1. Introduction
The transition from whole numbers to integers and rational numbers is one of the most significant conceptual leaps in a student’s mathematical development. The NCTM identifies this as a critical milestone: “In the middle grades, students should … develop an understanding of and fluency with rational numbers.” [1] NCTM, Principles and Standards, 2000, p. 214
1.1 Why Do We Need Negative Numbers?
Whole numbers (\(0, 1, 2, 3, \ldots\)) cannot represent many common situations:
- Temperature: “It’s 15 degrees below zero” → \(-15°\text{C}\)
- Elevation: The Dead Sea is 430 meters below sea level → \(-430\) m
- Finance: A debt of $200 → \(-\$200\)
- Football: A 7-yard loss → \(-7\) yards
- Time zones: UTC−5 means 5 hours behind Coordinated Universal Time
Mathematically, negative numbers arise from the need to solve equations like \(x + 5 = 3\), which has no solution in the whole numbers. Extending to integers gives us \(x = -2\). [2] Wu, H.-H., Understanding Numbers in Elementary School Mathematics, AMS, 2011, Ch. 19
1.2 Historical Development
Negative numbers were first used systematically in China around 200 BCE in the Jiuzhang Suanshu (Nine Chapters on the Mathematical Art), where red counting rods represented positive numbers and black rods negative ones. Indian mathematicians, notably Brahmagupta (628 CE), gave the first explicit rules for arithmetic with negative numbers, describing them as “debts” versus “fortunes.” European mathematicians resisted negative numbers well into the 18th century — even Euler and d’Alembert debated their validity.
[3] Katz, V. J., A History of Mathematics, 3rd ed., Pearson, 2009, pp. 41, 230–233
2. The Integers
2.1 Definition & Notation
The integers, denoted \(\mathbb{Z}\), consist of all positive whole numbers, their negatives, and zero:
\( \mathbb{Z} = \{ \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \} \)
The symbol \(\mathbb{Z}\) comes from the German word Zahlen (numbers). [4] Hungerford, T. W., Abstract Algebra: An Introduction, 3rd ed., Cengage, 2014, p. 14
Subsets of Integers
| Name | Symbol | Definition | Examples |
|---|---|---|---|
| Positive integers | \(\mathbb{Z}^+\) | \(\{1, 2, 3, \ldots\}\) | 1, 7, 42, 1000 |
| Negative integers | \(\mathbb{Z}^-\) | \(\{-1, -2, -3, \ldots\}\) | −1, −7, −42 |
| Non-negative integers | \(\mathbb{Z}_{\geq 0}\) | \(\{0, 1, 2, 3, \ldots\}\) | 0, 1, 5, 100 |
| Natural numbers | \(\mathbb{N}\) | \(\{1, 2, 3, \ldots\}\) or \(\{0, 1, 2, \ldots\}\) | Convention varies |
[5] CCSS, Common Core State Standards for Mathematics, 2010, 6.NS.5–6
2.2 The Integer Number Line
The number line extends infinitely in both directions. Positive integers are to the right of zero; negative integers are to the left.
On the number line, numbers increase as you move right and decrease as you move left. For any two integers \(a\) and \(b\): \( a < b \iff a \text{ is to the left of } b \)
[5] CCSS, 2010, 6.NS.7a
2.3 Opposites (Additive Inverses)
Two numbers are opposites (or additive inverses) if they are the same distance from zero on opposite sides. For every integer \(a\), its opposite is \(-a\), and:
\( a + (-a) = 0 \)
- The opposite of \(7\) is \(-7\), because \(7 + (-7) = 0\)
- The opposite of \(-13\) is \(13\), because \(-13 + 13 = 0\)
- The opposite of \(0\) is \(0\), because \(0 + 0 = 0\)
- The opposite of \(-(-4)\) is \(-4\), and \(-(-4) = 4\)
For any integer \(a\): \(\quad -(-a) = a\)
The opposite of the opposite returns to the original number. [2] Wu, 2011, p. 310
3. Absolute Value
3.1 Definition
The absolute value of a number \(a\), written \(|a|\), is its distance from zero on the number line. Distance is always non-negative.
