Mathematics: Number Sense and Operations – Grade 7

The Laws of Exponents are fundamental rules that govern how we work with exponential expressions. These laws emerge from patterns in multiplication and help us simplify complex expressions efficiently.

An exponential expression like $3^4$ consists of a base (3) and an exponent (4). The exponent tells us how many times to multiply the base by itself: $3^4 = 3 \times 3 \times 3 \times 3 = 81$. This notation becomes especially powerful when we need to work with very large or very small numbers.

When the base is a fraction, like $(\frac{1}{2})^3$, we apply the exponent to both the numerator and denominator: $(\frac{1}{2})^3 = \frac{1^3}{2^3} = \frac{1}{8}$. This pattern holds true for all rational number bases.

When multiplying expressions with the same base, we add the exponents: $a^m \cdot a^n = a^{m+n}$. Let's see why this works:

$2^3 \cdot 2^4 = (2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2) = 2^7$

Counting the total number of 2's being multiplied gives us $3 + 4 = 7$ factors, confirming that $2^3 \cdot 2^4 = 2^7$.

For example: $(\frac{1}{3})^2 \cdot (\frac{1}{3})^5 = (\frac{1}{3})^{2+5} = (\frac{1}{3})^7$

When dividing expressions with the same base, we subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$. This law helps us simplify division problems:

$\frac{5^6}{5^2} = \frac{5 \times 5 \times 5 \times 5 \times 5 \times 5}{5 \times 5} = 5^{6-2} = 5^4$

After canceling common factors, we're left with $5^4 = 625$.

When raising a power to another power, we multiply the exponents: $(a^m)^n = a^{mn}$. Think of this as repeated application:

$(3^2)^4 = 3^2 \times 3^2 \times 3^2 \times 3^2 = 3^{2+2+2+2} = 3^8$

Since we add the exponent 2 a total of 4 times, we get $2 \times 4 = 8$, confirming $(3^2)^4 = 3^8$.

The zero exponent rule states that any nonzero number raised to the power of zero equals 1: $a^0 = 1$. We can understand this by extending the quotient pattern:

$5^3 \div 5^1 = 5^{3-1} = 5^2 = 25$ $5^2 \div 5^1 = 5^{2-1} = 5^1 = 5$ $5^1 \div 5^1 = 5^{1-1} = 5^0 = 1$

This pattern shows why $5^0 = 1$.

Negative exponents represent reciprocals: $a^{-n} = \frac{1}{a^n}$. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$. This allows us to express very small numbers efficiently.

Exponent laws appear in many practical contexts:

• Scientific notation: $3.2 \times 10^5 \times 4.1 \times 10^3 = 13.12 \times 10^8$
• Compound interest: Money growing at 5% annually follows the pattern $P(1.05)^t$
• Population growth: Bacterial populations might double every hour, following $N \cdot 2^t$

When working with exponential expressions:

1. Identify the base and ensure bases match before applying laws
2. Expand expressions fully when patterns aren't clear
3. Check your work by substituting simple values
4. Look for patterns that suggest which law to apply
5. Work step-by-step rather than trying to apply multiple laws simultaneously

Rational numbers can be expressed in many equivalent forms – fractions, decimals, percentages, and mixed numbers. Understanding how to convert between these forms and when to use each one gives you flexibility in problem-solving and mathematical communication.

A rational number is any number that can be expressed as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$. This includes positive and negative integers, fractions, mixed numbers, terminating decimals, and repeating decimals.

Examples of rational numbers:

• $\frac{3}{4}$ (proper fraction)
• $2\frac{1}{3}$ (mixed number)
• $0.75$ (terminating decimal)
• $0.\overline{6}$ (repeating decimal)
• $150\%$ (percentage greater than 100%)

To convert a fraction to a decimal, divide the numerator by the denominator:

$\frac{3}{8} = 3 \div 8 = 0.375$ (terminating decimal)

$\frac{1}{3} = 1 \div 3 = 0.333...$ or $0.\overline{3}$ (repeating decimal)

Some fractions produce terminating decimals (finite number of digits), while others produce repeating decimals (infinite pattern of digits). Whether a fraction terminates depends on the prime factors of the denominator in lowest terms.

