It's our last challenge for 2020, which this time around was to write to the theme of "Wish I'd Been There," or to an historical event that incites wistfulness. I'm not sure the event I've chosen incites wistfulness, but this event reminds me of something my mother often said, "Oh, to be a fly on the wall." Or as Burr proclaims in Hamilton, "I wanna be in the room where it happens."
Let me take you back to 1872 in Rochester, NY.
Clipped from the Brooklyn Union newspaper, 5 Nov 1872
I'm not sure this poem is finished yet, but it's a start.
November 5th, 1872 - Election Day
I've always admired those resolute women corset wearing bustled and ruffled risk takers rule breakers history makers who cast their votes for Ulysses S. Grant pushed the boundaries of the 14th amendment determined to be heard who embraced fully the word citizen with all the rights it implied who broke the law then marched on
I do hope you'll take some time to check out all the wonderful poetic things being shared and collected today by Irene Latham at Live Your Poem. Happy poetry Friday friends! Sending you all wishes for a happy holiday season and much awaited new year.
It's another looking back month in our year of hindsight/foresight. I chose to revisit a poem from April 2019. While I didn't revise or talk back to that poem, I did take another stab at the form. In that challenge, we wrote anagram poems. The challenge was inspired by the poem A Garden to Gander, written by Linda Baie of Teacher Dance. Linda's poem included two anagrams in each line. I wanted to play with this idea a bit more. Instead of using words as I did the first time around, my new poem largely uses phrases.
This month's challenge was to write a naani with an autumn theme. A naani is a 4-line poem containing between 20-25 syllables. I had a lot of trouble with this one, but then I generally have difficulty with short forms that are rather open-ended, so I shouldn't have been surprised. In the end, I ended up with a longer poem composed of several naani strung together.
An Autumn Naani Story
my favorite maple tree blushes brilliant red with immodesty celebrating fall
oak stands quietly nearby cloaked in orange and gold dropping scads of acorns for every passing squirrel
both silent witnesses to a world in upheaval sentinels of change outlasting generations
If you’d like to write with us next month, the challenge is to pick one of your old poems to revise and/or write a new poem in conversation with it. We will be posting on the last Friday of the month (November 27th, the day after Thanksgiving) and would love to have you join us.
I do hope you'll take some time to check out all the wonderful poetic things being shared and collected today by Linda Baie of Teacher Dance. Happy poetry Friday friends!
The challenge this month was to write to the word ponderous or the image of a hippo. I decided to play around with an old quick write prompt issued by Nikki Grimes at Kate Messner's blog way back in 2014. Based on the examples she shared, I decided I wanted to begin a poem with "Hippo is a __ word." Here's where my poem went.
Hippo is a ponderous word a barrel-shaped word a semi-aquatic, unable to swim word a yawning, wide open word a gregarious, booming word an aggressive and unpredictable word Keep your hands in the boat while you float downriver, observing a bloat of hippopotamuses and hippopotami half-submerged ears twitching, eyes watching keeping the crocodiles at bay.
If you’d like to write with us next month, the challenge is to write a naani poem with the theme of foresight, or autumn, or both. We will be posting on the last Friday of the month (October 30th) and would love to have you join us.
I do hope you'll take some time to check out all the wonderful poetic things being shared and collected today by Jone MacCulloch. Happy poetry Friday friends!
This month's challenge was to pick one of our old poems to revise and/or write a new poem in conversation with it. I went back to a post from May of 2016. Here's how that challenge was defined.
This month the Poetry Seven crew wrote in the form of the tritina. The tritina is composed of 3 tercets and a final line (envoi) that stands alone. Similar to a sestina, though shorter, it uses a set of 3 alternating end words instead of six. The form is: ABC / CAB / BCA / A, B, and C (final line/envoi).
The words we chose from were selected by Tanita. They were: sweet, cold, stone, hope, mouth, thread
I wrote 2 poems back then, one for my father and one for sheer fun.
Looking back on these old challenges to find a poem to revisit, I was inspired by the words Tanita selected and decided to have another go at this one. Here's my new poem. It was written to this photo.
