Scale drawings and angle relationships
Finding actual lengths and areas from a scale drawing, reproducing a drawing at a new scale, and solving equations from angle facts
About three to four lessons of 45 to 60 minutes
A map is a lie you can trust: it shrinks the world, exactly
Every map, blueprint and floor plan shrinks reality by the same factor everywhere, which is exactly what makes it useful: 1 cm on the page might always mean 20 km in the real world. Read that ratio correctly and you can find real distances, real areas, or even redraw the whole thing at a different scale, without ever leaving your desk.
Angles have their own set of reliable relationships: angles on a straight line, angles around a point, and the matching angles formed where two lines cross, all obey exact rules that turn into simple equations. This unit builds both skills: scaling drawings up or down, and writing an equation from an angle fact to find a missing measure.
- A map scale: 1 cm represents 20 kmmultiply the map distance by the scale factor to find the real distance
- A blueprint of a room, redrawn at a new scaleconvert to actual size once, then to the new drawing size
- Two angles on a straight linethey always add to 180 degrees, giving an equation for a missing angle
- Two lines crossing in an X shapethe angles directly opposite each other (vertical angles) are always equal
What students will be able to do
Students will use a scale drawing to find actual lengths and areas, reproduce a drawing at a new scale, and write and solve equations using facts about supplementary, complementary, vertical and adjacent angles.
- I can use a scale to convert a length on a drawing to its actual real-world length.
- I can find an actual area from a scale drawing by converting both dimensions before multiplying.
- I can convert a real-world length to a new drawing at a different scale.
- I can write and solve an equation for a missing angle using facts about supplementary, complementary, vertical or adjacent angles.
Standards this unit teaches
- 7.G.A.1Common Core (US)Scale drawings
Solve problems using scale drawings, finding actual lengths and areas and reproducing a drawing at a new scale.
- 7.G.B.5Common Core (US)Angle relationships in figures
Write and solve equations using facts about supplementary, complementary, vertical, and adjacent angles.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Grade 6 ratios & rates teaching unita scale is a ratio, and scaling a drawing uses exactly the same reasoning as scaling any ratio
- Grade 7 ratios, proportions & percent teaching unitunit rates and proportional reasoning are the same idea applied to map and blueprint scales
- Angle in the glossarya refresher on what an angle measures, before the specific angle relationships in this unit
- Equation in the glossarya refresher on solving for an unknown, the skill angle problems in this unit apply
Words to teach and display
- Scale
- a ratio between a length on a drawing and the actual length it represents, such as 1 cm : 20 km
- Scale factor
- the number a drawing's length is multiplied by to find the actual length
- Supplementary angles
- two angles whose measures add to 180 degrees
- Complementary angles
- two angles whose measures add to 90 degrees
- Vertical angles
- the pair of equal angles directly opposite each other where two lines cross
- Adjacent angles
- two angles that share a vertex and a side but do not overlap
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Actual lengths from a scale drawing
ConcreteA scale such as '1 cm represents 20 km' is a ratio, scaled up by whole-number multiples exactly like a unit rate: multiply the drawing measurement by the real-world number the scale gives for every 1 unit on the drawing.
Two cities are 6.5 cm apart on a map with a scale of 1 cm : 20 km. Multiply: 6.5 x 20 = 130. The actual distance is 130 km.
A blueprint uses a scale where 1 cm represents 2.5 m. A wall measures 8 cm on the blueprint. Find the actual length of the wall.
- Multiply the drawing length by the scale's real-world value: 8 x 2.5.
- 8 x 2.5 = 20.
Answer: The actual wall length is 20 m.
- In a scale of 1 cm : 20 km, which number do you multiply the drawing length by to get the real distance?
- Why does a bigger 'represents' number in the scale mean the drawing is more shrunk down?
2. Actual areas from a scale drawing
PictorialFinding an actual AREA from a scale drawing takes one extra step: convert EACH linear dimension separately to its actual length first, then multiply those actual lengths together. Do not multiply the drawing's area by the scale factor directly, since area scales by the SQUARE of the linear factor, not the factor itself.
A scale drawing has a scale of 1 cm : 4 m. A rectangular room measures 3 cm by 5 cm on the drawing. Convert each side: 3 x 4 = 12 m, and 5 x 4 = 20 m. The actual area is 12 x 20 = 240 m². (Notice this matches 15 cm² x 4² = 15 x 16 = 240 as well, confirming that area scales by the scale factor SQUARED.)
A scale drawing has a scale of 1 cm : 4 m. A rectangular room measures 3 cm by 5 cm on the drawing. Find its actual area.
