Transcript Making Connections - University of Arizona

Making Connections
Mona Teitelbaum
Third grade
Lincoln Elementary School
Oak Park, IL
1
Classroom Demographics
• 15 students--6 boys, 9 girls
• 7 minority students, 1 ESL student
– 5 African American
– 1 Hispanic
– 1 bi-racial
• Young class-10 summer birthdays
• 2 low income
2
Classroom Procedure
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Whole class discussion of problem.
Break out into groups of 3- 4 students.
Mixed ability groups.
Whole class discussion of strategies and
solutions.
3
Problem 1
72 candies/even groups
• Used cubes to help with thinking.
• Students’ initial thoughts:
· “72/8= 9 (or 6) so it can’t work.”
· “It can’t work because 72 is an even number.”
4
Problem 1: “Star”
• (10 x 7) + 2 = 72
• (10 x 6) + 6 + 6= 72
• (10 x 4) + (8 x 4)=72
5
Problem 1: “Numbers”
• Use 8 x 9 =72 as a starting point and
move cubes around.
• “You need to move one cube from an
odd group to make it even.”
• “ Move two cubes from an even group
keeps it even.”
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Problem 1: “Superstars”
• Two groups of 30 and two groups of 6.
Then take the 6’s and divide them into
groups of 2.
• 7 6’s and a 30 make 72.
• 12 x 6 =72. Then move cubes around
to make 8 groups.
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Problem 2:
72 candies/odd groups
• “Superstars”
• 72 = 35 + 35 + 1 + 1. “You split the 35 in half (sic.) so you have 15+15
=30 so then you have 4 groups of 15 and you bring the 5 over to make
two other groups. 15+15+5 then 15+15+5. Then 2 groups of 1.
• 60 + 9 +1+1+ 1=72. Then: 15+15+15+15=60
• Comment: all your ways have 15. “Let’s try 21. 21+21 =42. Eventually
this leads to 2 groups of 21, 3 groups of 9, and 3 groups of 1.
• Teacher: “Last time you added up even numbers to get even.”
Student:“ This time we’re adding up odd numbers to get even.”
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Problem 2 “States”
• “Started with 3 in each group then added 2 to each group
except 1 group we subtracted 2. We did this because if we
added one it would make the groups even, but if we add 2
it will stay odd.”
• Picked 45 (because it is odd) then split up the cubes that
were left into groups of 5(because it’s odd). There were
two left so we made them be two groups of 1.
• Many other combinations courtesy of “Matt”.
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Problem 2
(continued)
• “Jelly Belly Chocolate Flavor”
· 72-21=51. So we took that 51 into a huge group
which is an odd number. Then there are 7
groups of 3.
• “No Name”
· Start with an odd number or two and used trial
and error.
· Moved cubes in twos to make groups odd.
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Student Comments on Odds and Evens
• Odd plus odd equals even.
• Even plus even equals even.
• If you add 1 to an even number you get an
odd number.
• You can count by 2 starting at 3 and you get
odd numbers.
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Problem 3: 73 candies/even groups
Problem 4: 73 candies/odd groups
• Predictions?
– “She counted wrong because last time it was 72
and if we add one more cube to any of our
groups then one of the groups will be even or
one group will be odd.”
– Most students could not understand what this
student was saying.
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Problem 3:
“Something” and “Tina and Leon”
• “7 groups of 2 and 59 left. That adds up to 73.
Terry counted wrong because every group we put
the cubes into there is a problem.”
• J.T.: “Can you say why?” No response. They keep
trying different combinations.
• Many combinations of even numbers but they
keep getting 70 so she must have counted wrong,
“because you can’t split up 3 and make it work.”
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Problem 4
• Students were content with one example to
show that it was not possible.
• Perhaps they could see ahead to tell that it
would not work.
• Many groups were unable to write anything.
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Problem 4: “Matt” and “Cally”
• Matt: you can’t add up odds and odds to make an
odd number. 3+3=6, 5+5=10, 7+7=14. (he shows
many, many examples).He concludes that
therefore this cannot possibly work.
• Cally: I don’t see how adding odd numbers and
getting even numbers relates to 73. (note: we had
discussed as a class that 73 was an odd number)
• Class: cannot understand how what Matt is saying
relates to the problem.
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