Module 12
When the English Tells You to Use Desmos.
計算方法を選ぶ （Keisan Hōhō o Erabu）
帰国子女・インターナショナルスクール生のための、デジタルSAT 数学の無料教材。独学で進められます。
Both digital SAT Math modules include the built-in Desmos graphing calculator. Certain phrases are effectively instructions to use it — and recognising them is a reading skill, not a maths skill.
| If the question says… | Do this |
|---|---|
| "the graph of…", "crosses the x-axis", "intercept", "vertex" | Graph it and read the point |
| "a system of equations", "the solution to the system" | Graph both, read the intersection |
| "which of the following is a solution to…" | Test the choices; don't solve |
| "for what value of x…" | Graph both sides; read where they meet |
| "in terms of", "which expression is equivalent" | Algebra by hand — Desmos will not help |
The rule of thumb. If the question gives you candidate answers and asks which one works, testing the four choices is almost always faster than solving — and it cannot go wrong in the way algebra can.
日本人がハマる理由
日本の数学教育は、自分の手で、きちんと最後まで解ききることを評価します。筆算の正確さは誇りの対象ですし、その訓練自体は本当に優れています。ただし、時間制限のある multiple-choice のテストでは、それが失点につながることがあります。4つの選択肢を一つずつ代入して確かめるのは、ずるをしているように感じます。しかし SAT では、それが速くて正しい解き方でしかありません。解くのではなく試す。それを自分に許してください。
練習問題
難易度は出題者による目安です。College Board の判定ではありません。
The graph of y = x² − 6x + 5 crosses the x-axis at two points. What is the distance between these two points?
Correct Answer: A
Explanation: "Crosses the x-axis" is the cue from the table: type it into Desmos and read the two crossings — they sit at x = 1 and x = 5, so the distance is 5 − 1 = 4. (By hand, if you prefer: x² − 6x + 5 = (x − 1)(x − 5) = 0. Same two roots, more steps.)
- (B) 5: One of the two roots, not the distance between them.
- (C) 6: From the coefficient of x, or from
1 + 5— the sum of the roots rather than their difference. - (D) 24: No meaningful route; typically from mis-expanding the factorisation.
Difficulty Justification
Rated Medium. "Crosses the x-axis" is the cue to set y = 0 — or simply to graph it. Note that the final step is a subtraction the question asks for and the algebra does not give you (Module 8).
For what value of x is y = 2x + 1 equal to y = −x + 7?
Correct Answer: B
Explanation: "For what value of x" with two equations is the cue to graph both lines and read the intersection: they meet at (2, 5), so x = 2. (By hand: 2x + 1 = −x + 7 → 3x = 6 → x = 2.) Check: 2(2) + 1 = 5 and −2 + 7 = 5. ✓
- (A) 1: From
3x = 3— the constants were combined incorrectly. - (C) 3: From dividing 6 by 2 instead of 3; the
−xwas not carried across. - (D) 5: The y-value at the intersection point. The question asked for x — Module 8 again.
Difficulty Justification
Rated Easy, but choice D is the other coordinate of the correct answer, which is the trap this question is built around.
Which of the following is a solution to x² + 3x − 10 = 0?
Correct Answer: B
Explanation: "Which of the following is a solution" means test the choices. (−5)² + 3(−5) − 10 = 25 − 15 − 10 = 0. ✓ (Factoring gives (x + 5)(x − 2) = 0, so the solutions are −5 and 2; only −5 is offered.)
- (A) −10:
100 − 30 − 10 = 60 ≠ 0. Taken from the constant term. - (C) 3:
9 + 9 − 10 = 8 ≠ 0. Taken from the coefficient of x. - (D) 10:
100 + 30 − 10 = 120 ≠ 0.
Difficulty Justification
Rated Easy and included to make the habit concrete: four substitutions take under thirty seconds and are immune to factoring errors. The phrase "which of the following" is your permission to stop solving.