Equivalent Fractions Lesson Fail (and Fix)

The week after vacation, as every teacher knows, is crunch time.  We know that the state math test is rapidly approaching and equivalent fractions lesson reviews are in order.  And we know that the pressure is ON.  As veteran teachers, the whole thing is old hat.

It's not old hat to our kids, however!

Usually my fourth graders tend to be well rested after a vacation (a benefit of teaching upper elementary kids; they don't forget about classroom expectations like the little ones).  But during Monday's equivalent fractions lesson, my kiddos were just not on their game. 

The Equivalent Fractions Lesson Fail

First, I was all excited to do an equivalent fractions activity involving stations and manipulatives and exploring different shapes and amounts of pieces that can equal one whole.  I was going to lead a discussion on patterns we notice and post photos of all their ideas and it was going to be collaborative and hands on and foster real mathematical thinking.

That is not what occurred.  At all.  

Similarly, my plan was to end the day with a fantastic board game I created that help kids practice how heating and cooling results in a solid, liquid, or gas.  I spent my vacation finding cute fonts for the game pieces, typing up directions, and looking forward to more thank yous like I got on 100s day for the fun activity that helped them practice a tricky concept until it was mastered.

Again...nope.  Most of the class was so off task that I called off the game.  I let the kids who had done focused work help me with some coloring/cutting prep, and left it for the next day.

It just confirms the fact that no matter how carefully you plan a lesson, no matter how fun you make it, and no matter how many classes it's worked for in the past, not every equivalent fractions lesson works for every group of children.  Otherwise there would be a single curriculum that everyone uses.

So here's how I turned a day 1 equivalent fractions lesson fail into a day 2 triumph!

Changing the Content Delivery for My Equivalent Fractions Lesson

The first problem was that the kids bicker, a lot.  Despite Morning Meeting and Responsive Classroom, and student generated rules and setting expectations for group work and reminders 5 minutes prior to starting, they still bicker.  

The second problem was that despite modeling each station, posting the model and directions on the board, and answering questions/the same question 10 times, lots of kids just did not understand concept and/or, more likely, the directions.  

So what I decided was to give them an extra day of break...from each other.  I love collaborative learning and I do lots of it.  Which is strange, because I'm an introvert.  Or maybe it's BECAUSE I'm an introvert, and it doesn't come naturally to me, so I want very much to impart the strategies I've learned in order to help other introverts in my class.  But when it's not working, there's no sense beating a dead horse.  When the concept is too hard AND they have to focus on collaborating well, it's too much for them to handle; one of those things has got to give, and today it was the collaboration.

The Equivalent Fractions Lesson Fix

So for Tuesday's equivalent fractions lesson reboot I came in with...a worksheet.  I know, I know, what a horrible teacher I am.  Antiquated methods and all that.  But you know what?  They did it.  They did focused work for the period, they noticed patterns during this revised equivalent fractions lesson, and they enjoyed the relaxing atmosphere that coloring patterns tends to evoke. 

Once they finished the worksheet, I pulled out the manipulatives once again with a different, higher level skill to work on.  But by starting out with a coloring worksheet (you can get it free from my TPT store) with the exact same skill that didn't come to them when they were collaborating with a hands on activity, suddenly they could do it.     

I continued to rethink the cooperative work hiatus in the afternoon.  I didn't let them play the science game in their groups.  Instead I told them they were going to "play with me."  I chose only 4 kids to get a game board, and I got one too.  I situated them so that kids could look on with those that got to play.  And then we each took turns, drew cards, read them aloud, talked about our move, practiced the good sportsmanship I was modeling, and applauded the first person who got to the finish line.  

I'd never tried a board game this way.

I liked it!

I'll definitely use this strategy at the beginning of the year, and whenever a class has a case of the "bickers."  Not only did it help, but the kids actually enjoyed it.  They started saying "us," instead of "her" for the person near them playing with the same board.  They formed informal teams.  It created a better atmosphere than having 5 separate groups with every man for himself.  Sometimes you just know:  It's time for plan B! 

So how do you use this in your own teaching?  My vignette was one example, but here are more general ideas.  