\( |a| = \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases} \)
[5] CCSS, 2010, 6.NS.7c–d
3.2 Examples
| Expression | Value | Reasoning |
|---|---|---|
| \(|5|\) | \(5\) | 5 is 5 units from zero |
| \(|-5|\) | \(5\) | −5 is also 5 units from zero |
| \(|0|\) | \(0\) | 0 is 0 units from zero |
| \(-|8|\) | \(-8\) | First \(|8|=8\), then negate: \(-8\) |
| \(|-3| + |-7|\) | \(10\) | \(3 + 7 = 10\) |
| \(|4 - 9|\) | \(5\) | \(|{-5}| = 5\) |
| \(-|-12|\) | \(-12\) | \(|-12|=12\), then \(-12\) |
| \(|(-2)^3|\) | \(8\) | \((-2)^3=-8\), then \(|-8|=8\) |
3.3 Distance Between Two Points
The distance between two numbers \(a\) and \(b\) on the number line is:
\( \text{distance} = |a - b| = |b - a| \)
- Distance between \(3\) and \(8\): \(|3 - 8| = |-5| = 5\)
- Distance between \(-4\) and \(6\): \(|-4 - 6| = |-10| = 10\)
- Distance between \(-7\) and \(-2\): \(|-7 - (-2)| = |-5| = 5\)
- Distance between \(-3\) and \(0\): \(|-3 - 0| = 3\)
The temperature dropped from \(8°\text{C}\) to \(-5°\text{C}\). How many degrees did it change?
\(|8 - (-5)| = |8 + 5| = |13| = 13°\text{C}\)
4. Adding & Subtracting Integers
Research shows that the “chip model” (positive/negative counters) and the “number line model” are the two most effective representations for integer operations. [6] Stephan, M. & Akyuz, D., “A Proposed Instructional Theory for Integer Addition and Subtraction,” Journal for Research in Mathematics Education, Vol. 43, No. 4, 2012, pp. 428–464
4.1 Adding Integers with the Same Sign
Add the absolute values and keep the common sign:
\( (+a) + (+b) = +(a+b) \qquad (-a) + (-b) = -(a+b) \)
- \((+4) + (+9) = +13\) — Both positive: \(4+9=13\), keep positive
- \((-3) + (-8) = -11\) — Both negative: \(3+8=11\), keep negative
- \((-15) + (-6) = -21\) — \(15+6=21\), negative
- \((+100) + (+250) = +350\)
- \((-1.5) + (-2.5) = -4.0\)
4.2 Adding Integers with Different Signs
Subtract the smaller absolute value from the larger, and use the sign of the number with the larger absolute value:
If \(|a| > |b|\), then \( a + (-b) = +(|a| - |b|)\text{ or }-(|a|-|b|) \) depending on the sign of \(a\).
