Computation Challenge: 2 Digit Multiplication (The Results)

Although I usually only post on Wednesdays, I wanted to continue sharing the results of yesterday's post about our 100s day math game.  Overall, the two digit by two digit multiplication game I created for my fourth graders was a big hit!  I had a couple kids thank me for two reasons: they thanked me because it was fun, and they thanked me because they could tell it was helping them hone their computation skills.  One girl told me, "I have had to do 8x9 so many times that I've got it now; I'm definitely not going to forget it!" 

There were a few problems with this 100s day math activity.  I had one girl who put way too much pressure on herself, so in the final 10 minutes of the game when she got one wrong she went over to a corner in the room and broke down in tears.  This is why I don't do competition very often.  But I simply said to her firmly, but in a caring voice, "Don't you dare cry.  You scored WAY too many points for your team today to cry.  You hold your head up high and be proud of yourself for that.  Understand?"  She smiled and wiped her tears away.  And one of the boys who had called her a good teacher earlier, because he was one of the kids who couldn't solve the problems independently until she helped him today, spent a couple minutes consoling her until she got up to start another problem.

Another problem was a boy on the other team who decided to try to win the game single-handedly for his team.  He tried to take on as many problems as he could instead of helping his teammates to do the problems.  He also did something in the end that felt a little bit ruthless.  The game was down to a tie.  Two girls were working together on a problem and double and triple checking it, afraid that they would get it wrong as their whole team watched on bated breath.  He solved his last problem and held onto it.  When asked why he didn't bring it over, he said, "Because their problem is easier.  I already know the answer to it.  If they get it wrong, they have to give it to me, and I'll hand that one in and win the game." 

I wasn't sure if I should allow that or not.  Was it a good strategy, or ruthless?  I decided to say nothing and see what his peers thought.  If it upset the other kids, I'd have put a stop to it.  But they seemed to take it in stride, so I allowed it.  I'm still not sure if I sent the right message by doing that, but I can always watch for other opportunities to help him learn about fair play.  Just as I did a few minutes later.

Because in the end:

It was a tie game!

The Tie-Breaker

I sat everyone down in front of the game board and asked:

"Why do you think we did this activity?"  We could have done lots of activities for 100s day.

The kids answered:

"It let us practice 2x2 digit multiplication."
"You let us pick the regular way or the lattice way."  
"It got us ready for the math MCAS.

"Yes, but what do you think was my non-math reason?"

"It made us work as a team."

"Yes," I answered.  "I wanted you to work as a team; that's why I made sure you knew who on your team needed your help, and let you know you needed to help them.  So for the tie-breaker, I want to see which did more to help out their members who needed it.  You will all have to solve one problem, by yourself.  Each correct answer gets 1 point, so those kids who were taught how to solve two digit multiplication problems by their teammates will push their team to victory."

And in the end, the difference between the winning and losing team was just one point.
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Computation Challenge: 2 Digit Multiplication (How To)

It seems like 2 digit by 2 digit multiplication is one of those procedures that kids HATE.  And STRUGGLE with.  And need tons of PRACTICE to master.  The question is, how do you get your fourth graders to sit down and practice all those steps enough times to really internalize them instead of raise their hand to go to the bathroom or get a drink of water, only to vanish for half of math time?

The answer: the 100s day computation challenge.  It can be done any time of year, but I thought it up on the 98th day of school this year, so I rolled it out on 100s day.  And it was a hit!

Step 1:  Set Up

I stuck 100 Post Its to our portable white board, then wrote the problems.  The thing is, my class is at the point where they really need to drill down on their 6, 7, 8, and 9s, not the easier facts.  So this time around, I was sure to get some 9s and 8s in each problem.  If I do an encore of this activity, you can bet it will feature 6s and 7s.

Step 2:   Foster Collaboration, Not Just Competition

I know that competition is healthy, but I believe it's healthy in small doses.  More important in a classroom setting is to foster social skills and teamwork.  So here's how I did it:  As usual, I gave my kids a "test" consisting of a single problem to determine who needs help, and who could give help.  They were allowed to use any method they want (my Multiplication by Color Task Cards were a great way to practice the standard algorithm).  I lined them up in two rows.  Then I alternated red and blue wristlets (stapled construction paper) so that the ability level of each team was even.  I let them know that if they "need help," they should check in with someone on their team who is a "helper" before bringing a problem up to me to be checked.  I also let them know that "helpers" should be sure to not only complete their own problems; they need to watch for their teammates who are looking for help with a step.

