Math Concept: Area

Target Grade Level: 4th Grade

CCSS.Math

4.MD.A.2: Apply the area and perimeter formulas for rectangles in real world and mathematical problems.

• Students will be applying the area formula for rectangles to determine the total area of each rectangle that they draw. Students will be applying the formula repeatedly with different combinations of dimensions.

CCSS Mathematical Practices

MP2: Reason abstractly and quantitatively.

• Students can work with the numbers generated by rolling the dice to find an answer, but can also understand that the numbers represent something concrete. Students must realize that the numbers being rolled represent the length and width of their rectangles rather than just numbers. They must also be able to attach the given measurement to the number. For example, a roll of 5 and 6 gives a rectangle with the dimensions of 5m x 6m and an area of 30m².

MP6: Attend to precision.

• Students must be able to determine that if each square on the graph paper represents one square meter, then each side is one meter. Students must then be able to apply this when translating the numbers on the dice to the dimensions of the rectangle and when calculating the area of the rectangle. Students can determine if their answer makes sense by looking that the actual rectangle that has been drawn and by paying close attention to the units. For example, a roll of 3 and 4 would have to translate to a 3m x 4m rectangle with an area of 12m². If students end up with 3m² x 4m² then the area would end up being 12m⁴. Students would need to realize that these units do not make sense when compared with their drawing that shows 12 square meters.

Learning Targets

Students will be able to explain the concept of square units.

Students will be able to apply the area formula for a rectangle.

Academic Language

• Dimensions (length, width)
• Area
• Square unit

Activity Purpose

This activity allows students to see how and why the area formula for a rectangle works. While playing, students can see that multiplying the dimensions will give them the total number of squares that are inside of the rectangle. Students will also be able to see where the idea of square units comes from. Students will be given the opportunity to not only solidify this understanding, but they will be given ample opportunity to practice applying the area formula. On every turn students will be drawing a rectangle and finding the area. Students will also continuously roll different numbers giving them practice with different dimensions, and showing that the area formula works consistently.

Activity: Capture the Area

Objective: To capture as much the “land” as possible

Time: 20-25 minutes

Players: 2-4  players

Materials

• A pair of dice (1 pair for each group)
• Centimeter graph paper
• Colored pencils (crayons and markers will also work)
• Copy of directions/discussion questions for each group (attached) Capture the Area- Directions

Technology

For this activity there is no need to use technology. A calculator could be used for students who need that adaptation.  However, a primary goal of this activity is to practice mental math and quick recall of multiplication math facts.

Problem

Neighboring farmers trying to acquire unclaimed farmland. The land can only be acquired in rectangular pieces. Instead of fighting for the land, the farmers have decided to take turns claiming pieces of land as determined by rolling dice. Each square plot of land is one square meter. Each farmer wants to get as much land as possible.

Directions/How to Play

1. Each player starts in a different corner and uses a different color colored pencil.
2. Players will roll a die to determine who will go first. (Highest roll goes first and game play continues clockwise).
3. Players roll the dice to find the dimensions of a rectangle that they will draw using their assigned color. Players assign one number to the width and the other number to the length of the rectangle. In addition to drawing the rectangle, players must write inside the rectangle the dimensions and the total area. Players start with the first rectangle in their own corner.
4. Each rectangle that follows must be drawn so that it is touching one of the sides of that player’s previous rectangles.
5. Game play ends when all players have met in the center or when no more rectangles can be drawn or at a specified time. To make games quicker use centimeter graph paper. 
6. Players then find the total amount of area they have acquired.
7. The player with the greatest total area is the winner.

Example

Discussion Questions

These questions are given to students in their groups or pairs to discuss. Students should be encouraged to use academic language (i.e. area, dimensions, square units) when answering the questions. Once students have discussed the questions in their groups, the teacher should initiate a whole class discussion.

1. As a farmer, why would you want all of your land together?
2. Why would you want rectangular plots of land?
3. Who won?
4. Why did they win?
5. What kind of dimensions do you want to role? Larger numbers or smaller numbers? Why?
6. What is the largest area that you can get from rolling the dice? What is the smallest area you can get?
7. How does the area formula work? Why do we multiple the length and the width?

Assessment

The answers to the discussion questions will serve as a formative assessment. The teacher should also be circulating the room and using observation to formatively assess students’ knowledge of multiplication facts and their ability to apply the area formula.  

Adaptions

There are a number of possible adaptions to make this lesson appropriate for younger or older students.

For younger students:

•  Add the two numbers rolled to determine the total number of squares that can be claimed.
• Give students multiple dice so they can practice adding three or more numbers at a time.

For older students (or for students who need an extra challenge):

·         Assign each student a particular crop. Each crop is worth a certain amount of money. In addition to determining the total area, students can also determine the total amount of money they could make.

• Convert the total area from square meters to square centimeters and/or square kilometers.
•  Determine the amount of fencing required to enclose all of the acquired land within one fence.
• Use triangles rather than rectangles. The numbers rolled would indicate the base and the height of a right triangle. Students would then have to determine the area of the triangles.

Additional Teacher Suggestions

We designed the modeling activity “Capture the Area” to allow students the opportunity to apply the area formula in real world problems to meet the CCSS 4.MD.A.2. We also felt that this could be a great activity to connect to students’ prior knowledge about area measurement.  This builds on their initial learning of the third grade standard CCSS 3.MD.C.5.A.

While teaching this activity we realized the importance of relating the activity to student interests.  To do this we discussed with students what they would like to farm on their land such as crops or animals.  This helped to increase their buy in for the activity.

We also noticed that some students struggled with the third grade square unit concept.  This means that this concept may need to be retaught.

Additionally, we learned that using larger graph paper makes it easier for people to see what they are doing and to complete the task. Smaller graph paper made the activity more tedious and made it difficult for students to follow their own work. 

Modeling Activity designed by Julie Murphy and Emily Phillips.