Mental Math curriculum

Split each number by place value and add each place separately. For 286 + 149, add the hundreds (300), the tens (80 + 40 = 120), and the ones (6 + 9 = 15), then combine: 300 + 120 + 15 = 435. Working from the left keeps the size of the answer in view and leaves the carrying to the final step.

When a number sits just below a round one, use the round number and correct afterward: 347 + 198 is 347 + 200 − 2 = 545. Subtraction works the same way: 512 − 297 is 512 − 300 + 3 = 215. Watch the direction of the correction; getting it backward puts the answer off by twice the correction.

Shift a little from one addend to the other so that one becomes round: for 496 + 237, move 4 across to get 500 + 233 = 733. The total is unchanged because the same amount was added and taken away. The method is quickest when one number is close to a hundred or a ten.

Before adding a list, look for pairs that make round numbers. In 36 + 58 + 64 + 42, pair 36 + 64 = 100 and 58 + 42 = 100 for a total of 200. Pairing removes most of the carrying, which is where list totals usually go wrong.

Multiplication and division come before addition and subtraction. In 12 + 15 × 4, multiply first, 15 × 4 = 60, then add: 12 + 60 = 72. Working strictly left to right, (12 + 15) × 4, gives 108, which is the most common wrong answer.

Anything in parentheses is done first, then multiplication, then addition and subtraction: (13 + 8) × 5 − 11 = 21 × 5 − 11 = 105 − 11 = 94. Doing the subtraction before the multiplication, as in 21 × (5 − 11), gives a negative number and is not what the expression says.

Multiplication and division share a level, as do addition and subtraction, and within a level you work left to right. So 48 ÷ 6 × 2 = 8 × 2 = 16, not 48 ÷ 12 = 4, and 20 − 7 + 3 = 16, not 20 − 10 = 10.

Before computing, rewrite the expression with the grouping the rules imply: 14 + 6 × 9 − 8 becomes 14 + (6 × 9) − 8 = 14 + 54 − 8 = 60. Making the hidden grouping visible prevents most order-of-operations slips, especially in a long expression.

Break one factor into parts that are easy to multiply, then add the partial products: 7 × 238 = 7 × 200 + 7 × 30 + 7 × 8 = 1,400 + 210 + 56 = 1,666. Starting with the largest part keeps the size of the answer in view from the first step.

When a factor is just below a round number, multiply by the round number and take off the extra: 24 items at $1.99 cost 24 × $2 − 24 × $0.01 = $48 − $0.24 = $47.76. Prices such as 99 cents or $1.99 are built for this: a dollar or two minus a cent.

Halving one factor and doubling the other leaves the product unchanged, and it can turn a hard product into an easy one: 18 × 35 = 9 × 70 = 630, and 16 × 125 = 8 × 250 = 4 × 500 = 2,000. It works best when one factor is even and the other ends in 5.

Multiplying by 5 is multiplying by 10 and halving: 286 × 5 = 2,860 ÷ 2 = 1,430. Multiplying by 25 is multiplying by 100 and dividing by 4: 64 × 25 = 6,400 ÷ 4 = 1,600. Both replace a hard multiplication with an easy one and a division.

Multiplying by 9 is multiplying by 10 and subtracting one copy: 9 × 147 = 1,470 − 147 = 1,323. Multiplying a two-digit number by 11 puts the sum of its digits between them: 11 × 63 = 693. When that sum passes 9, carry: 11 × 78 = 858, from 7 + 8 = 15.

Two numbers equally far from a round number multiply to the round number squared minus the distance squared: 102 × 98 = 100² − 2² = 9,996, and 17 × 23 = 20² − 3² = 391. The identity is (a + b)(a − b) = a² − b², and it applies whenever the midpoint is easy to square.

Round each factor to the nearest ten to estimate a product: 68 × 32 is about 70 × 30 = 2,100. The exact answer, 2,176, is close, so choices such as 21,760 or 1,176 can be rejected at once. When one factor rounds up and the other down, the errors partly cancel.

Treat a division as a missing factor: 144 ÷ 16 asks what times 16 makes 144, and 9 × 16 = 144, so the answer is 9. Multiplying back is also the fastest check for any division answer.

Split the dividend into parts the divisor divides evenly: 252 ÷ 7 = 210 ÷ 7 + 42 ÷ 7 = 30 + 6 = 36. Choose a first part that is a round multiple of the divisor, such as 7 × 30 = 210, so that what remains is small.

Dividing by 4 is halving twice, and dividing by 8 is halving three times: 304 ÷ 8 goes 304, 152, 76, 38. Each halving is easier than the full division, and an odd number partway through shows that the result will not be a whole number.

Dividing by 5 is doubling and dividing by 10: 345 ÷ 5 = 690 ÷ 10 = 69. Dividing by 25 is multiplying by 4 and dividing by 100: 1,175 ÷ 25 = 4,700 ÷ 100 = 47. Both turn the division into steps that are hard to get wrong.

A number leaves the same remainder on division by 9 as the sum of its digits does. The digits of 5,836 add to 22, which leaves 4 when divided by 9, so 5,836 leaves 4 as well. Keep adding digits until one digit remains; a final 9 means the remainder is 0.

The digit sum also tests divisibility by 3: 4,371 has digit sum 15, which 3 divides but 9 does not. So 4,371 is divisible by 3, and it leaves remainder 6 on division by 9. The test needs no division at all, only addition.

