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Build It = Deep Conceptual Understanding of Multiplication

Being a math interventionist is one of the hardest jobs I’ve ever had. It’s like a constant cycle of diagnose…teach…teach…teach…light bulb on…light bulb off…diagnose…teach…teach…teach…

I’ve been working with a group of fourth graders that were just struggling BIG TIME with multiplication. They weren’t able to see how a fact like 3×7 could help them solve 6×7 by just doubling. The numbers were too abstract, and they had nothing to connect it to.

Of course as I dug deeper I found that they simply had zero understanding of what 3×7 really means.  So as usual, I tried to find a way to connect it to real life, my favorite thing ever. I was trying to think of a way to help them remember the difference between rows and columns.  And then it occurred to me as I was waiting in line at the theater, no one wants to wait in line to sit in the first COLUMN for a popular movie. We all want the best seats in the house, so we wait to sit in the first ROW.

Enter the Movie Theater Multiplication Project.  I went home over the weekend, turned off all distractions and poured days and days into authoring this project. I needed to really think for myself what 3×7 represented.  I was not taught this way and I know that it doesn’t come easily to me.  I also had to think of a way that would be meaningful and that would STICK, since that seems to be the biggest challenge facing students in intervention. The last thing that I really needed to think about was the Concrete-Representational-Abstract instructional approach. Kids LOVE to build. The second they walk into my math lab, their hands are all over my cubes, blocks, counters, etc.  Starting with building means that they can usually connect a pictorial representation to it, and then connect that to numbers.

So I made a movie theater or two or three as I wrote the project. I made theaters that were 3 rows of 4 seats, making 12 seats total (3 x 4=12). I made tickets placing them in the correct row and seat number. And finally I made some mega theater designs so they could learn to use known facts to solve harder facts.  There is a fourth stage to the project also that involves some open ended challenges to calculate profits and revenue. Say hello to my little friends:

Multiplication-conceptual-understanding

Seemed like it might actually work!

So I brought the idea to school. I got out some tools and some little tiny bears and THAT got their attention.  Tiny bears! Seriously, that is all it took?!

The first three days of the project were brutal.  They were making columns instead of rows, they were making rows of 20 instead of 20 seats total.  With probing questions they started to see what was happening.  And THEN, light bulbs turned on…and for several days now the light bulbs have stayed on! Is it sticking?! I hope so, and we’ll find out when they get to the mega theaters and can break down more difficult multiplication. Wish me luck!

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If you want to try this out, in the preview you will find three parts of the four stages of the project that are free.  You could totally continue the project by giving them your own specifications!

 

 

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Oh My! The Progression of Multiplication

Well, I’ve watched this video three times now and I think I need to watch it at least five more times. I love, love, love how this presented to the audience.

 

My take aways for when I am teaching multiplication:

  1. I need to stop stealing the opportunity to let my students use concrete tools! They should be available every SINGLE DAY.
  2. Rushing to the traditional algorithm is a huge mistake. I am thinking we need to have some serious conversations about when to introduce this.
  3. I need to let the students explore. Let me say that one again, I need to let the students EXPLORE. So many times when they hit a struggling point I feel this need to jump in and tell…I need a muzzle for my mouth!

What did you take away from this?

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Math is Messy!

Today I filled in for a teacher who had to be out suddenly. It was so exciting! I’ve been coaching now for several months and I was anxious to slip back into teacher mode.

These were fourth graders who had been introduced to the distributive property the day before. The teacher was looking for something to both review and solidify their knowledge. I decided to try to connect their past experiences with multiplication to their new thinking from the day before. I wanted it to be a little messy!

The learning target:

I can solve multiplication sentences in more than one way.

I had the students rate themselves based on this success criteria:

0: “What is multiplication?”
1: “I’ve heard the word multiplication before but I do not have any strategies to solve it without help.”
2: “I can memorize facts, but I still don’t know what multiplication really looks like.”
3:  “I have more than one strategy to solve a multiplication sentence like 6 x 8: number lines, equal groups, repeated addition, etc.”
4:  “I can apply what I know for strategies to harder problems, like 16 x 8.”

Then, I put them in groups to solve the following two facts as many ways as they could on giant paper…6×8 and 17×5. The results were remarkable, AND messy!

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Math is messy!

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One student wanted a new paper, but I explained that mistakes feel kind of awesome. We should expect to make mistakes and try again, it is all about perseverance.

This messiness reflected the thinking of the students. I had to suppress the urge to organize their thinking into my own pretty little anchor chart. I had to remind myself that this wasn’t about an anchor chart, it was about letting the students know that math is messy and that they could learn from each other’s thinking.

After giving them about 10 minutes, it was time to share. We rated ourselves against the success criteria to see where each group was. Then, I asked them to think about which strategy was most efficient on their poster.  They laughed pretty hard about the inefficiency of drawing 85 circles/tallies.

Here were some of the favorite efficient strategies that students shared:

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A nice notation of the distributive property, where the student broke the 17 into 10 and 7.

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Classic repeated addition (bottom), and another way to note the distributive property on the top.

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This person used the distributive property, but broke the 17 into 8 and 9! Very cool. I also appreciated how they showed their thinking below the number sentence.

My personal favorite was this one:

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He broke the 10 into two fives! That is someone who really understands what he is doing.

