Aligns with the objective: " Students will apply their previous knowledge of area and perimeter to a real-world situation in order to develop a conceptual understanding of when and why we use these concepts"

Purpose: Before students can apply Area and Perimeter and access situational cognition, they must refresh their definitions.

Discussion: Ask class to refresh their memories, “We know the definition and formulas for Area and Perimeter”. Encourage students to look definitions up if they can’t remember these terms. Show/Post quick 1 minute video to connect the idea of tiles with area and perimeter. 

An Example of a Student Update:

~Area: The amount of space inside the boundary of a flat (2-dimensional) object.

Example: I used Area to figure out how much wallpaper I needed for the area of my wall.

~Perimeter: The length or distance around a shape.

Example: I used Perimeter to put up a fence along the boundaries of my yard.

Rubric: Use the following Rubric along with answer keys through out the learning module to grade and assess student work.

Rubric

Aligns with the objective: " Students will apply their previous knowledge of area and perimeter to a real-world situation in order to develop a conceptual understanding of when and why we use these concepts"

CCSS.MATH.CONTENT.HSG.MG.A.1
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).*

CCSS.MATH.CONTENT.HSG.MG.A.2
Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).*

Purpose: Now that students have thought of their own examples of why and when we use Area, provide them with a real-life situation. From personal teaching experience, motivation is increased in students when they have an answer to "when am I going to use this"? Researchers also agree with this method and, according to the social cognitive perspective, "students’ motivation is relatively situation or context specific (see Pintrich et al., 1993)". Therefore, this problem provides students with a context and real life materials. Encouraging students to ask more questions before starting the problem, allows them to critically think about the question being asked and all of the data they will need to collect before starting. This prepares them for the steps taken when problem-solving. 

Discussion: If teaching in person or over skype, you can pass around the tiles or show them on the screen. Otherwise, post the problem clearly for the students to see, and clear photos of the tiles you selected. Encourage critical thinking steps and questioning before starting to problem solve by creating a comment section asking them to think of more data they need in order to solve the problem. Also, ask them to create an update looking for sources that might help them solve this problem.

Examples of student questions to look for:

What are the dimensions of the tile?
What units are we measuring in?
What is the area of the tiles?
What is the perimeter of the tiles?
What are the dimensions of your bathroom?
How many tiles of each we would need?
What are the total costs for each tile?
Are you going to have to cut edges?
How do you tile a floor?

Aligns with the objective: " Students will apply their previous knowledge of area and perimeter to a real-world situation in order to develop a conceptual understanding of when and why we use these concepts"

CCSS.MATH.CONTENT.HSG.MG.A.3
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).*

Purpose: Providing students with answers to their questions on the post allows them to feel apart of the process. You are using scaffolding to guide students while encouraging critical thinking and problem solving skills. If all of this information was provided to them at the beginning they would miss involvement in questioning, critical thinking, and problem-solving. Students now have all of the information they need to fully solve the problem, and they most-likely have confidence starting the problem because they have given time to absorb the context and information. 

Discussion: "Now that you have had time to think about the problem, and you have all the information you need. Solve the problem showing all your steps". Encourage students to write down everything they are thinking throughout their problem-solving process. Ask students to use the worksheets provided or follow a similar outline so that their peers can easily analyze their work.

Common Mistakes to look for: Students might miss the conversion from feet to inches. Students might calculate using 50 or 20 instead of $0.50 and $0.20. Students will analyze eachothers work to look for these mistakes and peer review.

Key: This is an example of what students work should look like on their update. Their update might include additional sources or work.

Aligns with the objective: " Students will apply their previous knowledge of area and perimeter to a real-world situation in order to develop a conceptual understanding of when and why we use these concepts"

CCSS.MATH.CONTENT.HSG.MG.A.2
Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).*

CCSS.MATH.CONTENT.HSG.MG.A.3
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).*

Purpose: Problem 1 focused on Area, but now this problem relates to perimeter in a similar context. At this point students are innvolved in the situation of re-decorating a bathroom. However, they have learned that the tile does not effect the perimeter. This problem gives them an example of when we use perimeter in relation to this situation. It provides students with physical context that perimeter is just the distance around an object. However, it challenges students to think about how an open area (such as a door) effects the perimeter and will need to be subtracted from the problem. 