This was a hard lesson for SO many kids!  That's why I mentioned doing an encore above.  Not only do they need to practice their 6s and 7s, many of them had to find out for themselves that ignoring struggling teammates in order to do as many as they can on their own was not the way to go.  But more on that later.  At least some of the kids got this.  I had two students tell me during the course of the game that one of their peers is a good teacher.  "I needed help, but I don't anymore because she taught me how to do it.  I don't need to be checked now."  Huzzah!

Step 3:  Game-play and Management

To help with the pacing, that is, to prevent them from rushing in and grabbing the easy ones first, I drew a box around a big chunk of those problems in the top left corner.  I told them that they had to finish the problems in the box first, before moving on to the rest (this way those easy problems on the right and bottom edge were only accessible in the middle of the game, as a bit of an unexpected break).  I also pulled names to fairly decide who got to choose a problem first, second, third, and so on.  I had the two teams sit in different areas of the room as they worked, and then come over to me (sitting near the game board with a calculator in hand) to get in line and get their work checked.  If a student got the problem right, I colored the corner with their team color, and he/she got to stick it back up on the board.  If they got it wrong, they were required to hand the problem over to a member of the opposing team. 

Tomorrow I'll let you know how it all went down!
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More Equivalent Fractions: Discussing Nonexamples

Our last fraction lesson was about creating equivalent fractions.  I wanted the kids to discover for themselves what happened when they broke 1/2 of a colored figure down into smaller, equal sized pieces.

Today I had them vote on which they thought were right and wrong, and although the majority knew, not all of them did.  I separated out the majority of the correct figures and put them on the other poster.  I put only one correct example on this other poster so the kids could compare the non-examples to a correct one.

I wanted to hear students explain why their result was a correct fraction or not.  Again, I used the definition of a fraction of a whole as something that is cut into equal sized pieces.  Meg over at Fourth Grade Studio has been blogging about fractions and put so much emphasis on having kids explain their thinking that it made me realize I need to do more of that. 

For a few of these samples, the creators were able to explain, "No, I made a mistake.  This piece is bigger than that piece."  I was happy to see those kids say so, because they demonstrated their understanding of fractions, even if they aren't successfully folding them.

A few kids were too shy to "admit" to their sample, (I purposely did not ask them to put their names on them since this was an introductory lesson, and I planned on mistakes happening that we could learn from) so I asked for volunteers to talk about what they noticed in those figures instead.

Of the figures shown, (to the right) there were two that were the most obviously wrong to the class.  One was figure w (green, right, middle) which the creator quickly spoke up and said, "Yeah, I forgot to split up the boxes on the colored side."  So she had the concept; she simply needs to work on rechecking her work generally.

The other obviously wrong figure was figure v (bottom left).  The class mentioned that not only were the shapes different sizes, it also did not have half shaded.  The child who drew it stayed silent; he'd correctly created a different figure and this was an "experimental" and rushed attempt, so I knew he also had the concept already as well.

The other figures resulted in rich conversation.  Some of the kids were stumped with figure y.  They "wanted" it to be right, because it was such a clever way of creating 1/2, but when I wrote a number in each box and asked, "Is box 1 the same size as box 2," they conceeded that they were not.  A couple students pointed out that boxes 1 and 4 are equal however, which made me smile because I remembered doing proofs in high school geometry class about the exact same principle!

At that point more kids said, "then figure x is also wrong, because that [box 1] is smaller than that [box 3]!" I labeled those boxes with numbers as well so the rest of the class could see it and got more "ohhhhhh"s!

To finish the lesson, I asked students to draw fractions that are equivalent to 1/2.  There are a lot more who can, but still several who are either still drawing 1/2 using different shapes instead of dividing the boxes further, some who are starting with a "creative" shape that can't be divided into more than 2 equal pieces (shows left) and others who are stuck on where to begin to create a perfect array with a set number of boxes that they have in mind.

I can't wait to try Meg's next lesson on using fraction stations with my class after school vacation gets out.  I've got the manipulatives for 3 of the stations, but I'm planning to substitute one of them with my Trading Fractions product.  I'm excited about keeping most of the class busy, but still completely engaged in something that all ability levels can access, and only managing a small number of kids with the Trading Fractions.

I'm actually almost looking forward to this vacation being over just for this lesson!
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Starting Fractions: Math in Focus Chapter 6

We're in chapter 6 for Math in Focus, which is fractions.  I don't think it matters what math program you use, fractions is always an important chapter in fourth grade.  I love it and I hate it.  I love it because it's visual.  I hate it because so much of it is counter-intuitive (1/2 plus 1/2 equals...not 2/4, gah!).  So lots of kids get lured into a false sense of security.  That's why I'm linking up with some fellow New England teachers to clear up this common math misconception!