If you’d like to write with us next month, the theme is ponderous, or an image of a hippo. You may use any form you like. We will be posting on the last Friday of the month (September 25th) and would love to have you join us.
I do hope you'll take some time to check out all the wonderful poetic things being shared and collected today by Heidi Mordhorst at my juicy little universe. Happy poetry Friday friends!
Today I'm sharing another multiplication game. This is a variation on a bump game. It has a cow on it, so that's where Moo-ve it comes from.
In the game, players roll 2 dice and finds the product. They find that product on the board and cover it. If the product is already covered, the player may "bump" their opponent's piece off the board. The first player to get 3 in a row, horizontally, vertically, or diagonally, is the winner.
I have a number of multiplication games that students can play to practice basic facts. I really like Multiplication Four-in-a-Row. Students are give a number strip or table and begin by placing markers on any two numbers. The first player finds the product and covers that number. The second player moves one of the markers on the number table to create a new number sentence, finds that product, and covers the number. Play continues until one player gets 4 in a row, either horizontally, vertically, or diagonally.
I've been using this paper and pencil version for some time.
Have you every played Shut the Box? When my son was young, we used to eat at a little hole-in-the-wall restaurant/bar for breakfast. There was a Shut the Box game behind the bar and the owner would hand it to us when we walked in so we could play while we waited for our food.
Eventually, I made a portable version that fit in my purse. I took small poker chips and wrote the numbers 1-10 on them. I stuck them in a snack size baggie with two dice and we had a game we could play any time, anywhere.
I also made a paper and pencil version for use in class. We laminated some of these and placed additional copies inside sheet protectors for reuse.
In working to develop number sense in our second grade students, we spent time on activities that encouraged students to visualize numbers and think about number combinations. We did quick image activities with ten frames, solved Splat! questions (thank you Steve Wyborney!), played Shake and Spill, and conducted other related activities.
For quick images with ten frames we used both horizontal and vertical sets of ten frame cards. Cards were flashed for 2-3 seconds and then hidden. Students were asked:
What number did you see?
How did you see it?
How many more to 10?
Here's an example of the cards we used.
If you want to try this with your students, you can download a copy of each set of ten frame cards. Click here for the horizontal version. Click here for the vertical version.
Shake and spill is a game we played with two-color counters. Students would take a number of counters (6, for example) in their hands, shake them, and spill them on the desktop. Students then looked at the counters and determined how many of each color without counting. Eventually, the same combinations would keep coming up and students began to recognize that 6 could be made of 0+6. and 1+5, and 2+4, and 3+3, and so on. Once students were familiar with the game, we added in recording sheets. Here's the sheet for 6.
I first learned about Shake and Spill many years ago in an old video series (80s hair and clothes) created by Marilyn Burns. The videos focused on how different manipulatives could be used to teach a range of mathematical ideas. You can read about an updated version of this activity at the Math Solutions site.
I think every elementary teacher has played Race for a Flat, that game where students learn to regroup and exchange 10 ones for 1 ten, and 10 tens for 1 hundred, or a flat.
In the past, my students played on a legal-sized place value mat that looked like this.
Some of my students had difficulty with this, so I created this mat for them to use instead.
The blocks on this mat are composed of centimeter squares, so base-ten blocks fit perfectly. For whatever reason, students found it easier to make trades from ones to tens with the blocks in this format, which surprised me because we tried hard to get them to use ten frames to visualize numbers.
I like games that allow students to practice basic facts in a strategic way. Three to Win is a simple, quick game where students practice basic facts in addition and subtraction (and sometimes multiplication) while using problem solving strategies. The first player rolls two dice and says the numbers. They look at the empty spaces on the board and decide whether to ADD or SUBTRACT the numbers (or sometimes MULTIPLY). Once a choice is made, the player says the number sentence aloud and places their marker over the answer. If a player rolls doubles, they can cover one of the images on the board. Play alternates until one player gets three markers in a row, vertically, horizontally or diagonally.
Below are images of the digital version, which focuses on addition and subtraction.
