In plain English: 7.G.A.1 is the Common Core grade 7 math standard that asks students to solve problems with scale drawings of geometric figures, such as maps and floor plans. Students use the scale to find actual lengths and areas from a drawing, and they redraw a figure at a different scale. A key idea is that when every length is multiplied by a number, areas are multiplied by that number times itself.
Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
Common Core State Standards for Mathematics · Domain: Geometry (G) · Cluster: Draw, construct, and describe geometrical figures and describe the relationships between them. Also written as 7.G.1 · Official standard
A scale drawing is a drawing of a real object in which every length is the same fraction (or multiple) of the real length. Maps, floor plans and blueprints are scale drawings. The scale tells how drawing lengths and actual lengths (the real lengths) compare, for example 1 cm = 3 m on a floor plan. A scale can also be written as a ratio without units, such as 1 : 50, which means 1 cm on the drawing stands for 50 cm in real life.
In this lesson students use a scale three ways, as the standard asks: to find actual lengths from a drawing, to find actual areas from a drawing, and to redraw a figure at a different scale. The big idea for area is that each square on the drawing stands for a square in real life, so if lengths are multiplied by 3 (the length factor), areas are multiplied by 3 × 3 = 9 (the area factor). Students work on centimeter grid paper and measure real objects in the room.
Learning Objectives
By the end of this lesson, students will be able to:
Explain what the scale of a drawing means, written with units (1 cm = 3 m) or as a ratio (1 : 50)
Find actual lengths from a scale drawing, and drawing lengths from actual lengths
Find actual areas from a scale drawing and explain why the area factor is the length factor times itself
Find the scale of a drawing from one known length
Reproduce a scale drawing at a different scale on grid paper
Prior Knowledge Required
Students should already be comfortable with:
Using ratio tables and unit rates to solve problems 6.RP.A.3
Finding the area of rectangles, triangles and shapes made of rectangles 6.G.A.1
Converting units within one system, such as 1 m = 100 cm and 1 ft = 12 in 5.MD.A.1
Recognizing a proportional relationship, where one quantity is always the same number times the other 7.RP.A.2
Give each student a piece of grid paper and show this question:
Warm-Up Prompt
"A photo is 4 inches wide and 6 inches tall. A copy is made so that every side is twice as long. Is the copy twice as big? Sketch both on grid paper and count the squares."
The copy is 8 inches by 12 inches. The first photo covers 4 × 6 = 24 squares and the copy covers 8 × 12 = 96 squares, which is 4 times as many, not 2 times. Leave this surprise on the board; the lesson comes back to it when students find actual areas. Say that the copy is a scale copy: every length was multiplied by the same number, so the shape did not change.
Direct Instruction20 minutes
Define the words on the board. A scale drawing shows a real object with every length multiplied by the same number. The scale compares a drawing length with the actual length it stands for. The scale factor is the number you multiply drawing lengths by to get actual lengths, when both are in the same unit: at 1 : 50 it is 50. Then show the steps students use:
Read the scale: say it as a sentence, for example "every 1 cm on the plan stands for 1.5 m in the room."
Lengths: multiply each drawing length by the actual amount for 1 unit (or divide an actual length by it to go back to the drawing).
Areas: find the actual lengths first, then compute the area; or multiply the drawing area by the actual area of one drawing square.
New scale: find the actual lengths, then divide by the new scale to get the new drawing lengths.
Actual lengths from a floor plan
A classroom floor plan uses the scale 1 cm = 1.5 m. On the plan, the room is 6 cm long and 4.8 cm wide. How long and wide is the real room?
Equation: Length: 6 × 1.5 = 9 m. Width: 4.8 × 1.5 = 7.2 m. The room is 9 m by 7.2 m, a normal size for a classroom.
Actual area from the same plan
What is the actual floor area of the classroom in example 1?
Equation: Method 1: 9 × 7.2 = 64.8 m². Method 2: the plan area is 6 × 4.8 = 28.8 cm², and each 1 cm² on the plan stands for 1.5 m × 1.5 m = 2.25 m², so 28.8 × 2.25 = 64.8 m². Multiplying 28.8 by 1.5 would give 43.2, which is wrong because it scales only one direction.
