Find the Factors (ax±b)(cx±d)

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I liked making a puzzle using trinomials earlier today. This one will take more skill to solve even though it contains fewer trinomials. Some of the factors will have negative numbers, and the leading coefficients of the trinomials are not 1.

In this puzzle, you can see the number 24 twice. It needs to be factored to solve the puzzle. It might be 3 × 8 or 4 × 6, but it can’t be 1 × 24 or 2 × 12 because for this puzzle ALL of the factors of 24 have to be non-zero integers from -10 to +10.

Every factor must appear once in the first column and once in the top row. So if you put 2x + 5 in the top row, you will also have to put 2x + 5 somewhere in the first column as well.

Sometimes all of the terms in the trinomial have a common factor and can, therefore, be factored further, but don’t worry about that right now.

You will have to find all of the factors in the puzzle before you can figure out what the missing clue should be. That’s about all the mystery I can put in a puzzle like this. Good luck with it!

Since this is different than any other puzzle I’ve ever published, you can see the solution here:

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Positive Trinomial Puzzle

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Today on Twitter, Mr. Allen requested some good problem-solving resources for quadratics. He made up one himself.

I decided to make one as well. It is similar to my other Find the Factors puzzles. You will have to use logic to solve it, but in many ways, it will be easier to solve than most of my regular puzzles. Like always, there is only one solution.

Every term is positive so if you already know how to factor trinomials it should be relatively easy to solve. All the factors from (x + 1) to (x + 9) need to appear exactly one time in both the first column and the top row of the puzzle.  Once all the factors are found, the puzzle is solved, but you can find all the products of those factors and write them in the body of the puzzle if you want.

What You Need to Know About the Multiplication Game

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I recently wrote about Hooda Math’s Multiplication Game. There’s a couple of things I didn’t tell you in that post.

First of all, you don’t have to use a computer to play the game. (However, using one the first time you play will help you understand how to play). You can print a game board to play. I’ve created a game board below that you could use. Each player can use different items such as beads, pennies, nickels, and dimes as markers to mark the factors used and to claim the resulting products on the game board.

The second thing you should know is that getting four squares in a row, horizontally, vertically or diagonally is NOT equally likely every place on the board. If one particular number is all you need to get a win, you are less likely to get that number if it only has one factor (like the numbers marked in yellow have). As far as this game is concerned, the products have the number of factors that I’ve indicated, even though in reality most of them have more than that.

You can’t win unless your opponent gives you one of the factors you need to claim that winning space. If 4 of the 9 possible factors will get it for you, the odds are much better your opponent will give you what you need than if only 1 of the 9 possible factors will do it.

If you know which numbers have four possible factors, you may have an advantage over someone who thinks this game is really just a variation of tic-tac-toe. Of course, those products with four factors could also make you more likely to get blocked as well! And if you use my colorful game board, your opponent will know just as much as you do about how many ways they can get each square.

Hooda Math’s Multiplication Game

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The school year is almost over, and class periods were only twenty-five minutes long today. I went online looking for math games that would benefit my students and I found a winner with Hooda Math’s Multiplication Game.

If you count the multiplication facts in a 9×9 multiplication table, you will see 99 facts, but many of the products are duplicated in the table. Every yellow square below is also in white elsewhere in the table:

There are actually only 36 unique products in the multiplication table above. Hooda Math has cleverly arranged those 36 products in a 6×6 grid that becomes the game board. In this two-person game, students take turns moving one of two arrows to a number from 1 to 9 at the bottom of the screen and claiming the square that contains the product of the numbers. The catch is that players must keep one of the numbers chosen by the previous player and cannot claim a product that has already been claimed by either player. (Player 1 cannot score on his first turn.) One student is green and the other is purple and the first to claim four squares in a row is the winner. The rules on the website are VERY short and simple.

Students played this game today. I played it as well. Sometimes I won, and sometimes I lost, but the losses are more interesting than the wins:

In one game, my opponent took the square that I needed to get four in a row vertically for the win. All she was trying to do was block me from winning, however, when she took that square, the game declared her the winner. We were puzzled why she was the winner until she figured out that making that move gave her four in a row diagonally. That’s when we found out players can win by getting four in a row diagonally as well as vertically or horizontally.

