Tuesday, September 6, 2022

College Algebra: Quadratics

 I had my elementary ed class canceled for low enrollment this fall. Make of that what you will.

The replacement course is College Algebra. Ironically named, since it is mostly Algebra 2. Which is required in Michigan. Our sequence has been 097 (prealgebra) -> 110 intermediate algebra (algebra 1) -> 122 College Algebra.  It used to be + 123 (trigonometry) to go on to Calculus, but we have a nice precalc class now (124) so people needing to take calculus that don't place into it can just take 1 semester. The audience for 122 then, is now general education, and people who need courses that require it, like the basic chemistry, intro physics, and statistics. It's a 3 credit course, and my section meets twice a week.

The course has traditionally been quadratics -> polynomials -> rational functions -> exponentials -> logarithms -> light touch of statistics. So what do we want from the quadratics unit? This post is me trying to think out loud to get it straight for myself. The schedule is pretty packed, so I have 2-3 weeks per topic, 4-6 class periods.

The instructional sequence I have planned is visual patterns -> modeling (Penny Circle and Will It Hit the Hoop?) -> graphing/equation forms (Match My Parabola & Form Fix) -> solving equations (vertex form & graphing), mostly in a modeling context.

The visual patterns do a lot of work. They offer a hook, they give learners a chance to notice and wonder, they give us a chance to problem solve. They are also different from what most students have seen in algebra, sadly, so offer a way to let them know that this course might be different. I also have them read Elizabeth Statmore's post on math as a thinking class. I asked them, "What do you think the main idea is? How does this compare with your own ideas about learning math or your previous experiences?" and you can read their responses on this doc.  I think they get it. Mathematically, I think my main point is the use of variable as a relationship rather than an unknown. The transition from step number to x is very natural. Secondarily, they get to see multiple equivalent expressions. Which is one of those math ideas which many learners see as a bug, but mathematicians think is a central feature.  Part of the richness of these problems is what the old NCTM standards called the representation process standard. Tables, expressions, visual and the connections between them all move us forward. Here's a handout with four quadratic patterns. The bricks and the darts and kites are very difficult to visual make a symbolic rule for. I might have made them or might have found them at Fawn Nguyen's visualpatterns.org or it could be a mix.

Modeling is a key theme of the course, and Penny Circle and Will It the Hoop? are a good start to it. I was surprised how many learners went with an exponential form, and the reveal is the perfect way to settle it. We will be using Desmos activities a lot, and those are pretty slick introductions. The Penny Circle builds on the covariation use of variables, and the basketball leads into the graphing we'll be working on next. 

This is where we are as I write.

I'm convinced that one of the barriers for these students is understanding graphs. Thankfully, making them is easier than ever. But I don't think that many know how to think with them. Again with the representation standard, the connections between the symbolic expression and the graph is mostly taxonomical, and I want it to have meaning. Though this is a place where I could use some help. Regression supports this goal, as it brings tables into the web of connections. Activities where they vary parameteers and observe the effect on the graph help, at least in terms of taxonomy. Solving equations with graphs is an opportunity to build some of the understanding I want, as, especially for applications, the context is another piece of the representation. Writing this, I'm a little surprised by how hard it is for me to put my goal here into words. That would undoubtedly help with the teaching!

Solving equations is last for me, partly because it is so much what they perceived the focus to be in their previous math courses. I don't care especially for a lot of symbolic skill here. I don't teach solving by factoring, though the factored form in connection with graphs is something I emphasize. I do like the approach of solving from quadratic form, because it builds on a theme in math I love about doing and undoing. This leads better into exponentials and logarithms than it does polynomials and rational functions. The symbolic fluency that I want is being able to see a quadratic as series of steps. Take a number, subtract 2, square it, double it, add five is the same as  <math xmlns="http://www.w3.org/1998/Math/MathML" display="block" data-is-equatio="1" data-latex="2\left(x-2\right)^2+5=13"><mn>2</mn><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow><mn>2</mn></msup><mo>+</mo><mn>5</mn><mo>=</mo><mn>13</mn></math>. To find what numbers make 13 from that function, we can do by undoing.  I love Graspable Math for this, as the dragging to undo seems to really help get across the idea, though it doesn't work on the balance nature of equations. Here's an example GM activity with 3 quadratics to solve.

I'm very interested in your thoughts. What are the key ideas you want in a quadratics unit? What am I leaving out that you love? What understanding do you want your learners to develop or skills do you want them to have for graphing? Why?

P.S.

Probably violating some internet rule here, but really liking the Twitter discussion about this post.

@DavidKButlerUoA: This line was very interesting: "multiple equivalent expressions... is one of those math ideas which many learners see as a bug, but mathematicians think is a central feature". I'd love to hear more about that.

@joshuazucker: My interpretation is that beginners may want there to be only one answer and experts see how useful it is to have multiple representations that make different behaviors immediately visible.

@mathcurmudgeon: When 90% of calculus (and every math course, really) is rewriting expressions in an equivalent form that we can work with more easily.

@mathhombre: it starts with fractions. All these different ways to write the same thing. One of them must be right. (Often supported by teachers insisting on one.)  But that we can transform, rewrite and tinker leads to fluency, connections, and meaning.

