Mission 11 · Unit 5

Surface Area

6.G.A.4 · Unit 5
Today's objective: Use nets to find the surface area of 3-D figures.
Need a hint?
Re-read the problem and underline the numbers and the question. Pick one representation (model, table, or equation), show your steps, and check that your answer makes sense for the situation.

The art teacher is having students build gift boxes for the school fundraiser. Each box is a rectangular prism that is 10 inches long, 6 inches wide, and 4 inches tall. Students will cover every face of the box with decorative wrapping paper. Your team must figure out the total surface area so the art teacher can cut the right amount of paper for each box. She has a roll of paper that is 2,000 square inches. How many complete boxes can be wrapped with one roll?

3-D Box Front Top Side 10 in 4 in 6 in Unfolded Net Left 6x4 Front 10x4 Right 6x4 Back 10x4 Top 10x6 Bottom 10x6 Surface Area = 2(lw) + 2(lh) + 2(wh) = 2(10x6) + 2(10x4) + 2(6x4) = 120 + 80 + 48 = 248 in²

Investigation

The Problem: A gift box is 10 inches long, 6 inches wide, and 4 inches tall. (A) Draw and label the net of this rectangular prism. (B) Find the area of each face. (C) Calculate the total surface area. (D) How many complete boxes can be wrapped with a 2,000 in² roll of paper?

Visual Model: The Six Faces

Face Dimensions Area Formula Area x2 Top & Bottom 10 in x 6 in l x w = 10 x 6 60 in² 120 in² Front & Back 10 in x 4 in l x h = 10 x 4 40 in² 80 in² Left & Right 6 in x 4 in w x h = 6 x 4 24 in² 48 in² Total Surface Area = 120 + 80 + 48 = 248 in² 2,000 in² ÷ 248 in² per box = 8 complete boxes (with 16 in² left over)

Step-by-Step Investigation Guide

  1. Identify the 3 dimensions. A rectangular prism has length (l), width (w), and height (h). For this box: l = 10 in, w = 6 in, h = 4 in. Write these down. How many faces does a rectangular prism have? How many pairs of matching faces?
  2. Draw the net. Imagine cutting the box along its edges and unfolding it flat. You get a cross-shaped pattern with 6 rectangles. Label each face: Top, Bottom, Front, Back, Left, Right. Which faces are the same size? Why do they come in pairs?
  3. Find the area of each face. Top/Bottom: 10 x 6 = 60 in² each. Front/Back: 10 x 4 = 40 in² each. Left/Right: 6 x 4 = 24 in² each. Why do opposite faces always have the same area?
  4. Double each pair. Since there are 2 of each face: 2(60) = 120, 2(40) = 80, 2(24) = 48. Why do we multiply by 2 instead of just adding the three faces?
  5. Add all face areas. Total surface area = 120 + 80 + 48 = 248 in². This is how much paper covers one box completely. Does this formula work: SA = 2lw + 2lh + 2wh? Try plugging in the numbers to check.
  6. Solve the paper roll problem. Divide 2,000 by 248. You get 8 full boxes with some paper left over. Calculate the leftover amount. Why can you not wrap a 9th box? How much paper would you be short?

Language Support: Key Vocabulary

surface area — the total area of all the faces (outside surfaces) of a 3-D shape
face — one flat side of a 3-D shape; a rectangular prism has 6 faces
net — a flat pattern that can be folded up into a 3-D shape
rectangular prism — a 3-D box shape where every face is a rectangle
edge — the line where two faces meet on a 3-D shape
dimension — a measurement like length, width, or height
square inches (in²) — the unit for measuring area; a square that is 1 inch on each side
opposite faces — two faces across from each other; they are always the same size

Sentence Frames

"The net has _____ faces. The top and bottom each measure _____ by _____, so their area is _____."

"The total surface area is _____ in² because I added all _____ face areas together."

"With 2,000 in² of paper, we can wrap _____ boxes because 2,000 divided by _____ equals _____."

Multiple Representations

Net Drawing

Cut and fold a paper net. Label each face with its dimensions and area. Fold it to see the box. This physical model makes every face visible.

Best for: understanding what "surface area" covers and why nets work

Area Table

Make a table with columns: Face Pair | Dimensions | One Face Area | x2 Total. Fill in 3 rows (top/bottom, front/back, left/right). Sum the last column.

Best for: organizing the calculation and avoiding missed faces

Formula

SA = 2(lw) + 2(lh) + 2(wh)
SA = 2(10)(6) + 2(10)(4) + 2(6)(4)
SA = 120 + 80 + 48 = 248 in²

Best for: quick calculation once you understand the concept

Team Roles

Facilitator Keeps the group on track, watches the timer, makes sure everyone speaks. Mission 11 task: Ensure the team draws a net, calculates all 6 face areas, AND answers how many boxes can be wrapped.
Model Builder Draws the net, labels every face, and creates the area table. Mission 11 task: Draw a clearly labeled net of the 10x6x4 box. Color-code matching face pairs.
Precision Checker Verifies every multiplication, checks that all 6 faces are counted, and confirms units. Mission 11 task: Check that the formula SA = 2lw + 2lh + 2wh gives the same answer as adding individual face areas. Verify 2,000 / 248.
Reporter Prepares the defense: claim, evidence, and one mistake the team caught. Mission 11 task: Explain the connection between the net and the surface area formula. Show how the paper roll division works.

