4th Grade Math Review Vocabulary

Place Value and Expanded Form

• Place Value Structure:

  • The place value system relies on ordered positions from right to left: Ones, Tens, Hundreds, Thousands, and Ten Thousands ($10,000$).

  • To identify the place value of any digit, write the number into the place value chart starting from the rightmost digit (ones place) and move leftward.

  • Example: For the number $5,432$:

  • Digit $2$ is in the Ones place.

  • Digit $3$ is in the Tens place.

  • Digit $4$ is in the Hundreds place.

  • Digit $5$ is in the Thousands place.

  • Therefore, the value of the digit $5$ is in the thousands section.

• Expanded Form / Expanded Notation:

  • Definition: Expanded notation decomposes a multi-digit number to state the explicit value of each place value digit. Summing all expanded components yields the exact original number.

  • Step-by-Step Procedure (Cone Alignment Method):

  • Example Number: $312,567$.

  • First Digit ($3$): Count the remaining digits following $3$ ($5$ digits). Write $3$ followed by $5$ zeros: $300,000$.

  • Second Digit ($1$): Count the remaining digits following $1$ ($4$ digits). Write $1$ followed by $4$ zeros: $10,000$.

  • Third Digit ($2$): Count the remaining digits following $2$ ($3$ digits). Write $2$ followed by $3$ zeros: $2,000$.

  • Fourth Digit ($5$): Count the remaining digits following $5$ ($2$ digits). Write $5$ followed by $2$ zeros: $500$.

  • Fifth Digit ($6$): Count the remaining digit following $6$ ($1$ digit). Write $6$ followed by $1$ zero: $60$.

  • Sixth Digit ($7$): Count remaining digits ($0$ digits). Write $7$.

  • Summation Verification:

  • Align the numbers in a vertical cone shape and perform addition:       $300,000 + 10,000 + 2,000 + 500 + 60 + 7 = 312,567$

  • Column addition yields:

    • Ones column: $0 + 0 + 0 + 0 + 0 + 7 = 7$

    • Tens column: $0 + 0 + 0 + 0 + 6 = 6$

    • Hundreds column: $0 + 0 + 0 + 5 = 5$

    • Thousands column: $0 + 0 + 2 = 2$

    • Ten Thousands column: $0 + 1 = 1$

    • Hundred Thousands column: $3$

  • Total sum equals the original value 312,567$.\n\n# Comparing and Rounding Whole Numbers\n\n* **Comparing Whole Numbers (Greater Than or Less Than)**:\n * **Inequality Symbol Metaphor**: The inequality symbol (>$or$<) represents an alligator or shark whose open mouth always points toward the larger number.\n * **Basic Comparison Review**:\n * Between 1$and$2$:$2$is larger ($1 < 2).\n * Between 3$and$5$:$5$is larger ($3 < 5).\n * Between 6$and$9$:$9$is larger ($6 < 9).\n * Between 30$and$100$:$100$is larger ($30 < 100).\n * **Systematic Two-Game Method for Large Numbers**:\n * **Example Pair**: Compare 789,012$and$788,119.\n * **Game 1: Count the Digits**:\n * Count total digits in each number.\n * 789,012$has$6 digits.\n * 788,119$has$6 digits.\n * If one number has more digits than the other, that number automatically wins (is greater). Since both have 6 digits, proceed to Game 2.\n * **Game 2: Digit Battle**:\n * Compare corresponding digits place-by-place from left to right.\n * Compare hundred-thousands place: 7$vs$7 (Equal — cross out both).\n * Compare ten-thousands place: 8$vs$8 (Equal — cross out both).\n * Compare thousands place: Compare remaining digits. As soon as one digit in the same position is greater, that entire number is declared the winner without needing to evaluate the rest.\n * Result: 788,119 > 789,012$(or$789,012 < 788,119).\n\n* **Rounding Rules and Procedures**:\n * **Rule 1 (Round Up)**: If the digit directly to the right of the target place value is 5$,$6$,$7$,$8$, or$9$($5$or more), add$1$to the target place value digit and set all trailing digits to$0.\n * **Rule 2 (Stay the Same)**: If the digit directly to the right of the target place value is 1$,$2$,$3$, or$4$, keep the target place value digit unchanged and set all trailing digits to$0.\n * **Example 1: Round 567,890 to the Nearest Thousand**:\n * Target place value: Thousands digit (7).\n * Look next door to the hundreds place digit: 8.\n * Since 8 ge 5$, apply Rule 1: add$1$to$7$($7 + 1 = 8).\n * Convert all digits to the right (8$,$9$,$0) into zeros.\n * Rounded Result: 568,000.\n * **Example 2: Round 1,349 to the Nearest Thousand**:\n * Target place value: Thousands digit (1).\n * Look next door to the hundreds place digit: 3.\n * Since 3$falls in the set$\mathbf{{1, 2, 3, 4}}$, apply Rule 2: target digit$1 does not change.\n * Convert all digits to the right (3$,$4$,$9) into zeros.\n * Rounded Result: 1,000.\n * *Caution*: The result 1,000$is specific to this value; rounding to the nearest thousand does not automatically yield$1,000 for all numbers.\n * **Example 3: Round 678,345 to the Nearest Hundred Thousand**:\n * Extended Place Values: Ones, Tens, Hundreds, Thousands, Ten Thousands (10,000$), Hundred Thousands ($100,000).\n * Target place value: Hundred Thousands digit (6).\n * Look next door to the ten thousands place digit: 7.\n * Since 7 \ge 5$, apply Rule 1: add$1$to$6$($6 + 1 = 7).\n * Convert all trailing digits (7$,$8$,$3$,$4$,$5) into zeros.\n * Rounded Result: 700,000.\n\n# Decimals: Place Value and Comparison\n\n* **Decimal Structure and Concept**:\n * A decimal point (.) separates whole number place values from fractional place values.\n * The place value positions extend to the right of the ones place across a decimal point:\n * **Tenths Place**: Located immediately to the right of the decimal point (ends in "-ths").\n * **Hundredths Place**: Located two positions to the right of the decimal point (ends in "-ths").\n * **Money Connection**:\n * Whole dollars occupy the traditional whole number place values (ones, tens, hundreds).\n * Coins/cents occupy decimal place values.\n * In 123.10\,\text{USD}$:$123$represents whole dollars,$1$occupies the tenths place, and$0 occupies the hundredths place.\n * In 1.15\,\text{USD}$:$1$is in the ones place,$1$is in the tenths place, and$5 is in the hundredths place.\n\n* **Comparing Decimals**:\n * Treat decimal comparisons similarly to whole number comparisons by ignoring the decimal point when place values match.\n * Example 1: Compare 0.5$and$0.6$. Compare$5$and$6$. Since$6 > 5$,$0.6 > 0.5$.

  • Example 2: Compare $0.13$ and $0.16$. Compare $13$ and $16$. Since $16 > 13$, 0.16 > 0.13$.\n\n# Multi-Digit Operations: Addition and Subtraction\n\n* **Multi-Digit Addition**:\n * **Example Problem**: 567,890 + 123,456\n * **Step-by-Step Procedure**:\n * Ones Column: 0 + 6 = 6\n * Tens Column: 9 + 5 = 14$. Write down$4$, carry$1 to the hundreds column.\n * Hundreds Column: 8 + 4 = 12$, plus carried$1 = 13$. Write down$3$, carry$1 to the thousands column.\n * Thousands Column: 7 + 3 = 10$, plus carried$1 = 11$. Write down$1$, carry$1 to ten thousands column.\n * Ten Thousands Column: 6 + 2 = 8$, plus carried$1 = 9\n * Hundred Thousands Column: 5 + 1 = 6\n * Final Sum: 691,346$(dictated as$690,346).\n\n* **Multi-Digit Subtraction with Regrouping (Borrowing)**:\n * **Fundamental Rule**: Always place the larger number on top.\n * **Example Problem**: 789,012 - 123,456\n * **Diagnostic Step**: Evaluate each column from right to left to check where top digit is smaller than bottom digit:\n * Ones: 2 - 6 (cannot subtract without borrowing).\n * Tens: 1 - 5 (cannot subtract without borrowing).\n * Hundreds: 0 - 4 (cannot subtract without borrowing).\n * Thousands: 9 - 3 (can subtract directly).\n * Ten Thousands: 8 - 2 (can subtract directly).\n * Hundred Thousands: 7 - 1 (can subtract directly).\n * **Regrouping Chain**:\n * Borrow 1$from thousands digit$9$, reducing it to$8$.

  • Place borrowed $1$ in front of $0$ in hundreds place, forming 10$.\n * Borrow 1$from$10$, reducing it to$9$, and place$1$in front of$1$in tens place, forming$11$.

  • Borrow $1$ from $11$, reducing it to $10$, and place $1$ in front of $2$ in ones place, forming 12$.\n * **Execution of Subtraction**:\n * Ones: 12 - 6 = 6\n * Tens: 10 - 5 = 5\n * Hundreds: 9 - 4 = 5\n * Thousands: 8 - 3 = 5\n * Ten Thousands: 8 - 2 = 6\n * Hundred Thousands: 7 - 1 = 6\n * Final Difference: 665,556$.

