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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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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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Numeracy.

When students enter middle school, we hope they have mastered numbers. By mastering numbers, I mean, having strong numeracy skills and knowing the general processes for the operations.

Unfortunately, most students in the sixth grade are still struggling with numeracy. This is going to be a strong focus for the beginning of my year. (Yes, I know it’s awful to drill multiplication facts, but it has to be done.) I didn’t focus on these skills as much as I should have last year, and when we reached fractions some students just couldn’t keep up. They didn’t have the automaticity that was necessary to work with fractions.

I recently came across a book by Pamela Weber Harris, Building Powerful Numeracy for Middle and High School Students. This seems to be the answer to my problem. In it she outlines basic strategies and models, beginning with addition that students should know, and the number strings to teach them.For example:

39 + 41

23 + 31

Well, you get the idea. The question becomes how to fit it into 45 minute periods? Hmmm…

Some ideas I’ve been thinking of for drilling multiplication (it doesn’t look as bad in italics) :

A Friendly Game of Nubble

Multiplication War

Missing Number Problems

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Changes.

This year my room will not look like this. It already looks very different (I don’t have a picture of that). Tables have replaced individual desks and there are working nooks, created for students to work together uninterrupted. Each group will have a large whiteboard for collaborating on problems, a la @bowmanimal, and smaller individual whiteboards for working independently.

Group work has always been the norm in my room. I also feel that students used to be better at it. Now students believe group work is dividing up a problem and copying one student’s work. For group work to be successful there needs to be discussion about the problem before any work is done. I want to spend a lot of time teaching how to do group work at the beginning of the year. There needs to be a lot of processing while students are working. I think Dan Meyer’s beginning of the year activity  stacking styrafoam cups is a great way to start teaching group work. The marshmallow challenge is another idea for teaching group work. While it isn’t directly related to the curriculum, it is a great way to see how students work together. 

Technology! This is the first year that we have laptop carts to share for each grade, as well as projectors, wireless, and document cameras in each room.

1) Class Dojo is at the top of my list. This was a source of conversation at BLC12. The way that you can engage students in a discussion about behavior and what they want the classroom norms to be is very exciting. And who wouldn’t love the little monsters?

2) Edmodo is also going to be implemented this year in my classroom. I was using another website, but I want students to have the opportunity to interact with the site, ask questions, and answer each others questions. I also hope to start posting pencasts to begin flipping my classroom. I say begin, because I am not ready to commit. We’ll see how it goes.

Wow! I didn’t know I had so much to say.

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New year, New Blog!

I am inspired. Every year I have great ideas. I start out the year and I have really good ideas. I begin to implement them and then… I don’t really know what happens. I lose steam, I run out of time, or I don’t think it’s working. This year will be different, I think.

I have really good ideas this year. I could list them, but you would think I’m being too ambitious. I had the opportunity to attend BLC12 in July. I went last year for one day, but this year attended all three days. I was surrounded by amazing educators, who genuinely care about learning rather than test scores. People who are inspiring. I was told to do just one thing. That was the message over and over. Well, there are too many things I want to do, so this will be my one thing.