# Lesson 15

Weighted Averages

## 15.1: Part Way: Points (5 minutes)

### Warm-up

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.

### Student Facing

For the questions in this activity, use the coordinate grid if it is helpful to you.

- What is the midpoint of the segment connecting \((1,2)\) and \((5,2)\)?
- What is the midpoint of the segment connecting \((5,2)\) and \((5,10)\)?
- What is the midpoint of the segment connecting \((1,2)\) and \((5,10)\)?

### Student Response

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### Anticipated Misconceptions

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

### Activity Synthesis

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 \(\left(\frac12,1\right)+\left(\frac52,5\right)=(3,6)\).)

## 15.2: Part Way: Segment (15 minutes)

### Activity

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.

### Launch

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.

*Action and Expression: Develop Expression and Communication.*Maintain a display of important terms and vocabulary. During the launch, take time to review the following terms that students will need to access for this activity: midpoint, partition.

*Supports accessibility for: Memory; Language*

### Student Facing

Point \(A\) has coordinates \((2,4)\). Point \(B\) has coordinates \((8,1)\).

- Find the point that partitions segment \(AB\) in a \(2:1\) ratio.
- Calculate \(C=\frac 13 A + \frac 23 B\).
- What do you notice about your answers to the first 2 questions?
- For 2 new points \(K\) and \(L\), write an expression for the point that partitions segment \(KL\) in a \(3:1\) ratio.

### Student Response

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### Student Facing

#### Are you ready for more?

Consider the general quadrilateral \(QRST\) with \(Q=(0,0),R=(a,b),S=(c,d),\) and \(T=(e,f)\).

- Find the midpoints of each side of this quadrilateral.
- Show that if these midpoints are connected consecutively, the new quadrilateral formed is a parallelogram.

### Student Response

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### Anticipated Misconceptions

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.

### Activity Synthesis

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 \(B\).

*Speaking: MLR8 Discussion Supports.*As students share how they would calculate the horizontal distance between \(A\) and \(B\), press for details by asking students how they know that the distance is \(B-A\). Also ask how they know that the distance from \(A\) to \(C\) is \(\frac{2}{3}\) of the distance from \(A\) to \(B\). Show concepts multi-modally by drawing and labeling the horizontal lengths of \(A\), \(B\), and \(B-A\). This will help students justify why the value of \(C\) is equivalent to the expression \(A+\frac{2}{3}(B-A)\).

*Design Principle(s): Support sense-making; Optimize output (for justification)*

## 15.3: Part Way: Quadrilateral (15 minutes)

### Activity

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.

### Launch

*Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time.*Use this routine to help students improve their written responses for the last question. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “How do you know that \(AB’C’D’\) is a dilation of \(ABCD\)?” and “What is the center and scale factor of the dilation?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify why partitioning the segments of \(ABCD\) results in a dilation of the figure.

*Design Principle(s): Optimize output (for justification); Cultivate conversation*

### Student Facing

Here is quadrilateral \(ABCD\).

- Find the point that partitions segment \(AB\) in a \(1:4\) ratio. Label it \(B’\).
- Find the point that partitions segment \(AD\) in a \(1:4\) ratio. Label it \(D’\).
- Find the point that partitions segment \(AC\) in a \(1:4\) ratio. Label it \(C’\).
- Is \(AB’C’D’\) a dilation of \(ABCD\)? Justify your answer.

### Student Response

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### Anticipated Misconceptions

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.

### Activity Synthesis

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 \(D\).)

## Lesson Synthesis

### Lesson Synthesis

Ask students to use segment partitioning to dilate triangle \(END\) using center \(E\) and scale factor \(\frac 34\).

After 2 minutes of quiet work time, ask students to share their plans. Students should recognize that dilating by a scale factor of \(\frac34\) is equivalent to partitioning the figure’s segments in a \(3:1\) ratio. Give students a few more minutes of work time to complete the task. (\(N’=(2.25, 9), D’=(2.25, 3)\))

## 15.4: Cool-down - Part Way: Triangle (5 minutes)

### Cool-Down

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## Student Lesson Summary

### Student Facing

To find the midpoint of a line segment, we can average the coordinates of the endpoints. For example, to find the midpoint of the segment from \(A=(0,4)\) to \(B=(6,7)\), average the coordinates of \(A\) and \(B\): \(\left(\frac{0 + 6}{2}, \frac{4+7}{2}\right) = (3,5.5)\). Another way to write what we just did is \(\frac12 (A+B)\) or \(\frac12 A + \frac12 B\).

Now, let’s find the point that is \(\frac23\) of the way from \(A\) to \(B\). In other words, we’ll find point \(C\) so that segments \(AC\) and \(CB\) are in a \(2:1\) ratio.

In the horizontal direction, segment \(AB\) stretches from \(x=0\) to \(x=6\). The distance from 0 to 6 is 6 units, so we calculate \(\frac23\) of 6 to get 4. Point \(C\) will be 4 horizontal units away from \(A\), which means an \(x\)-coordinate of 4.

In the vertical direction, segment \(AB\) stretches from \(y=4\) to \(y=7\). The distance from 4 to 7 is 3 units, so we can calculate \(\frac23\) of 3 to get 2. Point \(C\) must be 2 vertical units away from \(A\), which means a \(y\)-coordinate of 6.

It is possible to do this all at once by saying \(C = \frac13 A + \frac23 B\). This is called a weighted average. Instead of finding the point in the middle, we want to find a point closer to \(B\) than to \(A\). So we give point \(B\) more weight—it has a coefficient of \(\frac23\) rather than \(\frac12\) as in the midpoint calculation. To calculate \(C = \frac13 A + \frac23 B\), substitute and evaluate.

\(\frac13 A + \frac23 B\)

\(\frac13 (0,4) + \frac23 (6,7)\)

\(\left(0,\frac43 \right) + \left(4, \frac{14}{3} \right)\)

\((4,6)\)

Either way, we found that the coordinates of \(C\) are \((4,6)\).