Critical Thinking

Critical thinking is the analysis of facts to form a judgement. This could be the answer to a scientific exploration, the response to a novel, or the answer to a math question. What ever the subject matter, students should be challenged to think about their thinking and put that thinking into an explanation.
Would You Rather...Math Problems and Answers!
Students have been given an open ended math situation from the website, www.wouldyourathermath.com. Here are just a few of the mathematical reasonings and critical thinking students have shown in the answers.
Coin Question Results:
Most Student who answered chose A!
Here's Some Great Student Responses:
 14+35+20+25 =94 but pile B only added up to 92.
 stack A has 94 cents, stack B 92 cents so i would rather have 94 cents because its more money and i like money. STACK A : 14+35+20+25 =94 STACK B : 2+20+20+50 =92
 I would chose A since having .14+.35+.20+.25=94 B is less since .2+.20 +.20 +.25 =.92
 I feel like B would be a better choice for me due to how many quarters there are. 2 quarters is 50 cent (25 x 2=50) then 2 dimes which equal 10 cent. Add those together you get 20 cent. (10 + 10=20), after you do that add together the 2 quarters to the 2 dimes you get 70 cent. (50 + 20=70) once you've done that time to look at the amount of nickles you have. Which would be 4, nickles are equal to 5 cent. You add up all 4 nickles you get 20 cent. Now add everything you got (70 + 20=90) then finally add up your 2 pennies. (90 + 2=92) Now all I have to do is look under a couple places and I can get a dollar.
 Option A"s total was $.94 and option B's total was $.92. I would choose option B because there's a lot less coins to carry.
 i pick A because A=94 B=92 and if i were to buy my mom a present i would take 94 sence so it wont take longer for me to find more money for her mothers day present
 I would choose stack A because in stack A 14, 35, 49, 20, 69, and 25 added all together equals 94 cent. But in stack B 2, 20, 22, 20, 42, and 50 added all together equals 92 cent. Stack A has more money than stack B.
Thank you for explaining your thinking! We can't wait to share Problem #2!

Would You Rather...Math Problem 2
There were lots of answers to this question! Be sure to include your mathematical thinking in your answer. Here's how the grade level participation looked.
And here's how many students chose choice A or choice B.
Some of the answers for Choice A were...
I would do this even it's a little more pricey it would be less messy and you would not have to throw away as many bags if you were to get the other eggs. Sam
I would choosen 18 eggs at the price of $8 because the more money there is the more it helps everyone Like the mayor,juges,and the teachers. Gabby
Some of the answers for Choice B were...
Each small bag is 97 cents each so it is cheaper. 18 eggs is 18 bags witch equals 11.64. Because you get six free with the offer. Charlie
I would rather buy option B because it is lower priced. Also, option B is buy 2, get 1 free and option A is 2 packs for 8 dollars. Option A would cost $12 for 18 eggs which is 3 packs. Option B would cost $6 for 18 eggs because it is buy 2, get 1 free. Brooklyn
I would like a because you get 18 eggs for 11.64 cents the other one cost 13.27 cents  Kylea
For option B, since they were buy one get one free, I figured out the cost of three eggs was .97x2= $1.94. Then, to get 18 eggs, you would have to do that 6 times, so I did $1.94 x 6 = $11.64. So the cost of 18 eggs was $11.64. For option A, the bag of 6 eggs were on sale 2 for $8.00. To buy 6 eggs, I did $8 divided by 2 = $4. To buy 18 eggs, I did $8(for 12 eggs) plus $4 (for 6) = $12 So the cost of 18 eggs is $12. That is more than option B, so I choose option B because I want to save money. Madeleline
Because option b has buy 2 get 1 free which i have to buy 6 packs for 12$ .97to reach the 18 eggs,but option A i have to buy 2 packs for 8$ and one pack for 5.27$ which is the total of 13.27$ to reach the 18 eggs for me for me option b is the best price option A equation $8+$5.27=$13.27 option B equation $1.94x6 =$12 Nader
a would cost 12 dollars and b cost 11.64 so i save $.36 Caleb L.
8.00+5.27=13.27 12x97 cents=11.64 So b is cheaper. Kaden R.
Thanks for sharing your thinking!! This week, problem 3!