- \((+8) + (-3) = +5\) — \(|8|>|{-3}|\); \(8-3=5\); positive wins
- \((-8) + (+3) = -5\) — \(|{-8}|>|3|\); \(8-3=5\); negative wins
- \((+12) + (-12) = 0\) — Equal absolute values → zero
- \((-50) + (+30) = -20\) — \(50-30=20\); negative wins
- \((+7) + (-19) = -12\) — \(19-7=12\); negative wins
- \((-2) + (+10) = +8\) — \(10-2=8\); positive wins
- \((-100) + (+75) = -25\) — \(100-75=25\); negative wins
4.3 Subtracting Integers
Subtracting an integer is the same as adding its opposite:
\( a - b = a + (-b) \)
This single rule reduces all subtraction problems to addition problems. [2] Wu, 2011, pp. 314–318
| # | Problem | Rewrite as Addition | Result |
|---|---|---|---|
| 1 | \(7 - 10\) | \(7 + (-10)\) | \(-3\) |
| 2 | \(-4 - 6\) | \(-4 + (-6)\) | \(-10\) |
| 3 | \(-9 - (-3)\) | \(-9 + 3\) | \(-6\) |
| 4 | \(5 - (-8)\) | \(5 + 8\) | \(13\) |
| 5 | \(0 - 15\) | \(0 + (-15)\) | \(-15\) |
| 6 | \(-20 - (-20)\) | \(-20 + 20\) | \(0\) |
| 7 | \(3 - 3\) | \(3 + (-3)\) | \(0\) |
| 8 | \(-1 - (-7)\) | \(-1 + 7\) | \(6\) |
| 9 | \(100 - (-50)\) | \(100 + 50\) | \(150\) |
| 10 | \(-30 - 45\) | \(-30 + (-45)\) | \(-75\) |
4.4 Multi-Step Worked Examples
Simplify: \((-8) + 5 - (-3) + (-2) - 7\)
Step 1: Rewrite subtractions as additions:
\((-8) + 5 + 3 + (-2) + (-7)\)
Step 2: Group positives and negatives:
Positives: \(5 + 3 = 8\)
Negatives: \((-8) + (-2) + (-7) = -17\)
Step 3: Combine: \(8 + (-17) = -9\)
At 6 AM, the temperature was \(-12°\text{F}\). By noon it rose 20 degrees. By midnight it dropped 15 degrees. What was the midnight temperature?
\(-12 + 20 - 15 = -12 + 20 + (-15) = 8 + (-15) = -7°\text{F}\)
Your bank balance is $45. You write a check for $60, then deposit $25. What is your balance?
\(45 - 60 + 25 = 45 + (-60) + 25 = 70 + (-60) = 10\)
Your balance is $10.
What is \(-15 - (-8) + 3\)?
5. Multiplying & Dividing Integers
5.1 Sign Rules for Multiplication
| Factor 1 | Factor 2 | Product | Memory Aid |
|---|---|---|---|
| \(+\) | \(+\) | \(+\) | “A friend of a friend is a friend” |
| \(+\) | \(-\) | \(-\) | “A friend of an enemy is an enemy” |
| \(-\) | \(+\) | \(-\) | “An enemy of a friend is an enemy” |
| \(-\) | \(-\) | \(+\) | “An enemy of an enemy is a friend” |
In short: same signs → positive; different signs → negative. [7] Gallian, J. A., Contemporary Abstract Algebra, 10th ed., Cengage, 2021, pp. 1–2
This is not just a rule to memorize — it follows from the distributive property. Consider \((-1) \times (-1)\):
\(0 = (-1) \times 0 = (-1)(1 + (-1)) = (-1)(1) + (-1)(-1) = -1 + (-1)(-1)\)
For the sum to be 0, we need \((-1)(-1) = 1\). This is a theorem, not an arbitrary convention. [2] Wu, 2011, pp. 323–326
5.2 Multiplication Examples
| # | Problem | Sign Analysis | Result |
|---|---|---|---|
| 1 | \(4 \times 7\) | \(+ \times + = +\) | \(28\) |
| 2 | \((-3) \times 5\) | \(- \times + = -\) | \(-15\) |
| 3 | \(6 \times (-8)\) | \(+ \times - = -\) | \(-48\) |
| 4 | \((-9) \times (-4)\) | \(- \times - = +\) | \(36\) |
| 5 | \((-1) \times 25\) | \(- \times + = -\) | \(-25\) |
| 6 | \((-7) \times (-7)\) | \(- \times - = +\) | \(49\) |
| 7 | \(0 \times (-99)\) | Zero factor | \(0\) |
| 8 | \((-2) \times (-3) \times (-4)\) | 3 negatives → negative | \(-24\) |
| 9 | \((-1)^4\) | 4 negatives (even) → positive | \(1\) |
| 10 | \((-1)^5\) | 5 negatives (odd) → negative | \(-1\) |
| 11 | \((-5) \times 2 \times (-3)\) | 2 negatives → positive | \(30\) |
| 12 | \((-10)^2\) | \((-10)(-10) = +\) | \(100\) |
When multiplying several integers:
- Even number of negative factors → product is positive
- Odd number of negative factors → product is negative
- If any factor is 0 → product is 0
5.3 Division Examples
Division follows the same sign rules as multiplication (since \(a \div b = a \times \frac{1}{b}\)): [5] CCSS, 2010, 7.NS.2b
| Problem | Sign | Result |
|---|---|---|
| \(24 \div 6\) | \(+ \div + = +\) | \(4\) |
| \((-24) \div 6\) | \(- \div + = -\) | \(-4\) |
| \(24 \div (-6)\) | \(+ \div - = -\) | \(-4\) |
| \((-24) \div (-6)\) | \(- \div - = +\) | \(4\) |
| \(0 \div (-5)\) | Zero divided by anything | \(0\) |
| \((-81) \div 9\) | \(- \div + = -\) | \(-9\) |
| \((-56) \div (-7)\) | \(- \div - = +\) | \(8\) |
| \((-100) \div 25\) | \(- \div + = -\) | \(-4\) |
5.4 Division by Zero
\( a \div 0 \) is undefined for any number \(a\). Here’s why:
If \(a \div 0 = c\), then \(0 \times c = a\). But \(0 \times c = 0\) for every \(c\), so there is no \(c\) that works (unless \(a=0\), in which case every \(c\) works, giving no unique answer). Either way, division by zero is impossible.
