For each of 8 varieties of hybrid turnips, the table shows the variety's average root diameter, in centimeters (cm), root...

OWNING THE DATASET

Let's start by understanding what we're working with. This table shows information about different turnip varieties with four key attributes:

• Root diameter (a numeric measurement in cm)
• Heat tolerance (categorical: weak, medium, strong)
• Cold tolerance (categorical: weak, medium, strong)
• Expected average yields (numeric, measured in t/ha)

Rather than reformatting the table, we can work directly with it as presented. The key to solving table problems efficiently is to quickly identify patterns and relationships between these attributes.

Note: With categorical data like tolerance ratings and numeric data like yields, sorting becomes our most powerful tool for revealing patterns instantly.

ANALYZING STATEMENT 1

Statement 1 Translation:
Original: "Each variety with an average root diameter of at least 12 cm is rated weak for heat tolerance."
What we're looking for:

• Find all varieties with root diameter \(\geq 12\text{ cm}\)
• Check if ALL of these have "weak" heat tolerance

In other words: Do ALL large turnips (\(\geq 12\text{ cm}\)) have weak heat tolerance?

Let's tackle this efficiently using sorting:

1. Sort by root diameter (descending) to group all the larger turnips together at the top.

When we do this, the turnips with diameter \(\geq 12\text{ cm}\) immediately appear at the top of our sorted table. Now we can visually scan just these entries to check their heat tolerance ratings.

Looking at these varieties with diameter \(\geq 12\text{ cm}\), we can see that all of them have "weak" heat tolerance. There are no exceptions – every variety with a root diameter of at least 12 cm shows "weak" for heat tolerance.

Teaching Callout: Notice how sorting eliminated the need to search through the entire table looking for varieties that meet our diameter criteria. Once sorted, the pattern became immediately visible, saving us from having to manually check each variety one by one.

Answer for Statement 1: YES

ANALYZING STATEMENT 2

Statement 2 Translation:
Original: "Each variety rated strong for cold tolerance has expected average yields of at least 60 t/ha."
What we're looking for:

• Find all varieties with "strong" cold tolerance
• Check if ALL of these have yields \(\geq 60\text{ t/ha}\)

In other words: Do ALL strong cold-tolerant turnips yield at least 60 t/ha?

Let's use our sorting strategy again:

1. Sort by cold tolerance to group all "strong" varieties together
2. Add a secondary sort by yield (ascending) to put the lowest-yielding varieties at the top of each group

This double-sort approach is powerful because:

• It immediately groups all "strong" cold tolerance varieties together
• Within that group, the lowest yield appears first – if it's below our threshold, we can immediately conclude the statement is false

After sorting, we can see that one of the varieties with "strong" cold tolerance – Shirayuki – has a yield of only 52 t/ha, which is less than our 60 t/ha threshold.

Teaching Callout: Here's where our efficiency shines! We found a counterexample immediately. There's no need to check all the other "strong" cold tolerance varieties – one exception is enough to disprove an "each" statement. This is the "Early Termination Principle" in action.

Answer for Statement 2: NO

ANALYZING STATEMENT 3

Statement 3 Translation:
Original: "Each variety rated medium for heat tolerance is rated medium for cold tolerance."
What we're looking for:

• Find all varieties with "medium" heat tolerance
• Check if ALL of these also have "medium" cold tolerance

In other words: Do ALL medium heat-tolerant turnips also have medium cold tolerance?

Let's sort effectively again:

1. Sort by heat tolerance to group varieties by their heat tolerance rating

After sorting, we can immediately see that there's only one variety with "medium" heat tolerance – Hakuju. This makes our verification extremely simple.

Looking at this single variety's cold tolerance rating, we can see that Hakuju indeed has "medium" cold tolerance.

Teaching Callout: This demonstrates "Data Sparsity Recognition" – sorting instantly revealed there was only one data point to check, making verification trivial. No need for complex tracking or multiple checks when the data is so sparse.

Answer for Statement 3: YES

FINAL ANSWER COMPILATION

After analyzing all three statements:

• Statement 1: YES (All varieties with diameter \(\geq 12\text{ cm}\) have weak heat tolerance)
• Statement 2: NO (At least one variety with strong cold tolerance has yield \(< 60\text{ t/ha}\))
• Statement 3: YES (The only variety with medium heat tolerance also has medium cold tolerance)

LEARNING SUMMARY

Skills We Used

• Sorting for Instant Pattern Recognition: We used sorting as our primary strategy to make patterns immediately visible
• Visual Scanning After Sorting: We scanned for patterns in our sorted data rather than checking each entry individually
• Statement Translation: We broke down complex "each" statements into clear criteria

Strategic Insights

• Sort Before Calculating: When dealing with tables, sorting should be your first instinct
• Double-Sort Power: Sorting by condition first, then by test variable creates instant visual verification
• Early Termination Principle: For "each" statements, one counterexample is enough – stop once you find it
• Data Sparsity Recognition: Sometimes sorting reveals very few data points match your criteria, making verification trivial

Common Mistakes We Avoided

• We avoided reformatting the table, which would waste time without adding value
• We didn't manually search for all relevant varieties, which would be time-consuming
• We didn't keep checking examples after finding a counterexample for Statement 2
• We recognized the power of visual pattern recognition after sorting instead of manual data interrogation

Remember: In table analysis questions, your goal is to find the most direct path to the answer. Sorting gives you this direct path by making patterns visible that would otherwise require tedious checking. When you see "each" statements, think about which column to sort by first to reveal the answer most efficiently.