A Day in the Life

5:30 am – Alarm…Snooze

5:53 – Peel myself out of the bed. Find something to wear. Go wake my boys. Dress Sam while he’s still sleeping and Ben’s talking to me from his crib. Dress Ben. Make lunch, pour cups of milk, make breakfast, make coffee, rush children out door…

6:30 – Take two trips to the car, one with bags, one with children.

7:15 – Drop both kids at daycare

7:30 – Get to work. Climb three flights to classroom.

7:40 – Finally get to classroom. Hmmm…maybe I should make some more copies, I never have enough copies. Go wait in line at the copier. Uh-oh, copier jammed, doesn’t look good. Oh well, hope I have enough for the morning.

7:55 – Change the date and day on the whiteboard, collect donations for Thanksgiving baskets.

8:10 – Listen to morning announcements, try to take attendance in Powerschool.  Won’t load, ugh, call the office with attendance. First class comes in…

Co-taught class – put on FM transmitter, make sure it’s on, and I begin. Collect homework, discuss multiplication of fractions, how do model one-half of one-third with the area model. Now with a length model…

Edcanvas of my lesson

8:55 – Bell rings, next class, “Uh, I’ve been absent for nine days, did I miss anything?” Multiplication of fractions.

9:46 – Bell rings, Prep. Tech guy comes in. Thank goodness, my LCD projector was knocked onto the floor Friday when I was out. It is working, but is it really okay. Seems to be.

Oh yeah, and my computer stopped printing fractions. I have to send all my files home in order to print them. That’s fun!

He finds the fix, no pictures on my problems…well that’s boring. When are we getting new computers, Christmas, maybe… Can he fix my clock? It was 40 minutes slow, but since daylight savings it’s 20 minutes fast. Nope.

10:37 – Next class…Multiply fractions

11:28 – Lunch, then directed study.

12:43 – Next class, multiply fractions

1:34 – Next class, multiply fractions

2:25 – End of the day. Students want to know if they are missing any work. Um, it’s the last day of the term.

Try to make more copies, do a little grading, start entering report card comments.

3:00 – Out the door to daycare. Pick kids up, both napped in good moods. Begin the drive home.

4:15 – Get home. Take two trips in, one with the kids, one with the stuff. Turn on the oven. Wash a chicken. Put chicken in the oven. Go play downstairs with the kids.

4:45 – What’s that noise? Oh yeah, the smoke alarm…

5:45 – Dinner’s ready, hubby walks in the door. Perfect timing.

7:00 – Ben to bed. Begin reading Twitter. Need to get caught up, need to get caught up.

8:00 – Sam to bed. Start writing….

9:00 – Go to bed.

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Excitement

We had a full day PD on Tuesday. They close down the schools to make it easier for everyone to vote. Some teachers dread these days… “What are they going to make us sit through now?” Not me…I secretly like them. I would never tell anyone this. I like listening to my colleagues teach me something they know and I don’t and I like listening to my other colleagues tell me about something they learned.

I have a renewed sense of excitement after PD. I’m excited to put into action whatever it is that I learned. I want to run back to my classroom and start teaching. This one was different. Ben Schersten taught us how to use Twitter for PD. I didn’t have to go back to my classroom… I could start right there. Twitter is an amazing tool for PD if only you know how to harness it. When Twitter was first introduced I couldn’t figure it out. What is this for? In the past three months I have become addicted I can’t stop reading. Thank you Ben for filling in the blanks. There is a lot to learn and there is a ton of information out there. Ben gave us people to follow, hashtags to search for and apps to use.

I have discovered another kind of PD as well. The kind I can sit on my couch in my PJs and enjoy. Thank you Global Math Dept I can learn something new every Tuesday night.

Blogging and twitter has changed the way I learn and teach. Thank you.

Ordering Fractions

I know. How dull! If I find it boring students must find it even more boring. As I mentioned before I’m kind of a number theory nut. After learning about the Farey sequence during my summer in PROMYS I knew it would be a great addition to my fraction bag of tricks. I tried it and students did it, but they weren’t getting where I wanted them. I don’t think I was asking the right questions.

I recently found this activity and tried the Farey sequence again. Students loved it. They wanted to try to get the next sequence. I shared it with a colleague and she couldn’t believe one of her students asked if it would be all right to do F9.

