SBA 6th Grade Math Practice Test Exhaustive Study Notes

SBA Practice Test Chunk #1: Basic Coordinates and Rational Numbers

• Calculator Usage Restriction: Please note that you will NOT have access to a calculator for questions 1 through 10.

• Question 1: Grid Coordinates and Distance

  • Task: Determine the distance, in units, between point A and point C on the provided grid.
  • Coordinates: The grid shows Points arranged in a coordinate plane.
  • Result: The distance between Point A and Point C is $5$ units.

• Question 2: Points on a Number Line

  • Task: Consider points A, B, C, and D plotted on a horizontal number line ranging from $-5$ to $5$.
  • Relative Positions: Moving to the right ("Right") on the number line means the value is "greater," while moving to the left ("Left") means the value is "less."

• Question 3: Truth of Statements regarding Number Line Points

  • Statement 1: The value of Point A is less than $-3$. (Evaluation based on visual placement on number line).
  • Statement 2: The value of Point B is greater than the value of point A. (True, because Point B is to the right of Point A: $-2 > -3$ and $-1 > -2$).
  • Statement 3: The value of Point D is $-4$. (False, visual assessment indicates Point D is at $+4$).

• Cookie Recipe Conversion Problem

  • Given: A recipe requires $1 \frac{1}{2}$ cups of flour for every batch of cookies.
  • Scenario: You have $16 \frac{1}{2}$ cups of flour.
  • Calculation: How many full batches of cookies can be made?
    • Method 1 (Division of Fractions): Calculate $16 \frac{1}{2} \div 1 \frac{1}{2}$.
    • Step 1: Convert to improper fractions: $\frac{33}{2} \div \frac{3}{2}$.
    • Step 2: Apply "Keep, Change, Flip": $\frac{33}{2} \times \frac{2}{3}$.
    • Step 3: Multiply: $\frac{66}{6} = 11$ batches.
    • Method 2 (Visual/Pictorial Logic): If $1 \text{ batch} = 1.5 \text{ cups}$, you can count the half-cups. $16.5 \text{ cups}$ contains $33$ half-cups. Since $1$ batch uses $3$ half-cups ($1.5 \text{ cups}$), then $33 \div 3 = 11$ batches.

Mathematical Operations, Inequalities, and Expressions

• Question 4: Inequalities and Number Comparison

  • Task: Determine if specific values of $x$ make the inequality $x > -1.5$ true.
  • Evaluations:
    • Value $x = -2 \frac{1}{2}$: No. On a number line, $-2.5$ is to the left of $-1.5$, meaning $-2.5 < -1.5$.
    • Value $x = 1.5$: Yes. $1.5 > -1.5$.
    • Value $x = 0.2$: Yes. $0.2 > -1.5$.
    • Value $x = 4$: Yes. $4 > -1.5$.

• Question 5: Algebraic Expressions and Distribution

  • Task: Find an expression equivalent to $7(10a + 3b)$ that shows the sum of exactly two terms.
  • Process: Apply the distributive property.
    • $7 \times 10a = 70a$
    • $7 \times 3b = 21b$
  • Equivalent Expression: $70a + 21b$.
  • Important Note: Do NOT combine terms after distribution because $a$ and $b$ are "not like terms."

• Question 6: Long Division

  • Task: Divide $21,900 \div 25$.
  • Execution:
    • $219 \div 25 = 8$ with a remainder of $19$ ($8 \times 25 = 200$).
    • Bring down $0$ to get $190$.
    • $190 \div 25 = 7$ with a remainder of $15$ ($7 \times 25 = 175$).
    • Bring down the final $0$ to get $150$.
    • $150 \div 25 = 6$ ($6 \times 25 = 150$).
  • Exact Quotient: $876$.

• Question 7: Equation Modeling and Variables

  • Scenario: Paul could play $16$ songs. He learns $s$ new songs. Now he can play $23$ songs.
  • Possible Equations:
    • $16 + s = 23$
    • $s = 23 - 16$
  • Solution: Paul learned $7$ new songs ($s = 7$).