5 Switcheroos for Plan B Activities:

1.  If a competitive game creates tension, take out the competition. 
2.  If a cooperative activity creates tension, try an individual activity (worksheets including coloring often go down well).
3.  If individual activities (like worksheets) create frustration, try a partner activity (Pick partners by personality, then sit with a couple partnerships who struggle with the content area).  
4.  If you don't have good working partnerships, play a teacher vs kids game.
5.  If a modality displeases the majority, pick a new one (drawing as opposed to music as opposed to acting out).

I originally posted this on All Things Upper Elementary.

Math in Focus: Bar Models with Manipulatives

At my school we use Math in Focus (Singapore Math) program, and with state testing coming up my fourth graders need a lot of review using bar models for problem solving.  This is by far the most challenging chapter (is it just us?).  For those of you who aren't familiar with MIF, one of the earlier fourth grade math word problems might be:

My sister had $8,359, and I had 3,596 less than her.  How much did we have altogether?

Well, my first year I think I was even more scared than they were (at first, since they didn't know better).  Even many adults, at first glance, estimate the answer to nearly $12,000 instead of the actual answer of over $13,000.  Although I'm more familiar with the content now, it's still a struggle for a lot of my students to master the sample problems as they become more complex than this one, and to complete them independently.  Like most programs, Singapore talks about a gradual release of responsibility.  And like most programs, we start with manipulatives before requiring abstract thinking.  Yet there were no manipulatives for bar models; just visuals.  Until now.

All I did was cut strips of paper cut into 2 different sizes.  The yellow paper is a Post It note with the sticky side holding the strips in place.  For my students, "Noun 1" and "Noun 2" were replaced with "Me" and "My sister." 

I told my class to remember these things as we progressed through the chapter:

1.  Bar models are just a tool to make solving math word problems easier; they are not there to make us crazy.  
2.  Bar models are at their most useful in fourth grade problems for helping us keep track of who has the bigger amount, who has the smaller amount, the difference, and the combined total.  

In the above problem, I told my students, "I have some money.  My sister has some more.  Decide where to put the little bar, and where to put the big bar."  Everyone was able to do that, so I told them that in this case, they'd know when it came time to draw the bars and put the bigger number with the right person, they could do it.

Below, I asked them to show me how much we had altogether.  Easy.  Indicate both.  That funny bracket in the book could be indicated with just their fingers.  Again, it totally took away the pressure of "I can't draw that weird mustache thing," (although they love when I draw it because hey, mustaches are cool, haha).

Below, I asked them to cover up all the extra money my sister had so that it looked like we had the same amount.  That covering up was "taking away" the extra. I told them that what I was doing for this sort of question was to subtract.

In the problem below, I told them, "I have some money.  My sister has some more.  Show me how much more."  Those funny brackets they show in the book were indicated with their fingers.  I told them if they were looking for the difference between the two bars, it's a smaller amount than the bigger bar, so they'd know they'd subtract.

The activity was a huge hit.  Notice that no numbers were mentioned today!  It was all about conceptualizing the process using manipulatives.  Kids whose computation is shaky could follow along at the same level as everyone else.  Even those kids who don't like to write were successful.  And an unexpected side effect to using these bars was that when it was time to start drawing them, all but two students had them lined up correctly on the left!  I wanted to go back to do this lesson all over again with my former students from last year who kept drawing the bars right after the words instead of lining them up correctly.

The next logical step for this lesson is, of course, to use numbers to introduce computation.  I start small (small numbers) and differentiate the numbers used for my class.  There's still minimal writing, and students move the bars around, but this time they also move around the numbers.  We followed up with a fun, free website I describe in this post.  I also created a self correcting bar model activity that you can preview in my store!

Readers, are any of you using Math in Focus/Singapore Math?  Do you have any tips you can share on how to survive chapter 3 with bar models?

Practicing Long Division with Money

Do you find that the topics that you struggled with the most in school are the ones that you love teaching the most?  That's how I feel about math, and in fourth grade the math topic I remember struggling with the MOST was long division.

