Ordering Integers - SHSAT Math

1. Which of the following lists shows the integers $-8,\;3,\;-2,\;5,\;0$ arranged from least to greatest?
   Answer: $-8,\;-2,\;0,\;3,\;5$ When comparing integers, you need to understand how positive and negative numbers relate to each other on the number line. Negative numbers are always less than zero and positive numbers, and among negative numbers, those with larger absolute values are actually smaller. Let's work through this systematically. Start with the most negative number: $-8$ is the smallest because it's farthest to the left on the number line. Next comes $-2$, which is negative but closer to zero than $-8$. Then $0$, followed by the positive numbers $3$ and $5$ in ascending order. So the correct arrangement from least to greatest is: $-8, -2, 0, 3, 5$. Choice A gives us exactly this correct ordering: $-8, -2, 0, 3, 5$. Choice B shows $5, 3, 0, -2, -8$, which is actually the numbers arranged from greatest to least—the complete opposite of what we want. Choice C lists $-8, 0, -2, 3, 5$, which incorrectly places $0$ before $-2$. Remember that zero is greater than any negative number, so $-2$ should come before $0$. Choice D begins with $-2, -8, 0, 3, 5$, putting $-2$ before $-8$. This is backwards because $-8$ is more negative (smaller) than $-2$. Remember this key principle: when ordering integers, negative numbers with larger absolute values are actually smaller. Think of the number line—the farther left you go, the smaller the numbers become.
2. Which of the following correctly orders the integers $-6,\;4,\;-1$ from greatest to least?
   Answer: $4>-1>-6$ When comparing integers, you need to understand how positive and negative numbers relate to each other on the number line. Positive numbers are always greater than negative numbers, and among negative numbers, those closer to zero are greater. Let's place these numbers on a number line: $-6$ is farthest left, $-1$ is closer to zero but still negative, and $4$ is positive. Moving from left to right gives us the order from least to greatest: $-6 < -1 < 4$. Therefore, from greatest to least: $4 > -1 > -6$. Looking at the answer choices: Choice A ($-6 > 4 > -1$) incorrectly places the most negative number first, showing a fundamental misunderstanding that negative numbers are smaller than positive ones. Choice C ($-1 > 4 > -6$) makes the error of thinking $-1$ is greater than $4$, forgetting that any positive number is greater than any negative number. Choice D ($4 > -6 > -1$) correctly identifies $4$ as the greatest but then incorrectly orders the negative numbers, placing $-6$ above $-1$ when $-1$ is actually closer to zero and therefore greater. Choice B ($4 > -1 > -6$) correctly orders all three integers from greatest to least. Strategy tip: When ordering integers, remember that positive numbers always beat negative numbers, and among negative numbers, the one with the smaller absolute value (closer to zero) is actually the greater number. Visualizing a number line can help you avoid common ordering mistakes.
3. Use the table to answer the question. Which city had the warmest temperature at noon?
   Answer: Albany The warmest temperature is the greatest integer. Albany recorded $6^{\circ}\text{C}$, Buffalo $-2^{\circ}\text{C}$, Rochester $1^{\circ}\text{C}$, and Syracuse $-4^{\circ}\text{C}$. Because $6>1>-2>-4$, Albany was warmest. B Wrong: $-2^{\circ}\text{C}$ is below both $6^{\circ}\text{C}$ and $1^{\circ}\text{C}$. C Wrong: $1^{\circ}\text{C}$ is warmer than the two negatives but cooler than $6^{\circ}\text{C}$. D Wrong: $-4^{\circ}\text{C}$ is the coldest of all.
4. The temperature at midnight was $-8°F$. By 3 AM, it had dropped by $5°F$. By 6 AM, it had risen by $11°F$ from the 3 AM temperature. If the temperatures at noon, 3 PM, and 6 PM were $4°F$, $-2°F$, and $-7°F$ respectively, which time had the lowest temperature?
