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Math Circles at Home: Adding Boxes

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Suppose you place two numbers in a row of boxes like so:

Then we generate the next number by adding those first two boxes together.

We continue in this fashion to complete the row:

 

 

On the other hand, suppose I give you this:

What numbers should be in the boxes between the 5 and the 16?

Some follow up questions:

  1. A simpler case: If there are only 3 boxes, and I give you the first and the last boxes, what goes in the middle?
  2. What happens if I switch the first and last number?
  3. Can you give a general rule for finding the numbers between the first and last box when there are 5 boxes?
  4. What if you had 4 boxes? Or 6? Or n? Can you come up with a general rule for finding the numbers in between when you have any number of boxes?
  5. If the first and last numbers are whole numbers, will the numbers in between be whole numbers? If not, under what conditions will those numbers be whole numbers?
  6. How can you change the rules? What questions could you ask then?

Box on!

Summer Opportunities & How to Multiply 9 Digit Numbers in Your Head

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Like MC2, a number of summer programs have now committed to running their camps online:

1. MC2’s Camps

Our lottery runs on Friday, 5/15. Camps will run the weeks of 7/13 and 7/20, serving rising 6th to 13th graders.

2. The Stanford Summer STEM Institute

This camp runs from June 21st to August 1st. It consists of a research and data science bootcamp, a masterclass lecture series, and  a guided research project. Rising 9th to 12th graders. Admission is on a rolling basis, so apply ASAP.

3. Wolfram High School Summer Camp

Wolfram has reached out to MC2, and they are looking to recruit more female students. The camp runs from July 5th to July 18th, and participating students will get an opportunity to learn the Wolfram programming language, engage in special computing topics like natural language and machine learning, while also having individual project time.

4. MyChiMyFuture

Mayor Lightfoot will be launching the MyChiMyFuture website on May 11th. You’ll find links to summer programs offered by Chicago out-of-school providers. This site will be a clearinghouse for the summer and beyond, from camps to online challenges to meet-ups!

 

Now, how do you multiply 9 digit numbers in your head?

First, and yes, this is a trick, one of the numbers has to be 142,857,143. But the other one can be anything you’d like! Here’s an example:

142,857,143 x 358,246,974

Steps below will be written out, but with practice you can do them all in your head.

Step 1: Write the chosen number twice: 358,246,974,358,246,974

Step 2: Take the number from step 1 and divide it by 7, moving left to right. That’s it! In this case:

35/7 = 5

8/7 = 1 with remainder 1, which you connect to the 2, the next digit
12/7 = 1, remainder 5, which you connect to 4, the next digit

54/7 =7, remainder 5

56/7 = 8

9/7 = 1, remainder 2, etc.

etc.

So, 142,857,143 x 358,246,974 =

51,178,139,194,035,282

Works every time!

 

Math Circles at Home: The Opposite of Times Tables

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When my children were in primary school, I found that multiplication times table tests were far less common than when I was their age. At least, teachers were less likely to give them. Other parents, however, worried that their kids wouldn’t get enough ‘practice’ and wouldn’t be ‘fluent’ in single digit multiplication. So they’d time their children completing times tables at home.

This, to me, encouraged flippancy, not fluency. In the short term, flippant memorization of multiplication facts might work, but in the long term it’s a disaster.

There is a place for (fluent) mastery of multiplication facts. So, if not times tables, what then?

Do math with your children instead. Give an actual problem to be solved, where the practice of ‘multiplication facts is built in. Try this:

Numbers can be partitioned in many different ways:

7 = 3 + 4 = 1 + 1 + 5 = 2 + 2 + 2 + 1 = 1 + 1 + 1 + 1+ 1 +1 + 1 = etc.

For a given number, take the numbers in one of its partitions and multiply them. Out of all the possible partitions of that number, what’s the biggest product you can get? Can you come up with a rule that guarantees the largest product?

In this investigation, you’ll find you have to do a lot of multiplication, more than if you completed a times table. But now you will multiply with a goal in mind. You’ll need to organize your work, make conjectures, consider simpler cases, connections between cases, and, most importantly, you will be doing this work with a purpose.

And all those things are sorely missing when you do times tables.