It was wonderful to get them talking and sharing.  The person doing the talking is the person doing the learning!

 

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A Team Approach to Learning Math Facts

(I wanted to title this post Ugh! %#$*% Math Facts, but somehow that didn’t seem very appropriate!) Sometimes I feel as though I am banging my head against the wall in an effort to help students learn math facts. There are many ways to practice in our room with a weekly set of facts (differentiated to their needs): building them, arrays, number lines, fact families, strategy work, drill, flashcards…

But we’ve hit a point in the school year where the students are less than happy to practice.  Whenever I mention math fact practice, I hear groans and moans about it.  IT IS DRIVING ME CRAZY. I started to look at their behaviors, noticing that when it was time to practice math facts, their efforts were half hearted as if they were on auto-pilot. I was having to track down who was practicing and who wasn’t.

On April 4th, I noticed only 7 people out of 25 students got 100% (10 out of 10) on their weekly math fact quick check.  I was completely at a loss because this number had been falling every week. My students know that I hold very high expectations, which means that somehow we were failing each other.

So I really started pushing them to practice at school, even twice a day at times for the next week.  I rewarded and recognized students who had completed their wok, and took photos of really cool strategies to put up on the interactive white board.

One week later right before we were about to start our quick check, I wrote this on the board:

Tips for Learning Math Facts

I explained that only seven people had gotten 100% last week.  I told the class that today I would keep track of the number of students who received 100%, and the difference between the two would be the number of extra minutes of recess given next Monday.  Here is what happened:

Tips for Learning Math Facts

 

We had a quick class chat after it was over. I offered this deal to them each week, April 4th was the baseline, the number of minutes of recess was up to them. I asked them if they could work together as a team to meet their goals. Here were the agreements that they came up with together after I told them it was a standing offer:

  1. Practice every morning when you get to school.
  2. Add 5 minutes of math fact practice at home.
  3. Practice when we finish our math assignment during math class
  4. Quiz each other on our math facts throughout the day

We will see how they do this coming Friday.  I am really interested in rewarding their hard work, and if it is a few extra minutes of recess, so be it!

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Drop Everything and Try it! Multi-Digit Multiplication Interactive Tool

I was told about this *amazing* online tool today.  It can be used in SO many ways. Here is one:

There is a core groups of students who have been absolutely flying ahead of the others with multiplication. They are ready to go deeper and play around with double digit by double digit numbers. The last thing I want to do is try to teach them the traditional algorithm I was taught in school at this point in their learning. I’d like them to understand what is happening conceptually first.

So I brought up the dreambox learning teacher tool page today during our computer lab time. I pulled that group of 7 students over to take a look at the Fourth Grade Multiplication and Division section.  You are looking specifically for the tool titled Multiplication: Open Arrays – Students compose arrays, use partial products, and develop understanding of the distributive property to mentally multiply up to 3-digit by 3-digit numbers.

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THIS IS AN AWESOME TOOL. My group of third graders went crazy over it. After I showed them a few examples (like below). They were so excited to be multiplying such huge numbers. Not to mention the fact that it helps solidify the concept of arrays and area.

This is a screenshot of the game. It totally walks them through the entire process.  Since they just recently learned how to multiply by multiples of 10, they were eating it up.

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The best part? It’s FREE!

P.S. Just as an FYI, we had a lot of problems with googling it, we had to actually type in http://www.dreambox.com/teachertools.

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Pre-Algebraic Thinking for Elementary Students

We came back from winter break with a two day week last week. I really wanted to do a review of multiplication concepts, because we had been away from school for twelve days. So I had planned some games to practice their facts.

That’s when I saw this on Pinterest:

Math Activity

Click the picture to go straight to Amber’s post!

Amazing! I’ve seen the “all about me in numbers” things on Pinterest, but this one gets them thinking deeper (because they have to make an appropriate number sentence that matches the number!) I dropped my lame game from my plans and whipped up my own version in about 5 minutes on the easel.  Clearly, I am not even half the artist that she is! (Also, I have WAY bigger feet!)

Elementary Math Project Idea

All of the number sentences could have been ___ x 1, but we brought it up a notch!

I showed them how to make it, and we talked about challenging ourselves with the number sentences.  The students who were comfortable could take this pretty far! This project totally differentiated itself naturally, AND we have the added bonus of trying to figure each other’s posters out.  This type of thing really helps develop pre-algebraic thinking, which can be very difficult for third graders to understand.  The activity was age appropriate and helped them see how multiplication and division were related!

Here are some of the student samples:

Elementary Math Project Idea

He *really* likes pigs.

Elementary Math Project

She actually measured her hair!

Thanks to Amber Thomas over at Shut the Door and Teach. I love that the more we share, the more we learn in education!

Shut the Door and Teach!
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You Call That a Multiplication Table? THIS is a Multiplication Table

Multiplication Table to Scale

Have you even seen this yet?! Click on the image to go straight to the PDF of this amazing multiplication table by David Millar of thegriddle.net. It is free to educators to print and use in the classroom. Even better if you go to the educator section, there is a black and white version that is both with and without numbers.

My mind is churning with the possibilities for using this in the classroom! I think it is the best multiplication table I’ve certainly ever seen. Talk about helping students conceptually understand multiplication, connect it to arrays as well as the concept of area.

Outstanding!