In another game, I had two possible moves that would have made me be the winner. I just needed my opponent to choose a 1, 6, or 8 as their other factor, and I would win with 1 × 1 = 1 or 6 × 8 = 48. Unfortunately, he knew to beware of the numbers that would make me win. One of the arrows was pointing to 5, and he made the other arrow point to 5. By now there were no other products left on the board that were divisible by 5, so I couldn’t win because I couldn’t move either of the arrows.

That’s how I didn’t win the game either of those times, but I had a lot of fun anyway, and you will, too!

 

 

127th Playful Math Education Blog Carnival

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Ladies and gentlemen welcome to the Playful Math Education Blog Carnival featuring the incredible number 127 of the famous Mersenne Prime family! Let’s give a big hand to. . . . .

2²-1 = 3, the smallest single Mersenne Prime;
2³-1 = 7, the smallest double Mersenne Prime;
2⁷-1 = 127, the smallest triple Mersenne Prime; and finally
39-digit 2¹²⁷-1, the smallest quadruple Mersenne Prime!

It took NINETEEN YEARS (1857 to 1876) for Édouard Lucas to test Mersenne Prime Number 2¹²⁷-1 BY HAND to successfully verify that it is indeed a prime number. It is the largest Mersenne Prime that has ever been verified by hand calculations!

Ladies and gentlemen, Prime number 127 has one more amazing feat up its sleeve that you will have to see to believe:

In this month’s blog carnival posts, the amazing Desmos will delight and astound young and old alike. There will even be an Easter egg hunt! The blog post links (in turquoise) are joined by several links from Twitter (in blue-violet) and a few from other places such as Youtube (in red). Stay as long as you like and ENJOY what the many carnival participates have to offer in 20 different amusement areas!

Art and Mathematics

  1. Stephanie showed off her colorful and impressive Tessellation Math Art Wall on twitter.
  2. David Petro used all 84 pieces of a 21st-century pattern block set to create a lovely symmetrical design.
  3. After this year’s very long winter, I’m especially glad Colleen Young collected some lovely and amazing springtime Desmos drawings and gifs in It’s Springtime. . . .
  4. Do you see mathematics everywhere? Continuous-Everywhere-but-Differentiable-Nowhere can and does, even on a student’s shirt. Read the story and see the t-shirt design replication in Desmos in Going Off the Beaten Path.
  5. Wanting to have your students create an art project in Desmos? 1ntegration-by-Parts has given that assignment many times and has links for student directions to help them meet your expectations in Desmos Art Project (Update).
  6. You must click on the turtle face link in Desmos Art! With just ONE equation, a magnificently detailed turtle face was produced in the Desmos calculator. I was so impressed. I tweaked that single equation by changing the number 16 to 7.29, added some color restrictions as well as equations to make a mouth and some eyes to produce my own Desmos Art piece I call Blue-eyed Beauty.

Classwork/Homework that is enjoyable

  1. Elementary-school-age students will enjoy Desmos’s Polygraph activity given to second graders that Matt Vaudrey shared.
  2. I-Speak-Math has a mathematics homework solution students LOVE. Read about it in Meaningful Homework and CPM.
  3. Jennifer Michaelailis has a pro tip on how to keep students who need a little extra help in class from feeling defeated.
  4. If you want a free math education gathering in your area, here’s how to get one started. Also, check out Denise Gaskins’s resources to keep the group going.

Creative Writing

  1. Many people have a personal story to tell that explains why they love mathematics. Through a Mathematician’s Eyes opens up and shares her experiences in My Story. What obstacles did she face? How did she feel about enjoying a subject so many others hated?
  2. Philip Jose Pacis played with some math vocabulary and wrote a poem he titled Fractions about a fractured relationship.
  3. Calendars and time are mathematical topics. How many other mathematical terms do you see in Maggie C.’s poem On Time? Do any of those words have more than one meaning?

Decimals

  1. Math Geek Mama has come up with Decimals on a Number Line Game and includes everything you need to teach this concept with complete confidence.
  2. Would you like to play Decimal Pickle? On Twitter, Mrs. Unger explained how to play it as well as a few of its variations.

Desmos First Aid Station?

