a “genre book club”. Instead of choosing one book for everyone to read, she chose a genre of literature. She collected as many examples of the genre as possible and let students pick from that collection. There wasn’t a lot more information, but it stopped me in my tracks.

The advantages of a genre book club flooded my thoughts. I felt that choosing a genre and offering students a selection of different titles within the genre would immediately solve some of my most pressing problems.

For our next meeting, I greeted students with a pile of books – literally! I chose the genre “Humor” to begin with and almost cleaned out our school library of joke books, comic-type books, and obviously funny, funny books. I explained to the book club members what I was doing and why. I very honestly shared why I was disappointed in how the club was going. I made it clear I was not disappointed in the students, but in how a one-book focus and a lack of resources seemed to be limiting our progress and enjoyment of reading.

I explained that we would start with Humor as our first genre study, but would gladly take suggestions for our next one. We would keep the time limit for each genre study open in case we wanted to extend one genre or cut short another one.

During that meeting, I briefly discussed the characteristics of Humor as a literary genre. I made it into a poster for later reference. Then, we dove into the stack of books. Hands reached over each other, searching for an interesting title. In just a few minutes, every student had a book. And…they read.

They read! They quietly, absorbedly read our books. For almost the first time, I saw students enjoying reading in our book club. To keep the momentum going, I kept the books checked out of the library under my name and had the club members check them out from me. They could come by my classroom at any time to return/check out these books as much as possible. When we met the next week, we shared what we enjoyed, what we didn’t enjoy, and how the books fit into the Humor genre.

The school year was almost over by this time and we didn’t have many more book club meetings. We were able to include the Poetry and Short Story genres. Most importantly, we ended the book club on a high note with at least some enthusiasm intact.

Lesson learned: If I ever sponsor another book club at the elementary level, I will seriously consider making it a Genre Book Club. I can’t guarantee this would be the best format for middle school or high school, but as an option for developing readers, the Genre Book Club is a seriously viable option.

In a nutshell, these are the advantages of a Genre Book Club. While offering the same benefits of a traditional book club, it can also:

•Eliminate or reduce the need to spend money on multiple copies of the same title
•Allow use of books from your school library
•Give students more choice in what they read
•Offer students a wide range of reading levels within the same genre
•Offer opportunities for students to read at their own speed
•Provide a feeling of ownership and responsibility over their own learning
•Discover personal preferences in books
•Solidify understanding of the characteristics of each genre

If I coached a book club again, I would also offer a way for students to respond privately about their reading and to encourage accountability. Follow this link to check out my Genre Book Report product.

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How To Start A Genre Book Club

If you are tasked with organizing a book club of any description, here is a list of steps to have it up and running successfully.

1.Always get approval from administration. It’s more efficient to have a plan to present before you approach them.
2.Decide what age/grade range you will include in the club.
3.Decide how many students you can comfortably accommodate.
4.Look at the length of time you’ll have. One semester? Two? A single 9-week grading period? How often will you meet? Weekly? Bi-weekly? Create a schedule – either a list-type or an actual calendar. I recommend a calendar because you can see where holidays and special school activities fall. It is helpful to parents when you send home a calendar with your club members.
5.Pick the days that your book club will meet. It’s certainly better to choose the same day each week or each meeting, but sometimes that’s just not possible. Plug these days into your schedule.
6.Decide how you will invite students to join the book club. Will it be a blanket invitation to a grade level? Will it be a personal invitation to selected students? Will there be an RSVP deadline? If parents must provide transportation, make sure it is clearly stated in the invitation. Consider how much advertising you will do to get the word out.
7.Decide how each meeting will progress. Will you serve refreshments? If so, decide how will you access them, via dedicated funds or parents volunteers.Where will you meet? Will you sit on the floor or at tables? Who will run the meeting? It is nice when students can begin to take on leadership roles of this type, but I recommend that the coach/mentor/sponsor begin leading so students can see how it is properly done.
8.Decide on a potential list of genres that you’d like to explore. Use the attached Basic Literary Genres list FREEBIE to get some ideas. Present your genre list to your club members at the first meeting. Consider letting students decide which genre to study next.
9.When you have your genre list, create a poster or other visual that lists the main characteristics of each one. These will be a handy reference tool. Consider using the 22 posters in my Genre Book Club product.
10.Go ahead and choose the first genre. “Humor” worked for me, but there are many others that would work.
11.Collect the books that you will offer to club members for your first genre. If your school library allows, use as many of their titles as possible. Include a wide range of reading levels in your selection of books.
12.Attend your first meeting. Discuss the nature of a Genre Book Club, your schedule,  and members’ responsibility. Start reading as soon as possible.
13.Enjoy your book club! Remember that you are helping students to enjoy reading and increase their literacy skills. What’s not to love?

Best of luck to any and all who sponsor/coach/mentor our readers! If you ever get the chance to run a book club, consider a Genre Book Club. It just might be right for your students.”

I hope you enjoyed that fabulous post about her book clubs!  Make sure to stop by her TPT store
to pick up a copy of her Genre Book Club product!

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How to Intervene with Children Who Don’t Know Their Addition Facts

I have worked with children from 2nd grade on up to help them learn their addition facts.  One common denominator exists among all of these students.  That is THEY DON”T SEE PATTERNS!  I remember having a difficult time learning my 9’s facts when I was growing up.  To help myself, I just took one off of the number I was adding to 9 in the ones place.  I noticed this pattern.  No one taught me this.  When I was growing up, learning facts was like, “Ok, Class, let’s learn all our 8s facts, let’s learn all our 6s facts and so on.”  This is not effective for students who don’t recognize patterns on their own. Now with the common core mathematical practices, we should be teaching children to explore patterns through thoughtful placement of number facts to help them recognize these patterns.  Giving students opportunities to see the patterns will result in more students who are fluent in their facts.  I have shown examples of this before such as in this post about using 10s to help with adding 9s.

