This week is GREAT practice for central tendency. If you have problems calculating central tendency make sure you talk with your teacher!
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Happy POTW-ing...
Grade 7/8 Solution POTW #2
Here is the answer for POTW #2 https://www.youtube.com/watch?v=ExUV3GOTDqE
Grade 7/8 Question POTW #3 (central tendency):
Grade 7/8 POTW:
ReplyDeleteInfo:
- 180,181,183,187,188,190,193,195,196
- Riky's height is also a whole number.
- With his height, all ten mean and median are increased by 0.5 cm.
Original median: 188
Original average: 181.1(recurring)
We need to find the relationship between the incorrect height and the correct height, because they are unknown. For this, we would need to use variables (x for incorrect, y for correct).
The average height of the team with x would be (1693+r)/10
The average height of the team with y would be (1693+y)/10.
To find the relationship between x and y, we would have to add the 0.5 to the first equation and simplify.
(1693+x)/10 +0.5=(1693+y)/10
1693+x+5=1693+y
x+5=y
This means that the incorrect height is 5 less than the correct height. Now, we can figure out the values of y (correct height)
We could use trial and error to figure out this problem. Since the original median is 188, we cannot add any numbers before 188, because the average would stay the same (187.5). We would need to start guessing at 188.
Using 188, we need to calculate the old median, then the new median, and then compare the two.
Old= (187+188)/2
Old=187.5
New=(188+188)/2
New=188
This is a possible height.
Using 189, we need to calculate the old median, and the new median and then compare the two.
Old= (187+188)/2
Old= 187.5
New= (188+189)/2
New= 188.5
This is not a possible height.
(Now you should get the pattern of my trial and error)
Possible height= 190
Old= (187+188)/2
Old= 187.5
New= (188+190)/2
New= 189
This is not a possible height.
Possible height= 191
Old= (187+188)/2
Old= 187.5
New= (188+190)/2
New= 189
This is not a possible height.
Possible height=192
Old= (187+188)/2
Old= 187.5
New= (188+190)/2
New= 189
This is not a possible height.
Possible height=193
Old= (188+188)/2
Old=188
New= (188+190)/2
New= 189
This is not a possible height.
Possible height=194
Old= (188+189)/2
Old= 188.5
New=(188+190)/2
New=189
This is a possible height.
Any value above from 194 and above would remain as the same median (189).
The only possible solutions are 188cm or 194 cm.
POTW
ReplyDeleteNumbers:180 181 183 187 188 190 193 195 196
Mean=181.1 repeated
Median=188
The 9 heights combined is 1693. In order to find Riky's height, I will make his height into a variable;R. (R+1693)/10. Since the coach made a mistake with Riky's height, I will have to add 0.5 to the equation. (R+1693)/10+ 0.5. R is the incorrect height so to make things easier, I will replace R with C to signify his correct height. C+1693)/10. O.5 times 10 = 5. 5 is how much less the incorrect height is compared to the correct height.
Finding the median is the easiest way to answering this question.
The old and correct median of this data set of heights is 187.5 as 188+187=375/2=187.5
I checked this by adding 0.5. 187.5 plus 0.5=188=incorrect median. This is how I know 187.5 is the old but correct median. Next, I will find the possible heights for Riky knowing that they can only be whole numbers.
I will start from 188 as if I start from a number below that, then the median would be below 188 and the average would stay the same. (188 and 187 are always the middle numbers and their average is 187.5 always).
Height=188
188 +188=376/2=188 188 is a whole number so 188 is a correct height for Riky.
Height=189
188 +189=377/2=188.5 188.5 is not a whole number and therefore not a suitable height for Riky.
Height=190
188 +190=378/2=189 189 is not a suitable height for Riky as there is already a height that is 90.
Height=191
188 +190=378/2=189 189 is not a suitable height for Riky.
Height=192
188 +190=378/2=189 189 is not a suitable height for Riky.
