For a certain community, the graph shows the total number of new homes built in each of the last 15...

Owning The Dataset

Table 1: Text Analysis

Table 2: Chart Analysis

Key Insights

The dataset describes annual home construction for 15 years in one community, calculated as both raw totals and annual changes. The first year with a negative annual change is Year 8. After that, there are exactly 5 consecutive years (Years 9-13) of positive annual change. There are only two years (Year 8 and Year 14) where the annual change is negative; all other years show increases or stable counts. The mathematical relationship of annual change is central to analyzing trends in this dataset.

Step-by-Step Solution

Question 1: Identifying the First Year with Negative Annual Change in New Homes Built

Complete Statement:

Among the Years 2-15, the first year for which the annual change in the number of new homes built was negative was Year ___.

Breaking Down the Statement

• Statement Breakdown 1:
  • Key Phrase: Among the Years 2-15
  • Meaning: We are to consider data only from Year 2 through Year 15, as Year 1 has no prior year to compare for annual change.
  • Relation to Chart: This tells us to ignore Year 1 in our search for the first negative change.
  • Important Implications: We start checking for negative changes beginning with Year 2.
• Statement Breakdown 2:
  • Key Phrase: annual change in the number of new homes built was negative
  • Meaning: The number of new homes built in that year was less than the previous year — a decrease.
  • Relation to Chart: This corresponds to any year where the bar (or value) is less than the year before.
  • Important Implications: We should compare each year's value to the prior year, looking for the first time it drops.

What is needed: The earliest year from Year 2 to Year 15 where the number of new homes built is less than in the preceding year.

Solution:

• Condensed Solution Implementation:
  Compare the number of new homes built in each year from Year 2 onward to the previous year, until a decrease is found.
• Necessary Data points:
  Year 7 had about 950 new homes; Year 8 had about 900, which is less than Year 7.
• Calculations Estimations:
  Year 8: 900 < Year 7: 950, first negative annual change occurs at Year 8.
• Comparison to Answer Choices:
  The choices are 5, 8, 12. The correct choice is 8.

FINAL ANSWER Blank 1: 8

Question 2: Counting the Consecutive Years with Positive Annual Change Following the First Negative

Complete Statement:

...and that year was immediately followed by exactly ___ consecutive years for which the annual change in the number of new homes built was positive.

Breaking Down the Statement

• Statement Breakdown 1:
  • Key Phrase: that year was immediately followed by
  • Meaning: We are looking at the sequence of years directly after the year found in question 1 (Year 8).
  • Relation to Chart: Start looking at Year 9 and onwards.
• Statement Breakdown 2:
  • Key Phrase: exactly ___ consecutive years for which the annual change...was positive
  • Meaning: Count how many years in a row the number of new homes built increased compared to the previous year.
  • Relation to Chart: This is represented by each year's bar or value being higher than the prior year's, and the sequence must not be broken by a year with no increase or a decrease.

What is needed: How many consecutive years after Year 8 had more new homes built than the previous year.

Solution:

• Condensed Solution Implementation:
  Start at Year 9, compare each year to its previous one, and count the number of consecutive years where the value increases, stopping at the first year that breaks the streak.
• Necessary Data points:
  Years 9 (1150) > 8 (900), 10 (1300) > 9 (1150), 11 (1350) > 10 (1300), 12 (1400) > 11 (1350), 13 (1600) > 12 (1400), 14 (1450) < 13 (1600).
• Calculations Estimations:
  There are increases from Year 9 through Year 13 (5 consecutive years), then Year 14 decreases (1600 to 1450), ending the run.
• Comparison to Answer Choices:
  Answer choices are 5, 6, 7. The correct count is 5.

FINAL ANSWER Blank 2: 5

Summary

To answer these questions, compare each year's number of new homes to the previous year. The first decrease happens at Year 8, and then five consecutive years of increases follow, ending at Year 13.

Question Independence Analysis

The questions are dependent. The second blank specifically refers to the outcome of the first question, requiring students to use their answer for the first negative change (Year 8) as the starting point for counting positive changes.