   Answer: 3 AM, when temperature was $-13°F$ Calculate each temperature: Midnight = $-8°F$, 3 AM = $-8 - 5 = -13°F$, 6 AM = $-13 + 11 = -2°F$, Noon = $4°F$, 3 PM = $-2°F$, 6 PM = $-7°F$. Ordering from lowest to highest: $-13°F < -8°F < -7°F < -2°F = -2°F < 4°F$. The lowest temperature was $-13°F$ at 3 AM. Students might incorrectly choose A by not calculating the 3 AM temperature, C by confusing 6 AM with 3 AM, or D by not comparing all temperatures systematically.
5. Which list shows the integers $-11,\;-9,\;-6,\;-3$ arranged from least to greatest?
   Answer: $-11,\;-9,\;-6,\;-3$ When ordering negative integers, remember that negative numbers follow the opposite pattern from positive numbers—the "larger" the negative number appears, the smaller its actual value. To arrange $-11, -9, -6, -3$ from least to greatest, visualize them on a number line. Moving left from zero, you encounter $-3$ first, then $-6$, then $-9$, and finally $-11$ farthest to the left. Since "least" means smallest value, and numbers get smaller as you move left on a number line, $-11$ is the smallest, followed by $-9$, then $-6$, and finally $-3$ as the largest. Choice A ($-11, -9, -6, -3$) correctly shows the progression from smallest to largest value. Choice B ($-3, -6, -9, -11$) arranges them from greatest to least—this is the trap answer for students who correctly identify the relative sizes but misread "least to greatest" as "greatest to least." Choice C ($-9, -11, -6, -3$) incorrectly places $-9$ before $-11$, suggesting confusion about which negative number is smaller. Choice D ($-6, -3, -9, -11$) shows a random arrangement with no clear ordering logic. The key insight is that $-11 < -9 < -6 < -3$ because each number is closer to zero than the previous one. Think of negative numbers as debts: owing $11 is worse than owing $3, so $-11 < -3$. Study tip: When comparing negative integers, the number with the greater absolute value is actually the smaller number. Practice visualizing negative numbers on a number line to avoid confusion.
6. If $-9<a<-3,$ which of the following could be the value of $a$?
   Answer: $-8$ When you encounter an inequality like $-9<a<-3$, you're looking at a compound inequality that describes all numbers between -9 and -3, not including the endpoints. Think of this as a range on the number line where $a$ must be greater than -9 AND less than -3 simultaneously. To find which value could be $a$, you need to check if each answer choice falls within this range. Since we're working with negative numbers, remember that numbers closer to zero are actually larger. For example, -4 is greater than -8. Choice B, $-8$, satisfies both conditions: $-8 > -9$ (true, since -8 is closer to zero) and $-8 < -3$ (true, since -8 is farther from zero than -3). This makes B the correct answer. Choice A gives us $-12$, which fails because $-12 < -9$, placing it outside our range on the left side. Choice C offers $-3$, but this is an endpoint that's excluded by the strict inequality ($<$, not $≤$). Choice D presents $2$, a positive number that's way outside our negative range and fails the condition $a < -3$. The key strategy here is to visualize the number line or systematically check both parts of the compound inequality. Remember that with negative numbers, the closer to zero, the larger the value. This type of question tests your understanding of inequality notation and negative number ordering—both fundamental skills that appear frequently on standardized tests.
