The table gives the types and measurements, in feet, for 20 trees that were nominated for the Big Tree Contest...

OWNING THE DATASET

Let's start by understanding what we're working with. This table contains information about different trees, showing their heights and circumferences.

A quick scan reveals:

• We have 20 different trees in our dataset
• Heights range from around 50 feet to over 100 feet
• Circumferences vary widely, with some under 10 feet and others over 18 feet
• There appears to be no consistent relationship between height and circumference at first glance

Key insight: When working with this type of comparative data, sorting will be our most powerful tool for revealing patterns quickly.

Let's analyze our three statements in the most efficient order—not necessarily in the order they're presented.

ANALYZING STATEMENT 2

Statement 2 Translation:
Original: "102 feet is the modal height."
What we're looking for:

• The height that appears most frequently in our dataset
• Whether that height is specifically 102 feet

In other words: Is 102 feet the most common height among these trees?

The most efficient way to determine the mode is to sort the data by height. This immediately groups identical values together, making the mode visually apparent.

Let's sort by height (ascending):

After sorting, we can quickly scan the height column for repeating values. Any height that appears multiple times will be grouped together in consecutive rows.

Looking at our sorted data, we immediately notice:

• Most heights appear only once
• Some heights appear twice
• 102 feet appears three times in consecutive rows

Since 102 feet is the only height that appears more than twice, it is clearly the modal height.

Statement 2 is Yes.

Teaching note: Notice how sorting eliminated the need to count occurrences manually for each height value. When looking for modes or frequencies, sorting should be your first instinct as it transforms counting from a tedious process into a quick visual scan.

ANALYZING STATEMENT 3

Statement 3 Translation:
Original: "The probability of randomly selecting a tree with a circumference of less than 10 feet is 20%."
What we're looking for:

• How many trees have circumference < 10 feet
• Whether that number divided by the total (20) equals 20%

In other words: Are exactly 4 out of the 20 trees (20%) less than 10 feet in circumference?

While we could scan the entire table checking each circumference, sorting makes this much more efficient.

Let's sort by circumference (ascending):

With this sort, all trees with the smallest circumferences appear at the top of our table. Now we can simply count from the top until we reach 10 feet.

Scanning down the sorted list:

• The first 4 trees all have circumferences less than 10 feet
• The 5th tree and beyond all have circumferences of 10 feet or greater

Therefore, 4 out of 20 trees have circumferences less than 10 feet.
Probability = \(\frac{4}{20} = 0.2 = 20\%\)

Statement 3 is Yes.

Teaching note: Sorting brings all relevant data points together, allowing us to count them in one continuous sequence rather than hunting through the entire dataset. This approach works wonderfully for any question about frequency or probability.

ANALYZING STATEMENT 1

Statement 1 Translation:
Original: "The circumference of a tree is proportional to its height."
What we're looking for:

• Whether the ratio of circumference to height is constant across all trees
• If not, the statement is false

In other words: Does the ratio of circumference/height remain the same for all trees?

For two variables to be proportional, their ratio must be constant. If we find even two trees with significantly different ratios, we can immediately disprove the claim.

While our table is still sorted by height, let's look at the extreme values to quickly test this proportionality claim:

Examining extreme cases:

• Smallest tree: Mulberry (52 ft height, 18.42 ft circumference)
• Tallest tree: Walnut (107 ft height, 13.33 ft circumference)

Immediately, we notice something striking: the tallest tree actually has a SMALLER circumference than the shortest tree! This visual observation already suggests the variables aren't proportional.

Let's calculate just two ratios to confirm:

• Mulberry ratio: \(\frac{18.42}{52} = 0.354\)
• Walnut ratio: \(\frac{13.33}{107} = 0.125\)

These ratios differ significantly (0.354 vs 0.125). If circumference were proportional to height, these ratios would be the same.

Statement 1 is No.

Teaching note: When testing proportionality claims, start with extreme values (highest and lowest). If proportional, the tallest trees should have the largest circumferences in the same proportion. This approach lets us disprove the claim with minimal calculation, rather than calculating ratios for all 20 trees.

FINAL ANSWER COMPILATION

Evaluating our three statements:

• Statement 1: No (Circumference is not proportional to height)
• Statement 2: Yes (102 feet is the modal height)
• Statement 3: Yes (Probability of a tree with circumference < 10 feet is 20%)

Our answer pattern is: No, Yes, Yes

LEARNING SUMMARY

Skills We Used

• Strategic Sorting: We sorted the data twice (by height and by circumference) to reveal patterns instantly
• Extreme Value Testing: We used the smallest and tallest trees to quickly disprove proportionality
• Visual Pattern Recognition: After sorting, we used visual scanning to identify the mode and count trees with small circumferences

Strategic Insights

1. Tackle statements in efficiency order, not numerical order: We started with Statement 2 because sorting by height would immediately reveal the mode.

1. Sort first, calculate last: Sorting transforms many calculation problems into simple visual observation tasks.

1. For proportionality claims, check extreme values first: If the tallest and shortest trees don't maintain the same ratio, we can immediately disprove proportionality without checking every tree.

1. When counting or finding probabilities, sort relevant data together: This eliminates the need to scan the entire dataset multiple times.

Common Mistakes We Avoided

• Calculating numerous ratios (5+) when only 2 were needed to disprove proportionality
• Manually counting occurrences of each height without sorting
• Scanning the entire unsorted table for trees with circumferences less than 10 feet
• Solving statements in numerical order rather than efficiency order

Remember: In table analysis questions, how you approach the data is often more important than the calculations themselves. Strategic sorting and visual pattern recognition can dramatically reduce the time needed to solve these problems accurately.