Browse course topics
Acuity, Hard Items, and Pacing
Worked Walkthroughs
Summary
Three real items, solved end to end. Nothing new here — only the sequence you already have, run in order: read the four orderings, fix the extremes, eliminate, and resolve the middle pair only if two choices are still standing.
Item 1: the extremes finish it
Rank the four angles from SMALL to LARGE.
Read the choices before you study the angles. A says 3-1-4-2, B says 1-4-2-3, C says 1-3-2-4, D says 2-3-1-4. Three different panels get nominated as the smallest — #3, #1 and #2 — and three get nominated as the largest — #2, #3 and #4. That is the whole item in one line: two questions decide it, and neither of them is about the middle.
Now the row. #1 is drawn with the longest pair of rays in the item and is still, unmistakably, the narrowest opening — a sliver, and the rays only make it a longer sliver. #4 is the most open, roughly three times #1's opening. Commit to both: smallest #1, largest #4.
Spend those two judgments:
- A opens with #3, so it claims #3 is the smallest. You have already decided otherwise. Gone.
- B opens with #1 — correct — but closes with #3, claiming #3 is the largest. Gone.
- D opens with #2. Gone.
- C opens with #1 and closes with #4. It is the only ordering that agrees with you on both ends.
Answer C, and notice what never happened: you never compared #3 against #2. The item did not ask you to, because no surviving choice disagreed about them.
In degrees: #1 = 20°, #3 = 33°, #2 = 47°, #4 = 60°. Ray length is a distraction — compare the openings near each vertex.
Item 2: two survive, and the middle pair decides
Rank the four angles from SMALL to LARGE.
Orderings first, again. A: 4-3-2-1. B: 3-2-4-1. C: 3-4-1-2. D: 3-4-2-1. Three of the four open with #3 and three close with #1, so the choices largely agree about the extremes. That is not permission to skip the extremes pass — agreement among the choices is not evidence, and assuming it is exactly how you inherit someone else's error — but it does warn you the item will probably come down to the middle.
The row is unhelpful in the usual way. #3 carries the longest single ray in the item and #2 the shortest, and ranking these four by how far apart their tips finish gets the smallest and the largest both wrong. Read the openings close to the vertices instead: #3 is the tightest of the four, #1 the most open.
- A opens with #4. Gone.
- C closes with #2. Gone.
- B and D both open with #3 and close with #1. Both survive.
Look at what B and D disagree about: whether #2 or #4 takes second place. Everything else in the two orderings is identical. The item has collapsed to a single comparison — #2 against #4, with the other two panels now irrelevant — and that is the comparison to make, in isolation, ignoring the rest of the row.
#4 is the narrower of the two, so #4 goes second and the answer is D. The explanation confirms it: 109, 115, 123, 129. The call you had to make was eight degrees wide, and the extremes you banked first were twenty apart — which is why they came first.
In degrees: #3 = 109°, #4 = 115°, #2 = 123°, #1 = 129°. Ray length is a distraction — compare the openings near each vertex.
Item 3: a hard item, the same sequence
Rank the four angles from SMALL to LARGE.
A: 2-3-1-4. B: 2-1-3-4. C: 1-3-2-4. D: 3-2-4-1. Three of the four choices close with #4 and only D disagrees, while the smallest is contested three ways between #2, #1 and #3. That tells you where the looking has to go — not what the answer is. A majority of the choices agreeing about something is not evidence; you still have to see it.
This row is tight — the four openings span nine degrees in total — and the ray lengths actively lie about it. #4, which is the widest opening in the item, is drawn with the least ray length of the four; #2, the narrowest opening, with the most. Rank this item by how far apart the ray tips end up and you get the two extremes exactly swapped: that ranking calls #4 the smallest and #2 the largest, and both are precisely backwards. Work near the vertices, where the openings are, and commit: #2 smallest, #4 largest.
- C opens with #1. Gone.
- D opens with #3 and closes with #1 — wrong at both ends. Gone.
- A and B survive, and they disagree about one thing: whether #3 or #1 comes second.
#3 is the narrower, which makes A the answer. Be honest about that last step: three degrees, and plenty of people will not separate them today. The extremes work still paid — a coin flip between two survivors is a fifty percent shot at an item that started as a twenty-five percent shot, and the eliminations that got you there were the easy comparisons, not the hard one.
In degrees: #2 = 97°, #3 = 100°, #1 = 103°, #4 = 106°. Ray length is a distraction — compare the openings near each vertex.
Name the step that failed
When you miss one of these, "I got it wrong" is not a finding. There are only three places the sequence breaks, and they need different fixes:
- Wrong extreme. You called the smallest or the largest incorrectly, and the elimination pass then threw away the right ordering. This is the expensive failure — it does not just cost the item, it makes you confident while losing it.
- Tip-gap ranking. You ranked by how far apart the ray tips ended up rather than by the opening at the vertex. The give-away is a miss on an item where the gaps were wide and the ray lengths were lopsided.
- Coin-flip middle. Extremes right, two survivors, and you lost the last comparison. On a narrow item this is not a mistake so much as the item working as designed.
Telling them apart takes about fifteen seconds after you answer: read the printed degrees, say which panel was actually smallest and largest, and find where your version of the story first went wrong. Do it on the items you got right too — a right answer that came from an impression of the whole row is a miss you have not had yet.
The takeaway
The sequence never changes with difficulty: read the four orderings to see which comparisons are contested, judge the extremes and commit, eliminate every ordering that disagrees with you, and only then — if two survive — decide the middle pair in isolation. Item 1 ended after the extremes; items 2 and 3 came down to one binary call each, made against a pair of survivors rather than the whole row. Name the step that decided each item you drill. That is what turns a stack of finished items into a sharper eye.
Practice Questions
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