Clare

> Oh! I don't know. Let's draw a picture and see if we can prove it.
>
> Clare, Geometry Unit 3, Lesson 10.2

**Clare** is popular amongst the characters of the Illustrative Mathematics workbooks, seeming to do math problems with all of them. This is likely because of her rich aunt, who may be famous, or may just be wealthy enough that people automatically want to be friends with Clare. Clare's only know hobbies are flying kites and riding Ferris wheels, but it is also implied that she is chronically absent from her classes. It may also be true that the popularity is getting to her head.

It is also important to note that Clare is having an affair with Andre. Although Andre is having several other affairs, Clare is perhaps the closest to him, challenged only by Mai for that title. Clare is also friends with Han, Diego, and Elena.

Clare's favorite subject is economics.

## Notable Appearances

**Geometry Unit 2**

* "Han, Clare, and Andre thought of a way to construct an angle bisector. They used a circle to construct points D and E the same distance from A. Then they connected D and E and found the midpoint of segment DE. They thought that ray AF would be the bisector of angle DAE." "Clare's rough-draft justification: Since segment DA is congruent to segment EA, triangle DEA is isosceles. DF has to be congruent to EF because they are the same length. So, AF has to be the angle bisector." (Lesson 14.2)

*Analysis: This is an example of Clare with Andre, furthering the theory that there is a relationship between them.*

**Geometry Unit 3**

* "Andre thinks these triangles must be congruent. Clare says she knows they might not be congruent." (Lesson 4, Practice Problems)

*Analysis: This is an example of Clare with Andre, furthering the theory that there is a relationship between them.*

* ""Andre remembers lots of ways to prove triangles congruent. He asks Clare, 'Can we use Angle-Side-Angle to prove triangles are similar?' Clare: 'Sure, but we don't need the Side part because Angle-Angle is enough to prove triangles are similar." Andre: 'Hmmm, what about Side-Angle-Side or Side-Side-Side? What if we don't know 2 angles?' Clare: 'Oh! I don't know. Let's draw a picture and see if we can prove it.' Andre: 'Uh-oh. If 'side' means corresponding sides with the same length, then we'll only get congruent triangles.'" (Lesson 10.2)

*Analysis: This is an example of Clare with Andre, furthering the theory that there is a relationship between them. However, this example specifically indicates a truly deep connection.*

**Geometry Unit 4**

* "Clare is flying a kite. She gets tired, so she stakes the kite into the ground." (Lesson 10, Practice Problems)

*Analysis: This may imply that Clare likes to fly kites.*

**Geometry Unit 7**

* Clare missed class and Jada is teaching her how to construct the circumscribed circle of a triangle. Here are the instructions Jada wrote. 'Construct all 3 perpendicular bisectors of the triangle’s sides. The point where the perpendicular bisectors intersect is called the circumcenter. Construct a circle centered at the circumcenter with radius set to the distance between the circumcenter and a vertex. If the triangle has a circumscribed circle, the circle you construct will go through all 3 vertices.'" (Lesson 13, Practice Problems)

*Analysis: This implies that Clare may be chronically absent.*

**Algebra 2 Unit 2**

* "At the end of 12th grade, Clare's aunt started investing money for her to use after graduating from college four years later. The first deposit was $300." (Lesson 2.3)

*Analysis: This shows that Clare has a rich aunt, which may be the reason for her apparent popularity.*

**Algebra 2 Unit 4**

* "On May 18, Clare visits the park. A park caretaker mentions to her that the pond will be completely covered in less than a week. Clare thinks that the caretaker must be mistaken." (Lesson 1.3)

*Analysis: Clare is an annoying know-it-all who thinks she knows more than the professional caretaker. The popularity is clearly getting to her head.*

**Algebra 2 Unit 5**

* "The table shows Clare's elevation on a Ferris wheel at different times. Clare got on the ride 80 seconds ago. Right now, at time 0 seconds, she is at the top of the ride." (Lesson 5.1)

*Analysis: This confirms that Clare enjoys riding Ferris wheels*

* "Today, Clare tried out two new Ferris wheels." (Lesson 9.2)

*Analysis: This reinforces Clare's love of Ferris wheels.*

* "Today, Clare tried out two new Ferris wheels." (Lesson 10, Practice Problems)

*Analysis: Clare doesn't just like Ferris wheels. She is a Ferris wheel connoisseur.*

