Can one problem become a live figure for class?

October 5, 2026

One math problem can become a live figure before the lesson starts. On GoodMaith you describe the problem in ordinary English. The tutor builds the objects on a board you can project, and you can still drag a point while you talk. Desmos and GeoGebra will graph this height too, once you have typed the expression yourself. The class version here is the expression, the marked points, and the sentences, kept on one board.

Say the flight

A ball leaves a hand 1 meter up. After $t$ seconds its height in meters is $- t^{2} + 4 t + 1$ . Ask when it meets the ground.

Read that sentence onto the board in pieces. At $t = 0$ the height is $1$ . That point is S, the release, fixed on the vertical axis. The path is the graph of the height for $t$ from $0$ until the curve meets height zero again. Completing the square rewrites the height as $- (t - 2)^{2} + 5$ . The highest point is therefore at $t = 2$ , and the height there is $5$ meters. That point is V. The landing is the later root of $- t^{2} + 4 t + 1 = 0$ , which is the same as $t^{2} - 4 t - 1 = 0$ . The positive root is $2 + 5$ , about $4.24$ seconds. That point is L, on the horizontal axis. P is a free point on the path. It starts at $t = 1$ , where the height is $- 1 + 4 + 1 = 4$ .

Time runs along the horizontal axis, height along the vertical. Drag P. S is the release, V the highest point, L where the height is zero again.

The horizontal axis is time in seconds. The vertical axis is height in meters. The curve is only the flight, from the release to just past the landing, so the class is not staring at a long tail of negative time. S, V, and L are labeled. P is the point a student can move.

What you say next, with the board still up

Leave the construction projected. The next sentence does not open a new figure.

Ask for the highest point. V is already there, at $(2, 5)$ . You can say why: the squared term $- (t - 2)^{2}$ is never positive, so the height is never more than $5$ , and it reaches $5$ when $t = 2$ . Ask for the landing. L is the later crossing. The earlier root of the same equation is negative, about $2 - 5$ , and it is not on this flight. The ball was released at $t = 0$ , already in the air. Students can see that the curve would have crossed the axis before time zero if you extended it, and that the extension is not part of the throw.

Then ask what changes if the ball leaves faster. On this page the drawn path stays $- t^{2} + 4 t + 1$ , so the numbers above stay checkable while you talk. In the chat, the next sentence can replace the $4 t$ with a larger coefficient, or add a second curve beside the first. The release height can stay $1$ . The new vertex and the new landing are new points. The first path can stay visible, the way the shifted parabola stays beside the original on describe a math problem and get an interactive graph. One pass through the chat asks a single question and waits, which is the version that keeps a class on one step. The other writes the path through and plots as it goes, which is the version you prepare before the lesson.

Dragging it in front of the room

Drag P along the curve. The point cannot leave the graph, so its height is always $- t^{2} + 4 t + 1$ at the time you have slid to. Near S the path is still climbing. Past V it is falling. At L the height is zero and the flight on this board ends. Students can call out the height at $t = 1$ before you move P there, and then see the point sit at $4$ . That is a different check from watching a finished animation, because the person at the board chooses the time.

You can try the tutor without an account, and sign in to keep the chats you used to prepare. The graphing workbench opens the same kind of board with no tutor when you want to type the height yourself and drag P with no lesson wrapped around it. Start from the problem in chat with a live board. GoodMaith is an AI math solver with live graphs.

If you teach and want a special price, email hello@goodmaith.com.