  1. There’s no need to call the paramedics when it’s time to learn about parametric functions. Suzanne Van Oy has come to the rescue! She sees a lot of value in parametric functions and their graphs. Why is she so excited about them? What’s all the fuss? Suzanne answers those questions and more in Why Parametrics?.
  2. Suzanne Van Oy also recently put an incredible spinning Desmos Birthday Cake on twitter. How does she make Desmos gifs that don’t look like they need to be put on life support? Six months ago she blogged about how to do it in Making a Great Desmos Gif. She certainly knows what she’s doing!
  3. Sometimes Desmos doesn’t do what you expect. Your work might need some first aid. DesmosGraph (Unofficial)’s post: Desmos Traps: Why Is It Not Working may have the diagnosis and cure you seek.
  4. Although this post from Gold & Ratios is two years old, I still wanted it included on this list. It teaches how to add color and texture to Desmos.

Eggstraordinary Mathematical Easter Egg Hunt from Twitter

  1. Cliff Pickover shared Don M. Jacobs equation for an egg laying on its side.
  2. Tamás Görbe shared an equation for an egg that is standing up on its wide end.
  3. Get out your markers and start coloring Paula Beardell Krieg’s graphic of an egg and Three eggs! Paula also shows how she colored one of her eggs.
  4. Mathigon shared a gif of a colorful, nine-piece, tangram-like egg puzzle that can also transform into a rabbit and a goose.
  5. Robert Bosch shared a TSP art depiction of an egg and a chick that have a few mathematical traits in common.
  6. Ha! Doesn’t this always happen? I found more fabulous eggs the day after the hunt: Daniel Mentrard’s eggs made in Geogebra,
  7. And these Eggsponenential eggs created by Traci Jackson!

Exponents and Exponential Functions

  1. Christopher Danielson posted a question about exponents on Twitter that generated a lot of thinking from adults. I’m sure it would do the same for kids who understand a little bit about exponents.
  2. Jongarland6 was able to get ELL students conversing with each other in English about exponential functions. How it was done is described in Desmos Sparking Academic Conversations.

Fractions

  1. Math Play Day gives ideas from 20SomethingKids and 1KookyTeacher about playful ways first-grade kids learned about fractions.
  2. Mathgeek Mama published some adorable free Equivalent Fraction Robot Puzzles.
  3. A recently released YouTube video has a little girl teaching about fractions in Maths 4 Kids’ Fractions of Shapes and Fraction Vocabulary.
  4. Here’s another one featuring the same little girl: Fractions of Amounts Using the Bar Model
  5. Every carnival has food concessions. You can have a lesson at breakfast on fractions based on CTSPEDMATHDUDE’s post Sausage Fractions: Real Life Example.
  6. When teaching fraction division, should you start with rules or diagrams? Filling the Pail speaks from experience in Fraction Division and explains the advantages and disadvantages of both approaches.

Games and Educational Toys

  1. On the spur of the moment, I came up with a very quick Yahtzee variation that I played with one of my students. We counted the number of rolls it took us to get a Yahtzee. Lowest score wins. He beat me badly every round as the graphic above this category attests, but on a different day, Lady Luck was with me more than him.
  2. Denise Gaskin also has a tried and true Yahtzee game variation that she calls Six Hundred. You only need to print her directions and scoring sheets, provide six 6-sided die and pen or pencils, and you’ll be ready to make memories in more ways than one.
  3. Autism Awareness Week was earlier in April. In this post, Special Educational Resources Blog reviewed three games made by Orchard Toys: Money Match Cafe (teaches about money), Look and Find Jigsaws (teaches number and letter recognition), and Bus Stop (uses processes like 3-2+4-1 to figure out how many kids are on the bus when the bus arrives at the bus station.)
  4. The Mathematical Tourist shared how to play a game called Clobber. The game has been around since 2001, but the best strategy to win the game is still a mystery. I’m sure getting clobbered will be just as much fun as winning.

Geometry

  1. What did an insightful five-year-old tell Paula Beardell Krieg about triangles?
  2. Simon Gregg showed some pictures demonstrating how students had fun exploring squares in several different ways.
  3. Similar geometric shapes line up and beg you to compare their dilations in Paula Beardell Krieg’s post About Halfway There.
  4. Robert Loves Pi creates wonderful 3-D rotating digital geometric models. This one he calls the Twelfth Stellation of the Triakis Tetrahedron.
  5. Desmos is a great tool, but sometimes I need to actually see how people use it to teach. Bearsemath.com does exactly that by sharing some pictures of Desmos Geometry being presented to a class of 10th graders.

