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# decimal grids

From Wikipedia

Multiplication table

In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system.

The decimal multiplication table was traditionally taught as an essential part of elementary arithmetic around the world, as it lays the foundation for arithmetic operations with our base-ten numbers. Many educators believe it is necessary to memorize the table up to 9 Ã— 9.

In his 1820 book The Philosophy of Arithmetic, mathematician John Leslie published a multiplication table up to 99 Ã— 99, which allows numbers to be multiplied in pairs of digits at a time. Leslie also recommended that young pupils memorize the multiplication table up to 25 Ã— 25.

In 493 A.D., Victorius of Aquitaine wrote a 98-column multiplication table which gave (in Roman numerals) the product of every number from 2 to 50 times and the rows were "a list of numbers starting with one thousand, descending by hundreds to one hundred, then descending by tens to ten, then by ones to one, and then the fractions down to 1/144" (Maher & Makowski 2001, p.383)

The traditional rote learning of multiplication was based on memorization of columns in the table, in a form like

1 Ã— 10 = 10 2 Ã— 10 = 20 3 Ã— 10 = 30 4 Ã— 10 = 40 5 Ã— 10 = 50 6 Ã— 10 = 60 7 Ã— 10 = 70 8 Ã— 10 = 80 9 Ã— 10 = 90

10 x 10 = 100 11 x 10 = 110 12 x 10 = 120 13 x 10 = 130 14 x 10 = 140 15 x 10 = 150 16 x 10 = 160 17 x 10 = 170 18 x 10 = 180 19 x 10 = 190 100 x 10 = 1000

This form of writing the multiplication table in columns with complete number sentences is still used in some countries instead of the modern grid above.

## Patterns in the tables

There is a pattern in the multiplication table that can help people to memorize the table more easily. It uses the figures below:

â†’ â†’ 1 2 3 2 4 â†‘ 4 5 6 â†“ â†‘ â†“ 7 8 9 6 8 â†� â†� 0 0 Fig. 1 Fig. 2

For example, to memorize all the multiples of 7:

1. Look at the 7 in the first picture and follow the arrow.
2. The next number in the direction of the arrow is 4. So think of the next number after 7 that ends with 4, which is 14.
3. The next number in the direction of the arrow is 1. So think of the next number after 14 that ends with 1, which is 21.
4. After coming to the top of this column, start with the bottom of the next column, and travel in the same direction. The number is 8. So think of the next number after 21 that ends with 8, which is 28.
5. Proceed in the same way until the last number, 3, which corresponds to 63.
6. Next, use the 0 at the bottom. It corresponds to 70.
7. Then, start again with the 7. This time it will correspond to 77.
8. Continue like this.

Figure 1 is used for multiples of 1, 3, 7, and 9. Figure 2 is used for the multiples of 2, 4, 6, and 8. These patterns can be used to memorize the multiples of any number from 1 to 9, except 5.

## In abstract algebra

Multiplication tables can also define binary operations on groups, fields, rings, and other algebraic systems. In such contexts they can be called Cayley tables. For an example, see octonion.

## Standards-based mathematics reform in the USA

In 1989, the National Council of Teachers of Mathematics (NCTM) developed new standards which were based on the belief that all students should learn higher-order thinking skills, and which recommended reduced emphasis on the teaching of traditional methods that relied on rote memorization, such as multiplication tables. Widely adopted texts such as Investigations in Numbers, Data, and Space (widely known as TERC after its producer, Technical Education Research Centers) omitted aids such as multiplication tables in early editions. It is thought by many that electronic calculators have made it unnecessary or counter-productive to invest time in memorizing the multiplication table. NCTM made it clear in their 2006 Focal Points that basic mathematics facts must be learned, though there is no consensus on whether rote memorization is the best method.

From Encyclopedia

Binary Number System Binary Number System

Answers:I think I can help with a few... 2/4 = 4/8 2/3 = ?/6 and 5/6 2/3 = 4/6 and 5/6 Brenda has the bigger piece. do a pie chart divided into 6...... 51 + 3 = 54 -26 + 3 = 29 ___________ 25 + 3 = 25 (subtract each set of numbers before and after the 3s) 4/100 and 25 2,542 6/100 and 2542.16

Answers:Most of these are standard definition and test taking techniques. If you need help here, there's nothing we can do to help you with your SAT.

Question:Ohkay. I need some MAJOR help on math. The title is "relateing rational numbers" and there is a box that says: FRANKS BUDGET: clothing - 1/8 entertainment - 1/20 food - 27.5% other - 0.4% transportation: 15% ----- then it says: "The grid below represtents 100% of Frank's monthly salary after taxes. Shade the grid to represent the information in the table above. [there is a grid with 100 little blocks] then there is questions: 12. What fraction of Frank's budget is used for TRANSPORTATION? explain the strategy you used to convert the perfect to a fraction. 13. hat percent of Frank's budget is used for CLOTHING? explain the strategy you used to convert the fraction to percent. 14. Use the shaded model to verify the percent given in problem 7 is equivalent to 1/8. 15. What decimal of Frank's budget is allotted for ENTERTAINMENT? explain the strategy you used to convert the fraction to decimal. 16. [same question, but with OTHER] 17. [same question but with FOOD] 18. What fraction of Frank's budget does he spend the least amount? Justify your response. 19. What catefory of Frank's budget does he spend the least ammount? justify your response. 20. Describe how to verify the given amounts in the table represent 100% of Frank's budget after taxes. What form of rational numbers did you choose to verify the total of the amounts represent 100% and why? 21. Without doing any calculations, estimate a resonable solution for the amount Frank spends on clothing each month if his salary is $3,500 after taxes. Describe the process you use. 22. Calculate the amount Frank spends on clothing each month if his salary after taxes is$3,500. Justify your answer. OKAY! :] i just need help, because i'm not that smart, i'd ask my dad but he is comeing back from Austin and is getting back late tonight. I NEED HELP right now, or i'll get into some trouble at school.. i'd do this earlier but i've been busy! pleaseee! -mandy