Middle School Mathematics UDL Instructional Unit- Lesson 3

Objectives

• Students will solve real-life and mathematical problems involving area, surface area, and volume.

Essential Questions

• How do we find the surface area of a rectangular prism?
• Where is surface area used in real life, and why is it important?

Vocabulary

Area - the measure of the size of the surface of a two-dimensional figure; measured in square units	Edge - a line segment joining two adjacent vertices in a two-dimensional figure or a shared boundary between two faces of a three-dimensional figure
Face - a flat surface of a three dimensional figure	Height - the perpendicular distance of a two- or three- dimensional shape from the base to the opposite vertex
Length - a measurement of the distance from one point to another	Net - a flattened three-dimensional figure that can be turned into a solid by folding it
Rectangle - a quadrilateral (four-sided figure) with four right angles	Rectangular prism - a three-dimensional figure of which all faces are rectangles
Surface area - the total area of all the faces or surfaces of a three-dimensional figure	Vertices - plural for vertex, the common point where two sides or edges meet
Width - the minimum distance between parallel lines of a figure

Materials:

Area formula chart including surface area formula

Graph paper

Introduction to an activity table

Ruler

Scissors

Sticky notes

Tape

Various rectangular prisms (e.g., cereal boxes, tissue boxes)

Worksheet

• Lesson 3 Practice
• Lesson 3 Exit Assessment

Introduction

Activate Previous Knowledge

1. Review with students how they have been finding the area of two-dimensional figures.
1. Ask students – Is a box a two-dimensional figure?
2. Lead a discussion about how two-dimensional figures and three-dimensional figures differ.
3. Have students locate two-dimensional figures and three-dimensional figures in the classroom.
2. Give each pair of students the net of a rectangular prism and some tape.
1. Have the students explore/problem solve how to construct a rectangular prism.
2. Once the rectangular prisms are constructed, discuss what a face is, what an edge is, and what vertices are.
3. Have the students complete a table that includes the following categories for their rectangular prisms.

Shape	# of faces	# of vertices	# of edges

4. After tables are filled out, discuss what the rectangular prisms have in common.
5. Ask students - Will different rectangular prisms have a different number of faces, vertices, or edges?

Multiple means of representation: Use models and/or drawings during large group instruction. Allow students to have a copy of a drawing or model at their desks.

Multiple means of expression: Provide a list of definitions to use in completing the table or provide options for using manipulatives and/or computer models.

Multiple means of engagement: Allow students to use paper/pencil, manipulatives, computer, etc.

Additional Considerations for Emerging Readers and Emerging Communicators

1. Provide a small box and a cutout or drawn figure (outlined in glue, etc., colored, or textured as necessary) the same size as the length and width of the box.
2. Provide concrete representations of several common shapes (rectangle/square, sphere, cylinder, pyramid, etc.) and cutout or drawn figures (outlined in glue, etc., colored, or textured as necessary) that correspond in exact size and shape.
3. Provide digital representations of 2-D and 3-D shapes that the student can manipulate on the screen (using AT). Add sound for interest.
4. Have the student use the shapes (2-D and 3-D) provided in 1. b. to find and categorize (by 2-D or 3-D, not by shape) figures in the environment.
• Record shapes by taking digital photos (using AT).
1. Provide net of rectangular prism with each face
• numbered
• colored differently
• textured differently
2. Provide a 3-D shape exactly corresponding to the net so the student has a more solid surface to construct the net around.
3. Have the student put a colored dot on each face (numbered sequentially from 1-6 if necessary).
• Have the student use a marker to highlight each edge.
• Use a different color marker or dots marked with "V"s to indicate the vertices.
• Have the student put a numbered face icon on each face.
• outline each edge with glue, yarn, etc.
• indicate each vertex with a colored or textured dot or sticker.
4. Provide table with symbol-based text.
5. Provide table with visual/tactile cues used on the actual net.
• put face icon on the face column of the table
• put yarn on the edge column
• put textured dot on the vertices column
6. Have the student fill out table after counting each characteristic (using the "+1" function on a calculator if necessary) by:
• writing the numbers;
• cutting and pasting pre-printed numbers;
• verbally stating the number, which the partner scribes;
• selecting the answer from two to several options) completing the table digitally (using AT), etc.
7. Let the student choose characteristics of rectangular prisms on the table to answer the question (using his /her preferred mode of communication).
8. Provide nets (adapted the same way as the first net was above) of several other, differently sized rectangular prisms that the student can use to complete similarly adapted tables so he/she can develop his/her answer.

Establish Goals/objectives for the Lesson

Inform students that they will solve real-life and mathematical problems involving area, surface area, and volume. Explain that they will:

1. Describe the characteristics of a rectangular prism.
2. Construct the net of a rectangular prism.
3. Determine the surface area of a rectangular prism from a net.
4. Determine the surface area of a rectangular prism using a formula.
5. Describe when surface area is used and why it is important in real life.

