TExES Mathematics 4-8 (115)

These ideas look familiar because they are the backbone of grades 4 through 8, but the exam tests them from a teacher's point of view and reaches beyond middle school. Expect questions about why an algorithm works, which model best shows a concept, what a student's error reveals, and how the number system extends to the real and complex numbers. The six chapters in this section build that understanding step by step, with worked examples, common traps, and classroom scenarios like the ones you will see on test day.

This chapter covers the base-ten place value system, the role of zero, expanded and exponential notation, other number bases, comparing and ordering real numbers, and the models teachers use to represent numbers (base-ten blocks, number lines, fraction strips, and area models). It serves Competency 001, which asks you to understand the structure of number systems and to represent and compare quantities in many forms.

Our number system is a positional, base-ten system. Positional means the value of a digit depends on where it sits. Base ten means each place is worth ten times the place to its right, and one tenth of the place to its left. In the number 5,555, every digit is a 5, but the four digits are worth 5,000, 500, 50, and 5. That single idea explains regrouping in addition, borrowing in subtraction, why multiplying by 10 shifts digits one place left, and why decimals work the way they do.

The exam often asks about this multiplicative relationship directly. In 3,333.33, the 3 in the hundreds place is worth 300 and the 3 in the hundredths place is worth 0.03. The first is 300 ÷ 0.03 = 10,000 times the second. You can also count places: the hundreds place is four places to the left of the hundredths place, so the ratio is 10^4.

Exam Tip: When a question asks how many times greater one digit's value is than another's, count the number of places between them and raise 10 to that power. Check by dividing the two place values.

Zero plays two roles. As a number, it is the additive identity: a + 0 = a for every a. As a digit, it is a placeholder that keeps the other digits in their correct places. The numbers 47, 407, and 470 use the same nonzero digits, but the zeros change the values completely. Ancient systems without a placeholder zero, such as tally systems or Roman numerals, needed new symbols for larger values and made written computation very difficult.

Students who do not understand the placeholder role of zero make predictable errors. They may write "four hundred seven" as 4,007 because they record 400 and then 7 side by side, or as 47 because they leave out the empty tens place. They may also drop a zero in a quotient during long division, which you will study in Chapter 1.3.

Common Trap: A student reads 3,050 as "three thousand five hundred" because the student skips the zero in the hundreds place. The error comes from reading digits left to right without attaching each digit to its place. Asking the student to name the value of each digit (3,000 + 0 + 50 + 0) exposes the problem quickly.

A number can be written in several equivalent forms, and the exam expects you to move among them easily.

Standard form: 42,608.35

Notice that the tens place is missing from the expanded form because the tens digit is 0. Also notice that the word "and" marks the decimal point in word form. Saying "four hundred and six" for 406 is common in everyday speech, but in mathematics classrooms teachers reserve "and" for the decimal point.

The exponential form makes the structure clear: the exponents decrease by one as you move right, and they keep going below zero after the ones place. That is why 10^0 = 1 is not an arbitrary rule. The pattern 10^3 = 1,000, 10^2 = 100, 10^1 = 10 divides by 10 at each step, so the next term must be 1, and the one after that must be 1/10.

Teachers study other bases because they make the logic of place value visible. In base b, the places are worth b^0, b^1, b^2, b^3, and so on, and the only digits allowed are 0 through b − 1. Base five uses digits 0, 1, 2, 3, 4 and places worth 1, 5, 25, 125, 625. Base two (binary) uses only 0 and 1 and places worth 1, 2, 4, 8, 16, 32.

The following table shows the place values in three bases.

Place Values in Several Bases

To convert 446 back, divide by the largest place value that fits. 446 ÷ 125 = 3 remainder 71.

71 ÷ 25 = 2 remainder 21. Then 21 ÷ 5 = 4 remainder 1. The last remainder is the ones digit, 1.

Read the digits in order: 3, 2, 4, 1.

Result: 3241 in base five equals 446 in base ten, and the reverse conversion returns the original digits.

