Krypto

Challenging card game available from Cuisenaire. Telephone 1-800-445-5985. Item CC010368. $7.00 + shipping ($5.00 on orders under $25.00. 10% on orders over $25.00.) A local school supply store may carry this product.

To play this game, you deal out five cards in a row and one underneath. Use addition, subtraction, multiplication and/or division to combine the top five numbers to end up with the last number. Each number can be used only once. For example: if you deal out 6, 1, 20, 7, and 8 on top and 5 underneath, you can combine the five numbers to get 5 in at least two ways:

1. 6 - 1 = 5
   20 divided by 5 = 4
   8 + 4 = 12
   12 - 7 = 5
2. 8 - 7 = 1
   6 - 1 = 5
   20 divided by 5 = 4
   4 + 1 = 5

Math Facts in Five Minutes a Day

In order to succeed with these Marvelous Math Tricks, students must know their facts, preferably to mastery. I have four books which can be used to practice facts to mastery in five minutes a day. There are many programs to drill these facts. Find one and use it consistently.

Adding by Endings

Use this technique with column addition. It will be helpful to use a colored pen. Begin adding each column from top to bottom. Whenever you get a two-digit sum, draw a line under the digit to represent the ten and keep the ending (the ones) in your head. At the bottom of the column, write down the number in your head (the ones) and count the lines for the number to carry. One caution: if you get to a two-digit number at the end of a column, such as 15, draw a line as before for the ten and write down the five. Do not carry just one ten. Count all of the marks in the column and carry.

Try using adding by endings to solve the following practice problem.

35,263
82,941
74,635
65,033
91,943
28,602
15,355

(The correct answer is at the end.)

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Gelosia Multiplication

The Gelosia method of multiplication predates the common algorithm. It is useful for solving very large multiplication problems. Students who can multiply large numbers feel quite confident to solve the smaller problems usually required of them.

1. Step 1. Using half-inch graph paper, make a grid with a square for each digit in the multiplicand across the top and a square for each digit in the multiplier down the right. Use a ruler to draw diagonal lines in each square from the top right to the bottom left.
2. Step 2. Fill in the grid with answers to each set of facts, tens in the top half of the intersecting square and ones in the bottom half.
3. Step 3. Add the columns diagonally. It is helpful to add by endings on big problems. Write the sum of each column below or beside where the column ends, wrapping the answer up the left side of the grid. Carry tens to the next column. Then copy the product, starting at the top left and going around the bottom. Add commas and check by casting 9s.

For practice, multiply 4,265 x 921. (Answer is at the end.)

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Casting Nines

Casting nines is a method of checking the accuracy of math calculation in addition, subtraction, multiplication and division. It is a "quick check" which confirms a correct answer and flags a wrong one. A great many mistakes in calculation are the result of careless errors and people's reluctance to spend time checking their work. Casting nines is quick and easy once learned, and many students consider it a game. In many cases, students who learn and apply this technique will get better grades and have more fun in math. When teachers don't have to deal with careless errors, it is possible to move more quickly to new math concepts. Warning: This checking method has one limitation. It can occasionally fail to flag a wrong answer. This happens when a zero is added or left out, when there is a transposition such as 3,528 instead of 3,582, and in a few other cases.

To cast nines from a number, add the digits:

17: 1 + 7 = 8 so 8 is the result of casting the nines

If the sum of the digits is nine, cast (throw away) the nine. It becomes a zero.

36:   3 + 6 = 9 and 9 - 9 = 0 so 0 is the result of casting the nines

If the sum of the digits is more than nine, cast the nines. You will always end up with just one digit when this is completed: 0, 1, 2, 3, 4, 5, 6, 7, or 8.

28:  2 + 8 = 10 and 10 - 9 = 1 so 1 is the result of casting the nines

Instead of subtracting the nine from a two-digit number as shown above, you may add the digits until only one digit remains. The result is the same.