Repeating decimals have digits that repeat in a pattern forever. We use bar notation to show the repeating part:

• $\frac{1}{3} = 0.\overline{3}$ means $0.333...$ (3 repeats)
• $\frac{1}{6} = 0.1\overline{6}$ means $0.1666...$ (6 repeats after the 1)
• $\frac{1}{7} = 0.\overline{142857}$ means the block "142857" repeats

When solving problems, using the exact fractional form often gives more precise answers than truncated decimals. For instance, calculating $3 \times \frac{1}{3} = 1$ exactly, while $3 \times 0.33 = 0.99$.

For terminating decimals, count the decimal places to determine the denominator:

$0.75 = \frac{75}{100} = \frac{3}{4}$ (reduced to lowest terms)

$0.125 = \frac{125}{1000} = \frac{1}{8}$ (reduced to lowest terms)

For repeating decimals, the process is more complex. A useful method involves algebraic manipulation:

To convert $0.\overline{6}$ to a fraction:

• Let $x = 0.\overline{6} = 0.666...$
• Multiply by 10: $10x = 6.666...$
• Subtract: $10x - x = 6.666... - 0.666... = 6$
• Solve: $9x = 6$, so $x = \frac{6}{9} = \frac{2}{3}$

Percentages represent parts per hundred. Converting between percentages and other forms:

Decimal to percentage: Multiply by 100 and add the % symbol

• $0.75 = 75\%$
• $1.25 = 125\%$

Percentage to decimal: Divide by 100 (move decimal point two places left)

• $45\% = 0.45$
• $150\% = 1.50$

Fraction to percentage: Convert to decimal first, then to percentage

• $\frac{3}{5} = 0.6 = 60\%$

Mixed numbers combine whole numbers and fractions: $3\frac{1}{4}$

Improper fractions have numerators greater than or equal to denominators: $\frac{13}{4}$

To convert mixed numbers to improper fractions: $3\frac{1}{4} = \frac{3 \times 4 + 1}{4} = \frac{13}{4}$

To convert improper fractions to mixed numbers: $\frac{13}{4} = 13 \div 4 = 3\frac{1}{4}$ (3 remainder 1)

Different forms work better in different contexts:

• Fractions: Best for exact calculations and when working with ratios
• Decimals: Useful for measurement, money, and calculator work
• Percentages: Clear for comparing parts to wholes, especially in statistics
• Mixed numbers: Natural for measurements and everyday quantities

Consider a store offering a 25% discount on a $\$48$ item:

• Percentage form: $25\% \text{ of } \$48$
• Decimal form: $0.25 \times \$48 = \$12$ discount
• Fraction form: $\frac{1}{4} \times \$48 = \$12$ discount

The sale price: $\$48 - \$12 = \$36$

Each form gives the same result, but decimals might be easiest for mental calculation while fractions show the relationship most clearly.

The order of operations provides a universal system for evaluating mathematical expressions consistently. When working with rational numbers, following this systematic approach becomes even more critical for accuracy.

Remember the order: Grouping symbols, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). This isn't just a memorized sequence – it reflects the mathematical relationships between operations.