7. Which of the following lists is already in descending order?
   Answer: $7,\;1,\;-3,\;-9$ When you encounter ordering problems, remember that descending order means arranging numbers from largest to smallest, moving down the number line from right to left. To check if a list is in descending order, verify that each number is greater than the one that follows it. For negative numbers, remember that numbers closer to zero are larger (so $-3 > -9$). Let's examine choice A: $7, 1, -3, -9$. Moving from left to right: $7 > 1$ ✓, $1 > -3$ ✓, and $-3 > -9$ ✓. Each number is indeed larger than the next, so this list is correctly arranged in descending order. Choice B ($-2, -5, 0, 4$) fails immediately because $-2 > -5$ is correct, but then $-5 < 0$, which breaks descending order. The list should continue decreasing, not increase. Choice C ($12, 9, 15, 3$) starts correctly with $12 > 9$, but then jumps up to $15$, which is larger than $9$. This violates descending order since $9 < 15$. Choice D ($5, -1, -4, 10$) begins properly with $5 > -1$ and continues correctly with $-1 > -4$, but then jumps to $10$, which is much larger than $-4$. This breaks the descending pattern. Strategy tip: When checking descending order, scan from left to right and ask "Is each number smaller than the previous one?" If you find even one place where a number increases instead of decreases, eliminate that choice immediately.
8. Which of the following sums is the greatest?
   Answer: $2+(-5)$ When comparing sums involving negative numbers, you need to carefully work through the addition to see which result is largest. Let's calculate each sum step by step. For choice C, $2+(-5) = 2-5 = -3$. This means you start at 2 and move 5 units left on the number line, landing at -3. Now let's check the other options. Choice A gives us $(-4)+(-1) = -4-1 = -5$. You're adding two negative numbers, so you move further left from -4, reaching -5. Choice B yields $(-7)+3 = -7+3 = -4$. Starting at -7 and moving 3 units right gets you to -4. Choice D results in $0+(-6) = -6$. Adding a negative to zero simply gives you that negative number. Comparing all results: A gives -5, B gives -4, C gives -3, and D gives -6. Since -3 is closest to zero on the number line, it's the greatest value. Remember that with negative numbers, the number closest to zero is always the largest. Choice A (-5) is smaller than C because -5 is further left on the number line than -3. Choice B (-4) is also smaller than C, though it's closer. Choice D (-6) is the smallest of all since -6 is furthest from zero. Strategy tip: When comparing negative numbers, remember that the number with the smallest absolute value (distance from zero) is actually the greatest. So -1 > -2 > -3, and so on.
9. In a certain code, positive integers represent floors above ground level, negative integers represent floors below ground level, and zero represents ground level. If an elevator starts at floor $-3$, goes up $8$ floors, then down $12$ floors, then up $4$ floors, on which floor does it end up, and how does this compare to the starting floor?
   Answer: Floor $-3$, which is the same as the starting floor Starting at floor $-3$: After going up 8 floors: $-3 + 8 = 5$. After going down 12 floors: $5 - 12 = -7$. After going up 4 floors: $-7 + 4 = -3$. The elevator ends up at floor $-3$, which is exactly where it started. Choice B incorrectly calculates the final position as $-5$. Choice C incorrectly gets $1$ as the final floor. Choice D makes multiple computational errors to reach floor $5$.
10. Which integer is BETWEEN $-5$ and $2$ on a number line?
    Answer: $-1$ When you encounter questions about numbers "between" two values on a number line, you're looking for numbers that fall strictly within that range, not including the endpoints themselves. To find which integer is between $-5$ and $2$, visualize or sketch a number line. The integers that lie strictly between these values are: $-4, -3, -2, -1, 0, 1$. Notice that $-5$ and $2$ themselves are not included since "between" means strictly inside the interval. Looking at the answer choices, option C gives us $-1$, which clearly falls within our range of acceptable values. You can verify this by noting that $-5 < -1 < 2$, confirming that $-1$ is indeed between the two given numbers. Option A ($-6$) is incorrect because $-6$ is less than $-5$, placing it to the left of our starting point on the number line. Option B ($-5$) is wrong because it represents one of the endpoints, not a number between them. Option D ($3$) fails because $3$ is greater than $2$, putting it beyond our ending point. Remember that "between" in mathematics typically means strictly between, excluding the boundary values themselves. When you see these types of questions on the SHSAT, quickly identify the range of acceptable integers and check which answer choice falls within that range. Drawing a quick number line can help visualize the problem if you're unsure.

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