Lesson 7

This warm-up reinforces students’ understanding of bases and heights in a parallelogram. In previous lessons, students calculated areas of parallelograms using bases and heights. They have also determined possible bases and heights of a parallelogram given a whole-number area. They saw, for instance, that finding possible bases and heights of a parallelogram with an area of 20 square units means finding two numbers with a product of 20. Students extend that work here by working with decimal side lengths and area. 

As students work, notice students who understand that the two identical parallelograms have equal area and who use that understanding to find the unknown base. Ask them to share later.

Give students 2 minutes of quiet work time and access to their geometry toolkits.

Students should be adequately familiar with bases and heights to begin the warm-up. If needed, however, briefly review the relationship between a pair of base and height in a parallelogram, using questions such as:

• “Can we use any side of a parallelogram as a base?” (Yes.)
• “Is the height always the length of one of the sides of the parallelogram?” (No.)
• “Once we have identified a base, how do we identify a height?” (It can be any segment that is perpendicular to the base and goes from the base to the opposite side.)
• “Can a height segment be drawn outside of a parallelogram?” (Yes.)

Some students may not know how to begin answering the questions because measurements are not shown on the diagrams. Ask students to label the parallelograms based on the information in the task statement. 

Students may say that there is not enough information to answer the second question because only one piece of information is known (the height). Ask them what additional information might be needed. Prompt them to revisit the task statement and see what it says about the two parallelograms. Ask what they know about the areas of two figures that are identical.

Students may struggle to find the unknown base in the second question because the area of the parallelogram is a decimal and they are unsure how to divide a decimal. Ask them to explain how they would reason about it if the area was a whole number. If they understand that they need to divide the area by 2 (since the height is 2 cm), see if they could reason in terms of multiplication (i.e., 2 times what number is 2.4?) or if they could reason about the division using fractions (i.e., 2.4 can be seen as \(2\frac{4}{10}\) or \(\frac {24}{10}\); what is 24 tenths divided by 2?).

Select 1–2 previously identified students to share their responses. If not already explained by students, emphasize that we know the parallelograms have the same area because they are identical, which means that when one is placed on top of the other they would match up exactly.

Before moving on, ask students: “How can we verify that the height we found is correct, or that the two pairs of bases and heights produce the same area?" (We can multiply the values of each pair and see if they both produce 2.4.)

In earlier lessons, students saw that a square can be decomposed into two identical isosceles right triangles. They concluded that the area of each of those triangles is half of the area of the square. They used this observation to determine the area of composite regions.

This activity helps students see that parallelograms other than squares can also be decomposed into two identical triangles by drawing a diagonal. They check this by tracing a triangle on tracing paper and then rotating it to match the other copy. The process prepares students to see any triangle as occupying half of a parallelogram, and consequently, as having one half of its area. To generalize about quadrilaterals that can be decomposed into identical triangles, students need to analyze the features of the given shapes and look for structure (MP7).

There are a number of geometric observations in this unit that must be taken for granted at this point in students' mathematical study. This is one of those instances. Students have only seen examples of a parallelogram being decomposable into two copies of the same triangle, or have only verified this conjecture through physical experimentation, but for the time being it can be considered a fact. Starting in grade 8, they will begin to prove some of the observations they have previously taken to be true.

It may not occur to students to rotate triangles to check congruence. If so, tell students that we still consider two triangles identical even when one needs to be rotated to match the other.

The discussion should serve two goals: to highlight how quadrilaterals can be decomposed into triangles and to help students make generalizations about the types of quadrilaterals that can be decomposed into two identical triangles. Consider these questions:

• How did you decompose the quadrilaterals into two triangles? (Connect opposite vertices, i.e. draw a diagonal.)
• Did the strategy of drawing a diagonal work for decomposing all quadrilaterals into two triangles? (Yes.) Are all of the resulting triangles identical? (No.)
• What is it about C and E that they cannot be decomposed into two identical triangles? (They don't have equal sides or equal angles. Their opposite sides are not parallel.)
• What do A, B, and D have that C and E do not? (A, B, and D have two pairs of parallel sides that are of equal lengths. They are parallelograms.)

Ask students to complete this sentence starter: For a quadrilateral to be decomposable into two identical triangles it must be (or have) . . .

If time permits, discuss how students verified the congruence of the two triangles.

• How did you check if the triangles are identical? Did you simply stack the traced triangle or did you do something more specific? (They may notice that it is necessary to rotate one triangle—or to reflect one triangle it twice—before the triangles could be matched up.)
• Did anyone use another way to check for congruence? (Students may also think in terms of the parts or composition of each triangle. E.g. “Both triangles have all the same side lengths; they both have a right angle”).

Previously, students decomposed quadrilaterals into two identical triangles. The work warmed them to the idea of a triangle as a half of a familiar quadrilateral. This activity prompts them to think the other way—to compose quadrilaterals using two identical triangles. It helps students see that two identical triangles of any kind can always be joined to produce a parallelogram. Both explorations prepare students to make connections between the area of a triangle and that of a parallelogram in the next lesson.

A key understanding to uncover here is that two identical copies of a triangle can be joined along any corresponding side to produce a parallelogram, and that more than one parallelogram can be formed.

As students work, look for different compositions of the same pair of triangles. Select students using different approaches to share later.

When manipulating the cutouts students are likely to notice that right triangles can be composed into rectangles (and sometimes squares) and that non-right triangles produce parallelograms that are not rectangles. Students may not immediately recall that squares and rectangles are also parallelograms. Consider preparing a reference for students to consult. Here is an example:

As before, students make generalizations here that they don't yet have the tools to justify them. This is appropriate at this stage. Later in their mathematical study they will learn to verify what they now take as facts.   

For students using the digital activity, an applet can be used to compose triangles into other shapes.

Students may draw incorrect conclusions if certain pieces of their triangles are turned over (to face down), or if it did not occur to them that the pieces could be moved. Ask them to try manipulating the pieces in different ways.

Seeing that two copies of a triangle can always be composed into a parallelogram, students might mistakenly conclude that any two copies of a triangle can only be composed into a parallelogram (i.e., no other quadrilaterals can be formed from joining two identical triangles). Showing a counterexample may be a simple way to help students see that this is not the case.

The focus of this discussion would be to clarify whether or not two copies of each triangle can be composed into a rectangle or a parallelogram, and to highlight the different ways two triangles could be composed into a parallelogram. 

Ask a few students who composed different parallelograms from the same pair of triangles to share. Invite the class to notice how these students ended up with different parallelograms. To help them see that a triangle can be joined along any side of its copy to produce a parallelogram, ask questions such as:

• Here is one way of composing triangles S into a parallelogram. Did anyone else do it this way? Did anyone obtain a parallelogram a different way?
• How many different parallelograms can be created with any two copies of a triangle? Why? (3 ways, because there are 3 sides along which the triangles could be joined.)
• What kinds of triangles can be used to compose a rectangle? How? (Right triangles, by joining two copies along the side opposite of the right angle.)
• What kinds of triangles can be used to compose a parallelogram? How? (Any triangle, by joining two copies along any side with the same length.)