Grouping symbols include:

• Parentheses: $(3 + 2)$
• Brackets: $[4 - 1]$
• Absolute value bars: $|{-5}|$
• Fraction bars (which group numerator and denominator)

Let's work through an example: $\frac{1}{2}(3^2 - 4) + |7 - \frac{1}{6}|$

Step 1: Handle grouping symbols first

• Inside parentheses: $3^2 - 4 = 9 - 4 = 5$
• Inside absolute value: $7 - \frac{1}{6} = \frac{42}{6} - \frac{1}{6} = \frac{41}{6}$, so $|\frac{41}{6}| = \frac{41}{6}$

Step 2: Now we have: $\frac{1}{2} \cdot 5 + \frac{41}{6}$

Step 3: Multiply: $\frac{1}{2} \cdot 5 = \frac{5}{2}$

Step 4: Add: $\frac{5}{2} + \frac{41}{6} = \frac{15}{6} + \frac{41}{6} = \frac{56}{6} = \frac{28}{3}$

Remember that subtraction is addition of an opposite: $a - b = a + (-b)$. This helps clarify expressions like:

$-\frac{3}{4} - (-\frac{1}{2}) = -\frac{3}{4} + \frac{1}{2} = -\frac{3}{4} + \frac{2}{4} = -\frac{1}{4}$

Similarly, division is multiplication by a reciprocal: $a \div b = a \times \frac{1}{b}$

Consider this expression: $18 - 3(4.12 + 7.6 \div 2)$

Step 1: Parentheses first, but follow order of operations inside

• $7.6 \div 2 = 3.8$
• $4.12 + 3.8 = 7.92$

Step 2: Multiplication

• $3(7.92) = 23.76$

Step 3: Subtraction

• $18 - 23.76 = -5.76$

Sometimes parentheses indicate multiplication rather than grouping:

• $(\frac{4}{6} + 9) \times 87$ – parentheses for grouping
• $(\frac{4}{6} + 9)(87)$ – parentheses indicate multiplication
• $(\frac{4}{6} + 9)(-87)$ – parentheses indicate multiplication by negative number

Context and spacing help distinguish these uses.

Fraction bars group both numerator and denominator:

$\frac{(12-|8-5|)^3}{36} = \frac{(12-3)^3}{36} = \frac{9^3}{36} = \frac{729}{36} = \frac{81}{4}$

The fraction bar tells us to evaluate the entire numerator and entire denominator before dividing.

When using calculators or computers, parentheses become crucial for ensuring correct order of operations. For example:

• To calculate $\frac{3+4}{2+5}$, enter: $(3+4)/(2+5)$
• Without parentheses, $3+4/2+5$ would be calculated as $3 + 2 + 5 = 10$

Misconception: Multiplication always comes before division Reality: Multiplication and division have equal priority and are performed left to right

Example: $20 \div 4 \times 2 = 5 \times 2 = 10$ (not $20 \div 8 = 2.5$)

Misconception: Parentheses always mean multiplication Reality: Parentheses primarily indicate grouping; multiplication is implied by context

1. Color-code different operation levels when learning
2. Work step-by-step rather than trying to do multiple operations mentally
3. Check your work by substituting simpler numbers
4. Estimate first to catch major errors
5. Use technology to verify complex calculations

Mastering order of operations with rational numbers builds the foundation for algebra, where expressions become more complex but follow the same fundamental rules.

Procedural fluency with rational numbers means performing operations accurately, efficiently, and flexibly. This fluency builds from understanding the underlying concepts and developing reliable methods for each operation.

To add or subtract fractions, we need common denominators. The key is finding the least common multiple (LCM) of the denominators:

$\frac{2}{3} + \frac{1}{4} = \frac{8}{12} + \frac{3}{12} = \frac{11}{12}$

For mixed numbers, you can either:

1. Convert to improper fractions first: $2\frac{1}{3} + 1\frac{1}{4} = \frac{7}{3} + \frac{5}{4} = \frac{28}{12} + \frac{15}{12} = \frac{43}{12} = 3\frac{7}{12}$
2. Add whole and fractional parts separately: $2\frac{1}{3} + 1\frac{1}{4} = (2+1) + (\frac{1}{3} + \frac{1}{4}) = 3 + \frac{7}{12} = 3\frac{7}{12}$

With negative rational numbers, follow the same sign rules as with integers:

$-\frac{2}{5} + \frac{3}{10} = -\frac{4}{10} + \frac{3}{10} = -\frac{1}{10}$

$\frac{1}{6} - \frac{2}{3} = \frac{1}{6} - \frac{4}{6} = -\frac{3}{6} = -\frac{1}{2}$

Remember: subtracting is the same as adding the opposite.

