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# Finding an unknown angle

## Task

In the figure, $ABCD$ is a rectangle and $\angle CAD = 31^{\circ}$. Find $\angle BAC$.

## IM Commentary

The purpose of this task is to give 4th grade students a problem involving an unknown quantity that has a clear visual representation. Students must understand that the four interior angles of a rectangle are all right angles (4.G.2) and that right angles have a measure of $90^\circ$ and that angle measure is additive (4.MD.7). In a teaching scenario, students may be allowed to verify the computations using a protractor to measure the angles. However, care should be taken beforehand to ensure that the measurements of the printed figure match the stated measurements.

The task may also be viewed as preparation for later work when, in 6th grade, students are introduced to algebraic expressions. In that context, unknown angle problems will use variables to label missing angles, and students will write and solve equations to find the missing angle measures.

This task includes an experimental GeoGebra worksheet, with the intent that instructors might use it to more interactively demonstrate the relevant content material. The file should be considered a draft version, and feedback on it in the comment section is highly encouraged, both in terms of suggestions for improvement and for ideas on using it effectively. The file can be run via the free online application GeoGebra, or run locally if GeoGebra has been installed on a computer.

This applet allows the teacher to first enter the any angle. After setting the angle, the problem appears and can be given to the student.

## Attached Resources

## Solution

All four angles in a rectangle are right angles, so $\angle BAD$ is $90^\circ$. Since $\angle BAC + \angle CAD = \angle BAD$, we have that

$$\angle BAC + 31^\circ = 90^\circ$$

which is the same as saying

$$\angle BAC = 90^\circ - 31^\circ.$$

Thus, $\angle BAC = 59^\circ$.

## Finding an unknown angle

In the figure, $ABCD$ is a rectangle and $\angle CAD = 31^{\circ}$. Find $\angle BAC$.

## Comments

Log in to comment## David Woodward says:

over 5 yearsI have heard some debate about whether the intention of this standard includes the symbolic language included in this illustration. I guess the big question is how does PARCC see it (or Smarter Balanced if you are on that side of things.) Does anyone know? Is there any offical reference to the details of this standard that would help us to understand?

## Kristin says:

over 5 yearsI assume you are asking about the angle notation like $\angle BAC$. I can't speak for PARCC or Smarter Balanced, but if you read the K-5 geometry progressions document at

http://commoncoretools.me/wp-content/uploads/2012/07/ccss_progression_gm_k5_2012_07_21.pdf

You will see on the right lower corner of page 24 that there is an example about what a student might be asked that uses this notation. Also, Jason Zimba suggested we add this task to this standard and Bill McCallum agreed. So (at least) two members of the writing team think this notation is appropriate.

I actually wasn't convinced at first that this notation was the best idea to use with 4th graders, but the more I thought about it, the more I realized that any other notation could easily cause confusion about which angle is meant. So I've come around to thinking it is a good idea myself.