Rectangles with Fractional Side Lengths
5 min
Teacher Prep
Setup
Narrative
The purpose of this Warm-up is to elicit observations about the areas of squares with whole-number and fractional side lengths, which will be useful when students reason about rectangles with fractional side lengths in a later activity. While students may notice and wonder many things about these images, ideas about finding areas by tiling are the important discussion points.
When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.
Launch
Arrange students in groups of 2. Display the image of squares for all to see. Give students 1 minute of quiet think time, and ask them to be prepared to share at least one thing they notice and one thing they wonder. Give students another minute to discuss their observations and questions.
Student Task
What do you notice? What do you wonder?
Sample Response
Students may notice:
- There are 3 squares.
- The side lengths of the squares are 21 inch, 1 inch, and 2 inches.
- Each side length is twice the side length of the square before it.
- Two of the squares are partitioned into 4 smaller squares.
- Four 21-inch squares can fit in the 1-inch square. Four 1-inch squares can fit in the 2-inch square.
Students may wonder:
- Are the squares part of a pattern?
- Will the side length of the next square be 4 inches?
- What is the area of each square?
Activity Synthesis (Teacher Notes)
Consider telling students that we can call a square with 1-inch side length “a 1-inch square.”
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If reasoning about the area of a region by covering or tiling it with squares of known area does not come up during the conversation, ask students to discuss this idea.
Highlight the following points:
- A square with a side length of 1 inch (a 1-inch square) has an area of 1 in2.
- A 2-inch square has an area of 4 in2, because 4 squares with 1-inch side length are needed to cover it.
- A 21-inch square has an area of 41 in2 because 4 of them are needed to completely cover a 1-inch square.
Standards
- 5.NF.4.b·Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
- 5.NF.B.4.b·Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
20 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Rectangles with Fractional Side Lengths
5 min
Teacher Prep
Setup
Narrative
The purpose of this Warm-up is to elicit observations about the areas of squares with whole-number and fractional side lengths, which will be useful when students reason about rectangles with fractional side lengths in a later activity. While students may notice and wonder many things about these images, ideas about finding areas by tiling are the important discussion points.
When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.
Launch
Arrange students in groups of 2. Display the image of squares for all to see. Give students 1 minute of quiet think time, and ask them to be prepared to share at least one thing they notice and one thing they wonder. Give students another minute to discuss their observations and questions.
Student Task
What do you notice? What do you wonder?
Sample Response
Students may notice:
- There are 3 squares.
- The side lengths of the squares are 21 inch, 1 inch, and 2 inches.
- Each side length is twice the side length of the square before it.
- Two of the squares are partitioned into 4 smaller squares.
- Four 21-inch squares can fit in the 1-inch square. Four 1-inch squares can fit in the 2-inch square.
Students may wonder:
- Are the squares part of a pattern?
- Will the side length of the next square be 4 inches?
- What is the area of each square?
Activity Synthesis (Teacher Notes)
Consider telling students that we can call a square with 1-inch side length “a 1-inch square.”
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If reasoning about the area of a region by covering or tiling it with squares of known area does not come up during the conversation, ask students to discuss this idea.
Highlight the following points:
- A square with a side length of 1 inch (a 1-inch square) has an area of 1 in2.
- A 2-inch square has an area of 4 in2, because 4 squares with 1-inch side length are needed to cover it.
- A 21-inch square has an area of 41 in2 because 4 of them are needed to completely cover a 1-inch square.
Standards
- 5.NF.4.b·Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
- 5.NF.B.4.b·Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.