A field is every solution at once, which is exactly what a printed curve cannot show.
Let them tap, and watch the curves refuse to cross.
Drop three or four curves on the first field and the family appears. Then ask whether two could ever meet. They crowd together and never touch, because each point has exactly one gradient.
The logistic field is the one to finish on: everything levels off at y = 1, and the constant solutions along y = 0 and y = 1 are visible as rows of flat dashes.
Students are asked to use and interpret slope fields. Sketching a solution on a printed field, and describing long-term behaviour, is what comes up.
So spend the time on following dashes accurately and naming what a curve tends towards, not on generating fields by hand.
| dy/dx = x − y | Flat dashes along y = x. Solutions bend towards the line y = x − 1. |
| dy/dx = y(1 − y) | Constant solutions at y = 0 and y = 1. Everything between rises to 1; anything above falls to it. |
| dy/dx = −x/y | Circles, since x² + y² is constant. |
| 1. Along y = x | A, horizontal, since x − y = 0 there. |
| 2. Why curves cannot cross | B. One gradient per point. |
| 3. Starting at y = 0.3 | B, levels off at y = 1. |
1 markStarting the sketch at the given point. A correct shape in the wrong place earns nothing.
1 markStaying tangent to the dashes along the whole curve.
1 markDescribing long-term behaviour: which line, and from which side.
| They give | What it means |
|---|---|
| Gradient 1 along y = x (Q1) | Confused the gradient of the LINE with the gradient the field prescribes. Very common, and worth a board sketch. |
| "Curves are parallel" (Q2) | Plausible from a tidy field, and wrong. Drop two curves and watch them diverge. |
| "Grows without limit" (Q3) | Did not look at the gradient above y = 1, where it turns negative. |
| A sketch crossing y = 1 | The clearest possible signal that the field was not being followed. Constant solutions are walls. |
"Do I have to solve it?" No, and that is the point. The field gives the shape with no solving at all, which is why it exists for equations that cannot be solved.
"Why are some dashes nearly vertical?" Because the gradient is large there. On a printed field they are drawn the same length regardless, which is why they look like a texture rather than a scale.
"What if my curve runs off the page?" Fine. Draw it to the edge. Running off is itself a description of the behaviour.
| What is happening | |
|---|---|
| 1 | Show a field with no equation. What is it telling us? Let them work out that each dash is a gradient. |
| 2 | Tap to drop curves. Build the family and the no-crossing rule. |
| 3 | Sketching by hand on a printed field, starting from a marked point. |
| 4 | The logistic field and long-term behaviour in words. |
| 5 | Questions 1 to 3, and a past-paper style sketch. |
| 6 | Link forward: Euler is this, done numerically. |
Do not have them generate a field by hand beyond a handful of points. It is slow, it is not what is assessed, and it eats the lesson.
Do not accept a sketch that crosses a constant solution. It is the quickest way to see that a student is drawing a shape they expect rather than following the field.
Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.
The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.
The same skill inside a real situation, where the first job is working out what is being asked.
Reasoning, working backwards, or spotting an error. These are where the top grades are decided.
Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.