\( \frac{5}{0} = \text{undefined} \qquad \frac{0}{0} = \text{indeterminate} \qquad \frac{0}{5} = 0 \)
[2] Wu, 2011, p. 95
6. Rational Numbers
6.1 Definition
A rational number is any number that can be expressed as a ratio (fraction) of two integers \(\frac{p}{q}\), where \(q \neq 0\). The set of all rational numbers is denoted \(\mathbb{Q}\) (from the Italian quoziente, quotient).
\( \mathbb{Q} = \left\{ \frac{p}{q} \;\middle|\; p, q \in \mathbb{Z},\; q \neq 0 \right\} \)
[4] Hungerford, 2014, p. 17
| Number | As \(\frac{p}{q}\) | Rational? |
|---|---|---|
| \(\frac{3}{4}\) | Already a fraction | Yes |
| \(-7\) | \(\frac{-7}{1}\) | Yes (every integer is rational) |
| \(0\) | \(\frac{0}{1}\) | Yes |
| \(0.6\) | \(\frac{6}{10} = \frac{3}{5}\) | Yes |
| \(2.75\) | \(\frac{275}{100} = \frac{11}{4}\) | Yes |
| \(-0.\overline{3}\) | \(-\frac{1}{3}\) | Yes (repeating decimals are rational) |
| \(1.\overline{142857}\) | \(\frac{8}{7}\) | Yes |
| \(\sqrt{2}\) | Cannot be expressed | No (irrational) |
| \(\pi\) | Cannot be expressed | No (irrational) |
A number is rational if and only if its decimal expansion either terminates or eventually repeats. [8] Hardy, G. H. & Wright, E. M., An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, Theorem 159
Number System Hierarchy
Natural numbers ⊂ Integers ⊂ Rationals ⊂ Real numbers. Each set extends the previous. [9] Bloch, E. D., The Real Numbers and Real Analysis, Springer, 2011, Ch. 1
6.2 Forms & Conversions
Converting a Repeating Decimal to a Fraction
Let \(x = 0.363636\ldots\)
\(100x = 36.363636\ldots\)
Subtract: \(100x - x = 36\)
\(99x = 36\)
\(x = \frac{36}{99} = \frac{4}{11}\)
Let \(x = 0.8333\ldots\)
\(10x = 8.333\ldots\)
\(100x = 83.333\ldots\)
Subtract: \(100x - 10x = 83.333\ldots - 8.333\ldots = 75\)
\(90x = 75\)
\(x = \frac{75}{90} = \frac{5}{6}\)
6.3 Rational Numbers on the Number Line
Every rational number has a precise location on the number line. Between any two rational numbers, there are infinitely more rational numbers.