Here is F4:

Fourth Farey Sequence

from nrich.maths.org

I love lessons like this one. There is so much mathematics packed into it. We have mixed ability classes so students that are struggling with multiplication are sitting next to students that can solve equations in there heads. This problem allows students to move through at their own pace. I want students to practice comparing fractions. The fact that they are discovering patterns and symmetry along the way is a bonus.

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LCM, GCF, and Fractions.

We had a great time last week working with cogs and I think the mathematical thinking my students did is the kind I want them doing all year. Unfortunately, this lesson is not one of them.

My colleagues and I discussed whether we should even teach these topics. They are definitely not sixth grade topics, but we need somewhere to start and once we start simplifying fractions, we need some common language. I start out by reminding students how they usually find the greatest common factor, by listing out all common factors. I’m always hoping that someone will have a better method, using prime factors, maybe? I then show students the ladder method we used for prime factorization and how it can help us here.

Once again we ask ourselves the same question, “What is the smallest prime that goes into…?”

Using the same method we used for prime factorization we divide out prime common factors. What we are left with is the prime factorization down the left side of the chart, the LCM can be found by multiplying all the outside factors together and there are several other patterns in the chart as well. The bottom factor on the right multiplied by the number on the top left also equal the LCM. Here is the same page with a few completed examples.

There are obvious implications here for finding common denominators. Students can quickly find common denominators and the chart will give the factors that the denominators need to be multiplied by to get to that common denominator. There are a lot of patterns for students to see here and I think it gives a real understanding of how prime numbers can help us work with much larger numbers.

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Cogs

I was really looking forward to this problem. I couldn’t wait to see what my classes would do with it and it did not disappoint. I mixed up my classes (assigned seats) and gave them very little direction. I wanted to see what kind of mathematical thinking they would do on their own.

nrich.maths.org

The Counting Cogs is very specific and Nrich has even included step-by-step group directions so students can easily manage the problem in a group. Students need to discover which pairs of cogs will allow a colored tooth on one cog to go into every gap on the other cog. Students cut out the cogs, colored one tooth, and started spinning cogs. Nrich even has an interactive that helped me give some classes an idea of what they should be doing. Thank you Nrich for a great problem.

I was thrilled with the thinking. Some students quickly realized they needed some way to keep track of pairs and created great tables and charts. Some students wanted to immediately begin making conjectures. We talked about proving what we believe to be true, and they began to see that one example is not enough to be a proof. Some students even started creating other cogs than the ones supplied to see if their conjecture really worked. I heard a lot of conjectures! Tomorrow when we begin discussing this problem students will quickly discover the cogs that are relatively prime to each other work and the others do not. I did hear this conjecture today, but students just didn’t have enough time to really prove it.

This problem let students extend their thinking about factors, primes, and relatively prime numbers. They were actually applying these concepts rather than completing another worksheet. I really like the way multiple concepts are intertwined in this problem.

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Primes

I spent a few summers working in the PROMYS program for teachers at BU a few years ago. You spend the six weeks working with other teachers on number theory topics. Somehow the beauty of number theory had escaped me until this time. Ever since that summer I have really enjoyed teaching prime numbers.

Here are my thoughts on the next week. My sixth graders have seen prime numbers before. I always start teaching it as if they’ve never seen them before. I’m going to have them start by completing a frayer model with four vocabulary words, prime, multiple, factor, and composite. I’m taking the foldable frayer model from here, but we’re not going to fold it.  My thought is once their memories are jogged most students already know these concepts. Why am I reteaching them?

Next up, a little DIVISIBLE Lab. Students once again have seen the divisibility rules in the fifth grade. It drives the faster students crazy when I make them sit and outline all the rules again for the students that have forgotten them. This lab allows students to get up and move around the room. They cannot leave a station until they have the teacher sign off that they are correct. Each letter in DIVISIBLE is a station and students have a lab sheet with questions they complete at each station. I will give students a sheet with all rules written out to keep in their binders after the activity.

Now, the Sieve of Eratosthenes. I used to think it was a waste of time, but I think it is a good activity for students to really see the power of primes. Students go through and cross out all the multiples of two, all the multiples of three, etc. In the end they are left with only the primes. Students then see that there really is no pattern to the primes, that not all odd numbers are primes, all the common misconceptions.