I remember when I was a kid, it would take me FOREVER to solve a page of 10 long division problems.  I was one of the last ones done, sitting off to the side to finish up.  I'd stare blankly at 658 divided by 7 and try multiplying every number by 7 in the attempt to get 65.  I felt so frustrated that I had to do all those "extra" math problems and hope that they were right too.

So when it comes time every year to teach my students long division, I've tried lots of ways.  I had them make their own mnemonic devices to remember the steps (similar to the Do My Scissors Cut Bricks type acronym for Divide, Multiply, Subtract, Check, Bring Down).  I've tried partial quotient.  But for me it always comes back to practicing the physical process of dividing.  Base ten blocks are a great way to do it, but my favorite manipulative when it comes to math is money.  There's just something about counting money that makes people happy! 

Previously I described how I introduce the concept of dividing hundred dollar bills, ten dollar bills, and one dollar bills into "wallets" over in this post.  This activity helps get the kids used to the manipulatives involved and has a high success rate; with numbers that are carefully chosen to be evenly divisible (such as 486 divided by 2) the kids are able to compute in their heads before long.  So to up the ante the next day, I give the kids numbers that are not as simple to divide.

In the TOP PHOTO, you can see that the child knew he couldn't divide a hundred dollar bill 4 ways, so he had to regroup it for ten tens.  He was then able to put 2 ten dollar bills in each wallet, and he had 2 ten dollar bills left as a remainder.

Of course this does not complete the problem, but he is learning that 20 is a reasonable answer to 100 divided by 4, which is pretty powerful!  I'm much happier with him knowing this than memorizing DMSCB and not understanding what numbers to compute at each step. 

The child in the MIDDLE PHOTO is working on the same problem.  She also knows that the answer to 100 divided by 4 will be about $20.  However she is developing her understanding of the regrouping process.  She is trading in those 2 ten dollar bills for one dollar bills.  Although she has written a bunch of zeroes in the ones place on her place value mat, she will soon be able to refine her answer to include the ones place.

The child who completed the problem on the BOTTOM PHOTO has completed the process of dividing 100 by 4.  She is also able to accurately record the steps she has taken to arrive at an answer.  She regrouped the one hundred to become ten tens (although she forgot to erase the hundred).  She also knew she could tally up the 4 twos in the tens place when writing out the long division algorithm.  She used the difference (10 - 8 = 2) to find how many tens to regroup into ones, and she divided the twenty ones by 4 to get 5.  Although the photo does not show her paper, she transferred the process she used with the manipulatives to the traditional long division algorithm. 

I love this activity because pretty much everyone has play money they can use in their own classroom.  But just in case you don't, you can get play money here (affiliate link). And if you need more examples of numbers that increase in difficulty when it comes to long division, I have 3 different levels of long division task cards with numbers similar to these two examples available as a bundle.  I use the cards as review throughout the year because long division is one of those concepts that kids need to see many times before they can master it.  I never feel guilty leaving these in my sub plans because the kids know what to do and actually enjoy long division when they get to use the money.

Fourth Grade Test Prep: Math Review Centers with Both Classes!

Our state testing is rapidly approaching!  My fourth graders will take their math MCAS right in the middle of Teacher Appreciation Week.  Sad, but true, so what else can I do?  Pretend I didn't realize/care to my kids, whine to my colleagues, and have my fun this week instead with some of their favorite math review centers!

When determining activities we would use for our centers, I look at data from our unit tests.   Each year there tends to be a trend, so my games and activities usually target 2 digit multiplication, long division, lowest common multiple, prime and composite numbers, equivalent fractions and fraction of a set.  Every so often I pull out these centers and group kids according to which skills they need (keeping a strong student at each group unless I plan to work with the lowest kids while the strong kids have a challenge to do). 

Then one year I had another idea to really motivate the kids and keep the centers from feeling stale.  I asked the other fourth grade teacher in my building what he planned to do for review, and when he said with a sigh, "just keep going over old MCAS tests," I asked if he'd like to join forces!  My idea was to "mix up" the classes, and have half of his class come to my room, half of my class go to his, and have a mix of half and half in each activity station.  Being the epitome of easy-going, he said sure! 