- Convert the length: 3 x 4 = 12 m.
- Convert the width: 5 x 4 = 20 m.
- Actual area = 12 x 20 = 240.
Answer: The actual area of the room is 240 m².
- Why is it wrong to find the drawing's area (3 x 5 = 15 cm²) and simply multiply that by 4?
- If the scale factor is 4, what number would you multiply a drawing's area by directly to get the actual area?
3. Reproducing a drawing at a new scale
PictorialRedrawing at a new scale is two conversions, not one: first convert the original drawing to the actual real-world size using the FIRST scale, then convert that actual size to the new drawing using the SECOND scale.
A garden path is 6 cm long on a drawing at scale 1:50 (1 cm represents 50 cm). Actual length: 6 x 50 = 300 cm. Redrawn at a new scale of 1:25 (1 cm represents 25 cm), the new drawing length is 300 / 25 = 12 cm.
A scale drawing of a garden uses a scale of 1:50. A path is 6 cm long on this drawing. Redraw the path at a new scale of 1:25. How long will the path be on the new drawing?
- Find the actual length using the original scale: 6 x 50 = 300 cm.
- Find the new drawing length using the new scale: 300 / 25 = 12 cm.
Answer: The path will be 12 cm long on the new drawing.
- Why does this problem need two separate conversions instead of one?
- If the new scale shrinks the drawing MORE than the original (a bigger 'represents' number), will the new drawing length be longer or shorter than the original 6 cm?
4. Angle relationships: writing and solving equations
AbstractFour angle facts turn straight into equations: angles on a straight line sum to 180 degrees (supplementary), angles that complete a right angle sum to 90 degrees (complementary), angles directly opposite where two lines cross are equal (vertical), and angles sharing a vertex and a side are adjacent.
Two angles are supplementary: one measures (2x + 20) degrees, the other 70 degrees. Since supplementary angles sum to 180: 2x + 20 + 70 = 180, so 2x + 90 = 180, so 2x = 90, so x = 45.
Two angles are supplementary. One measures (2x + 20) degrees and the other measures 70 degrees. Find x.
- Supplementary angles sum to 180: (2x + 20) + 70 = 180.
- Combine constants: 2x + 90 = 180.
- Subtract 90: 2x = 90.
- Divide by 2: x = 45.
Answer: x = 45 (the two angles are 110 degrees and 70 degrees).
- How would the equation change if the two angles were complementary instead of supplementary?
- Why are vertical angles always equal, rather than needing to add to some total?
Common misconceptions and how to address them
MisconceptionTo find an actual length from a scale, the drawing measurement is DIVIDED by the scale's real-world number instead of multiplied.
Why it happens: It is easy to reverse the direction of the conversion when the scale is written as 'X represents Y'.
How to address it: Identify which number in the scale belongs to the DRAWING (usually the '1') and which is the real-world multiplier, then always multiply a drawing measurement by the real-world number to scale UP to actual size.
MisconceptionA scale drawing's area is found by multiplying the drawing's area directly by the linear scale factor, e.g. a drawing area of 15 cm² at scale factor 4 gives an actual area of 15 x 4 = 60 m².
Why it happens: It feels consistent with how a single length is scaled, but area is a two-dimensional measurement.
How to address it: Area scales by the SQUARE of the linear scale factor (here 4² = 16, giving 15 x 16 = 240, matching converting each side separately). Always convert each linear dimension to its actual length first, then multiply.
MisconceptionAdjacent angles are assumed to always be equal to each other.
Why it happens: Students confuse 'adjacent' (sharing a vertex and a side) with 'vertical' (the equal, opposite pair formed by two crossing lines).
How to address it: Adjacent angles just share a corner and a side; they can be any size. Only VERTICAL angles, the pair directly opposite each other where two lines cross, are guaranteed to be equal.
MisconceptionComplementary angles are assumed to sum to 180 degrees, the same total as supplementary angles.
Why it happens: Supplementary angles are usually taught and drilled first, so their 180-degree total gets applied to complementary problems by mistake.
How to address it: Keep the two totals clearly separate: complementary sums to 90 degrees (think 'c for corner', a right angle), supplementary sums to 180 degrees (think 's for straight line').
MisconceptionWhen reproducing a drawing at a new scale, the SAME original scale is used for both conversion steps instead of switching to the new scale for the second step.
Why it happens: Having just used one scale, it feels natural to reuse the same number rather than switching to the new one.
How to address it: There are always two separate conversions: original drawing to actual size (use the FIRST scale), then actual size to new drawing (use the SECOND, NEW scale). Write both scales down before starting either calculation.
