Each week will now also include the previous POTW answers. See below. Ask me if you have any questions! Happy POTW-ing...
Grade 8 Solution POTW #1
Grade 7 Solution POTW #1
Grade 7/8 Question POTW #2
There is a math riddle involving the missing piece of the triangle below.
The area seems to remain the same yet there is a piece missing when you reorganize the
Right triangle. BOTH triangles make a 13 x 5 shape.
All you've done is reconfigure the parts of the triangle, the sizes of each piece
have not changed. How can there be a missing space if it makes the same shape?
Explain with details.
The area seems to remain the same yet there is a piece missing when you reorganize the
Right triangle. BOTH triangles make a 13 x 5 shape.
All you've done is reconfigure the parts of the triangle, the sizes of each piece
have not changed. How can there be a missing space if it makes the same shape?
Explain with details.
Grade 7/8 POTW:
ReplyDelete(Fancy Schmancy GIF, by the way)
Since the red triangle is an 8 x 3 triangle, it would take up the whole space before the reconfiguration. But, since the blue triangle is only a 5 x 2, it would take up less space, therefore leaving more space (specifically 3 blocks in length) for the yellow irregular hexagon to take up. But since the hexagon's bottom side is only a 2 block length, there would be one block that isnt taken up. Once the red triangle is moved to the top section, since it is one block shorter than the blue one, it would have to take up some of the yellow hexagon's space, forcing the hexagon to move in a space that it does not fit. This is why there is a missing space even though the same pieces are being used and the same amount of space is being taken up. This question reminds me of shapes that have the same area, but different perimeters. This configuration uses the same shapes, but different rearrangements of the shapes, making them seem different, yet the same at the same time (sorry if that's a little confusing).
(and all this action is happening while the green piece stays still)
Oh wow Fiona!
DeleteGrade 7/8 POTW
ReplyDeleteI first thought that there was probably a missing space because the triangle was arranged wrong; maybe the yellow and green shapes should have been flipped or something. So, to start off solving the POTW, I tried rearranging the green and yellow pieces to see if anything could work. Nothing did, and then I realized there's probably something more to the problem, as my method didn't use any math at all. So then I thought, "what if this is a trick question and this big triangle isn't even a triangle at all?" and decided to calculate the area of all of the shapes, then the area of the whole "triangle".
BLUE TRIANGLE : measurements= 5 by 2 -> b x h divided by 2 = A -> 5x2 divided by 2 = A -> 10 divided by 2 = A -> 5 units squared = A.
RED TRIANGLE : measurements= 8 by 3 -> b x h divided by 2 = A -> 8x3 divided by 2 = A -> 24 divided by 2 = A -> 12 units squared = A.
YELLOW IRREGULAR SHAPE : I split this up into two parts, the 2 by 2 square and the 3 by 1 rectangle. lxw(square) + lxw(rectangle) = A -> 2x2 + 3x1 = A -> 4 + 3 = A -> 7 units squared = A.
GREEN IRREGULAR SHAPE : I also split this up: a 3x2 rectangle and a 2x1 rectangle. lxw(rectangle 1) + lxw(rectangle 2) = A -> 3x2 + 2x1 = A -> 6 + 2 = A -> 8 units squared = A.
Add them all together and you get 32, which is supposedly the area of the big triangle. But, let's calculate the total area of the triangle just in case.
The big triangle's dimensions are 8+5 by 3+2 (13 by 5) .
bxh divided by 2 = A -> 13x5 divided by 2 = A -> 65 divided by 2 = A -> 32.5 units squared = A.
Now we can see that there is definetely 0.5 square units missing, but why?
I examined the GIF a bit longer, until I noticed that the blue triangle doesn't fit exactly into the bigger red triangle. The blue triangle ends at exactly two squares high after 5 squares to the left, while the red triangle doesn't and ends up at a little under two squares high at the 5 square checkpoint (this is hard to explain) . What I can conclude from this is that the big triangle will not have an exact diagonal like other triangles not composed of different parts do. I think that maybe the excess in the diagonal is equal to the area of the missing square, and therefore is the reason there is a missing square in this adjusted triangle.
Esther - Gr. 7/8 POTW
ReplyDeleteAll of the shapes together create a triangle. The longest side of the triangle is what the red and blue smaller triangles create. The blue triangle is 5 spaces long and 2 spaces high, while the red is 8 spaces long and 3 spaces high.
When these two triangles switch spots, the side length of the larger triangle remains the same, however because the blue triangle is only 2 spaces in height, it is merely a fraction of what the red triangle covered. The red triangle is 3 spaces in height, which is taller than the blue triangle's initial spot. Because of this, the extra 5 spaces of the yellow hexagon overlap with the red triangle. This pushes the hexagon to move to the remaining space where it can fit without overlapping. This extra space is a 2 by 3 rectangle, but the hexagon does not contain any 2 by 3 rectangle (there is only a 2 by 2 square) which is why there is an extra space left.
