TSI Math Study Guide
Number Sense
Number sense is the ability to understand how numbers behave and to work with them accurately in different forms. These skills appear in direct computation questions, but they’re also essential for algebra, geometry, ratios, and percents.
Positive and negative numbers
On a number line, numbers increase as you move to the right.
−5 < −2 < 0 < 3 < 8
Don’t assume the negative number with the larger numerical part is greater.
−8 < −3
Think temperature: −8°F is colder — and therefore lower — than −3°F.
Adding and subtracting signed numbers
Same sign
Add the absolute values and keep the sign.−7 + (−4) = −11
Different signs
Subtract the smaller absolute value from the larger; use the sign of the larger.−9 + 5 = −4
For subtraction, rewrite as adding the opposite:
6 − (−3) = 6 + 3 = 9 −4 − 7 = −4 + (−7) = −11
Multiplying and dividing signed numbers
Same signs → positive
(−6)(−3) = 18
Different signs → negative
(−6)(3) = −1824 ÷ (−6) = −4
Fractions
Fractions appear throughout TSI math, so fraction operations are especially important.
Adding and subtracting
Fractions need a common denominator before they can be added or subtracted.
Don’t add the denominators. The denominator identifies the size of the pieces and must stay consistent.
Multiplying
Multiply the numerators and multiply the denominators. Simplify before or after when you can.
35 × 27 = 635
Dividing
To divide by a fraction, multiply by its reciprocal.
Remember
Keep the first fraction, change division to multiplication, flip the second fraction.
Fractions, decimals, and percents
The same number can be written in several forms. Moving between them is useful for comparing values.
34 = 0.75 = 75%
Fraction → decimal
Divide numerator by denominator.38 = 3 ÷ 8 = 0.375
Decimal → fraction
Use place value, then simplify.0.45 = 45100 = 920
Decimal → percent
Multiply by 100.0.62 = 62%
Percent → decimal
Divide by 100.7% = 0.07
Percent calculations are covered fully in Percents and Finance.
Rational and irrational numbers
Rational
Can be written as a fraction of two integers, a/b with b ≠ 0. Includes integers, fractions, terminating and repeating decimals.−4, 23, 0.75, 0.333…
Irrational
Can’t be written as a ratio of two integers. The decimal continues forever without repeating.2, 7, π
Not every square root is irrational.
49 = 7
7 is an integer, so it’s rational.
Exponents
43 = 4 × 4 × 4 = 64
Exponent rules
The exponent applies to −3.
The exponent applies only to 3; the negative comes after.
Roots
64 = 8 because 8 × 8 = 64
Perfect squares worth knowing:
Estimating a root
Absolute value
|5| = 5 and |−5| = 5
First evaluate the absolute value (5), then apply the negative sign.
Order of operations
Example
Comparing and ordering numbers
When numbers appear in different forms, convert them to a common form first.
The same strategy works for ordering several numbers from least to greatest or greatest to least.
TSI strategy: estimate before you calculate
Before a lengthy calculation, estimate what the answer should look like.
If your calculation gives 4, 400, or −40, something went wrong. Estimation catches misplaced decimals, sign errors, fraction mistakes, and calculator-entry slips — even when you use a calculator, judging whether a result is reasonable prevents avoidable errors.
When a problem has several number forms or operations, simplify one step at a time. Don’t do the whole calculation mentally. Rewrite fractions with common denominators, convert to comparable forms, and follow the order of operations. A few extra written steps are faster than fixing an avoidable mistake.