Reproducing at a different scale
A community garden is shaped like an L. On grid paper at 1 square = 3 m, it is 6 squares long and 4 squares wide, with a 2-square by 2-square corner cut out. Redraw it at 1 square = 2 m.
Equation: Actual lengths: 6 × 3 = 18 m, 4 × 3 = 12 m, and the cut-out corner is 2 × 3 = 6 m on each side. New drawing: 18 ÷ 2 = 9 squares, 12 ÷ 2 = 6 squares, and 6 ÷ 2 = 3 squares. Every length in the new drawing is 3/2 of the old one (see Diagram 1).
Actual area of a triangle
A park shaped like a triangle is drawn with a base of 4 cm and a height of 3 cm. The scale is 1 cm = 40 m. What is the actual area of the park?
Equation: Actual base: 4 × 40 = 160 m. Actual height: 3 × 40 = 120 m. Area: (1/2) × 160 × 120 = 9,600 m². Check with the drawing: its area is (1/2) × 4 × 3 = 6 cm², and each cm² stands for 40 × 40 = 1,600 m², so 6 × 1,600 = 9,600 m².
Finding the scale
On a house plan, a wall that is really 12 m long is drawn 8 cm long. What is the scale? How long is a 5 cm hallway on the plan in real life?
Equation: 12 ÷ 8 = 1.5, so the scale is 1 cm = 1.5 m. The hallway is 5 × 1.5 = 7.5 m long. As a ratio without units: 1.5 m = 150 cm, so the scale is 1 : 150.
Show Diagram 2 and go back to the warm-up. At 1 cm = 3 m, a 1 cm by 1 cm square on the drawing stands for a 3 m by 3 m square, which holds 9 squares of 1 m². So areas are multiplied by 9 while lengths are multiplied by 3. Then use Diagram 1 to show redrawing: the actual garden is the same in both drawings (180 m²), but Drawing B uses more, smaller-value squares. Ask: "Which drawing is larger on the paper, and why?" (Drawing B, because each square stands for less of the garden.) A drawing can also be larger than the real object, for example a drawing of an ant at 10 : 1.
Guided Practice15 minutes
Pairs solve four problems. One partner writes the scale as a sentence, and the other does the computation; they switch for each problem. Ask each pair to label every answer with units.
Guided practice problems with answers
Problem
Answer
A map uses 1 cm = 5 km. Two towns are 7.4 cm apart on the map. How far apart are they?
7.4 × 5 = 37 km
A plan uses 1/4 inch = 1 foot. A bedroom is 3 inches by 2 1/2 inches on the plan. Find its actual length, width and area.
1 inch = 4 feet, so 12 ft by 10 ft, and the area is 120 ft²
A basketball court is drawn 14 cm by 7.5 cm at 1 cm = 2 m. Find its actual area.
28 m by 15 m, so 420 m² (the drawing area 105 cm² times 4 m² per cm² also gives 420)
A rectangle is drawn 6 cm by 4 cm at 1 cm = 10 m. Redraw it at 1 cm = 8 m.
Actual 60 m by 40 m, so the new drawing is 7.5 cm by 5 cm
Listen for these errors: multiplying the drawing area by the length scale only once, forgetting to change 1/4 inch = 1 foot into 1 inch = 4 feet, and dividing by the old scale instead of the new one when redrawing. Ask: "Should the new drawing be bigger or smaller than the old one? Each centimeter now stands for 8 m instead of 10 m."
Independent Practice15 minutes
Students solve six problems on their own and write one sentence for each answer, with units. They check each area answer both ways: from actual lengths and from the drawing area.
Independent practice problems with answers
Problem
Answer
Scale 1 in = 10 ft. A drawing length is 2.25 in. Find the actual length.
22.5 ft
Scale 1 : 50. A drawing length is 7 cm. Find the actual length in meters.
350 cm = 3.5 m
A bike trail is 3.2 cm long on a map at 1 cm = 25 km. How long is the trail?
80 km
A garden is drawn 5 cm by 3.5 cm at 1 cm = 2 m. Find its actual area.
10 m by 7 m = 70 m² (or 17.5 cm² × 4 = 70 m²)
A rectangle is drawn 2 in by 3 in at 1 in = 6 ft. Redraw it at 1 in = 4 ft.
Actual 12 ft by 18 ft, so 3 in by 4.5 in
A 15 m wall is drawn 6 cm long. What is the scale?