Would You Rather...Math Problem 3
There were 89 responses to the question! Great Job!
Now we need to work on mathematical reasoning. That means using a math equation to explain your thinking. Here are some excellent responses!
Students who chose bedroom B said..
"Option B means the room is 14x14 for 196 sq feet is bigger than A."  Gabriel
"I chose option B because it is bigger. Option A is 180 square feet, option B is 196 square feet. 196180= 16 square feet bigger. A. 2 sides are 20 feet, the other 2 sides are 9 foot. 582020=18 18 divided by 2=9 20x9=180 square feet b. 2 sides are 14 foot, the other 2 sides are 14 foot 561414=28 28 divided by 2=14 14x14= 196."  Delaney
"I would rather have a length of 14ft and a perimeter of 56ft . It will give me an area of 196ft which is bigger than having a length of 20ft and a perimeter of 58ft ."  Nahla
"I choose B because it is bigger this is how it has a area of 196[14x14] and A has only 180[20x9]"  Nader
"A bedroom length of 20ft with a perimeter 58ft would be to thin, but 14ft with a perimeter of 58 is good." Grayson
"BECAUSE ALL LENGTH AND WIDTH ARE 14." Josan
"B. becuase its big and i have lots of space."  Edgar
"It seems to be bigger when you do the math and it is nicer."  Charlie
"B becasue it has a larger area." Caleb
"I chose B because the area of B is 196 square feet while the area of A is 180 square feet and that is why I chose B.." Francisco
"room a= 20 x 9=180A room b= 14 x 14=196A." Brady
"A bedroom with a length of 20ft and a perimeter of 58 seems just the slightest bit too big and then to make it look like its actually decorated then you would probably need a lot more furniture." Bryce
"the first room has a length 20 feet and a width of 9 feet totaling 58 feet which is pretty narrow. The second room has a length of 14 feet and a width of 14 feet which totals 56 feet which is square. I personally prefer a square room."  Eric
"To figure out the area of bedroom A, first I added 20 ft. + 20 ft. = 40 ft. Then I subtracted 40 ft. from the perimeter, which was 58 ft. The answer was 18 ft. Since the room was a rectangle, I knew that there was two sides that I did not know the measurement for. So I split 18 in half, which gave me the answer 9 ft. Then I solved 9 ft. times 20 ft. and got the answer 180 ft. which was the area of the entire room. Then to figure out the area of bedroom B, first I added 14 ft. + 14 ft. =28 ft. So then I subtracted 28 ft. from the perimeter, 56 ft. which equaled 28. I split it in two because there was two missing sides. It equaled 14 ft. and that was when I realized that it was square! Then I solved 14 ft. times 14 ft. and it gave me the answer 196 ft. I choose answer B because it has a greater area the answer A because I have a lot of stuff."  Madeleine
Students who chose A said...
"I want this room because i like small rooms still and its one of my favorite." Zoe
"I pick A. A = L=20 W=9 P=58. 20+20+9+9=58. L=14 W=7 P=56. 14+14+14+14=56" Rhylee
"The view is better and the room is smaller and I like small places." David
Good Luck on Problem #4!

Would You Rather...Math Problem 4
It looks like our 4th graders have taken a slight lead in responding to Would You Rather this week! And most students chose option B.
Here's why most students chose B as their answer:
"158=7, 156=9. more money with option 2."  Austin
"it is a trick question $8.00 to make 30 cookies is more than $6.00 to make 30 cookies." Travis
"156=9 choice b means they made $9." Kendon
"Choice B , because in total for the cookies and costing would be 9$.It would be one dollar more than choice A ." Natalie
"Choice B, because 30 cookies for 15 dollars with a cost of 6 is more money than Choice A." Eric
"I chose choice B because choice A, if you multiply 30 by 0.50 you get 15.00 then minus that with 8.00 you get 7.00 so you get a profit of $7.00 if you do choice A and if you minus$15.00 by $6.00 you get a profit of $9.00 if you do choice B so that is the reason I chose choice B because you make more profit."  Francisco
"CHOICE (B) 15.006.00= 9.00 CHOICE (A) 30.00 X .50 = 15.008.00=7.00 I would rather be choice (B) because you make more money." John B.
