From Parallelograms to Triangles

5 min

Narrative

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 corresponding 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.

Launch

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.)

Student Task

Here are two copies of a parallelogram. Each copy has one side labeled as the base bb and a segment drawn for its corresponding height and labeled hh. 

Two triangles, base b, height h. On right, b is slanted side and height is outside of triangle, perpendicular to the slanted side.

  1. The base of the parallelogram on the left is 2.4 centimeters; its corresponding height is 1 centimeter. Find its area in square centimeters.
  2. The height of the parallelogram on the right is 2 centimeters. How long is the base of that parallelogram? Explain your reasoning.

Sample Response

  1. 2.4 square centimeters. (2.4)⋅1=2.4(2.4) \boldcdot 1=2.4
  2. 1.2 centimeters. Sample reasoning: The area of the second parallelogram is also 2.4 square centimeters. Since the base and height must multiply to the same area of 2.4, the base must be 1.2 centimeters because (1.2)⋅2=2.4(1.2)\boldcdot2=2.4.

Synthesis

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.)

Anticipated Misconceptions

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 know what to do to find the unknown base in the second question but be unsure how to divide a number containing a decimal. Ask them to explain how they would reason about it if the area were a whole number. If they understand that they need to divide the area by 2 (because the height is 2 cm and the area is 2.4 sq cm), encourage them to reason in terms of multiplication, for instance by asking, “2 times what number is 2.4?” Or, urge them to consider dividing using fractions, for instance, by seeing 2.4 as 24102\frac{4}{10} or 2410\frac {24}{10}. Ask, “what is 24 tenths divided by 2?”

Standards
Addressing
  • 6.G.1·Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
  • 6.G.A.1·Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

15 min

15 min