Plot \(-\frac{3}{4}\) on the number line:
- It is between \(-1\) and \(0\)
- Divide the segment from \(-1\) to \(0\) into 4 equal parts
- \(-\frac{3}{4}\) is 3 parts to the left of \(0\)
7. Rational Number Operations
7.1 Adding & Subtracting Rational Numbers
The same rules for integer addition/subtraction extend to all rational numbers. The key skill is combining sign rules with fraction arithmetic. [5] CCSS, 2010, 7.NS.1
| # | Problem | Work | Answer |
|---|---|---|---|
| 1 | \(\frac{2}{5} + \frac{1}{5}\) | Same denominator: \(\frac{2+1}{5}\) | \(\frac{3}{5}\) |
| 2 | \(-\frac{3}{7} + \frac{5}{7}\) | \(\frac{-3+5}{7}\) | \(\frac{2}{7}\) |
| 3 | \(-\frac{2}{3} - \frac{1}{4}\) | LCD = 12: \(-\frac{8}{12} - \frac{3}{12}\) | \(-\frac{11}{12}\) |
| 4 | \(\frac{5}{6} + (-\frac{7}{6})\) | \(\frac{5-7}{6}\) | \(-\frac{2}{6} = -\frac{1}{3}\) |
| 5 | \(-\frac{3}{8} - (-\frac{1}{8})\) | \(-\frac{3}{8} + \frac{1}{8}\) | \(-\frac{2}{8} = -\frac{1}{4}\) |
| 6 | \(\frac{1}{2} + (-\frac{2}{3})\) | LCD = 6: \(\frac{3}{6} - \frac{4}{6}\) | \(-\frac{1}{6}\) |
| 7 | \(-\frac{5}{9} + \frac{2}{3}\) | LCD = 9: \(-\frac{5}{9} + \frac{6}{9}\) | \(\frac{1}{9}\) |
| 8 | \(-0.75 - 1.25\) | \(-0.75 + (-1.25)\) | \(-2.0\) |
7.2 Multiplying Rational Numbers
Apply the sign rules, then multiply numerators and denominators:
\(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\)
- \(\frac{2}{3} \times \left(-\frac{4}{5}\right) = -\frac{8}{15}\) — \(+ \times - = -\)
- \(\left(-\frac{3}{4}\right) \times \left(-\frac{2}{7}\right) = +\frac{6}{28} = \frac{3}{14}\) — \(- \times - = +\)
- \((-5) \times \frac{3}{10} = -\frac{15}{10} = -\frac{3}{2} = -1.5\)
- \(\left(-\frac{1}{2}\right)^3 = -\frac{1}{8}\) — odd power, negative
- \(\left(-\frac{2}{3}\right)^2 = +\frac{4}{9}\) — even power, positive
- \((-0.4) \times (-0.5) = +0.20\)
7.3 Dividing Rational Numbers
Dividing by a fraction is the same as multiplying by its reciprocal:
\(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \qquad (c \neq 0)\)
[2] Wu, 2011, Ch. 14
The reciprocal (or multiplicative inverse) of \(\frac{a}{b}\) is \(\frac{b}{a}\), provided \(a \neq 0\). Their product is 1: \(\frac{a}{b} \times \frac{b}{a} = 1\).
- \(\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1\frac{7}{8}\)
- \(-\frac{5}{6} \div \frac{1}{3} = -\frac{5}{6} \times \frac{3}{1} = -\frac{15}{6} = -\frac{5}{2} = -2.5\)
- \(\frac{7}{8} \div (-\frac{7}{8}) = \frac{7}{8} \times (-\frac{8}{7}) = -1\)
- \((-12) \div \frac{3}{4} = (-12) \times \frac{4}{3} = -\frac{48}{3} = -16\)
- \(\frac{0}{5} \div \frac{2}{3} = 0 \times \frac{3}{2} = 0\)
- \(-\frac{9}{10} \div (-\frac{3}{5}) = -\frac{9}{10} \times (-\frac{5}{3}) = +\frac{45}{30} = \frac{3}{2}\)
- \((-2.4) \div 0.6 = -4\)
- \((-0.5) \div (-0.25) = +2\)
7.4 Operations with Mixed Numbers
Compute \(3\frac{1}{4} - 5\frac{2}{3}\):
Step 1: Convert to improper fractions: \(\frac{13}{4} - \frac{17}{3}\)
Step 2: LCD = 12: \(\frac{39}{12} - \frac{68}{12}\)
Step 3: Subtract: \(\frac{39 - 68}{12} = -\frac{29}{12} = -2\frac{5}{12}\)
Compute \(-1\frac{1}{2} \times 2\frac{2}{5}\):
Step 1: Convert: \(-\frac{3}{2} \times \frac{12}{5}\)
Step 2: Multiply: \(-\frac{36}{10} = -\frac{18}{5} = -3\frac{3}{5}\)
What is \(-\frac{2}{3} \div \frac{4}{9}\)?
