Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
Display an image of the flag of Nepal for all to see:
Tell students this is the flag of Nepal, which is a country in South Asia that borders India and China. Ask students: “What do you notice? What do you wonder?”
Students may notice:
It looks like it’s made of triangles.
I could break it up into a rectangle and three triangles.
The bottom left corner looks like a right angle.
Students may wonder:
What shapes make up the flag?
Why does it look like this?
How could we break up the white symbols into simpler shapes?
If it does not come up during the discussion, ask students how they might break up the shape of the flag into simpler shapes so that they could, for example, find its area.
Then tell students that, in this task, they will see how to construct the flag, and examine its shape. They also will examine the shapes that make up a flag of their choosing.
Task Statement 1
Teacher Instructions
Students need to trisect a line segment in order to construct segment \(AC\). They can measure first to see where the points that trisect the segment should go. Instructions for trisecting a line segment can be found online. There are many ways of doing it, and some are easier than others. If students find the instructions themselves, they should be encouraged to compare several methods. If students try different methods, it could be beneficial to invite selected students to show the class what they did. The other steps for constructing the flag should be familiar. Providing students access to geometry software can streamline the construction process.
If students have trouble finding the angles, encourage them to give segment \(AB\) a specific value and calculate the other side lengths.
Once students have completed the task, invite them to share the flags they chose and to show how they adapted them to fit on the triangular banner.
The Constitution of Nepal includes instructions for constructing the flag.
Step 1:
Draw a line segment \(AB\) of the required length. Segment \(AB\) will be the bottom edge of the flag.
Step 2:
Draw a line segment \(AC\) perpendicular to segment \(AB\) so that the length of segment \(AC\) is \(\frac43\) the length of segment \(AB\).
Step 3:
Mark point \(D\) on line segment \(AC\) so that segment \(AD\) is the same length as segment \(AB\). Then join points \(B\) and \(D\) with a line segment.
Step 4:
Mark point \(E\) on line segment \(BD\) so that segment \(BE\) is the same length as segment \(AB\).
Step 5:
Draw a line segment, \(FG\), that goes through point \(E\) and is parallel to and the same length as segment \(AB\). Point \(F\) should be on line segment \(AC\).
Step 6:
Join points \(C\) and \(G\), with a line segment.
Construct the flag according to the instructions. You can make it any size you want.
Find the measurement of each angle in the flag.
Imagine there is a parade featuring the flag of Nepal. There are small flags and large flags in the parade. Decide on the size of each type of flag.
How much material is needed to make each type of flag?
If the border of a flag is made by sewing a ribbon along its edge, how long is the ribbon for each type of flag?
Imagine a parade featuring a flag that is meaningful to you. (Choose a flag or design your own.) At the parade, there are small flags, large flags, and triangular banners such as this:
How can you adapt your flag to fit on the triangular shape?
How much material of each color do you need for each kind of flag?
Task Statement 2
Teacher Instructions
Students who want an extra challenge can find the areas and the perimeters of both the small flag and the large flag in the second question, and then compare the areas to see the effect of the scale factor. The relationship between scaling lengths and scaling areas will be a topic in the next unit, so students do not need to have a full understanding of it at this time, but this is a good opportunity to preview the concept.
Another opportunity for an extra challenge is to describe the exact ratios between the side lengths, using radicals. An especially interesting detail students may notice is that the ratio between segments \(AF\) and \(AD\) is the same as the ratio between segments \(AD\) and \(BD\), \(\tfrac{1}{\sqrt2}\).
Providing students access to geometry software can help them visualize the shapes.
Once students have completed the task, invite them to share the flags they chose and to show how they adapted them to fit on the triangular banner.
The Constitution of Nepal includes instructions for constructing the flag.
Step 1:
Draw a line segment \(AB\) of the required length. Segment \(AB\) will be the bottom edge of the flag.
Step 2:
Draw a line segment \(AC\) perpendicular to segment \(AB\) so that the length of segment \(AC\) is \(\frac43\) the length of segment \(AB\).
Step 3:
Mark point \(D\) on line segment \(AC\) so that segment \(AD\) is the same length as segment \(AB\). Then join points \(B\) and \(D\) with a line segment.
Step 4:
Mark point \(E\) on line segment \(BD\) so that segment \(BE\) is the same length as segment \(AB\).
Step 5:
Draw a line segment, \(FG\), that goes through point \(E\) and is parallel to and the same length as segment \(AB\). Point \(F\) should be on line segment \(AC\).
Step 6:
Join points \(C\) and \(G\), with a line segment.
Clare is constructing the flag, and she starts with segment \(AB\) of length 6 inches.
Here is her finished flag shape. Use the instructions to fill in as many lengths and angles as you can.
Imagine there is a parade featuring the flag of Nepal. There are small flags and large flags. Decide on the size of one type of flag.
How much material is needed to make the flag?
If the border of a flag is made by sewing a ribbon along its edge, how long is the ribbon?
Imagine a parade featuring a flag that is meaningful to you. (Choose a flag or design your own.) At the parade, there are rectangular flags, and triangular banners such as this:
How can you adapt your flag to fit on the triangular shape?
How much material of each color is needed for each kind of flag?