Giving Back

  1. When Women Inspire gives Three Worthy Reasons to Teach Charity to Your Kids. One of those reasons is that they will naturally learn the mathematics of money management.
  2. Read the impressive CBS News account of how a STEM Robotics team made a toddler wheelchair for a  two-year-old whose family couldn’t afford one.
  3. LMS Life Skills was practically speechless! Her class designed quilts blocks by using linear equations. Then the class made two quilts and donated them!

Linear Equations

  1. Wheeler’s Thoughts on Teaching used a bank balance problem to teach about solving a system of linear equations. The students were able to think about the problem and work on it with much fewer hints from their teacher. That makes teaching math much more fun for the students as well as the teacher.
  2. Jeff Lay created an Easter egg hunt activity to review linear equations, and he is happy to share the google docs he made with you.
  3. Ms. Wheeler exclaimed that sometimes crayons and paper do the trick while her glass created stain class art.
  4. Coincidentally, Ian Maclellan also had his class produce some stain glass art with linear equations.
  5. Alicia Phillips shared one of her student’s projects that used only linear equations and was made on Desmos.

Literature and Mathematics

  1. Imagine this carnival ride: a catapult that will send you flying through the air! Lana Pavlova and Meredith Wilkes have assisted Math Book Magic in creating the perfect design of an unforgettable carnival ride in Play with Your Math with Little Pea. How far will this catapult take you?
  2. Erikson Institute writes how Anno’s Flea Market by Mitsumasa Anno, Which Would You Rather Be? By Willaim Steig, and Whose Shoes? By Stephen R. Swinburne are Three Books That Encourage Simple Graph Explorations with Young Ones.
  3. Life Through a Mathematicians Eyes loves to curl up with a good book that features mathematics. Find out which books she has gathered and plans to read in her MathReadathon.
  4. Kelly Anne Garner received several must-have mathematics in literature book suggestions from Twitter to build a fabulous math library. Check out the whole thread.

Museum of Mathematics

  1. Chirag Mittal took charge of April’s birthday celebration of Leonard Euler. Did you know that Euler is credited with being the first to use letters from our alphabet and the Greek’s alphabet to represent some very important functions and numbers: Σ, f(x), e, i,  and π?
  2. Alan Paar of Established 1962 has put together a tour of Wendover School and the way teachers taught and students there learned several mathematical topics and other subjects from 1868 to 1930.
  3. Jo Morgan retweeted a tweet that caught my eye and was, therefore, the catalyst for bringing  MathigonOrg’s expansive and interactiveTimeline of Mathematics to this month’s carnival.

Number Theory

  1. The number 127 is a centered hexagonal number as demonstrated by the graphic above.
  2. Ramblings of a Writer recently marveled about how many things come in fours in her post Exploring the Number Four.
  3. Dr. Helen J. Williams has pictures from a very playful session on “Fiveness”.
  4. Science Switch had a few things to write about Belphegor’s Prime number, 1000000000000066600000000000001, in The Most Evil Number.

Optical Illusions

  1. When there was a day off from school in the middle of the week, BMore Energy found plenty of kid’s activities in Manhattan’s Museum of Illusions.
  2. Love Travelling takes us on a trip to see the fun-filled Vilnius Museum of Illusions. There is so much to see there!
  3. While Matematickcom shows how to make a paper optical illusion that you can make yourself in very little time.

Puzzles


  1. Math with P. Nik gives instructions and several examples of his Three Elastic Bands puzzles. He said they were easy to make, so I made the one at the top of this category. Follow P. Nik’s instructions and you probably won’t need to click on the tiny answer key under the puzzle.
  2. When Simona Prilogan of Fiat Lux writes a number puzzle, it is much more than it appears.  You have to study patterns inside the puzzles to figure out what the relationship really is. Give her Wednesday Math Puzzle a try!
  3. Simona included a bonus, information about Bolsover Castle, in her Monday Math Puzzle. You will find two different Math Puzzles in the middle of reading about the Castle!
  4. This clever tie matching exercise from Math with P. Nik feels more like a puzzle than a worksheet. Can you match the graph families with the correct equation families?