8. Ordering & Density
8.1 Ordering Rational Numbers
To compare rational numbers, convert them to the same form (common denominator or decimals).
Arrange: \(-\frac{3}{4},\; 0.5,\; -\frac{2}{3},\; \frac{1}{8},\; -0.8\)
Convert all to decimals:
- \(-\frac{3}{4} = -0.75\)
- \(0.5 = 0.5\)
- \(-\frac{2}{3} \approx -0.667\)
- \(\frac{1}{8} = 0.125\)
- \(-0.8 = -0.8\)
Order: \(-0.8 < -0.75 < -0.667 < 0.125 < 0.5\)
So: \(-0.8 < -\frac{3}{4} < -\frac{2}{3} < \frac{1}{8} < 0.5\)
Which is greater: \(-\frac{5}{8}\) or \(-\frac{7}{12}\)?
LCD = 24: \(-\frac{5}{8} = -\frac{15}{24}\) and \(-\frac{7}{12} = -\frac{14}{24}\)
Since \(-14 > -15\), we have \(-\frac{7}{12} > -\frac{5}{8}\).
(Remember: with negatives, the one closer to zero is greater.)
8.2 The Density Property
Between any two distinct rational numbers, there exists another rational number. In fact, there are infinitely many rational numbers between any two.
One way to find a rational between \(a\) and \(b\): take their average: \(\frac{a+b}{2}\)
[9] Bloch, 2011, Theorem 1.7.7
Find three rational numbers between \(\frac{1}{3}\) and \(\frac{1}{2}\):
Method 1 — Averaging:
- Average of \(\frac{1}{3}\) and \(\frac{1}{2}\): \(\frac{\frac{1}{3}+\frac{1}{2}}{2} = \frac{\frac{5}{6}}{2} = \frac{5}{12}\)
- Average of \(\frac{1}{3}\) and \(\frac{5}{12}\): \(\frac{\frac{4}{12}+\frac{5}{12}}{2} = \frac{9}{24} = \frac{3}{8}\)
- Average of \(\frac{5}{12}\) and \(\frac{1}{2}\): \(\frac{\frac{5}{12}+\frac{6}{12}}{2} = \frac{11}{24}\)
Method 2 — Common denominator: \(\frac{1}{3} = \frac{4}{12}\), \(\frac{1}{2} = \frac{6}{12}\). Need more room? Use \(\frac{1}{3} = \frac{40}{120}\), \(\frac{1}{2} = \frac{60}{120}\). Then \(\frac{41}{120}, \frac{42}{120}, \frac{43}{120}, \ldots, \frac{59}{120}\) all lie between them.
9. Real-World Applications
A submarine is at \(-200\) meters (below sea level). It ascends \(75\) meters, then descends \(120\) meters. What is its final depth?
\(-200 + 75 + (-120) = -200 + 75 - 120 = -245\) meters
A stock changes value over five days: \(+2.50, -1.75, -3.25, +0.50, +1.25\). What is the net change?
\(2.50 + (-1.75) + (-3.25) + 0.50 + 1.25\)
Positives: \(2.50 + 0.50 + 1.25 = 4.25\)
Negatives: \(-1.75 + (-3.25) = -5.00\)
Net: \(4.25 + (-5.00) = -0.75\). The stock dropped $0.75 overall.