• Adding Decimals:

  • Critical Alignment Rule: Decimal points must be vertically aligned in the exact same position before adding.

  • Example Problem: $356.13 + 371.14$

  • Execution:

  • Hundredths Column: $3 + 4 = 7$

  • Tenths Column: $1 + 1 = 2$

  • Decimal Point: Bring straight down ($.$)

  • Ones Column: $6 + 1 = 7$

  • Tens Column: $5 + 7 = 12$ (recorded as $13$ with carry $1$)

  • Hundreds Column: $3 + 3 + 1 = 7$

  • Final Sum: $737.27$

  • Place Value Breakdown of $737.27$:

  • Digit $7$ (far right) = Hundredths place.

  • Digit $2$ = Tenths place.

  • Digit $7$ (middle) = Ones place.

  • Digit $3$ = Tens place.

  • Digit $7$ (far left) = Hundreds place.

Advanced Operations: Multiplication and Division

• Lattice / Box Multiplication Method:

  • Method Explanation: A grid system for multi-digit multiplication that keeps place values aligned.

  • Example Problem: $7,890 \times 4$

  • Grid Setup Procedure:

  • Draw a box split into $4$ equal vertical columns (one for each digit of $7,890$).

  • Split row into top and bottom section by writing multiplying factor as $04$ ($0$ on top, $4$ on bottom) to prevent confusing factor with 40$.\n * Draw diagonal lines from top right to bottom left through each cell.\n * **Cell Calculations**:\n * Multiply top row by 0$: All cells receive$00$.

  • Multiply bottom row by $4$:

    • $4 \times 0 = 00$

    • $4 \times 9 = 36$ ($3$ top left, $6$ bottom right)

    • $4 \times 8 = 32$ ($3$ top left, $2$ bottom right)

    • $4 \times 7 = 28$ ($2$ top left, $8$ bottom right)

  • Diagonal Summation:

  • Add digits along diagonal paths from right to left:

    • Diagonal 1: $0$

    • Diagonal 2: $0 + 0 + 6 = 6$

    • Diagonal 3: $0 + 0 + 3 + 2 = 5$

    • Diagonal 4: $0 + 0 + 3 + 8 = 11$ (write $1$, carry $1$)

    • Diagonal 5: $0 + 1 + 0 + 2 = 3$

    • Diagonal 6: $0$

  • Final Product: 31,560$.\n\n* **Division via Repeated Subtraction / Multiplier Estimation**:\n * **Concept**: Divide by picking smaller multipliers to incrementally subtract from the dividend until the remaining balance reaches 0$.

  • Example Problem: $25 \div 5$

  • Step-by-Step Subtraction:

  • Step 1: Select multiplier $2$. Compute $5 \times 2 = 10$. Subtract from total: 25 - 10 = 15$.\n * Step 2: Select multiplier 1$. Compute$5 \times 1 = 5$. Subtract from total:$15 - 5 = 10$.

  • Step 3: Select multiplier $2$. Compute $5 \times 2 = 10$. Subtract from total: 10 - 10 = 0$.\n * **Final Quotient Calculation**: Sum all selected multipliers: 2 + 1 + 2 = 5$. Therefore,$25 \div 5 = 5$.

  • Inverse Check: Verify quotient by multiplying result by divisor: 5 \times 5 = 25$.\n\n# Fraction Fundamentals, Comparison, and Conversions\n\n* **Fraction Representation**:\n * **Denominator (Bottom Number)**: Represents total equal parts in the whole.\n * **Numerator (Top Number)**: Represents number of shaded or selected parts.\n * Example: A shape divided into 4$equal parts with$2$shaded parts represents the fraction$\frac{2}{4}.\n\n* **Comparing Fractions (Butterfly Method / Cross-Multiplication)**:\n * **Purpose**: Determines which fraction is larger without drawing models.\n * **Example**: Compare \frac{2}{4}$and$\frac{3}{4}.\n * Team 1: Multiply numerator of first fraction by denominator of second fraction (2 \times 4 = 8).\n * Team 2: Multiply denominator of first fraction by numerator of second fraction (4 \times 3 = 12).\n * Compare products: 12 > 8$.

  • Conclusion: Team 2 is larger, so $\frac{3}{4} > \frac{2}{4}$.

• Equivalent Fractions:

  • Definition: Fractions that represent equal numerical values or equal proportions of a whole.