Math Circles at Home: Trapezoids, Hexarights, and Reptiles

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With school back in session (online), here’s a simple math circle activity to try at home when you need a break. 🙂

  • A trapezoid is a quadrilateral with at least one pair of parallel sides.
  • A hexaright is a hexagon with all sides at right angles.
  • A reptile is a polygon that can be divided into four congruent pieces, all similar to the original.

This figure at right is a trapezoidal reptile; note that the horizontal sides of the trapezoid–both the ‘big’ original along with the four ‘little’ copies–are in a 2:1 ratio.

 

Show that the trapezoids and hexarights below are also reptiles. From there, try to make up a new reptile of your own!

Math Circles at Home: Two Tricks to Teach Your Parents

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I used to be hesitant in building a math circle session around tricks, but I’ve come to appreciate their place. On MC2 Parent Surveys, we often get responses like this: “I can see my child’s curiosity about math is sparked–she tells me about what she did afterwards and has to tried to have me play along with games she learned.” There’s a place for puzzles, tricks, and games that students can share with a parent or sibling.

For this ‘At Home Math Circle’, I encourage the kid to read the directions and then try it out with a parent! Note: You’ll need a standard deck of cards for each trick, and a standard die for the 2nd.

1. Piles to King

Note: It’s essential that your deck of cards be a complete deck with 52 cards–no cards missing, no jokers!

Step 1: Deal a card face up. Say it’s a 6. Continue to deal the cards face up on top of that 6, and in your head count up to king. Count in your head–so you’d deal the 6, then count 7,8, 9, 10, jack, queen, king to make a pile (to be clear, the value of the cards will likely not be be 7, 8, 9, etc.; it’s just a way to keep track of how many cards should be in your pile). Note that if the pile starts with 6, it will have 8 cards. If you start with a 3, you’ll get a stack with 11 cards; a stack that starts with a king will only have that one card!

When your count gets to king, that pile is done. Keep making other piles in the same manner. Make 4 or 5 piles.

Step 2: Turn all of the piles upside down. Ask the subject/parent/victim of your trick to hand you all but three piles. Take the cards that you are handed and put them with the unused remainder of the deck.

Step 3: Have the subject turn over the top card of two of the remaining piles. Take the deck and count out the number of cards corresponding to the cards that are revealed. If you see a 3 and a queen, count out 3 cards, then count out 12 cards (a queen is a 12, a king is 13, and a jack is 11; aces are 1’s). Important: After that, deal out 10 more cards.

Step 4: Count the remaining cards in your hand. Suppose there are six; announce that the remaining card on top of the third pile is a six (if there are 11, you’d say jack). Then turn the top card over on the final pile, to the amazement of all (assuming you’ve done your arithmetic right!)

Tips:

  • Do all of the counting in your head. This will make the trick more mysterious.
  • You can make as many piles as you want, just don’t run out of cards. For example, if you are close to running out, and you deal out an ace, that pile would have 13 cards–if you run out, the trick won’t work if you make that pile!

2. The 3 and a Half of Clubs

Note: You can buy a three and a half of clubs card online, but, at the moment that may be hard to do. Alternatively (ask your parents if this is ok), draw on the three of clubs and make it a three and a half. 🙂

Step 1: Put the three and a half of clubs 9th from the top of the deck. Shuffle the deck, but don’t disturb the 9 cards at the top!

Step 2: Invite the subject to deal out 20 cards, face down, one on top of the other.

Step 3: Have the subject then remove from 1 to 9 cards–their choice–from the top of this new stack. (They shouldn’t tell you the number they chose).

Step 4: Have the subject figure out how many cards remain in the pile. This is a two digit number. Have them add those digits together, and then remove that many more cards. (If there were 14 cards left in the pile, they would remove 1+4=5 cards).

Step 5: Have the subject roll the die but not show you the result. Tell them that the average of the top and bottom of the die will be the same number as the card at the top of the stack.

If they laugh at you because the average is 3 and a half, great, because you know what’s going to happen!

Extensions:

  • Try to think about why each trick works.
  • For Piles to King, how would you have to adjust the trick if there were four piles and the top card were revealed for three of them?