SBA Practice Test Chunk #2: Expressions, Equations, and Ratios

• Question 8: Equivalent Expressions Distributive Analysis

  • Target Expression: $4(5x + 2y)$.
  • Correct Selections:
    • $4(5x) + 4(2y)$
    • $20x + 8y$
  • Incorrect logic examples:
    • $9x + 6y$ (Incorrectly added the multiplier $4$ to the coefficients instead of multiplying).
    • $20x + 2y$ (Forgot to distribute the $4$ to the second term).
    • $4(7xy)$ (Cannot combine $5x$ and $2y$; they are NOT like terms).

• Question 9: Solving for Variable n

  • Task: Identify all equations where $n = 6$ is a valid solution.
  • Evaluations:
    • $2 + n = 6 \rightarrow 2 + 6 = 8$ (Not a solution).
    • $n + 6 = 12 \rightarrow 6 + 6 = 12$ (Correct solution).
    • $4 \cdot n = 24 \rightarrow 4 \cdot 6 = 24$ (Correct solution).
    • $n \cdot 3 = 2 \rightarrow 6 \cdot 3 = 18$ (Not a solution).

• Question 10: Ratios and Proportions

  • Scenario: A quiz has history and geography questions. The ratio of history questions to total questions is $4 : 5$.
  • Data Derivation:
    • History questions = $4$
    • Total questions = $5$
    • Geography questions = $5 - 4 = 1$
  • True Statements:
    • The ratio of total questions to history questions is $5 : 4$.
    • The ratio of history questions to geography questions is $4 : 1$.
  • False Statement: "There is 1 more history question than geography questions." (False; there are $3$ more history questions because $4 - 1 = 3$).

Calculator Allowed Operations: Data and Proportions

• Question 11: Ratio Tables (Punch Recipe)

  • Task: Fill in missing values for Ginger Ale (L) and Fruit Juice (oz).
  • Known Ratio: $3 \text{ L}$ of Ginger Ale to $18 \text{ oz}$ of Fruit Juice.
  • Unit Rate: $1 \text{ L}$ of Ginger Ale per $6 \text{ oz}$ of Fruit Juice ($18 \div 3 = 6$).
  • Missing Table Entries:
    • For $5 \text{ L}$ Ginger Ale: $5 \times 6 = 30 \text{ oz}$ Fruit Juice.
    • For $9 \text{ L}$ Ginger Ale: $9 \times 6 = 54 \text{ oz}$ Fruit Juice.

• Question 12: Creating a Histogram

  • Data Set: Texts per day: $0, 5, 39, 29, 0, 30, 14, 23, 25, 22$.
  • Frequency Distribution:
    • Range $0 - 9$: $3$ values ($0, 5, 0$).
    • Range $10 - 19$: $1$ value ($14$).
    • Range $20 - 29$: $4$ values ($29, 23, 25, 22$).
    • Range $30 - 39$: $2$ values ($39, 30$).

• Question 13: Multi-Step Financial Planning

  • Scenario: Allison saves for a $\$500$ bicycle. She earns $\$8$ per hour, works $20$ hours per week, for $4$ weeks.
  • Step 1 (Weekly Earnings): $\$8 \times 20 = \$160$ per week.
  • Step 2 (Total Earnings): $\$160 \times 4 = \$640$ total.
  • Step 3 (Surplus): $\$640 - \$500 = \$140$ remaining.

• Question 14: Tape Diagrams and Ratios

  • Scenario: Sam and Tyrone built $42$ model cars altogether. The ratio of Sam to Tyrone is $3$ units to $4$ units.
  • Step 1: Total units = $3 + 4 = 7$ units.
  • Step 2: Calculate value per unit square: $42 \div 7 = 6$.
  • Conclusion: Each square in the tape diagram represents $6$ cars.

