Triangular Numbers, More on Pascal's Triangle and Odd/Even Numbers

Remember when we played with squares and square numbers?

I thought maybe it was time to look at triangular numbers.

We took out circles because triangular number is the number of dots in an equilateral triangle uniformly filled with dots. 

Okay, one is the smallest way to make a triangle. How many circles does it take to make the next size larger triangle? We played and built.

Three.

Next size up?

Six.

Can you predict the next size up triangle?

Nine? Is it multiples of three?

Let's see.

No, ten.

Let's build one more.

Fifteen.

Now let's take them and make squares out of them. How many more do you need?

One more to the three.

Three more to the six.

Can you predict the next amount it will take to make the triangles a square?

Six?

Let's try and see.

Why did you say six?

Because it was the same sequence of numbers...1...3...and 6.

The sum of two consecutive triangular numbers is a square number.

Then I told them a bit about Karl Gauss. He lived in the 1800's and he proved that you can make any whole number by adding no more than three triangular numbers and that triangular numbers never end in 2, 4, 7 or 9.

What triangular numbers add up to your age.

Ten is already a triangular number but it can also be made by1 +3+6!

Seven is 6+1!

Thirteen can be made by adding 10 + 3.

What about you? Hmmmm...49 can be made by adding 28+28+3.

Lastly, remember Pascal's Triangle?

Can you find triangular numbers in it?

Then I found this investigation of number patterns and Pascal's Triangle and I had to try it out too.

So, I printed out a copy of Pascal's Triangle and we went to work coloring multiples of two and three.

We tag teamed this activity by starting with Quentin until he wanted to pass the marker to James and then James to Sam.

You may find it helpful to know that the sum of the digits of any multiple of 3 is itself divisible by 3. For example, 252 = 2+5+2 = 9, and 9 is divisible by 3; 924 = 9+2+4 = 15, and 15 is divisible by 3—so both 252 and 924 are colored, but 560 = 5+6 = 11, is not.

You could also look at it in terms of odd and even numbers and if two cells above are of the same type (both odd, both even), you color the cell below even. If the cells above are different (odd + even, or even + odd) you color it odd. Why does this work?

Because of the principle that an odd + an odd always equals an even. Same is true with the sum of two even numbers is itself even. If your student doesn't know this, he will need to play with numbers to prove it to himself. With every step of the way, he needs to see for himself what the rules of math are, and not just take it from you. 

Advanced Multiplication (3-6)

Multiplication with Beans and Cups

For cups and beans, I begin with two cups and a stockpile of beans. I first tell them to start with putting 1 bean in each cup, and I begin recording while they count the beans. 2 (cups) x 1 (bean in each cup) = 2

Then I ask them to put 1 more bean in each cup and I record while they count.

2 x 2 = 4

We continue on like this for a few minutes.

Somewhere along the way, I just give them 1 instead of a set of two beans and ask them what do we do about this?At this point we talk about some numbers not coming out evenly, and then we add 1 more to make a set he can add to the cups, and continue on from there. We can add more beans or more cups as long as there is interest in this activity. In this way, young children can bridge the gap between addition and multiplication seamlessly.

Lattice Multiplication and Napier's Bones

As problems increase in size, the use of these materials becomes impractical. Traditionally an abstract system of long multiplication the distributive process has been taught.
"Research has shown, however, that the lattice system of multiplication allows students to compute multi-digit multiplication problems in significantly less time and with greater accuracy than is possible using the distributive method. "- A Comparison of Two Methods of Teaching Multidigit Multiplication [University of Tennessee, Frank George Hughes]

The lattice method came out of a calculating aid made by John Napier called "Napier's Rods."
"John Napier was a Scottish nobleman who loved mathematics. He invented logarithms, worked in spherical trigonometry and designed "Napier's rods," a mechanical calculating aid...These were an assortment of rods marked off with numbers. When these rods were arranged correctly, they could be used for multiplication and division...They were a sort of movable multiplication table -an early type of slide rule, which is what people used before pocket calculators. Because they were made of bone or ivory strips they were sometimes called "Napier's Bones."

-Mathematicians Are People, Too, Reimer and Reimer

To use them, you take out the rods that have at the top the first number you are working with.

For example if your problem is 298 x 7, you take out the 2 rod, the 9 rod and the 8 rod and lay them down on the table. You can see, if you go down to the 7th row of this set (because 7 is the second number you are working with), on the top of the slashes is 165. Write that down. These are your tens. Below the slashes are the numbers 436. Since these are your ones, you will write them under the 165, but you will indent out one column, to make the number above start in the tens column. Add these two numbers together, and you will get 2086, which is the answer to 298 x 7.