@mathforge: The belief that out of all the ways of writing it there must be a RIGHT way is SUCH an interesting belief. I've never thought before that people might believe this.

I suspect that this is more prevelant than we might admit. As experienced mathematicians we might chuckle at people who think that there is a "best" way to write, say, a quadratic or a fraction. But we probably fall into the same trap with ideas.

I might, to take a random example, think that there is a "right" way to think about differentiation, or Pythagoras theorem, or a topology, or the category of smooth functions. What I mean is, "this is the way I find most intuitive".

===

@KarenCampe: Love the visual patterns start & modeling focus. 

When you do graphing equation forms & use match my graph/form fix you will surely cover symmetry of the graphs & how factoring gives x-ints. I like how graphing & alg manipulation of quadratics are interconnected...

Use graphing as tool to support any algebraic rearranging we might want. Look for hidden parabola that shows complex roots. Axis of symm hidden in quadratic formula.

===

[In response to "Quadratics unit in a college algebra course. What goes in, what's left out? "]

@theresawills: Probably too vague, but worth saying: RICH PROBLEM SOLVING.

 

Sunday, September 4, 2022

Fraction Reaction

Some years I'm fortunate to be able to lead a capstone seminar where future teachers research math games and develop a math game of their own.

Gretchen Zeuch developed Fraction Reaction to be a simple to learn, easy to play, fast game that works on fraction magnitude and mixed number fraction equivalence. 

She writes:

The process of making this game had many stages. The first stage was deciding what kind of materials I wanted to use in my game. I decided to use a standard deck of cards because I really wanted to make a game that was accessible to every classroom. I then had to pick the mathematical content I wanted my game to be based on. I started by just laying out all the cards in a standard deck and brainstorming different mathematical content. I finally landed on fractions because I liked the students being able to physically see it. I then decided that making the connection between improper fractions and mixed fractions would be the most helpful. I then went through a lot of trial and error by playing the game with a variety of people. This helped me decide how points would work, specialty cards, and general playing rules.

This game is great to teach in a classroom when students are learning about improper and mixed fractions. It is very easy to teach to students as well as all students will be able to play at the same time because of the accessibility of the materials. This game will help students make the connection between an improper fraction and a mixed number. They will also be able to compare the sizes of mixed numbers and improper fractions so identify which is larger and which is smaller. Overall, this game is simple to understand and helps to solidify students' understanding of improper fractions and mixed numbers.

There are a few different uses for this game in a classroom. The first use is that, while students play, you can have them record all of their improper fractions turned into mixed numbers and then have them sort them on a number line. Another use is for students to record their answers during the game and then answer some comparison questions at the end. Lastly, another in class use for this game is to have students discuss the differences between fractions and mixed numbers and how they relate to each other.


Rules - https://bit.ly/FractionReactionRules

In addition, Gretchen made a video to promote the integer game, Zero Rummy. She  writes: This is a great game to use with young children to get them working on their addition and subtraction or to help introduce the concept of negative numbers. This game should be used as a fun exercise rather than to teach a skill. The great thing about this game is that it is stimulating for children so that they are doing math without knowing they are. It is very easy to use in the classroom with minimal materials and does not take up a large chunk of time. Children really enjoy this game and it is a very easy game to play for many ages with multiple variations.


Rules: https://bit.ly/ZeroRummy-rules


polyGONE

Some years I'm fortunate to be able to lead a capstone seminar where future teachers research math games and develop a math game of their own.

Melanie Hanko came into seminar with a vision of making a math game inspired by Bohnanza, a collecting and trading game with a lot of strategy and a fair amount of luck. She really worked on the details for this game. Often times we focus on making games that use minimal materials, but this is much like a commercial game, with a lot of necessary components. For a teacher wanting to give it a try, I would love to see the learners get involved with making the cards. 

Melanie writes:

In the hopes of making an exceptional game, I set off looking for game structures that were simple but had a lot of potential. Then, interested in the structure of Bohnanza: The Bean Game, I started looking at mathematical content that involved some sort of sorting. Eventually, I landed with organizing shapes into hierarchies - specifically quadrilaterals. This is largely based on a 5th grade standard. polyGONE: The Shape Game is the sort of game that engages students with mathematical discourse and reasoning minus the negative attitudes about math. While players need to have a good base understanding of the hierarchy of quadrilaterals and the different types of triangles, this game will help players to create more connections between shapes and gain a broader understanding of what gives a shape its name. 

A lot of the pieces of the game are designed with specific purposes, either to clear up misunderstandings or confusion in early versions or to clear out some of the underlying confusion. The part of the game with the most meaningful design, is the deck of cards. These cards are created to broaden player’s understanding of shapes. Included in the cards are traditional and non-traditional shapes. Different cards show different attributes of a family, like parallel lines, congruent lines or angles, and even lines of symmetry. Different cards show different looking shapes - for example both a concave and a convex kite. This differentiation within the cards, will broaden player’s understanding of shapes and relationships between shape families.