Sure enough, on the day of the pretest (really a check up of what they remember from third grade, not a pretest of what is to come in fourth) a large portion of my class said, just before taking the pretest, "Oh yay!  Fractions!"  And then as they were taking it, the hands started going up.  "What does equivalent mean?"  "What does express mean?"  "I don't even know what fraction of a set means.  Is my answer going to be a fraction?"  I wrote down the questions they had on the board and let them know we'd be answering all of them over the next several weeks.

Meanwhile, just as I started digging through the extra practice from third grade, picking and choosing what to hone in on during school hours, I happened to start reading more teacher blogs in the evenings.  And coincidentally, one of my co-collaborators over at ATUE, Meg over at Fourth Grade Studio has been working on fractions for the past week or so, and her blog is AMAZING. 

Some of it is stuff I've done in the past, and will continue to do, like paper folding.  But she doesn't just stop there.  Every lesson is so purposeful in where she wants students to end up, yet she encourages the kids to do their own exploration, gives them room to make mistakes, and guides them toward self correcting their own misconceptions.  Talk about the "guide on the side, not sage on the stage."  Plus I'm learning new things, like I'd have never thought about having a debate about 2/4 like in this post.  

Seriously, Meg:

So with this challenge in mind, I planned my week.  First I challenged my class to draw several fractions.  Most of them were able to accurately draw some fractions, since we do "Calendar Math" fractions every day.  And yet, surprise, surprise.  They all drew the exact same shape:  circles.  Not one of my students had the fluidity of thought to extend what they knew beyond circles.  EEK!  I knew that some paper folding with rectangles was just what they needed.  I warmed them up a little after drawing all their circle fractions by asking, "who thinks they can show 2/4 with a rectangle," and so on, and most of them could do it.  Some had very creative ideas with prompting.  But they clearly needed practice to be really comfortable with other ways of representing fractions besides the circles they were used to seeing.

At this point I wanted them to explore like Meg's students did.  I wanted to give them room to make mistakes that they could learn from, but within a framework.  I paid close attention to Meg's definition of fractions as having equal sized pieces, not equal shaped pieces, which I emphasized to my class. 

I decided that for our paper folding exploratory activity, I wanted to stick with 1/2.  Why?  Because Math in Focus focuses on a half as a "benchmark fraction."  One way the kids are encouraged to figure out which fraction is greater is to tell if it's greater than a half first.  Then, if the other fraction is less than a half, they know the answer without having to find common denominators.  So I figured the more practice they get with fractions that are equivalent to 1/2, the better.

I started out by asking them to fold the rectangle in half, and each half had to be exactly the same shape.  As expected, some folded it vertically, some horizontally, and some diagonally.  What I didn't expect was that last one on the top right in the photo!  One of my hyperactive, tough to get his ideas down on paper sort of kids has a real knack for seeing how things work (a free thinker instead of a logical thinker) had the idea of making opposite corners meet.  I could not dispute that it was exactly in half when he showed me how he did it!

The next steps were to trace the fold so it was more visible, and finally, color one half.  Once each kid had at least one made, I asked those with more than one to stick one up on the board.  After we critiqued each one, I told them I wanted them to fold their other half colored rectangle more times, but the catch is, when they open it up, all the pieces should be the same size.  I did 2/4 to demonstrate, and I told them they could copy me for the first one, but after that, experiment.  "If you open it up and feel like it's wrong, leave it on the floor in front of you, get a new one from one of the piles near you (I had them scattered around) and try a different way.  This is your chance to test out your ideas!"

I put them all on the board, and I chose a couple of the accurate models to talk about the fact that they all still had just 1/2 colored, but the number of pieces changed (This poster came later, after all the fractions had been critiqued).

I told them tomorrow they'd "defend" why they thought their fraction was correct by focusing on the size of the pieces.  Then we had a blizzard that cancelled school for a couple days.  But after that, we were on our way again, hah.

