For each of the years 2012, 2013, 2014, and 2015, the table gives the total number of businesses that had...

OWNING THE DATASET

Let's start by understanding what we're working with. This table shows business holdings data across several years (2013-2015), comparing them to a baseline of January 1, 2012. For each year, we can see:

• Number of businesses with increases in value and their average increase
• Number of businesses with decreases in value and their average decrease

Key insight: The table doesn't directly tell us the total value change - it gives us the components we need to determine direction of change. This means we can potentially avoid detailed calculations if we spot clear patterns.

For example, looking at 2014's row, we can immediately notice that more businesses decreased (39) than increased (28), AND the average decrease ($41,742.33) is significantly higher than the average increase ($30,517.86). This visual pattern suggests a net decrease without needing exact calculations.

ANALYZING STATEMENT 1: 2013

Statement 1 Translation:
Original: "As of December 31, 2013, the value of holdings had increased compared to January 1, 2012."
What we're looking for:

• Did the total value of holdings increase from the 2012 baseline by the end of 2013?
• We need to determine if positive changes outweighed negative changes

In other words: Were the gains in 2013 greater than the losses?

To determine this efficiently, let's estimate the total increases and decreases:

For increases:

• 34 businesses with an average increase of approximately $39,000
• Quick estimation: \(34 \times \$39,000 \approx \$1,326,000\) in total increases

For decreases:

• 33 businesses with an average decrease of approximately $38,000
• Quick estimation: \(33 \times \$38,000 \approx \$1,254,000\) in total decreases

Comparing these estimates:

• Total increases: \(\sim\$1,326,000\)
• Total decreases: \(\sim\$1,254,000\)
• Estimated net change: \(\sim\$72,000\) (positive)

The increases slightly outweigh the decreases, indicating a positive net change from the 2012 baseline.

Statement 1 INCREASED FROM 2012.

Note how we used rounded numbers for quicker mental math rather than calculating to the penny. When we only need to know if a value increased or decreased, approximation is much more efficient than precise calculation.

ANALYZING STATEMENT 2: 2014

Statement 2 Translation:
Original: "As of December 31, 2014, the value of holdings had increased compared to January 1, 2012."
What we're looking for:

• Did the total value of holdings increase from the 2012 baseline by the end of 2014?
• We need to determine if the net change was positive

In other words: Were holdings worth more at the end of 2014 than they were at the start of 2012?

Here's where pattern recognition becomes powerful. Looking at the 2014 data, we can spot two critical patterns:

1. More businesses decreased (39) than increased (28)
2. The average decrease ($41,742.33) is significantly higher than the average increase ($30,517.86)

When both factors (quantity and magnitude) point toward decreases, we can be certain that the overall change is negative without performing detailed calculations. This is a key pattern to recognize for efficiency!

Let's verify with a quick magnitude check:

• Total increases: \(28 \times \sim\$30,500 \approx \$854,000\)
• Total decreases: \(39 \times \sim\$41,700 \approx \$1,626,000\)
• Estimated net change: \(\sim-\$772,000\) (a substantial decrease)

This negative change clearly exceeds 2013's modest positive change, meaning that holdings have fallen below the 2012 baseline.

Statement 2 DID NOT INCREASE FROM 2012.

Teaching callout: Notice how recognizing the double-negative pattern (more businesses decreasing AND higher average decrease) allowed us to quickly determine the direction of change without precise calculations.

ANALYZING STATEMENT 3: 2015

Statement 3 Translation:
Original: "As of December 31, 2015, the value of holdings had increased compared to January 1, 2012."
What we're looking for:

• Did the total value of holdings increase from the 2012 baseline by the end of 2015?
• We need to determine if there was a net positive change since 2012

In other words: Were the cumulative changes across 2013-2015 positive overall?

Here we can leverage a powerful inference chain from our previous finding. Since we already know that holdings were below 2012 levels at the end of 2014, we only need to determine if 2015 showed enough positive change to overcome that deficit.

Looking at the 2015 data, we see the same pattern as 2014:

1. More businesses decreased (35) than increased (32)
2. The average decrease ($36,500) is higher than the average increase ($31,800)

With both factors again pointing toward a negative change within 2015 itself, it's impossible for holdings to have recovered back above 2012 levels. In fact, the gap would have widened further.

Quick verification:

• Total increases: \(32 \times \sim\$31,800 \approx \$1,017,600\)
• Total decreases: \(35 \times \sim\$36,500 \approx \$1,277,500\)
• Estimated net change for 2015: \(\sim-\$259,900\) (another decrease)

Since 2014 already put us below the 2012 baseline, and 2015 shows another negative change, holdings remain below 2012 levels.

Statement 3 DID NOT INCREASE FROM 2012.

Here's where strategic thinking saves time: Once we know we're below the baseline after 2014, we only need to confirm 2015 doesn't show a massive recovery to answer the question.

FINAL ANSWER COMPILATION

After analyzing all three statements:

• Statement 1 (2013): INCREASED FROM 2012 - Holdings increased compared to Jan 1, 2012
• Statement 2 (2014): DID NOT INCREASE FROM 2012 - Holdings decreased below Jan 1, 2012 levels
• Statement 3 (2015): DID NOT INCREASE FROM 2012 - Holdings remained below Jan 1, 2012 levels

The correct answer pattern is: INCREASED FROM 2012, DID NOT INCREASE FROM 2012, DID NOT INCREASE FROM 2012

LEARNING SUMMARY

Skills We Used

• Pattern Recognition: Identifying when both quantity and average amount point in the same direction
• Strategic Inference Chains: Using previous answers to simplify subsequent questions
• Approximation: Rounding numbers for faster mental calculations
• Direction vs. Precision Focus: Focusing only on whether values increased or decreased, not by how much

Strategic Insights

1. Know what the question is really asking: We only needed to determine if holdings increased or decreased compared to 2012, not calculate exact values.
2. Look for double indicators: When both the number of businesses and the average amount point in the same direction, you can quickly determine the overall direction without detailed calculations.
3. Build on previous findings: Once we knew 2014 dropped below 2012 levels, we only needed to check if 2015 showed recovery or continued decline.
4. Round numbers strategically: Using ~$39K instead of $38,836.51 makes mental math much faster while still giving accurate directional results.

Common Mistakes We Avoided

1. Unnecessary precision: We didn't calculate values to the penny when we only needed to know direction of change.
2. Failing to leverage patterns: Many students would calculate each year independently rather than using visual patterns and inference chains.
3. Redundant calculations: We didn't recalculate 2013 and 2014 data when analyzing 2015, instead building on what we already knew.

Remember: On data questions, always ask yourself "What's the minimum I need to determine to answer this question?" Often, it's much less than you think!