Another purpose of the design of the cards is to increase their usefulness. With all of the cutting and printing, the cards better be usable for multiple occasions. Since there is so much differentiation between the cards, you can easily use them in a sorting or a matching activity. Even before playing the polyGONE, you could match cards based on if they have certain attributes. For example, matching cards that have two pairs of congruent sides. The cards can be used in explorations of the “rules” for each shape family. For example, deciding if a right angle is necessary for a trapezoid, or if it is something that only occurs in some trapezoids. These and other activities can be easily supported with these cards and will help to broaden students’ understanding of shapes and the shape hierarchy. 


The teachers also make a video for a good math game which they would like to promote. Melanie found one of Kent Haines' games that is a very good Nim variant. She writes: 

The 100 Game is a part of the math game genre of nim, which are mathematical strategy games in which players take turns removing objects from distinct piles or groups. Not only does the 100 Game require almost no materials and setup, but it is a fun game full of mathematical reasoning. In the forefront, the game makes practice subtracting within 100 enjoyable. Behind this practice, players strategize how to not be the last person to take away from the total. This requires deductive reasoning, an important mathematical skill. Besides the math, this game is quick to learn and engages players quickly - even unwilling players. 



Mathematical Applications: practice subtraction, strategy and deductive reasoning

Materials: paper and pencil, two players

Object of the Game: Players start at 100 and subtract any number 1-10 from the total. The goal is to NOT be the last person to subtract a number. So you want to subtract the second to last number from the total.

How to Play:

  • Player one will start the game by saying “100 minus [blank] equals [insert new total]. You can only subtract numbers from 1-10.
  • Then both players will write out the subtraction sentence player one just said out loud.
  • Now, it’s player two’s turn. This player will pick a new number to subtract, say the subtraction sentence, and both players will write down the sentence.

Example Play: Here is an example of what each player would say for a few turns. Remember that BOTH players are writing down the subtraction sentences as well.

  • Player One (P1): “100 minus 5 equals 95”
  • Player Two (P2): “95 minus 10 equals 85”
  • P1: “85 minus 7 equals 78”
  • P2: :78 minus 9 equals 69”
  • ...
  • P2: “23 minus 9 equals 14”
  • P1: “14 minus 3 equals 11”
  • P2: “11 minus 10 equals 1”
  • P1: “1 minus 1 equals 0”

In this game, player one lost because they were the last person to subtract a number from 100.

Notes: After you play this game a few times, you might start to develop a sure strategy. In fact there is something special about the number 12. Finding this strategy is what engages players in deductive reasoning. Some questions you might want to ask yourself or your students/children include the following:

  • What should your strategy be?
  • How can you ensure that you will win?
  • At what point in the game do you need to start using your strategy?
  • Does it matter who goes first?

Be sure to check Kent's blogpost for more ideas.

Sorry, It's Fractions

 Some years I'm fortunate to be able to lead a capstone seminar where future teachers research math games and develop a math game of their own.

This is Alaina Murphy's game, Sorry, It's Fractions. She was really persistent in the playtesting for this game, and did a lot of work to make it fun while keeping the math content front and center in a natural way.

She writes:

When coming up with this game, I knew I wanted to make a game that dealt with some aspect of fractions. In my opinion, fractions are one of the first aspects of math that students begin to lose interest, lack understanding, and start to hate this subject. So fractions it was. Next, I wanted the game to peak their interest, while having some mechanics that they might be familiar with. Thus, I chose to utilize a board game that many kids have played at some point in their life - SORRY. This would allow kids to focus more on learning the math of this game in comparison to first trying to figure out how to play the game. So, I had the content area and the mechanics. The next step was deciding how this was going to work. I wanted to make sure that thirds and fifths were included in this game because I believe these are the scary fractions to students. I find that students have an easier time with even numbers, but give them an odd denominator and they are out. The best denominator for including halves, thirds, fourths, and fifths was 60. So what better way to help students understand the numerical value of fractions and become more comfortable with them than using a clocklike numberline! 

The rest of designing this game involved play testing to decide how exactly I would apply the mechanics and actually designing the game board. The best way to get students to want to do the math and find the most reduced fraction was to make the fractions they landed on special, rather than the cards. I wanted to ensure that the materials of this board game would be resources a teacher could acquire. So, the board can be printed or they can have students make their own, place markers can be anything - sticky notes, erasers, beads, paper clips it doesn’t really matter - and I either wanted to use dice or playing cards to move around the board. By using a deck of playing cards, students would be able to draw larger numbers and make it further around the board to larger fractions, because the probability of getting a card with a higher value is higher than if they were to roll dice. Plus, the probability of getting any value is equivalent between cards where it is not when rolling dice. In order to make the game faster for classroom use, I incorporated four entrances to home that all players can enter and reduced the number of place markers to two, requiring only two pawns to make it home for the game to end. I incorporated a lot of DRAW AGAIN fractions as a way to make it further around the board and as a catch-up mechanic. Bumping, swapping and sorry’s are also catch-up mechanics and they make the game more competitive, creating more interaction and discussion. Lastly, I wanted to use the colors of SORRY, but I also wanted to create a board similar to Prime Climb where the colors have meaning. So based on the factors of 60 I wanted to color coordinate the prime denominators.

