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. 

Thanksgiving Week Activity: Turkey Glyph

Today when I announced to my boys that we were only going to do Thanksgiving related activities this week, I don't think they had a math activity in mind! This is another activity from Mathwire.

You have all seen the paper turkeys with multi-colored tail feathers.

Well, the spin on these is that each color represents something on our dinner menu and they had to pick the colors for their turkey's tail feathers that match up to the foods they like and plan to eat from our Thanksgiving menu.
Light Brown- Stuffing
Black- Gravy
Red- Cranberry/Orange Relish
White-Mashed Potatoes
Yellow-Corn on the Cob
Green-Pickles & Olives tray
Light Blue-Deviled Eggs
Orange-Sweet Potato Casserole
Orange Striped-Pumpkin/Cranberry Bread
Tan-Turkey (shaped) Bread
Pink-Tortellini Salad
Dark Blue-Jello
Dark Brown-Pumpkin Pie
Purple-Cake

For the turkey body, you could choose light brown for white meat and dark brown for dark meat.
James wanted both, so we made his turkey body with both light and dark brown.

Quentin's.



Sam's.
Even though this is an early elementry activity, I had everyone in the family do it, so we could have more to graph.



James'

Alex's

Mom's

Katie's

And our vegetarian Dad just made a clutch of feathers.

You can use these to graph preferences.

On a practical level, it can help you to plan how much of each item on the menu to make. And lastly, serve as cute decorations for Thanksgiving Day.
Just as James said, "Each of the turkeys tells a story."

Interest-Led Study: Origami and Math

Katie has been enjoying teaching James how to make origami creations. They became inspired by watching Between the Folds (on Netflix instant play).

Between The Folds

The artistic value of origami was pretty instant and obvious to them, but once I became interested in their project, I was amazed and fascinated by all the mathematical concepts that could be taught along the way, in a very pleasurable and engaging way.

To get you started:

symmetry

angles

mirror images

 area

 congruence

 volume

 geometric terms: faces, edges and vertices.

Finding area and volume

Percentages (What % of the exterior surface is (a certain color)?

And once you have explored these topics, you might want to go a bit further...

 Origami and Fractions

The Origami Cube with some math explorations (algebraic formulas)

Origami and Math

Origami Proof of the Pythagorean Theorem

Paper Folding Geometry

And a Little Further (Lesson Plans and Books)

• Origami Solutions: figures that will be useful for teaching geometry concepts, including reflections, rotations, and geometric solids.
• Symmetry Questions in a Wreath and Pinwheel
• The Geometry Junkyard

Books: 

• Origami and Math: Simple to Complex, John Montroll

And, by the way, Katie is half-way through her project of folding 1000 paper cranes.

La Fiesta Restaurante: Designing Floor Plans

La Fiesta Restaurante

The Rosada family has just purchased a new restaurant that is larger than their La Tostada Sabrosa restaurant. Since they are moving into a brand new building, they are excited about creating a floor plan. Your assignment is to assist them by coming up with a plan for them to look at.

The Floor Plan

This is the floor plan. It has windows on two sides, and a corner entrance. The doorway cannot have any objects placed in it. They need to place the kitchen, the bathroom and 11 tables (4 tables that seat up to six people, 4 tables that seat up to four people and 3 tables that seat one or two people.) We discussed determining dimensions, perimeter and area.

The City Codes

There are city building codes that the Rosadas need to follow in the design of their restaurant.

1. The Kitchen can:
1. share 2 sides with the perimeter of the restaurant, OR
2. share 1 side with the perimeter of the restaurant, OR
3. be freestanding within the restaurant
2. Bathroom must share one OR both sides with the perimeter of the restaurant
3. Tables can only have their shortest side touch the perimeter OR tables can be freestanding.
4. There must be one empty space (square) between any two objects in the restaurant.
5. Nothing may be placed on the shaded entryway squares.

The Assignments

Students need to able to show how their plans meet all the building codes. If the floor plan passes inspection,  they can record their Restaurant Floor Plan. You can also have your student write a persuasive paragraph to the Rosadas explaining why they should choose his plan over any others. Alternatively, they could write in their math journals about what strategies they used to situate the kitchen, bathroom and tables. Did he have to change the strategy as he went along? They need to explain their problem-solving process in detail.

related posts:

• La Tostada Sabrosa
• Money Matters: Tostada Cost Analysis
• Money Matters: Setting Tostada Prices
• Combination Platters: Making Combo Plates

source:

• GEMS guide, Math on the Menu

1. 
   

Combination Platters: Making Combo Plates

The Rosada family has had tremendous success selling tostadas at their restaurant and now they want to expand their menu. They want to feature combination plates with a choice from six Mexican dishes: quesadillas, enchiladas, burritos, tamales, tacos and chiles rellenos. They want their customers to be able to choose three different items on their combination plates. The Rosada family wants to know how many combination are possible. Juanita, who is one of their four children, came up with the answer of 18 different combination plates. She used a grid just like you had used. Do you agree or disagree with Juanita's answer?