 

The Two Most Important Things to Know About Online Learning

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Twitter is the place where people go to complain. I saw a tweet where someone shared a teacher’s rules for her online classroom. The tweet disparaged the teacher’s insistence that students not eat during class.

The problem with most of the discourse about online learning is no different than the problems we had a month ago when online learning wasn’t the focus of everyone’s attention. And that problem is that without a sense of focused principles, all we have is a lot of complaining without a sense of what is important. It might sound silly to restrict eating in class when the child in question is sitting in their own kitchen, and I’d agree; nevertheless, it’s far more important that we turn our attention to what does matter.

1. The Task

Alfie Kohn is best known for his book Punished by Rewards, but another favorite of mine is Beyond Discipline. His main point–there are hundreds of books about controlling children in the classroom, and they almost never have anything to say about what children are being asked to do.

Current discourse about online learning often focus on the technology, not the task. If we don’t ask children to engage in worthwhile learning tasks, the whole enterprise is misguided. We can list rules, teach kids protocols for online hand raising, and introduce them to Zoom; but if we don’t have a great task, all they will learn are rules, protocols, and how to use Zoom.

2. The Community

Yesterday I taught my first online math circle, and my main take-away is that recreating the kind of community you build in an in person classroom is challenging! A strong classroom community makes it possible for students to feel comfortable sharing their mistakes–and thinking about your thinking is where learning happens. That comfort gives teachers the chance to do formative assessment effectively, which is the most important feature of a powerful classroom. (Formative assessment–collecting information about what kids do and don’t understand and acting on that information.)

As you evaluate the learning experiences you or your children having, start by asking yourself, what’s the task, and what opportunities are we given to make personal connections to support learning? Teaching decisions follow from there.

 

In addition to our online math circles, I’m also making a weekly post with a math circle task that can be done at home. Check out these posts for what I believe are some worthwhile tasks!

1: 1D Tic Tac Toe

2: Knight Moves

3: Domino Variations–Get It Covered

Math Circles at Home 3: Domino Variations–Get it Covered

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Here are two classic questions about dominoes, with variations for each.

1. The classic: Imagine a standard 8×8 chess board. Remove the opposite corners of the board; now there are 62 spaces. Can you cover the board with 31 dominoes, if each domino covers exactly two spaces?

Variations:

  • What is the removed corners are adjacent?
  • What if you remove 4 spaces at each corner of the board?
  • Go back to a full chess board; it can be easily covered with 32 dominoes. When I did this, I found that 16 were placed vertically, and 16 horizontally. Can you place them in any other combination?

2. The classic: Imagine a strip 8 squares long, covered with dominoes and monominoes (monominos are also called ‘squares’).

In this figure, the strip is covered by two brown dominoes and four green monominoes. You could, of course, just use four dominoes, or 1 domino and 6 monominoes, etc. In how many ways can this you cover the strip using only these figures?

Variations:

  • What if the strip were 100 squares long?
  • What if you had brown and blue dominoes, and yellow and green monominoes?
  • What if you had monominoes, dominoes, and triominoes (formed by connecting three squares in row)?
  • What if these figures were made by linking together equilateral triangles rather than squares?

The variations are endless!

PS. Online math circles start next week. Learn more here!

And in case you missed it:

Math Circles at Home Take 1: 1d Tic-Tac-Toe

Math Circles at Home Take 2: Knight Moves

 

Math Circles at Home: Knight Moves

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Idea #3: Knights of all Kinds

The initial puzzle: The most interesting chess piece is the knight because of how it moves. A knight moves along an L-shape: two spaces up and one space over. Or one space up, and two spaces over. For visual learners and those who like overkill–here’s a two minute and 14 second video of someone moving a knight on a chessboard.

If a single knight is on an infinite chessboard, can it reach every space (eventually!)? How do you know? Can you use the rotational symmetry of the grid to make a concise argument that all of the spaces can be reached? [Suppose you can show you can reach the space directly to the left of the starting square, and the space above that; I claim that’s all you need to show. Why?!]

 

After you explore this initial question, extend!

Chess has a problem, and it’s a problem that’s 100’s or 1,000’s of years old–the rules never change (I imagine serious chess players would disagree. If that’s you, write your own blog). Thankfully, this is a (virtual) math circle, so we will immediately change the rules.