  • ½ is blue which is a primary color because 2 is a prime number.
  • Thirds are red which is a primary color because 3 is a prime number.
  • Fourths are a dark blue because 4 = 2 x 2 so it is the combination of two blues, producing a darker shade.
  • Fifths are yellow which is a primary color because 5 is a prime number.
  • Sixths are purple because 6= 2 x 3 so it is the combination of blue and red, producing purple.
  • Tenths are green because 10 = 2 x 5 so it is the combination of blue and yellow, producing green.
  • Twelfths are a dark purple because 12 = 6 x 2 = 3 x 2 x 2 so it is the combination of red and two blues or red and a dark blue, producing a dark purple.
  • Fifthteenths are orange because 15 = 3 x 5 so it is the combination of red and yellow, producing orange.
  • Twentieths are teal because 20 = 10 x 2 = 5 x 2 x 2 so it is the combination of yellow and two blues or green and blue, producing teal.
  • Thirtieths are gray because 30 has many factors so it is a combination of many colors but one less than 60 making it gray.
  • Sixtieths are black because 60 also has many factors so it is a combination of many colors and they are irreducible so I wanted it to be the same color as the outline. 

This is a great game for all types of learners to become more comfortable with fractions. Visual learners will be able to utilize the clock model and color scheme, hands on learners will be able to use the structure and game aspect, auditory learners will be able to use the discussions and verbal addition and reducing, and if teachers had students make their own boards it would be useful for those who learn from writing. 

This game is a great way to get students excited about adding and reducing fractions while becoming more familiar with factors of 60, exploring prime numbers, and ultimately improving their understanding of fractions. Other applications of this game would be to refine subtracting fractions skills by playing the game counter clockwise and subtracting the value of the drawn card, rather than adding. In order to incorporate more unlike denominators, the game board could be labeled in the most reduced form (i.e. rather than 30/60, label it as ½) and the students would add the cards in the same way. This board could be used at a younger age range to better understand adding or subtracting and number sense by labeling the board with whole numbers and playing in a similar way - this variation could be useful for learning to read a clock as well! Lastly, this game could be modified to the unit circle with pi/12 radians or 15 degrees and played with dice - here it would be beneficial for students to create their board as they go using trig to come up with the value of each position. 

Some problems that apply to this context:

  • Reduce 24/60
  • Reduce 13/60
  • Which fraction is closer to one, ⅔ or ⅗? 
  • If there are 60 people at a party and 12 are vegetarian and 4 have a nut allergy what fraction of people at the party have a dietary restriction.
  • If it takes me ⅚ of an hour to get ready for school and the bus leaves in 48 minutes, do I have time to make it to the bus if it takes me 1/15 of an hour to walk to the bus stop? If not, how much time do I have to get ready?
  • If I am 3 minutes away from the bus stop and it takes the bus 1/10 of an hour to get to my stop, and my sister walks 11 minutes home from school. Who will get home first? What fraction of an hour will it take each of us to get home?


Rules: https://bit.ly/SorryItsFractions-rules

Board: https://bit.ly/SorryItsFractions-board


The teachers also made a video for a math game they wished to promote. While there are other videos for games called Guess My Rule, Alaina wanted to share her own take. I heartily endorse this, and have used it myself from 2nd grade to university. She writes:

There are various reasons why Guess My Rule should be used in your classroom. First of all, this game requires little to no materials - no printing, cutting, or random pieces needed. As long as students have a way to record numbers they will be set. Games, such as this one, will get students thinking about math in a fun, hands-on way that encourages collaboration and critical thinking. With this version of the game, students are encouraged to explore functions and identify patterns that will allow them to predict outputs and eventually deduce a rule. This game will give students an opportunity to experiment with expressions, practice solving equations, and familiarize themselves with symbolic representations. 

If you are not convinced yet, there are so many ways that we can apply the framework of this game to learn and practice math!  If you plan to use this game in an algebra class you will not be wasting your time, because it can be applied to any algebraic function and even graphs. In geometry this game could be used for guessing what axiom a figure or statement applies to or for learning terminology by grouping correct shapes. It can also be used with younger kids to learn simpler arithmetic. Lastly, we can extend this problem to higher level learners and explore various rules at the same time, not limiting the rule keepers to linear functions but allowing them to pick from any range of functions. So why not use this game?

Standards: 

  • CCSS.MATH.CONTENT.8.F.B.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear).
  • CCSS.MATH.CONTENT.8.F.B.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change  and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
  • CCSS.MATH.CONTENT.8.F.A.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). 
  • CCSS.MATH.CONTENT.8.F.A.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.1



Rules:

  • Rule Keeper makes a rule
  • Rule Guessers take turns giving an input
  • Rule Keeper records input, calculates output (secretly), and records the output
  • Rule Guessers continue to one by one give inputs until they feel they have found the rule
  • ON THEIR TURN, Rule Guessers must say I would like to guess, then they must give an input AND predict the output of their given input.
  • Rule Keeper informs the guesser if the output is correct
  • If the output is CORRECT, the Rule Guesser guesses the rule
  • If the output is INCORRECT, the next Rule Guesser continues giving an input or they can choose to guess.
  • If the Rule Guesser successfully guesses the rule, they will become the next Rule Keeper and the current Rule Keeper becomes a Rule Guesser

Link to John's version of the game.







Binomial Battleship

Some years I'm fortunate to be able to lead a capstone seminar where future teachers research math games and develop a math game of their own.