We decided to have a nice long block of time, since kids would have a new environment, half of them would be learning a new activity (Mr. E. had his own favorite fraction activities that differed from my usuals) and there would be a whole new group dynamic trying to work cooperatively with kids they hadn't worked with in nearly a year.  We also decided on only having kids visit 2 stations in that time so that they got a taste of transitioning (and nearly all could have a chance to visit the "other" teacher's room) but they didn't feel too rushed and frustrated to have to leave an activity just when they were getting the hang of it.  So we settled on 45 minutes on the first activity (since it would include some introduction time) and 30 minutes on the second activity, with 5 or 10 at the end to debrief and clean up before lunch.

Next, I created a table for the groups.  Names of centers (like Least Common Multiple) on the vertical, and teacher's names on the horizotal.  I put those kids who could use a challenge into the activities Mr. E. had created, then used my test data analysis to place the other kids into activities that they needed to practice.  I made sure I had at a child with a leader-type personality in each, but with my class there is no shortage of those!  I knew that if they could teach a center to someone else, they would feel much more confident come test time.  Once that was finished, I filled in a second copy of the sheet for session 2, making sure I had different kids in each, and I handed it off to Mr. E.

Once he got me a copy of the finalized roster, I wrote the 7 centers on the board (sorry no photo of this) and a piece of colored construction paper got taped under each.  I copied the 6 names for each center underneath, then flipped the papers over and wrote the next six names on the back.  This way once we were ready to switch groups, it was as easy as flipping over 7 papers.  It went pretty easily!  They did have to pack up and bring their centers into the other room at the end of the first session (to help everyone get their change of scenery) but that also ensured couldn't blame any mess on the previous session, since they had to have everything they needed in order to play for the half hour, hehe.  

Overall, it was a huge success!  The kids seemed very invested.  They seemed to enjoy showing new people how to do the games they were familiar with, and they concentrated hard on learning a new game that their peers were already familiar with, like Lattice Multiplication.  Then they relaxed and had fun until it was time to move on.  It really helped keep math practice fresh and interesting during a time of the year when test anxiety can make school feel like the last place you want to be.  At the end of the second session, Mr. E.'s kids were thanking me!  The following week we repeated the process, but had 3 sessions instead of 2 since they were familiar with the process.  I never thought I'd say this, but now I love MCAS review day! 

How do you keep review interesting prior to your state exams?  If you'd like a set of games to cover those easy to learn and easy to forget math skills, I've created a Math Test Prep Game Bundle for you.  It covers prime, composite, LCM, GCF, estimating, and mathematical decision making.  Check it out! 

Constructing 3 Dimensional Figures

In geometry, 3 dimensional figures is one of those areas of math that is a lot of fun to teach, but can be tricky for kids who are usually great at math!  

Instead of mathematical reasoning, it involves more spatial awareness, and the ability to mentally turn shapes around.  As a result, you may find some kids who are not used to coping with frustration in math class suddenly feeling overwhelmed.  The upside is that some kids who feel like they are not good at math are suddenly your helpers.  Use this as an opportunity to teach that no one is good at everything, but everyone is good at something!  

To foster cooperation and appreciation for others' unique talents, I like to partner students up to complete sets of 3D models.  There are two activities that help kids with different objectives for 3 dimensional figures.

Tips for folding and taping our 3 dimensional figures

The first activity involves nets.  These are great for helping kids see all the faces in a model at a glance.  Students can count the faces more easily on a flattened net than on the drawing in their workbook or test.  

When preparing materials for this activity, consider the needs of your class.  The thicker the paper, (cardstock, coverstock, 32 pound paper, or copy paper) the harder it is for some students to cut and fold clean creases.  I find it's more difficult than it used to be, since younger kids have more screen time and less color/cutting/fine motor time.  On the other hand, once the creases are in place, it's easier to tape the thicker the paper.  If you spread this project over two days (one cutting/folding and the other taping/counting) consider if and when you will have extra hands in the class to help out.  

