Lesson 2

This Number Talk encourages students to use their understanding of mixed numbers and properties of operations to mentally solve problems. The strategies elicited here will be helpful as students develop their fluency in performing operations on fractions.

• 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 is each expression related to the one before it?”
• “How can the first three expressions help us find the value of the last 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?”

The purpose of this activity is for students to describe and sort triangles based on the length of their sides and the size of their angles. Students are asked to find all the triangles (from the same set of cards used in the previous lesson) that have specific attributes.

Along the way, students notice attributes that all triangles seem to share (for instance, having three angles and at least two acute angles), attributes that some triangles share (for instance, having two equal angles or two equal sides), and those that no triangles seem to have (for example, parallel sides or multiple obtuse angles). With these repeated observations, students deepen their understanding of the properties of triangles and their sub-groups (MP8).

In the synthesis, students are introduced to right triangles.

• Each group needs a set of shape cards from the previous lesson. If time permits, separate the triangle cards from each set in advance.
• Gather the Collect and Display chart from the previous lesson for display in the activity synthesis.

• Groups of 2
• Give each group a set of cards from the first lesson.
• Provide access to rulers and protractors.
• Make available the chart with vocabulary from the previous lesson for reference during this lesson.
• “Use only the cards with triangles for this activity.”
• “Complete the triangle hunt with a partner. When your group is done, compare responses with another group.”

• 5 minutes: group work time
• 3 minutes: discussion with another group
• 2 minutes: individual work time on the last question

If students place each triangle in only one category, consider asking:

• “How did you make sure you found all the triangles that have each attribute?”
• “How would you check if a triangle has only one of the listed attributes or more than one?”

• Add new vocabulary and sketches to the chart from the previous lesson.
• “What do you notice about triangles H, L and V?” (They all have right angles.) 
• “Triangles that have a right angle are right triangles.”
• “How did you know whether an angle in a triangle is a right angle?” (I measured it with a protractor or an object with a square corner.)
• “Can you find other right triangles in the set?” (Triangle Y)
• Invite students to share their completed sentences for the last question. Record their responses and invite the class to agree or disagree.
  • Some triangles . . .
  • All triangles . . .
  • No triangles . . .
• If time permits, consider asking: “In our set of triangles, we didn’t see any with more than one right angle or more than one obtuse angle. Do you think it’s possible for a triangle to have:
  • Two right angles?” (No, two of the sides won’t meet to make a triangle.)
  • Two obtuse angles?” (No, two of the sides won’t meet to make a triangle.)

In this activity, students identify right triangles. They also explain why certain given shapes are not right triangles. As they determine whether a triangle is a right triangle, students may use the corner of an index card or a piece of paper, use a protractor, or trace an angle in a triangle to compare with an angle elsewhere that they know to be a right angle. In doing so, students practice choosing tools strategically (MP5).

• Groups of 2
• Provide access to index cards, patty paper, and protractors.

• 5 minutes: independent work time
• 2 minutes: partner discussion
• Monitor for students who use tools to check angles and those who estimate them visually. Ask the latter: “How might we check that the angles that look like right angles are \(90^\circ\)?”

Students may not identify triangles C or D because of the orientation. Consider asking:

• “How can we show that a triangle has a right angle?”
• “Where else do you see triangles that may have perpendicular sides?”
• “What tools could you use to see if each triangle has any right angles?”

• Invite students to share their responses and how they found the right triangles in the set.
• “Were there any triangles that appeared to be right triangles but are not, or don’t appear to be right triangles but are?” (Triangle H looks like one but isn’t.)