  • Think of the traditional chess knight as a (1,2) knight. A (1,3) knight is able to move one up and three over (or vice versa). Can a (1,3) knight reach every space on an infinite chess board? If not, which spaces can it reach?
  • What about the (1,4), (1,5), and (1,6) knights?
  • What about (2,5)? (4,8)? (3,9)? (7,11)? Explore.

Generalize and Extend More

  • Try to answer the question in general–how can you tell whether an (m,n) knight can reach every square on an infinite board, just by look at m and n?
  • What other chess pieces that don’t exist can you imagine, and what questions can you ask about them?

 

Idea #4: Math and Music

The connections between mathematics and music are at the heart of the newly released WFMT Explainer Series. The series features Eugenia Cheng, Scientist in Residence at the Art Institute, and leader of the Math Circles of Chicago’s summer experience for high school students. These videos are short, informative, and of interest to anyone who enjoys math or music. Highly recommended!

For more ideas, see Math Circles at Home: Take 1.

 

 

Math Circles at Home: Take 1

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I know parents and caregivers are struggling to find activities for their children to engage in as we wait to see when school will relaunch. I thought MC2 could provide some ‘curated’ suggestions for giving your child a math circle like activity that you can try at home. Rather than providing a laundry list of links, I feel concise and specific suggestions might be most helpful. So here’s a list…of two ideas:

Idea #1 1d Tic-tac-toe

The words in italics describe the progression of a typical math circle. Initially, engage in a game that is accessible. Next, try to ‘solve’ the game, describing a strategy that works for the initial version of the game. Then generalize and create!

The initial rules of the game: Draw seven blank spaces in a row. Take turns putting x’s (and only x’s; no o’s!) in open spaces. The person who gets three or more x’s in a row wins. It’s important to understand that no one ‘owns’ an x–what matters is who was the person who added that last x that led to at least three x’s being consecutive.

Play a few times and start to ask questions:

  • Are there particular moves for player 1 or player 2 that are ‘bad’?
  • Can player 1 or player 2 always win?
  • Can you describe what moves will ensure that victory? Is there more than one choice of moves that will lead to victory no matter what?

Generalize!

  • How can you change the rules of the game? The first, perhaps most obvious generalization is to change the number of spaces. I encourage you to explore that first.
  • What’s the fewest number of spaces that a game can have? [3] Who wins in that case?
  • Who can always win with games of 4, 5, and 6 spaces? Organize your thinking in a table, and continue to explore bigger numbers. Is there a fundamental difference between games with an even and an odd number of spaces?
  • At some point, jump to a big number, like 171. Can you give a winning strategy for player 1 or player 2 there?
  • What big mathematical ideas are at play? How does symmetry support your thinking?

Invent & QED

Now it’s time to make up your own variation and explore. Here’s an old post that can help support your brainstorming if you need it, but I believe there is something particularly valuable about kids generating and exploring their own ideas. That’s what QED is all about.

Idea #2 Tangrams

My initial love of mathematics was seeded by toys like soma cubes, rubik’s cubes, and, plain old cubes. I also have early memories of playing with tangrams.

Tangrams are a collection of seven shapes–5 isosceles triangles, a square, and a parallelogram. Play with tangrams generally involves arranging these shapes into a figure shown as a silhouette–the number of possible tangram puzzles is limitless.

  • You can find instructions here for creating your own tangrams if you don’t have any on hand.
  • There are any number of websites that give tangram challenges–here’s one that has links to a few 100 at a variety of  difficulty levels.

I believe that many children will be given school work to do at home in the upcoming weeks to continue their academic studies. I hope you’ll help them continue their enrichment studies too–we’ll keep the posts coming. 🙂

 

 

Camps: Ours and Others

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Registration for MC2 Summer 2020 is open! We’ve also launched a new webpage that lists our summer sites (much like our Locations page does for the Academic Year). This new site also explains our summer fees–please know that we have a sliding scale, and we expect that many of our campers will attend for free.

To enter the summer lottery, login to your account on our registration page and sign up as usual–the summer lottery will run on March 31st.

 

While we hope you’ll join us this summer, what’s most important is that you do fun math somewhere! Some other suggestions: