Logic Puzzle - Fun With Isosceles Triangles and some Good Books to Boot

Summer has completely gotten away from me, between the younger girls' continual cycle of church camps and Vacation Bible Schools (this week has been a church-based basketball camp, that they've really enjoyed), and my search for the perfect "text" books for the fall - a fun but completely distracting pursuit for any bibliophile/homeschool mom, and one of the hazards of homeschooling high school students, when the texts become absorbing to teacher as well as (or hopefully for) the students.

This week I've found myself lost in geometry (Head First),

physics (...of Superheroes),

and Die Tribute Von Panem 1. Tödliche Spiele (The Hunger Games in German - for me...not so much for the children, but what a kick).

Anyway, while I haven't had a great amount of time left over to post about our summer fun, this year, I have tried to keep up with fun for the children - mainly by proving an endless supply of cookies, scattering games around the house, and keeping fresh puzzles on our board.

I'm especially happy with this week's puzzle - found (with the solution included in the comments - pictured at top of page) at smart-kit.com.  It's a perfect follow up to Dragon Box Elements - more puzzle than math, needing some basic algebra and geometry skills (I provided some hints for my own children - worded with them, rather than proper math rules, in mind), but more thought than anything else.

I worked through the problem first - just to be sure it wasn't too hard.  And, to be honest (truth in blogging, you know) had to offer up a milkshake to anyone willing to take on the puzzle...

...before I had takers.

But, since it is summer, and I like milkshakes too - I figured it wasn't a bad trade off for getting to see my teens wrestle through this one. 

23 Free Christmas Multiplication Worksheets and Printable or Online Games

While we're on Christmas "break" I thought it might be a good time to have my youngest two work on speeding up their multiplication facts. They both know them, but could be a lot faster at bringing up the answers.

We could drill the facts - but I always hated drills in school.  Since they know their times tables, but are just a little slow pulling up answers, I thought a quick (Christmas themed) daily review might do the trick.

I've been trying to have a worksheet or game printed out for the girls each morning, but its been tricky.  There are a lot of Christmas themed multiplication worksheets in the Google search, but not all the links lead to free worksheets, or multiplication worksheets, and some lead to...well...places I don't want to go.

Rather than battle it out each day, I took some time this afternoon to seek out and find enough free worksheets, and games (printable and playable online) to last from now until Christmas.  Many of them are freebies from Teachers pay Teachers.  They may not stay free forever, but once they're downloaded, they're yours.  I didn't run into any problems with any of the sites linked below, they all seem to be what they promise.

1. Math Mystery Multiplication - a multiply and color by answer sheet from teach-nology.com (double digit by single digit).

2. Math Salamander's Christmas Codebreaker - single digit multiplication, solve problems to decode message.

3. Polar Bear Multiply and Color - very simple, single digit from Classroom Jr.

4. Christmas Bunny Multiply and Color - multiply by 5's from Classroom Jr.

5. Mitten Math - match and color problems and answers, multiplying by 10.

6. Color by Number find the Missing Factor - a Christmas tree worksheet.

7. Santa Claus Mystery Picture - single digit, fill-in-the-graph.

8. Christmas Multiply - a free printable game from Teachers pay Teachers (you have to log-in to download freebie), a game from 2 players, needs counters to cover numbers.

9. Online Game - Holiday Fun, practice number facts, and decorate a gingerbread house.

10. Online Game - Merry Christmas Math Facts from ABCya. Choose a level, answer multiplication problems, and earn lights to decorate a house.

11. Online Game - Holiday Puzzle Pics - drag and drop puzzle pieces onto the correct answer spots to solve puzzle.

12. Online Game - Christmas Math Facts from A Kid's Heart - practice multiplication facts and uncover a Christmas puzzle.

13. Gingerbread Bump - a free printable game from Teachers pay Teachers (you need to log-in to download) a game for 2 or more players, needs counters to cover numbers, and 2 dice.

14. Peppermint Products - a free printable game from Teachers pay Teachers (you need to log-in to download)  a game for 2 or 3 players, needs counters to cover numbers.

15. Santa's Coal - a free printable card game from Teachers pay Teachers (you need to log-in to download) a game for 3-4 players.  You might want to laminate the cards.

16. Christmas Math Mystery Picture - Christmas Owl - a free, multiply and fill in the graph printable from Teachers pay Teachers (you need to log-in to download).

17. Christmas Tree Fact Triangles - from Teachers pay Teachers (you need to log-in to download)

18.  Multiples Light Up the Tree - free printable game for 2-3 players from Teachers pay Teachers (you need to log-in to download), requires three different colored crayons or colored pencils, a metal fastener, and a paper clip to make a spinner.

19. Christmas Mystery Picture - Mario, multiply and fill in the graph to reveal picture, from Teachers pay Teachers (you need to log-in to download).

20. Christmas Find a Line - A free print and play game from Teachers pay Teachers (you need to log-in to download).

21-23. Multiplication Trees - multiple free worksheets from Itsy Bitsy Fun.

Candy Corn, Peanuts, Salted Nut Rolls, and a Venn Diagram

When I was in the grocery store checkout, buying candy corn for National Candy Corn Day (it's coming up this weekend, you know) and the woman behind me in line mentioned if you mix candy corn and peanuts together in a bowl to pop by handfuls at a time into your mouth, they taste exactly like Salted Nut Rolls, the first thing I thought was, "What a great opportunity for reviewing Venn diagrams!"

Okay, not really.  What I really thought is, "Well, now I need to buy some Salted Nut Rolls."

I really love Salted Nut Rolls.  If it weren't for their obscene amount of calories, they'd be my favorite snack.

But, after I mixed up a batch of candy corn and peanuts (which is not bad - in fact much better than a bowl of plain candy corn - but not exactly like Salted Nut Rolls), the children had a hard time agreeing in what ways it differed from the candy bars.

So...

...we (meaning I) traced out a quick diagram...

...so we (meaning they) could explore the ways the candy and nut treats were alike and different from the candy and nut bars.

In the end, while the ingredients of the two candies were, apart from sugar, corn syrup and artificial flavoring, significantly (or maybe that was slightly) different - and the textures and looks of the two were nothing alike...

...the taste might have been quite similar, if our candy bars, which were on the stale side (despite coming straight off the shelf) had been as fresh as our candy corn and peanuts.
 
For now, hoping to avoid sugar induced comas, we've tabled the discussion until next year when, if we can obtain a fresher sampling of the candy bars, we might take up the comparison again. 

Science and math are all about persistence and accuracy, after all.  Somebody has to take the fall, and eat the sugar.

McMinion Math

Update:  Below is the contents of the email I received back from McDonald's.  At this point I'm assuming the puzzle is a misprint.  Even so, if you happen to see 96 triangles, I'd love to know the solution.
  
Thank you for contacting McDonald's and sharing your comments with us. We greatly appreciate this opportunity to address your concerns.

I'm sorry you were disappointed recently with the puzzle on our Minion themed Happy Meal box . At every McDonald's we strive to deliver 100-percent Total Customer Satisfaction. However, it appears as though we've failed to deliver these standards to you. I apologize that I do not have a solution to the puzzle to share with you.

Because you are a valued customer, I have shared your comments with our Quality Assurance team. They work closely with our suppliers to ensure that all of our products meet our quality standards. Your complete satisfaction is one of our top priorities.

Again, thank you for bringing this matter to our attention. We hope to have the opportunity of serving you again.


Well, I have to admit to being  Mc-stumped.  I ordered the tweens, McDonald's Happy Meals for lunch - sort of as a joke, since they outgrew Happy Meals some time ago.  (Okay, maybe it was because I wanted to hear what the Minion toys actually had to say).


Original Post:
Anyway, the puzzle at the top of this page was printed on the side of the box.  The kids took a couple of seconds counting triangles, and then turned back to their burgers and fries, content to trust the Minion (Phil? or Tom?) that the "pyramid" was made up of 96 triangles.

I wanted proof.  So, I copied the triangle, made up of 16 small triangles, 96 times to make a worksheet...

...to color in, so it would be easy to see that we weren't getting any duplicate triangles in our count.  I figured I'd work on it alone, find the solution, and then challenge the tweens to take an real stab at it.  The only problem was that I didn't find 96 triangles, I only found 27.

Of course, the Minion said there were 96 triangles in the "pyramid" not the "big triangle".  So, I tried again working with the shape as a four sided pyramid (or tetrahedron)...

...which would flatten out as in the picture below.  However, 27 triangles on each side, times 4 sides, equals 108 - not 96.

And, I briefly considered a five sided, square bottomed, pyramid, but that would take figuring the area of the square bottom divided by the area of the smallest triangles, and not only did that come out uneven, and greater than 96 (something like 101 triangles), it seemed a little convoluted for the target age of children normally ordering Happy Meals.

My children are content to believe it is probably supposed to be more of a "how many licks does it take to get to the center of a Tootsie Pop" sort of question - one meant to be enjoyed, and fudged, rather than solved.  Just to be sure, I emailed McDonald's and ask for a solution.

