MTEL Mathematics (Secondary) (63)

The questions in this subarea look simple on the surface, but they are rarely one-step recall. Expect applied, multi-step problems: a markup followed by a discount, a formula evaluated in scientific notation, a greatest common factor hidden inside a packaging problem, a matrix acting on a square, or a property that justifies a missing step in an argument. The two chapters in this section build the number sense that the rest of the test assumes, so time spent here pays off in every other subarea.

This chapter covers Objective 0001, the real number system. It reviews the structure of the real numbers and their subsets, the properties that govern arithmetic, exponents and roots, scientific notation, the Fundamental Theorem of Arithmetic and the Euclidean algorithm, divisibility rules and their proofs, and the ratio, rate, and percent problems that appear in nearly every MTEL form.

The real numbers are built up in nested subsets. Each set is closed under more operations than the one before it.

Subsets of the Real Numbers

A rational number is any number that can be written as p/q, where p and q are integers and q ≠ 0. Every rational number has a decimal expansion that either terminates or eventually repeats, and every terminating or repeating decimal is rational. An irrational number has a decimal expansion that neither terminates nor repeats. The rationals and irrationals together make up the reals, and they do not overlap.

A fraction p/q in lowest terms has a terminating decimal exactly when the prime factorization of q contains only 2s and 5s. So 7/40 = 0.175 terminates (40 = 2^3 · 5), while 7/30 = 0.2333... repeats (30 contains a factor of 3).

Common Trap: The irrational numbers are not closed under any of the basic operations. For example, sqrt(2) · sqrt(2) = 2 and sqrt(2) + (-sqrt(2)) = 0, both rational. A test item that asks "which set is closed under multiplication" will often list the irrationals as a tempting distractor.

The real numbers form an ordered field. The properties in the following table are the ones MTEL expects you to name when it asks which property justifies a step.

Properties of Real-Number Operations

Two more properties matter. The density property says that between any two distinct real numbers there is another real number; in fact there are infinitely many rationals and infinitely many irrationals between them. For example, (a + b)/2 is always a rational number between two rationals a and b. The integers are not dense: there is no integer between 3 and 4.

The order properties (trichotomy, transitivity, and the rules for inequalities) also come from the field structure. Adding the same number to both sides of an inequality preserves it; multiplying by a negative number reverses it.

Exam Tip: When a question shows a chain of equalities and asks which property justifies one step, compare the expressions on either side of that step and ask what actually changed. If only the grouping changed, it is associative. If only the order changed, it is commutative. If a product was expanded or factored, it is distributive. If something like x + 0 became x, it is the additive identity.

The absolute value |a| is the distance from a to 0 on the number line, so |a| = a if a ≥ 0 and |a| = -a if a < 0. The distance between a and b is |a - b|. Two important facts are |ab| = |a| |b| and the triangle inequality |a + b| ≤ |a| + |b|.

To order numbers given in mixed forms (fractions, decimals, percents, radicals), convert everything to decimals with the calculator, or compare squares when all values are positive. For example, to compare 3 sqrt(5) and 2 sqrt(11), square both: 45 versus 44, so 3 sqrt(5) is larger.

Estimating irrational numbers relies on perfect squares. Since 49 < 55 < 64, sqrt(55) is between 7 and 8, and since 55 is 6 units past 49 out of the 15-unit gap to 64, a reasonable estimate is a little less than 7.5. (The calculator gives 7.416.)

Place value in base ten means each position is worth ten times the position to its right. Questions about place value often appear as reasonableness checks: is a product of two three-digit numbers about 10^4, 10^5, or 10^6?

Inverse operations undo each other: addition and subtraction, multiplication and division, squaring and taking a nonnegative square root (for nonnegative inputs), and raising to a power and taking the corresponding root. Recognizing inverse pairs is the basis of equation solving.

To convert a repeating decimal to a fraction, multiply by a power of 10 that shifts one full period, then subtract.

Worked Example: Write 0.3181818... (the block 18 repeats) as a fraction in lowest terms.

Let x = 0.31818.... The repeating block has length 2 and starts after one nonrepeating digit.

Area models give a visual meaning to multiplication. The product (2/3)(3/5) is the overlap when a unit square is cut into thirds one way and fifths the other way: the overlap is 2 × 3 = 6 of the 15 small rectangles, or 6/15 = 2/5. The same model explains (a + b)(c + d) = ac + ad + bc + bd as the sum of four rectangular areas, and it is how many teachers introduce the distributive property and multi-digit multiplication.

For a ≠ 0 and integers m and n, the exponent laws are:

Laws of Exponents

The laws for rational exponents hold when a > 0. With negative bases, care is needed: (-8)^(1/3) = -2 is fine, but (-4)^(1/2) is not a real number. The definition a^(1/n) = nth root of a is forced by the power-of-a-power law, since (a^(1/n))^n must equal a^1 = a.

The principal square root sqrt(a) is always nonnegative, so sqrt(x^2) = |x|, not x. Radicals simplify by pulling out perfect-power factors: sqrt(72) = sqrt(36 · 2) = 6 sqrt(2). To rationalize a denominator such as 1/(3 - sqrt(2)), multiply numerator and denominator by the conjugate 3 + sqrt(2), giving (3 + sqrt(2))/7.

A number in scientific notation has the form a × 10^n, where 1 ≤ |a| < 10 and n is an integer. Multiply or divide the coefficients and add or subtract the exponents, then renormalize.

