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Last week, my colleague Kami Wenning and our math coach, McKenzie Zimmerman conducted an informal morning PD session on the site
Estimate 180. Kami has been using the site with her 3rd graders and the conversations around it have been astounding so they wanted to share the resource.
Estimate 180 is a website created by Andrew Stadel (@mr_stadel). According to his website, he is a middle school math teacher and coach. He began the site in October 2012 with estimating activities he uses with his students each day of the school year.
After the PD, McKenzie and I talked about how I could use this site She facilitated the class while I transcribed and listened to her language with students. She went through the 4 day Lego Estimations and I watched from the back of the room to learn what I could about how best to use this resource and to listen to and record my students' thinking. The goals for the lesson were from the math practice--explaining your mathematical reasoning and understanding someone else's math reasoning. So that was the focus of the talk over the four days.
The conversations across days went so far beyond the typical estimation activities I've seen. The way that the site is built, the learning builds from one day to another and kids have information to build from. The talk around numbers was incredible and the engagement was high. Knowing the standards so well, McKenzie was able to take advantage of the last day's conversation to create a number sentence with a number to solve for. I am finding that oral language and conversation is such a huge part of math learning and Estimate 180 definitely supports this.
There are so many amazing things about the Estimate 180 site. There is a huge variety on the site. So many math concepts are covered in the over 200 estimation activities on the site. In a few weeks, I am going to use a series of lessons designed around estimating height and I am looking at another that estimates the amount of money in coins. You can browse the site or search estimations based on math topic. I also love that these are multi-day activities that are built to help kids think across time and to use understandings from one day to solve the next day's challenge.
Mr. Stadel must think about estimation all day every day because so many of these estimations come from real, daily life and I think kids will start seeing estimation opportunities everywhere after a few weeks of these.
I loved this site so much that I just had to share. I am excited to jump into another estimation with my kids next week (Cheeseball Estimations) and see where the conversations go!
STRING
YARN
ROPE
Our guidance counselor did a fabulous growth mindset lesson last week on how neural pathways are built in our brains. She talked about how new information is as tenuous as string or thread, but that with repetition and learning, pathways become as strong as yarn and as durable as rope.
On Friday, my math class did a pre-assessment on 5.NBT.2 -- understanding patterns of place value in numbers that are multiplied/divided by powers of ten, exponents, and metric measurement.
We're only two months into the school year, but my students understand that pre-assessments are to show what they know so that I can better meet them at their level. They have learned to approach them with a sense of curiosity -- a pre-assessment is a sneak peak at what they'll learn in the coming weeks. But these concepts on Friday were so far out of their realm of background knowledge that one student told me his neurons weren't string, they were spider webs! Not to be outdone, another student said, "Mine aren't even spider webs...they are CLOUDS!"
My response was, "That's okay, because in two weeks -- **finger snap** -- you'll understand all this!"
This will be a fun two weeks in math, and we'll keep a close watch on the way our learning grows as the strength of our understanding progresses from clouds to webs to string to yarn to rope.
I picked up a great new math picture book from Cover to Cover last week. It is called
Charlie-Piechart and the Case of the Missing Pizza Slice. Charlie has some friends over and they are ordering pizza. There are 6 pizza eaters so everyone will get 2 slices. But then one slice goes missing so they are not sure what to do!
This is a great book about fractions and the one thing I think it explains well is parts in a group--for example when Charlie is trying to solve the mystery of who stole the piece of pizza, he knows that every person is 1 out of 5 or 1/5 of his suspects but as people are cleared, the fraction changes to 1/4 and 1/3 etc. as he looks at each suspect.
This is a fun story that will start lots of good conversations about fractions. I am finding with 3rd graders, stories like this read over and over help kids make sense of some of the more complex math concepts. So I am happy to have this one for our classroom.
I have been reading lots of
math professional books as we've implemented Math Workshop in our district. This summer, I read
Well Played: Building Mathematical Thinking Through Number Games and Puzzles, Grades 3-5. This was a perfect read for me this summer as we move forward with Math Workshop. I love that Stenhouse provides such an amazing online preview on their site. I was able to dig in and know that I wanted to own a copy of this book. I know it is one I will revisit throughout the school year.
The book is more than just a set of games. As Kassia Omohundro Wedekind states in her Foreword,
"This is a book about math games and puzzles, but it is also a book about building communities of mathematicians who work together to problem solve, talk about math and figure things out."
