## Friday, September 30, 2022

### Playful Math Carnival 159

Welcome one and all! Come on in and have a ball. 159 is semiprime and that's just fine.

Lucky that I'm hosting this, or is it just that 159 is lucky. How do you get lucky? Start with the counting numbers. Delete every 2nd number, leaving 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45... odd.  The 2nd number remaining is 3, so delete every 3rd number, leaving 1 3 7 9 13 15 19 21 25 27 31 33 37 39 43 45... now that's interesting in and of itself. Next delete every 7th number, leaving 1 3 7 9 13 15 21 25 27 31 33 37 43 45 ...; now delete every 9th number; etc.  How far do we have to go before we know 159 is lucky? Does knowing 151 is the previous lucky number help? Interesting to look at the gaps in each step, and the cutlist for each step.

Is it rarer to be a semiprime or a number with only odd digits? Odd increasing digits? Linear pattern in its digits?  Alyssa would like it as is.

Pat Bellew's 159 facts are that 159 is the sum of 3 consecutive prime numbers (which?) and can be written as the difference of two squares in two different ways (don't you want to find them?).

He also has that __ __ •159 = __ __ __ __ using all 9 nonzero digits. Of course, you can brute force it, but can you deduce this digitally complete product?

What #playfulmath have you seen this month? Here's some of what I have noticed.

September started with Math on a Stick in full swing. Doesn't get more playful than that!

Katie Steckles and Jimi went over the math in the Spider-Man No Way Home end credits. I lost my mind when watching it in the theatre, and am so glad someone's sharing it. SO MANY MC Escher references.

And as if the visuals weren't sufficient, the song is De La Soul's great cover of the Schoolhouse Rock classic, Three is a Magic Number.

 Live human scale Prime Climb at NCTM-LAphoto Liesl McConchie

Is that Howie Hua? (Yes - He and Annie Forest won.)

 as yet undiscovered unpentennium
Christine Thielen tweeted about her class' enjoyment of the Mathigon puzzle of the day.

Speaking of puzzle of the day, Michael Pershan wrote about this new Beast Academy (upper elementary and higher) daily arithmetic puzzle, Make Ten. I enjoy his PershMail newsletter each week.

The Erikson Institute is a great source for early math insights, and here they cover four playful number books.

Charlotte Sharpe shared a quick, rich early math game with dice and subitizing cards.

Michael Minas & helpers are back with an inequality game, Big Bad Wolf and the Three Little Pigs.

Australian Math Circles shared this online interactive math game with lots of nice number recognition and sense images.

Libo Valencia tweeted about his class playing this angle game, Daniel Mentrard's Polar Battleship

He also shared his daughters catmathart... a perfect transition to the next section.

Zarah Hussain shared her icosahedron statue on public display in London.

Paula Beardell Krieg is always busy with something creative and beautiful. For instance, her Rather Strange Solids. (But while you're there, poke around.)

Sophia Wood does programming, teaching and art. Her latest bird is perched on an unorientable branch...

Sam Hartburn sang to some Ayliean artwork for a recent Clopen Mic Night.

SimonLav with a Marvel-ous Desmos animation.

David Reimann nods to Magritte with this piece, related to his Bridges article.

Last but not least, I'm very happy to be a part of David Coffey's newest project: the Teaching Like Ted Lasso Podcast. Episode 1 is out, and it's on... PLAY! Check the show notes for scoonches of resources on play in math class.

As long as I'm on the pitch... just after this post on this blog are some very fun, well developed math games from my students.

And what's next? #Mathober! Sophia Wood has put together a list of prompts.

Each day there's a theme. Share a bit of math, a doodle, a comic, some art on the theme. Play along one day, or all 31. Tweet or send it to Sophia or myself and we'll share.

 Ferarri 159S

See you next month at Denise Gaskins' place, the founder of this here blog carnival. Info there on how to ask to host. I highly recommend it! So much playful math to celebrate. While you're there, check out her weekly Math Game Monday.

Vroom!

## Tuesday, September 6, 2022

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  . 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.

@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.

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.

### 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?

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.

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.