1 cm = 2.5 m
Closure5 minutes
Exit ticket: (1) A plan uses 1 cm = 4 m. A room is 3.5 cm by 2 cm on the plan. Find the actual length, width and area. (Answer: 14 m by 8 m, 112 m².) (2) Redraw that room at 1 cm = 2 m. (Answer: 7 cm by 4 cm.) (3) In one sentence: why is the area not multiplied by the same number as the lengths?
Differentiation Strategies
For Struggling Students
Give a two-row ratio table for each scale (drawing cm in the top row, actual m in the bottom row) so students can scale up step by step
Use grid paper where 1 square = 1 cm, and have students count squares for areas before they multiply
Start with whole-number scales (1 cm = 2 m) before scales with decimals or fractions (1/4 inch = 1 foot)
For Advanced Students
Give a map with a scale bar instead of a written scale, and ask students to measure the bar and write the scale in words and as a ratio
Ask students to choose a scale so that a real place, such as the school field, fits on one sheet of paper, and to justify the choice
Ask how the volume of a scale model would compare with the real object if every length is 1/10 as long (a preview of later grades, beyond this standard)
Assessment Guidance
What to Look For
Check that students say what the scale means before they compute, and that every answer has units. For areas, look for the square of the length factor, either by finding actual lengths first or by using the actual area of one drawing square; multiplying the drawing area by the scale once is the error to catch. When redrawing, students should find actual lengths first (or the ratio of the two scales) and should predict whether the new drawing is larger or smaller.
02
Classroom Activities
3 Activities
1
Classroom Floor Plan
20 minGroups of 3-4
Groups measure the classroom with tape measures, to the nearest 0.1 m, and draw a floor plan on centimeter grid paper at 1 cm = 0.5 m. They add the door, the board and the teacher's desk, then use the plan to answer questions about actual lengths and areas.
Procedure
Measure the length and width of the room, and the width of the door and the teacher's desk
Convert each measurement to plan centimeters: divide by 0.5 (or multiply by 2)
Draw the room and label each side with both the plan length and the actual length
Find the floor area two ways: from the actual lengths, and from the plan area (each cm² stands for 0.5 m × 0.5 m = 0.25 m²)
Sample Measurements (answer key)
Room 8.6 m by 6.4 m: plan 17.2 cm by 12.8 cm
Floor area: 8.6 × 6.4 = 55.04 m², and 17.2 × 12.8 = 220.16 cm² on the plan, times 0.25 = 55.04 m²
Door 0.9 m wide: 1.8 cm on the plan. Teacher's desk 1.5 m by 0.75 m: 3 cm by 1.5 cm
Discussion Questions
Why does the floor area on the plan (in cm²) have a bigger number than the actual area (in m²)?
If you redrew the plan at 1 cm = 1 m, would it still fit on your paper? How big would it be?
Modification for Distance Learning
Students measure one room at home, such as a bedroom, in feet, and draw it at 1 inch = 2 feet. They share a photo of the plan and one area computation.
2
Scale Card Match
15 minPairs
Pairs match 6 drawing cards with 6 of the 8 actual-size cards. Two actual-size cards are traps: each one comes from a common error. Pairs must explain the error behind each trap card.
Drawing Cards
A1: A fence is 5 cm long at 1 cm = 4 m
A2: A wall is 3.5 in long at 1 in = 6 ft
A3: A rectangular lot is 3 cm by 2 cm at 1 cm = 5 m (find the area)
A4: A bench is 8 cm long at 1 : 25
A5: A rectangular patio is 1.5 in by 1 in at 1 in = 10 ft (find the area)
A6: A road is 4.5 cm long on a map at 1 cm = 20 km
Actual-Size Cards (answer key)
20 m (A1), 21 ft (A2), 150 m² (A3), 2 m (A4), 150 ft² (A5), 90 km (A6)
Trap cards: 30 m² (A3 with the drawing area multiplied by 5 only once: 6 × 5) and 15 ft² (A5 with the drawing area multiplied by 10 only once: 1.5 × 10)
Discussion Questions
Cards A3 and A5 both have the answer 150. Are the two areas the same? Why not?
Card A4 has no units in its scale. How did you decide what 1 : 25 means?
Which trap card did you almost choose, and what check would have caught it?
