Angles are a thorny concept to teach because of the fundamentally different ways in which they can be used and understood. In the article What’s Your Angle on Angles?, the authors divide the concept of angle into three main groups: angle-as-figure, angle-as-wedge, and angle-as-turn.
In the Web Sketchpad game below, we focus on angle-as-turn. Given an angle, students enter an estimate for its measurement. When they press Check, a ray sweeps out their estimated angle. Not only does this provide a dynamic image of an angle-as-turn, but it also injects some drama into the game as students watch the angle grow, wondering whether the turning red ray will come to rest atop the stationary blue ray.
We created five levels of the game that start concretely and gradually become more abstract. You can navigate from level to level by using the arrows in the lower-right corner of the websketch. Here is a description of the levels:
- Level 1: Tick marks appear at every 10° along the circle to aid in students’ estimation. Each randomly generated simple angle has one of its rays in a vertical or horizontal position, making it easier for students to estimate its measurement.
- Level 2: This level is identical to Level 1, but now the rays of the angle are not restricted to vertical or horizontal positions.
- Level 3: This level is identical to Level 2, but now the angles can be either simple or reflex.
- Level 4: This level removes the tick marks, but retains the segments that divide the circle into quarters.
- Level 5: This level leaves only the unadorned angle.
In all the levels, the sides of the angle are of different random lengths. The intent is to get students to focus on the turn needed to sweep out the angle and to not pay attention to the lengths of the angle’s sides.
By creating this sequence of angle games, we’re putting into practice the theory of progressive abstraction. Put simply, this theory suggests that abstract mathematical ideas can be made more accessible by presenting them in a carefully sequenced set of experiences that begin very concretely and gradually strip away the support structures. Said another way, the “concreteness fades” as the ticks marks, horizontal and vertical rays, and the circle and segments are removed one by one.