Also consider whether or not your nets will have "flaps" for taping.  For some kids, it's an extra folding step, which is a disadvantage.  But again, when it comes to taping, it can be helpful.  My advice is to use nets with flaps because you can always cut them off to differentiate for some kids.  

Collecting and Analyzing Data from our 3 Dimensional Figures

Once the folding and taping is complete, further students learning by counting the faces AGAIN.  This can be challenging for students because there's no obvious starting or stopping point, so it's easy to lose count.  Model the strategy of drawing a dot on each face to help keep track of what's been counted.

Another activity that is always a hit is building models with toothpicks and gum drops (or marshmallows).  This activity helps students count the edges and points in 3 dimensional figures.  I would NOT have students count the faces on these models, since they are "empty." 

Tips for building our 3 dimensional figures

Before tackling this challenge, I teach students the difference between prisms and pyramids.  

A prism can start with any 2D shape for a base.  Next, build up with vertical toothpicks.  Put gumdrops on the top of each, and connect the top gumdrops so there is a face with an identical shape and size as the base.  The resulting sides are squares.  Explain to students that if they had longer toothpicks for the sides, those faces would have been rectangles instead of squares, and it's a prism either way.

A pyramid, like prisms, can start with any 2D shape for a base.  However, instead of building up with vertical toothpicks, tilt them slightly toward the center.  Those toothpicks need to come together in the center to be joined by a single gumdrop (or marshmallow).  The top center vertex is the defining quality of a pyramid.  Notice that the sides are all triangles.  

As an enrichment activity, students can try creating shapes with all triangle or all pentagon faces, similar to the nets for dodecahedrons or octahedrons, tetrahedrons.  Just be aware these will require an abundance of materials!  If you're short on time, you can of course analyze premade geometric solids (affiliate link).

If you'd like recording sheets for your students as they design and test their own nets and constructed 3 D figures, check out my Geometry Unit.  

Measuring Angles Art

Geometry is my favorite thing to teach in math and that's probably because I can turn anything geometry related into an art project!

To help kids understand measuring angles, we start with quarter turns.  Since I teach it after a unit on time measurement, I make the connection to starting at 12:00, and the minute hand stopping at 15, 30, 45, and back to 1.  I explain that one difference between the angles and the clock is the hour hand moves slightly, while the angles stay at multiples of 90 degrees.

The first art project happens once we make the connection that two right angles, each 90 degrees, add up to 180.  And the result is a straight line.  We use drawing triangles to draw right angles and straight lines.  But to get a feel for the movement and amount of space from one position to the next, we trace circles.  Teaching tip:  use yellow paper and at least half your students will be psyched to be making Pac Man during math class.

Measuring Angles Art

Paper or plastic cups are good options for tracers.  I tend to have plenty of plastic takeout containers on hand so we can reuse them for painting from year to year and those work great for this project as well.  By filling in the arcs, kids can see that even though the arcs can vary in size, they have a defined start and end point if it is 90, 180, 270, or 360 degrees.

For our second art project, I give students two "lines" (strips of paper) and we practice rotating one of them around in a clockwise direction.  Once they are glued we trace the the arcs again, and this time we label the degrees after measuring angles.

By introducing these 4 angle sizes, students were able to make pretty good initial estimates about angles that were "near 90," "near 180," "near 270" and "near 360."  Before I even started teaching them how to line up and read a protractor, they were getting a sense of reasonable measurements of angles.

Measuring Angles Task Cards for Additional Practice

These art activities get kids excited before the hard work of using protractors for measuring angles to the nearest 5 degrees.  My class used to struggle with this skill for one silly reason:  laying a protractor across the spine of the workbook.  It just didn't work!  And since tearing out pages was another fine motor challenge they weren't prepared for, I realized in needed an alternative.  So I made my own task cards!

These cards are great for measuring angles in isolation as well as within polygons.  I even threw in a bonus for students to "discover" when it comes to adding up all the angles of like colors (hint:  each set of like colored angles form a complete circle).  I've had great success with these task cards since there are lots of examples for kids who need repeated practice, and adding up to 360 degrees is engaging enrichment for students who need more stimulation.  And if you need protractors, you can get a cheap class set here (affiliate link).