Challenge Variation
Pairs write two new drawing cards and one trap card for each, then trade with another pair.
3
Resize the Cabin
15 minPairs
Each pair gets the front view of a cabin on grid paper at 1 square = 2 ft: a rectangle 8 squares wide and 6 squares tall, with a triangle roof on top that is 8 squares wide and 3 squares tall. Partner A redraws it at 1 square = 1 ft, and Partner B redraws it at 1 square = 4 ft.
Procedure
Find the actual width of the cabin, the height of the wall and the height of the roof, in feet
Divide each actual length by your new scale to get the new number of squares, and draw the cabin
Count or compute the squares inside your drawing, and multiply by the actual area of one square
Compare with your partner: do both drawings give the same actual area?
Answer Key
Actual: 16 ft wide, wall 12 ft tall, roof 6 ft tall
At 1 square = 1 ft: 16 by 12 squares with a roof 16 wide and 6 tall; 192 + 48 = 240 squares of 1 ft² = 240 ft²
At 1 square = 4 ft: 4 by 3 squares with a roof 4 wide and 1.5 tall; 12 + 3 = 15 squares of 16 ft² = 240 ft²
The original at 1 square = 2 ft: 48 + 12 = 60 squares of 4 ft² = 240 ft²
Discussion Questions
Your three drawings have 240, 60 and 15 squares. Why is the actual area the same every time?
Partner A's drawing is 4 times as wide as Partner B's. How many times as many squares does it have?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: One Garden at Two Scales
The L-shaped garden from example 3, drawn on the same grid paper at two scales. At 1 square = 3 m it is 6 by 4 squares with a 2 by 2 corner cut out; at 1 square = 2 m it is 9 by 6 squares with a 3 by 3 corner cut out. Every length grows by 3/2, and both drawings give the same actual area, 180 m². Drawn exactly on the grid.
Diagram 2: Why Areas Scale Differently
At a scale of 1 cm = 3 m, one square centimeter on the drawing stands for a 3 m by 3 m square, which holds 9 square meters. Lengths are multiplied by 3, but areas are multiplied by 9. The right-hand square is drawn 3 times as long on each side as the left-hand square.
04
Homework Assignment
~30 min
7.G.A.1 Homework: Scale Drawings
Directions: Show all work. Write the scale as a sentence before you use it. Label every answer with units, and use square units for areas.
Part 1: Actual Lengths (Problems 1-2)
A map uses the scale 1 cm = 12 km. Find the actual distance for each map distance: (a) 3 cm (b) 5.5 cm (c) 0.75 cm.
A floor plan uses the scale 1/2 inch = 3 feet. A kitchen is 2 1/2 inches long and 2 inches wide on the plan. Find the actual length and width of the kitchen.
Part 2: Actual Areas (Problems 3-4)
Use the kitchen from Problem 2. Find its actual floor area. Then explain why multiplying the plan area (in square inches) by 6 does not give the actual area.
A soccer field is drawn 21 cm by 13.6 cm at the scale 1 cm = 5 m. Find the actual length, width and area of the field.
Part 3: Reproducing at a Different Scale (Problems 5-6)
A rectangular patio is drawn 9 cm by 6 cm at 1 cm = 0.5 m. Redraw it at 1 cm = 1.5 m. Give the new drawing lengths, and say whether the new drawing is larger or smaller than the first one.
A triangular lawn is drawn on grid paper at 1 unit = 5 ft, with a base of 6 units and a height of 4 units. Redraw it at 1 unit = 2 ft. Give the new base and height in units, and show that both drawings give the same actual area.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Using the Scale
Scale written as a sentence and used correctly for every length
Scale used correctly with one error
Scale not used or used backward
Areas
Actual areas correct, with the length factor used in both directions
Correct method with a computation error
Drawing area multiplied by the scale only once
Redrawing
New drawing lengths correct and the size change explained
Lengths correct without the explanation
Lengths incorrect
Units
All answers labeled, square units for areas
Some labels missing
No units
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Write the scale as a sentence before you answer. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
A plan uses the scale 1 cm = 6 m. A gym wall is 4.5 cm long on the plan. How long is the real wall?
Answer: C
Each centimeter stands for 6 m, so 4.5 × 6 = 27 m. Choice A divides 4.5 by 6 instead of multiplying. Choice B adds 4.5 and 6. Choice D puts the decimal point in the wrong place (45 × 6).