One such is this high school algebra game from Lucas Pohl. He writes about this in what follows.

When thinking about creating a math game myself, I knew I had a couple goals in mind. We had done multiple readings about what makes a good classroom game, and obviously I wanted to fulfill those criteria. Things such as being engaging, strategic, and grounded in coursework were very important to me. I had two initial thoughts: at first, I wanted to do a game that is based on statistics. Statistics is one of my favorite areas of math, and I think that it could lead to a great board game. However, I ended up going to my second thought, which was an adaptation of Battleship.

The initial idea was that the coordinate system used in battleships reminded me of different methods I had seen to multiply polynomials together. In school I remember myself and classmates having trouble multiplying polynomials together, so I thought that would be a good context of the game. Luckily, making an adaptation of a game checks some game design criteria for you. Because of this, I felt like I could focus on the subject area of the game. After trial and error, I had figured out the best setup for the game. Each team gets two grids, an attack and defense grid. The attack grid had the binomials on the sides, and the attackers would have to calculate the trinomials to attack, however, the defense grid was completely filled out. The sequence and fluidity of gameplay was then discovered through playtester feedback.

I think that teachers should want their learners to play this game because it is very effective at its job. Even creating the game, I became much more efficient multiplying binomials together. There is very little to suggest that playing this game is off topic, or unuseful. The game essentially is essentially getting students to do homework level repetitions, but in a context that makes it more competitive and fun. Another reason for teachers to implement this game is the opportunity for variations, and classroom connections. I feel this game has great flexibility and potential to be implemented in not only a lesson plan, but even lecture, or assessment questions. For example, teachers could use this game to get into conversations about common factors, and factoring trinomials. The game could become more engaging by letting students choose their own binomials for the grid.

These are just a few examples of the advantages of implementing Binomial  Battleship into the classroom. The truth is, this game is very young, but the potential it has to advance student learning is very high.


Handout: https://bit.ly/BinomialBattleship-handout 
Game board: https://bit.ly/BinomialBattleship-board 


These teachers also make a video to promote an excellent math game they found. I couldn't agree more with this one, a classic from Joe Schwartz. I first saw it in this blogpost.

Lucas writes: The hundreds chart game is a great game for you to bring into your classroom for many reasons. I am going to give you three reasons why you should adopt this game into your classroom. First of all, it is incredibly engaging for students. This game will have students thinking of math in a more fun way, and they will likely find themselves enjoying math. Second, it encourages strategic thinking, and helps students develop that part of their brain. Developing this type of critical thinking will not only help them in your class, but all of their classes. Thirdly, it is incredibly easy to set up. There are almost no required materials for it. All you need is a 10x10 grid, and two different color pens. This game is the definition of minimal time and setup for the teacher, and maximum benefit for the students.




Saturday, April 30, 2022

Playful Math 155

 Welcome to the Playful Math Carnival, 155th edition!

155, tell us your secrets.


Via Pat Bellew, 155 is the sum of the prime numbers between its smallest and largest prime factors, 5 and 31. 5+7+11+13+17+19+23+29+31=155. How would you go about finding more of these? What would you call them? Pat also notes that 155 is the number of primitive permutation groups of order 81. Which is odd, because it is more than double the number of groups for any order less than 81. And there's not another larger (than 75 even!) until you get to order 256 (which has 244). Do 81 and 256 have anything in common?

Wait, 5 and 31? That means 155 is semiprime. What is the previous and what is the next semiprime? (They're both even...) Are there more primes or semiprimes smaller than 100?

The coolest thing I found is that 155 is a toothpick number. You start with a toothpick, then add a perpendicular toothpick anywhere there is an exposed endpoint. Here is 1, 3, 7, 11, 15, 23, 35, 43, 47, 55, 67. How many more steps to 155? Is it a fractal? Is it a cellular automaton? Mathematicians have also studied T(n)/n^2. Does it have a limit? Does it have an extremum? Here's some GeoGebra to make your own.


155 is also a generalized pentagonal number. The pentagonal numbers have a rule n(3n-1)/2, usually for n =1, 2, 3... , giving 1, 5, 12, 22, 35, ... But there are also positive outputs for negative integers, 2, 7, 15, 26, 40 ... which pleasantly fit between the usual pentagonal numbers. What patterns do you notice? Which negative number gives 155? I've been trying to think about how to visualize these negative pentagonals, to no avail so far. Have you got any ideas?


Maybe the toothpick was a little too crazy of a visual patten? Here's one I was trying to make to have 155. Did it work? If so, which step? Fawn always asks for the 43rd step... what's that? Is there a rule? What if step 1 had -1 square, what would the rule be?




I also found this pattern over at OEIS from Ilya Gutskovskiy. Which step is 155? How would you write the rule? What is a Fibonacci polynomial? From where did that question come?

On to the goodies...

Blogger of the Month
Jenna Laib is killing it. Creator of Slow Reveal Graphs, she has so much good writing on so many different topics, it is amazing. For example, THIS MONTH, planning (with a great pattern/multiplication activity), the Ramadan calendarlearning progressions,  mathematizing children's literature plus part 1 and part 2 examples. In addition, she edits the Illustrative Math blog, where she also sometimes writes gems like this on instructional routines Plus Slow Reveal Graphs, which just this week included How Loud is Too Loud?, Amazon Worker Injuries, and Australian Housing.



