Higher Level only. It needs 3.14's vocabulary and 3.9's matrix multiplication, and it is where the calculator starts doing real work in this block.
Write A on the board, ask the class to predict A²'s diagonal, and collect the prediction before stepping the figure. Nearly everyone says zeros. The reasoning is good and the conclusion is wrong, which is the most useful shape a misconception can have.
Step to view 2. The diagonal is shaded, reading 1, 2, 1, and the walk 1 → 2 → 1 is drawn as two arcs above the path so the out-and-back is visible as motion rather than asserted in words.
Then the second observation, which is the one worth taking away: the diagonal of A² is the degree sequence. 1, 2, 1 are the degrees of the three vertices. One two-step return per edge at that vertex, and that holds on every graph without loops.
| Question | Answer |
|---|---|
| 1. (A²)1,1 | 1, the walk 1 → 2 → 1. |
| 2. (A²)2,2 | 2, which is vertex 2's degree. |
| 3. Walks of length 3 from 1 to 2 | 2: 1→2→1→2 and 1→2→3→2. |
| 4. Walks of length 2 from 1 to 2 | 0. Every 1-to-2 walk here has odd length. |
| 5. Why A²'s diagonal is not zero | B. A² counts two-step walks, and out-and-back is one. |
Questions 3 and 4 are the same pair of vertices at two different lengths, and the answers are 2 and 0. On a path graph the parity of the walk length decides everything, and noticing that is worth more than either answer.
1 markA written down with the vertices in the question's order.
1 markThe right power.
1 markThe right entry, described as a count of walks.
The last mark wants a sentence. "2" alone leaves the examiner unable to tell which entry was read, and on a symmetric matrix that is forgiving but on a directed one it is not. Train the phrasing: there are 2 walks of length 3 from vertex 1 to vertex 2.
| They wrote | What happened |
|---|---|
| 0 on question 1 | The diagonal, tidied away. They read A's diagonal, or they decided a non-zero diagonal must be a mistake and corrected it. Some will have computed 1 and then crossed it out, which is worth asking about directly. |
| 2 on question 1 | Read the 2,2 entry. Vertex 1 has degree 1, so its entry is 1. |
| 4 on questions 1 or 2 | The trace, 1 + 2 + 1. They have added the diagonal instead of reading one entry of it. |
| 1 on question 3 | Gave A's own 1,2 entry, which counts walks of length 1. The question is about length 3. |
| 3 on question 3 | Gave the length as the count. Easy to catch and worth catching, because it means "number of walks of length k" has not been parsed as a phrase. |
| 1 on question 4 | Counted the edge itself, which is a walk of length 1. |
| 2 on question 4 | Gave the length-3 answer, or the 2,2 entry. The 1,2 entry of A² is 0. |
| Anything but 0 on question 4 | Worth a sentence: zero is an answer here, and in this block zero is nearly always informative rather than empty. |
"Can a walk really go back along the same edge?" Yes. That is what makes it a walk rather than a path. The four words are worth distinguishing once, because 3.16 needs them: a walk may repeat anything, a trail repeats no edge, a path repeats no vertex, and a circuit or cycle returns to its start. Matrix powers count walks, the loosest of the four, which is exactly why they are easy to count.
"Why do the probabilities go down the columns?" Convention, and it is the one the syllabus and both calculators use, so that the next state is T times the current state vector. A class that has met the row convention elsewhere should be told the two are transposes and then asked to stick to columns here. The check is the same either way: whichever direction sums to 1 is the direction the probabilities live in.
"Can I take powers of a weighted matrix?" You can compute them and they mean nothing useful, because matrix multiplication multiplies weights along a walk and adds across walks, which is the opposite of what a route length does. Weighted tables are read, not powered. Say it before someone builds a method on it.
"What is A0?" The identity, and it is consistent: there is exactly one walk of length 0 from a vertex to itself, namely standing still, and none to anywhere else. A pleasing thirty seconds for a strong class and safely skipped otherwise.
| Stage | What to do |
|---|---|
| Demonstrate | Store A, then show a^2, a^3, a^4 in sequence. The diagonal alternates between non-zero and zero as the power alternates between even and odd. Three key presses and the parity structure of the whole graph is on the screen, which is a better route to it than any explanation. |
| Where they stick | Reading a 3 by 3 result off the Casio, which does not fit on one screen and needs arrowing. Students read the first visible entry and assume it is the one asked for. Also the fraction display: a transition matrix with thirds in it shows as fractions in Math mode, and a student then reports 1/3 where 0.333 was wanted. The fix is Input/Output set to Linear, or the S↔D key on the answer. Do NOT touch the SET UP item called Mode, which is the number base: choosing Dec there puts the machine into integer mode and a decimal cannot be entered at all. |
| The check | Label the matrix. Vertex names along the top and down the side, on paper, before anything is entered. It is the only defence against reading the 3,1 entry for the 1,3 one, and on a directed graph that error is both likely and invisible. |
No angle mode involved. Set the Casio's Input/Output to Linear before transition matrices and leave it there for the rest of the block.
| Step | What |
|---|---|
| 1 | Build A from the path, with the vertex labels written round the outside. |
| 2 | Note the three properties: symmetric, zero diagonal, rows sum to degrees. |
| 3 | Predict A²'s diagonal. Collect the prediction. |
| 4 | Figure view 2. The walk drawn as two arcs. |
| 5 | Name it: the diagonal of A² is the degree sequence. |
| 6 | A³, and the parity observation. a^4 on the machine to confirm it. |
| 7 | Walk, trail, path, circuit, as four words, with the figure still up. |
| 8 | Transition matrices, with the column-sum check done out loud on each column. |
Do not say "the diagonal is always zero". It is true of A for a graph without loops and false of every even power of it, and it is the sentence that makes students delete a correct answer. Say "A has a zero diagonal because there are no loops", with the reason attached, so that the statement cannot travel to A².
Do not say "A² squares the connections". It is vague enough to sound right and it gives no way to interpret a 2. Say that the i, j entry of Ak is the number of walks of length k from i to j, in those words, and then read entries off against it until the class can do it without the sentence.