Following are some activities relating to arithmetic.
The items marked with are the contributions of the Summer 2000 participants.

 Things Could Be Worse Krypto Making Napier's Bones. Using the Mayan Number System. Using the Greek Number System. Things Could be Worse!

This activity is designed to help your student’s understand the history of long division. Most students do not like to do long division so by showing them the Galley method they will realize how easy the modern technique is and they will, hopefully, never complain about long division again.

Background: The Galley method or Scratch method was thought to have originated in India by the 12th century. Different sources stated different time frames; one source stated that Fibonacci invented this method in the 15th century. So as you can see the time frame is anywhere from the 12th to the 15th century. The Galley method derived its name from old-fashioned galley with sails set since upon completion of the problem it resembled the galley of a boat. The terminology, scratch method, was used since they did not have tools that allowed them to erase; therefore, they scratched out mistakes and kept on going with the division problem.

Time Allowance: One to two class periods depending on mathematical skills of the students.

Example: We will use the scratch method to solve 73485 divided by 214.

First, take a look at the modern technique vs. the scratch method: Now lets take a look at how it is done, keeping in mind that you must work in columns:

STEP 1: Write 73485 followed by a ( , underneath write the number 214 under the numbers 734. The placement of the number 214 is important, 2 is smaller than 7 so it can be placed in the same column. If it would’ve been greater than the 2 would be placed in the 2nd column. STEP 2: 2 times what number gets you close to 7? 3, place the 3 to the right of the ( . STEP 3: 2x3 = 6 and 7 - 6 = 1. Place the 1 directly above the 7. Scratch out the 2 and 7 since you have used them. STEP 4: 1x3 = 3 and 3 - 3 = 0. Place the 0 directly above the 3. Scratch out the 1 and 3. Step 5: 4x3=12 but since 4 - 12 is a negative number you need to borrow one from the column to the left. Since the number in the column on the left is 0 you will need to borrow one from the 1st column. Leaving 0 in the 1st column and 9 in the 2nd column. Once you borrow the 4 turns into a 14, then subtract 12 from 14 to give you 2. Place the 2 above the 4. Scratch out both 4’s. Now that you are done with 3 you need to add the number 214 again. Place the 2 underneath the 2nd column, 1 in the 3rd column and 4 in the 4th column next to the other 4.

STEP 6: 2 times what number gets you close to 9? 4, place the four to the right of the 3 in your answer spot. Continue with the same process as above but this time multiple by 4. 4x2 = 9 and 9 - 8 = 1, place the 1 above the 9 and scratch out the 2 and 9. STEP 7: 1x4 = 4 since you cannot subtract 4 from 2 then borrow from column to the left to make it 12. 12 - 4 = 8. Place the 8 above the 2, scratch out the 1 (from borrowing), 2 and 1. STEP 8: 4x4 = 16 and 18 - 16 = 2. Borrow 1 from 8, scratch out 8 leaving 7, scratch out 8 and write 2, scratch out 4. Place 214 again, 2 in the 3rd column, 1 in the 4th column and 4 in the 5th column.

STEP 9: 2 times what numbergets you close to 7? 3, place next to 4 in the answers spot. 2x3 = 6 and 7 - 6 = 1, scratch out the 7 and place a 1 above it. Scratch out the 2 that you used at the bottom. STEP 10: 1x3 = 3 and 2 - 3 you can’t have so borrow from 1 and the 2 becomes a 12, now 12 - 3 = 9. Place the 9 above the 2 and scratch out the 2, 1 and the 1 used for borrowing. STEP 11: 4x3 = 12 and 5 - 12 you can’t have so borrow 1 from the 8 leaving 8 and the 5 becomes a 15 15 - 12 = 3. Scratch out the 4 and 5 and place the 3 above the 5. The numbers remaining unscratched is the remainder and number following the ( is your quotient.

Fun huh? It will take students awhile to catch on to the scratch or Galley method but it is fun, once you get the hang of it. Enjoy!

Contributed by Carey Eskridge-Lybarger

References:

1. Smith, D. (1955). Number Stories of Long Ago. United States of America.
2. Freitag, A., & Freitag, H. (1960). The Number Story. United States of America.
3. Boyer, C., & Merzbach U. (1989). A History of Mathematics. New York, NY: John Wiley & Sons, Inc.
4. Eves, H. (1983). An Introduction to the History of Mathematics. New York, NY: CBS College Publishing.

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Krypto

Grade Level:
Middle School

Purpose:
To test the students ability to add, subtract, multiply, and divide quickly and accurately.

Resources:

1. Pack of krypto cards (the teacher may have the students make their own deck of cards. They can do this by taking 3"x5" index cards and having the students write the numbers:
• 1-4, five times
• 5-8, 3 times
• 9-11, 2 times
• 12-15, 1 time,
and for higher level students add
• 16-20, 1 time.)
2. A sheet of paper and pencil for each participant, and one to keep score for each group.
3. Three to five students per group and 1 set of krypto cards per group.

Activity:
After the students have got into groups, have one student lay 5 cards face up in a row so all the students can see them. The first four are the numbers you are working with, and the fifth card is the anwser the participants are trying to come up with. The participants of the game may add, subtract, multiply, or divide any which way her or she can to make the first four cards equal the fifth card. They may put the numbers in any order that works. The first to come up with problem correctly anwsering the fifth card is the point. They may play to a certain number, or for a certain time period.