Elementary and Middle
Math for Love shared their 40 Face Puzzle. 100% will try, as I've loved the 100 Face activity, too.

Brian Bushart got playing Heads and Tails, a game/probability exploration.

Andrew Fenner made a hundred chart game in KnowledgeHook. (Free account but you have to log in to see it.)

Karen Campe wrote about special number pairs in math. The game I love adapting for these is Go Fish. For example, my preservice teachers were playing 1s Go Fish with some fraction cards they made with 4th and 5th graders. (2 cards each of: ½, ¼, ¾.⅓, ⅔, ⅙,⅚, 1/12, 5/12, 7/12, 11/12, one choice or can make two more different 1/2s, or a 0 and a 1.) I also made these fraction card blanks, but they might be more middle school...

Not this month, but there is a collection of tiny elementary math games here on this blog. Pointed for specific content, but low effort, low materials. As wih the fractions above, I love playing them with student made cards.

Wow. Rajeev Raizada made paper pool in Desmos!




High School and Beyond
Henri Picciotto shared a blogpost from Liz Caffrey using his Lab Gear for algebra. 

Deana Sample shared a fun bodyscale similar triangles activity.

Matt Enlow shared his progress on a crazy problem cutting up spheres to get different surface areas.

Also 3D, Sophia Wood shared her learners' work making nets for some interesting polyhedra in Polypad. (Which lets you fold them! Select all the tiles in the net, and a fold option appears. Select a polyhedron and an unfold option is there.)

Erin and Taylor, two of my seniors, put together a sweet 1 week graph theory unit for high school, which ends with a math game built on some pretty cool discrete ideas.

Mathigon shared their timeline scavenger hunt, using their excellent timeline of math and mathematicians.

Dave Richeson investigates Möbius strips with zippers with his learners.

James Propp applies proof by contradiction constructively in this month's post.

Math Art & Puzzles
Melynee Naegele sent the hexaflexagons from Sarah at Math Equals Love. These are always amazing! Sarah is also the queen of classroom puzzles, so check them out while you're over there.

Margie Pearse collected a bunch of math puzzles for May. (Gdoc)

Via James Propp and Daniel Kline, the Jumping Julia puzzle

Speaking of puzzles, Ms. Messineo sent Justin Aion's pride in solving Will M Dunn's puzzle. Feels like some kind of planar Ramsey Theory problem... Keep reading, the #mtbos discussion was pretty cool.

Patrick Vennebush wrote & joked about I Don't Know Puzzles.

Obviously I love using Polypad at Mathigon. Well they're having an art contest! For the under 18 crowd, but I'm planning to go gawk. HT Sophia.

Speaking of art, Paula Beardell Krieg sent Celeste Bancos' Origami Pockets post, which also had some great informal measurement investigation and what if thinking. Paula has been blowing me away with her #mathsartmonday tweets, like this one.

Lee Trent was playing with fractal cats. Fracatals? Not her first...

Speaking of tumblr, this poster described this viral video as stochastic continuous nim. Spot on.



Tik Tok?
Howie Hua is the king of math TikTok. Check out gems like his mixture puzzle.

The undisputed master of math tech, Steve Phelps is there.

Ms. Callahan is the funny math teacher.

Math Letters is shooting for a Math with Bad Drawings vibe for TikTok. 

But there must be more! Help us find them...

Off Ramp
Karen Campe reminded me to promote Ben Orlin's new math game book, the epitome of playful math. I am loving it. Somehow it's even better than I expected. Karen also pointed out a pretty sweet hexagon tessellation at La Guardia of all places, so you know she has an eye for fun.

The previous Playful Math Carnival was at Denise Gaskins' blog, the founder of the carnival. Be sure to check her site weekly for the Math Game Mondays which are only up one week! Other goodies, too, though. Next up is at Nature Study Australia.  Contact Denise if you're interested or willing to host. It really impresses me every time I do just how much good stuff is out there.







PS. I've been working all year with Xavier Golden (yes relation) a preservice art teacher on a math graphic novel. And we're starting to see some inked and colored pages... I'm so excited!







Friday, February 25, 2022

Early El Math Games

As my preservice teachers have had the opportunity to work with a K/1 classroom this year, I've been thinking a lot more about early math games. Mostly I'm trying to tie these to the components of number sense. 

Number Sense

In our class we discuss these as: 

  • one-to-one correspondence - as learners count, they have one (and only one!) number assigned to each object being counted.
  • hierarchical inclusion - (worst name candidate) the idea that a number contains smaller numbers. If you have 6 you also have 5, etc.
  • subitizing - visual recognition of quantities. Perceptual subitizing is immediate recognition of quantities, most commonly up to 5 or 6. Conceptual subitizing is visual chunking of a collection into smaller groups that can be perceptually subitized.
  • cardinality - the center and core. Recognition of numbers as quantities, a characteristic of a collection that doesn't change with rearrangement. Kids can have most of these other concepts but still not have assembled them into cardinality.
  • magnitude/comparison - both being able to directly compare quantities, and identify relative size - like locating where 7 is between 5 and 15.
If possible, my favorite thing for many of these games is for kids to have number cards which they have a hand in making. Similar to Tiny Polka Dot cards, which are a great commercial version. The idea is to make four suits, 0 or 1 to 10, where the suits are different representations of the numbers. Ten frames, symbols or shapes organized into patterns, randomly placed or groups of shapes to encourage subitizing, etc. You can have numerals or tally marks or number words if that's something you want your learners working on. I tend to prefer cards that involve counting and supportive structures. I used to have my own cards I'd print, but the opportunity for creativity, ownership and doing mathematics is strong with kids making the cards. (Not to mention some sneaky assessment.)