Guided practice (with answers)
1. A map scale is 1 cm : 20 km. Two cities are 6.5 cm apart on the map. Find the actual distance.
Answer: 130 km, because 6.5 x 20 = 130.
2. A scale drawing has a scale of 1 cm : 3 m. A rectangular deck measures 4 cm by 6 cm on the drawing. Find its actual area.
Answer: 216 m², because the actual dimensions are 4 x 3 = 12 m and 6 x 3 = 18 m, and 12 x 18 = 216.
3. A drawing at scale 1:100 shows a fence 5 cm long. Redraw it at scale 1:20. Find the new length.
Answer: 25 cm, because the actual length is 5 x 100 = 500 cm, and the new drawing length is 500 / 20 = 25 cm.
4. Two angles are complementary. One is 27 degrees. Find the other.
Answer: 63 degrees, because 90 - 27 = 63.
5. Two angles are supplementary. One is (4x + 10) degrees, the other is 90 degrees. Find x.
Answer: x = 20, because 4x + 10 + 90 = 180, so 4x = 80, so x = 20.
6. Two vertical angles are (2x + 8) degrees and (4x - 14) degrees. Find x.
Answer: x = 11, because vertical angles are equal: 2x + 8 = 4x - 14, so 22 = 2x, so x = 11.
Independent practice worksheets
Practise scale drawings, angle relationships and enlargement with computed, never-wrong answer keys.
Differentiation
- Start scale problems with whole-number scale factors (x2, x10) before introducing decimal ones (x2.5).
- For scale-area problems, always have students write both converted side lengths down BEFORE multiplying, so the squaring effect is visible rather than skipped.
- Give a one-page reference card of the four angle facts (supplementary 180, complementary 90, vertical equal, adjacent share a vertex/side) to check against while solving.
- Model writing the equation from the angle fact as its own step, separate from solving it, so the two skills are not tackled at once.
- Introduce a scale given as a fraction or in mixed units (e.g. 1 inch represents 1/4 mile) rather than a clean whole number.
- Ask students to design their own scale drawing of a real room, choose a sensible scale, and calculate the actual area it represents.
- Pose a multistep angle problem combining two facts, such as an angle that is both part of a straight line AND a vertical-angle pair with an unknown elsewhere in the same figure.
- Investigate: if a shape is enlarged by scale factor k, by what factor does its PERIMETER scale, and by what factor does its AREA scale? (Perimeter by k, area by k².)
Assessment: exit ticket
A three-question exit ticket sampling scale area, supplementary angles, and vertical angles.
1. A blueprint scale is 1 cm : 1.5 m. A room is 5 cm by 4 cm on the blueprint. Find the actual area.
Answer: 45 m², because the actual dimensions are 5 x 1.5 = 7.5 m and 4 x 1.5 = 6 m, and 7.5 x 6 = 45.
2. Two angles are supplementary: one is (3x - 6) degrees, the other is 96 degrees. Find x.
Answer: x = 30, because 3x - 6 + 96 = 180, so 3x + 90 = 180, so 3x = 90, so x = 30.
3. Two vertical angles are (6x - 4) degrees and (4x + 10) degrees. Find x and the measure of each angle.
Answer: x = 7, and each angle is 38 degrees, because 6x - 4 = 4x + 10 gives 2x = 14, so x = 7, and 4(7) + 10 = 38.
Teacher notes and timings
- Rough timing across three to four lessons: Lesson 1 scale lengths (section 1), Lesson 2 scale areas and reproducing at a new scale (sections 2 and 3), Lesson 3-4 angle relationships plus the exit ticket (section 4 and assessment).
- This unit assumes comfort with ratios and unit rates (Grade 6 ratios/rates unit, and this Grade 7 ratios/percent unit). Revisit those first if scaling by a non-whole-number factor feels shaky.
- Language to keep repeating: multiply a drawing length by the scale's real-world number to scale up; area scales by the scale factor SQUARED; complementary is 90, supplementary is 180, vertical angles are equal.
- The double-number-line figure in section 1 deliberately reuses the exact ratio-scaling model from the Grade 7 ratios/percent unit, since a map or blueprint scale is mathematically the same idea as any other ratio scaled by whole-number multiples.
- Curriculum note: 7.G.A.1 (scale drawings) and 7.G.B.5 (angle relationships) are both grouped under Geometry, but are mathematically distinct skills, ratio reasoning applied to length/area versus equation-writing from angle facts; the unit treats them as two related-but-separate halves.
- Present mode and print both work: project the double-number-line figure to build the scale relationship live with the class in section 1, then print the worksheets for independent practice.