Let's try another problem; 31 x 24. This system is sometimes called the "Lattice System" because when you get to working with larger numbers, it becomes easier to make a grouping of boxes. You make a graph with the number of boxes across as there is in your first numeral; in this case two. You make the number of boxes down as you have numerals in your second numeral; in this case also two. So you have a graph with two boxes going across and two boxes going down for this problem.
Now put diagonal lines going from one corner of each box diagonally to the other corner. I extend my diagonal to make it clear where the answers go. The drawing of the boxes may seem complicated, but it is easy once you have done it a few times and kids find it easy to do on any blank piece of paper. Next, write the digits of your problem along the sides of your boxes. Now, get out your rods for the first two digits of your problem; in this case the 3 rod and the 1 rod. Go down the number of boxes according to the numerals along the side of your graph; in this case, 2 first. Copy down the boxes just as they are on the rods, the numerals above and below the slashes will correspond to the boxes and slashes you have in your graph; in this case 0/6 and 0/2. Continue this way with the next digit; in this case, 4. Now you have numerals all around your boxes. Ignore the numerals along the top and right sides now, and add only the numbers within the boxes on the diagonal. Starting at the bottom corner the one diagonal triangle box has the numeral 4, so I write 4 down below it. The next row of diagonal triangle boxes contain the numerals 2,0 and 2, which if you add them together equal 4, so I write 4 below them. The next diagonal row of numerals are 1,6 and 0, which equals 7, so I write 7 below it. The corner diagonal box contains 0, which I chose to just leave out since it won't affect the answer, but you could have your students write a 0 there just to be consistent and get in the habit of always writing down the numerals so as not to forget any by accident. It depends on how old they are and their understanding of the concepts.
The answer to this problem, reading the numerals from right to left is 744.

This method can be used for as large a problem as you want. Just make sure you draw the correct number of boxes according to the numerals that are in your problem. Here is the problem 123 x 12. We got out our 1, 2 and 3 rods and made our graph 3 boxes by 2 boxes and wrote the numerals around the top and right hand edges. We then copied what the first (1) and second (2) boxes of the bones had in them. We then added the numerals on a diagonal, getting the correct answer; 1,476.
You have to use a slightly different method of adding up the numbers if you get numbers that have carrying in them. For this problem, for example, 123 x 67, you will get the numerals from left to right, 7 1,14,11 and 7. If you get numerals over 9, you must carry them over into column addition. You write down the numeral(s) and then add the number of 0's after it that corresponds to the number of columns after it. Since 1 is in the first row, it gets no 0's after it. 14 is in the next row, and it has only 1 row after it, so you write down 14 with 1 0 behind it, or 140. The next row has 11 in it (pardon the odd looking numerals; my son made a mistake in his addition the first time and got 10, but then changed it to an 11) so you write 11 with two 0's behind it for the two columns, or 1100. The last row has a 7 in it and it has three 0's behind it for the three rows, or 7000. Add these numbers together and you will get 8,241; the correct answer. This is something that is complicated to explain but easy to use once you understand how it goes. All a child needs to be able to do is add two digit numbers.

 More about how it works here at Math is Good For You!

"The lattice method produces the same kind of understanding as the distributive method but is easier to teach, faster to use, and less prone to error. " -Mathematics...A Way of Thinking, Robert Baratta-Lorton

How to Make Napier's Bones

If you would like to make some Napier Rods, just get 9 wide craft sticks (like tongue depressors) and divide them into 9 fairly even sections. I just eyed mine; I did not measure them. If you are using these with children younger than 3rd or 4th grade, I would divide them into 10 sections and use the top section to put the number of the rod on the top. My Kindergartner can use these, but sometimes has difficulty reading the top number to identify which rod he is using. I ended up putting the number of the rod on the back for him. He chooses the rods by the numbers on the back and then turns them over to use them. My severely dyslexic son has trouble sometimes counting down the blocks correctly. Perhaps using different colors for the block divisions than for the numbers would help with this. (It would also be possible to make a thin column down one side to mark the rows with numbers.) Divide each of these sections with a diagonal line and copy the numbers above. The numbers are just the traditional multiplication tables. I used an Ultra-fine point Sharpie to write with, but it did bleed into the wood a little, making the numerals fuzzy. I am not sure if there would be a better writing instrument to use for this.