Height=193
188 +190=378/2=189 189 is not a suitable height for Riky.
All heights over 194 would not be suitable as the same median would keep getting repeated over and over again(189). 194 would be a suitable height as 189 and 188 would have to be added for the old but wrong heights. The average of these two numbers are 188.5. The new height would be 189 and 189 is 0.5 greater than 188.5 and it is a whole number which means it is a possible height.
The possible heights for Riky are 188 and 194 cm.
Maksim done by hand
ReplyDeleteAnswers- The possible correct heights for Riky are 188cm and 194cm
All of the team members heights excluding Ricky's= 1693 cm
ReplyDeleteI started with finding the median
If Ricky's original number is 187 or less than the original median would be 187.5, the new median would have to be 188 meaning new height would be 188.
188+1693=1881/10=(188.1-0.5)x10-1693+183 so the original number would be 183 and the new number would be 188.
If our original number was 188 the original median would be 188 the new median would have to be 188.5 meaning the new number would have to be 189, although it works for the median but between the numbers there isn't enough space for the 0.5 gap.
Finally if the original number is 189 the original median would be 188.5 the new median would be 189 and the new number would be 190 or greater, (189+1693)/10= (188.2+0.5)x10-1693=194
So the only two possibility are 188cm and 194cm
Math Journal #2
ReplyDeleteNew Heights
Introduction
We are learning to
Calculate measures of central tendency
Solve for a missing variable
In order to be successful, we need to:
Have an understanding of basic algebra
Given Information
Question
New Heights
The heights of the ten members of the Riky’s basketball team were revealed. The heights, in CM, of his teammates were: 180, 181, 183, 187, 188, 190, 193, 195, and 196. Riky’s height is also a whole number.
The coach made a mistake when measuring Riky’s height. After it was corrected, both the mean and median increased by 0.5. Find all possible values for Riky’s height.
Breakdown
The mean and median of Riky’s team and the incorrectly recorded height is 0.5 less than the mean and median of his team and actual height. What are all possible values of Riky’s real height?
Required to Find
We need to find possible values for Riky’s old and new height. To do this, we need to find the mean and median of the teams before and after the height is corrected.
Application
Let’s apply variables to Riky’s wrong and right height. R=Riky’s real height r=Riky’s wrong height. Using some basic algebra and the calculations for mean, we get this:
(180+181+183+187+188+190+193+195+196+r)/10+0.5= (180+181+183+187+188+190+193+195+196+R)/10
Let me explain. The formula for mean says that “Every number from the data set divided by the number of numbers in that data set is the average of that data set.” That means that (180+181+183+187+188+190+193+195+196+x)/10 will be at the beginning of each mathematical statement. Plugging in what we know about their height differences, we get the equation. So what do we do next? We remove the denominator.
1693+r+5=1693+R
You might be wondering what just happened. Both sides of the equation were multiplied by 10! It might seem illegal, but it falls under algebraic rules that “to remove a denominator, or multiple denominators, simply multiply the whole equation by the denominator or LCM of the denominators.” Therefore, multiplying by 10 is a legal action.
1693+r+5=1693+R
We can then eliminate 1693 from both sides, as it is a constant on both sides. That leaves us with this formula;
r+5=R
This narrows down the options and makes the search for Riky’s height much easier.
(I had to split this into to parts srry)
ReplyDeleteSolution
We need to try different values of r we find reasonable. I believe all values from 182 to 194 should be tried. These were the results:
If r is 182 or less: The median would stay at 187.5 with the height correction, but the previous median would also be 187.5. Impossible. (This is because both terms are less than the median).
If r is 183: This would be a possible solution, as Riky’s correct height during this scenario would increase the median from 187.5 to 188. This would make Riky’s correct height 188, and thus gives us one solution.
If r is any number from 184 to 188: These all alter the median more than intended (an increase in the median of either 1 or 1.5. Impossible.