Statistics and Probability

  1. Yes, you can do statistics in Desmos! You can make Normal, Poisson and Binomial distributions and even graph box and whisker plots in Desmos! Colleen Young shows you what that looks like in her post Statistics with Desmos.
  2. Does El Niño play much of a part in rising global temperatures? In New Kid in Town, Open Mind answers that question and includes line graphs to help us visualize global temperature data collected since 1979.
  3. This year Easter occurred on April 21st. That seemed rather late to me, but it isn’t the lastest it could be. In Joseph Nebus’s post, What Dates Are Most Likely for Easter?, he’ll direct you to a post he wrote two years ago where all the data is lined up to figure out the probability.

Telling Time

  1. What time is it? There is more than one valid way to give the correct time, and one way should not be labeled as a smarter way to give the time than the others. That’s the message given in Dan Meyer’s Don’t Teach Math “the Smart Way”. He even suggests a lovely game from Desmos to get kids talking about telling time.
  2. After a long winter with snow causing several school days to be replaced with “e-learning days,” Educational Technology in Action wrote about using that same Desmos talking time activity in Desmos for meaningful e-learning days.

Vocabulary

  1. On Twitter, Jo Morgan shared a photo that truly enhanced the 1679 definition of a Rhombus.
  2. Joseph Nebus of NebusResearch regularly writes about mathematics-themed comics. Here is a comic about the difference in definitions of vertex and apex. It also has a graph theory puzzle and three other comics about story problems involving addition and subtraction, slope intercept form, and paradoxes.
  3. What does the word Asymptotic mean? Hazel Clementine shared a catchy musical definition.

Thanks for coming to this month’s carnival! I hope you enjoyed it. I had a wonderful time hunting for goodies to put in the carnival and organizing it. I felt like I was on an Easter egg hunt looking for the best eggs!

Math Misery? will host May’s Playful Math Education Blog Carnival. Perhaps YOU will consider contacting Denise Gaskins and volunteering to host a future carnival! There are two open dates in the summer still available this year.

You can also visit The 126th Playful Math Education Blog carnival hosted by Math Mama Writes. . . or the 157th Carnival of Mathematics hosted by Lines Curves Spirals for more mathematical adventures!

Easter Basket Challenge

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Occasionally,  we hear that the number of Easter eggs that are found is one or two less than the number of eggs that were hidden. Still most of the time, all the eggs and candies do get found. You really have no trouble finding all those goodies, and the Easter Egg Hunt seems like it is over in seconds.  You can find Easter Eggs but can you find factors? Here’s an Easter Basket Find the Factors 1 – 10 Challenge Puzzle for you. I guarantee it won’t be done in seconds. Can you find all the factors? I dare you to try!

Let’s Get Ready for the Playful Math Carnival!

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Too many people think that mathematics is a house of horrors, but there are plenty of bloggers out there, who know that done right, math is actually ALL fun and games. It is like a carnival! Every month, you can play at the Playful Math Education Blog Carnival, and it really is play! What does a playful math carnival look like? Go on over to see how Math Mama Writes… and puts on a fabulous March carnival!

I will be hosting this monthly carnival the last week of April! Why do I get to host it? I sent a message on twitter to Denise Gaskins who coordinates the carnival, and I requested the privilege. If you would like to host it in the future, let her know. She is always looking for blogs to host, and she will be very happy to hear from you.

In the meantime, you can help me with my carnival. If you blog about mathematics in a playful way that could benefit children who are somewhere between preschool to high school age, I would love to include your post in my carnival. The carnival is a FREE way to promote your post, so if you would like more traffic to your blog, submit a post using the link from Denise Gaskins’ website by Friday, April 19. Then before the end of the month, you will be able to enjoy the carnival even more because of your participation!

Yahtzee How-Many-Rolls Variation

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Today on the spot I made up a quick variation of Yahtzee, and one of my students played it with me.