SBA Practice Test Chunk #3: Geometry and Logic

• Question 15: Coordinate Park Map (Area Calculation)

  • Part A: Rectangle $ABCD$ coordinates: $A(-4, 4)$, $B(3, 4)$, $C(3, -2)$, $D(-4, -2)$.
  • Part B (Total Area):
    • Base ($b$) = $7$ units (from $-4$ to $3$).
    • Height ($h$) = $6$ units (from $4$ to $-2$).
    • $Area = b \times h = 7 \times 6 = 42 \text{ square yards}$.
  • Part C (Red Flowers Area):
    • Triangle $ABD$ is half the rectangle: $Area = \frac{42}{2} = 21 \text{ square yards}$.
    • Red flowers occupy $\frac{1}{3}$ of triangle $ABD$.
    • $\text{Area with red flowers} = \frac{1}{3} \text{ of } 21 = 7 \text{ square yards}$.

• Question 16: Decimals and Place Value Logic

  • Claim: Will states, "There is no number greater than $0.59$ that has a $5$ in the tenths place."
  • Counterexample: The statement is False. A number like $0.591$ is greater than $0.59$ and correctly maintains a $5$ in the tenths place. The thousandths place can be any digit other than zero to satisfy the "greater than" condition.

• Question 17: Volume of Compound Solids

  • Task: Calculate the volume of a solid made of two right rectangular prisms.
  • Prism 1 Volume: $12 \, \text{cm} \times 5 \, \text{cm} \times 4 \, \text{cm} = 240 \, \text{cm}^3$ (Calculation note: $12 \times 5 = 60, 60 \times 4.5 = 270 \, \text{cm}^3$ if interpreted as $4.5$, but visual notes suggest splitting to $90$ and $270$).
  • Segmented Calculation according to Transcript Notes:
    • Volume 1: $5 \times 4 \times 4.5 = 90 \, \text{cm}^3$.
    • Volume 2: $12 \times 5 \times 4.5 = 270 \, \text{cm}^3$.
  • Total Volume: $90 + 270 = 360 \, \text{cm}^3$.

Unit Rates and Proportial Reasoning

• Question 18: Constant Rate of Cost

  • Variable $s$: Number of songs.
  • Variable $d$: Cost in dollars.
  • Ratio: $2 \text{ songs}$ cost $\$2.58$.
  • Unit Price: $\$2.58 \div 2 = \$1.29$ per song.
  • Calculated Table values:
    • $4 \text{ songs} = 4 \times 1.29 = \$5.16$
    • $7 \text{ songs} = 7 \times 1.29 = \$9.03$
    • $11 \text{ songs} = 11 \times 1.29 = \$14.19$
    • Correct number of songs for $\$21.93$: $21.93 \div 1.29 = 17 \text{ songs}$.

• Question 19: Percentages

  • Given: Ethan answers $80\%$ (or $32$ questions) correctly.
  • Find: Total questions ($x$).
  • Equation Setup:
    • $\frac{80}{100} = \frac{32}{x}$
    • $80x = 3200$
    • $x = 40$ questions.

• Question 20: Volume and Dimensions

  • Given: Volume $V = 243 \, \text{in}^3$, Height $h = 3 \, \text{in}$.
  • Formula: $V = l \times w \times h \rightarrow 243 = l \times w \times 3 \rightarrow l \times w = 81$.
  • Part A (Equal Dimensions): Length = $9$, Width = $9$ ($9 \times 9 = 81$).
  • Part B (Unequal Dimensions): Length = $3$, Width = $27$ ($3 \times 27 = 81$).

• Question 21: Absolute Value Claims

  • Claim: "If $|x| > |y|$, then $x > y$."
  • Definition: Absolute value is always positive.
  • Supports Claim: Not shown for negative integers.
  • Does Not Support (Counterexamples):
    • $x = -15, y = 14$: $|-15| > |14|$ ($15 > 14$) is true, but $-15 > 14$ is false.
    • $x = -0.9, y = -0.8$: $|-0.9| > |-0.8|$ ($0.9 > 0.8$) is true, but $-0.9 > -0.8$ is false (it is smaller on number line).

SBA Practice Test Chunk #4: Statistics and Applications

• Question 22: Statistical Histograms and Measures of Center

  • Data Shape: The histogram is "skewed to the right" (tail on the right).
  • Calculations: Total students surveyed = $5 + 8 + 6 + 1 = 20$.
  • Median: For $20$ pieces of data, the median is between the $10^{th}$ and $11^{th}$ values. These fall in the $2.0 - 3.9 \text{ mile}$ interval.
  • IQR Logic: Since $Q_3$ must be less than $6$, the Interquartile Range cannot be greater than $6.0$.

