Higher Level only. It follows 3.10 directly and feeds 3.12, where the parameter stops being a dial and becomes the time.
Put view 1 up, write both equations on the board, and ask for a show of hands on whether they are the same line. Collect the vote before stepping to view 2. Most classes split, and a few will be confident they are different because no number matches.
Then view 2. One line on the screen. The second is drawn dashed on top of the first so it reads as an overlay rather than a replacement.
View 4 is the control and it matters as much as view 2: a line through the same point in a different direction. Same point, different line. Having both on the same figure stops the class concluding that any two equations sharing something are the same line.
| Question | Answer |
|---|---|
| 1. x = 2 + 3t at t = 4 | 2 + 12 = 14. |
| 2. 4x − 3y at (7, 10) | 28 − 30 = −2, so the point is on the line. |
| 3. 4x − 3y at (5, 5) | 20 − 15 = 5, so it is not. |
| 4. t at (7, 10) | 2, from both components. |
| 5. What shows two equations are one line | B. Parallel directions and a shared point. Either alone is not enough. |
Questions 2 and 3 are deliberately the same arithmetic with different numbers, one on the line and one off it. Doing both in immediate succession is what builds the habit; doing only the one that works teaches nothing, because the student never finds out what failure looks like.
1 markThe direction, from two points or read off the equation.
1 markThe equation as a point plus t times a direction.
1 markThe parameter found, or the point test done on both components.
On a "show that P lies on the line" question, a single component is not a show-that. The markscheme wants both, or the cartesian form, and a student who checks x alone has found only that SOME value of t puts the line at the right x, which any line with a non-zero x component in its direction does somewhere.
| They wrote | What happened |
|---|---|
| 12 | On question 1, dropped the starting value. They are treating the direction as the whole equation. |
| 20 | Computed (2 + 3) × 4. Order of operations, not vectors, and worth separating from the vector work so it does not get blamed on the topic. |
| 58 | On question 2, added instead of subtracting. Watch for it again on question 3, where it gives 35. |
| 2 instead of −2 | Computed 3y − 4x. Right size, wrong sign, and the sign is what the test compares. |
| −2 on question 3 | Did not compute it. They assumed the answer must match the line, which is the opposite of testing. Ask what the arithmetic actually gives. |
| 6 | On question 4, 7 − 1 without dividing by 3. The direction's length is the thing they have skipped, and it is the same omission as treating (6, 8) and (3, 4) as different directions. |
| 2.33 | 7/3, using the destination without subtracting the start. |
"So which answer do I write for the equation?" Any correct one. If a markscheme gives (1,2) + t(3,4) and a student writes (4,6) + t(6,8), that is right and examiners accept it. Saying so early prevents a term of students trying to guess the official point.
"Why bother with vector form in two dimensions?" Because 3.12 needs the parameter to be time, and y = mx + c has nowhere to put it. Also because a vertical line has no gradient and does have a perfectly ordinary direction vector (0, 1). Both reasons are worth giving; the second is the one that convinces.
"What if the components give different t values?" The point is not on the line, and that is the answer, not a sign of a mistake. Students often redo the arithmetic three times looking for an error. Tell them the disagreement IS the result.
"Are these lines parallel or the same?" Two questions, in order. Parallel is about the directions only. Same needs parallel plus a shared point. If a class learns to ask them separately, the whole family of questions becomes mechanical.
| Stage | What to do |
|---|---|
| Demonstrate | Graph both lines parametrically and let the second draw over the first. Then the point test on the Nspire, one component at a time: nSolve(1+3t=7,t) and nSolve(2+4t=10,t) both give 2. Run the same pair on (5, 5) and they give 1.33 and 0.75. Two numbers that disagree is what "not on the line" looks like, and seeing it on the screen is worth more than being told. The permitted Nspire is non-CAS, so there is no single command that takes both equations at once; that limitation is the lesson. |
| Where they stick | The parameter range. Both machines have a separate T window from the x and y windows, and on the Casio it is on the second page of V-Window. A line that appears as a short stub, or not at all, is almost always a T range of 0 to 1 rather than a problem with the equation. |
| The check | Substitute the answer back. If t = 2 is claimed, compute (1,2) + 2(3,4) and see (7, 10). It is four multiplications and it catches every arithmetic slip in the sub-topic, including the 6 and the 2.33. |
No trigonometric function appears here, so the angle mode is irrelevant on this page. It becomes relevant again at 3.13, where the angle between two lines is asked for.
| Step | What |
|---|---|
| 1 | Build the equation from a point and a direction, with t as a dial. Read off t = 0, 1, 2, −1. |
| 2 | Both equations on the board. Vote on whether they are the same line. |
| 3 | Figure view 2. One line. |
| 4 | Why: the point is on the other line, and the directions are multiples. |
| 5 | View 4, the control: same point, different line. |
| 6 | Eliminate t to get 4x − 3y = −2, then test (7,10) and (5,5) back to back. |
| 7 | The same equation in three dimensions, component by component. Nothing new, which is the selling point. |
Do not say "t is just x". It is tempting in two dimensions and it is false the moment the direction's first component is not 1, which is most of the time, and in three dimensions it has no meaning at all. Call t a parameter and let it be its own thing from the first minute.
Do not say "put it in y = mx + c to compare two lines". It works in two dimensions and it teaches a method that dies in three, which is where this sub-topic is heading. The parallel-and-a-point test works in both, so teach that one and nothing else.