“In this lesson we are going to continue to explore the concept of measurement. We will specifically look at the area of two-dimensional shapes. Area is the number of square units needed to cover a flat surface. Small shapes can have their areas measured in square centimeters, square inches, or smaller square units. Larger shapes can have their areas measured in square feet, square yards, square meters, or larger square units. Calculating the area of a shape is helpful when determining how much carpeting or tile is necessary to cover a floor or how much wallpaper is necessary to cover a wall. Remember, area is two
dimensional because it is a measurement of two dimensions: length and width.”
Show students models of what square units look like. You can draw various square units on chart paper or construct them with rulers or yardsticks. Provide examples of square inches and square centimeters (M-4-1-2_Square Centimeter Grid Paper.doc and M-4-1-2_Square Inch Grid Paper.doc). You can measure out a square foot on chart paper for students to see or tape four rulers together. Repeat the same process for students to see what a square yard and a square meter look like. Ask students why a model of a square mile or a square kilometer cannot be shown in the classroom. Once students have seen the visual models for square units, have students choose an appropriate unit to measure the areas of different shapes in the classroom. Examples of classroom shapes to measure include the area of a bulletin board, the top of a student desk, the top of your desk, a computer screen, and the classroom floor.
On construction paper, draw two shapes that are different in area. Regular or irregular shapes are possible examples. Show the figures to students. “How can we figure out which of these two shapes has the larger area? Remember area is the number of square units needed to cover a two-dimensional surface.” Post students’ suggested methods on the board and model how each method might be used to determine which shape has the larger area. Students might suggest covering both shapes with square tiles, counting the number of square tiles, and comparing the quantities. At this point it would be a good idea to remind students that square tiles are a standard unit of measure. This type of method also gives an estimated area. Another strategy might consist of cutting one shape into pieces and arranging them on top of the other shape. Students can then compare the areas of the shapes.
Have each student draw a two-dimensional shape on a piece of construction paper. Collect them. Randomly pick two shapes and have partners determine which shape has the larger area. Have students generate as many varied methods as possible to use in verifying their answers. While students are working, monitor their progress. Ask questions similar to the ones listed below.
- What is area? (the measure of the surface of an object)
- Which square units would be the best choice to measure the area of these shapes?
(square inches, square centimeters, square millimeters, etc. . . .)
- What method did you use to determine which shape has the greater area?
- Is there another method you could use?
If necessary have students exchange shapes with partners and repeat the process to reinforce understanding.
“If you wanted to find the area of this shape would you be able to find an exact answer or an estimated answer? Be prepared to share your thoughts and discuss with students around you.” (M-4-1-2_Shape 1.doc) After students have time to
think-pair-share, have students share their thoughts with the class. Guide student toward the idea that calculating the area of an irregular shape will be an estimated answer. Discuss which standard unit of measure, square inches or square centimeters, would give the more accurate estimate and why. (Square centimeters would give the more accurate estimate because the pieces are smaller and fit better into the surface of the shape.)
“Remember, our standard unit of measure is a square, such as square inches. We can't find whole squares, so we will need to estimate the area, that is, the number of whole squares we think would cover the two-dimensional shape. How can we do this and get a reasonable result?”
Model a method for students to find the estimated area of the shape using square inches. A transparency can be made; chunks of square inches can be cut to best fit the shape. Alternatively, the shape can be cut out and traced onto square inch paper (M-4-1-2_Shape 1 with Unmarked Square Inches.doc). “We can count the square inches.” (There are 20.) “If I say the area of this shape is 20 square inches, is that an accurate estimate? Discuss with students around you, and be prepared to share your thoughts.” After students have time to think-pair-share, have students share their thoughts with the class. “Twenty square inches is not a good start for our estimate of the area of the two-dimensional shape, but we can be more accurate. This number would show an overestimated area. Look at all the square units that are not on the surface of the shape. In order to get a more accurate estimate of the area we have to combine partial pieces until they equal the area of one square inch.” Show students Shape 1 with Square Inches handout (M-4-1-2_Shape 1 with Square Inches.doc). “Look at how I numbered the pieces to show which partial pieces would make approximately one square inch. My new estimated answer is about 11 square inches. I can cut out the shape and trace it on square inch paper if I have to. Now you and a partner will use a similar strategy to find the estimated area of this shape using the standard unit of measure of square centimeters. Remember to combine partial pieces until they equal the area of one square centimeter in order to get a more accurate estimated area.” Give each pair of students the Shape 1 with Square Centimeters handout (M-4-1-2_Shape 1 with Square Centimeters.doc). Allow students time to calculate the estimated area. Monitor student performance and guide student understanding when necessary. While students are working, ask students questions similar to the ones listed below to assess understanding.
- What does the area of a shape represent? (the measure of the inside of the shape)
- What are some different standard units of measure you can use to measure the area of a two-dimensional shape? (square inches, square centimeters, square feet, square meters, etc.)