Below are the images of the print version, which focuses on addition, subtraction, and multiplication.
I like place value games that ask students to think about the value of a digit. One of my favorite games is called Closest to 100. In this game, students must consider how the position of a digit in a number changes its value.
Two versions of the game are presented. In each version, students roll dice and determine whether the numbers rolled will be used to make tens or ones. The goal is to be the player closest to 100 at the end of five rounds.
I have created both a hands-on version and a digital version. Below are images of the Google slides and the print version.
I love to teach kids to play strategy games. They're great for indoor recess and for encouraging mathematical thinking. They require kids to predict, plan ahead, analyze moves and mistakes, and reflect on their strategies. I generally print and laminate copies of game boards and directions and provide simple markers for kids to play. In this time of COVID-19, I've been working on digital editions of games.
Alquerque is a strategy game that is thought to have originated in the Middle East. It is considered to be the parent of draughts (checkers). In Alquerque, two sides of twelve pieces face each other on a board of 25 points. These are joined by horizontal, vertical and diagonal lines, though not every point has diagonal connections. The goal of the game is to capture all of the opponent's pieces.
The game begins with 24 pieces, divided between the two players, 12 black for one player and 12 white for the other.
The player who begins is decided at random, or at the agreement of the players. (Note that the first player is at a slight disadvantage.)
A piece can be moved from its starting point, along any marked line, to an adjacent empty point.
A piece can capture an adjacent enemy piece, if a marked line joins their respective points, by leaping over the enemy onto the empty point beyond. The enemy is then removed from the board.
If such a capture can be made, then it is REQUIRED. If there is a choice of such captures, then the player may choose which capture to make.
When a piece has captured an enemy and is in a position to capture another in the same manner, then the further capture MUST be made. In other words, the capturing piece must perform all possible captures in its turn.
The game ends when one player has lost all his pieces. The opponent is the winner.
The game is considered a draw if both players are equal, and neither can safely engage their opponent without losing the game. This most often happens when both players are reduced to one or two pieces each.
Several years ago I made a set of place value strips to help students think about expanded notation and decomposing numbers. I created three variations of the strip sets so that they could be used to differentiate instruction. The first form includes the words for place value on the strip (ones, tens, hundreds, etc.). The second form includes numbers to show the value of the digit in a particular place. The third form includes no additional information. Each set includes whole numbers from ones through thousands and comes in both color and black and white versions.
This year I had the opportunity to put the strips to work in second grade. The students seemed to really enjoy working with them. It was clear as we worked on connecting the strips to base-ten blocks and written numbers that students were making connections and understanding the value of a digit in a particular place, and how a 3 in the tens place is different from a 3 in the ones place.
Here are some photos of the students at work.
Here are a few sample pages from the file.
These strips were designed for use in activities that meet the following Common Core Standards for Math:
1.NBT.2. Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:
a. 10 can be thought of as a bundle of ten ones — called a “ten.”
b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).
1.NBT.3. Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.
2.NBT.1. Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:
a. 100 can be thought of as a bundle of ten tens — called a “hundred.”
b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).
2.NBT.2. Count within 1000; skip-count by 5s, 10s, and 100s. 2.NBT.3. Read and write numbers to 1000 using base-ten numerals, number names, and expanded form. 2.NBT.4. Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.
A few days ago I found myself down an internet rabbit hole and came across this reply to a tweet about random facts:
Of course, this tweet is referring to the geologic time scale, a perfect vehicle for consider large numbers.
This got me thinking about magnitude and wondering how we can help students move beyond basic place value knowledge to a true understanding of large numbers. As we work to provide opportunities for students to build number sense, developing an understanding of magnitude and how large quantities relate to one another can be a challenge for some students. The fact that teachers and textbooks often emphasize digits (place value names) to the exclusion of understanding relative size makes ideas of magnitude challenging for students.
For many years I have asked preservice teachers to place 1 million on a number line with 0 and 1 billion marked. The results are always eye-opening. A few years ago, Mark Chubb wrote a post entitled How Big is Big? in which he described asking teachers to show where 1 billion would go on a number line with 0 and 1 trillion marked. Go read it. He describes all the issues that come to mind when grappling with these ideas.
