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Creating a Common Space


Grade Level: 8-10

Description: In this project, students learn about about scale, surface area, and volume, and their ideas about community building to design a common space to encourage student interaction.

How it Works: Students perform investigations that lead to formulas for area of circles and polygons, surface area, and volume of cylinder and prisms. They use maps to estimate distances, discuss error in measurement, and create scale drawings from measurements of actual objects.

Final Project/Product: The final product of the unit is a presentation that takes the format of a share fair, where students post their work and peers evaluate and make comments as they view the work. It includes a scaled drawing, the cost for materials, and a letter explaining benefits of the design and the maximum number of students allowed in the space at the same time.

Overall Value: This project gives students an opportunity to use math in context as they prepare for the Integrated Algebra Regents. Students are presented with a “real” issue and have to create a product that may help resolve that issue. The creation of the product involves application of content skills, research, mathematical reasoning, communication of ideas, and awareness of community building.

English Language Learners: The strategies used within this unit to support the needs of English Language Learners are: collaboration, visuals, explorations, explicit vocabulary instruction within context, and informal and formal writing.

Tips for the Teacher: If students don’t usually work in groups, it may be worthwhile spending time on community building activities, so that they become comfortable with each other and learn how to work together.


 Standards Addressed
Find the area and/or perimeter of figures composed of polygons and circles or sectors of a circle.
  Grade: 9-12 Subject: Integrated Algebra
Use formulas to calculate volume and surface area of rectangular solids and cylinders.
  Grade: 9 -12 Subject: Integrated Algebra
Calculate the relative error in measuring square and cubic units, when there is an error in the linear measure.
  Grade: 9-12 Subject: Integrated Algebra
Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid.
  Grade: 9-12 Subject: Integrated Algebra
Students locate and use school and public library resources for information and research. They read and follow written, complex directions and procedures to solve problems and accomplish tasks, and skim texts to gain an overall impression and scan texts for particular information.
  Grade: 9-12 Subject: English Language Arts

Day 1: Area and Error in Measurement
Students find the area and perimeter of polygons.
Students solve problems involving area.
Students explain their process for solving problems and discuss different ways of solving problems.
Students convert from one unit of measurement to another.
Students find error of measurement.
Tape measure or trundle wheels
Handouts for each student/group of students
Overhead projector and transparency copies of all handouts
scissors, chart paper, markers
Area, square unit, base height, dimensions, perimeter, substitute
Procedure 1
Tell students that they are going to explore the idea of area.
a. Ask students if they are familiar with the word and develop a definition with their input. Draw a picture to show the area of a shape.
b. Distribute handout titled “Area of a Rectangle” to students. Allow 10 minutes to complete. Circulate to ensure students understand task.
c. Distribute handout titled “Problem Solving 1”. Allow 30 minutes to complete. Ask students to share answers. Stress the fact that different methods presented are all correct.
Procedure 2
Introduce students to new project. (Handout titled “Creating a Common Space.)
a. Distribute measuring tapes of trundle wheels.
b. Explain to students that they will work in groups to measure field. Distribute handout titled “Project Site.”
c. Ask students to work in groups to measure the field and complete handout.
Procedure 3
As a Do Now, ask students how to find the area of a rectangle. Also ask if this works for all rectangles.
a. Draw a rectangle on the board. Ask students for the area of the entire figure. Divide the rectangle into two triangles. Ask for the area of one triangle. Ask how they can use the area of a rectangle to find the area of a triangle. Ask if this will work for all triangles.
b. Distribute handout titled “Investigation for Area of a triangle A” and scissors. Allow 15 minutes for students to complete their investigation.
c. Discuss findings, formula, and vocabulary (base and height of triangles that appear different). See handout 3b.
Procedure 4
Tell students that they will be solving a problem involving area and perimeter. Review area and perimeter.
a. Distribute handout titled,“Problem Solving 2”. Read scenario aloud to students.
b. Ask students to work on problems individually (5 minutes). Ask students to share within their groups (20 minutes) and prepare to present solutions to the class. Distribute chart paper and marker.
c. Ask groups to present solutions. Discuss solutions.
Journal Entries: Area of parallelograms, trapezoid, and triangle. Task: Explain how to find the area of the figures above. (You may use pictures to help with your explanation.) Handouts: Area 1, Area 2
Observation, homework