The book begins with thoughtful chapters around the use of games in the math classroom. Early on in the book, the authors state, "...many students experience games or puzzles as fun activities or time fillers, but do not consider them as essential to their learning or as an important part of a lesson for which they are accountable." The authors go on to show us how to make games a more critical piece of our workshop and to help students have ownership of the games, their goals and the conversations they have while playing.
There is a great section about discussions and the authors give lots of practical tips for teaching kids to have productive conversations while playing game. There are so many examples of these conversations, questions that push thinking and ways to differentiate throughout the book.
Much of the book is organized in chapters by math concept and there are many games that support kids across levels and operations. The authors give great games and give great variations of several of the games. The games focus on engagement and problem solving and give kids ways to use math vocabulary throughout.
The games throughout the book are introduced in a way that you can really visualize how they might look in your classroom. Directions and materials are given as well as an example of how one teacher introduced the game in a real classroom. (The appendix is large and provides blackline masters for all of the games, directions, etc.) The game pages include Tips, What to Look For when observing kids play the game, Exit Card ideas and Extension of ways to change up the game.
An amazing resource for intermediate math teachers!
It's Math Monday!
for the Math Monday link up!
We were going to begin a big estimation problem (How Many Books Are There in Ms. Hahn's Classroom?), so I chose my math workshop opener from
Estimation 180 -- days 28-30, a sequence of toilet paper estimations. I knew exactly what I wanted to get out of this opener, and I expected it to be quick.
What I didn't expect what that my students would get mired down in a dis-remembering of what exactly perimeter, area, and volume are, and why the square footage fact we jotted down from the packaging shown in the answer of day 28 could not be used as the total length of the toilet paper on the roll on day 30. Maybe it's because we were talking about squares of toilet paper that their brains convinced them that square feet would be okay as a unit of length.
I let them struggle through misconceptions like squares and cubes are the same and you use 3D measurement for square feet. It was one student's tentative sharing of a rhyme she learned at her old school, "Perimeter goes around, but area covers the ground" that finally turned the tide away from the confident assertion of another student that square feet is a measure of length. You should have seen the lightbulbs go off above the heads. Boom. They had it back. Area is LxW (2D), volume is LxWxH (3D), and perimeter is S+S+S... (a measurement of length). Whew.
I've written often about the difference between leading the learning and following the learning. The importance of following is something I have to remember over and over again.
It's Math Monday!
for the Math Monday link up!
Dividing whole numbers by unit fractions and unit fractions by whole numbers are 5th grade standards. (CCSS.MATH.CONTENT.5.NF.B.7)
I've never taught division of fractions, but when you are struggling to understand something yourself, you often do a better job explaining it to someone else.
In my classroom, dividing fractions is all about pans of brownies.
If you have four pans of brownies and you want to divide them each into fourths, how many fourths will you have?
4 ÷ 1/4 = 16
You will have sixteen one-fourth-sized pieces to share with your friends.
But what if you you share 15 of those one-fourth-sized pieces and realize you forgot to share with 4 other friends?
If you chop a one-fourth-sized piece into four pieces, what size of piece will each of those friends get?
1/4 ÷ 4 = 1/16
They will get a tiny little piece, but at least you didn't completely forget them!
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| Creative Commons photo from Wikimedia Commons |
I got the best compliment ever last week: "That math problem was really fun! That was the best day in math so far this year!"
It was this problem from Robert Kaplinsky: "How Much Money IS That?!"
I put the pertinent information (photos, link to the video, questions to ask) into
Google Slides, and printed the above picture for individuals and small groups to mark up. (We did the Coinstar problem the day before.)
I wish you could have been there when I started the slide show with the above picture! Excited conversation ERUPTED all around the classroom! Questions, predictions, estimates, scenarios...leave it to money to get kids excited to solve a problem!
We worked on this problem over the course of two days, and our final answer was in the ballpark of the actual amount, but not at all spot-on. That's okay. We had already determined that we were not going to be able to aim for precision with this problem.
This week, I'm going to try some of the problems from
Inside Mathematics. I like how they come tiered with different levels on the same theme.
Happy Problem Solving, and Happy Math Monday!
It's Math Monday!
for the Math Monday link up!
It's Math Monday!
Well, spring break is well and truly over. The Problem Solving With Fractions math quiz that my 5th graders took on the Thursday before break finally got graded Saturday night. Because of the i
mportance of assessment driving instruction, I needed to grade that quiz before I knew what I would be teaching today in math.