It's not just school age students who struggle with issues of magnitude. In the 1982 article entitled On Number Numbness, Douglas Hofstadter wrote:
It is fashionable for people to write of the appalling illiteracy of this generation, particularly its supposed inability to write grammatical English. But what of the appalling "innumeracy" of most people, old and young, when it comes to making sense of the numbers that run their lives?
. . . . .
This kind of thing worries me. In a society where big numbers are commonplace, we cannot afford to have such appalling number ignorance as we do. Or do we actually suffer from number numbness? Are we growing ever number to ever-growing numbers?. . . . .
. . . . .
Combatting number numbness is basically not so hard. It simply involves getting used to a second set of meanings for small numbers-namely, the meanings of numbers between say, five and twenty, when used as exponents. It would seem revolutionary for newspapers to adopt the convention of expressing large numbers as powers of ten, yet to know that a number has twelve zeros is more concrete than to know that it is called a "trillion".
. . . . .
The world is a big place, no doubt about it. There are a lot of people, a lot of needs, and it all adds up to a certain degree of incomprehensibility. That, however, is no excuse for not being able to understand, or even relate to, numbers whose purpose is to summarize in a few symbols some salient aspects of those huge realities. . . . . For people whose minds go blank when they hear something ending in "illion" all big numbers are the same, so that exponential explosions make no difference. Such an inability to relate to large numbers is clearly bad for society. It leads people to ignore big issues on the grounds that they are incomprehensible.
I'm seeing a lot of this "incomprehensibility" in articles and data visualizations related to COVID-19, though I worry using these examples may be too traumatic for some students. I want to use authentic problems and real-world contexts in examining large quantities. There is a nice overlap here with science, as considering the solar system leads to examinations of size and distance. Here are a few resources I like to use to consider large quantities and measures.
Powers of 10 Video (1977) - It's old, but oh so powerful.
The Powers of Ten video takes viewers on an adventure in magnitudes. Starting at a picnic by the lakeside in Chicago, viewers are transported to the outer edges of the universe. Every ten seconds the starting point is viewed from ten times farther out until the Milky Way galaxy is visible only as a speck of light among many others. Returning to Earth with breathtaking speed, the view moves inward- into the hand of the sleeping picnicker- with ten times more magnification every ten seconds. The journey ends inside a proton of a carbon atom within a DNA molecule in a white blood cell.
Pair this video with the book Looking Down by Steve Jenkins.
Powers of 10 - Interactive Java Tutorial Soar through space starting at 10 million light years away from the Milky Way down through to a single proton in Florida in decreasing powers of ten (orders of magnitude). Explore the use of exponential notation to understand and compare the size of things in our world and the universe.
How Big Are Things? This site tries to make size something you can remember, use, and optionally calculate with.
Is That a Big Number? A web site designed to "restore some number sensitivity," you can enter any number and receive a bunch of relevant comparisons to put the number in context.
I few years ago I heard a story on NPR. It began with the question "How many trees are there on the planet?" I started to think about how one would make such an estimation. My guess was 100 billion. Was I close? No. The actual answer is closer to 3 trillion. That's TRILLION, or 3 x 1,000,000,000,000. You can hear the story at NPR in the post entitled Tree Counter is Astonished By How Many Trees There Are.
And while this sounds like a huge amount (no, we don't have enough), the researchers found that the Earth has lost nearly half its trees since the start of human civilization. We also know that we are losing 10 Billion trees every year. All of this is pretty disturbing.
These numbers teach us a lot about habitat loss, how much carbon dioxide is being absorbed from the atmosphere, how water is recycled in an ecosystem, and how we can preserve and replenish our forests. Take a minute to learn more in this video.
"So why do we learn mathematics? Essentially, for three reasons: calculation, application, and last, and unfortunately least in terms of the time we give it, inspiration.
