Day 3 — Thursday, May 14, 2026

Yesterday's exam: 14/18 on A and 14/18 on B. 28/36 total.

You took both versions — that was your call, and it gave us double the data. Read this before you study. The plan for today is built directly on what we saw.

What's solid: periodic table 8/8, wars 6/6, evolution 6/6. Memorization is working. Notes were honest, sometimes funny ("didn't we do this last time?"). Good.

GCF — 5 / 6. You stuck with listing factors. That's fine when you list carefully. The one you missed was (20, 25): your factor list for 25 was 1, 2, 5, 10, 15? 25. — both 2 and 10 and 15 are wrong (they don't divide 25), and the question mark was you knowing something was off. When you hesitate, stop and test: does 10 divide 25? 25 / 10 = 2.5 — no. Cross it off.

LCM — 3 / 6. Three different mistakes.

1. You confused LCM with GCF. On (10, 15) you wrote 10- 1, 2, 5, 10. 15- 1, 3, 5, 15. — those are factors, not multiples. You picked 5, the largest common factor. That's GCF, not LCM. LCM is about lists going up forever (10, 20, 30, 40…) until they meet. GCF is about lists going down to 1.
2. You listed multiples correctly but picked the wrong common. On (4, 6) you wrote 4: 4, 8, 12, 16, 20, 24… and 6: 6, 12, 18, 24, 30…. You said 24. But 12 is in both lists first. The LCM is the first match, not any match.
3. A typo / arithmetic slip on (5, 7) — you wrote 34, not 35. Read your own list back before you submit.

Fraction addition — 0 / 4. This is the headline. You don't have a model yet. You wrote: "if I have half an apple and a 4th of an apple I'd have a third of an apple so 2/3." That's a reasonable guess from a kid who hasn't learned the rule, but it's wrong. The actual rule is only below — read it, do it, drill it.

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No subtraction. No multiplication. No row 3. Today is a drill day. We're not adding new things while addition is at 0. Today's exam is 22 questions — six of them are addition, four are LCM, two are GCF. The other ten are periodic / wars / evolution review (you have those cold).

Practice, practice, practice.

Do not assume you can solve a fraction-add problem because you read the lesson once. Do not start the exam until you have done at least 20 addition problems on paper or on /fractions. Same for LCM — do 20 LCM problems before the exam. The lesson explains the rule. Practice makes the rule yours.

This week is to start. Next week we will give you better tools and more ideas about how to study. For now, the tool is your paper, the /fractions drill page, and time.

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The deal stays the same: 9:00 – 13:30, no phone, no PC outside this window. Plan on 3 to 4 hours. Two days down, two to go.

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Today's exam covers

• Math — GCF refresh (2 questions). Listing or Euclid, your call.
• Math — LCM (4 questions). Three of these are direct repeats of yesterday's misses.
• Math — Adding fractions (6 questions). Heavy practice. Two are same-denominator warmups, four require LCM.
• Periodic table — rows 1 and 2 review (4 questions). No new elements today.
• History — wars 1-8 review (3 questions). No new wars today.
• Biology — active list review (3 questions). No new events.

22 questions per version. Two versions; you can take both.

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Adding fractions — the rule, with no shortcuts

There is one rule and it has three steps. Memorize it, then practice it 20 times today.

Step 1 — find a common denominator

The common denominator is the LCM of the two denominators.

1/2 + 1/4 Denominators are 2 and 4. LCM(2, 4) = 4.

Step 2 — rewrite each fraction with that denominator

To rewrite a/b so it has denominator c:

• compute c / b (this gives you a multiplier m),
• multiply both top and bottom by m.

1/2 → multiplier is 4 / 2 = 2 → top × 2 = 2, bottom × 2 = 4 → 2/4 1/4 → multiplier is 4 / 4 = 1 → unchanged → 1/4

Step 3 — add the numerators only

The denominators are now equal. Just add the tops.

2/4 + 1/4 = (2 + 1) / 4 = 3/4

That's the answer. It's already in lowest terms.

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A second worked example

2/3 + 1/4

1. LCM(3, 4) = 12.
2. 2/3 → mult is 12/3 = 4 → 8/12. 1/4 → mult is 12/4 = 3 → 3/12.
3. 8/12 + 3/12 = 11/12.

A third — same denominator

3/7 + 2/7

1. LCMs are equal already. No rewriting needed.
2. (skipped)
3. (3 + 2) / 7 = 5/7.

Where you went wrong yesterday

You added the numerators (good), then invented a denominator from the two original denominators (bad). The denominator is not averaged, not added, not randomly picked. It is the LCM of the two starting denominators. Period.

Full lesson: lessons/fraction-add. The /fractions page has an Add mode — drill until you can do five in a row right.

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LCM vs GCF — one-line difference

GCF is the largest number that divides into both. It comes from listing factors of each number (1, 2, 3, 6 for 6 — small numbers). The answer is at most the smaller input.

LCM is the smallest number that both numbers divide into. It comes from listing multiples of each number (6, 12, 18, 24 for 6 — big numbers, going up forever). The answer is at least the larger input.

If you ever catch yourself listing 1, 2, 3 — you're solving GCF. If you're listing 6, 12, 18, 24 — you're solving LCM.

Full lesson: lessons/lcm.

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Suggested study path

Total time: 3 to 4 hours.

1. ~25 min — Re-read the LCM lesson. Practice 10 LCM problems on paper, including (10, 15), (4, 6), and (5, 7) — yesterday's failures. Verify each.
2. ~60 min — The fraction-add lesson. Read it twice. Do every worked example yourself first, then compare to the answer.
3. ~30 min — /fractions drill page, Add mode. Do 20 problems. Don't move on until you can get 5 in a row right.
4. ~10 min — Quick refresh: GCF, periodic table rows 1–2, wars 1–8, evolution active list. You have these cold; just touch them.
5. break — ~30 min off-screen.
6. ~30 min — Take one (or both) versions of the exam.

Reminder: the score reads only the answer field, not your note. Type the number in the answer box. The note is for your reasoning.

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Begin the exam

Pick A or B. After you submit, you can take the other version too.