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Making Connections

Mona Teitelbaum

Third grade

Lincoln Elementary School

Oak Park, IL

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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

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Classroom Procedure

• Whole class discussion of problem.

• Break out into groups of 3- 4 students.

• Mixed ability groups.

• Whole class discussion of strategies and solutions.

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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.”

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Problem 1: “Star”

• (10 x 7) + 2 = 72

• (10 x 6) + 6 + 6= 72

• (10 x 4) + (8 x 4)=72

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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 groupsProblem 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.