After this, we’ll start prime factorization. I used to teach factor trees, but I had students missing factors when they went to write out the factorization. Last year I tried the box method. Students only have to ask themselves one question, What is the smallest prime that goes in to _________? It looks something like this:

2

24

2

12

2

6

3

3

1

The prime factorization is easy for kids to pull out down the left side. They know when they are done because they get one in the bottom. This makes everyone’s life easier.

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More PEMDAS

I love being a connected educator. I have learned so much from other teachers in other parts of the world. I just discovered Angie at Coefficients of Determination. She was having the same struggles with the order of operations.

She has created a perfect foldable that I completed with my class this week. They loved it, and I think it finally solidified the order of operations. They are really starting to understand multiplication and division go together and that addition and subtraction go together. Here it is.

Then she played the game risk with them. I had a hard time imagining how this game would work, but I gave it a try and the kids loved it. Check it out.

I’m going to give a quick quiz on the order of operations and then move on. Even my struggling students have an excellent understanding much better than in past years. Thank you everyone!

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PEMDAS

I have resisted this acronym in the past. I just don’t think it is as useful as everyone else. I am constantly frustrated by students multiplying before dividing regardless of the problem. I gave in this year. I went for it. I circled the M and D. I put arrows between the M and D as well as the A and S.

I hoped… They have all done this before I told myself.

It didn’t matter. The same mistakes happened, they are still happening. Each time I find a student doing all multiplication before the division they look at me like I’m the one that’s confused when I try to correct them. When I show them the circled letters with the arrows between them they say, “Oh.”

 

Regardless, it was very clear today that we need some more practice with the order of operations. We have listened to the PEMDAS song, done a PEMDAS relay (scroll down), and today we did a Treasure Hunt (found this great idea here at mr barton maths).

To do the treasure hunt I cut out and laminated (not necessary, but now I have them for next year) each card. Students can start with any card do the problem, then they have to find their answer on another card and complete the problem on that card. This continues until they get back to where they started.

The treasure hunt definitely got everyone up out of their seats doing math. Everyone was focused doing twenty-four math problems on a Friday! Students discovered mistakes right away because they couldn’t find their answer on another card. This is an activity I’ll modify for other concepts throughout the year. I really liked the conversations that students were having. No one was giving answers, but helpful hints were happening all over the room.

Here’s the Treasure Hunt Template if anyone wants to change it for their own class.

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Long Division

Long division is always a struggle at the beginning of the year. Many students do not know the process and even more students do not understand what they are doing. I’m afraid that long division will join algebra soon and mathematics education will only require students to learn how to multiply.

Long division is a multi-step process that many students struggle with to the point of making up rote memorization mnemonics. Students that have been shown how to repeatedly subtract multiples of the divisor don’t understand how this connects to the more efficient method.

My sixth grade classes today looked at me stunned when I asked them to divide 32 into a ten digit dividend. Does it matter? Have they never been asked to do something that is not printed on a worksheet? Something that might be challenging?

Several students asked if they could just do short division. This is something that really bugs me! Am I the only one?

I think there is so much math in long division that short division loses. Remainders are so important. Which remainders are repeating? Can we have a remainder larger than our divisor? What are the possible remainders when we are dividing by 32? These are questions that sixth graders don’t know the answers to. These are questions that are important to their understanding of division.

When I divided 132 by 11 today students couldn’t tell me what it meant to say 11 goes into 13 once. Why didn’t I put the 1 over the two? Why don’t I just subtract the 11 from the 32? These students have all seen their fourth grade teacher repeatedly subtracting multiples of 11’s. What is missing? Is it repetition? How do we help students “really” understand what they are doing?

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Partial Products

Every year I spend too long reviewing basic algorithms. This year I’m going to jump in feet first and begin by doing partial products to develop both the multiplication algorithm, and then the division algorithm. I know the majority of my students know the multiplication algorithm but only a handful of them can explain why it works.

Last year I spent some time doing partial products with base ten blocks and students really seemed to expand their knowledge of these basic algorithms. I found a great activity to expand on what I did here.

Once students understand this activity they can then begin to draw generalized rectangle models to lead into the distributive property.

 

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