Day 2: Circumference and Area of a Circle
Students will explore the relationship between the circumference of a circle and its diameter.
Students will “discover” area of a circle.
Students will solve problems involving area and circumference.
Circluar objects (4 per group)
rulers and string (2 per group)
handouts for all students, overhead projector and transparency copies of all handouts
scissors, glue
circumference, diameter, radius, pi
Procedure 1
Distribute handout “Relationship between circumference and diameter."
a. Distribute circular objects, rulers, strings, and handouts to each student.
b. Review task and vocabulary words with students. Draw pictures and use objects to illustrate. Write the definition and draw pictures on the board.
c. Ask students to complete handout (20 minutes).
d. Ask every group to share measurements and ratio for one object. Record their responses on an overhead transparency. Ask groups to share conclusion. Formally define pi as an approximation of the ratio of the circumference of a circle to its diameter.
Procedure 2
As a Do Now, ask students to find the circumference given the radius of a circle and the area of a rectangle. They will need to review this in order to complete today’s activity. Allow 5 minutes to complete.
a. Review Do Now.
b. Tell students that they will complete an activity that will help them find the area of a circle.
c. Distribute handout titled “Area of a Circle.” Allow 20-25 minutes to complete.
d. Ask students to share their conclusion and formula for area of a circle. Write the formula for area on the board.
Journal Entry: Create and solve two problems where you have to apply your understanding of the circumference and area of a circle. Handout: Circumference and Area 1.
Observation, journal entry, Do Now

Day 3: Surface area of prisms and cylinders
Students will explore surface area of three-dimensional figures.
Students will create three-dimensional figures using nets.
Students will solve problems involving surface area.
construction paper, scissors, tape, rulers
two- and three-dimensional objects
empty boxes (1 per pair)
Surface area, net, face, edge, vertices, two dimensional, three dimensional
Procedure 1
Group objects based on dimension and ask students to write why they think you have arranged the objects in this manner. Ask them to share their responses. Hopefully, they mention two and three dimensions. If not, introduce and define the terms.
a. Tell students that today they will be finding the surface area of three-dimensional figures.
b. Distribute empty boxes and rulers to each group. Ask students to complete handout titled “Surface area of rectangular prism.” (Allow 20 minutes to complete.)
c. Ask each group to share their method and formula for finding the surface area. Ask them explain their methods. Listen for opportunities to introduce the words: net, edge, face, and vertex.
d. Write all formulas on the board, and check with the class for accuracy. Draw a rectangular prism and label with words: edge, face, and vertex.
Procedure 2
Distribute construction paper, ruler scissors, compass, and tape to students.
a. Distribute handout titled “Surface area of a cylinder.”
b. Review directions. As you review, ask students for the definition of net. Write definition on the board. If students don’t know, give them the definition. Use a rectangular prism (box) to show a net.
c. Allow 20-25 minutes for students to work on handout titled ”Surface area of a cylinder.”
d. Discuss the conclusion of the exploration: “How can we find the surface area of a cylinder? How do the dimensions of the rectangle relate to the circumference of the circles (top and bottom faces)?
Journal Entry: Surface Area of a Rectangular Prism. Task: Create and solve a problem where you have to find the surface area of a rectangular prism. Explain your steps. Journal Entry: Surface Area of a Cylinder. Task: Create and solve a problem where you have to find the surface area of a cylinder. Handouts: Surface Area 1.
Journal entry, observations