You can instead go to Mathwire and print and cut out these Napier's Bones. They can be used on their own or cut out and glued to large craft sticks.
"Students who have difficulty reasoning abstractly with numbers are frequently unable to grasp the numberical logic behind either the distributive or the lattice approach to long multiplication. Knowledge of why an abstract system of producing answers works is not as important as the knowledge that it does wok. For this reason, answers to the initial problems students work using a lattice method are checked against the materials, ususally chips on the chip trading boards."
-Mathematics...A Way of Thinking, Robert Baratta-Lorton

source: 

• Mathematics Their Way, Mary Baratta-Lorton

Eratosthenes Sieve

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Since we have been working with fractions, and therefore prime numbers, I decided to teach James about Eratosthenes Sieve. Eratosthenes was a Greek mathematician who, among other things, created a simple way of finding prime numbers. A prime number is a whole number greater than 1 that has no positive divisors other than 1 and itself.You can just use a 100's chart for this exercise. First cross out the numeral 1 because it cannot help be used to build any other numbers. Circle the numeral 2 as it is the first prime number and cross out any numeral that can be made with a 2. Circle 3 as the next prime and cross out any numeral that can be made with a 3.Continue with the numerals, crossing out numerals in this manner.Through this process, you will sieve out all numbers, leaving just prime numbers remaining. We will be working with factoring numbers into their primes as a way of finding equivalent forms in fractions next week.

Fun with Pascal's Triangle

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We had a lot of fun today playing with Pascal's Triangle, invented by Blaise Pascal, a French Mathematician in 1653. First I started out giving them the following math problem to solve.

Imagine you're buying an ice cream cone. If there are 5 flavors, how many possible combinations are there? How many ways of having no flavors? How many ways of having 1 flavor? How many ways of having 2 flavors? How many ways of having 3 flavors? How many ways of having 4 flavors? And, how many ways of having 5 flavors?

This was a good word problem for my 10-year old to solve. While he was working it out, I cut out squares of colored paper and glued them on another sheet of paper to form a pyramid or triangle. Once he finished solving the problem, I read to him some about Pascal and then I had him fill in all the outside squares on our triangle with the numeral 1. I then asked him to fill in the rest of the squares by adding together the numerals in the two squares that joined above each square. We only had seven rows, including the top 1, before we ran out of room. If I do this again, I might get a larger piece of paper to work from.

Now I told him to count down five rows, not counting the top 1, and look at the numbers that ran across this row. Did he see any similarities to the answers he had for the problem he had worked on earlier?

They, of course, matched.

We counted down five rows because the possible flavors we had to work with in the problem was five. Pascal's triangle can be used to find the answers to any of such type problems with ease.

I then asked him to add up the numbers in each row.

What pattern do you see?

He began saying the numbers out loud, "2, 4, 8...they double each time!"

I then read to him about something called a Galton board. The board is made with nails arranged like Pascal's triangle. Marbles are then poured through it. The probability of an marble ended up in a particular column is easy to work out by looking at the numbers in Pascal's triangle. The final pattern, too, is in the shape of a bell curve.

Galton board, photo from Wikipedia

There are so many fascinating patterns to be found in Pascal's Triangle. What patterns do you see?

Why not build one yourself and see?

Everything is Numbers

"Math is the real world, okay it's everywhere, okay. Can I show you? You see how the petals spiral?

The number of petals in each row is the sum of the preceding two rows,

the Fibonacci Sequence. It's found in the structure of crystals and the spiral of galaxies and a nautilus shell.

What's more, the ratio between each number in the sequence to the one before it is approximately 1.61803, what the Greeks call the Golden Ratio. It shows up in the pyramids of Giza and the Parthenon at Athens, the dimensions of this card.

And it's based on a number we can find in a flower.

Math is nature's language... its method of communicating directly with us.

Everything is numbers."

- Charlie Epps, NUMB3RS

Who was Fibonacci and what is the Fibonacci sequence?

Leonardo of Pisa, (yes, the same Pisa know for its Leaning Tower) known as Fibonacci, was born about 1175 AD and was probably the greatest European mathematician of the middle ages. Leonardo grew up with a North African education under the Moors. He traveled extensively around the Mediterranean coast meeting with many merchants learning their systems of doing arithmetic. Europe was still using the Roman number system and Fibonacci realized the importance of the Hindu-Arabic number system with its ten digits, decimal point and a symbol for zero, and so was one of the first people to introduce it into Europe. Another important discovery which was named after him is called the Fibonacci Sequence. You can arrive at this sequence by beginning with number one add one to it, and you get two, now add these two numbers together and you get three, now add the last two numbers together and you get 5, and the next two added together is eight. Keep going, and you will see that the ratio that occurs after the number 3, is the number 1:618, which is the Golden Ratio.

