Project Math Access

Teaching Mathematical Concepts

Basic Number Facts and Operations

Collaborative and Inclusive Strategies

The Personal Perspective of Abraham Nemeth

The ability to calculate mentally with efficiency is a very important skill for all students, but especially for visually impaired and blind students. Using the braillewriter, and the abacus can be very labor intensive and time consuming, and calculators have their own limitations (see the discussion on calculators). The more efficiently students can estimate, calculate, and check the reasonableness of answers using mental math techniques, the more facile they will be at using numbers, in both schoolwork and independent living skills. These strategies should be taught to students as soon as they begin to count and work with simple numbers.

In order to manipulate numbers and calculate mentally, students must understand the concept of “complements” or “partners” of numbers. For example, in addition and subtraction, the student needs to know that the number 5 is made up of addends of 2 and 3, or 1 and 4 (complements, partners). Likewise, the number 12 is made up of 3 and 9, or 6 and 6, or 10 and 2. In multiplication and division, the student must know that the number 24 is made up of factors of 2 and 12, or 6 and 4, or 8 and 3.

While there are many individual techniques for estimating and calculating mentally, most strategies involve one of the following four basic approaches:

- decomposing numbers — breaking apart numbers into meaningful and useful units or groups that can be easily recomposed
- making easier numbers to work with — putting numbers together that are easier to use, often by changing the order of numbers
- substituting numbers — replacing values with equal values that are easier to manipulate
- compensating — rearranging numbers so they are easier to work with, either by changing a number and then adjusting the answer, or by adjusting both numbers so there is no need to change the answers

Following are several examples of strategies which may help students develop skills in counting and using the basic operations.

- Using the idea of complements, the student can adjust numbers to make adding a lot easier.
- Students can handle larger, more complicated numbers by starting their addition by adding the largest place values first, then next largest, etc.
- Students could also simplify their addition by adding the tens or hundreds together first, and then adding the units. (The student could write down subtotals of 120,15,135 as needed as he or she calculated mentally.)
- Students who have difficulty remembering many of the facts, can use the additive principle, doubles (2+2, 3+3) or facts or “partners” for numbers up to 10 (7+3, 6+4), and derive other facts from these (4+3=1 less than 4+4).
- For the addition of nines, the student can keep in mind that the one’s digit in the sum is always one less than the number added to the nine.

- The ability to understand the concept of partners or complements comes in handy in mental subtraction, as with other operations.
- Start by practicing subtracting partners from numbers up to 10.
- Continue the process by subtracting partners from numbers 20, 30, 40, etc.
- Continue practicing subtracting two digit numbers from 100, and then numbers larger than 100.
- Another approach involves subtracting numbers from smaller units which are closer to the actual subtrahend, and then adding the remaining portion. Always start by subtracting digits from the same number of digits immediately above it, then deal with the remaining amounts.
- When subtracting a number from a number that is a power of 10, use the complements that make up the numbers 9 and 10.
- Students can also “balance” numbers by adding the “same difference” to both to make them easier to work with.
- “Balancing” can also be done with decimals.

- Students can learn to think in patterns or arrays by using a “thinking model” with naturally occurring arrangements like those occurring in egg cartons, pop bottle cases, buttons on cards, cookies or candies packaged in rows, etc; then children can develop their own arrays. This approach can also be used with auditory cues; for example, how many times do you hear 2 taps, 3 rings, etc.?
- Emphasize the associative properties of the factors in multiplication. For example, remind the student that 3 fours is the same as 4 threes, 2 sixes is the same as 6 twos (rotate an egg carton 90 degrees to illustrate).
- Multiplication is repeated addition—if the child knows that 2 fours is 8, then 3 fours is 8 plus another 4, or 12.
- Use the concept of doubles—if the child knows that 2 sixes are twelve, then 4 sixes is twice as much, or 24.
- When multiplying by multiples of 10, students can just remember to add zeros-add one zero for each time that a number is multiplied by a multiple of 10.
- As with addition, students can also think of numbers as quantities rather than digits, and start by multiplying the largest units first.
- Another way of using this “front end” multiplication follows:
- When multiplying mentally by 5, 50, or 500, the student can simply multiply by 10,100, or 1000 and then divide by 2.
- When multiplying mentally by 9, 99 or 999, students can multiply by 10, 100, or 1000 and then subtract one multiplier.
- Another application of this “rounding” and adjusting approach could be used for many other numbers.
- When multiplying decimals by .1, .01 or .001, students can keep in mind that the number of decimal places is equal to the total number of decimal places in the factors.
- Whenever multiplying decimals by tens, the decimal point “slides” over one place to the right; when multiplying decimals by hundreds, the decimal point slides over two places to the right, etc. In the same manner, whenever dividing decimals by tens, the decimal point slides over one place to the left; when dividing decimals by hundreds, the decimal point slides over two places to the left, etc.
- A game called “Buzz” (Petreshene, 1985) can help students practice reviewing the products of a given number times many different multipliers. Specify a number, and have the student start counting from 1 to 100. Whenever he or she comes to a multiple of the specified number, the student says “buzz” instead of the actual number. For example, if 4 were the designated number, the student would count “1, 2, 3, buzz, 5, 6, 7, buzz, 9, 10, 11, buzz” and so on. Extra challenge could be added by calling any number which contains the designated number, “busy”: “1, 2, 3, busy-buzz, 5,6,7, buzz, 9,10,11, buzz, 13, busy, 15, buzz, 17,18,19, buzz, 21, 22, 23, busy-buzz” and so on. Variations might include time restraints, or brailling the numbers and words.

Example:

27x138 (think of 138 as 100 and 30 and 8, etc.)

20x100 equals 2000

20x30 equals 600, added to 2000 equals 2600

20x8 equals 160, added to 2600 equals 2760

7x100 equals 700, added to 2760 equals 3460

7x30 equals 210, added to 3460 equals 3670

7x8 equals 56, added to 3670 equals 3726

- Again, the student can use his or her understanding of partners or complements-how numbers are made up of other numbers-to calculate mentally.
- Just as it is sometimes easier to calculate multiplication mentally by dealing with the larger units within the factors, and then adding the progressive products, it can also help to use a similar approach when dividing larger numbers.

The student can keep a running record of his or her intermediate quotients and add them progressively (e.g., 100+30+8, or 138)

- Practice mentally calculating number chains; these can be used for any operation, or they can be combined to work on a variety of basic facts.
- Practice with familiarity in the sequence of numbers.
- Play games such as “secret number” where the student draws a card with a numerical relationship on it and another student must provide the answer to that relationship.
- Play a form of Jeopardy where the student provides the answer and other students provide different descriptions of that number.

Example:

5+3-4+3-5

6+4-2x3÷3

Example:

What comes after?

10, ___

23, ___

199, ___

268, ___

Example:

Skip counting ahead

2, 4, 6, __

3, 6, 9, __

5, 10, 15, __

15, 30, 45, __

Example:

What comes before?

__, 16

__, 25

__, 200

Example:

Skip counting backwards

10, 8, 6, __

15, 12, 9, __

350, 300, 250, ___

Example:

2 less than 87

11 more than 40

7 below 81

Example:

What is 2? (factor of 4, square root of 4, remainder of 10-8)

Example:

What is 100? (product of 10x10, half of 200, 1 more than 99)

A thorough search of the literature will reveal several books which provide numerous additional ideas for developing mental math skills at different grade levels.

Petreshene, S. S. (1985). *Mind joggers! 5 to 15 minute activities that make kids think*. West Nyack, NY: The Center for Applied Research in Education, Inc.