Mathematics is the science of patterns, and we study it to learn how to think logically, critically and creatively, but too much of the mathematics that we learn in school is not effectively motivated, and when our students ask, "Why are we learning this?" then they often hear that they'll need it in an upcoming math class or on a future test. But wouldn't it be great if every once in a while we did mathematics simply because it was fun or beautiful or because it excited the mind?"
Learn more about these ideas and the beauty of Fibonacci numbers in Benjamin's talk.
I've been a puzzle-solver and game player for as long as I can remember. I love to do math for fun. I want more students to do math for fun and experience the joy of mathematics. I want them to experience the rush of exuberance and confidence that comes with finding the solution to a challenging problem. I want more elementary teachers to see the value of puzzling through non-traditional problems and the long-term benefits it brings. I want them to make this a part of regular classroom instruction. Perhaps instruction is too strong a word. Maybe we need more general language about learning time, because students learn in more ways than just from traditional instruction.
This notion of math as inspiration and joy and beauty is exemplified in one of my favorite videos from Cracking the Cryptic. Yes, I love math and puzzles so much that I watch others solve problems. And honestly, I believe that I have become a better problem solver since playing along and listening to Simon Anthony talk through his thinking and solution strategies.)
If you haven't seen this video, you should take some time to watch it. It's 25 minutes that will astound you. And I'll bet when it's over you'll want to go out and start solving these non-traditional sudoku.
That's it for day 11 of #MTBoSBlaugust. I hope you'll come back tomorrow to see what else I have to share.
In my work with preservice teachers I find myself thinking a lot about the language of math and the models we use. I spent even more time thinking about these issues as I watched them play out in second grade last year. Today I want to discuss the differences between the 100 board and 0-99 chart.
In some circles, the 0-99 chart is also called a 100 board. In old-school language it is called a counting chart. I don't particularly like either of these names. Here's why.
Even though 100 numbers are represented on the 0-99 chart, it does not extend to 100. To be called a 100 board, it should include this number.
A counting chart should begin with the first counting number, which is 1. Zero is not a counting number. Therefore, a chart that begins with 0 should not be called a counting chart.
I know these are really minor points, but they speak to the issue of precision and the language we use to describe mathematical tools.
Beyond the semantic issues of what we call these charts, I have greater concerns about the use of 100 board (and the 120 board we now see to support Common Core standards).
Our system of numeration is a base-10 system. This means we have 10-digits (0-9) and that every number can be made using one or more of these digits in combination. Ten and powers of ten are used to construct the system. Larger numbers are built by repeatedly bundling ten: 10 ones make one ten, 10 tens make one hundred, 10 hundreds make one thousand, and so on. In simpler language, every time we reach ten of a particular unit, it is regrouped and renamed. Here's what the tens column looks like on these charts. Note the placement of each.
This is where my problem with the 100 board comes in. When we reach 10 ones on a hundred board, they still remain in the ones row (or column, depending on how the chart is arranged), but they belong at the beginning of the next decade. On a 0-99 chart all the numbers in a decade appear in the same row. For example, on a 100 board, the decade row for thirty begins with 31 and ends with 40. On a 0-99 chart, the decade for the thirty row begins with 30 and ends with 39.
This representation on the 0-99 chart is much clearer and more accurately represents the way our number system operates.
The other idea the 0-99 chart makes explicit is that zero is an even number. Even though we don't begin counting with 0, placing it on the board shows students that one less than 1 is 0 and that zero IS a number! On a 100 board we recognize numbers ending in 0 as even, but because zero does not appear, students don't view it as a number (just a placeholder) and often question its classification, wondering if it is even, odd, or neither. The 0-99 chart can help students overcome these misconceptions.
In the final analysis, both charts can be used for developing skills in counting up and counting back, skip counting, finding one more/one less and ten more/ten less than a number, recognizing patterns, place value, addition, subtraction, finding multiples, prime numbers, and more. However, the 0-99 chart does this while helping students work within the structure of our base-10 system.
Now that I've articulated the reasons that I feel the 0-99 chart is a better tool than the 100 board, here's a set of charts for you to use. One is a traditional 0-99 chart (oriented horizontally), while the other displays the numbers vertically.