We won't be starting with dividing fractions just yet, that's for sure. So many stitches were dropped in that quiz that we'll start off with a healthy dose of review. We need to go all the way back to reading a problem carefully to understand what it's asking, and paying attention to key words and phrases like "product," "how much more/less/farther/bigger," and "in all/total." Carefully reading the problem will tell us how to label our answers, or better yet, write the answer in a sentence.
We'll remember what we learned about multiplying a fraction by a whole number or a mixed number by a fraction, and how to take an answer that's an improper fraction and simplify it into a mixed number.
Then I'll give them their quizzes back, marked with the problems that need a second look, and we'll see if they can fix their mistakes, or finish their work by simplifying or correctly labeling answers.
And for the three students who got everything correct, I will give them this problem that NO ONE got correct (and that confounded me for a minute or two when I started grading). It doesn't make sense when you're solving for area to wind up with an answer that's less than the length of one of the sides. How can you solve this so your answer makes sense...and answers the question?
Jen makes a rectangular banner that is 1 3/4 yards long and 6/12 yards wide. What is the area, in square yards, of the banner?
It's Math Monday!
for the Math Monday link up!
"That doesn't seem right..."
How I love those four little words!
Saturday was our appointment with the tax lady. When she was all done and told us that our return would be xxxx, my heart sank a little -- the amount was almost a third of what we have been getting back in recent years. But I didn't say anything.
Lucky for me, our brilliant tax lady said, "That doesn't seem right..." and poked around until she found a default setting that doesn't fit for us. She worked and worked to get the online form to reflect our particular reality, and when she was done, she said, "That's better! Your return is XXXX! That's more like it!"
Indeed! Now we can get new siding for the house AND have some left over!
This story is brought to you by Math in the Real World. You can bet I'm going to tell this story to my students today, and the moral will be to ALWAYS think about whether your answer makes sense!
Yes, I know it's a stretch to share my monthly mosaic as a Math Monday post, but #arraychat is a real thing on Twitter! Math in the real world. It doesn't get any better.
Row 1 -- The first three are from North Market. The last one in this row and
Row 2 -- the first one in this row are a glimpse of hope for spring! The next three are William and his sunbeam, what a kitty has to do when his sunbeam gets too warm, and the face of a contented cat.
Row 3 -- #DubLit15 -- my Tech Kids, Chris Lehman learning from Franki's Tech Kids, Lisa Graff signing, the cookies donated by Wonderopolis for our afternoon snack.
Row 4 -- The walkway to Tucci's for the after-conference author dinner -- a winter wonderland. In contrast, don't get me started about the over-plowing of our street. Why do so many streets go unplowed, and yet the Snow Warriors come back again and again to our street, plowing shut every driveway on our street
repeatedly and throwing slush up onto cleared-off sidewalks. There's no good reason for it. (deep cleansing breath) The third shot is a jazzy shot of a jazz band at Natalie's. Next is a science shot -- the dark leaf got warm enough to melt down into the snow beneath it.
You can see all these pictures larger and un-cropped on Flickr
here.
It might not happen so much for primary teachers, but I am humbled on about a weekly basis by students in my 5th grade math class who are smarter than I am.
Case in point, this pizza problem. Do whatever you need to do to enlarge that picture. The work you will see there is flat-out brilliant.
In this problem, a class has won the PTO's pizza party for bringing in the most Boxtops For Education™. Each student gets their own personal pizza and eats a different fraction of the pizza. They eat thirds, fourths, eighths, twelfths, and sixteenths. The challenge was to put the fractions in order from greatest to least to find out which student(s) ate the most pizza, and then find out which table group ate the most pizza.
The pair of students who made this poster demonstrate two different ways to create equivalent fractions with a common denominator of 48: the "Bring to 48" table at the top in the center of the page, and the longer version on the right side of the page. (I didn't teach them either of these methods. They came up with them on their own. Brilliant, right?)
On the left side of the poster, they show their work finding an equivalent fraction for each of the children in the problem. They add each column to find out which table group ate the most, and they put all of the fractions/students in order (below the "Bring to 48" table in the center of the page).
Differentiation is important. While these two were engaged in solving this problem and demonstrating their work on this poster, I was working with a group of students who still can't independently make equivalent fractions in order to add and subtract with an unlike denominator. Others in the class were working on solving the pizza party problem, but they never got to the demonstration stage, or else their demonstrations were not nearly as elegantly organized.