Day 4: Volume
Students will solve problems involving volume of cylinder and rectangular prisms.
Students will explore the relationship between area and volume.
Students will explore the relationship between radius and volume of a cylinder.
construction paper
tape, scissors
rice or beans
Procedure 1
Display handout titled ” Creating a Common Space” on the overhead. Remind students that they will be applying everything they have been studying to complete the project. Ask them to take out handout titled “Project Site.”
a. Ask them to discuss within groups the dimension of one of their pools. Allow 5-10 minutes to complete. Ask them to share; then ask them how much space the pool occupies. Use their responses to transition to the introduction of volume.
b. Introduce the word "volume". Show volume using a stack of paper, etc. Include that volume is represented in using cubic units.
c. Distribute cubes to students. Ask them to complete handout titled ”Volume of Rectangular Prisms.”
d. Ask students to share their work. Discuss the relationship between area and volume.
Volume, cubic units, rectangular prism, cylinder
Procedure 2
Draw a rectangular prism next to a cylinder on the board.
a. Review how to find the area of a rectangular prism on the board (area of the base and height).
b. Ask students to think about and write how they can use what they know about finding the volume of a rectangular prism to finding the volume of a cylinder. (Allow 5 minutes.) Ask students to share their responses. Use their responses to explain the formula for volume of a cylinder.
c. Take 2 sheets of construction paper and make 2 cylinders, one with a height of 8.5 inches and the other 11 inches. Ask students to make a prediction about which cylinder has the most volume. Record students’ predictions on the board.
d. Distribute handout titled “Exploring volume of a cylinder.” Ask groups to share conclusions. Discuss as a class.
Journal Entry: Volume. Task: Explain what the word “volume” means to you. Handouts: Surface area and Volume 1, Effects of linear measurement
Observation, journal entry, homework

Day 5: Designing a Common Space
Students will create a scale drawing of their design of a common space.
Students will apply mathematical reasoning and problem-solving skills.
Students will conduct research for cost of materials for their design.
Students will present their work to the class.
Rulers, trundle wheels
Graph paper, grid chart paper, pencils
Computers with Internet access
Maps with a scale (map of U.S. or transportation map)
scale, scaled drawing
Procedure 1
Distribute map and ask students to use the scale to approximate the distance between two points. Before they begin, define scale. Students may know about scales used to measure mass. Discuss different uses for the word. Ask them to share how they used the scale.
a. Tell them that the map is scaled drawing. Ask “What does that mean?” Use their response to write definition for scaled drawing.
b. Model to students how to create a scaled drawing of an object.
c. Distribute graph paper and rulers and ask students to create a scaled drawing of their desk. Ask students with different scales to share and explain their scales.
d. Distribute grid chart paper to students. Ask students to use their measurements from the field to create a scaled drawing. Each group member will have the same dimensions but may choose to use a different scale.
Procedure 2
Distribute handout titled “Designing a Common Space 1.”
a. Allow 20-30 minutes to complete.
b. Ask students to share their ideas. Record on board.
c. Distribute handout titled “Designing a Common Space 2.” Ask students to complete.
Procedure 3
Ask students to use the handout titled “Designing a Common Space 2” to complete their scale drawing. It may be necessary to take students outside so that they get a better sense of the size of smaller spaces within their design.
a. Circulate to ensure students understand task. Ask questions to assess understanding.
Procedure 4
Distribute handout titled “Cost of Materials.”
a. Assign each group two computers.
b. Ask students to complete the handout.
Procedure 5
Tell students that they have to write a letter to Ms. Alex to convince her to choose their design.
a. Distribute Handout titled “Writing Your Letter.” Allow students time to complete letter.
b. Ask students to peer-edit in class and revise their letters at home.
c. Have students display their work around the classroom and have group members evaluate their work and provide feedback. Use handout titled “Rubric: Creating a Common Space.”
d. Ask students to complete handout titled “Reflection” for homework.
Edit letters, Reflection handout
Scaled drawing, letter, cost of materials

Vadewatie Ramsuchit


International High School at Prospect Heights
883 Classon Avenue
Brooklyn, NY 11225

Vadewatie is a math teacher and mentor at the International High School at Prospect Heights. Since the school’s inception, she has been working with her fellow teachers to create and adapt the curriculum to make it more accessible for English Language Learners. She strives to create curricular units that are in context and supportive of students’ language development skills and their ability to think mathematically. Her goal has been to create a learning environment where students feel supported and challenged; where they work to help each other improve.

Important documents for this lesson plan.

Creating a Common Space.doc
Designing a Common Space 1_ 2.doc
Rubric_Creating a Common space.doc


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