Multiple means of representation: Along with posting lesson objectives in the classroom, provide individual copies for students.

Multiple means of expression: Allow students to record lesson objectives in different formats: mathematics journal, computer, graphic organizers (premade or original), etc.

Multiple means of engagement: Brainstorm ideas of how and when these skills might be relevant

Additional Considerations for Emerging Readers and Emerging Communicators

1. Provide rectangular prisms and tables the student used in the first part of the Introduction.
2. Provide a net of a rectangular prism.
3. same as 2.
4. Provide a formula using symbols and textures.
5. Provide examples of situations where determining surface area is important to "me".

• Examples may be captioned picture representations of situations
• Examples may be digital animations of situations
• Examples may be tactile representations with auditory captions

Body

Form student groups and provide each group with a rectangular prism/box of the same size. Tell the students that they will be turning the rectangular prism/box into a net like the paper nets they just folded in the previous activity. Students may work as a whole group with the teacher or in small groups. Note: There is more than one net for a rectangular prism.

1. Make a sketch of the rectangular prism/box.
1. Ask the students: For the rectangular prism/box, what are the length, width, and height?
2. Measure and record each of these dimensions.
2. Label the faces of the rectangular prism/box before flattening it to make a net.
1. Use sticky notes labeled: Front Face, Back Face, Left Side Face, Right Side Face, Top Face, and Bottom Face.
2. Work together to label the rectangular prism's/box's faces.
3. Explain that by labeling the faces, it will be easier to visualize how the net makes up the rectangular prism/box.
3. Make the net.
1. Cut the rectangular prism/box so that each face of the rectangular prism/box is still connected.
2. Brainstorm with the students about where to cut before making cuts.
3. Once the cuts are made, show the students the flattened rectangular prism/box which is now a net.
4. Have the students draw the net on paper or graph paper.
5. Ask the students: Can you see how the net drawing could be cut out and taped together to form the rectangular prism/box?
6. Have the students include the fold lines on the sketch.
7. Have the students label the faces on the net sketch.
8. Measure the lengths and widths of the rectangles of the flattened rectangular prism/box and have the students record these on the net sketch.
1. See pages in Lesson 3 Resources.
4. Discuss with students how they have been finding the area of two-dimensional figures. Ask the students: Since the net is a two-dimensional figure, what is its area? Students may use an area chart to find the area.

Shape/Face	Area Formula	Formula with values	Area (unit2)

Total Area =

5. Explain to the students that when a figure is three-dimensional, surface area is found. Ask the students: How would the surface area of the rectangular prism/box compare to the area of the net?
6. Instruct students that there is a formula for finding the surface area of a rectangular prism: Surface Area = $2 (lw + hl + hw)$.
1. Ask the students: From the measurements made on the rectangular prism/box, what is the surface area using the formula?
2. From the net, find the area of each face using the chart.
1. See pages in Lesson 3 Resources

Prism Surface	Face Area
Top Face	l x w
Bottom Face	l x w
Front Face	h x l
Back Face	h x l
Left Side Face	h x w
Right Side Face	h x w

7. If we know the area of each face, how can we find the surface area of the figure? (Add together all of the faces.)
8. How does the surface area found by the formula compare to the surface area found by adding together the areas of the faces? (They are the same.)
9. Since they are the same, let's see how the formula was developed.
1. First, horizontally list the faces and put the area formula below.
2. $Top + Bottom + Front + Back + Left + Right = Surface\ area$
3. $lw + lw + hl + hl + hw + hw = Surface\ area$
10. Use algebra to simplify:
1. $2lw + 2hl + 2hw = Surface\ area$ (Combine like terms.)
2. $2(lw + hl + hw) = Surface\ area$(Factor out a 2.)

Multiple means of representation: Use models and/or drawings during large group instruction. Allow students to have a copy of a drawing or model at their desks. Provide a list of area formulas.

Multiple means of expression: Allow students to solve problems using formulas and/or models and record information into tables using various formats (paper and pencil, computer, etc.) Allow students to use a reference of formulas.

Multiple means of engagement: Allow students to use paper/pencil, manipulatives, models, computer, etc.

