Measuring Angles Inside Shapes

Today's Number Talk

For a detailed description of the Number Talk procedure, please refer to the Number Talk Explanation. For this Number Talk, I am encouraging students to represent their thinking using a number line model. For each task today, students shared their strategies with peers (sometimes within their group, sometimes with someone across the room). It was great to see students inspiring others to try new methods and it was equally as great to see students examining each other work for possible mistakes!

Getting Started

Prior to the lesson, I placed magnetic money and fractions on the board to help students conceptualize our number talk today. 

I invited students to get a Student Number Line and Hundred Grids. I then drew a Number Line on the Board and marked 0, 1, and 2 on the line. I asked students to do the same on their own number lines.

Task #1: Add 1/4 + 0.6

To begin, I asked students to add 1/4 + 0.6 on their number lines and hundreds grids. During this time, some students chose to work alone while others worked with a partner in their math groups. I took this time to conference with students.

Next, some students volunteered to explain their reasoning out loud while I modeled their thinking on the board: 1:4 + 0.6 Teacher Demonstration Number Line

Others watched carefully, checking their own number lines and hundreds grids to make sure they agreed with the thinking of other students. Here are a few examples of student work during this time:

• 1:4 + 0.6 Student Hundreds Grid
• 1:4 + 0.6 Student Number Line

Task #1: Add 1/10 + 1.5

Next, we moved on to adding 1/10 + 1.5. Most students converted 1.5 to 1 5/10 and then used their number lines to take a jump of 1 5/10 and then a jump of 1/10. 

Here is an example of student number line and hundreds grid: 

• 1:10 + 1.5 Student Hundreds Grid
• 1:10 + 1.5 Student Number Line

Again, a few students explained how they solved this problem while I modeled their thinking on the board. Here's the end result of the last number talk task: 1:10 +1.5 Teacher Demonstration Number Line.

You'll notice a list of patterns off to the right side of the picture listed above. One student had pointed out 1 6/10 = 1 3/5 which is equal to 1.6 because 1.6 = 1 6/10. Then, we discussed how 3/5 is equivalent to 0.6.

I asked  a few students to grab a calculator to divide 3 by 5 (3/5). They got 0.6. I asked: Can anyone else think of an equivalent fraction to 6/10? I wonder if all fractions equivalent to 6/10 are also equal to the decimal number, 0.6? Students took turns providing equivalent fractions. The students with the calculators would then check the decimal equivalency by dividing the numerator by the denominator. Each time they got 0.6! This was a fun and exciting moment for students! 

Reasoning

For today's lesson, I wanted to provide students with more practice using the protractor and with an opportunity to discover the pattern that all angles in a circle add up to 360 degrees. So I created shape puzzles for students to investigate. For each shape puzzle, I drew two lines intersecting in the middle of the shape. This resulted in four angles that would always have a sum of 360 degrees. After measuring and adding all four angles of several shapes, students began to realize that they always equal 360 degrees! 

Review

To connect today's lesson with previous lessons, we began by singing our fun Angles Song. 

Next, we reflected upon the Complementary & Supplementary Angle Poster from yesterday. Students added on the following observations on the Supplementary Angles side, "One angle is acute and the other is obtuse," and, "They can also be two right angles." 

Then, I reminded students of our current goal: I can find the sum of angles. 

Activity Explanation

I pointed out the following shapes on the counter: Shape Puzzles. I wanted groups of 2-3 students to be able to choose one shape at a time from the counter to investigate, so I printed three copies of each shape to make sure students didn't have to wait on other groups to finish in order to continue their investigation. 

Using the Group Chart, I modeled how to record the figure in the first column. Using the pentagon, I then modeled how to measure each angle on the inside and showed students how to write an addition equation using the measurement of all the angles. 

I continued: Once you're done investigating all the figures on the back counter, write down your observations, and then it's your turn! Pointing to the rectangle at the bottom of the page, I explained: You get to split this rectangle up in a similar manner using a ruler or protractor. 

Students were ready to investigate! 

Attending to Precision

For today's lesson in particular, students will be engaged in Math Practice 6: Attend to precision. All the angles of inside each shape is supposed to add up to 360 degrees. When students are off by just a degree or two, they'll end up with sums close to 360, such as 362 or 359. 

With time, students will see the pattern and will realize how important it is to attend to precision in order to get the sum of 360 degrees exactly!

Choosing Partners

Picking math partners is always easy as I already have students placed in desk groups based upon behavior, abilities, and communication skills. 

Monitoring Student Understanding

While students were working, I conferenced with every group. My goal was to support students by providing them with the opportunity to explain their thinking and by asking guiding questions. I also wanted to encourage students to construct viable arguments by using evidence to support their thinking (Math Practice 3). 

1. What are you lining up? (encouraging vocabulary use)
2. Can you explain what you are noticing?
3. What do you think? Why do you think that?
4. How do you know it's not ______? (non-example)
5. Is this acute or obtuse? How do you know?
6. What are you finding out about the sum of angles inside each figure?
7. Can you explain your thinking to your partner to make sure he agrees?  
8. What is the sum of all the angles? 
9. What did you notice about the two angles across from each other? 

Conferences

Here, Examining Angles Closer, a student explains how the length of angle arms does not effect the angle measurement. I also like watching them use the protractor to measure two angles at once. 

Here, Connecting the Sum of Angles with 360 Degrees, two students explain why all the angles add up to 360 degrees. 

Student Work

Here's an example of student work during this time: Example of Student Work.

After today's investigation, I invited students to join me on the front carpet to discuss observations. One student pointed out that the arms of an angle can keep going on forever and it doesn't change the angle measurement. I drew a picture to help other students understand this student's thinking: Comparing Angle Arm Sizes.

I drew a picture of two intersecting lines and labeled the angles A, B, C, and D. I then asked: What did you notice when measuring angles the result from two lines intersecting? I labeled the drawing, Sharing Observations, as students shared the following observations:

1. Angles A and C are congruent.
2. Angles D and B are congruent too!
3. Angles A and B have a sum of 180 degrees.
4. Angles D and C have a sum of 180 degrees too!
5. Altogether, A + B + C + D = 360 degrees.   

I then asked: Based on your investigation today, do you think this is true with all intersecting lines?Most students said, "Yes!"