Multiplying fractions is straightforward: multiply numerators together and denominators together:

$\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}$

Before multiplying, look for common factors to simplify:

$\frac{6}{7} \times \frac{14}{9} = \frac{6 \times 14}{7 \times 9} = \frac{84}{63}$

Or more efficiently: $\frac{6}{7} \times \frac{14}{9} = \frac{\cancel{6} \times 14}{7 \times \cancel{9}} = \frac{2 \times \cancel{14}}{\cancel{7} \times 3} = \frac{2 \times 2}{1 \times 3} = \frac{4}{3}$

With mixed numbers, convert to improper fractions first:

$2\frac{1}{3} \times 1\frac{1}{5} = \frac{7}{3} \times \frac{6}{5} = \frac{42}{15} = \frac{14}{5} = 2\frac{4}{5}$

To divide fractions, multiply by the reciprocal:

$\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1\frac{7}{8}$

The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$. For whole numbers, the reciprocal of $n$ is $\frac{1}{n}$.

A complex fraction has fractions in the numerator, denominator, or both:

$\frac{\frac{2}{3}}{\frac{4}{5}} = \frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$

For more complex cases:

$\frac{\frac{1}{2} + \frac{1}{3}}{\frac{2}{5} - \frac{1}{10}}$

Step 1: Simplify numerator and denominator separately

• Numerator: $\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$
• Denominator: $\frac{2}{5} - \frac{1}{10} = \frac{4}{10} - \frac{1}{10} = \frac{3}{10}$

Step 2: Divide: $\frac{\frac{5}{6}}{\frac{3}{10}} = \frac{5}{6} \times \frac{10}{3} = \frac{50}{18} = \frac{25}{9}$

For addition and subtraction, align decimal points:

$7.24 - 5.01 - 78.4 = 7.24 - 5.01 - 78.40 = 2.23 - 78.40 = -76.17$

For multiplication, multiply as if they were whole numbers, then place the decimal point:

$1.2 \times 0.35 = 1200 \times 35 = 42000$

Total decimal places: $1 + 2 = 3$, so the answer is $0.420 = 0.42$

For division, move decimal points to make the divisor a whole number:

$4.8 \div 0.12 = 480 \div 12 = 40$

Consider: $\frac{15}{6} \times (-1.2)$

Method 1: Convert decimal to fraction $-1.2 = -\frac{12}{10} = -\frac{6}{5}$

$\frac{15}{6} \times (-\frac{6}{5}) = -\frac{15 \times 6}{6 \times 5} = -\frac{90}{30} = -3$

Method 2: Convert fraction to decimal $\frac{15}{6} = 2.5$

$2.5 \times (-1.2) = -3$

Both methods give the same result, demonstrating the equivalence of different forms.

1. Practice estimation before calculating exactly
2. Look for simplification opportunities before computing
3. Check answers by working backwards or using different methods
4. Develop number sense by recognizing when answers seem unreasonable
5. Use mental math for simple computations to build speed
6. Practice with and without calculators to develop both computational skills and technology fluency

Real-world problems with rational numbers require you to identify which operations to use, set up calculations correctly, and interpret results in context. These problems help you see the practical value of mathematical skills.

Develop a systematic approach:

1. Read carefully – What's the situation? What are you asked to find?
2. Identify given information – What numbers and relationships are provided?
3. Choose operations – Which mathematical operations will help solve the problem?
4. Estimate first – What should a reasonable answer look like?
5. Calculate step-by-step – Work systematically and show your reasoning
6. Check your answer – Does it make sense in the context?

Example: Daliah purchases eggs by the dozen for her two children. Each day, Zane eats $\frac{1}{4}$ carton and Amare eats $\frac{1}{6}$ carton. A carton of 12 eggs costs $\$1.65$. How much does Daliah spend on eggs in 30 days?