**Algebra 2 Unit 7**

* "Clare analyzes oil prices in her economics class." (Lesson 15, Practice Problems)

*Analysis: Clare takes and enjoys an economics class.*

## Miscellaneous Appearances

**Geometry Unit 1**

* "Clare used a compass to make a circle with the radius the same length as segment AB." (Lesson 1, Practice Problems)
* "Clare says, 'Last class I thought the parallelogram would have reflection symmetry. I tried using a diagonal as the line of symmetry but it didn't work. So now I'm doubting that it has rotation symmetry.' Lin says, "I thought that too at first, but now I think that a parallelogram does have rotation symmetry. Here, look at this.'" (Lesson 16.3)

**Geometry Unit 2**

* "Clare says that ABEF is congruent to CDFE because sides AB and CD are corresponding." (Lesson 4, Practice Problems)

**Geometry Unit 3**

* "Clare and Diego are discussing quadrilaterals. Clare thinks the quadrilaterals are similar because the side lengths are proportional. Diego thinks they need more information to know for sure if they are similar." (Lesson 16, Practice Problems)

**Geometry Unit 4**

* "Clare and Han are discussing how to find the missing lengths. Clare says she is using similarity. Han says he is using the Pythagorean Theorem." (Lesson 2, Practice Problems)

**Geometry Unit 5**

* "Clare says, 'We know that if we dilate a cube by a factor of k, the cube's volume is multiplied by k^3. It seems like that must apply to all solids, but I'm not sure how to prove it.' Elena says, "Earlier in the unit, we showed that we can cover any two-dimensional shape with rectangles, so the property that area changes by k^2 when we dilate a figure by k applies to all shapes, not just rectangles. Can we do something similar here?'" (Lesson 6.3)

**Geometry Unit 7**

* Clare and Diego are discussing inscribing circles in quadrilaterals. Diego thinks that you can inscribe a circle in *any* quadrilateral since you can inscribe a circle in any triangle. Clare thinks it is not always possible because she does not think the angle bisectors are guaranteed to intersect at a single point. She claims she can draw a quadrilateral for which an inscribed circle can’t be drawn." (Lesson 9, Practice Problems)

**Algebra 2 Unit 1**

* "Clare takes a piece of paper with length 8 inches and width 10 inches and cuts it in half. Then she cuts it in half again, and again..." (Lesson 8.2)
* "Tyler has a piece of paper and is sharing it with Elena, Clare, and Andre. He cuts the paper to create four equal pieces, the hands one each to the others and keeps one for himself." (Lesson 11.2)

**Algebra 2 Unit 2**

* "Clare wants to make an open-top box by cutting out corners of a 30 inch by 25 inch piece of poster board and the folding up the sides." (Lesson 4, Practice Problems)

**Algebra 2 Unit 3**

* "Clare said, 'I know that 9^2 = 9 x 9, and 9^0 = 1. I wonder what 9^1/2 means?' First, she graphed y = 9^x for some whole number values of x, and estimated 9^1/2 from the graph." (Lesson 3.2)
* "Diego saw Clare's work and said, 'Now I'm wondering about 3^1/2.'" (Lesson 3.2)
* "Clare was adding √4 and √9, and at first she wrote √4 + √9 = 2 + 3. But then she remembered that 2 and -2 both square to make 4, and that 3 and -3 both square to make 9." (Lesson 6.2)

**Algebra 2 Unit 4**

* "Clare says that log(0.1) = -1. Kiran says that log(-10) = -1." (Lesson 12, Practice Problems)
* "Andre says that g(x) = 0.1x(0.1x - 5)(0.1x + 2)(0.1x + 5) is obtained from f by scaling the inputs by a factor of 0.1. Clare says this graph is a vertical shift of the graph of f down 100 units. Diego says the graph of k(x) = -x(x - 5)(x + 2)(x + 5) is the reflection of the graph over the y-axis." (Lesson 9, Practice Problems)

**Algebra 2 Unit 7**

* "Clare is designing an experimental study. She says that it is important to randomly assign people to random groups in an experimental study because this helps reduce the likelihood of grouping subjects into groups that may differ on some characteristic that is related to the response of interest." (Lesson 6, Practice Problems)
* "Elena and Clare are working on a project about how high school students are having trouble finding jobs." (Lesson 11.2)
* "Noah and Clare are studying the noise level, in decibels, outside of their school in the morning." (Lesson 12, Practice Problems)