I saw this graphic on Lester Laminack's FB page and started thinking about the ditties and tricks we teach. When are they a good support, and when does a learner spend so much time on the trick that they might as well just learn the fact or concept?
The one above is catchy and fun, but we don't teach all of those concepts in the same grade. So, by the time a student is learning to find the mean, and therefore might be ready for the rhyme, one would hope that they had already internalized median, range, and mode.
You know that trick for the nines table in multiplication that you can do with your hands? ("Holding both hands in front of you, number the digits from left to right so the left pinkie is 1. Then bend down the finger you want to multiply by -- so if you're multiplying 9x4, bend down the fourth finger. The fingers to the left of the turned-down finger are the "tens" digit of the answer (3), and the fingers to the right are the "ones" digit of the answer (6)."
description found here) Kids love knowing that trick, but if you have to put your pencil down and hold both hands in front of you to solve a multiplication problem, you might as well just learn the facts.
What are your favorite tricks or ditties to teach as you help your students internalize the fact or process?
What tricks or ditties drive you crazy because they have become more important than the underlying fact or concept?
Today I'd like to share one of my favorite brain games. It's a game that, now that I think about it, reinforces many of the Standards for Mathematical Practice.* I think I'll teach it at indoor recess today!
I have 4-6 decks of SET cards, but I was thrilled to find SET dice on sale at United Art and Education this past weekend.
You can play one free game per day online
here, or you can play a multiplayer game in realtime
here. There is also SET for
iPhone and
iPad.
In the game of SET, players must find combinations of color, shape, shading, or number that are either all alike in some way, or all different in some way. On the game board above, you can see that I made a SET with all purple, all solid, all three, all diamonds. I also made a set with all the same color (purple) and shading (solid), but all different number and shape. I have a set with all the same shading and shape, but all different color and number, and another with all the same number, shape and shading, but all different colors.
It's a great game. If you haven't played it yet (or recently)...what's stopping you?!?!
*CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2 Reason abstractly...
CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP6 Attend to precision.
CCSS.MATH.PRACTICE.MP7 Look for and make use of (patterns and) structure.
You might think
ASSESSMENT THAT DRIVES INSTRUCTION
is a trendy catchphrase
(syn: slogan, motto, catchword, buzzword, mantra)
that you can afford to ignore because it will eventually go away.
Sorry.
I'm here to tell you that if you are just teaching standards because
they are in your pacing guide
or on the next page of your math book
and you have no idea
whether or not your students already know those concepts,
then chances are
you will be wasting your time and theirs.
Yes,
it's a pain to give a pretest
and grade it
AND
go through the results child by child
to see who does and doesn't know which concepts.
But then your teaching path spreads before you
and you can clearly see
which students
need
which concepts,
what to teach whole class
and what to teach to just those few.
It's a pain
but it's worth it
and it's good teaching
so it's not going away anytime soon
and you might as well get on the
bandwagon*.
(or...in the words of a beloved former curriculum director...*the clue bus)
Because of the holiday, I have some extra time to play around with my favorite math -- baking.
I love the precision of measuring all of the ingredients to begin the dough, and then, when it's time to add the rest of the flour to the butter-milk-yeast-salt-sugar-flour starter, knowing exactly how much I can
not measure, and instead rely on the feel of the dough.
When do kids get the joy of using math to
make something?
Mid-month payday was last Thursday. On Saturday, I got to do another of my favorite maths -- balancing my checkbook. This is a bi-monthly game of (again) precision: Can I be accurate enough in my accounting to match my online bank statement to the penny? You'd think at my stage in life that I would be able to do this without a problem every single time. How hard can it be? Well, that's the point -- it's not hard, but it does take attention to detail. Constantly.
When do kids get the joy of using math in a way that really, really matters?
Somewhere along the line at the end of last year, the
iPhone app Elevate caught my eye. This "brain training" app was Apple's 2014 App of the Year. It was free, so I downloaded it. I am rocking all the games that tap into my reading, writing and vocabulary skills. None of those feel like training to me! But, when one of my three free games for the day is Math Conversions or Math Discounting, I groan out loud...but still play the game. I often make so many mistakes that I "lose all of my lives," or I run out of time because I can do it...just not quickly enough. (For comparison's sake -- when I get the game where I have to look at faces and hear names and facts about people and then remember that information...I actively AVOID that game because it is such a weakness for me that the game causes the same kind of anxiety I have in real life about names and faces!)