  • Testing Equivalence (Butterfly Method):

  • Example: Check if $\frac{2}{5}$ and $\frac{4}{10}$ are equivalent.

  • Cross-multiply diagonal pairs:

    • Left cross: $2 \times 10 = 20$

    • Right cross: $5 \times 4 = 20$

  • Comparison: Both cross-products equal 20$.\n * Conclusion: \frac{2}{5} = \frac{4}{10}, confirming they are equivalent.\n\n* **Mixed Numbers and Improper Fractions**:\n * **Definitions**:\n * **Mixed Number**: Combines a whole number and a proper fraction (e.g., 3\frac{2}{3}).\n * **Improper Fraction**: A fraction where numerator is greater than or equal to denominator (e.g., \frac{11}{3}$or$\frac{3}{2}).\n * **Converting Mixed Number to Improper Fraction**:\n * Example: Convert 3\frac{2}{3} to an improper fraction.\n * Step 1: Multiply whole number by denominator (3 \times 3 = 9).\n * Step 2: Add original numerator to product (9 + 2 = 11). This forms the new numerator.\n * Step 3: Retain original denominator (3).\n * Result: \frac{11}{3}.\n * **Converting Improper Fraction to Mixed Number**:\n * Example: Convert \frac{3}{2} to a mixed number.\n * Step 1: Draw total dots equal to numerator (3 dots).\n * Step 2: Group dots into sets equal to denominator (2 dots per group).\n * Step 3: Count complete groups (1$group of$2$) and leftover dots ($1 leftover).\n * Step 4: Form mixed number: Whole number = 1$, Numerator =$1$, Denominator =$2$.

  • Result: $1\frac{1}{2}$.

Geometry, Lines, Angles, and Symmetry

• Geometric Vocabulary:

  • Point: An exact location represented by a dot.

  • Line: A straight continuous path extending infinitely in opposite directions.

  • Line Segment: A straight line path bounded by two distinct endpoints (or arrow ends).

  • Straight Angle: An angle forming a completely straight path.

  • Obtuse Angle: An angle with a large open opening (described as "fat").

  • Acute Angle: An angle with a small narrow opening (described as "tiny").

  • Parallel Lines: Two straight lines that run side-by-side in the same direction, remaining equal distance apart forever without ever crossing, intersecting, or touching. Lines that crash or cross are not parallel.

• Symmetry and Lines of Symmetry:

  • Definition: A shape exhibits symmetry if it can be divided into two halves that match identically when folded along a line of symmetry.

  • Shape Symmetry Analysis:

  • Circle: Symmetric when divided straight through the center.

  • Triangle: Symmetric when cut vertically down middle; not symmetric when cut horizontally if top and bottom halves differ.

  • Square: Contains multiple lines of symmetry (vertical, horizontal, diagonal top-left to bottom-right, diagonal bottom-left to top-right).

  • Irregular Shapes: May contain zero ($0$) lines of symmetry if no fold yields matching halves.

Personal Financial Literacy

• Core Financial Terminology:

  • Financial Literacy: The study and management of personal money, banking, and expenses.

  • Lender: An institution (such as a bank) that supplies money to a borrower.

  • Borrower: An individual who receives money from a lender and agrees to repay it over time.

  • Interest: An additional fee charged by a lender to a borrower for the service of borrowing money.

• Interest Calculation Example:

  • A borrower requests $100,000\,\text{USD}$ from a bank to purchase a house.

  • The bank acts as lender and grants $100,000\,\text{USD}$.

  • The bank charges $10,000\,\text{USD}$ as an interest fee.

  • Over the loan term (paid via monthly installments over $10$ to $30$ years), the borrower repays a total of $110,000\,\text{USD}$ ($100,000\,\text{USD} + 10,000\,\text{USD}$ interest).

• Fixed vs. Variable Expenses:

  • Fixed Expense: An expense where the cost remains constant and identical every payment period without changing.

  • Example 1: House Rent ($500\,\text{USD}$ in January, $500\,\text{USD}$ in February, $500\,\text{USD}$ in March).

  • Example 2: Flat-rate cell phone bill ($40\,\text{USD}$ fixed monthly fee).

  • Variable Expense: An expense where the cost changes or fluctuates based on daily/monthly usage or individual choices.

  • Example 1: Electricity / Light bill ($0\,\text{USD}$ on day one without usage, $20\,\text{USD}$ on day two with heavy usage, $5\,\text{USD}$ on day three).

  • Example 2: Dining out at restaurants ($20\,\text{USD}$ for one meal, $30\,\text{USD}$ for a second meal, $5\,\text{USD}$ for a third meal).