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But now I have actually put all of my work with struggling learners into a packet which could be used whole group for grades 1 or 2.  At the 3-5 level this could be used for students in intervention or as part of the RTI process.  Here is a look at the packet  that I have put together to help students become fluent with all of their addition math facts.  It is on TPT !

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You can also try out a little sample of this product for FREE here.

Don’t Do This When Teaching Graphs

To enjoy the week before Christmas break, I decided to try out a lesson idea I had about pictographs. I built this lesson around misconceptions that students always have about keys with numbers larger than one. The day before I taught this lesson, I was asked to teach math at a moment’s notice in another third grade class. When this third grade class interpreted a pictograph with a key of four, I noticed that this was really way too easy for these smart students. I decided to change my planned lesson idea into something more challenging. This is how the lesson went…

First, I did a quick lesson for understanding. I surveyed the class about their favorite ice cream: vanilla, chocolate, or strawberry. I then made a frequency table with this information on the board. With some yellow circle die-cuts with one circle to represent two children’s votes, I asked the class how many votes needed to stand for each flavor. Students did this very well even if this required half of a circle for one person. Then I had each flavor group stand in a different area of the room and I passed out the number of circles the children told me to each group. When I passed out the circles to each group and instructed the students to write their name on a circle, some disequilibrium arose. Students said they didn’t get a circle. I told them to talk to their group and they finally understood that they would need two names on each circle. I couldn’t have been more proud of this part of the lesson and the mathematical practices that arose from their discussion. Then came the next part of the lesson…

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I previously gathered real life data about students’ favorite Christmas candy.

  • I chose four types of candy: fudge, candy canes, M&M’s, and chocolate Santa’s. I gave three third grade classes each a roll list with a blank beside it for the students to write their favorite candy on. I collected these vote (data) sheets right before the lesson (mistake #1 don’t do this). I made six copies of the data sheets for each of the groups.
  • In the lesson I presented the children with the data and told them to create a tally chart/frequency table with the data. Students waded through this for a long time without consistent results. I should have counted the data before I passed it out even though I had estimated the results to around a total of 66 students. I ended up having to total it up while the students were counting (mistake #2 don’t do this). Have your data counted before you pass it out, so you know when your students are way off. (But, hey, in my defense, I have been pulled in all directions lately, substitute, fill in secretary, test coordinator, you name it.)
  • I made one student in each group of four responsible to count each class’s data since there were three classes and one student to record the data on a table. This worked very well to keep every child involved in the group.
  • We stopped the lesson to ensure the children had the correct data before we proceeded with graphing. I wanted to be respectful of the teacher’s time while I was in her room, but I may have had them check to see if their data was correct in a different way, such as, compare your data with a different group and see if you agree before I just revealed the correct answers.

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Next, I told students their key had to be 5 or more (I really had 6 in mind), but the teacher was concerned that it wouldn’t come out evenly for the students. The data had to be slightly altered for the die-cut circles to come out with wholes and halves. She didn’t want the students to have to deal with smaller fractional parts. I was actually okay with the students having the data not come out evenly because I wanted this to be a challenge for them to grapple with. In the real world, data doesn’t come out all nice and clean. Third graders had already proven themselves with a key of four, and I felt like they needed a challenge. I wanted them to discuss how much of a circle to cut apart. Again, I wanted to be respectful of this teacher’s time, so I allowed this and the students’ graphs all turned out pretty much perfectly. No fun :/ There was no argument over how much of a circle to cut apart. (mistake #3, don’t make your data work out perfectly). Needless to say here, I had many reflections after this lesson. This was definitely not my best moment as a math coach (I am at a new school this year). BUUUUUT!!! I loved this lesson, it was so great, even though it wasn’t! I know exactly what to do next time to make it better, and it would be an exceptional lesson with lots of mathematical practices involved! Next steps for students after this lesson–ask students what would have happened if the key was 5 or 6 to at least get them to think about this.

Are You Taller Than a Third Grader?

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With the combination of special programs and snow days, our time to teach all of the standards before our 3rd graders’ PARCC test is running out.  With this in mind, I made a graph to help third grade out using the data from the whole 3rd grade with a fraction line plot.  This type of  graph and fractions are not as familiar to third graders because they haven’t been exposed to line plots in earlier grade levels.  I put the graph in a central location where other grade levels could see it.  That way other students could experience measurement and interpreting graphs as well.

 

I started out with an area by the water fountains for repeated exposure to student traffic.

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Next I put up a strip of this amazing ruler like tape that I got at Office Depot when they had all of their special masking tape at back-to-school time.  The tape counts every 12 inches.  So in the picture below, I marked off every twelve inches with little triangles that mentioned that each 12 inches was 1 foot.  Next, I marked off the fractions of an inch with stickers.  I marked off the halves, thirds, and fourths so that students could easily see the relationship between the graph and the tape measure.Tape

Then I had students come a few at a time and measure themselves to the closest fraction of a foot.  Students recorded their X’s on sticky notes.  The only reason I had them record their X’s on sticky notes is because this ensured having them all the same size.  Line plots can make data look skewed if students don’t draw their X’s the same size.  Plus on the PARCC assessment when students drag X’s on the line plot graph questions, students drag the X’s into little boxes which makes test question boxes resemble sticky notes.  Students got to initial their X.  Also, if students in the least bit chuckled about anyone else’s height because they were short, I immediately told them they wouldn’t even get to put an X on the graph.  After I graphed most of the students from two classes, I only had two students who didn’t get to put their X on the graph because of this reason.

Here is the whole picture of everything with a more than willing model :)…Slide5

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