Lesson 9

The purpose of this warm-up is to draw students’ attention to the first few multiples of 45, which will be helpful as students continue to work with benchmark angles and use a protractor to measure angles. Students have the skills to perform the multiplication in each equation, but computing each product may be time-consuming. Students can more efficiently tell if the equations are true or false if they consider properties of operations and look for and make use of structure.

• Display one equation.
• “Give me a signal when you know whether the equation is true and can explain how you know.”
• 1 minute: quiet think time

• Share and record answers and strategy.
• Repeat with each statement.

• Some students may notice that it is handy to think in terms of \(2 \times 45\) because it would mean dealing with multiples of 90 rather than multiples of 45. Highlight their explanations.
• If no students decomposed expressions such as \(3 \times 45\), \(4 \times 45\), and \(6 \times 45\) into sums of \(1 \times 45\) and \(2 \times 45\), discuss how this could be done. (See sample responses.)

This activity develops an understanding of the degree as a unit used to measure angles and introduces students to the protractor.

By now students have encountered many angle measurements and have some intuitive awareness of the sizes relative to a full turn (\(360^\circ\)), half of a full turn (\(180^\circ\)), and a quarter of a full turn (\(90^\circ\)). In this activity, students learn that a \(1^\circ\) angle is \(\frac{1}{360}\) of a full turn and that an angle that is composed of \(n\) \(1^\circ\) angles has a measurement of \(n^ \circ\). For example, a \(7^\circ \) angle is made up of seven \(1^\circ\)angles.

• Groups of 2
• 2 minutes: independent work time to find the fractions of a full turn
• Record students’ responses. Solicit some ideas on how large a \(1^\circ\) angle is. Emphasize that a \(1^\circ\) angle is \(\frac{1}{360}\)of a turn and that it’s the size of the angle formed by the sides of the pieces of the circle created if we cut a full circle into 360 equal parts.
• “To measure angles in degrees, we can use a protractor.” (Consider displaying different types of protractors.)
• Give each student a protractor.
• “Compare this tool to the one you used in the previous lesson. How are they the same? How are they different?” (They are both semi-circles. They both show angles like \(0^\circ\), \(30^\circ\), \(45^\circ\), and \(90^\circ\). The protractor is transparent or has a hole, while the paper version is solid. The protractor shows many more lines or tick marks and more numbers around the curve.)

• 2–3 minutes: group work time on the second set of questions. 
• Monitor for students who look for structure to find the number of \(1^\circ\) increments on the protractor (for example, noticing that every group of ten \(1^\circ\) increments are marked and counting those instead of counting individual tick marks).
• Pause for a discussion. Make sure students see that a \(1^\circ\) angle on the protractor results when we draw rays through a pair of neighboring tick marks.
• 2–3 minutes: individual or group work time on the remaining questions

• “How did you find out the size of the two angles when the protractors show no numbers or scales?” (We know that the turn from one tick mark to the next is \(1^\circ\). We can use the tick marks to count the number of \(1^\circ\) turns. We can imagine the tick marks split the angle into a number of \(1^\circ\) angles. We can count each \(1^\circ\)angle to find the measurement.)

In this activity, students learn how to use a protractor. They align a protractor to the vertex and a ray of an angle so that its measurement can be read. The given angles are oriented in different ways, drawing students’ attention to the two sets of scales on a protractor. Students need to consider which set of numbers to pay attention to and think about a possible explanation for when or how to use each scale. Moreover, one of the scales is only marked in increments of 10 degrees so if students use this scale they need to reason carefully about the precise angle measure (MP6).

This activity uses MLR1 Stronger and Clearer Each Time. Advances: reading, writing.

• Groups of 2–4

• 5 minutes: independent work time
• 2–3 minutes: group discussion
• Monitor for students who find each measurement by:
  • looking for the scale with a ray on \(0^\circ\) and counting up on that scale
  • subtracting the two numbers (on the same scale) that the rays pass through

MLR1 Stronger and Clearer Each Time

• “Share your response to the last problem with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
• 3–5 minutes: partner discussion
• Repeat with 2–3 different partners.
• “Revise your initial draft based on the feedback you got from your partners.”
• 2–3 minutes: independent work time

• Select students who used different strategies to share their responses and reasoning.
• Display the image of Elena and Kiran’s angle. If no students mentioned that \(80^\circ\) is not possible because the angle clearly appears greater than a right angle (\(90^\circ\)), consider asking how Elena can use what she knows about right angles to think about her measurement.