(I don't know if any of this is right or makes sense)
(Brian)
ReplyDeleteThe shape is not actually a triangle. You can see that at the top of the first frame the blue triangle isn't fully complete. When the red shape is moved the slope of the triangle is slightly changed, changed enough that the one square is removed.If you measure the first and last frame you can tell that the shape is slightly changed.To find the slop you do slope=rise/run. To prove that the slope has changed. The area under the blue triangle at first is 20 and the red's 12. Then after the change the area under the blue triangle is only 5 and then red is 28. That means the ratio is 32:33. Then 33 is where the extra 1 square comes from.
Grade 7/8 POTW
ReplyDeleteAllen(In case I'm not signed in)
Since the yellow shape(I don't know what shape it is) is an irregular shape, it can only fit with the green shape in order to not leave an missing white block open. So once the yellow block has to move, thanks to the swap of the blue triangle with the red triangle, the yellow is forced to move into an area that it can't completely fit. The red triangle is 8x3 and the blue is 5x2. Since the blue triangle won't fit into the 8 space of the red triangle, it causes the white squares. And the red triangle is too tall for the original blue triangle area, it forces the yellow to move into the blank area beside the blue triangle and it won't fit into it. That's why the there will be 1 white square missing.
(Brian)My first answer was wrong
ReplyDeleteThe slope isn't changed, if it was then the area would be different
The area under the blue triangle at first is 20 and the red's 12. Then after the change the area under the blue triangle is only 5 and then red is 28. That means the ratio is 32:33. Then 33 is where the extra 1 square comes from.
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ReplyDeleteAfter watching the GIF multiple times I focused on certain points along the hypotenuse such as point (5,2). The Red triangle fails to reach the point but the Blue triangle does suggesting there's been an increase in area. Since it is impossible to change the area of a set perimeter with given measurements, the only way this is possible is because the riddle's given area could be more than what the actual area is allowing them to fit in an extra square, so the first triangle may not be a triangle at all, but the second image is.
ReplyDeleteHerald: Messenger bringing news
EON by Alison Goodman
Grade 7/8 POTW
ReplyDeleteBefore I start solving the problem, I first want to identify the parts of the "right triangle".
Red right triangle = 3 x 8
Blue right triangle = 2 x 5
Yellow irregular trapezoid
Green irregular trapezoid
The original shape is a "right triangle". The blue triangle is translated 3 units south and 8 units west leaving a 2 x 5 space. The red triangle is translated 5 units east and 2 units north leaving a 2 x 6 space. The yellow irregular hexagon still has to cover a 1 x 2 space and is translated 1 unit south and 3 units west. Since the yellow hexagon only has 5 available unit^2 space, it can't cover the last unit which is why the "right triangle" has that square space.
However, this only covers the translations and how the shapes showed the extra square, but it doesn't really explain at all why that space is there. The answer involves the 2 right triangles.
Red right triangle = 3 x 8
Blue right triangle = 2 x 5
The real reason why the "right triangle" has that 1 square difference is because it isn't even a triangle. If the entire "right triangle"'s hypotenuse were really a straight line, then the dimensions of both smaller right triangles would be proportionate making a straight line. One is 3 x 8 while the other is 2 x 5. In order to be proportionate (or in ratio), the red triangle would have to be 3 x 7.5 or the blue triangle would have to be 2 x 5.333..., but since it isn't, the "triangle"'s hypotenuse isn't straight or technically not a hypotenuse at all. In other words, the original shape is an irregular concave quadrilateral turned into an irregular octagon. The "hypotenuse" from before is originally an obtuse angle turned into a reflex angle. So really, this entire question is like a trick question. The "triangles" are really a quadrilateral and octagon and the area isn't 13 x 5 or higher, it's 31.5 units squared. It can make a missing space because it doesn't make the same shape. The missing square is between the extra space of the reflex angle and how it is actually supposed to be (straight).
(All of this might have been confusing so ask what needs clarification if anything)
When you rearrange the triangles you can see that at the edge it takes up some partial square space. When the change takes place you can see that the partial space shifts and takes up a little more space. Thats why one square is missing from the middle.
ReplyDeleteMaksim
ReplyDeleteWhen you rearrange the shapes shown on the diagram there is a space left because if you count the length of the red triangle it is one square longer than the blue triangle and the yellow shape. That's why when the red takes place of the yellow and blue shape the red fits perfectly and the others leave one square blank.
Leah's POTW
ReplyDeleteThe space that the two right triangles do not take up in the full right triangle is 5x3 in the beginning but in the end of the gif, the space is 8x2. 5 times 3 is 15 and eight times two is 16. 16 minus 15 is one, leaving the one square that is empty in the full right triangle.
The angle of the slope is different when the gif changes and there's one less space than there were before
ReplyDeleteMaruti
ReplyDeleteThe two triangles have different dimensions 8x3 and 5x2, so when they move they don't fit into the right dimensions, and they force the yellow shape into a space with 8 spaces while it only takes 7 spaces which is why there is 1 space not filled in.
