You know I love math! Today I’m sharing some of the best parts of math. So if you and your child are struggling with math, add these in. Don’t dare say, “We are taking a break from math today to do this project.” What I am showing you is real math. What we teach our kids is mostly calculations and algorithms. What I’m going to show you is the reason mathematicians become mathematicians, so just fold it into your curriculum and say, “This is the math we are doing today,” and then watch you and your family fall in love with math. Enjoy, friends!!
Prime and Composite Numbers
This Website – Dancing Numbers – is an animation of counting number, as far as I can tell indefinite, that shows the relationship of numbers through the patterns with dots. I hesitate to tell you the secret, because it’s so fun to discover all for yourself. Be sure to pause the animation often and discuss what you see. Some comments and questions that might prompt these discoveries:
“Oooh, look at that pattern.”
“Have we seen that pattern before? Let’s go back and look. Oh yes, we saw it in the 3. How is this pattern different from the pattern in 3? Why do you think that is?”
“I notice that every so often there is just a circle of dots. What’s going on there?”
“What do you notice?”
Be sure you channel your inner most “Della” and say these things with the enthusiasm of someone who just discovered a new planet or a new dinosaur fossil.
The Math Book
The Math Book by Clifford Pickover. This book is full of short little stories of math history and concepts in chronological order with some gorgeous pictures for each page, and it’s fascinating. It starts with counting ants (an experiment that I found cruel) and then on to how some animals count and then to more complicated theorems. We used this at the end of math each day in late about twice a week in elementary years and through middle school. My son choose whatever page he liked, and we read it. Some of the stories were over our heads, but most were intriguing. A few had us going down rabbit holes to see if we could repeat their experiments. Those days were awesome. This is a great way to add some fun math into your day.
There is so much to do here from the youngest age of just putting dots on a circle and connecting the dots to using a counting wheel to dividing a circle to more complicated multiplication circles. See the following blog post and videos for each. If you are just putting dots on a circle be sure to play around and skip count the dots seeing what different patterns that you get. These are in my Quality of Numbers Study Guide.
You can play with dividing a 12-point circle from my free printables. It doesn’t have to be 12 points. You can play with a circle divided by any number.
You can also play with multiplication circles. These are so much fun and great for older kids.
This is a fascinating one. (Do I say that about them all?) You can just start with adding counting numbers and the previous sum. 0+1=1 and 1+1=2 and 1+2=3 and 2+3= 5 and 3+5=8 …so that gives us 1, 1, 2, 3, 5, 8…are you seeing the pattern here. This is Fibonacci sequence, and it’s seen all through nature, and it’s related to the Golden Triangle and the irrational number Phi (did you know there was another famous number similar to Pi?) but it’s not the only digit sums that mathematicians have played with.
Take a 100 chart or your 100 block chart (Here’s a blog to make your own) and find the multiples of nine on the chart. What’s the physical pattern? Add the digits of each single multiple, for instance for 18, add the 1 and the 8. What do you find? Do the next multiple for 9, 27 (2+7=???) What do you notice? How cool is that?!? You can do this for 3’s as well. First find the pattern for 3’s in a 100 chart. What do you notice? They run in a diagonal. Now add the digits of each of the products (12 is 1+2=3, 15 is 1+5=6, 18 is 1+8=9.) Keep going? Notice anything? This is where the divisibility rule for 3’s come from– if the sum of a number’s digits is divisible by 3, it is divisible by 3.
Tessellations are so much fun to learn about! Not only are there no calculations, but they are just visually stunning. The Ancient Islamic tessellations are full of geometry as well as reflective and rotation symmetry. The Kid Citizen Site has a few free tessellation activities. One of my favorite tessellation activities is playing with tiles from Talking Math with your Kids. I think we now have all of his tiles, but my favorites are the Versitiles and the 21st Century Pattern Blocks. If you play with tiles be sure to do a “Notice and Wonder” activity using the prompts “I notice…what do you notice?” and “I wonder…What do you wonder?”
Fractals I first learned about fractals when registering for this self-paced, self-priced course called Multiplication Explorers. (I’d highly recommend, by the way.) Fractals complex patterns that’s small parts look like the largest parts. The most well-known fractal is Sierpiński triangle, but there are all kinds of fractals and several found in nature – mountain ranges and tree branches.
I learned of these in the multiplication course as well. These can be made of legos or fruit loops or any kind of colored blocks. The first set of staircases is a multiple of 1 all in one color. Then build the next tower next to it doubling each of the first with the first of the double the same color that you used for 1 and the second a different color. These give a great visual of how multiplication builds quickly over time.
Who loves spirals?!? This is where you say, “We do!!!” Did you know there are all kinds of spirals? There are spirals that rotate evenly such as the Archimedes spirals. Then there are those that are a little more sophisticated. Fibonacci’s sequence makes a beautiful spiral with a golden rectangle (a ratio of the side to the number Phi.) You can explore spirals found in nature with the book Swirl by Swirl by Joyce Sidman. You can also make a spiral from nesting shapes like triangles or squares. For triangles after drawing the initial, draw the next triangle inside the first with vertices of the new triangle at 1/3 mark of the sides of the outer triangle. Continue with the next inscribed triangles in the same way. Then begin to color the portions making a spiral. You can do the squares the same way or just use the midpoint of the outer square for the vertices of the inner square. Color certain sections to make your spiral.
Parabolas are found everywhere, especially in physics. When you throw or hit a ball, it travels in the path of a parabola. The headlights on a car – a parabola. There are several definitions of a parabola in math depending on what perspective you are looking from. It’s a cross-section of a cone. It’s the curve in which all light shined into it shines into a single focal point. It is the curved formed from the equidistance of a line and a point not on the line. Any definition you use they are fun to explore. You can explore them through origami by place a line along the edge of a sheet of paper and then a dot about an inch from the line in the center of the paper. Then starting at one end of the line, take the line to the dot and fold that crease. Move along the line just a little further down the line bringing that point on the line to the point not on the line and fold the crease again. Continue to do this so that points all along the line are folded to meet the point and creased. When you unfold your paper, you will see the shape of of all the folds form a parabola.
My favorite way to explore parabola is through its tangents. You do this by drawing two lines of equal length on a piece of paper – a vertical line along the left side of the paper and a horizontal line at the bottom of the paper. Then take your ruler and mark off intervals along those lines. Older children will be able to do the markings in centimeters, younger will do better with inches. Then take a pencil and draw a straight line with a ruler from the top most point on the vertical line to the left most point on the horizontal line. Then connect the second point in the vertical to the second point from the left on the horizon. Continue to draw these tangent lines until you have connected all available points. These tangent lines for a curve and that curve is a parabola.
Okay, your turn? What is your favorite part of math?
Postscript: The picture for this post is an exercise done from Jamie York’s Middle School level Making Math Meaningful exploring the relationships between the lengths of lines of the pentagram as Phi.