If r is 189: The median will increase by 0.5, as the corrected height will be 1 higher than normal. ½ of 1 is 0.5.
If r is 190 or more; the median will remain unchanged at 189. This would make the scenario impossible.
Statement
“The possible values for Riky’s correct height are 188 and 194.”
Reflection
I believe that this math journal was harder than usual because it involved elements not normally seen in grade 7, like cancelling out the denominator and altering the equation. While it isn’t necessarily a hard algebra problem, it may have been a struggle or some of the other students to overcome. Personally, knowing a bit of algebra, I would rate it a 5 (Ready for grade 8) question. I believe that, personally, this was a fun way to make use of Algebra in this problem, as well as using the measures of Central Tendency.
I learned that sometimes, when you think there’s not enough info, there actually is. The hardest part of the problem, to me at least, was how to go about solving it. When I saw how nicely everything fit together at the end, I was genuinely surprised.
You can use this in the real world to… extract hidden info, I guess? If someone from the deep web were to leak info on a certain spy organization, but not straightforwardly, you could decrypt the message… that’s a little far fetched. The only thing I could think of using this knowledge in real life is predicting the stocks during an uncertain time.
That certainly was… a math journal. Again, not much more to say.
THE END… or is it?
Done by hand however the possible heights for Riky are 188 and 194.
ReplyDeleteGrade 7/8 POTW
ReplyDeleteIn order to solve this problem, I first had to find out the relationship between the incorrect height (variable a) and the correct height (variable b). To do this, we can use the other 9 heights and the general concept of the mean (average) in an equation.
Mean:
(180 + 181 + 183 + 187 + 188 + 190 + 193 + 195 + 196 + a)/10 + 0.5 = (180 + 181 + 183 + 187 + 188 + 190 + 193 + 195 + 196 + b)/10
Simplified to:
(1693 + a)/10 + 0.5 = (1693 + b)/10
This can be further simplified by multiplying both sides by 10 simplifying to:
1693 + a + 5 = 1693 + b
This is even FURTHER simplified by subtracting 1693 from both sides.
a + 5 = b
In other words, the relation between the incorrect and correct Riky height is that the correct height is 5 more than the incorrect height. Now we have to use this requirement when solving the actual values using the median.
Median:
Instead of trying out every possible way (even in the range of the values), there is a much quicker and more efficient way of doing this. First I'm going to list the heights from least to greatest:
180, 181, 183, 187, 188, 190, 193, 195, 196
After a glance at these values, you can realize that the incorrect median and correct median only have certain possible values. If variable a is less than or equal to 187, the incorrect median would be 187.5. In this case the correct median would have to be 188. The only way to do this is with the value 188 (due to the order of the numbers). Therefore 188 is a possible value and there are no other incorrect values equal to or below 187. If variable a is more than or equal to 190, the incorrect median would be 189 meaning the correct median needs to be 189.5. 188 is ALWAYS influencing the median and due to this and the order, it is obvious to assume that there are no possible values when the incorrect median is above or equal to 190. If the incorrect value was 188, the incorrect average would be 188 and the correct average would be 189. This doesn't fit the requirements. If the incorrect value was 189, the incorrect median would be 188.5 and the required correct median would be 189. The only value available is 194 which by default provides a correct median of 189. By only doing 4 calculations for the median (using a sort-of shortcut),
I can conclude that Riky's correct height can be (in cm) either 188 or 194.
did it in my math journal but the possible heights for Ricky are 188 and 194
ReplyDeleteSo we can find an algebraic equation:
ReplyDelete(180+181+183+187+188+190+193+195+196+r)/10+0.5= (180+181+183+187+188+190+193+195+196+R)/10
Do we know that the correct height is 5 more than the wrong height.
Now, we find the median and to do this, we have to list it from least to greatest.
180, 181, 183, 187, 188, 190, 193, 195, 196
Now it's about trial and error because: With his height, all ten mean and median are increased by 0.5 cm. So after I did my trial and error, I can conclude that the possible heights for Ricky are 188cm and 194cm.