"With A 30 cookies times .50 is $15 and B is the same $15 cost for cookies. I chose B because I realized I made a profit of $9 vs $7 " Rhylee
"First for A, one cookie costs 50 cents so two cookies would have the cost of 1$. Then I counted by 2s, all the way to 30. The answer I got was fifteen dollars. Then I subtracted 8$ from 15$, which equaled 7$. So, the answer to A, is 7$. For B, since I got 15$ from selling the 30 cookies, I subtracted 6$ from 15$ because 6$ was the amount used to make the cookies.That left me with a total of 9$. I choose B because I would have gotten more money because 9$>6$." Madeleine
"I make more money. 158 is 7 but 156 is 9." Jaxon
"Because i will have 9 dollars left over but if i choose choice a i will have 7 dollars left." Janiya
"Since they both sell for $15 I would rather make choice B because they are $2 cheaper to make." Rachel
"I would rather sell a batch of 30 cookies altogether for $15.00 with a cost to make of $6.00 to make $9.00 profit, Choice B (15 minus 6 is 9) than Choice A, Sell a batch of 30 cookies each for 50 cents with a cost to make $8.00 to make $7.00 profit, (15 minus 8 is 7). And $9.00 is more than $7.00 and I want to make more money!" Brooklyn
"I would rather sell choice B because I have more money left over after I pay the cost. Answer A is selling 30 cookies for 50 cents each when you multiply those two number you $15 and then you subtract $8 because of the cost to make the cookies, and you have $7 left. While B you get $15 dollars for a batch of 30 which is the same you get for the other one, and then you subtract $6 for the cost and get $9 left over." Kristal
" i chose answer B because i get more money Answer A 30 x .50 = $15.00 $15  $8 =$7 Answer B $15.00  $6 = $9." Delaney
"You will have 9.00 dollars left for the 2 one and have 7.00 left for the 1 one."Payton
"Both equal $15 but B you pay $6 in A you pay $8." Lucas
"i pick choice B because 6 dollars are less than 8 dollars."Anthony
"i spent less on suplies wich means i made more money."Micaella
"u earn more money and lose less." Jayanna
"I would pick option B, because of how I would still make more money. If I had picked option A, I would have gotten less money for each cookie and the cost to make is higher so I would make less money (7$). But if I had picked choice B, I would get more money. If I sell every cookie for 15$ and they had cost 6$ to make I would have to subtract 15 by 6 which would equal 9. Making choice B a better answer (9$)." Sam
158=7 =choice A 156=9=choice B" Kaden R.
Students who also agreed with Answer B stating making more money and spending less and profits were: Zion, Jayde, Andrei, Gabby, Khylee, Jeremia and David.
GREAT JOB Everyone!! Keep answering those questions with your mathematical justifications!! Let's see what you think about Problem 5!

Would You Rather...Math Problem 5
We had great participation on our math problems again! Thank you for sharing your thinking! Here's the graph of grade level participation:
We had a larger amount of students pick choice B! But it was close! Here's the graph of answer choices:
Why Did You Choose That Answer?
There were many reasons the different cakes were chosen.
Let's start with choice A.
Many students noted the area of the cake being larger.
"I would choose A because we get more cake! 15x15=225 while 12x17=204." Kristal
"Choice A because then I would have a bigger cake and would have enough slices of cake for me and my friends." Kaden P.
"I pick this answer because I would get more inches of cake. I figured this out by multiplying 12 x 17 (choice B) and getting 204, (204 inches of cake). Then I multiplied 15 x 15 (option A) and got 225, (225 inches of cake). This means I would get more cake if I picked option A, so that's why I picked option A." Sam
Some noted number of friends and calories.
"I GET MORE FRIENDS." Josan
"I like squares and am on a diet." Caleb
Some commented on the shape.
"i choose choice A because its me plus my other 8 friends and if u split choice a up with two lines going horizontal and one line vertical there will be 9 slices." Zyasia
Why did students choose choice B?
Let's see what they figured out.
Using logic and measurement helped some students decide.