In the meantime, the kids' homework was to draw 1/2 five different ways.  They were able to draw it using lots more shapes than they did before this lesson, and everyone who followed the directions knew how to draw the sections in equal sizes (a couple kids misread the directions and drew a bunch of 1/5s and could fix it on their own once I asked them to reread the directions).  Only 2 kids started to make the connection between today's lesson and what is coming next, which is equivalent fractions.  Those two kids drew shapes with 2/4, 5/10, 4/8, and so on.  Those two kids internalized the lesson after one day; the rest would need another look after our debate.  But not to worry: we haven't even TOUCHED the algorithm for the denominator being twice the numerator yet.  My dream is for every kid to "discover" that on their own...we'll see how many I can get there before spilling the beans!
Here are more ways to help clear up some common math misconceptions:

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Math Dice Games

Today on Teachers Pay Teachers I got a comment that said, "Thanks; I'm always looking for more dice games."  And it made me think back to years past.

At the beginning of every new school year I give my class a survey about themselves and school experiences, and one of the questions asks, "What's something fun you did in third grade."  Well, for the first couple years, a lot of kids wrote, "We played games in math."  And right away I said, "Ah, you like math games, eh?  Well, I don't really have many, but I'll see if I can make at least a couple this year."

And I did.  At first, they were simple ones, like "roll the dice and multiply the numbers you get" instead of a worksheet.  But even those were appreciated!  Then I started altering the dice if I wanted zeroes, or numbers larger than 6.  The games in my arsenal increased as time went on.  And finally, thanks to TPT, I've had the motivation to get my games are typed up, printed in color, and laminated.

We use dice for estimating.  Lots of dice for numbers with high place value, directions, and a recording sheet for accountability.
We use dice to practice finding the least common multiple of two numbers.  

And just to change it up, instead of dice we use a board game to practice prime and composite numbers!  Because sometimes flipping cards gives you the engineered numbers/odds that you need to make a game more playable.  

I think the kids liked the games just as much when they were hand written as they do now that they're of publishable quality.  But for me it's exciting that not only am I giving the (young) people what they want, I'm able to get my math games into other classes, thanks to TPT.

And I might not have had as much success with my math games if my former students hadn't requested them, or if their third grade teacher hadn't introduced them to the idea that math can be a game.  So thanks Mrs. D.  Enjoy retirement; I'll keep making math fun!
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Math in Focus Chapter 3: Bar Modeling for Word Problems

I attended a Math in Focus/Singapore Math workshop on using bar models for math problem solving this past week.  It was a good training in that the speaker was very engaging.  In the afternoon I got to show off my fourth grade bar modeling skills (some people prefer algebra for problem solving, but my brain is without a doubt wired for bar modeling). 

Post Its and Paper Strips
I even got to show off a system I developed for making the bar modeling strategy more concrete for kids using just Post It Notes and paper strips.  It really helped get my kids into the problem solving mindset at the start of the unit this year.  You can read that at this post.

The problem I had with the training (other than the fact that we had already just finished the chapter) is I didn't feel like I learned any strategies for helping struggling students.  It's been emphasized that we need to give kids time and space to learn how to persevere with novel problems.  And I understand the gradual release of responsibility; at some point they need to hear, "You try it now." 

But is it reasonable to give novel problems to every nine year old?  I keep thinking back to my college professor who said, "Telling kids 'think harder' is not going to help them."   

The missing link was a way to get kids practicing more in an engaging way, to build up their skills for those kids who just haven't arrived when it comes to preserving through novel math word problems.  And then I found Thinking Blocks.

I love this program because it's free, it's differentiated, and it gives so much immediate feedback to kids as they are practicing.  

 For fourth grade, I just clicked on "Addition" near the top (don't click on "Start" at the bottom; I didn't find that activity useful).    

The following page is all about differentiating the numbers involved in the problem, as well as how many steps are involved:
At that point, kids start solving a problem.  At several points in their process, they have to make decisions and then check to see if they are on the right track:

This problem has 5 separate points in which kids are deciding where to move pieces, with a hint at each step, and then click "check" to make sure they are on the right track before proceeding.  They start to see that they aren't going to get an answer in the first screen; a single problem requires a fairly lengthy procedure, even though each step in the procedure is not terribly difficult. 

They have to solve 5 similar problems to "beat the level."  I think that's plenty of practice, and they love seeing their achievement page filled with a row of stars. 

When I introduced this site to my class, they were excited about the overall look of it.  Once they actually sat down to play it, I saw kids who felt confident, not sitting stuck.  I saw kids who are not proficient in computation excitedly working through the numbers before typing in their final answer.  And when I said, "Okay kids, our computer lab time is half over.  If you'd like to exit this activity to have some time to read and reply to our classroom blog you may," most kids asked, "Can we stay on this site instead?"  

Our school still only has computers once per week, so we continue our practice on paper during math using this bar model activity

Do any of you teach Singapore Math?  What tricks to you have to help kids become more comfortable with demanding word problems? 
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