This ratio was first discovered by the Greek mathematician Pythagoras. Later, an Athenian architect using the Golden Section in building design came up with Phi, the number 1.618. Fibonacci made the next leap when he published a book in 1202 called “Liber Abaci”. He introduced a math problem where a pair of rabbits were placed in a field with the provision that they could not escape or die. At the age of 1 month the female gives birth to 2 new rabbits (1 male, 1 female). The female rabbit does this each month for 1 year. How many rabbits would there be at the end of the year? The answer to this question contains a series of numbers (1, 1, 2, 3, 5, 8, 13, 21, 34, 55…..). I am sure you are getting used to this sequence by now.

So, what does this have to do with mathematics for children?

We have spent a lot of time playing with, looking at and discovering patterns in our math. This is because math is really the study of patterns, and looking at the patterns in math can inspire in children the respect and love that math deserves. I encourage you to spend as much time as you can inspiring your students to make, look at and make discoveries with patterns. This appreciation for patterns can overlap into nature study, science and art.

This week, for example, I had my boys look at the outside of a purple cabbage, a banana, an onion, an apple and a pomegranate.

We have done an activity like this a few years ago. We looked at a kiwi, an avocado and an artichoke. I picked those particular fruits because I was pretty sure they were not familiar with them and so had to think about what they might look like inside based solely on what they saw on the outside. I had them sketch the outside of the fruits so that they would pay close attention to their shape, size, texture and color. Then they sketched what they guessed the inside would look like. If the outside was tough, would that make the inside soft? Would their be cavities inside? Would the color be the same or different? If the color was the same, would the shade be different? What would the seeds look like? Would they be small or large? Hard or soft? I made sure that they knew that I did not expect them to get the correct answers, but that I just wanted them them wonder, to infer. Then we cut them open to see how they looked.
And they drew them again, noting all the things that they had wondered about before.

We did this exercise again this week, using different fruits and vegetables. This time it was ones they were familiar with. Did it make a difference in their ability to accurately predict? Not necessarily, as people don't tend to take the time to notice details. And that was what I was asking them to do. I encourage them to describe the shape, texture and markings of each of them. Note the color. Can you make your sketch have the exact color of the fruit or vegetable? What is the texture like -smooth, bumpy, crackly like paper? Does it pull apart? Does it have spots or other markings? How many sections does it divide into?

Using a magnifying glass can help for closer examination.

Can they predict what each piece will look like when it is cut open? Will the color be the same on the inside as on the outside? Do you expect to find a few seeds or a great many or none at all?

Discuss the lines and the proportion and the different shades of color.
Cut it open and talk about what you discover. Look at the different patterns inside each piece. Have them sketch them, or make prints of them to look at in the future. Do you see spiraling patterns?
Sketch the patterns...

and perhaps watercolor them.

My youngest son became so interested he asked for other fruits and vegtables to examine, so we examined a cucumber and a tomato as well.
At another time you could have them examine a branch of leaves and ask them to describe whatever they notice. Do you see different shades of color? Do you see any places where there are different shades of color? Do you see any places where there are different thicknesses? Do these always occur in the same place? what do you notice about the spaces between the leaves? Are all the leaves the same size? How are they attached to the branch? Are they like your arm and are straight across or do they zig-zag? Do you see any diagonals? Are all the parts the same texture? The leaves on many plants are staggered in a spiral pattern to permit optimum exposure to sunlight. If we apply the Golden Ratio to a circle we can see how it is that this plant exhibits Fibonacci qualities. Cut the leaves off the branch. Can they sort them?

Make an effort to encourage them to notice patterns during nature study. Encourage them to make as accurate a sketch as they are capable, or take photographs to examine later for patterns.

In the case of tapered pine cones or pineapples, we see a double set of spirals – one going in a clockwise direction and one in the opposite direction. When these spirals are counted, the two sets are found to be adjacent Fibonacci numbers.Similarly, sunflowers have a Golden Spiral seed arrangement. You can see the beautiful pattern easily. This provides a biological advantage because it maximizes the number of seeds that can be packed into a seed head.
Notice the numbers within other flowers. Lillies and Iris tend to have 3 petals. Buttercup, Wild rose, Larkspur, and Columbine tend to have 5. Delphiniums have 8. Marigolds have 13. Aster, Black-eyed susan, and Chicory have 21. Count Daisy petals; do their petals count out as a Fibonacci number?

Another great activity involving art and Fibonacci Sequence can be found here at Teach Kids Art.

Or here, at Almost Unschoolers.

More about Fibonacci can be found in these books: