In my capstone class for future secondary math teachers, I ask my students to come up with ideas for *engaging* their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Chi Lin. Her topic, from Geometry: deriving the midpoint formula.

**What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)**

To let students engage in the topic, as teachers, we want to create some good examples for students to let them interested in doing it. We need to know what students are interested in or students realize they can use this knowledge in the real life. For example, if students like eating pizza, then I will create some examples about pizza or some delicious food and using pizza representation to raise their attention. In this topic, since we are going to talk about the midpoint formula, one of the real-world problems that I can come up with is using Google Maps. I will show a big Google map of the US in the class, and I will ask students question that “Miss Lin is planning a road trip from Dallas to Arizona on Thanksgiving. However, she wants to split the driving into two days. Now Miss Lin needs your help to figure out what is the middle city (midpoint) between Texas (Dallas) to Arizona.” After students talk with their groupmates, I will invite students to come to the map and circle the city that their think is the middle city between Texas (Dallas) to Arizona and explain their thoughts as well.

**How does this topic extend what your students should have learned in previous courses?**

Khan Academy shows that what students show know before we learn how to derive the midpoint formula. It gives some details which help the teacher to prepare the lesson. First, students should know points in the coordinate plane. Students should require describing every point on the plane with an ordered pair in the form correctly. Second, students have learned how to use addition, subtraction, and square with negative numbers. Students need to know the distance and slope between points on the coordinate plane, how to represent points on the left or below the original point. Third, students have learned the distance and displacement between points to calculate the slope. Students need to understand what absolute value is as well. The last thing I think students should have learned in the previous class is the slope and square root.

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**How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.**

Khan Academy is a good resource for students to study themselves when they want to study this topic. It tells students what they need to know and why this topic is important before they get to learn. Students might think about deriving the midpoint formula is just figuring out some points in the coordinate plane. However, Khan Academy shows that knowing the midpoint formula is not only for figuring out the points in the coordinate plane but also related to the distance formula. Also, Khan Academy provides online tutoring videos to help students understand the materials. If students don’t understand or forget the materials, they can always go back to check the videos. Khan Academy also provides practices for students to do after each topic, it helps students do the self-checking. I recommend this website because, since the covid, we realize that online learning is also one of the ways for students to learn. However, sometimes it is hard for teachers to check students’ understanding through the screen, and we couldn’t make sure that every student is on the same page with us. Khan Academy does provide detailed explanations on their website, so I will suggest students check this website with this topic if my class is online.

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