TX PACT Math 703 • Competency 001 • Chapter 1

Number Properties and Operations

Everything in this competency, shown as a diagram first. Five skills: (1) place value and magnitude, (2) the four whole-number operations, (3) equivalent forms, (4) operations with positive and negative numbers, (5) word problems with ratios and proportions.

1. Place Value and the Magnitude of Numbers

A digit's worth depends on its seat. Each seat to the left is worth ten times the seat on its right, all the way up to the ten-millions and beyond.

The whole-number board for 52,748,603

TEN MILLIONS	MILLIONS		HUNDRED THOUS	TEN THOUS	THOUS		HUND REDS	TENS	ONES
5	2	,	7	4	8	,	6	0	3

← each seat ×10 going left each seat ÷10 going right →

Read it: fifty-two million, seven hundred forty-eight thousand, six hundred three. The 5 alone is worth 50,000,000.

To the right of the decimal point

ONES		TENTHS	HUND REDTHS	THOUS ANDTHS
8	.	5	2	6

Tenths, hundredths, thousandths shrink by ten each step. 0.526 is just past one half.

Comparing and ordering: line up the seats

Pad with trailing zeros so every number has the same number of decimal seats, then read left to right. The first seat that differs decides which is larger.

Order 0.4, 0.41, 0.409 from least to greatest

0 . 4 0 0

0 . 4 1 0

0 . 4 0 9

Tenths tie at 4. The hundredths seat breaks the tie: 0 beats 1.

Common trap: "more digits means bigger." False for decimals. 0.409 has more digits than 0.41 but is smaller.

2. The Four Whole-Number Operations

Addition: carry when a column reaches ten

3,576 + 2,849

111

3576

+2849

6425

Ones: 6 + 9 = 15, write 5 carry 1. Tens: 7 + 4 + 1 = 12, write 2 carry 1. Keep going left.

Subtraction: borrow when the top is too small

742 - 568

61312

742

-568

174

Ones: 2 is less than 8, borrow to make 12 minus 8 = 4. Tens become 13 minus 6 = 7. Hundreds: 6 minus 5 = 1.

Check by adding back: 174 + 568 = 742. If it returns to the top number, the subtraction is right.

Multiplication: shift each partial product one seat left

246 × 23, colored by which digit you multiply

246

× 23

738

4920

5658

246 × 3 = 738 246 × 20 = 4920 placeholder 0 add = 5,658

The grey 0 is the point: the 2 in the tens seat shifts its partial product one place left.

Long division: divide, multiply, subtract, bring down

4,728 ÷ 6 = 788

788

6 4 7 2 8

quotient digits multiply 6 × quotient digit bring down the next digit

6 into 47 is 7 (42), remainder 5, bring down 2 to get 52. 6 into 52 is 8 (48), remainder 4, bring down 8 to get 48. 6 into 48 is 8, remainder 0.

On the exam: whole-number items are framed as inventory, seating, or fundraising totals. A wrong choice usually drops a placeholder 0 or forgets a carry, landing the answer off by a factor of ten.

3. Equivalent Ways to Write a Number

Exponents: repeated multiplication

2 to the 4th power

One amount, three outfits: fraction, decimal, percent

The bar shows they fill the same space

Convert: fraction → divide top by bottom → decimal → multiply by 100 → percent. Reverse the arrows to go back.

Scientific notation: where does the dot land?

Big: 53,400 becomes 5.34 × 10 to the 4

Tiny: 0.0072 becomes 7.2 × 10 to the negative 3

On the exam: to compare two values in scientific notation, compare the exponents first. Only if the exponents tie do you compare the front coefficients.

4. Operations with Positive and Negative Numbers

Add and subtract on the number line

Start at -3, add 5: land on +2

Adding moves right, subtracting moves left. Subtracting a negative, like 4 minus (-3), flips to 4 + 3 and moves right.

Multiplying and dividing signs

Same signs give positive, different signs give negative

× or ÷	+ value	- value
+ value	+	-
- value	-	+

(-6) × (-4) = +24. (-20) ÷ 5 = -4. 8 × (-3) = -24.

Common trap: two negatives multiplied make a positive. A wrong choice keeps the answer negative.

5. Fraction and Decimal Operations

Add: rebuild in a common seat size

Subtract: same idea, take away

Multiply: area model, 2/3 × 3/4

2 of 3 rows by 3 of 4 columns shades 6 of 12 cells = 6/12 = 1/2. Multiply straight across: 2×3 over 3×4.

Divide: keep, change, flip the divisor

6. Word Problems: Percents, Ratios, and Proportions

Percent of a number

30% of 250 students

part = percent × whole. Each tenth of 250 is 25, so 0.30 × 250 = 75.

Percent change

A quiz average rises from 60 to 75 points

original = 15 ÷ 60 = 25% increase

Common trap: divide the change by the original (60), not the new value. Dividing by 75 gives 20%, the classic wrong answer.

Ratio 3 blue : 2 white, scaled to 6 : 4

Proportion: cross-multiply

On the exam: word problems are framed as recipes, maps, discounts, and mixtures. Identify whether it is a percent, ratio, or proportion, set matching units across from each other, and solve.

Quick Reference Card

Each place is 10 times the one on its right, from ten-millions down through thousandths.
Pad decimals with zeros before comparing. More digits does not mean larger: 0.409 < 0.41.
Add and subtract by columns (carry past 10, borrow when the top is too small). Check subtraction by adding back.
Long multiplication: shift each partial product one seat left. Long division: divide, multiply, subtract, bring down.
Exponent = repeated multiplication. Scientific notation: coefficient 1 to under 10, exponent counts decimal moves (left positive, right negative).
Signs: same signs multiply or divide to positive, different signs to negative. Subtracting a negative adds.
Fractions: common denominator to add or subtract; multiply straight across; divide by keep, change, flip the divisor.
part = percent × whole. Percent change divides by the original. Cross-multiply to solve a proportion.

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