• Question 23: Rate and Time

  • Rate: $36 \text{ miles}$ in $45 \text{ minutes}$.
  • Simplified Rate: $\frac{36}{45} = \frac{4 \text{ miles}}{5 \text{ minutes}}$.
  • Prediction for 60 mins: $60 \div 5 = 12 \text{ sets of 5 minutes}$. $12 \times 4 \text{ miles} = 48 \text{ miles}$.

• Question 24: Factory Efficiency

  • Production Rate: $1,200 \text{ shirts} \div 6 \text{ hours} = 200 \text{ shirts per hour}$.
  • Total Production Needed: $12,600 \text{ shirts}$.
  • Total Hours Needed: $12,600 \div 200 = 63 \text{ hours}$.
  • Workday Conversion: $63 \text{ hours} \div 9 \text{ hours/day} = 7 \text{ workdays}$.

• Question 25: Geometric Expressions for Area

  • Scenario: Field width = $5.5 \, \text{m}$. Length = $2 \, \text{m}$ (tulips) + $x \, \text{m}$ (lilies).
  • Valid Equations for Total Area:
    • $A = 5.5(x + 2)$
    • $A = 5.5x + 11$ (distributed form).
  • Invalid Forms: $A = x^2(5.5)$ or $A = 2x + 5.5x$.

• Question 26: Budget Constraint Analysis

  • Goal: Spend between $\$18$ and $\$20$.
  • Prices:
    • Potato Salad: $\$2.25$ per $1.5 \, \text{lb}$ ($\$1.50 \, \text{lb}$ calculated from $3.5 \, \text{lbs} \rightarrow \$5.25$).
    • Ham: $\$2.00 / \text{lb}$.
    • Cheese: $\$0.60 / \text{lb}$.
    • Turkey: $\$3.50 / \text{lb}$.
    • Shrimp Salad: $\$5.50 / \text{lb}$.
  • Combination 4 Example: $2 \, \text{lbs}$ Shrimp Salad ($\$11$) + $\$2$ Ham ($\$4$ for $2 \, \text{lbs}$?) + Turkey + Potato Salad results in $\$19.50$, which is within range.

• Question 27: Spatial Packing (Volume Optimization)

  • Containers: $19.5 \, \text{in} \times 39 \, \text{in} \times 19.5 \, \text{in}$.
  • Tissue Boxes: $6.5 \, \text{in} \times 6.5 \, \text{in} \times 6.5 \, \text{in}$.
  • Boxes per dimension:
    • Width: $19.5 \div 6.5 = 3 \text{ boxes}$.
    • Length: $39 \div 6.5 = 6 \text{ boxes}$.
    • Height: $19.5 \div 6.5 = 3 \text{ boxes}$.
  • Total Capacity: $3 \times 6 \times 3 = 54 \text{ tissue boxes}$.

Final Chunk: Number Comparisons and Error Analysis

• Question 28: Comparing Fractions

  • Inequality: $\frac{n}{5} > \frac{4}{20}$.
  • Simplification: $\frac{4}{20} = 0.2$. $\frac{n}{5} = n \times 0.2$.
  • Iteration:
    • If $n=1$, $0.2 = 0.2$ (Not greater).
    • If $n=2$, $0.4 > 0.2$ (True).
  • Lowest Whole Number Numerator: $2$.

• Question 29: Ratio Error Analysis

  • Analysis: The table shows apples in relation to boxes. The relationship is $1 \text{ box} = 15 \text{ apples}$.
  • Check:
    • Row 2: $5 \times 15 = 75$ (Correct).
    • Row 3: $7 \times 15 = 105$ (Correct).
  • Error Detection: In Row 1, the table indicates $2 \text{ apples}$ in $2 \text{ boxes}$.
  • Correction:
    • Part A (Error Row): Row $1$.
    • Part B (Correct Value): $2 \times 15 = 30 \text{ apples}$.