calculating area, math people a question

[athomeontheprairie]

My dd10 needed to calculate the area of a hexagon 3 different ways. She split it into two trapeziods, 6, equilateral triangle, and 4 triangles. She measured to a tenth of a centimeter. Her calculations are correct, but the areas are slightly different.

 

1. Why are the areas different? (Even in the solution manuals, the areas are slightly different, why? How? Can someone explain this to me please)

 

2. When calculating area, how much difference should be allowable? If she measures 4.5, and the book says 4.6 for a given side-I'm okay with that. Maybe I shouldn't be? But when two of the measurements have a difference of 3.5 centimeters I want her to remeasure. How much of a difference should I allow?

[luuknam]

1. Rounding errors can cause slight differences.

2. If she's supposed to measure to 1 mm (a tenth of a cm), I'd be okay with 4.5-4.7 if the books says 4.6, but no bigger error than that. 

 

ETA: but, if she measures a square to have sides of 4.5 her area will be 20.25, whereas the book's 4.6 would cause an area of 21.16, so for the final answer you'd just care that the calculations are correct with the numbers she got (I know you weren't doing squares, that was just an example).

 

ETA: wrt 1, to further clarify, suppose you measured a line to be 1.0 cm. Now, suppose you divided that line into 3 segments and measured the segments to a mm precision - they'd be 0.3cm each. 0.3+0.3+0.3=0.9, even though the line was 1.0. If you were measuring to a tenth of a mm, then you'd have less of a rounding error... you'd measure 0.33 cm, so 0.33+0.33+0.33=0.99cm. Still not quite 1.0, but much closer.

Edited November 25, 2015 by luuknam

[athomeontheprairie]

^{Thanks for the explaination-I assumed it was something along those lines.}

 

^{WRT2... Using the square as an example. 3 identical squares: She measures them at 4.5, 4.6, and 4.7. The areas are now respectivily: 20.25, 21.16, and 22.09. The difference between the two extremes is 1.84=is this too great of a difference? Should she be required to go back and remeasure? (This is for more complex shapes, not squares where, hopefully, she would notice the difference prior to calculating) What is the greatest difference I should be allowing?)}

[EKS]

Is this an exercise in how to measure with a ruler or how to find area? Because if it is the latter, and she is having trouble with the slight differences between her measurements and the book's getting magnified by subsequent area calculations, I'd separate the two issues and just give her the book's measurements and tell her to use those. If she needs practice using a ruler, have her do that separately.

[luuknam]

^{Thanks for the explaination-I assumed it was something along those lines.}

 

^{WRT2... Using the square as an example. 3 identical squares: She measures them at 4.5, 4.6, and 4.7. The areas are now respectivily: 20.25, 21.16, and 22.09. The difference between the two extremes is 1.84=is this too great of a difference? Should she be required to go back and remeasure? (This is for more complex shapes, not squares where, hopefully, she would notice the difference prior to calculating) What is the greatest difference I should be allowing?)}

 

No, I would not have her go back and remeasure. I'd either check her work to see she did the calculations correctly based on the measurements she got, or, if I'm lazy, I'd give her the measurements in the book (after she measured the things herself, since at 10 she probably could benefit from measuring as well as calculating area) and have her use those measurements to calculate the area. And, it'd also make a great little discussion about math vs real life. Math is precise... but if you throw in real life measurements, you're going to have measurement imprecisions that are going to make the answers from your math less precise.

 

In other words, as long as each of the individual measurements is within 1mm of the book's answer, I would not have her remeasure. It doesn't matter what the final answer's difference is... if she did a million squares at 4.5 instead of 4.6 the area would be 910,000 sq cm too small (4.6*4.6*1,000,000-4.5*4.5*1,000,000), but that doesn't mean she'd have to remeasure a million squares.

 

Hope that makes sense.

[luuknam]

Btw, if you have a nice big tile floor, say 1,000 tiles long and 1,000 tiles wide, you wouldn't want to measure a tile and multiply by 1,000,000 to find the area of your floor, because if you were just 1 mm off you'd have a pretty big error in your final result (910,000 sq cm too few if you measured 4.5cm instead of 4.6cm, which would be a pretty big problem if you bought carpet to put over the floor). You'd want to measure all 1,000 tiles in the width at once (and then also the 1,000 tiles in the length, just because). Your error should be less than 910,000 sq cm I'd hope (e.g. if you measured 4599 cm by 4599 cm, your error would be only 9199 sq cm).

[Paige]

Maybe I'm wrong, but when my kids have had minor measurement differences compared to the answer key, and when I check and their measurements are correct, I assume that it's a printing error in the book. Depending on how wide the lines are and how well the book printed, I think small errors can be expected occasionally. 

[kateingr]

Luuknam, I love your tile example. Understanding where error creeps in is really important, and this example with area is a great example of how squaring a number magnifies the error.

 

OP, I don't think it matters at least how far off your daughter's answers are as long as she understands how to measure with a ruler, how to calculate area, and why small differences end up mattering. The other thing I'd want to make sure your daughter understands is that the shape does in fact have one, true area--it's just our inexact measuring tools that make it difficult to find precisely.

[EndOfOrdinary]

I think this might be an ideal place to talk about how we are humans, and humans are fallible. Math is just a tool to help us, not a savior. It is only as helpful as we can make it, so we have to use or brains not use our math. Measure as closely as you can, calculate as well as you can, but in the end know this is all very messy and imprecise