Additional Considerations for Emerging Readers and Emerging Communicators

1. Allow the student to work directly from the rectangular prism rather than the sketch.
1. Mark the length, width, and height directly on the prism.
2. Allow the student to use the same supports for measuring and documenting measurements as provided in Lesson 1- Body.
2. Add extra visual (symbols or color) or tactile cues (texture, etc.) to the sticky note labels.
3. Outline where the cuts should be made.
1. Allow the student to use adaptive or electric scissors.
2. Have the student trace the flattened prism/net onto graph paper graph paper enlarged to 1-inch squares.
3. Have the student label the faces on the net using the same supports used in labeling the faces of the prism.
4. Have the student determine the area of each face, by measuring and using the formula or by counting using a grid and/or manipulatives, and record in the table. Allow the student to use whatever supports he/she has been using for measuring, calculating, and recording.
5. Allow the student to use either formula or manipulatives to find area of the entire net (surface area of rectangular prism).
1. Have the student compare the net to the folded out prism by laying one on top of the other.
• Have the student compare the net to the rectangular prism by reassembling the prism and then folding the net around it. Hook-and-loop tape could be used instead of sticky tape. (Pre-teach or review the concept of "same" if necessary using systematic instruction techniques.)
2. Allow the student to use a calculator (handheld, digital, big button, talking, etc.) or manipulatives.
• Show the two calculations and ask if they are the same. (Pre-teach or review the concept of "same" if necessary using systematic instruction techniques.)
6. Provide choices in the student's preferred communication mode.
7. Model how the top and bottom are represented by length and width, two sides are represented by height and width, and two sides are represented by height and length.
8. Provide a template for the formula.
1. Allow the student to use a calculator.
2. Student may use manipulatives instead of the formula.

Practice

1. Have the students work in pairs on a variety of problems using different rectangular prisms and nets.
1. Give each student Lesson 3 Practice worksheet.
2. Once the groups have completed the problems on the worksheet, ask one student from each pair to discuss the results.
2. Demonstrate to students that there is more than one possible way to create a net for a rectangular prism, but the surface area remains the same.

Multiple means of representation: Provide students with a copy of the word problem, a template for the formulas for area and a table to find composed areas. Have drawings and manipulatives available for the students to use.

Multiple means of expression: Provide a list for formulas to determine area or provide options for using manipulatives and/or computer models, etc. Students may draw or use manipulatives or use the computer to model solutions.

Multiple means of engagement: Ensure that each student is actively involved in his/her pair. Allow students to use paper/pencil, manipulatives, computer, etc.

Additional Considerations for Emerging Readers and Emerging Communicators If the student is struggling with the concept of surface area after the completing the introduction and body of this lesson, it might be helpful to provide additional instruction using the Lesson 3 Concept Reinforcement Activity (CRA) prior to prior to this practice activity of Lesson 3.

1. Provide all supports used in Lessons 1-3 up to this point including those for measuring, calculating, and recording.
1. Provide worksheet with supports used in Lesson 3 – Introduction.
1. Put formula with correctly measured numbers on each face.
2. Provide rectangular prisms and nets representing the problems.
3. Provide digital representations and access for manipulation using AT.
2. Allow the student to use all supports and materials provided to contribute to discussion.
3. Allow the student to use his/her preferred mode of communication.
4. Preplan one contribution to the discussion.
2. Provide one rectangular prism with several possible nets that can be folded around the prism to illustrate the concept.

Closure

Revisit/Review Lesson and Objectives

Remind students of the lesson's objectives and expected student outcomes. Have students discuss which of the lesson's problems addressed the objectives below and what they learned from their experiences. Students will solve real life and mathematical problems involving area, surface area, and volume.

1. Describe the characteristics of a rectangular prism.
2. Construct the net of a rectangular prism.
3. Determine the surface area of a rectangular prism from a net.
4. Determine the surface area of a rectangular prism using a formula.
5. Describe when surface area is used and why it is important in real life.

Ask students: How are area and surface area connected? Brainstorm with students when they might use area and surface area in everyday life.

Multiple means of representation: Along with posting lesson objectives in the classroom, students may refer to their individual copies.

Multiple means of expression: Students can share what they have learned in different formats through writing, drawing, creative expression, discussion, etc.

Multiple means of engagement: Brainstorm ideas of how and when these skills might be relevant to "me".

Additional Considerations for Emerging Readers and Emerging Communicators

1. As review is conducted, provide the student with all previously used materials, strategies, and supports.
2. Preplan information for the student to contribute to the discussion using his/her preferred mode of communication.

Exit Assessment

Give the students a new word problem to solve that includes finding surface area. Have students work independently to find the surface areas of the two boxes and solve for the situation. Students should justify their answer.

• See Lesson 3 Exit Assessment Worksheet for drawings of the wooden boxes.

Above are two sample wooden boxes they have for sale in a shop. You want to buy one to cover with carpet to make your cat a scratching post. However, you only have 40 square feet of carpet left over from having new carpet put in the house. Do you have enough carpet to cover either box or only one? Support your decision by showing your work.

Multiple means of representation: Provide students with a copy of the word problem and a template of the formulas for area. Have drawings and manipulatives available for students to use.

Multiple means of expression: Allow the students to solve the problem using formulas and/or models.

Multiple means of engagement: Create situations that include areas of interests to the students.

Additional Considerations for Emerging Readers and Emerging Communicators Provide all supports previously used in Lessons 1-3.

• Summarize the word problem.
• Highlight important information in the word problem.
• Provide a number line representing square feet with the acceptable range of square feet of carpet highlighted.

As this is now assessment as opposed to all previous activities in this lesson, the student's first independent attempts solving the two problems and answering the question should be observed and recorded for accuracy.

Return to Lesson 2 Proceed to Lesson 4