To answer this question, Quentin made a flip-book. Each third has all six dishes, so he could use them to make all the possible combinations. All he had to do was make sure that he didn't repeat any dishes, because all combinations must have only one serving of each dish. He wrote down the combinations, using abbreviations for each of the dishes, such as Q-T-B for Quesadilla-Tamale-Burrito. He disagreed with Juanita's solution. Do you?

related posts:

• La Tostada Sabrosa
• Money Matters: Tostada Cost Analysis
• Money Matters: Setting Tostada Prices

source:

• GEMS guide, Math on the Menu

Money Matters: Setting Tostada Prices

The Rosadas decided to sell each of their tostadas for the same price, regardless of the combination of toppings a customer selected Now that the business has been going along successfully  they want to know how much each of the different tostadas actually cost to prepare so that they can analyze their profit margin and adjust the cost as necessary. Based on the cost of the ingredients they want to know how much each tostada costs to prepare The based on what the students advise them to do , the Rosadas may need to change their prices.

Last week the boys determined the prices of each tostada, given the cost of the ingredients. This week they calculated the average (or mean) cost. The Rosadas have determined that they sell almost an equal number of each combinations every day. They also let us know that in the restaurant business owners need to charge three times the cost of the ingredients in order to pay for such things as rent utilities, salaries and to make a profit They decided to triple the cost of each tostada combination, add them together and then divide that number by the number of combinations. There are other ways of coming to this average, but their way did work. They came up with the average cost of $3.32, which they wanted to suggest to the Rosadas that they should make the price $3.35 as it is a more usual price. (Rounding down to $3.30 would also be a good answer.) Next week the Rosadas will expand their menu to include combination plates, and will need more help from the boys to figure out what to charge for them.

related posts:

• La Tostada Sabrosa
• Money Matters: Tostada Cost Analysis

source:

• GEMS guide, Math on the Menu

Money Matters: Tostada Cost Analysis

For the next part of La Tostada Sabrosa math project, the boys learned about averages and why averaging can be useful.

I gave each of the tostada toppings from our last math session a price and had them add them up to get a price for each topping combination.

Olives 35 cents

Lettuce 15 cents

Salsa 25 cents

Cheese 65 cents

Beans 25 cents

Fried Tortilla 10 cents

Quentin learned how to use a calculator for this lesson because he has already worked on adding double digit numbers but he has never worked with a calculator to solve math problems before.

Next I gave them a lesson about averages. I had Quentin measure the hand spans (the distance between the tip of the thumb to the tip of the pinkie, across the hand) of every member of the family using yarn (as we have made non-standardized measurements before) and then measure them using Unifix cubes.

He taped the yarn pieces to a piece of paper and labeled them according to whose hand span it was and how many Unifix cubes long it was.

We then discussed averages. We took the seven numbers from the seven measurements he took and put them in order from the smallest to the largest, repeating numbers if appropriate. We then made observations about the data. I told him that the amount of difference between the smallest number and the largest number is called the range.  

I told James and Quentin about the differences between median (number in the center of the data) and mode (the most frequently occurring number or numbers).

Quentin also determined the average by adding all the measurements and dividing by the amount of numbers in the data. This can be done by comparing the cube trains for each measurement, if the student needs a more visual way of doing this. They can then even out the cube trains by taking some cubes off the longer trains and adding them to the shorter trains until most trains are the same length.

James was able to determine the averages of the tostada costs for tostadas with three topping combinations. He estimated that the average was $1.15, and was pleased that his estimate was spot on.

These averages will be used in future math lessons to determine one standard price for the tostadas.

related posts:

• La Tostada Sabrosa

source:

• GEMS guide, Math on the Menu

La Tostada Sabrosa

We had tostadas for lunch, too.

I gave my boys the task of helping the (imaginary) Rosada family who are opening a new restaurant, La Tostada Sabrosa (The Delicious Tostada) that will feature, you guessed it, tostadas. The Rosadas plan to offer a choice of five different toppings to go on the fried tortillas, beans, cheese, salsa, lettuce and olives. 

I gave Quentin (age 8) the task of figuring out how many combinations could be made if they allowed two different toppings from the choice of five toppings and I gave him paper manipulatives to help him figure it out, and I showed him how to transfer that information to a chart.

I gave James (age 12) the task of figuring out how many combinations could be made if they allowed three different toppings from the choice of five toppings and I gave him just a chart to work with. I let them do it any way they wished this time so that they could make their own organizational discoveries. After they had solved their problems, we discussed any patterns they had noticed. They noticed that each ingredient was used an equal amount of times. We also discussed how it would be easier to determine all the combinations if the ingredients were chosen in a systematic way. We also talked about how it was easier for Quentin to use the paper ingredients than the chart, but how James didn't want to use them because he didn't need them and considered working with the paper manipulatives as an extra, unneeded step. 

Spy Mission

Later on I gave them a spy mission to solve and I was pleased that they used a similar chart to organize the clues they were given to solve the case.

math skills:

• problem-solving strategies
• combinations
• using charts

source:

• GEMS guide, Math on the Menu

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