Question 2 of 20 · Multiple Choice
A map uses the scale 1 inch = 15 miles. Two cities are 105 miles apart. How far apart are they on the map?
Answer: A
Divide the actual distance by 15: 105 ÷ 15 = 7 inches. Choice B subtracts 15 from 105. Choice C multiplies 105 by 15, which goes the wrong way. Choice D divides 15 by 105.
Question 3 of 20 · Multiple Choice
A rectangle is 4 cm by 2 cm on a drawing with the scale 1 cm = 4 m. What is the actual area?
Answer: B
Actual lengths: 4 × 4 = 16 m and 2 × 4 = 8 m, so the area is 16 × 8 = 128 m². Choice A multiplies the drawing area (8 cm²) by 4 only once. Choice C keeps the drawing area and changes only the units. Choice D adds the drawing lengths and multiplies by 4: (4 + 2) × 4.
Question 4 of 20 · Multiple Choice
On a scale drawing, every actual length is 5 times the drawing length. How many times the drawing area is the actual area?
Answer: C
A 1 by 1 square on the drawing stands for a 5 by 5 square, which is 25 times the area. Choice A uses the length factor for area. Choice B doubles the length factor instead of multiplying it by itself. Choice D multiplies 5 × 5 × 5, using three directions, but area has only two (length and width).
Question 5 of 20 · Multiple Choice
A rectangle is drawn 12 cm by 10 cm at the scale 1 cm = 2 m. It is redrawn at 1 cm = 4 m. What are the new drawing lengths?
Answer: D
Actual lengths: 12 × 2 = 24 m and 10 × 2 = 20 m. New drawing: 24 ÷ 4 = 6 cm and 20 ÷ 4 = 5 cm. Each new centimeter stands for twice as much, so the drawing is half as long. Choice A doubles the lengths instead of halving them. Choice B divides the old drawing lengths by 4 without finding the actual lengths first. Choice C subtracts 2 from each length.
Question 6 of 20 · Multiple Choice
On a house plan, a wall that is 36 feet long is drawn 4 1/2 inches long. What is the scale?
Answer: B
Divide the actual length by the drawing length: 36 ÷ 4.5 = 8, so 1 inch = 8 feet. Check: 4.5 × 8 = 36. Choice A multiplies 36 by 4.5. Choice C subtracts 4.5 from 36. Choice D divides 4.5 by 36, which gives inches of plan per foot of wall, not feet per inch.
Question 7 of 20 · Multiple Choice
A drawing of a table uses the scale 1 : 40. The table is 3.5 cm long on the drawing. How long is the real table?
Answer: A
A scale of 1 : 40 means each drawing length is multiplied by 40 in the same unit: 3.5 × 40 = 140 cm, or 1.4 m. Choice B adds 40. Choice C multiplies by 4 instead of 40. Choice D divides 40 by 3.5 (about 11.4).
Question 8 of 20 · Multiple Choice
Which of these is NOT a scale drawing of a rectangle that is 10 m long and 6 m wide?
Answer: D
In a scale drawing, both lengths use the same scale. For 4 cm by 3 cm, 10 ÷ 4 = 2.5 m per cm but 6 ÷ 3 = 2 m per cm, so the shape is stretched. Choice A uses 1 cm = 2 m, choice B uses 1 cm = 4 m, and choice C uses 1 cm = 1 m, the same for both lengths.
Question 9 of 20 · Multiple Choice
A triangle on a drawing has a base of 6 cm and a height of 4 cm. The scale is 1 cm = 10 m. What is the actual area of the triangle?
Answer: C
Actual base 60 m and height 40 m, so the area is (1/2) × 60 × 40 = 1,200 m². Choice A multiplies the drawing area (12 cm²) by 10 only once. Choice B forgets the 1/2 in the triangle area formula. Choice D is the drawing area with the units changed.
Question 10 of 20 · Multiple Choice
A square meeting room has an actual area of 49 m². On a plan, the room is drawn as a square 2 cm on each side. What is the scale?
Answer: A
The actual side is 7 m, because 7 × 7 = 49. So 2 cm stands for 7 m, and 1 cm = 7 ÷ 2 = 3.5 m. Choice B divides the area 49 by the side 2. Choice C divides 49 m² by the drawing area 4 cm², which gives the area of one drawing square (12.25 m²), not a length. Choice D divides 2 by 7 (about 0.29), which turns the scale upside down.
Question 11 of 20 · Multiple Choice
On a map, 1/4 inch = 5 miles. How many miles do 2 1/2 inches on the map stand for?
Answer: B
1/4 inch = 5 miles means 1 inch = 20 miles. So 2 1/2 × 20 = 50 miles. Another way: 2 1/2 inches holds 10 quarter-inches, and 10 × 5 = 50. Choice A treats the scale as 1 inch = 5 miles. Choice C also multiplies by 1/4. Choice D counts the quarter-inches but forgets to multiply by 5.
Question 12 of 20 · Multiple Choice
Drawing A of a garden uses 1 cm = 2 m, and drawing B of the same garden uses 1 cm = 5 m. Which statement is true?
Answer: A
In drawing A each centimeter stands for less of the garden (2 m), so A needs more centimeters for the same garden: A is 5/2 as long as B in every direction. Choice B mixes up the two scales. Choice C would be true only if the scales were equal. Choice D is wrong, because the comparison works for any garden size.
Question 13 of 20 · Multiple Choice
A figure is redrawn so that every length is 2/3 as long as before. The first drawing has an area of 45 cm². What is the area of the new drawing?
Answer: D
Areas change by (2/3) × (2/3) = 4/9, so the new area is 45 × 4/9 = 20 cm². Choice A multiplies by 2/3 only once. Choice B multiplies by 3/2, which enlarges the drawing. Choice C multiplies by 9/4, which also enlarges it.
Question 14 of 20 · Multiple Choice
A drawing of a phone is made at the scale 3 : 1, so the drawing is larger than the phone. The drawing is 44.1 cm long. How long is the real phone?
Answer: C
At 3 : 1, every drawing length is 3 times the real length, so the phone is 44.1 ÷ 3 = 14.7 cm long, a normal phone size. Choice A multiplies by 3, as if the drawing were smaller than the phone. Choice B subtracts 3. Choice D divides by 9, which is the area factor.
Question 15 of 20 · Short Answer
A park map uses the scale 1 cm = 80 m. A path is 6.5 cm long on the map. How long is the real path, in meters and in kilometers?
6.5 × 80 = 520 m. Since 1 km = 1,000 m, the path is 0.52 km long.
Question 16 of 20 · Short Answer
A rectangular swimming pool is drawn 5 cm by 2.5 cm at the scale 1 cm = 5 m. Find the actual length, width and area of the pool.
Length 5 × 5 = 25 m, width 2.5 × 5 = 12.5 m, and area 25 × 12.5 = 312.5 m². Check: the drawing area is 12.5 cm², and each cm² stands for 25 m², so 12.5 × 25 = 312.5 m².
Question 17 of 20 · Short Answer
A deck is drawn 3 inches by 2 1/4 inches at the scale 1 inch = 4 feet. Redraw it at 1 inch = 3 feet. What are the new drawing lengths?
Actual deck: 3 × 4 = 12 ft and 2 1/4 × 4 = 9 ft. New drawing: 12 ÷ 3 = 4 inches and 9 ÷ 3 = 3 inches. The new drawing is larger, because each inch now stands for less (3 ft instead of 4 ft).
Question 18 of 20 · Short Answer
A bedroom covers 60 cm² on a plan with the scale 1 cm = 0.5 m. What is the actual floor area? Explain your method.
Each 1 cm by 1 cm square on the plan stands for 0.5 m × 0.5 m = 0.25 m². So the actual area is 60 × 0.25 = 15 m², a reasonable size for a bedroom.
Question 19 of 20 · Short Answer
Kai says: "At 1 cm = 4 m, a rectangle that is 2 cm by 5 cm on the drawing has an actual area of 10 × 4 = 40 m²." Find his mistake and the correct area.
Kai multiplied the drawing area by the scale only once, which scales just one direction. The actual lengths are 2 × 4 = 8 m and 5 × 4 = 20 m, so the area is 8 × 20 = 160 m², which is 10 × 16.
Question 20 of 20 · Short Answer
A school garden is 14 m long and 9 m wide. Choose a scale so that a drawing of the garden fits on a sheet of paper 20 cm by 15 cm, and give the drawing lengths. Explain why your scale works.