28:  2 + 8 = 10 and 1 + 0 = 1 so 1 is the result of casting the nines

Here is an example of casting nines from a large number, 5,628:

5 + 6 = 11 and 1 + 1 = 2
2 (from 5 + 6) + 2 = 4
4 (from last line) + 8 (next digit) = 12
1 + 2 (from 12) = 3
3 is the result of casting nines from 5,628

To speed up casting nines from large numbers, look for and lightly cross out nines or combinations which equal nine. Add the remaining digits. For example, in 5,629,384,272.

Cross out the nine.
Cross out the combinations which equal nine: 5 + 2 + 2, 6 + 3, 7 + 2
Then add any digits still not crossed out: 8 + 4 = 12, 1 + 2 (from the 12) = 3
3 is the result of casting nines from 5,629,384,272.

Once you are able to cast nines from any number, use it to check problems as shown below.

For practice, cast nines from the following numbers:

___ 47
___255
___819
___3,212
___4,531
___9,306
___38,267
___846,787

(Answers are at the end.)

Checking Addition

Cast nines from both addends and add the one-digit results. Continue to cast nines until a one-digit "check number" remains. Then cast nines from the sum. The results must match or the answer is wrong. If the results do not match, thoroughly check the calculations to locate and fix the error.

For practice, check 468 + 327 = 795 to see if the answer is correct.

Checking Multiplication

Cast nines from the multiplicand and multiplier and multiply the results. Continue to cast nines until a one-digit "check number" remains. Then cast nines from the product. The results must match or the answer is wrong. If the results do not match, thoroughly check the calculations to locate and fix the error.

For practice, check 833 x 532 = 443,156 to see if the answer is correct.

Checking Subtraction

Cast nines from the minuend and the subtrahend. Subtract for the "check number". Then cast nines from the difference. The results must match or the answer is wrong. If the results do not match, thoroughly check the calculations to locate and fix the error.

For practice, check 692 - 325 = 367 to see if the answer is correct.

In subtraction, the one-digit result of the subtrahend may be larger than that of the minuend. If this occurs, add a nine to the top number so subtraction is possible. Continue as before.

For practice, check 1,243 - 746 = 497 to see if the answer is correct.

Checking Division

Cast nines from the dividend to get the "check number". Cast nines from the quotient, the divisor, and the remainder. Multiply the one-digit results as is done normally in checking division: quotient times divisor plus remainder. Cast nines as you go. The result must equal the check number or the answer is wrong.

For practice, check 36,225 / 75 = 483 to see if the answer is correct.

Answers:

• Adding by endings answer: 393,745
• Gelosia multiplication answer: 3,928,065
• Casting nines answers: 2, 3, 0, 8, 4, 0, 8, 4
• Checking addition: Check number is 3, answer is correct
• Checking multiplication: Check number is 5, answer is correct
• Checking subtraction: Check numbers are 7 and 2, answers are correct
• Checking division: Check number is 0

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Recommended Resources

• Krypto card game, available from Dale Seymour, 1-800-872-1100. ISBN 076908720, $7.95.
• Susan's books:
  • Addition Facts in Five Minutes a Day
  • Subtraction Facts in Five Minutes a Day
  • Multiplication Facts in Five Minutes a Day
  • Division Facts in Five Minutes a Day
  • Casting Nines
• Teach Timer, a "stopwatch" to use on the overhead in classrooms. (Click "Index of Products" then "Teacher Tools" then "Teach Timer.") From Kagan Cooperative Learning, $45. "Just put the TeachTimer on the overhead projector and let it keep track of time for you! The TeachTimer counts either down or up, and sounds and flashes a "Time's Up" alarm. Great for giving students five minutes to write in their journals, two minutes for pairs to discuss a topic, or even 10-15 seconds of think time. Use the built-in chronograph to see how quickly students can line up, make a formation, quiet down, or solve a challenging problem. When it's not in use, the TeachTimer doubles as an overhead clock. A great management tool viewed anywhere in the class. Includes hard protective case and attachment to wear on neck for outdoor use.