Step 1: Find daily egg consumption $\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12} \text{ carton per day}$

Step 2: Find 30-day consumption $\frac{5}{12} \times 30 = \frac{150}{12} = 12.5 \text{ cartons}$

Step 3: Calculate total cost $12.5 \times \$1.65 = \$20.625 ≈ \$20.63$

Daliah spends approximately $\$20.63$ on eggs in 30 days.

Example: All 7th grade homeroom classes collected recycling, with the top three classes splitting a $\$800$ prize proportionally. Mr. Brogle's class turned in 237 pounds, Mrs. Abiola's class turned in 192 pounds, and Mr. Wheeler's class turned in 179 pounds. How should they divide the money?

Step 1: Find total recycling $237 + 192 + 179 = 608 \text{ pounds}$

Step 2: Find each class's fraction

• Mr. Brogle: $\frac{237}{608}$
• Mrs. Abiola: $\frac{192}{608}$
• Mr. Wheeler: $\frac{179}{608}$

Step 3: Calculate each class's share

• Mr. Brogle: $\frac{237}{608} \times \$800 = \$311.84$
• Mrs. Abiola: $\frac{192}{608} \times \$800 = \$252.63$
• Mr. Wheeler: $\frac{179}{608} \times \$800 = \$235.53$

Check: $\$311.84 + \$252.63 + \$235.53 = \$800$ ✓

Example: Anjeanette's cupcake recipe calls for 2 cups of flour, $\frac{1}{2}$ cup of butter, and $\frac{3}{4}$ cup of sugar per batch. She has 8 cups of flour, 2 cups of butter, and 2 cups of sugar. How many batches can she make?

Step 1: Find how many batches each ingredient allows

• Flour: $8 \div 2 = 4$ batches
• Butter: $2 \div \frac{1}{2} = 2 \times 2 = 4$ batches
• Sugar: $2 \div \frac{3}{4} = 2 \times \frac{4}{3} = \frac{8}{3} = 2\frac{2}{3}$ batches

Step 2: The limiting ingredient determines the answer Sugar limits her to 2 complete batches (since $2\frac{2}{3}$ means she can't make a full third batch).

Example: Kari and Natalia each have $\$20$ to spend at the Fun Warehouse. After paying a $\$3$ entry fee each, they both play laser tag ($\$4.50$) and mini-bowling ($\$3.25$), then 4 video games each ($\$0.75$ per game). How much money is left?

Step 1: Calculate spending per person

• Entry fee: $\$3$
• Laser tag: $\$4.50$
• Mini-bowling: $\$3.25$
• Video games: $4 \times \$0.75 = \$3$
• Total spent: $\$3 + \$4.50 + \$3.25 + \$3 = \$13.75$

Step 2: Find remaining money $\$20 - \$13.75 = \$6.25 \text{ each}$

Together they have $2 \times \$6.25 = \$12.50$ left.

Visual representations help understand complex problems:

• Draw diagrams for measurement problems
• Use tables for multi-step calculations
• Create fraction models for part-whole relationships

Estimation strategies:

• Round numbers to check if your answer is reasonable
• Use benchmark fractions ($\frac{1}{2}$, $\frac{1}{4}$, $\frac{3}{4}$) for quick approximations
• Compare your result to familiar quantities

Common error prevention:

• Read the problem multiple times to understand what's being asked
• Identify the units in your answer and make sure they make sense
• Consider whether your answer is too large or too small for the context
• Check your arithmetic by working backwards or using a different method

These problem-solving skills prepare you for:

• Financial literacy: budgeting, calculating discounts, understanding loans
• Cooking and baking: scaling recipes, measuring ingredients
• Sports and fitness: calculating statistics, measuring progress
• Science: working with data, measurements, and proportions
• Career preparation: any job requiring quantitative reasoning