What’s the Angle (Measure)? Appreciating the ProtractorSami Briceño, sbriceno@carnegielearning.com & Sue Hamilton, sue.Hamilton@carnegielearning.com

http://www.carnegielearning.com/nctm2017

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Session presentation and handouts are located @

http://www.carnegielearning.com/nctm2017

Come by and visit us in the Exhibit Hall @ the Carnegie Learning Booth #743!

What is an angle?

Partner Turn and Talk

What is an Angle?

• Angles are composed of two rays that are infinite in length with a common vertex. The only difference in their size is how widely or narrowly the two rays are spread apart.

• How do we determine or describe how much one ray has rotated away from the other ray to create a specific angle?

“…have students think (of angles as) how much one ray has rotated away from the other. Do you think of the angles in a triangle as one side being rotated away from the other?”

Van de Walle 2007

Angle Match

• Partner up!

• Create a wall using the folder

• Partner A will create an angle by rotating one ray away from the other.

• Partner A then has to describe to Partner B how to create an angle that matches theirs without using adult knowledge of degrees or protractor

• Compare! How close did your angles match?

• Switch roles

TABLE TALK

• What type of language

did you use to get

your partner to create

an angle that matched

yours?

• What proved to be

difficult about this

task?

Creating a Unit Angle for Measuring

• Using a straight edge, draw an angle on an index card and cut it out (like a pizza slice).

• Name your unit angle.

• Partner Talk

– How could you use your unit angle to measure or describe other angles?

• Remember, we are measuring and describing how far one ray is being rotated from another ray to create a specific angle.

“Measuring an angle is the same

as measuring length or area; unit

angles are used to fill or cover

the spread of an angle just as

unit lengths fill or cover a length.”

VAN DE WALLE 2007

Creating a Unit Angle for Measuring

• We discussed strategies for using our unit angles to measure other angles. Let’s put the strategies to test for #1 on the provided page.

• Use your unit angle to measure the remaining angles on the provided page and record the measurements on the line marked Round 1. (Independent work)

• Partner up and compare your measures.

– What accounts for any differences?

How do we compare in measuring with our unit angles?

• Let’s compare our results as a whole group.

• What proved difficult about this activity?

• How can our process be improved?

PARTNER TALK

How do you introduce standard

protractors to students and how

to use them?

Creating a Measuring Tool

For this activity you will need either a writing utensil, a straightedge, and either a piece of wax paper or patty paper.

Creating a Measuring Tool

Identify a unit angle in this measuring tool.

We will call our unit angle a “WEDGE”.

TABLE TALK

• If we are using different sized paper at your table or around the room, how do we know we are really using the same size unit angles (WEDGES) to measure?

Creating a Measuring Tool

How many WEDGES construct the whole circle or 1 full circle rotation of 1 ray from another?

What fraction does each WEDGE represent of the whole circle?

Using our Measuring Tool

Return to the Measuring Angles page.

Measure each angle using our new measuring tool and record your measurements on the line marked Round 2

• How many wedges fill each angle?

TABLE TALK

• What did we improve

in our process from

measuring with our

own unit angles?

• What still proved to be

difficult and could be

improved?

Partner Talk

What similarities and differences exist between a standard protractor and our own wax/patty paper measuring tool?

Non-standard and Standard Units

• The only real difference is the size of the unit angle.

– What is the standard unit angle in a protractor?

– What fraction does each degree represent of a circle?

Understanding the Design of a Protractor

Using our Measuring Tool

Return to the Measuring Angles page.

Measure each angle using the standard protractor and record your measurements on the line marked Round 3

• How many degrees fill each angle?

Activity Idea:

Creating and Evaluating Models

• Take a slip of paper with an angle measure written on it; keep it secret from your group members.

• Construct the angle measure using spaghetti (rays), marshmallow (vertex) and a protractor (measuring tool).

• When finished, you will trade models with a group member.

• Complete the handout when evaluating your groupmate’smodel.

How is measuring an angle with a protractor different from sketching a specified measure

with a protractor?

Have students write their own steps for measuring and sketching angles using a protractor. Then have student partners test out each other’s steps and give feedback. Teachers can evaluate and comment on student written

steps as a final round of feedback.

How Close Can You Get?

• Everyone create 3 little slips of paper. On each one, write a different angle measure between 0 and180 degrees. (Don’t get too crazy!)

• Fold the papers up and mix them in the center of the table. Each person randomly select 1 slip of paper.

• Using spaghetti pieces (snap them into smaller pieces), construct the angle measure as accurately as you can WITHOUT a measuring tool. Tape the spaghetti down onto the provided paper and write the angle measure you were creating next to the model where designated.

• Trade papers with someone at your table. For the paper you receive, verify how close they were with a protractor. Mark where the actual angle measure should be, and label what exact degree the angle was that the spaghetti created. Return paper to owner.

How Close Can You Get?

What were your strategies in creating specific angle measures without a measuring tool?

Angle Estimation

• Why is it important for students to have a good understanding of 90 degree and 45 degree angles? How can knowing those be useful?

• Knowing a 30 degree angle, which is a third of a right (90 degree) angle, is useful for better estimates. Think of seeing angle subdivisions to help estimate angles.

• Some may call these angles that “students need to have a good understanding of” as BENCHMARK ANGLES.

“Estimation of angle size is sometimes overlooked with too much attention given to measurement with protractors. Student should have a good understanding of 90 degrees and 45 degrees.”

Van de Walle 2007

How Close Can You Get?—Round 2!!!

• Each person randomly select 1 more slip of paper from the center of the table.

• Using spaghetti pieces (snap them into smaller pieces), construct the angle measure as accurately as you can WITHOUT a measuring tool. Tape the spaghetti down onto the provided paper in the section marked Round 2 and write the angle measure you were creating next to the model where designated.

• This time, try and use the idea of a “Benchmark Angles” strategy to be more precise.

• Trade papers with someone at your table. For the paper you receive, verify how close they were with a protractor. Mark where the actual angle measure should be, and label what exact degree the angle was that the spaghetti created. Return paper to owner.

Activity Idea:

Angle Estimation Scavenger Hunt

• Put students into small groups with 1 picture taking device.

• Students are given a page of angle measures and have to find different size angles in the real world.

• They take pictures of them to post in a google folder or share with the teacher.

• You could ask students to place the protractor next to the object or angle they found in the real world to prove that it is close to the desired angle measure.

TABLE TALK

What makes these session activities worthwhile?

• Thank your partners and tablemates today for discussion and collaboration opportunities!

• Please return materials to the table bags and complete the NCTM session evaluation before you go!

http://www.carnegielearning.com/nctm2017