A division with a remainder satisfies dividend = divisor × quotient + remainder, and the remainder is always smaller than the divisor. For 5,836 ÷ 9 the quotient is 648 and the remainder 4, since 9 × 648 = 5,832. A remainder as large as the divisor means the quotient was too small.

Find 10% by dividing by 10, then build from it. For 520, 10% is 52, so 30% is 156, 5% is 26, and 35% is 156 + 26 = 182. Each step is a doubling, halving, or addition that is easy to check.

Several percents are simple fractions: 50% is a half, 25% a quarter, 20% a fifth, and 75% three quarters. So 75% of 96 is three quarters of 96: 96 ÷ 4 = 24, and 3 × 24 = 72. One division and one small multiplication replace the percent arithmetic.

A percent of a percent multiplies the two rates: 35% of 60% of 500 is 0.35 × 0.6 = 0.21 of 500, which is 105. Multiplying the rates first keeps the numbers small, and the order does not matter, since 60% of 35% gives the same result.

When one percent of a number is known, scale it instead of finding the number first. If 35% of a number is 28, then 5% is 28 ÷ 7 = 4, so 15% is 12 and the number itself is 80. Dividing down to a convenient step and multiplying up avoids fractions.

Successive percent changes multiply rather than add. A $1,040 price raised 25% and then cut 20% ends at 1,040 × 1.25 × 0.8 = $1,040, because 1.25 × 0.8 = 1. Adding the percents, +25 − 20 = +5%, would predict $1,092, which is wrong.

Raising by a percent and then cutting by the same percent always ends lower, because the cut applies to the larger amount. $250 raised 40% becomes $350, and a 40% cut then removes $140, leaving $210, not $250. In general the result is 1 − p² of the start, here 1 − 0.16 = 0.84.

To find a price before a change, divide by the change's factor. A coat selling for $117 after a 35% discount was 117 ÷ 0.65 = $180, and one selling for $92 after a 15% markup was 92 ÷ 1.15 = $80. Adding the percent to the new price gives the wrong base.

Square a number near a round one by writing it as a sum or difference: 97² = (100 − 3)² = 10,000 − 600 + 9 = 9,409. The middle term is twice the product, 2 × 100 × 3 = 600, and forgetting to double it is the usual slip.

To square a number ending in 5, multiply its tens part by one more than itself and write 25 after the result: 95² starts with 9 × 10 = 90, so it is 9,025, and 105² starts with 10 × 11 = 110, so it is 11,025. The pattern comes from (10n + 5)² = 100n(n + 1) + 25.

The last digit of a power depends only on the last digit of the base, and it repeats in a short cycle. For 13, the last digits of 13, 13², 13³, and 13⁴ are 3, 9, 7, and 1, then the cycle restarts, so 13^12, whose exponent is a multiple of 4, ends in 1. Divide the exponent by the cycle length and use the remainder.

A unit fraction such as one sixth means dividing into that many equal parts: one sixth of 78 is 78 ÷ 6 = 13. Check by multiplying back: 6 × 13 = 78. The larger the denominator, the smaller each part, which is a quick check on the size of the answer.

For a fraction with a numerator above 1, divide by the denominator, then multiply by the numerator: three eighths of 96 is 96 ÷ 8 = 12, and 12 × 3 = 36. Dividing first keeps the numbers small; multiplying first reaches the same answer through larger ones.

To count how many fractional pieces fit in an amount, divide by the fraction, which means multiplying by its reciprocal. Bottles that each hold 5/8 of a liter, needed for 15 liters, number 15 ÷ 5/8 = 15 × 8/5 = 24. Pieces smaller than 1 always give a count larger than the amount.

A sum of evenly spaced numbers is the number of terms times the average of the first and last term. For 2 + 4 + … + 70 there are 35 terms, and the average of 2 and 70 is 36, so the sum is 35 × 36 = 1,260. Count the terms carefully; it is the step most often wrong.

The numbers 1 to n add to n(n + 1) ÷ 2: 1 + 2 + … + 60 = 60 × 61 ÷ 2 = 1,830. Pairing 1 with 60, 2 with 59, and so on gives 30 pairs of 61, the same result. A row count that rises by one each row is this sum.

An average times its count gives the total. If 7 scores average 18, they total 126; if the first 6 add up to 104, the last is 126 − 104 = 22. Working through the total is faster and safer than comparing averages.

The minute hand moves 6° per minute, since it turns 360° in 60 minutes. The hour hand moves 30° per hour and also 0.5° per minute, because it drifts toward the next number as the minutes pass. Forgetting that drift is the most common error in clock-angle questions.

Find each hand's position measured from 12, then subtract. At 4:50 the minute hand is at 50 × 6 = 300° and the hour hand at 4 × 30 + 50 × 0.5 = 145°, so they are 155° apart. If the difference is more than 180°, subtract it from 360° to get the smaller angle.

The last digit of a product depends only on the last digits of the factors: 7 × 238 must end in 6, because 7 × 8 = 56. A choice ending in any other digit is wrong, whatever its size, so this check often settles a question on its own.

Digit sums give a quick check on multiplication. For 7 × 238 = 1,666, the digit sums of the factors are 7 and 4 (from 13), and 7 × 4 = 28 reduces to 1. The answer reduces the same way: 1,666 gives 19, then 10, then 1. A mismatch proves an error; a match is strong but not certain evidence.

Check the size of an answer before its digits. Two two-digit numbers multiply to three or four digits; a percent below 100% of a number is smaller than the number; a discounted price is lower than the original. An answer of the wrong size is wrong whatever its digits.