While I'm waiting, what do you think?  Is the pyramid made of 96 triangles, or not?

Exploring Perimeter and Area

While we still had the cardboard tubes out, we used them for a quick lesson on perimeter and area to go along with an episode of PBS' Cyberchase...



...exploring the possibility of shapes with the same area having different perimeters.

Sewing Card String Art Heart Pattern - Valentine's Math Meets Art

I used instructions for a sort of hybrid cardioid/concentric circle/string art heart from Math Craft, to map out a pattern in Paint for a sewing card craft for the girls.

After printing a few out, I taped them down (lightly, at the corners) to pieces of card stock, and "punched" out the holes with a thumb tack, followed by the sharp point of a pencil.

Then, I cut about three feet worth of yarn (the amount would vary, depending on the size of the print-out). I taped one end of the yarn to the back of the card stock, and threaded a darning needle onto the other, and gave the girls instructions...

...to follow the numbers, sewing up, through from the back, on the odds...

...and down, through from the front, on the evens, being careful to pull all the yarn through.

When they got to the end, we removed the needle, tied the loose ends together on the back...

...tore away the paper, and decorated the card stock...

...with crayons for a finishing touch.

The project was a lot of fun, and easy enough for my 8 year old to do on her own, with only a little double checking of the numbers.  If string art isn't your thing, don't worry, I also made up a blank pattern (for myself, really)...

...to play with in Paint.

Happy Valentines Day!

Valentine Math - Simple Steps into Multi-Step Problem Solving

Looking at the heart shaped tangram type puzzles, floating around this time of year, I spotted an easy opportunity to practice multi-step problem solving skills with my younger children (ages 8-11).

One of the first hurtles students hit in high school geometry is the switch over from simple equations to multi-step problem solving.  This transition can be made easier by introducing younger children (long before they reach high school geometry) to the concept in simple ways through puzzles, or games, using the math they already know, or can easily grasp.

So, back to the tangram - if you ask a child to find the area of a heart shape, like the one below, they will most likely be stumped.

It is not one of the usual shapes found in math books.  There is no simple formula for the area of heart. 

So, we come to the first step of multi-step problem solving, determining what we already know.  Depending on the age of the children, they will probably know the formal for finding the area of a square.

length x width = area

They might also know how to find the area of a triangle.

1/2 base x height = area

And, most will be able to put the two together to find the area of a right trapezoid (even if they don't know what it's called, or realize they have just taken a baby step into the kind of  problem solving we're talking about).

Once they've mastered the trapezoid, finding the area of a parallelogram should be a snap.

height x base = area

 Some children will even already know the formula for the area of a circle.  If they don't, they can learn it pretty easily, even if they don't fully understand its meaning.  It might be a good time to pull out Cindy Neuschwander's Sir Cumference books.

πr² = area

So, that's what we know.   Be what we need to know is the area of a heart.

? = area

How can we use what we know, to find out what we need to know?  That's where the tangram comes into play.  By matching the colors of the shapes within the tangram to the shapes the children already know how to solve for area, the problem becomes simple.

And hopefully, at this point, children will be able to see on their own, that for this particular heart, the problem can be solved with fewer steps, as well.

We will do each of the steps above, with paper cut-out of the shapes, too.  Just to add an element of familiar, hands on learning to the process.

A Tensegrity Icosahedron Model (or that cool collapsible thing from Little Man Tate)

A tensegrity icosahedron model, is one in which a 20 faced polynomial is outlined by free hanging struts, suspended in the tension of elastic bands. 

I'll be honest, I don't really know what its purpose would be.  But, I can tell you, a tensegrity model of a icosahedron is the cool collapsible shape Fred Tate made in the 1991 movie Little Man Tate.

And, not only is a lot of fun...

...to twist...

...and squish...

...and flatten, as you explore a three dimensional plane in motion...

...it turns out, it's also very easy to make.

All you need is -

• 6 unsharpened pencils
• 6 heavy duty rubber-bands, long enough to fit comfortably around the length of the pencils (elastic string tied securely into bands, will work as well).
• 12 pencil erasers
• and instructions.

We followed YouTube instructions made for a Los Angeles Trade and Technical College's Architectural Department project.




Don't worry about having slotted dowels, and plastic caps, like in the video.  All you have to do is loop the bands over the pencil ends, and hold them in place with the erasers.

Oh, and be warned, if you go looking for (and manage to find) a copy of Little Man Tate to watch with your children - there is a lot of great math, engineering and art to be found...but also a good dose of language, and a touch of adult subject matter to keep in mind.