Lesson 15

Students review the definition of midpoint by finding the midpoint of a horizontal segment, the midpoint of a vertical segment, and the midpoint of the diagonal formed by these two segments. Identify students who found the midpoint visually as well as students who found the midpoint algebraically.

If students don’t remember the term midpoint, remind them that it is the point that partitions the segment exactly in half.

The purpose of the discussion is to introduce notation for segment partitioning.

Display a graph of the three segments for all to see. Invite the previously selected students to share their methods. If anyone used the midpoint formula, ask them to share last. (This isn’t the “best” method, but it’s the most connected to the new notation introduced here.)

Ask students how the word average relates to finding midpoints. (The coordinates of the midpoint are the averages of the corresponding pairs of the coordinates of the endpoints of the segment.)

Now ask students to consider what the notation \(\frac 12 (A+C)\) or \(\frac 12 A + \frac 12 C\) could mean. (These both represent the midpoint. The midpoint is halfway between the 2 \(x\)-coordinates and halfway between the 2 \(y\)-coordinates. It is like the average of the two points.)

Invite students to describe the calculations indicated by the notation \(\frac 12 A + \frac 12 C\). (First, calculate half of each coordinate for both points. Point \(A\) has coordinates \((1,2)\), so \(\frac12 A = \left(\frac12, 1\right)\). Point \(C\) has coordinates \((5,10)\), so \(\frac12(C)=\left(\frac52, 5\right)\). Now add them together to get \(\left(\frac12,1\right)+\left(\frac52,5\right)=(3,6)\).)

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 \(2:1\) ratio. Next, students compute a weighted average and connect that to the first prompt. Finally, they generalize the process for a \(3:1\) ratio.

Display this image for all to see.

Ask students what point would partition segment \(AB\) in a \(1:2\) ratio. That is, if we call the point \(C\), the ratio \(AC:CB\) should be \(1:2\).

If students struggle, remind students that this notation means that the whole segment is divided into 3 equal sections. To the left of point \(C\) will be 1 of those equal sections, and to the right of \(C\) will be 2 equal sections. The point \((4,2)\) meets this description.

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 \((0,2)\) and \((12,2)\) 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 \(AB\) in a \(2:1\) ratio can be expressed as \(\frac13A + \frac23B\). Invite students to consider just the \(x\)-coordinate, or the horizontal component. Display this image for all to see.

Tell students that the midpoint notation \(\frac12A + \frac12B\) represents an average. The expression \(\frac13A + \frac23 B\) is called a weighted average because the 2 points have different weights. Ask students these questions:

• “What is the horizontal distance from \(A\) to \(B\)?” (6 units)
• “How would you calculate this distance if you couldn’t just count it?” (Subtract the coordinates. We can write \(B-A\).)
• “What fraction of this distance do we need to add to \(A\) to get to \(B\)?” (We need to add \(\frac23\) of this distance to get to \(C\).)
• “In light of the previous answers, what does \(A+\frac23(B-A)\) mean?” (Take \(A\), and add \(\frac23\) of the distance from \(A\) to \(B\).)
• “How can we rewrite this expression to look more streamlined?” (Distribute the fraction and combine like terms to get \(\frac13A + \frac23 B\).)

Now invite students to describe all of this in common sense language. The key idea that should surface is that point \(B\) needs to be more heavily weighted in the calculation, because \(C\) is closer to \(B\) than it is to \(A\). On the segment connecting \(A\) and \(B\), point \(C\) is \(\frac23\) of the way towards \(B\).

Students apply partitioning to a quadrilateral to see that segment partitioning is another method for building similar figures on the coordinate plane. Monitor for 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 \(1:4\) (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 since they are repeating the same ratio three times.)

Tell students this process was a second coordinate version of dilation. Ask students how the process they completed here matches the definition of a dilation on their reference chart. (We did exactly what the definition says: We found the points along each ray \(AB,AC,\) and \(AD\) whose distance from \(A\) was \(\frac15\) the original distance from \(A\) to points \(B,C,\) or \(D\).)