Scale Drawings – Quick Review

• What is a Scale Drawing? A proportional picture of an /.

• It uses a fixed ___________ (_) to link drawing size to real size.

• Reasons for using scale drawings: show very /_ things; used in , , _.

• Core skills for scale drawings:

– Build _ ratios to switch between _ ↔ _.

– Apply perimeter formula P=2l+2wP = 2l + 2w and area formula A=l×wA = l \times w for real lengths.

– Changing the scale changes every length _.

• Example (Central Park): Given scale 1 in=0.5 mi1\text{ in} = 0.5\text{ mi}.

– If drawing is 5 in5\text{ in}, actual is __ mi.

– If drawing is 1 in1\text{ in}, actual is __ mi.

– Calculate the perimeter: P = 2(_
____) + 2(_
____) = _
____ \text{mi}

– Calculate the area: A = _
____ \times _
____ = _
____ \text{mi}^2

• Example (Tappan Zee Bridge): Changing scale from 1 cm=100 m1\text{ cm}=100\text{ m} to 1 cm=50 m1\text{ cm}=50\text{ m} __ drawing lengths.

• Complete the equivalent-ratio table (scale 1 in=0.5 mi1\text{ in}=0.5\text{ mi}):

\begin{array}{c|ccccc}\text{In.}&1&2&3&4&5\hline\text{Mi.}&0.5&1&1.5&2&2.5\end{array}

• Scale change rule: If the denominator in the scale , drawing lengths (and vice-versa).

Quick Conversions

• To find actual length, _ drawing length by the _ _.

– 8\text{ cm} \times 50\text{ m/cm} = _
____ \text{m}

– 12\text{ in} \times 0.25\text{ mi/in} = _
____ \text{mi}

• Extended-response idea: Keep _ scale when enlarging/reducing. For example, mural height on wall found via proportion: drawing inactual ft=mural in10 ft\frac{\text{drawing in}}{\text{actual ft}} = \frac{\text{mural in}}{10\text{ ft}}.

Solving Scale-Drawing Problems

  1. Identify the _ (e.g., 1 in=0.5 mi1\text{ in}=0.5\text{ mi}).

  2. Set up _ ratios: \frac{_
    _______}{_
    _______}=\frac{_
    _______}{_
    _______}.

  3. Solve for missing lengths; keep _ clear.

  4. Use P=2l+2wP = 2l + 2w and A=l×wA = l \times w for perimeter/area once _ lengths are known.

Typical Homework Patterns

• Map distances: _ inches by _-per-inch.

• Fields/rectangles: convert _ sides, then find AA and PP.

• Bridges/models: _ drawing cm by _-per-cm.

Assessment Focus

• Multiple-choice test speed: Know how to do -step conversions .

• Short-answer: Clearly state _ with answer.

• Extended-response: Set up _ equations and _ each step.

• What is a Scale Drawing? A proportional representation of an object or real-world item.

• It employs a fixed ratio (the scale) to establish a consistent relationship between the dimensions on the drawing and the actual dimensions of the object.

• Reasons for using scale drawings: depicting very large or very small objects effectively; commonly applied in fields such as architecture, engineering, and cartography (map-making).

• Core skills for scale drawings:

– Construct equivalent ratios (proportions) to easily convert between drawing dimensions and actual dimensions.

– Apply the perimeter formula P=2l+2wP = 2l + 2w and the area formula A=l×wA = l \times w to calculate actual perimeters and areas once real lengths are determined.

– Understand that changing the scale proportionally alters every length on the drawing relative to the actual object, and vice-versa.

• Example (Central Park): Given scale 1 in=0.5 mi1\text{ in} = 0.5\text{ mi}.

– If a drawing dimension is 5 in5\text{ in}, the actual dimension is 2.5 mi2.5\text{ mi}. (Calculation: 5 in×0.5 mi1 in=2.5 mi5\text{ in} \times \frac{0.5\text{ mi}}{1\text{ in}} = 2.5\text{ mi}. The actual length is "0.5"×5="2.5""0.5" \times 5 = "2.5"

– If a drawing dimension is 1 in1\text{ in}, the actual dimension is 0.5 mi0.5\text{ mi}. (This confirms the scale itself.)

