Lesson 6

The purpose of this warm-up is to encourage students to reason about properties of operations without evaluating each expression. Encourage students to think about the meaning of the operations in each question.

Display each problem one at a time. Tell students to give a signal when they have a response. Give 30 seconds of quiet think time. Ask students to share their reasoning for each. Record and display their thinking for all to see. Leave each problem displayed as you move onto the next problem.

Some students may try evaluating each side of each equation. Encourage them to look for patterns or shortcuts that would help them answer each question without doing all the calculations.

Ask students to share their reasoning. Record and display the responses for all to see. To involve more students in the conversation, use some of the following questions:

• “Do you agree or disagree? Why?”
• “Who can restate ___’s reasoning in a different way?”
• “Did anyone reason about the problem the same way, but would explain it differently?”
• “Did anyone reason about the problem in a different way?”
• “Does anyone want to add on to _____’s reasoning?” If there is time, ask students how they could rewrite the false equations to be true.

See MLR 7 (Connect and Compare) for more examples.

The purpose of this activity is for students to explore a physical representation of polygons and make observations about triangles and quadrilaterals. This introductory activity serves to familiarize students with the tools and definitions they will use in future activities. Students may notice that some sets of 3 strips cannot make a triangle, but formalizing rules about what lengths can and cannot be used to form a triangle is not the goal of this lesson.

This is the place to notice that pairs of triangles with 3 matching lengths make identical triangles, whereas pairs of quadrilaterals with 4 matching lengths do not necessarily make identical quadrilaterals. Students will express this observation more formally in the next activity.

As students work, select at least one group’s triangle and another group’s quadrilateral to recreate for the whole-class discussion.

Some students may try to bend the strips to make shapes with curved sides. Remind them that a polygon has all straight sides.

The purpose of this discussion is to establish what is meant when we say two shapes are identical copies. While students don’t use the word congruent until grade 8, they should recognize that two shapes are identical only when they can match perfectly on top of each other by movements that don’t change lengths or angles.

Invite a group to share the side lengths they chose for their triangle. Create a copy of the triangle with another set of strips and fasteners. Display it for all to see alongside the group’s original triangle, but oriented differently. Ask students, “Are the two shapes identical? How can you tell?” (Yes, because you can put them on top of each other and they match up exactly.) Demonstrate turning or flipping the triangle, as needed, to place one copy on top of the other and show that they match.

Repeat the demonstration with a group’s quadrilateral. Create a copy that has the same side lengths as what they used, but different angles. Demonstrate the “floppiness” of the quadrilateral (that is, the angles can change even though the side lengths remain the same). Make sure students realize that the two quadrilaterals are not necessarily identical copies, even though they have the same side lengths.

The purpose of this activity is to reinforce that some conditions define a unique polygon while others do not. Students build polygons given only a description of their side lengths. They articulate that this is not enough information to guarantee that a pair of quadrilaterals are identical copies. On the other hand, triangles have a special property that three specific side lengths result in a unique triangle. Students should notice that their recreation of Jada’s triangle is rigid; the side lengths and angles are all fixed.

Monitor for students who try putting the side lengths together in different orders to build different polygons and invite them to share during the whole-class discussion.

Students may think that their triangle is different from Jada’s because hers is “upside down.” Ask the student to turn their triangle around and ask them if it is now a different triangle. While there is a good debate to be had if they continue to insist they are different, let the students know that, for this unit, we will consider shapes that have been turned or flipped or moved as identical copies and thus “not different.”

Select previously identified students to share their constructions and explanations. Display each student’s example for all to see.

If desired, reveal Diego and Jada’s shapes and display for all to see along side students’ work. 

Ask students:

• “Is this what you thought Jada and Diego’s shapes looked like?”
• “Which shape did you make an identical copy of?” (Jada’s triangle.)
• “Why did you not make an identical copy of Diego’s shape?” (Because you can make quadrilaterals with the same side lengths but different angle measure.)

Consider explaining to students how this finding is applied in construction projects. For stability, the internal structures of many buildings (and bridges) will include triangles. Rectangles or other polygons with more than three sides often include triangular supports on the inside, to make the construction more rigid and less floppy.

The purpose of this activity is for students to see that sometimes it is impossible to build a polygon with certain conditions, but also that they need to think carefully about the information they are given before making assumptions.

In this case, students are given 3 side lengths that cannot form a triangle and are told to build a polygon. Students may assume that because they were given 3 side lengths, their shape is supposed to be a triangle; however, they will only succeed in building a polygon using the specified side lengths if it has more than 3 sides. Formalizing rules about what lengths can and cannot be used to form a triangle is not the goal of this activity.

As students work on the task, monitor for students who realize that the shape cannot be a triangle and for students who realize it can be a polygon with more than 3 sides.

Students may say that there is no way Han could have built this shape, because they are assuming it must be a triangle. Ask students if the question specifies that the shape is a triangle. If needed, remind students of the definition of polygon and prompt them to consider what they could do to finish building a closed shape with all straight sides.

Select previously identified students to share their explanations. Make sure students realize that this shape cannot be built as a triangle; however, it is possible to build a polygon with more than 3 sides. Tell students, “As the unit progresses, you will be asked to create or draw shapes that include some conditions, but there may be some flexibility with the pieces that are not mentioned. Be aware of what must be included and what is not mentioned.”

If time permits, consider asking some of the following questions:

• “What length did you choose to use for your fourth side? Would another choice have worked?”
• “If another group used the same length for their fourth side as you, does their polygon have to be an identical copy of yours? How do you know?”
• “Did any group choose to have more than 4 sides on their polygon? Does such a shape fulfill the given conditions?”