Learn on PengiIllustrative Mathematics, Grade 5Chapter 7: Shapes on the Coordinate Plane

Lesson 10: Perimeter and Area of Rectangles

In this Grade 5 Illustrative Mathematics lesson from Chapter 7, students explore how the length and width of rectangles with a fixed perimeter or fixed area relate to each other by plotting coordinate pairs on a grid. Students calculate perimeter using 2l + 2w and area using multiplication, including with fractions and decimals, then graph the results to visually compare how each relationship behaves differently. The lesson addresses standards 5.G.A.2, 5.NBT.B.7, and 5.OA.B.3.

Section 1

Finding Perimeter and Area on a Grid

Property

For a rectangle on a grid:

  • Perimeter is the total number of unit lengths around the outside of the rectangle.
  • Area is the total number of unit squares inside the rectangle.
P=2l+2wP = 2l + 2w
A=l×wA = l \times w

Examples

  • A rectangle with a length of 55 units and a width of 33 units on a grid has a perimeter of 2(5)+2(3)=162(5) + 2(3) = 16 units and an area of 5×3=155 \times 3 = 15 square units.
  • A square with a side length of 44 units on a grid has a perimeter of 4×4=164 \times 4 = 16 units and an area of 4×4=164 \times 4 = 16 square units.

Explanation

You can find the perimeter of a rectangle on a grid by counting the number of unit segments along its boundary. To find the area, you can count the number of unit squares that fill the inside of the rectangle. These counting methods provide a visual way to understand the formulas for perimeter and area, where length (ll) and width (ww) correspond to the number of units along the sides.

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Finding Perimeter and Area on a Grid

Property

For a rectangle on a grid:

  • Perimeter is the total number of unit lengths around the outside of the rectangle.
  • Area is the total number of unit squares inside the rectangle.
P=2l+2wP = 2l + 2w
A=l×wA = l \times w

Examples

  • A rectangle with a length of 55 units and a width of 33 units on a grid has a perimeter of 2(5)+2(3)=162(5) + 2(3) = 16 units and an area of 5×3=155 \times 3 = 15 square units.
  • A square with a side length of 44 units on a grid has a perimeter of 4×4=164 \times 4 = 16 units and an area of 4×4=164 \times 4 = 16 square units.

Explanation

You can find the perimeter of a rectangle on a grid by counting the number of unit segments along its boundary. To find the area, you can count the number of unit squares that fill the inside of the rectangle. These counting methods provide a visual way to understand the formulas for perimeter and area, where length (ll) and width (ww) correspond to the number of units along the sides.