I know that the height difference is 5 cm between the fake and real heights, because 0.5 times 10 is 5. Since the rule of the mean and median don't work if the fake height is below 183, I know it is impossible. However, when you look at 183, the fake median is 187.5 and the real height is 188, that is correct. So one real height is 188. The next real height is 194, which I know because the fake height is 189, so the median is 188.5. When the real height is introduced, the median becomes 189, which follows the rule. Anything above 194 keeps the median the same and the mean continually gets higher. So the real heights that Riky can be are 188cm and 189cm
ReplyDeleteDear Mr. Milette,
ReplyDeleteI didn't understand the POTW but I still tried. Maybe after you could explain it to me?
x=Ricky’s height
ReplyDelete180+181+183+187+188+190+193+195+196 = 1693
Find Median for original
Case 1: If x is 188 then median = median = 188 + 189/190/190+
If 188 +189 then median = 188.5 works but not no numbers with x will make a original median 189 expect 90+
Orignal mean (1693 + 189) /10 = 188.2
(188.2+0.5)x10-1693= 194
Numbers can only be 188 and 194
To find the solution, we would have to add up the 9 known heights, and Riky’s unknown height, and then divide it by 2 to find the average. In this case x will be Riky’s incorrect height, while y can show Riky’s correct height. This could be shown as:
ReplyDelete(180 + 181 + 183 + 187 + 188 + 190 + 193 + 195 + 196 + x) / 2
You would add 0.5 to the solution, which would give you the correct average of Riky’s height.
You can change this to:
1693 + x + 5, or, 1692 + y
Therefore,
x + 5 = y
So, as long as y is 5 more than x, the mean is 0.5 more.
Now, let's look at the solution for the median
To find this, you can go through many different numbers, but in the end these are the 2 I found that had the median increase by 0.5.
If the incorrect height was 189, the correct height would be 194. If we find the median of the correct height, it would be 189, but with the incorrect height the answer would be 188.5, which is 0.5 less than 189. Therefore, one possible answer is 194
Now, if the incorrect height is 183, then the correct height would be 188. The median with 183 would be 187.5, but with 188, the median would now be 188, which is also 0.5 cm more than 187.5. Therefore, another possible answer would be 188.
So, the 2 possible answers would be 188, and 194.
My POTW was done on paper.
ReplyDeletePossible heights for Riky include 188 and 194
So ricks hight is 5 more than what was put down.
ReplyDelete180, 181, 183, 187, 188, 190, 193, 195, 196= median is 188. 188 + .5 is 188.5. if this is the median then one possible answer would be 194. another way is 188.
Using algebra you can find the solution. In order to use algebra we first have to determine the variable and the equation if the variable is Riky's height then x=Riky's height and m=Riky's incorrect height, now we need to find the equation which would be
ReplyDelete(180 + 181 + 183 + 187 + 188 + 190 + 193 + 195 + 196 + x) / 10 = y + 0.5
performing this formula with variables you will eventually come across the fact that the heights of everyone plus Riky's with 5 or 1693 + m + 5 = y + 0.5
then using this formula multiple times checking mean and median we then come across 2 numbers that would fit the answer 194 and 188
This was way to late but anyways if we break down the equation from what we learned in algebra we can find that the wrong height (wh) and Ricky's height (rh) is 5 numbers apart(from wh to rh). Then you can elminate all rh numbers below and equal to 187 as 187 + 188 wouldn't be 0.5 higher(wh numbers below 182 too). Then to get 188 which is 0.5 higher than 187.5 you would have to add 188 as rh or as wh. After that all numbers are invalid as they are all more than 0.5 above the median.
ReplyDeleteThis is way too late but Ricky heights are 188 and 194 as 188 and 188 can only make 188 as a median and the wrong and Right height only reach 188 at Ricky's height of 188 and 194.
ReplyDelete