Once you have the cards, familiar games create terrific mathematical opportunities. Go Fish and Memory/Concentration create counting opportunities, and set up future games using those structures, like 10s or equation Go Fish or Concentration.


General Educational Game Advice
Many traditional games have a rule that when you're successful, you go again. I recommend against this because it increases wait time for other players, works against catch up, and can discourage the kids we want most to engage.

Similarly, I try to avoid games that emphasize speed, or require correctness to score and advance. I love for games to be an opportunity for collaboration and discussion, not a stand in for a quiz.

Divvy Up (Counting, Hierarchical Inclusion) Materials: Number Cards

Put about ten objects in the middle for each player. Using your number cards or dice, a player flips over a card and takes that many objects from the pile. Then counts up how many they have total. If appropriate, can have a score sheet where they write down that number. Game has two winners - one who takes the last object, one who has the most things.

Optional, arrange the 10 objects in two rows of five to sneak in some 5s structure and complements of 10.
Variation: if there are not enough to take, you have to pass. Encourages comparison, but can make the end take a while.

More or Less (Comparison, Strategy)
Materials: Number Cards

Idea: instead of War, which is not bad, in the math game sense, try this game. Draw 3 cards and teams take turns. The team whose turn it is chooses more or less. Both teams choose a card and hold it face down, then reveal. If more was chosen, the larger number wins, if less, the smaller. If it's a tie, you chose a 2nd card from your hand with the same rule.

More Together (Counting on, addition, hiearchical inclusion, decomposition)
Materials: Number cards mixed up in four piles.

Two teams: each turn over a card. Who has more? Then the teams turn over their 2nd card. Who has more together?

If learners are ready to count on, can just count from the first number. (6,5) Had 6, then 7, 8, 9, 10, 11 - pointing to pips on cards. If students would benefit from counting out blocks for how much (6 for this, 5 for that, count together), use blocks. Can introduce counting on here, too.

A tie? Flip over one more. No need for an overall winner, just who wins each turn.

Staircase  (Counting, counting on, hiearchical inclusion)

Materials: optional gameboard, a lot of stacking cubes and a die.


Play: roll a die, and build a stack of that many cubes, then roll another (or reroll) and add that many, with the two summands in different colors. Put them on your team’s track on the sum. If you already have that number, that’s okay, put it on the same space. Winner is the first to get three spaces in a row (make a staircase). Some students lay them down, some stand them up. Variation 1: If the three step game is too short, play to four or five steps. Variation 2: if you roll a sum you already have, you can choose to remove the same sum from your opponents’ board. (Increases interaction.) Variation 3: Playing with number cards 1-10. If you get a 1 or a 2 first card, you must take another. Otherwise it’s your choice. Bigger than 12 is a bust, you lose your turn. Probably best with a four or five step win condition, and can be combined with variation 2 as well. Lots of opportunity to notice and wonder. Notice the different ways to get the same sum, wonder how much you have together, notice that 2+5 is the same as 5+2, ask what you hope to get on that second die roll…


How many behind? (Decomposing, hiearchical inclusion, part part whole stories) Materials: 10 (or 12!) unifix cubes.
Show and count how many cubes in the stack. Now put the whole stack behind your back, and bring 1 cube out front. Ask: how many cubes behind my back? Next time, keep 1 behind your back, then show the rest. (If your partner’s there, have them go.) Learners and teachers take turns being the hider. If you want, you can always start with the same amount shown in front, or let people show a different number, then hide some behind. If the learners haven’t got the one less idea, try that one a few more times.

Big Three (Magnitude) Materials: deck of number cards. Idea: Players start with 3 face down cards. On your turn, draw a card from the deck or the top card of the discard pile. Replace one of your face down cards with it. No peeking! The goal is to find the biggest cards you can. The card you replace is then discarded, even if it was a high card. When someone thinks they have the biggest cards, they call “Last Turn” and everyone else takes one more turn. Players add up their cards to see who has the Big Three. Option: need more challenge? Play Big Four!
(Riff on Rat-a-Tat-Cat, a great commercial math game.)


Moving to Story & Operation
As kids have started to acquire number sense, we move into stories that provide the context for operations. The Cognitively Guided Instruction Framework, based on research analyzing how children acted out elemental math stories.
  • Join. One quantity, increasing over the story. Unknown could be the start, the change or the result.
  • Separate. One quantity, decreasing over the story. Unknown could be the start, the change or the result.
  • Comparison. Two quantities, related by the difference between them. Unknown could be the referent, the difference or the compared quantity.
  • Part Part Whole. Two quantities that are part of a group. Unknown could be either part or the whole.
  • Grouping. A number of groups, each group with a number of things, and a total. If the total is unknown, it's multiplication; if the number in the group is unknown, it's fair share/partative division; if the number of groups is unknown, it's measure/quotative division.

