Students find the point that partitions a segment in a given ratio. The activity starts with an introduction to the concept of partitioning a segment. Then students use informal methods to find a point that partitions a particular segment in a ratio. Next, students compute a weighted average and connect that to the first prompt. Finally, they generalize the process for a ratio. As students attempt a variety of partitioning problems, using multiple methods, they are making sense of the relationship between weighted averages and partitioning a segment in a ratio (MP1).

Some students might write the ratios for the fourth question with the fractions reversed. Ask them to choose a pair of points—perhaps the points and from the activity launch—and use their expression to calculate the coordinates of the desired point to see if it’s correct. It’s okay if students continue to struggle; the activity synthesis will help all students gain intuition about the placement of the fractions.

The goal of the discussion is to make sure students understand why the point that partitions segment in a ratio can be expressed as . Invite students to consider just the -coordinate, or the horizontal component. Display this image.

Tell students that the midpoint notation represents an average. The expression is called a weighted average because the 2 points have different weights. Ask students these questions:

• “What is the horizontal distance from to ?” (6 units)
• “How would you calculate this distance if you couldn’t just count it?” (Subtract the coordinates. We can write .)
• “What fraction of this distance do we need to add to to get to ?” (We need to add of this distance to get to .)
• “In light of the previous answers, what does mean?” (Take , and add of the distance from to .)
• “How can we rewrite this expression to look more streamlined?” (Distribute the fraction and combine like terms to get .)

Now invite students to describe all of this in their own language. The key idea that should surface is that point needs to be more heavily weighted in the calculation, because is closer to than it is to . On the segment connecting and , point is of the way towards .

Students apply partitioning to a quadrilateral to see that segment partitioning is another method for building similar figures on the coordinate plane. Identify students who use a weighted average.

If students struggle to begin, ask them how many parts total we have if we are looking at a ratio of (5 parts total). Ask students how dividing by this number might help them. Tell students that it’s okay if their answers don’t come out to integers. Decimal values are valid answers.

Invite students to share their approaches, including the previously selected student(s) who used weighted averages. Ask the class which method seems easiest. (Any answer with support is valid. Students might find weighted averages most efficient in this case because they are repeating the same ratio three times.)

Here are some questions to help students connect partitioning with dilation:

• "How is partitioning the segments in a ratio similar to dilating a figure?" (Figuring out what the fraction is for the partition is like figuring out what the scale factor is for a dilation.)
• "How can you know that the new figure created with the partitions is a dilation of the original figure?" (Since the partition ratios for each side were all the same, each side was multiplied by the same fraction to get the side length, which is the scale factor.)
• "How is using the weighted average method helpful if the figure is in the coordinate plane?" (It gives the new coordinate for each point rather than just a side length, so we can plot the points directly in the plane.)
• "How does partitioning segments match the definition of a dilation on your reference chart?" (We did exactly what the definition says: We found the points along each ray and whose distance from was the original distance from to points or .) 

Tell students that partitioning a segment using weighted averages is another way to define dilations when we are using coordinates.