"i like rectangle shaped cakes because if i,m sharing with only six people then the slices are bigger." Symphony
"The math will give you that the second cake has a smaller area but the cake has less friends to share with and need only 7 slices with you included." Charlie
"There will be more cake if we chose cake b with 6 friends ." Aleigha
"because b is longer for cutting." Aniya
Calculating the size of each piece of cake helped other students decide.
"I would rather choose Cake B because it is divided between less friends so we get more cake each! Since 12 by 17 equals 204, 6 people will get 34 in. of cake each. " Brooklyn
"With cake B I would get 34 square inches. With cake A I could only get 28 square inches if we divide it evenly." Eric
"First I found the area of cake A by multiplying 15 x 15 = 225. Then I divided 225 /8=28. So the area of one slice of one cake is 28 square inches. Next I found the area of cake B by multiplying 12 x 17 = 204. Then I divided 204 / 6 = 34 square inches. There for I would choose cake B because it has a greater area then one slice in cake A." Madeleine
"I picked cake b because cake a has smaller pieces because 15 x 15 = 225 / 8 18. 125 and cake a is 12 x 17 = 204 / 6 =34 cake b has bigger pieces then cake a so I pick cake b." Rhylee A.
"I chose b because it is bigger this is how [12x17=204/6=34] and cake a has 28 so cake b is bigger. " Nader
"with a you will get a smaller piece of cake. 28.125. with b it is 34."  Anthony R.
And some noted the distributive property of multiplication they used to solve the problem.
"if you give 2 inches to 12 you have 14 by 15 and you split it with 6 animals." Andrei
Great Job and GREAT mathematical reasoning! We can't wait to see what you think about Problem 6.

Would You Rather...Math Problem 6
This is our final math question for the school year! We still have lots of mathematicitians participating in the discussions!
Most students chose A as their icecream choice. But there were lots of reasons given.
Here's what students said about Choice A:
"in a you get more for less and a=128, b =59." Sara
"Option B is only 6 lb 11oz option A is 8 lbs you get more ice cream with a different 96 cent difference." Kendon
"I would rather buy Option A because dividing the total amount of money ($24.56) by the total amount of ounces (128) equals paying only .19 per ounce. Option B costs more because the total amount of money ($23.60) divided by the total amount of ounces (70) equals .33 per ounce which costs more." Brooklyn
"8 Ib is more than 3 Ib 11oz x 2." Eric
"It is less than a dollar more than choice B and you get 10 oz. of ice cream more than choice B." Bryce
"A. A is .19 an ounce because 8x16=128. 128/24.59 = .19. " Rylee
"The first option has more ice cream because math states that 3.11 x 2 = 6.22 witch is less than the first amount." Charlie
"First for option A, I had to figure out what 8x16 was. So I did the math and I got 128 oz. Knowing that the price was 24.56, I divided 24.56 cents / 128 oz. =.19 cents per oz. First for option B, I knew that 2x3=6, so then I figured out 6x16=96oz. Then I added 11oz. to 96 = 107 oz. Next I divided 23.60/ 107=.22 cents per oz. I Choose A because It would give me more ice cream for a lower price per ounce." Madeleine
"I thing A because i get more than B .But if you are speding money I would choose option B." Jhony
"3.11 + 3.11=6.22 (7lbs 6oz) 24.5623.60=.96 You pay .96 more for almost 1/2 lb more of ice cream." John B.
"because you get more and only pay 96 more cents." Andrei
"because a has more ice cream and its only one dollar and 4 cense more than b." Nader
"option a is cheaper than option b a 8x16 =128 oz 24.56/128 = .19 b 3lbs x 16 oz = 48 48+11=59 59 x2 =II8 23.60/118=.20." Delaney
"I would choose choice A because it cost less per unit for more ice cream. Choice A is $3.07 per pound, and choice B is more than $3.37 per pound for less ice cream." Amaya
"I choose this answer because 16 ounces is a pound. Making choice B smaller, it might be lower priced but still smaller. While choice A is 8 pounds and is only about a dollar higher then choice B. Which isn't bad." Sam
Choice B was chosen for some logical reasons too.
"Choice B because it costs less and I already eat a lot of sugar." Zaila
"Choice b because choice b has less money and choice a has more money that's why I picked choice b." Andriya
Thank you for sharing your thinking and justifying your answers!!
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