A second method for converting from base ten uses repeated division by the base and reading the remainders from bottom to top. For 446: 446 ÷ 5 = 89 r 1; 89 ÷ 5 = 17 r 4; 17 ÷ 5 = 3 r 2; 3 ÷ 5 = 0 r 3. Reading up gives 3241. Both methods rely on the same structure.

Exam Tip: On the exam, a base question is really a place value question. If you see a digit equal to or greater than the base (such as a 6 in a base-five numeral), that numeral is not valid, which is a useful way to eliminate choices.

A useful fact about any base: the largest numeral with n digits is one less than the next power of the base. In base ten, 999 + 1 = 1,000. In base five, 44 (five) + 1 = 100 (five), which is 25 in base ten. In base two, 11 (two) + 1 = 100 (two), which is 4. In base eight, 77 (eight) + 1 = 100 (eight), which is 64 in base ten. Students who understand this "rollover" understand regrouping in every base.

The TEKS expect students to represent numbers with concrete and pictorial models before working abstractly. You should know what each model shows well and where it falls short.

Common Number Models

Base-ten blocks are flexible. If the large cube is 1, then the flat is 0.1, the rod is 0.01, and the small unit cube is 0.001. Changing which block is "one" helps students see that decimals follow the same ten-to-one pattern as whole numbers.

A 10-by-10 grid is the classic area model for decimals and percents. Shading 37 small squares shows 0.37, 37/100, and 37 percent at once. Shading three full columns and seven more squares shows the same quantity as 3 tenths and 7 hundredths, which connects the grid to expanded form.

The number line is the most important model for middle school, because it works for every real number. A number's position shows its order, and the distance between two points shows the absolute value of their difference. On a number line, |−3 − 5| = 8 is the distance from −3 to 5.

To compare real numbers in different forms, the most reliable approach is to convert them all to decimals with enough places to separate them. A calculator helps, and the exam gives you an on-screen scientific calculator. For negative numbers, remember that the number farther left on the number line is smaller, so −0.8 is less than −0.75 even though 0.8 is greater than 0.75.

Worked Example: Order these numbers from least to greatest: −3/7, −0.43, −√0.18, −4/9.

Convert each to a decimal: 3/7 ≈ 0.42857, √0.18 ≈ 0.42426, 4/9 ≈ 0.44444, and 0.43 is exact.

The numbers are therefore about −0.42857, −0.43000, −0.42426, and −0.44444.

For negatives, the one with the largest absolute value is the least. The absolute values in order from largest to smallest are 0.44444, 0.43000, 0.42857, 0.42426.

So the order from least to greatest is −4/9, −0.43, −3/7, −√0.18.

Result: −4/9 < −0.43 < −3/7 < −√0.18.

Two strategies help when a calculator is not handy. First, benchmarks: compare each fraction to 0, 1/2, or 1. The fraction 7/15 is slightly less than 1/2 because 7.5/15 would equal 1/2. Second, cross-multiplication: for positive fractions, a/b < c/d exactly when ad < bc. To compare 5/8 and 7/11, compare 5 × 11 = 55 with 8 × 7 = 56. Since 55 < 56, 5/8 < 7/11.

Common Trap: Students often believe that a decimal with more digits is larger, so they say 0.405 > 0.45. This is "whole number thinking" applied to decimals. Writing both with the same number of places (0.405 and 0.450) or shading them on hundredths and thousandths grids shows that 0.45 is larger.

A related trap goes the other way. Some students learn "shorter is larger" after being corrected and then say 0.3 > 0.35. Neither length rule works. Only place value comparison does: compare tenths first, then hundredths, then thousandths.

Every rational number can be written in many equivalent forms: 3/8 = 6/16 = 0.375 = 37.5% = 3 × 2^-3. Competency 001 asks you to choose an appropriate form for a situation, not just convert. Some guidelines follow.

Fractions are best for exact values and for multiplying or dividing (2/3 × 9/10 is easier than 0.666... × 0.9).

Decimals are best for comparing values and for money and measurement.

Percents are best for comparing parts of different-sized wholes, such as 18 of 24 (75 percent) and 30 of 42 (about 71 percent).