One answer: 1 cm = 1 m gives a drawing 14 cm by 9 cm, which fits because 14 < 20 and 9 < 15. Other scales work too, such as 1 cm = 2 m (7 cm by 4.5 cm). A scale such as 1 cm = 0.5 m does not work, because the drawing would be 28 cm long. Scoring: any scale where 1 cm stands for at least 0.7 m works (14 ÷ 20 = 0.7 and 9 ÷ 15 = 0.6). Full credit: a working scale, both drawing lengths correct, and a comparison with both sides of the paper. Partial credit: a working scale with one length wrong or no comparison with the paper.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 7.G.A.1 mean?
7.G.A.1 means students can solve problems with scale drawings, such as maps, floor plans and blueprints. They use the scale to find real lengths and real areas from the drawing, and they redraw a figure at a new scale. For example, at 1 cm = 1.5 m, a plan 6 cm long stands for a 9 m room.
What grade is 7.G.A.1, and what comes after it?
7.G.A.1 is a grade 7 geometry standard, and it uses the proportional reasoning of 7.RP.A.2. In grade 8, students describe scale copies as dilations (resizing a figure from a center point) and similar figures (8.G.A.3 and 8.G.A.4). In high school geometry, students use similarity transformations to decide whether two figures are similar (HSG.SRT.A.2).
What is the difference between a scale and a scale factor?
A scale compares a drawing length with an actual length, often with two units, such as 1 cm = 3 m. A scale factor is the single number that multiplies lengths when both are in the same unit. Since 3 m = 300 cm, the scale 1 cm = 3 m has a scale factor of 300, which is also written as the ratio 1 : 300.
Why does the area change by a different amount than the lengths?
Area measures two directions, and each direction is scaled. At 1 cm = 3 m, a 1 cm by 1 cm square stands for a 3 m by 3 m square, which is 9 m². So if lengths are multiplied by k, areas are multiplied by k × k. Students can see this by counting squares on grid paper, as in Diagram 2.
How do you redraw a scale drawing at a different scale?
Find the actual lengths first, then divide each one by the new scale. In the garden example, 6 squares at 1 square = 3 m is 18 m, and at 1 square = 2 m that is 18 ÷ 2 = 9 squares. A shortcut is to multiply every drawing length by the old scale divided by the new scale, here 3/2. If each unit stands for less, the new drawing is larger.
Can a scale drawing be larger than the real object?
Yes. Drawings of small things, such as insects, cells or machine parts, are often enlarged. A scale of 5 : 1 means the drawing is 5 times as long as the real object, so a honeybee about 1.2 cm long is drawn 6 cm long. The same steps apply, but you divide to go from the drawing to the real object.
What mistakes should teachers watch for in 7.G.A.1?
A common mistake is multiplying a drawing area by the length scale only once, which gives an area that is too small. Other frequent errors are using the scale backward (dividing when you should multiply), mixing units such as centimeters and meters, and redrawing at a new scale without first finding the actual lengths. Ask students to predict "bigger or smaller" before they compute.
How does 7.G.A.1 connect to ratios and proportional relationships?
A scale is a ratio, and actual length is proportional to drawing length. At 1 cm = 1.5 m, the equation actual = 1.5 × drawing has 1.5 as the constant of proportionality, the number that every drawing length is multiplied by. Students can use ratio tables, unit rates or equations from 7.RP.A.2 to solve any scale problem.
Do students need to convert units in scale problems?
Often, yes. A scale like 1 : 50 has no units, so 1 cm stands for 50 cm, which is 0.5 m. A scale like 1/4 inch = 1 foot is easier to use as 1 inch = 4 feet. Students should write the scale as a sentence with units before they compute, and give the final answer in a sensible unit, such as meters for a room.
How can parents help with 7.G.A.1 at home?
Look at maps together and use the scale bar to estimate real distances, such as from home to school. Measure a room with a tape measure and draw it on grid paper at a scale such as 1 square = 1 foot. Then ask how many squares the floor covers and what that means in square feet.
07
Related Standards
6 standards
These standards connect to 7.G.A.1: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
6.RP.A.3Prerequisite
Use ratio and rate reasoning, such as ratio tables, to solve real-world problems
