Tag Archives: Exploration

The Emperor and the Chess Board

Explore MathTo complete this activity, you’ll need some rice and a checkerboard. You can download the checkerboard… I can’t fit the rice through the slot in my computer, though.

There is a famous story about the meeting between a Chinese emperor and the inventor of the game of chess. There is a less famous story about the meeting between the emperor and the inventor of the game of checkers.

However, the emperor was so overjoyed by both games that he offered each inventor anything he wanted in the kingdom. Both inventors said they wanted some grains of rice.

The inventor of chess said, “I would like one grain of rice on the first square of the board, two grains on the second, four on the third, eight on the fourth, and so on. I would like all of the grains of rice that are placed on the board in this way.”

The inventor of checkers, however, was greedy and said, “ I would like one million grains of rice for every square on the board. I would like all of the grains of rice that are placed on the checker board in this way.”

The Chinese emperor laughed in the face of the inventor of checkers and said he wouldn’t be able to afford such an amount of rice. Instead, the emperor said he would give the checkers inventor the same amount he gave to the chess inventor.

  1. What do you think of each inventor’s proposition and the emperor’s response?
  2. What do you notice about the amount on a square of the chessboard and the amount on the previous square?
  3. What do you notice about the amount on a square of the chessboard and the amount on all the previous squares put together?
  4. What do you notice about the amount on a square of the checkerboard and the amount on the previous square?
  5. How many grains of rice will be on the entire checkerboard? The entire chessboard?

The Legend of Chess and Checkers

Printable Checkerboard

Exploring Bivariate Data

Explore MathWe’re trying a Math Nspired™ lab provided by Texas Instruments today.

Would you like to try this at home? If you have a TI nSpire calculator you can download the tns file to your handheld or to the Nspire app on your iPad and try this any time. You can also try the student software free for 30 days using the link on the Resources page.

Maybe your parents would like to give it a try!

Exploring_Bivariate_Data_Student

Exploring Bivariate Data.tns (For Nspire Calculator)

Even if you don’t have access to an Nspire calculator, you can now examine the plotted data from this exercise! The plots (with any luck even numbered correctly) are posted below.

1.2
1.2
2.1
2.1
3.1
3.1
4.1
4.1

Reflections and Rotations!

90º CCW rotation of quadrilateral DEFG
90º CCW rotation of quadrilateral DEFG

If you’d like to play more with the Geogebra models we used in class today, here are links to the model for rotations and reflections.

I modified the rotations model with a suggestion from Cooper so that it is less tedious to move points in the pre-image. I apologize for not having the reflection model ready to use during class, but do recommend you take some time to experiment with it and believe what your new-found algebraic rules tell you.

Pythagorean Challenge—Extra Credit Opportunity

Stretch your brain, master the pythagorean theorem, and earn some extra points in the process. You’ll need to show your work and be able to explain your methods for credit.

You may need some formulas…

The area of a triangle can be expressed as:

{{A}_{triangle}}=\frac{1}{2}bh

Where b is the base and h is the height.

The area of a circle can be expressed as:

{{A}_{circle}}=\pi {{r}^{2}}

Where r is the radius of the circle or half the diameter.

You can ask questions before school, during lunch, after school, or by sending a tweet @rudunn.

Here’s the challenge…

Pythagorean Challenge

Determining the Interest rate based on payments

HomeworkDuring class today as we were working the problem of the day, several people proposed possible interest rates to go with the interest paid by Henrí in paying off his loan.

Using a calculator similar to the one we used on Friday, it is possible to determine the interest rate.

Interest rate calculation from calculator.net. The formula for this calculation isn't exactly simple.
Interest rate calculation from calculator.net. The formula for this calculation isn’t exactly simple.

I like this calculator, but I’m sure there are plenty more to choose from. Plugging in our example of $274/month for 36 months returns an interest rate of 6.05% for a total interest cost of $864.00. Several suggested that you could simply divide \frac{864}{9000}=9.6%?

Why does an interest rate of 6.05% per year end up with a total cost of 9.6%? Why would it not be .0605\cdot 3=18%?

Ponder those questions for next time.

Remember to take a look at the rest of the exploration for 16.3 (page 453, green book) and get started on the homework, but the homework won’t be due until Thursday. I do expect you to complete the exploration before class on Wednesday.

Feeling adventurous? Remember the equation I mentioned in the last post about this topic?

R=\frac{P\cdot i}{1-{{(1+i)}^{-n}}}

You can use a function on the Nspire handheld to solve that equation for i to find the interest rate. On the handheld, you’d input something like this:

Using nSolve on the Nspire handheld, you can solve the equation to find the interest rate. How could you modify the input to solve for the number of months as in problem 13 from the 16.1 homework?
Using nSolve on the Nspire handheld, you can solve the equation to find the interest rate. How could you modify the input to solve for the number of months as in problem 13 from the 16.1 homework?

Note that the variable you solve for is the interest per month. To get the equivalent annual rate, you must multiply by 12.

 

Repaying Loans

Explore MathAs the end of the year approaches rapidly, we’ll now shift our focus toward understanding money and finance. Our first exploration will look at how interest rates and the duration of loans combine to impact the cost of borrowing money.

As you complete your homework, you’ll need to use an online calculator to find the monthly payment for a given set of loan terms. One calculator can be found at Bankrate.com. Alternatively, you can use an Nspire calculator with the function tvmPmt. The required parameters for the function are: number of months, interest rate (leave off the percent), loan amount, 0 (zero), 12, 12. For a loan of $5,000 for 36 months at 15% interest, your function would be entered as tvmPmt(36,15,5000,0,12,12) and would return -173.327. You should get the same result entering those criteria on a web-based calculator.

If you want to calculate the old-fashioned way, you can use the following formula.

\text{R=}\frac{\text{P}\cdot \text{i}}{\text{1-(1+i}{{\text{)}}^{\text{-n}}}}

\begin{array}{l}\text{R=monthly}\ \text{payment}\\\text{P=loan amount}\\\text{i=monthly interest rate in decimal}\\\text{n=number of payments}\end{array}

You will have to convert the annual interest rate in percent to decimal and divide by 12 to get the monthly interest rate.

Personally, I’d stick with the web calculator, but I thought you might like to see the alternatives.

In class we mostly discussed short-term loans. You may have heard the President encouraging people to convert to 15 year mortgages. Compare these two options.

Joanne purchases a $150,000 home using a 30 year mortgage at 5.5% interest.

Bob purchases a similar $150,000 home with a 15 year mortgage at 5.5% interest.

What is the cost of each loan? How much more interest will Joanne pay compared to Bob?

Exploring Bivariate Data

Explore MathWe’re trying a Math Nspired™ lab provided by Texas Instruments today and tomorrow.

Would you like to try this at home? If you have a TI nSpire calculator you can download the tns file to your handheld or to the Nspire app on your iPad and try this any time. You can also try the student software free for 30 days using the link on the Resources page.

Maybe your parents would like to give it a try!

Exploring_Bivariate_Data_Student

Exploring_Bivariate_Data.tns

Even if you don’t have access to an Nspire calculator, you can now examine the plotted data from this exercise! The plots (with any luck even numbered correctly) are posted below.

1.2
1.2
2.1
2.1
3.1
3.1
4.1
4.1