Scientific notation is best for very large or very small quantities, such as 6.02 × 10^23 or 3.2 × 10^-6.

Radicals and exponents are best for exact irrational values: √50 = 5√2 is exact, while 7.071 is only an approximation.

Exam Tip: When answer choices mix forms, convert everything to one form, usually decimals, before comparing. When a question asks for an exact value, an answer such as 5√2 or 2π/3 is expected, and a rounded decimal choice is a trap.

Large whole numbers are read in groups of three digits called periods: ones, thousands, millions, billions, trillions. In 4,207,035,600 the periods are 4 (billions), 207 (millions), 035 (thousands), and 600 (ones). The number reads "four billion, two hundred seven million, thirty-five thousand, six hundred." The commas mark the periods, which is one reason teachers insist on writing them.

The same pattern helps with scientific notation. A billion is 10^9 and a million is 10^6, so a billion is 10^3, or 1,000, times a million. Students often underestimate that difference. A useful classroom comparison: a million seconds is about 11.6 days, while a billion seconds is about 31.7 years.

Rounding replaces a number with a nearby number that is easier to use, and it depends entirely on place value. To round 46,382.759 to the nearest hundred, look at the hundreds digit (3), check the digit to its right (8), and round up to 46,400. To the nearest hundredth, look at the hundredths digit (5), check the next digit (9), and round up to 46,382.76.

Teachers should know that rounding is about distance on a number line. 46,382.759 is between 46,300 and 46,400, and it is closer to 46,400. Students who learn only the "5 or more, round up" rule may round 3.449 to the nearest tenth by first rounding to 3.45 and then to 3.5. That double rounding is an error; the correct answer is 3.4 because 3.449 is closer to 3.4 than to 3.5.

Classroom Scenario: A sixth grader says that 0.3 is less than 0.25 "because 25 is bigger than 3." The student is applying whole number reasoning to decimals and is not attending to place value. The best next move is not simply to restate a rule. The teacher should have the student shade 0.3 and 0.25 on two 10-by-10 grids (30 squares and 25 squares) or locate both on a number line marked in hundredths, then ask the student to explain which shows more. Following up with "How many hundredths is 0.3?" connects the model to the notation and helps the student rename 0.3 as 0.30.

The concrete-representational-abstract (CRA) sequence appears throughout the exam. A student first handles base-ten blocks (concrete), then draws place value sketches or grids (representational), then works with numerals alone (abstract). When a student shows a misconception at the abstract level, a good teacher move often steps back to a representation, then reconnects it to the symbols.

The exam reaches above grade 8, so be ready for these extensions.

Any rational number has a decimal that either terminates or repeats. Any irrational number has a decimal that neither terminates nor repeats. Chapter 1.2 explores this in depth.

Numbers in any base can have "decimal" (radix) places. In base five, 0.2 (five) means 2/5 = 0.4 in base ten, and 0.13 (five) means 1/5 + 3/25 = 8/25 = 0.32.

Complex numbers a + bi are represented as points (a, b) in the complex plane. The horizontal axis is the real axis and the vertical axis is the imaginary axis. The absolute value (modulus) of a + bi is its distance from the origin, √(a^2 + b^2). So the number line is just the real axis of a larger plane.

Memory Aid: "Place, then face." To find a digit's value, first name its place, then multiply by its face value (the digit itself). For the 7 in 47,215: place = thousands (1,000), face = 7, value = 7,000.

Base ten is positional: each place is worth ten times the place to its right, so the ratio between two digit places is a power of 10.

Zero is both the additive identity and a placeholder; many student errors come from losing a placeholder zero.

Other bases (base two, base five, base eight) follow the same structure with powers of the base, and they make regrouping and rollover visible.

Compare real numbers by converting to a common form, usually decimals; for negatives, the number with the larger absolute value is smaller.

Choose forms on purpose: fractions for exact work, decimals for comparison and money, percents for different-sized wholes, scientific notation for very large or small values.

Models such as base-ten blocks, grids, number lines, and fraction strips support the CRA sequence and are often the best response to a place value misconception.