- What did you do first? Why?
- Do you have to count every square to find the area of the two-dimensional shape? Is there another strategy you can use? (You can use multiplication.)
- What do you think would be a good estimated area? What are you basing your answer on?
- What if you have leftover parts of square centimeters? How do you use these pieces to calculate a more accurate area? (Use square millimeters.)
- How are square centimeters similar to square inches? (They both measure areas of surfaces.)
- Which would give a more accurate representation of the shape’s area? Why?
For students who finish quickly and demonstrate understanding, have them create irregular shapes and exchange with a partner to calculate the area. “This activity showed us how we can find the area of a shape using standard units of measures. The shapes we used were irregular two-dimensional shapes. Now let’s look at finding the area of two common
two-dimensional figures: rectangles and squares.”
Students are given a pair of rectangles that are either the same or very close in area. These rectangles can be drawn out on paper or cut from construction paper. Some suggested pairs are as follows: 4 × 10 and 5 × 8, 5 × 10 and 7 × 7, 6 × 8 and 12 × 4, 4 × 5 and 12 × 2. Students also are given a drawing of a single square unit and a ruler that measures the appropriate unit (a square inch unit and a ruler with inches or a square centimeter unit and a centimeter ruler). “Your task is to use your rulers to determine, in any way that you can, which rectangle has a greater area or whether they have the same area. You are not permitted to cut out the rectangles. You may draw on them if you wish. Use words, pictures, and numbers to explain your conclusions.”
Not all students will use a multiplicative approach. To count a single row of squares along one edge and then multiply by the length of the other edge, the first row must be thought of as a unit that is then repeated to fill in the rectangle. Many students will attempt to draw in all the squares. To help transition from the need to visualize all the square units, one option may be to cut out strips (rows of squares) to fill in the rectangle. At this point giving students grid paper with the units might be helpful. In the rectangle below, the row has eight square units and three strips of eight squares are needed to fill in the rectangle. A visual that will help students focus on what the dimensions of the rectangle represent is the following “L”.
Notice how the “L” further encourages students to visualize the square units not drawn.
Have students exchange rectangle pairs and repeat the process. After multiple attempts, students may be able to refine their thinking and adjust strategies to become more proficient.
To assess student understanding, have students complete the Area Four Square activity (M-4-1-2_Area Four Square and KEY.doc). “We have explored the concept of area of two-dimensional shapes in this lesson. The area of a shape can be determined using standard and nonstandard units of measurement. Sometimes an area is an estimated value because of the units used and the actual shape of the object. Choosing the appropriate unit of measurement is important not only when determining area but when calculating any measurement.”
Use the following strategies and activities to meet the needs of your students during the lesson and throughout the year.
- Routine: Have a large piece of graph paper hanging in one part of the room. Throughout the year, occasionally have a student draw a shape on it. Then have students estimate the area of the shape using estimation by counting squares. Then have a small group determine the area and model the process they used to the rest of the class.
- Small group: Ask students to help generate a list of some two-dimensional, non-curved objects in the classroom whose areas they could determine using a standard unit of measure. Remind students that area applies only to two-dimensional shapes. Explain to students that they will work in pairs to select two objects on the list and measure the objects’ areas. Students will measure each object with several different standard and nonstandard units. The units can include square tiles, triangles, and rhombi from
pattern-block sets, and/or 3 × 5-inch index cards. Distribute a chart (M-4-1-2_Small Group Area Chart.doc) to each student for recording.
Number of 1-inch Squares
Number of Index Cards
sheet of notebook paper
Choose one item from the list of shapes students generated and model the process for the group. Discuss why some measurements on the chart will be more accurate estimates than others. Give students time to complete the task. Monitor student interaction and clarify misunderstanding when necessary. Gather the small group back together and discuss findings. Ask each student to verbally complete the following stem: Area is . . .
- Expansion: Have students who demonstrate proficiency explore finding various shapes with the same area. Students can be given a specific area like 12 square units and be asked to find as many varied shapes with the area of 12 square units as they can. Students can use grid paper or square tiles (pattern-block pieces). Once students become proficient at finding as many varied shapes as possible, encourage them to focus only on finding as many varied rectangles as possible with that area. For example, if students are given the area of 12 square units they can make the following rectangles:
1 × 12, 2 × 6, and 3 × 4. Try numbers such as 24 square units and 36 square units to start. Ask students to explain how they know if they found all the possible rectangles with a given area.
This lesson is designed to continue to explore the concept of measurement with a specific focus on area of two-dimensional shapes. Students will be exposed to the various units of measure that can be used to find the area of shapes. Students will begin to realize the importance of choosing a reasonable unit with which to measure the area of chosen objects. Estimated areas will be discussed, showing students how the attributes of the shape along with the units used may result in an approximated area. Using various strategies like counting square units and multiplication, students will calculate the area of irregular shapes along with rectangles and squares.