The area of a right angle triangle with the height of 5 units, and the base of 13 units would not have space for an extra square, but the overall shape is not a real right triangle. If you find the angle of the overall triangle, you would have to do 5 / 13 (height divided by width) which equals about 0.38. Then, you could find the angle of the red triangle by doing 3 / 8 = 0.375. Finally, we can do 2 / 5 for the blue triangle, which equals 0.4. This proves that each of the shapes have a different angle, and therefore the shape is slightly concave, allowing an extra square when the shapes are rearranged.
ReplyDeleteThe Red Triangle is 8x3 so when the blue piece is moved, it takes up less space. So therefore, when the red is moved to the top, more space it there. So there is one extra space available. The blue one is 5x2 and to prove that this statement is true, the blue right triangle is 5x2 at the start but in the final formation, the space is 8x2. 5 times 3 is 15 and 8x2=16. But it is 15 so 16-15=1 so there is 1 empy square
ReplyDeleteSince the red right triangle is 8x3 (the bottom of the triangle being 8) and the blue right triangle is 5x2 (the bottome of the triangle being 5), it takes up less space when they are switched around and leaves 3 extra blocks (since 8-5= 3) and then the bottom of the yellow irregular hexagon takes up 2 of these spaces so there is one empty square remaining. Because the red right triangle was larger than the blue right triangle, it took up more space at the top and had more space remaining at the bottom for the blue right triangle.
ReplyDeleteCarina
ReplyDeleteThe space that the 2 right triangles don't cover up in the full right triangle would equal 5 by 3 in the beginning , and then after the gif moves, the dimensions would be 8 by 2. Since 5 times 3 is 15 and 8 times 2 is 16, you woul then subtract 15 from 16 and that would equal 1 which represents the space left in the ending of the gif.
The space that the two right triangles don't take up in the full right triangle is 5 by 3 in the beginning, but then afterwards when the triangles have moved, the dimensions would be 8 by 2. 5 multiplied by 3 is 15 and 8 multiplied bye 2 is 16. Then you would subtract 15 from 16 and you would get 1, which would then represent the blank space in the end of the gif.
ReplyDeleteAnya (wasn't able to sign on)
ReplyDeleteRed triangle= 8x3
Blue triangle= 5x2
so when the smaller blue triangle takes the place of the big red triangle it is likely that there would be left over space.
the triangle has a side of 6 and the side of the blue triangle is 2 so 2x6=12 and the area of the red triangle= 13 so 13-12= 1 therefore one space left
POtW Edward
ReplyDeleteThe triangles are different sizes which means when they move, they make a different shape. After the triangles have moved, the yellow irregular hexagon moves. According to the gif, there are 6 spaces for the hexagon to cover to become a full triangle. In any way the hexagon is rotated, it cannot cover the 6 spaces (3 by 2) without interfering with other shapes. This means that it can only cover 5 spaces which means one space is left empty. The green hexagon does not move which means this is the final look of the shape. This is why there is one space open.
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ReplyDeleteThe reason the shapes match is because the angles of the triangles are the same, but when moving the shapes, moving the green and yellow next to each other can occupy the missing length. However, due to the nature of the rectangle green and yellow create (one of them has one more square in mass), doing this will cause a missing space.
ReplyDeleteThe reason you can create that shape is because the 3 by 8 red triangle can fill up the whole space however the 5 by 2 and the 2 by 2 + 3 by 1 cannot as when you put them in to replace the 3 by 8 that moved it leaves one empty space when placed together as yes it has the amount of squares necessary and even more but those squares cover separate squares so if you take out the unnecessary squares there are actually not enough to cover up all the space
ReplyDeleteNaumana's POTW:
ReplyDeleteThe total area of the figure is 32.5 square. The area of the blue triangle is 5, and the red triangle is 12. Both triangles fit perfectly in the shape. However, both triangles have an irregular slope. So, when rearranged, the triangles will take up more space near the area of the irregular edge, creating a missing square.
POTW's Worksheet :P
ReplyDeleteWell, my first reaction to the gif is that the blue triangle doesn't necessarily fit in the exact same position as the red one. It's slightly lower and shorter. As a result, the yellow piece can't really slide into it's space without a block not being filled because the blue shape is 3 blocks shorter in width than the red. If we figure out the measurements, maybe things would clear up?
(Note: It's hard to count squares from a moving gif)
Red: 8 x 3
Blue: 5 x 2
(Considering 8 and 5 is the base)
8 - 5 = 3
These 3 squares have been left out when the blue replaced the red. In order to make the end result the same triangle without overlapping and switching, the yellow irregular shape has to move. However, it's "base" is only 2 blocks wide. (3-2 =1) so when it goes into it's position, it leaves the empty white space.
The only way this workd is if both of the angles of the triangles are the exact same but when you move the shapes because the green and yellow rectangles is 15 cm 2 when you move the yellow one because the red and blue triangles aren't the same it changes the shape
ReplyDelete