Finding Area by Decomposing and Rearranging
5 min
Narrative
This is the first Notice and Wonder activity in this course. Students are shown four drawings and asked: “What do you notice? What do you wonder?”
Students are given time to write down what they notice and wonder about the images and then time to share their thoughts. Their responses are recorded for all to see. Often, the goal is to elicit observations and curiosities about a mathematical idea that students are about to explore. Pondering the two open questions allows students to build interest about and gain entry into an upcoming task.
The purpose of this Warm-up is to elicit observations about squares that tile a region, which will be useful when students think about the meaning of “area” later in the lesson. While students may notice and wonder many things about these images, the important discussion points are observations about equal-size squares covering a region completely without gaps or overlaps.
When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language that 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
Student Task
What do you notice? What do you wonder?
Sample Response
Students may notice:
- A, B, C, and D are all the same shape.
- All four drawings are filled with squares or parts of squares on the inside.
- A, C, and D are filled with squares that are the same size. The squares in B are of two different sizes.
- In C, there are some gaps between the squares, and some squares overlap.
- There are many more squares in D compared to the other figures.
Students may wonder:
- Why are three squares in C rotated to follow the direction of the slanted side?
- Why is B filled with different-size squares?
- What is the size of each large square and each small square?
- Can the partial squares in A, B, and D be put together to make whole squares?
Activity Synthesis (Teacher Notes)
Ask students to share the things they noticed and wondered. Record and display their responses for all to see, without editing or commentary. If possible, record the relevant reasoning on or near the images. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
If the gaps and overlaps in C don’t come up during the conversation, ask students to discuss this idea.
Math Community
Display the class Math Community Chart for all to see and explain that the listed “doing math” actions come from the sticky notes students wrote in the first exercise. Give students 1 minute to review the chart. Then invite students to identify something on the chart they agree with and hope for the class or something they feel is missing from the chart and would like to add. Record any additions on the chart. Tell students that the chart will continue to grow and that they can suggest other additions that they think of throughout today’s lesson during the Cool-down.
Standards
- 3.MD.5.b·A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.
- 3.MD.C.5.b·A plane figure which can be covered without gaps or overlaps by <span class="math">\(n\)</span> unit squares is said to have an area of <span class="math">\(n\)</span> square units.
- 6.G.A·Solve real-world and mathematical problems involving area, surface area, and volume.
- 6.G.A·Solve real-world and mathematical problems involving area, surface area, and volume.
10 min
20 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Finding Area by Decomposing and Rearranging
5 min
Narrative
This is the first Notice and Wonder activity in this course. Students are shown four drawings and asked: “What do you notice? What do you wonder?”
Students are given time to write down what they notice and wonder about the images and then time to share their thoughts. Their responses are recorded for all to see. Often, the goal is to elicit observations and curiosities about a mathematical idea that students are about to explore. Pondering the two open questions allows students to build interest about and gain entry into an upcoming task.
The purpose of this Warm-up is to elicit observations about squares that tile a region, which will be useful when students think about the meaning of “area” later in the lesson. While students may notice and wonder many things about these images, the important discussion points are observations about equal-size squares covering a region completely without gaps or overlaps.
When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language that 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
Student Task
What do you notice? What do you wonder?
Sample Response
Students may notice:
- A, B, C, and D are all the same shape.
- All four drawings are filled with squares or parts of squares on the inside.
- A, C, and D are filled with squares that are the same size. The squares in B are of two different sizes.
- In C, there are some gaps between the squares, and some squares overlap.
- There are many more squares in D compared to the other figures.
Students may wonder:
- Why are three squares in C rotated to follow the direction of the slanted side?
- Why is B filled with different-size squares?
- What is the size of each large square and each small square?
- Can the partial squares in A, B, and D be put together to make whole squares?
Activity Synthesis (Teacher Notes)
Ask students to share the things they noticed and wondered. Record and display their responses for all to see, without editing or commentary. If possible, record the relevant reasoning on or near the images. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
If the gaps and overlaps in C don’t come up during the conversation, ask students to discuss this idea.
Math Community
Display the class Math Community Chart for all to see and explain that the listed “doing math” actions come from the sticky notes students wrote in the first exercise. Give students 1 minute to review the chart. Then invite students to identify something on the chart they agree with and hope for the class or something they feel is missing from the chart and would like to add. Record any additions on the chart. Tell students that the chart will continue to grow and that they can suggest other additions that they think of throughout today’s lesson during the Cool-down.
Standards
- 3.MD.5.b·A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.
- 3.MD.C.5.b·A plane figure which can be covered without gaps or overlaps by <span class="math">\(n\)</span> unit squares is said to have an area of <span class="math">\(n\)</span> square units.
- 6.G.A·Solve real-world and mathematical problems involving area, surface area, and volume.
- 6.G.A·Solve real-world and mathematical problems involving area, surface area, and volume.