A prime is an integer greater than 1 whose only positive divisors are 1 and itself. An integer greater than 1 that is not prime is composite. The number 1 is neither.

The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be written as a product of primes, and this factorization is unique apart from the order of the factors. For example, 360 = 2^3 · 3^2 · 5.

From prime factorizations you can read the GCF (take each common prime to its smallest exponent) and the LCM (take every prime to its largest exponent). For any positive integers a and b, GCF(a, b) · LCM(a, b) = ab.

To test whether n is prime, you only need to check divisibility by primes up to sqrt(n). If n = ab with both a and b greater than sqrt(n), then ab > n, a contradiction, so at least one factor is at most sqrt(n). For 221, check primes up to 14: 221 = 13 · 17, so it is composite.

There are infinitely many primes. Euclid's proof is a classic indirect proof that MTEL may ask you to follow or complete.

Proof: There are infinitely many primes.

Suppose, for contradiction, that there are only finitely many primes p_1, p_2, ..., p_k. Form N = p_1 p_2 ··· p_k + 1.

N is greater than 1, so it has at least one prime factor p.

Dividing N by any p_i leaves remainder 1, so no p_i divides N. Therefore p is not on the list.

This contradicts the assumption that the list contained every prime.

This completes the proof.

Divisibility Rules

The rules for 3 and 9 have a short proof based on the fact that 10 leaves remainder 1 when divided by 9, so every power of 10 also leaves remainder 1.

Proof: A positive integer is divisible by 9 exactly when the sum of its digits is divisible by 9.

Write the number as n = d_k 10^k + ... + d_1 10 + d_0, where the d_i are its digits.

Therefore n and its digit sum differ by a multiple of 9, so one is divisible by 9 exactly when the other is.

This completes the proof.

Because 9 is a multiple of 3, the same argument proves the rule for 3. The rule for 11 works because 10^k leaves remainder 1 or -1 (mod 11) as k is even or odd.

Square roots connect to prime factorization: an integer is a perfect square exactly when every exponent in its prime factorization is even. That fact gives the standard proof that sqrt(2) is irrational. If sqrt(2) = p/q, then 2q^2 = p^2. The prime 2 appears an even number of times in p^2 but an odd number of times in 2q^2, contradicting unique factorization.

For large numbers, factoring is slow. The Euclidean algorithm finds the GCF by repeated division, using the fact that if a = bq + r, then GCF(a, b) = GCF(b, r). Any common divisor of a and b also divides r = a - bq, and any common divisor of b and r also divides a = bq + r.

Worked Example: A teacher has 462 red tiles and 1071 blue tiles. She wants to pack them into identical kits with no tiles left over, each kit having the same number of red tiles and the same number of blue tiles. What is the greatest number of kits she can make, and what is the LCM of 462 and 1071?

Result: She can make 21 kits; LCM(462, 1071) = 23,562.

Running the algorithm backward writes the GCF as a combination of the original numbers: 21 = 7 · 462 - 3 · 1071.

Memory Aid: For GCF, "smallest exponents, shared primes only"; for LCM, "largest exponents, every prime." Then check with GCF × LCM = product.

A percent is a ratio with denominator 100. Most percent problems reduce to "part = percent × whole." Percent change is (new - old)/old. Successive percent changes multiply: an increase of r% followed by a decrease of s% multiplies the original by (1 + r/100)(1 - s/100).

Percent Change Multipliers

Worked Example: A store marks up a calculator 30% over its cost, then later puts it on sale at 30% off the marked price. As a percent of the cost, what is the store's profit or loss on the sale?

Let the cost be C. The marked price is 1.30C. The sale price is 0.70 × 1.30C = 0.91C.

The sale price is 0.91C, which is less than C, so the store loses money.

To reverse a percent change, divide by the multiplier. If a price after a 20% discount is $64, the original price was 64/0.80 = $80, not 64 × 1.20 = $76.80.

A ratio compares quantities of the same kind; a rate compares quantities with different units, such as miles per hour. A proportion states that two ratios are equal, a/b = c/d, and cross-multiplying gives ad = bc. A unit rate (a rate with denominator 1) is the most efficient way to compare options: at $4.29 for 15 ounces versus $5.49 for 20 ounces, the unit prices are about $0.286 and $0.275 per ounce, so the larger package is the better buy.

Ratio problems with a total are solved with a common multiplier. If boys and girls are in the ratio 4:5 and there are 72 students, then 4k + 5k = 72, so k = 8 and there are 32 boys and 40 girls.

Exam Tip: The calculator handles arithmetic, but MTEL tests whether you set up the right multiplier or proportion. Write the multiplier chain on the scratch paper before touching the calculator, and do a quick reasonableness check: a discount should lower the price, a markup should raise it.

Know the nested subsets N, W, Z, Q, and R, which operations each is closed under, and that the irrationals are closed under none of the four basic operations.

Identify the field property that justifies a step by asking what changed: order (commutative), grouping (associative), expansion (distributive), or an identity or inverse.

Use prime factorization or the Euclidean algorithm for GCF and LCM, and use GCF · LCM = ab as a check.

Be ready to follow or complete proofs of divisibility rules, the infinitude of primes, and the irrationality of sqrt(2).

Chain percent changes by multiplying multipliers; reverse a percent change by dividing.