Comparison Game
Materials: number cards, especially if you have organized ones like dice face, hashmarks (if those are good for your kids), or ten frames. Plus 50-60 unifix cubes. Both players flip a card and build a stack that tall. Compare the stacks. Count the difference and take it off the taller stack. The player with more scores the difference. First player to 20 scored cubes wins. If it’s a tie, no score. Afterwards be sure to describe the score as 8 is 3 more than 5, or 5 is 3 less than 8. You could write down 5+3=8 (or 8-5=3 if they seem familiar with subtraction and super-comfortable with addition number sentence already.) Transition to them writing the number sentences and saying which is how many more than the other. If they are able to find the difference without counting blocks, make sure to have them describe their thinking. If they need challenge, don’t put the stacks together as they try to figure out how much more and less.

Making a Difference Materials: unifix cubes or counters about 30, number cards. Play: Both players have three cards. Choose a card to play. The lower card scores how many blocks it takes to make it equal to the other card - let the learners know that low cards are better.. If students can do with just numbers, that’s fine. But at least the first couple plays, build both numbers and count up how many cubes to make the difference. The person with the lower card scores those blocks. If it’s a tie, you have to play a second card from your hand. Draw back up to three cards. Winner is the first player or team to 12 cubes.

Facts
I feel like this is a place where games have made an inroad. But still, there's plenty of fun to be had.

10s Go Fish and Concentration Make 10
Pretty self explanatory. Remember to not let kids take extra turns. Both games I like to have kids score by counting their 10s.

Double Time (Doubles and counting on)
Materials: a game track, which can be numbered. 1 to 40 or 50 makes a good length with number cards, 30 is okay with dice. Bonus if you color or design the track in alternating spaces, to hint at the counting by 2s connection.

Play: students roll one die and move that plus the same. First to the finish line wins. I like to have students write down what they rolled and how far they went. 3+3=6, etc. If the track is numbered, you can start sneaking in some questions like 'Oh, you're on 24 and moving 8? Where will you end up?' For students working on counting on, this game provides lots of practice, since you don't start with 24, 1 is 25.

Ten Penny Game (Fives structure, sums to 10)
Have two ten frames out, the blocks, and some pennies or chips for scoring. Put a penny on the tenth spot of each. Players take turns rolling a die, and adding that many blocks to one of the ten frames. If they fill up the last spot, you get the penny as a point. Clear all the blocks and put on a new penny. There will be lots of opportunities for counting, counting on, and using the fives structure. "How many on this ten frame? How many more to fill it?" Are good questions here.

Cover All (Addition, decomposing)
This is the classic math game Shut the Box.

Cover All gameboard, but really all students need is a track from 1 to 10.
Play: roll two dice, and cover up any combination of numbers that add to the same amount.

With some kids, blocks help. If they set out how many they rolled, they can break them up in different ways. Consider questions to ask: what would be a good roll? What numbers might be harder to cover? What are different ways to split up our roll? (Helping them realize they have a choice.) What really makes this game a classic to me is that it really generates problems. Not how do you make 10, but how do you make 10 if I already used 7, 6 and 5. Is it even possible?

Dice Squares (adapted from Illustrative Math)
Materials: Gameboard, dice. This is a clever variation on dots and boxes. Roll two dice and fill in an edge next to that number. The player who puts the fourth edge on a box scores it! Mark with your symbol (X or O) or initials. 


Play with your students, thinking aloud at how you get your sums. For most of the kids, counting on would be a good strategy. 3 & 5, 5 -> 6,7,8. If students could benefit from using manipulatives to count, have them take as many as each roll, then find the total.


Make Your Own


Notice how simple some of these are? Really, some of these tiny math games are just born from thinking what do I want learners experiencing, and then adding dice or cards. Competition is fine - and a reason to engage for some learners, but try to avoid rewarding speed and correctness. Add in a representation (cards or the gameboard or a manipulative) and you probably have a classic in the making. (Then send it to me!) The easy wrinkle to add to the strategy and thinking required is to add choice. Much like More or Less above is basically War - with two layers of choice added in. Instead of flip a card, have a hand of two or three and choose one. Try to make choices real choices though. In More or Less, the choice of more or less makes the choice of the card much more significant.


Give Me More

Just two resources to end.

  • One of my favorite YouTube channels is Michael Minas, who makes up tiny math games with his kids and then demonstrates them. A lot of good games, but what's better is the spirit of invention.
  • Jenna Laib has a few easy, high leverage games. She writes about making games and then shares her favorites. We've used Number Boxes a lot this year, from 1st to 5th grade, just altering for what content the kids are thinking about. (Really, just read everything she writes.)

Just this week we were using ___ x ___ – ___ with a trash can ___ with 3rd graders. I wanted it not to be just who gets the biggest numbers, so added in the subtraction. I like having a trash can because it adds some choice, which gives even kids who have all their facts something to think about. There is so much thinking you can see and assessing you can do even just watching kids play these, and if you get to play with them... forget about it!


Game on!