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Floor Plan


Alignments to Content Standards: 7.G.A.1

Task

Mariko has an $80 : 1$ scale-drawing of the floor plan of her house. On the floor plan, the dimensions of her rectangular living room are $1 \frac78$ inches by $2 \frac12$ inches.

What is the area of her real living room in square feet?

IM Commentary

The purpose of this task is for students to translate between measurements given in a scale drawing and the corresponding measurements of the object represented by the scale drawing. If used in an instructional setting, it would be good for students to have an opportunity to see other solution methods, perhaps by having students with different approaches explain their strategies to the class. Students who can only solve this by first converting the linear measurements will have a hard time solving problems where only area measures are given.

Solutions

Solution: Scaling up and then finding the area

The dimensions of the real living room in inches are $$80 \times 1\frac78 = 150$$ and $$80 \times 2 \frac12 = 200.$$

The area of the real living room is $150\text{ in} \times 200 \text{ in} = 30,000\text{ in}^2$.



To convert square inches to square feet, you have to divide by $12^2=144$. $$30,000\div144=208\frac13$$

So the area of the living room is $208\frac13\text{ ft}^2$.

Solution: Finding the area and then scaling up

The area of the living room in the floor plan is $$1\frac78 \text{ in}\times 2 \frac12 \text{ in} = \frac{75}{16} \text{ in}^2.$$ Since the lengths scale by $80$, the area scales by $80^2$. So the area of the real living room in square inches is $\frac{75}{16} \times 80^2=30,000$.

To convert square inches to square feet, you have to divide by $12^2=144$. $$30,000\div144=208\frac13$$ So the area of the living room is $208\frac13\text{ ft}^2$.

Cat says:

over 2 years

In the world of architectural drawings, the scale is the scale. Units are not usually specified. Your students got stuck because they did not realize the relationship between square feet and square inches, not because they don't understand scale.

Shelly Lawson says:

over 4 years

My class struggled with the conversion between sq inches and sq feet. I realized that they did not have a sound understanding of the concept. I suggest that the ratio 80:1 be explicitly stated in inches. Having a discussion of the square footage of an average house prior is also helpful. It is a great problem as it requires a variety of skills that were previously taught in isolation to be used together in order to solve.