At Desmos, we hope our professional learning models good student learning. Teachers can help students learn math through effective comparisons and contrasts, so this past week we asked our Desmos Fellows to compare and contrast two similar activities. Here is the first and the second. We asked which one they preferred and why. Their answers were illuminating.

Activity 1 starts off with a screen that asks students to put two points on a line. Students know the task is completed because they see a line pass through all of the points.

While it’s helpful for students to know when they are correct, fellows were
quick to point out that this type of instant feedback could lead to a number
of complications. **Mark Alvaro** wonders if students will guess and check
for this task. **Nolan Doyle** has been focusing this year on holding back
feedback to avoid a guess-and-check situation: “Some students might move that
point until the line appears without thinking about what they are doing.”
Others echoed this sentiment, with **Anna Scholl** reminding us that the
immediate feedback might not challenge students to think about the strategy
for finding slope through the relationships between points.

We asked ourselves what we could do ensure that students are thinking about
the slope instead of guessing and checking on this screen. Several of the
fellows suggested modifications to the screen, follow up questions, and
teacher moves that could help. **Jenn Vadnais** and
**Anna Scholl** suggested giving students an opportunity to make a
prediction about where the points would be, followed by a chance later in the
activity to check their prediction. **Linda Saeta** offered a similar idea
of adding a field where students could explain how they placed the points. In
this case, it might be helpful for a teacher to highlight several student
responses for the class. We can model for students the types of explanations
that we hope to see, giving students an opportunity to improve on their
explanations when they try a similar task later in the activity.

Activity 2 starts similar to Activity 1, asking students to place a point on
an imaginary line. **Stephanie Blair**, **Nolan Doyle** and
**Paul Jorgens** appreciated that this activity started without numbers,
building up the need for the coordinate grid.

This introduction allows for estimation, and perhaps students wishing they had a ruler to help them put the point on the line. This informal introduction doesn’t require precision but in screen 2 we can use the coordinate grid to help us make a more accurate placement of the black point on the line.

**Nathan Kraft**, **Lisa Bejarano**, and **Suzanne von Oy** agreed
that this activity builds on student intuition, using it as a starting point
for understanding the slope formula when it is introduced at a later time.

As with the first activity, there are potential challenges with asking students to think of slope in terms of proportionality on screens 4 and 5.

**Bob Lochel** says, “The staircase idea in both cases requires a solid
understanding of proportion and perhaps similarity - ideas which may distract
from a student’s intuitive ideas concerning slope. I’d like to see
students develop and communicate the rise/run idea on their own before I give
it to them.”

Neither activity is perfect. Another consideration is that context always
matters. It all comes down to how and when in a lesson you use the activity.
Choosing between these two activities might come down to the needs of your
students as well as at what point in the year you use the activity, as
**Nolan Doyle **noted. Not only that, the different activities may help
with developing different ideas or working on different skills.
**Nick Corley** considers how activities help students connect prior
knowledge to new learning as they explore new topics.
**Sarah Vandivort** considers the type of discussions that might emerge
during an activity.

Which activity do you prefer and why?