CCSS-M G-GPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

When I started writing this lesson last year, I did a Google search for ideas. What I found felt out of control to me, to say the least. Should our focus for this standard be on the formula that I found at this site?

Or the explanation from this link?

My search also led me to the curriculum site of the government of India, where I found some good and interesting tasks.

We started our lesson with partitioning vertical and horizontal segments.

And we talked about the need for directed distance.

When we were ready to move to an oblique segment, it occurred to me that maybe we should estimate where the point would be before we tried to calculate the coordinates of the point. I set up a Drop Points Quick Poll.

75% of my students had the correct answer, although they didn’t all just estimate. (I had not marked a correct answer yet on purpose so that we could look at the graph and decide whether (3,-1) was the desired point.

How did you get your answer? What did you notice? Some had calculated using the Pythagorean Theorem. They realized that the given segment was 3√13. So we needed to find Q such that QR was 2√13 and QS was √13. Another student was trying to come up with what he called a “modified midpoint” formula. Instead of finding the mean of the x coordinates of the endpoint, he wanted some kind of weighted mean to get the 2:1 ratio.

I asked students what they saw in the picture.

Endpoints. A segment.

What else?

A triangle.

What kind of a triangle?

A right triangle. The segment is the hypotenuse of a right triangle.

We found the lengths of the vertical and horizontal sides of the right triangle. Can that help us find the point that we need?

Students began to recognize several right triangles in the diagram that would be helpful. Each with the hypotenuse on the segment. Each with a vertical length of 2 (which is one-third of the whole length 6) and a horizontal length of 3 (which is one-third of the whole length 9).

We tried another problem, for which I had them enter the coordinates instead of dropping a point on the graph.

Only 7 students have the correct response. What happened? Fractions, of course.

Those who got it correct talked about what they did.

And so we tried again.

This time we have 10 with the correct response until the students were dismissed to the faculty-student basketball game

At some point during class, I found what looked like a modified midpoint formula on another site.

I included it in the students’ notes, but as I reviewed their summative assessments, I notice that no one used it. Instead, they made sense of partitioning a segment using similar triangles and proportional relationships. My hope, as the journey continues, is that they have a better chance of remembering those connections than they would a formula …

julieriggins

April 9, 2014 at 8:53 pm

Amen on your statement “My hope, as the journey continues, is that they have a better chance of remembering those connections than they would a formula” I wish I had found your post before I taught this topic, we are doing our formative assessment tomorrow. But I thought as you did about the students remembering the concept better than a formula. I showed them a formula after we explored the concept of similar triangles and the students shook their heads and said “I will never remember that!”

jwilson828

April 20, 2014 at 1:46 am

I don’t think I’d remember the formula either, Julie! I actually like making sense of it with triangles. Who cares how fast you can do it if you don’t know what you are doing?!

Jen Spencer

January 26, 2016 at 9:15 am

Hi Ms. Wilson,

I came across this blog post and would like permission to write your problems into our curriculum here at Match (small charter school network in Boston, MA). The progression you have here for partitioning along a number line is perfect for us! Also, if any of the sources of the screen shots above are from somewhere else (other than your brain), do you have the source? I have the India source written in.

Obviously am crediting you in the curric.

Thank you!

Jen

jwilson828

January 26, 2016 at 12:59 pm

Hi, Jen.

I’m glad the post is helpful.

I think all of the resources that I used are linked above. Some of the problems came from “this site” and “this link” that are linked in the first paragraph. Otherwise, I’m pretty sure I made them up!

Good luck!

Jennifer