Do kids choose to play video games that improve their math skills?
It's on my weekend to-do list to finish gathering and organizing everything for 2014 taxes. I'm avoiding that item. There's still time; it can wait. And about taxes themselves -- I used to stubbornly do them on my own. I wanted to believe that an American citizen with decent math and literacy skills should be able to manage their own Income Taxes. Yeah. I made a few mistakes about a decade ago. Didn't get audited, but now I pay a professional to do the taxes.
Paying someone else to do the math for you is one of the privileges of adulthood. Sorry, kids! For now, you have to do your own math homework!!
I am joining Mandy at Enjoy and Embrace Learning for
Math Monday.
I am joining Mandy at
Enjoy and Embrace Learning for Math Monday.
I had been struggling with an authentic reason to introduce Google Presentation to my students. I needed a time when we would collaborate on a presentation rather than each student doing his/her own. And I needed a way for collaboration to happen without students revising each other's work.
When we were working on irregular volume in math, I found a way to use Google Preso! I created a slideshow with a page for each student and shared it to their Drives. I demonstrated how to use the drawing tools to make rectangular prisms. Their job was to first build two rectangular prisms using manipulatives, then combine them into one shape, and finally represent them and solve for volume on their slide. If they got finished early, they could add an additional slide and tell the three most important things about volume. For the sake of privacy, I have taken the students' names out.
I am joining Mandy at
Enjoy and Embrace Learning for Math Monday.
Franki has resolved to join Alyson Beecher's Nonfiction Book Challenge in order to stretch herself to read more nonfiction. I'm going to stretch myself in a different direction and try to focus on what's working (or not working) in my 5th grade math workshop.
This week (in between a snow day and a windchill day) we began working towards a deep understanding of division. Our standards in 5th grade do not require students to be able to do long division with the algorithm. We will be exploring multiple strategies for division.
Mandy wrote this week about
the importance of play. What I discovered was the importance of manipulatives...even for fifth graders.
Students were in groups of 4 or 5 on the floor in the meeting area. Each group had different manipulatives (beans, dominoes, pattern blocks, tiles). We modeled what addition looks like (combining groups) and what subtraction looks like (starting with a big group and taking some away from it).
Then we moved to modeling multiplication, which was surprisingly hard for them. After I gave them a problem to model (3x4), they realized/remembered that they needed to make equal groups or an array. We spent a lot of time thinking about what a multiplication problem SAYS -- "Three TIMES" tells you will be repeating a process three times, or making three groups.
Modeling division was as challenging as modeling multiplication. We started with a problem that they could easily solve with mental math so that they could check to make sure their model made sense (22 ÷ 2). Knowing that partial products is one of the first strategies we'll work on once we move to paper-pencil, I also gave them problems like 68 ÷ 5 so we could talk about efficient ways to share 68 into 5 equal groups rather than counting one by one. (Starting with 10 in each of the five groups, for example, and then sharing the leftover 18 into the 5 groups.)
Our math block is cut 10-15 minutes short by related arts, which we have actually come to love, because we can come back to our work and share, or students can complete an exit ticket or formative assessment that will inform my instruction for the next day. I gave each student a sheet of notebook paper and asked them to draw a model for 19 ÷ 3 and then write three things they know about division. What an eye opener! I've got a group of 5-6 who modeled 19 x 3, and another 4 or so who modeled 19 ÷ 3, but didn't demonstrate complete understanding by giving an answer. There were students who could model, but not write anything they know about division, and there were students who could write three things about division but not model.
So, now it's time for differentiation. I need to get some students to that deep understanding of what division means (modeling), and I need to move others along to applying that understanding to various strategies! This is the tricky part! This is the FUN part!
The first few days of math are always so interesting as I listen into conversations. On the third day of school, we used our math time to do a "Numbers About Me" project. I've seen this often on Pinterest and blogs and wanted to make sure we started the year thinking about math in our world. It was an interesting conversation as their eyes lit up each time they realized the things in their lives that involved numbers. They were simple things but making the connection to math made for a good conversation. We combined this with self-portrait work and the kids had a great time creating themselves with their Numbers About Me information.
*Please note that the 3rd boy in the top row made himself wearing an "I Love Mrs. Sibberson" shirt. Hysterical. Gotta love 3rd grade :-)
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