This high level job is an instance of using geometric methods to solve architecture problems and also satisfy physical constraints. This job is easily accessible to all students. In this task, a typographic grid device serves together the background for a standard file clip. A metric measurement scale is drawn across the bottom of the grid and the document clip extend in both direction slightly beyond the grid. Students are offered the approximate size of the paper clip and determine the number of like document clips do from a provided length the wire. Extending the paper clip beyond the grid gives an chance to include an estimation ingredient in the problem. In the attention of open-ended difficulty solving, no scaffolding or additional questions room posed in this task. The record clip modeled in this difficulty is one actual big standard file clip.

You are watching: Length of a small paper clip

This open-ended task has a multitude of entrance levels. College student will apply prior knowledge to solve this problem and their knowledge of concepts such as separating a composite number into familiar geometric representations, circumference of a circle, one of a semi-circle, symmetry, estimation, and metric measurement will recognize their entry level and solution route of choice. Numerous different solution paths are possible and must not be restricted to the options provided. Offered the estimate component developed into this problem, we encourage friend to expropriate all reasonable responses. Materials made accessible for open-ended trouble solving rises the likelihood of varied responses.

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## Solutions

Solution:Solution 1

**Basic Level Responses**

One method is to division the document clip into vertical regions, and then to use the measurement grid to determine the length of the directly sections and also estimate the length of the curved sections using a string or thin wire in conjunction v the measurement range provided. One such division is accomplished using 3 vertical dividers dividing the file clip right into four unique regions together shown.

RegionsNumber of direct SectionsNumber of bent SectionsRegion A | 0 | 1 |

Region B | 4 | 0 |

Region C | 2 | 1 |

Region D | 0 | 1 |

The lengths the the straight sections were identified using the gridlines. The estimations the the lengths the the curved sections were identified using a cable or slim wire in conjunction through the measurement range provided.

RegionsMeasurement of straight Sections(listed from height to bottom)Estimated measure of curved SectionsRegion A | 2.5cm | |

Region B | 1.8cm, 1.5cm, 1.8cm, 1.5cm | |

Region C | 0.8cm, 0.8cm | 1.6cm |

Region D | 2.5cm |

The length of wire required to produce one document clip is currently approximately:$$1.8 ext cm + 1.5 ext cm +1.8 ext cm + 1.5 ext cm + 0.8 ext cm + 0.8 ext cm + 2.5 ext cm + 1.6 ext cm + 2.5 ext cm = 14.8 ext cm$$

The size of the straight piece of cable is 10 meters. Because 1 meter is the very same as 100 centimeters, 10 meter is $10cdot 100=1000$ centimeters. Finally, we uncover that in ~ $14.8 ext cm$ per paper clip, 1000 centimeters will develop approximately$$frac100014.8approx 67.6 ext file clips.$$Since we can only make a totality number of paper clips, us conclude that roughly 67 document clips might be manufactured from a straight piece of wire 10 meter in length.

Note: Students may or may not notification the two various units of measure provided in this task. The paper clip defined in terms of centimeters, and the piece of wire is explained in terms of meters. This will certainly be noticeable if 10 meters is no converted come 1,000 centimeters once determining the number of paper clips that have the right to be manufactured. The use of cable to estimate the length of the bent sections that the paper clip create a much less than precise outcome. There is a difference in the size of the curved ar in an ar A and the curved section in an ar B. This distinction is not always obvious when using a cable measurement strategy due to the fact that the distinction in the length of the curved sections is tiny (0.2cm). Second source for miscalculation is where students actually location the string when attempting to map the curvatures. Each bent section has actually an outer and also inner radius as result of the thickness that the cable which may result in potential overestimation or underestimation.

*Alternate Strategy used to division the record Clip right into Regions*

This strategy uses only designated tick marks on the measure up scale and also divides the record clip vertically into five regions while isolating the bent portions.

Solution:Solution 2

**Advanced Level Responses**

One strategy is to divide the paper clip right into vertical regions, use the measurement network to recognize the size of the right sections and also use geometric formulas to recognize the size of the bent sections. This division can be accomplished, for example, by using 3 vertical dividers separating the document clip right into four distinctive regions as shown. The external two extr lines prolong the grid and also serve as borders for the two external regions. These extr lines assist in the estimation of the size of the file clip drawn past the gridlines.

The vertical department lines were purposely put such that each curved ar of the document clip shows up to it is in in the shape of a semi-circle. The radius of the semi-circle is identified using the gridlines and also the gridline extension to accommodate the part of the file clip beyond the grid. The approximated lengths that the curved sections were established using the semi-circumference formula, $C = pi r$. There is, yet the same problem as in the previous solution; namely, deciding even if it is to usage the external radius, the within radius, or a radius somewhere between the two. The listed below table reflects the results of making use of in each instance a radius an extremely close come the external radius.

RegionsNumber of curved SectionsLength the Radius in bent SectionsEstimated one of CircleEstimated circumference of Semi-CircleRegion A | 1 | $r = 0.8 ext cm$ | $5.0 ext cm$ | $2.5 ext cm$ |

Region B | 0 | |||

Region C | 1 | $r = 0.6 ext cm$ | $3.8 ext cm$ | $1.9 ext cm$ |

Region D | 1 | $r = 0.9 ext cm$ | $5.6 ext cm$ | $2.8 ext cm$ |

The lengths of the direct sections were figured out using the gridlines.

RegionsNumber of direct SectionsMeasurement of direct Sections (listed from top to bottom)Region A | 0 | |

Region B | 4 | 1.8cm, 1.5cm, 1.8cm, 1.5cm |

Region C | 2 | 0.8cm, 0.8cm |

Region D | 0 |

The length of wire necessary to to produce one paper clip:$$1.8 ext cm + 1.5 ext cm +1.8 ext cm + 1.5 ext cm + 0.8 ext cm + 0.8 ext cm + 2.5 ext cm + 1.9 ext cm + 2.8 ext cm = 15.4 ext cm$$

The size of the straight item of wire is 10 meters. Due to the fact that 1 meter is the same as 100 centimeters, 10 meters is $10cdot 100=1000$ centimeters. Dividing, we find that at $15.4 ext cm$ per paper clip, 1000 centimeters will develop approximately$$frac100015.4approx 64.9 ext document clips.$$Since we deserve to only make a whole number of file clips, us conclude that approximately 64 record clips might be manufactured from a straight piece of wire 10 meters in length.

Note: making use of geometric formulas to calculate the bent sections in each region are a bit more dependable than making use of a wire strategy, however accuracy is tho compromised. Students may miscount the gridlines as soon as determining the radius of the semi-circle, and also there is a round off off worry when pi is provided in the formula come compute the circumference. Climate to link the level of error, the length of the document clip i beg your pardon is an approximation is separated into the 10 meter or 1,000 centimeter piece of wire.

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*Alternate Strategy offered to division the document Clip right into Regions*

This strategy divides the file clip horizontally through the middle right into two regions taking benefit of horizontal symmetry. Each ar of wire over the line can be mapped onto the precise section of wire below the line. Students calculation lengths associated with the top or lower fifty percent of the document clip and dual the results.