Lesson 10

The purpose of this Number Talk is to encourage students to make use of structure to perform division involving increasingly larger dividends and divisors (MP7). The reasoning here helps students to develop fluency in finding quotients of multi-digit numbers. It also reinforces students’ familiarity with factors of 180 and 360, which will be helpful as they continue to work with angle measurements.

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

• Record answers and strategy.
• Keep expressions and work displayed.
• Repeat with each expression.

• “How are the expressions related?”
• “How did finding the value of one expression help you find the value of the next expression?”
• Consider asking:
  • “Who can restate _____’s reasoning in a different way?”
  • “Did anyone have the same strategy but would explain it differently?”
  • “Did anyone approach the expression in a different way?”
  • “Does anyone want to add on to _____’s strategy?”

In previous lessons, students learned to read the measurement of an angle with a protractor already in position. In this activity, students practice using a protractor to measure angles. They decide where to place the tool, how to align it with the vertex and rays of the angle, and which set of numbers on the protractor to use.

Some of the figures in the activity explicitly show angles formed by two rays. In others, students are asked to find and measure the angles within polygons. In both cases, students may find it necessary to extend one or both rays of an angle so that it can be measured more effectively or precisely (MP6). Doing so reinforces the idea that the size of an angle is not determined by the length of the segments that frame it, but by the rays that compose the angle.

• Groups of 2
• Give each student a protractor and access to rulers or straightedges.

• 5 minutes: independent work time
• 1–2 minutes: partner discussion
• Monitor for students who:
  • align the rays of the angle to tick marks on the protractor and count from one ray to the other
  • don’t align either ray of an angle to \(0^\circ\) or \(180^\circ\) on the protractor and instead find the difference of the numbers where the two rays land on the protractor
  • always align one ray of an angle with the \(0^\circ\) or \(180^\circ\) line on the protractor and always read from the scale that starts with \(0^\circ\)

Students may carefully line up rays on the protractor, but find angle measurements that are unreasonable. Ask students to explain how they used the protractor to measure. Consider asking:

• “Based on the angles you have measured in previous lessons, does _____ degrees make sense as the measure of this angle? Why or why not?”
• “What is another way you could use the protractor to check your measurement?”

• Display the angles and figures. Invite previously identified students to share their methods for measuring angles, in the order shown in the monitoring notes. 
• “What are some benefits of aligning one ray of an angle to \(0^\circ\) on the protractor?” (The measurement can be identified right away: it’s the number where the second ray lands on the protractor. Aligning the first ray to a non-zero number means having to count or subtract two numbers before finding the measurement.)  
• “How was measuring the second set of angles like measuring the first set of angles?” (They both involve aligning the center point of a protractor to the vertex of the angle, and matching the \(0^\circ\) line on the protractor to one ray or segment of the angle. It can be helpful to extend one or both lines framing the angle.) 
• “How was it different than measuring the first set?” (The two-dimensional shapes have other segments and angles nearby, so more attention was needed when placing the protractor and reading the measurement.)
• “How can you tell if your measurement was reasonable? How can you make sure your measurements make sense?” (Compare them against a familiar angle like \(90^\circ\). If an angle looks larger than a right angle, it can’t be less than \(90^\circ\).)

In this activity, students fold paper to form right angles and learn that intersecting lines that form \(90^\circ\) angles are perpendicular lines. They then identify perpendicular segments in two-dimensional figures and explain how they know the segments are perpendicular.

To create four right angles that share the same vertex by folding generally means making two folds through the same point. The first fold, which can be done in any way (as long as it goes through point \(P\), in this case), creates two straight angles. The second involves folding through the point again such that the edges formed by the crease of the first fold match up exactly, creating two equal halves or two \(90^\circ\) angles.

While students have experience with folding paper to partition a shape or an angle, some may need support in folding precisely. Consider providing a straightedge to facilitate the folding.

This activity uses MLR2 Collect and Display. Advances: conversing, reading, writing

• Prepare at least 2 pieces of paper (or sticky notes) for each student.

• Groups of 2–4
• Give each student 2 pieces of paper and colored pencils. Provide access to straightedges or rulers, in case requested.
• Read the opening prompts and the first question.
• “What do you think Lin did with her paper? Mark a point on a piece of paper and try folding it as Lin might have done.”
• 2–3 minutes: quiet think time on the first problem
• Pause for a discussion. Invite a couple of students to share how they think Lin met the challenge.

• 6–7 minutes: group work on the remaining questions
• Circulate, listen for, and collect the language students use to define perpendicular lines. 
• Record students’ words and phrases on a visual display and update it throughout the lesson. 
• Monitor for students who:
  • reason that their folded lines form right angles because the first fold makes two \(180^\circ\) angles through the point and the second fold splits each into halves, making \(90^\circ\) angles
  • use a protractor (or a square corner) to verify perpendicularity of the sides of shapes in the last problem (rather than relying on appearance)

When making their second fold, students may not align the two edges formed by the crease of the first fold, resulting in a pair of angles of one size and another pair of a different size (instead of four right angles). Consider asking: “How might you adjust your folding to create two equal angles?”

Students may say they do not see perpendicular sides in the last problem because, unlike the intersecting lines in the folded paper, the sides of the shapes do not cross. Consider asking:

• “Where do the sides meet? What do you notice about where the sides meet in this figure?”
• “What if you extend each side to show the line that each segment is a part of? What do you notice?”

• “What statement could we write to explain to another student what perpendicular lines are?” (Lines that intersect and create right angles.)
• Remind students that they can use words or phrases from their personal word walls in their responses.
• Invite students to share the perpendicular lines they created by folding.
• “How can you be sure that the creases from your folding are perpendicular or created \(90^\circ\) angles?” (My first fold makes two \(180^\circ\) angles. My second fold splits each of those into two equal angles, so each one must be \(90^\circ\). We can measure each angle with a protractor.)
• “In the last problem, how did you know which shapes have perpendicular sides?” (By measuring the angles with a protractor, or by comparing them with a square corner.)