The object of the game was to get all five dice to show the same number of dots at the same time, but instead of only being allowed to have up to three rolls, we took as many rolls as need. To take a turn, one of us would roll the dice then look to see if any of the dice were the same. Any die that didn’t match would be included in an additional roll until it did match. We counted each roll we took and got one point for each roll. The lowest score would determine the winner. The student and I played four rounds. He was elated because he won EVERY round so, of course, he was the overall winner, too.

Usually, when we play a game together the scores are much closer. Sometimes I win, sometimes he wins. Today I couldn’t believe my bad luck! Sometimes none of the dice matched after my first roll. And what about my student’s very good luck getting five of a kind in just one roll? I’m sure some good probability discussions could result from this game.

Our data might suggest that 9 rolls is the most that a person could get, but I rolled the dice for the picture included in this post, and it took me 19 rolls to get that Yahtzee! And I actually had four 4’s after just 6 rolls before I took those last 13 rolls.

You never know for sure what will happen when it comes to games of chance. If you study probability, you can have a good idea about what is most likely to happen, but you cannot guarantee it will happen. If we had taken the time to play more rounds, maybe my student would have needed 10 or more rolls to get at least one of his Yahtzees, and the game would have been more competitive. (At least, that was what I was thinking before he rolled on rounds 3 and 4.)

I’d like to encourage you to try playing this game, too. I thought it was a lot of fun even though I lost miserably.

 

Today is a Good Day to Review Proof by Induction

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0² = 0
1²  = 1

Does that pattern hold for all natural numbers? Could we claim that n²  = n?

Yes, we can, and I’ve written a proof to prove it! The proof uses a valuable concept in mathematics called induction. I remember being introduced to proofs by induction when I was in Junior High. Nowadays, if it is not part of Common Core, it wouldn’t be taught much anymore. Nevertheless, I will use it here to prove that n² = n.

Using a similar proof, we can also prove that n³ = n, n⁴ = n, n⁵ = n, n⁶ = n, and so forth!

Today is the perfect day to review how to use proof by induction so try your hand at proving at least one of those mathematical statements on your own. Use the same steps in my example: prove true for n=1, assume true for n = k, prove true for k + 1, write your conclusion. then have a very Happy April Fools’ Day, Everyone!

Today is also a very good day to review that (x + y)² = x² +2xy + y²  and NOT x²  + y², a very common error students make. Confession: I remember making that exact error in high school when I definitely should have known better. Using induction to prove something in mathematics is a valid technique, but if you use invalid equations like
(x + y)³ = x³ + y³, you will make invalid conclusions. Thus, today might also be a good day to review the binomial theorem and Pascal’s triangle. (Pascal’s triangle has numbers in its interior, not just 1’s going down the sides, after all.)

My post today was inspired by a post written by Sara Van Der Werf titled Why I’ve Started Teaching the FOIL Method Again. In her post, she not only plays a great April Fools’ joke on her readers, but she explains a tried and true way to multiply binomials and other polynomials.

I read her post exactly one year ago today, and since then, I have been waiting for April Fools’ Day to roll around again so that I could share this post with you. It is my hope that you will enjoy my little prank and learn a little mathematics from it as well.

Detail Left Out of the History Books

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Today I was indexing some July 1944 death records from Budapest, Hungary and noticed that Boldizsár Klein and his wife, Regina Leichtmann, died only one day apart from each other. We don’t index causes of death, but I looked at their causes of death because their deaths were so close to each other. The same word was used for both causes of death. I wasn’t sure of all the letters in the word, but it started the same as a word I had seen before, öngyilkos, which literally means self-murder.

First I consulted my hardback Hungarian dictionary, but I didn’t find the word. Next, I looked at two online Hungarian genealogy dictionaries. Finally, I typed what the letters most looked like to me into Google Translate. After a few trials and errors with different letters of the alphabet with and without the prefix, ön, I found the word and their cause of death, önmérgezés, which means self-poisoning or intoxication.

Why did this happen to them?!!

From the record, I knew that both 74-year-old Boldizsár and 66-year-old Regina were Jewish. I googled and learned that the Nazis invaded its previous ally, Hungary, only a few months earlier on 19 March 1944 and mass evacuation of Jews to death camps began immediately. Since this couple lived in Budapest, the horrors of this occupation must have been felt most intensely. I cannot imagine what they went through, but trying to put ourselves in their shoes may help prevent history from repeating itself.