– To calculate the perimeter (e.g., if the actual park dimensions are 2.5 mi2.5\text{ mi} by 0.5 mi0.5\text{ mi} as derived from corresponding drawing lengths of 5 in5\text{ in} and 1 in1\text{ in}): P=2(2.5 mi)+2(0.5 mi)=5 mi+1 mi=6 miP = 2(2.5\text{ mi}) + 2(0.5\text{ mi}) = 5\text{ mi} + 1\text{ mi} = 6\text{ mi}

– To calculate the area using the same actual dimensions: A=2.5 mi×0.5 mi=1.25 mi2A = 2.5\text{ mi} \times 0.5\text{ mi} = 1.25\text{ mi}^2

• Example (Tappan Zee Bridge): Changing the scale from 1 cm=100 m1\text{ cm}=100\text{ m} to 1 cm=50 m1\text{ cm}=50\text{ m} increases the drawing lengths (the drawing becomes larger to represent the same actual object, as each unit on the drawing now covers a smaller real distance).

• Equivalent Ratio Table: Students should be able to complete tables based on a given scale, demonstrating understanding of proportional relationships. (Example for scale 1 in=0.5 mi1\text{ in}=0.5\text{ mi}):

\begin{array}{c||ccccc}\text{In.}&1&2&3&4&5\hline\text{Mi.}&0.5&1&1.5&2&2.5\end{array}

• Scale Change Rule: If the actual distance represented by a unit on the drawing decreases (e.g., from 100 m100\text{ m} to 50 m50\text{ m} for 1 cm1\text{ cm}), the corresponding drawing lengths increase (to show the same actual object at a larger scale). Conversely, if the actual distance represented by a unit on the drawing increases, drawing lengths decrease.

Quick Conversions

• To determine the actual length, multiply the drawing length by the scale factor (the actual unit per drawing unit).

– Example: 8 cm×50 m/cm=400 m8\text{ cm} \times 50\text{ m/cm} = 400\text{ m}

– Example: 12 in×0.25 mi/in=3 mi12\text{ in} \times 0.25\text{ mi/in} = 3\text{ mi}

• Extended-Response Application: Emphasize maintaining a consistent scale when enlarging or reducing figures. For instance, determining mural height on a wall using a proportion based on a known scale: drawing inactual ft=mural in10 ft\frac{\text{drawing in}}{\text{actual ft}} = \frac{\text{mural in}}{10\text{ ft}} (Students must identify the corresponding parts of the proportion correctly).

Solving Scale-Drawing Problems

  1. Identify the scale provided (e.g., 1 in=0.5 mi1\text{ in}=0.5\text{ mi}).

  2. Set up proportional ratios or equivalent fractions. Ensure consistent units and placement (e.g., drawing lengthactual length=known drawing lengthknown actual length\frac{\text{drawing length}}{\text{actual length}} = \frac{\text{known drawing length}}{\text{known actual length}}). Example: drawingactual=drawingactual\frac{\text{drawing}}{\text{actual}}=\frac{\text{drawing}}{\text{actual}}.

  3. Solve for missing lengths by cross-multiplication or other ratio methods; always keep units clear on both sides of the equation.

  4. Use the perimeter formula P=2l+2wP = 2l + 2w and the area formula A=l×wA = l \times w for calculations once actual lengths have been accurately determined from the scale drawing.

Typical Homework Patterns

• Map distances: Students typically convert drawing inches by using the miles-per-inch scale factor.

• Fields/rectangles: Convert both drawing sides to actual lengths, then use these actual lengths to find area (AA) and perimeter (PP).

• Bridges/models: Convert drawing centimeters by applying the meters-per-centimeter scale factor (or appropriate units).

Assessment Focus

• Multiple-choice assessments: Students should develop efficiency in performing quick, one-step conversions to determine actual or drawing lengths.

• Short-answer questions: Require students to clearly state the units along with their numerical answer.

• Extended-response questions: Students must demonstrate their understanding by setting up proportional equations or clear conversion steps, and justifying each step of their calculation process.