A recipe calls for \(2\frac{1}{3}\) cups of flour. You want to make \(1\frac{1}{2}\) times the recipe. How much flour do you need?
\(2\frac{1}{3} \times 1\frac{1}{2} = \frac{7}{3} \times \frac{3}{2} = \frac{21}{6} = \frac{7}{2} = 3\frac{1}{2}\) cups
The formula to convert Celsius to Fahrenheit is \(F = \frac{9}{5}C + 32\).
Convert \(-15°\text{C}\) to Fahrenheit:
\(F = \frac{9}{5}(-15) + 32 = \frac{-135}{5} + 32 = -27 + 32 = 5°\text{F}\)
In golf, scores are given relative to par. A player’s round scores over 4 days are: \(-3, +1, -2, -4\). What is the total score relative to par?
\(-3 + 1 + (-2) + (-4) = -8\) (8 under par)
- Compute: \((-18) + 7 - (-4) + (-9)\) → Answer: \(-16\)
- Compute: \((-6) \times (-3) \times 2\) → Answer: \(36\)
- Compute: \(-\frac{5}{8} + \frac{3}{4}\) → Answer: \(\frac{1}{8}\)
- Compute: \(\frac{7}{10} \div (-\frac{14}{15})\) → Answer: \(-\frac{3}{4}\)
- Order from least to greatest: \(-1.5, \; -\frac{4}{3}, \; -1.4, \; -\frac{3}{2}\) → Answer: \(-\frac{3}{2} = -1.5 < -1.4 < -\frac{4}{3} \approx -1.333\)
- Find two rational numbers between \(\frac{2}{5}\) and \(\frac{3}{5}\) → Answer: e.g. \(\frac{1}{2}, \frac{11}{20}\)
References & Citations
- National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, VA: NCTM, 2000. ISBN: 978-0-87353-480-2
- Wu, H.-H. Understanding Numbers in Elementary School Mathematics. Providence, RI: American Mathematical Society, 2011. ISBN: 978-0-8218-5260-6
- Katz, V. J. A History of Mathematics: An Introduction. 3rd ed. Boston: Pearson, 2009. ISBN: 978-0-321-38700-4
- Hungerford, T. W. Abstract Algebra: An Introduction. 3rd ed. Boston: Cengage Learning, 2014. ISBN: 978-1-111-56962-4
- National Governors Association Center for Best Practices & Council of Chief State School Officers. Common Core State Standards for Mathematics. Washington, DC: Authors, 2010.
- Stephan, M. & Akyuz, D. “A Proposed Instructional Theory for Integer Addition and Subtraction.” Journal for Research in Mathematics Education, Vol. 43, No. 4, 2012, pp. 428–464.
- Gallian, J. A. Contemporary Abstract Algebra. 10th ed. Boston: Cengage Learning, 2021. ISBN: 978-0-367-65178-7
- Hardy, G. H. & Wright, E. M. An Introduction to the Theory of Numbers. 6th ed. Oxford: Oxford University Press, 2008. ISBN: 978-0-19-921986-5
- Bloch, E. D. The Real Numbers and Real Analysis. New York: Springer, 2011. ISBN: 978-0-387-72176-7
- Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. Connected Mathematics 2: Accentuate the Negative — Integers and Rational Numbers. Boston: Pearson, 2009. ISBN: 978-0-13-366140-8
- Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. Elementary and Middle School Mathematics: Teaching Developmentally. 10th ed. New York: Pearson, 2019. ISBN: 978-0-13-480208-5
- National Research Council. Adding It Up: Helping Children Learn Mathematics. Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). Washington, DC: National Academies Press, 2001. ISBN: 978-0-309-06995-3
- Bishop, J. P., Lamb, L. L., Philipp, R. A., Whitacre, I., & Schappelle, B. P. “Using Order to Reason About Negative Numbers: The Case of Violet.” Educational Studies in Mathematics, Vol. 86, No. 1, 2014, pp. 39–59.
This page aligns primarily with CCSS 6.NS (The Number System) and 7.NS (Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers). It also supports the NCTM’s Number and Operations strand for grades 6–8.