Discussion: Provide students with adequate information and a photograph of what baseboarding is. Ask students is baseboarding would effect the area or perimeter of my bathroom? Get students involved in calculating the perimeter of the bathroom for the baseboard. Challenge the students by providing them with the obstacle of the door. 

Key: This is an example of what students work should look like on their update. Their update might include additional sources or work.

Aligns with the objective: "Students will be provided with a physical context that explains where the area of a triangle formula is derived from."

CCSS.MATH.CONTENT.HSG.MG.A.1
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).*

CCSS.MATH.CONTENT.HSG.MG.A.3
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).*

Purpose: This problem addresses where the area of a triangle is derived from. The focus of these problems has been on the Area and Perimeter of a rectangle and square, but it is important to discuss triangles, as well. Providing students with this physical picture and context gives them a better understanding of why the Area formula for a triangle is 1/2 the length times width. Also it will bring up previous knowledge on pythagorean theorem and how students can find the third side for the perimeter.

Discussion: Provide students with a photograph of the tiles cut diagonally. Ask students to reason through the effect this has on the tiles area and perimeter. Encourage students to draw on previous theorems they have learned in Geometry to solve for the new perimeter.

Key: This is an example of what students work should look like on their update. Their update might include additional sources or work.

(detailed work)

Update answers: We get triangles when we cut the tiles in half and this effects the area by half. The conclusion we can draw is that the Area of a Triangle formula is: 1/2 x base x length. This effects the perimeter because now there are only three sides. To find the third side we can use pythagorean theorem. I know the formula of pythagorean theorem and backed it up with the following link below. I also know that I can apply the pythagorean theorem to find the third side because you need a right triangle and both of these are right triangles. I know this based upon my previous knowledge of squares and rectangles and how all four angles are 90 degree angles (or right angles). 

Example of Source backing up my information: 

(Khan Academy, June 20, 2010). Pythagorean theorem 2 | Right triangles and trigonometry | Geometry | Khan Academy. https://youtu.be/O64YFlX1_aI

Aligns with the Objective:  Students will be able to apply the area and perimeter formulas for squares, rectangles, and triangles.

CCSS.MATH.CONTENT.HSG.MG.A.3
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).*

Purpose/Theory: This part of the lesson removes the context and situational learning to assess if students can apply the knowledge they gained from the three problems. This lesson is based around situated cognition, but now we are changing switching to applied knowledge with think-pair-share. Situational cognition should have provided an experience for the students that will allow them to understand the formulas and apply them with understanding. They started with three more challenging problems and are going to first conclude what they learned, and then apply it to simpler math problems. Students should still be able to solve the problems with the context removed, and it should be easier for them. They no longer have to critically think or problem-solve, but now just apply , and reflect on what they have learned. Then you will implement think-pair-share, by having students complete a peer review project on this update. This will allow them to solidify their conclusions and assess each others' work, providing them with feedback on their conclusions.  

Discussion: Provide students with a table to organize their conclusions from the three problems. Encourage students to apply their conclusions to simpler math problems, now removing the context. However, they will remeber the context that was initiated in these problems and they might still see these problems as tiles. Encourage students to reflect on their learning, apply it, and think-pair-share. Once they have provided feedback to eachother they will submit a final copy to you for grading based on the rubric below. Finally you will ask the students to complete a google form as a survey.

Survey: Provide the following google form survey to students to collect data and do your own reflecting as a teacher on what went well in the lessons and what can be improved upon in the future. There are no correct answers to this, just a collection of data. https://forms.gle/NfxZPPC9vzLfgYaq7

Key: This is an example of what students final work should look like on their update. The answer key for the problems are also provided to help assess students conclusions, and if they can apply what they learned.

Conclusions for the teacher: Use the Rubric and keys to grade students on each update, comment, and math work. Students are peer reviewing throughout the module so this will help with grading.

Purpose: Flip the role onto the students and have them design their own project with their own problems. This allows them to explore something they are interested and apply situational cognition. Students can peer review eachothers work and see the range of ways Area and Perimeter can be applied in the real-world. This also works for any applicable math and does not just have to be on a lesson for Area and Perimeter.

Tips for Teacher: If you do not have enough time for students to complete the work, just have them complete one more update with an extension to problem 2.