Timed Lab Phases

Ready
Click a phase, then press Start.
03:00

Read the scenario out loud. Assign roles. Underline the 3 dimensions: 10 in, 6 in, 4 in.

  • How many faces does a box have? Can you name them?
  • Which faces are the same size? (Think about opposite sides.)
  • What does "net" mean? Imagine unfolding a cereal box.
Checkpoint: Every teammate can name all 6 faces and identify the 3 pairs.

Draw the net and label each face. Calculate the area of each face.

  • Draw a cross-shaped net with 6 rectangles.
  • Write dimensions on each rectangle: 10x6, 10x4, or 6x4.
  • Calculate the area inside each rectangle.
  • Color-code matching pairs (e.g., top/bottom same color).
Checkpoint: The net is drawn with all 6 face areas calculated.

Add all face areas for total surface area. Then divide the paper roll by the surface area.

  • SA = 2(60) + 2(40) + 2(24) = 120 + 80 + 48 = ?
  • Also check using the formula: SA = 2lw + 2lh + 2wh.
  • Divide: 2,000 / 248 = ? boxes. What is the remainder?
  • Could you wrap part of another box with the leftover paper?
Checkpoint: SA = 248 in², and the team knows how many boxes can be wrapped.

Prepare your presentation. Show the net, the calculation, and the paper roll answer.

  • State your claim: "The surface area is 248 in²."
  • Point to the net as evidence showing all 6 faces.
  • Explain how you know you did not miss any faces.
  • Share one mistake your team caught (forgot a face, wrong pair, etc.).
Checkpoint: Reporter can walk through net, formula, and answer in under 60 seconds.

Challenge Zone

Extension: The art teacher wants to add a ribbon around the middle of each box (the perimeter at height = 2 in). The ribbon goes around the long way: 10 + 6 + 10 + 6 = 32 inches per box. If ribbon costs $0.05 per inch, what is the ribbon cost for all 8 boxes?
What If? What if the box were a cube with 6-inch sides instead of 10x6x4? Would the surface area be more or less than 248 in²? Calculate it and compare. Which shape uses less paper per cubic inch of space inside?
Real-Life Connection: Think about wrapping a birthday present. You always need more paper than the surface area because of overlaps and folds. If you need 10% extra paper for overlaps, how much total paper do you need per box? How does this change how many boxes you can wrap?

Defense Preparation

Questions Your Team Must Answer

  1. How does the net connect to the surface area? Point to each face on your net.
  2. Why does a rectangular prism always have exactly 3 pairs of matching faces?
  3. What is the total surface area? Show the calculation.
  4. How many boxes can be wrapped with the paper roll? What happens to the leftover paper?
  5. If one dimension changed (say height becomes 5 inches), which faces change and which stay the same?

Sentence Starters for Your Defense

"The surface area is _____ in² because the box has _____ pairs of faces with areas of _____, _____, and _____."

"Our net shows all 6 faces. We verified by checking that 2(60) + 2(40) + 2(24) = _____."

"With 2,000 in² of paper, we can wrap _____ boxes because 2,000 / 248 = _____ with _____ left over."

"One mistake we caught was _____. We fixed it by _____."

Accuracy (4 pts) All face areas are correct. SA = 248 in². Paper roll division is correct.
Net (4 pts) Net is drawn with all 6 rectangles correctly sized and labeled with dimensions.
Reasoning (4 pts) Team explains why faces come in pairs and connects net to formula.
Communication (4 pts) Reporter uses vocabulary (surface area, net, face, edge, dimension) and teammates can explain.

Exit Product

Deliver: Gift Box Wrapping Report

Your team submits a one-page report that includes:

  • A labeled net of the 10 x 6 x 4 rectangular prism
  • The area of each face pair (top/bottom, front/back, left/right)
  • Total surface area using both adding faces and the formula
  • Division showing how many boxes can be wrapped from 2,000 in²
  • Explanation of what happens to leftover paper
  • A 3-sentence defense: claim, evidence, and reasonableness check

Self-Assessment

  • I can draw and label the net of a rectangular prism.
  • I found the area of all 6 faces and added them for surface area.
  • I can explain why opposite faces have equal area.
  • I used correct units (in²) throughout my work.

Student Work Space

Net Drawing (label all faces with dimensions and areas):

Face Area Table:

Total Surface Area Calculation:

Paper Roll Division:

Defense (Claim, Evidence, Reasonableness):