Contributed by Jennifer Goodwin

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Making Napier’s Bones

Lesson Objective:
Students will be able to multiply using Napier’s Bones

Supplies Needed:

• 3/4 in. craft sticks (jumbo craft sticks)
• ruler
• ink pen
Directions:
• Each student will need 11 craft sticks.
• Have the students mark off the stick in 1.5 cm increments. These lines can be drawn in pencil first and then traced with ink pen.
• This should mark off the stick into 10 even sections.
• Place the sticks so they are vertical on your desk.
• At the top of each stick, write the numbers 0 through 9. On the eleventh stick, write the word Index.
• On the index stick, write the numbers 1 through 9 going down.
• On the ten other sticks, the students need to draw a diagonal line in each section going from upper right to lower left. (the letter ‘Z’ will be formed)
• On each of the ten sticks, write the multiples of that number putting the tens on the left and the ones on the right.
Example:
The two stick (multiples of two) How to use your sticks to multiply:

Problem:
 834 x 72

• Place the 8,3, and 4 sticks side by side in that order.
• Take your index stick and place beside the eight stick. • Since the two is in the ones place, add up the diagonal sections that correspond with the index stick at 2. You should see 1668.[1(6+0)(6+0)8]
• Now go down to the seven and add the sections that correspond with the index stick at 7. You should get 5838. [5(6+2)(1+2)8] Because this was actually multiplied by 70 you would need to add a zero to the end of the number making it 58380.
• Now simply add 1668 and 58380.
• You should get 60048!
On your own:

Try these problems using your Napier’s Bones.
 7462 512 809 x 25 x 6 x 83
Ideas and Suggestions:

• You could give a brief overview of John Napier either before or after this activity.
• Instead of using calculators in class the students could use their Napier’s Bones.
• Use the sticks when working with fraction. They are great for finding the least common denominator!
• Have upper grade students make sets of Napier’s Bones and teach lower grade students how to use them.
• The jumbo sticks are easier to read than the smaller craft sticks, but you could use the smaller ones.
• Have your students show an adult how to use Napier’s Bones and write a reflection on what the adult thought.
• The list is endless, so have fun!

Contributed by Linette Liby

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Using the Mayan Number System

The Mayas flourished on the Yucatan peninsula of Mexico about 1300 years ago.

The Mayas had a base 20 number system instead of the base 10 that we are used to. There were variances in their place-value system. The priestly class used one system, and another was used by the common classes. The following is a place-value chart. Their numbers are vertical rather than horizontal like our own.

They wrote numerals from 1-19 as follows, with a dot for 1 and a horizontal bar for 5. The powers of 20 get large very fast as the table below shows.
 12,800,000,000 = 207 64,000,000 = 206 3,200,000 = 205 160,000 = 204 8,000 = 203 400 = 202 20 = 201 1 = 200

I will pass out beans for the one’s, toothpicks for the five’s and macaroni for the zero. The Mayan’s were one of the only ancient civilizations with a zero, . We’ll use these to count to 15.

Using their place value system we will change the following base 10 numbers to the Mayan’s base 20 number system.

 Problem Solution 48 = (2 x 201)+(8 x 200) 134 = (6 x 201)+(14 x 200) 286 = (14 x 201)+(6 x 200) 1849 = (4 x 202)+(12 x 201)+(9 x 200) The following are Mayan numerals. What is the value of the following in base 10.

 Problem Solution (15 x 201)+(6 x 200) = 306 (17 x 202)+(6 x 201)+(7 x 200)= 6,927 (6 x 203)+(0 x 202)+(0 x 201)+(3 x 200) = 48,003 (6 x 202)+(1 x 201)+(0 x 200) = 2,420

Adding using Mayan numbers:

Verify the following addition problem. Did you get 112,229 + 8,297 = 120,526?

Contributed by Pamela Nye

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Using the Greek Number System The table above is the ancient Greek number system. But that’s not all. It’s also the Greek alphabet. It may have been hard to tell if the ancient artifacts were writings or if they were mathematics. • I have replaced the Greek letters with our own alphabet, in the above chart. When you see ‘h’, it will be equivalent to our 8. Notice that we lack enough letters in our alphabet so I used a ‘#’ for 900.
• Number your papers 1-5 using the Greek number system (a, b, c, d, e in our alphabet). Did you number your paper? Of course, not only does it look like we numbered our papers it also looks as if we put a-e.
• For the first exercise (a), write the number 174 using the table above.
• For the second exercise (b), write the number 573 using the table above.
• For(c) write 967
• For (d) write 82
• For (e) write 4,783. Oops! So far, we have only given the numbers through 999. How do you write larger numbers? For the number 1,000 the Greeks wrote (a’) or “(a) prime”. So, for 4,783 we will write d’yqc. d' = 4000, y = 700, q = 80 and c = 3.
• Now let’s try adding. I am going to add in our number system, then in the Greek system for the sake of familiarity and to check our work. • Subtraction in Greek is our next adventure with Greek numbers. • Multiplication is really a challenge. Let’s just try multiplying these two numbers. You will probably understand how the expression “It’s all Greek to me!” came about.

548 x 62 = 1096 + 32,880 = 33,976

wmh x ob = 1’rf + cMb’zq = cMc’?ph

The Greeks didn’t like doing multiplication with their number system either, so they came up with counting boards to do their math on. When we look at other number systems of the past, it helps us appreciate what we ordinarily take for granted. Like medicine, transportation, and many other things, mathematics has made many advances since ancient times.

Contributed by Pamela Nye

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