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.
This is an old (2010) video, but I keep returning to it because I love the notion of a math salon.
A while back I wrote a post describing the components of elementary school homework I believe are important. Here's a description of one of those components.
Puzzle - When was the last time you sat down to solve a puzzle and did it for fun? I do this all the time. Sudoku, crossword puzzles, jigsaw puzzles, logic problems, tangrams ... I could go on. Puzzles are good for the brain. They develop critical thinking and problem solving skills. They teach kids to persevere, guess and check, collaborate with others, and try a whole host of new strategies. Can you think of a better training ground for mathematical thinking than puzzling? Now imagine if your teacher encouraged you to do this for homework.
This is exactly the kind of think happening in this math salon--kids exploring ideas in meaningful ways that just happen to touch on aspects of mathematical thinking. Just imagine what you could do with this idea. What might it look like in a classroom? How can we provide a safe space in mathematics instruction for exploration, experimentation, and failure without judgment? I know there are committed teachers already doing this, but how do we make it the norm and not the exception?
In October of 2011, one of my students, now a highly successful classroom teacher, created an annotated bibliography on connecting math and art in the teaching of multiplication. I've updated her original work and included a number of new resources. My thanks to Christine Mingus for the foundational work developed in this class assignment.
In the interest of finding new and artful ways to help teach children about multiplication, this post highlights resources that use art to enhance math instruction, increase student motivation and engagement, and help students tap into fun and creative ways to think about math concepts.
Visual learners, and even ELL students, benefit from making projects that allow them to explore mathematical concepts in new ways -- and art is all about looking at the world from multiple perspectives. Art simply feels like play for elementary aged students, and using it to explore multiples and place value helps foster their confidence in their math abilities and definitely works to keep a multiplication lesson percolating. Many children see art as a non-threatening subject matter, and using it to ground a lesson is a great way to generate ideas and excitement among your young mathematicians. Getting them to see the visual and creative aspects of math can go a long way in teaching the algorithms of multiplication and can create a lot of math-positive, math-rich dialog in the process.
In third grade, students begin to explore basic multiplication by utilizing strategies, algorithms, and appropriate methods of computation. As they progress beyond simpler whole number operations into this new realm of mathematical thinking (learning the multiplication tables through the twelves), art can lead the way and ease the transition into higher level reasoning and problem-solving.
Web Resources Artful Connections With Math: Multiplication Array Prints - In this lesson, students explore positive and negative space and create a stamp on top of a watercolor wash. They repeatedly print the stamp as an array representing a fact family of multiplication and division number sentences. The video below provides additional information about this lesson.
Multiplication City Art - See how one teachers helps her students learn array multiplication by creating city blocks where the buildings are created by differing arrays of windows.
Art Inspired Math - This brief article by Michael Naylor describes some activities that will get students looking for mathematics in artwork and also creating their own artwork to show off geometric ideas.
Origami Multiplication Flash Cards - If cootie catchers aren't your thing, you may want to make these basic origami shapes and encourage your students to do the same. These are found on Rick Morris's New Management website, and you can use them with his multiplication triangle tests (downloadable as a PDF file at the bottom).
String Art and Math: A Project in Multiplication - While I normally associate string art with bad 1970s home decor, I think this is a really interesting way to play artfully with multiplication. Instead of using nails to create the string art, students will have to be advanced enough to know how to use embroidery thread and needles to pull the thread through cardstock printed with the design template. This could be a good art/math center activity for those students with strong fine motor skills, and could produce very colorful work for display in the classroom. (The sample worksheet is downloadable as a PDF file at the bottom of the page.)
National Gallery of Art: NGA Kids - While this is just a wonderful arts site for kids to create their own artistic compositions, you can bookmark this at your classroom computer center and invite students to create online works that demonstrate multiplication concepts. Basic multiplication can be expressed through art through image repetition, and here kids can try all kinds of cool programs: BRUSHTER (an interactive painting machine); NGAkids Photo Op (an introduction to digital photography and image editing); the Collage Machine; Mobile Maker; FLOW (a remarkable interactive motion painting program); and Wallovers (which uses a 6 x 4 grid for students to create symmetrical compositions and can reinforce the interrelatedness between multiplication and addition). Students can create the multiplication masterpieces and then write out what kinds of images are multiplied in their work (and express them as equations).
This book contains 22 different math activities focused on geometry and measurement, though multiplication and division concepts can be found woven throughout the lessons. Fun and engaging, all of these activities promote mathematical thinking and creativity through problem solving.
This book shows you how to successfully integrate arts into any elementary math curriculum. Each of the math activities ultimately result in an aesthetically-pleasing project that reinforces a basic math concept. Inside are two great projects to help teach multiplication to third grade students: a project about multiples (p. 14 - 17), which demonstrates how to skip count and how to tell the difference between common and uncommon multiples, and another that shows how to represent multiplication facts as arrays (p. 26 - 27). All the lessons list teaching objectives, what materials are required, full project descriptions, and black line masters of reproducible assessment sheets (which require students to explain their mathematical thinking in writing).
This is such a brilliant idea - what a fantastic way to blend art, math, reading,and writing. Using comics to explore math encourages students to use their visual, verbal, spatial, and logic skills in interpreting situations. While most of the multiplication exercises in the book incorporate multi-step problems (some of which also ask children to take the next step into division), there is a nice section in the book with lots of fun activities for them to solve (p. 12 - 19). Each lesson features a comic at the top, which concludes with a "figure it out" question, and then the follow-up questions listed below take on a more-familiar worksheet format. There are "super challenges" at the bottom of each page to help students extend what they've learned and some encourage them to create their own pictures, diagrams, charts and models. Consider setting up a math/art center for students to make their own multiplication comic strips. You could use graph paper to help them better design the more traditional story-board layout, or take larger paper and fold it to create frames for students to draw in. Encouraging them to think about and ask multiplication problems through art and humor and then asking them to share them with classmates could be incredibly fun for everyone.
This book is full of good ways to connect math to simple art projects, and inside there are two very fun multiplication projects: multiplication constellations (p. 38 -41), and my favorite idea, multiplication houses to reinforce basic multiplication facts for numbers 1 to 12 (p. 40 - 41). Using the template, students can create their houses - or if you are more artistically inclined, you can encourage them to create their own houses (or skyscrapers, or other buildings). For example, if a student creates the house of nine, the number 9 would be written on the door. The numbers 1 to 12 would be written on the house's twelve windows. When you open the window to any given number, you see the product of nine and that number revealed. Creating these houses for kids to use as lesson guides would be very enjoyable, and if you made large ones for your classroom, you could create your own Multiplication Neighborhood that your students could look at for review.
Here is another great art-math resource offered by Scholastic, chock full of great, inexpensive, and simple projects that anyone can do. There are a number of very cute multiplication activities (p. 14 - 22): Be Mine! Multiplication (which you can see in color on the cover of the book), Multiplication Menageries, Fall Factor Trees, and Teeny-Tiny Times Table Books. The author makes it clear that educators should test out these projects before introducing them to the class - as any art teacher can tell you, figuring out how to do them on your own first can help you determine exactly how you need to model instruction, what steps are important to emphasize, and will help you figure out how complicated (and possibly time consuming) it is going to be. Many of these multiplication projects would be suitable to use at math centers.
That's it for day 8 of #MTBoSBlaugust. I hope you'll come back tomorrow to see what else I have to share.