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BIG IDEA 1 GRADE 5

BIG IDEA 1 Develop an understanding of and fluency with division of whole numbers

536 ÷ 4 =

• Page from an 1860s 6th grade book:Long Division—ugh!!Long Division—ugh!!

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BIG IDEA 1 BENCHMARKSMA.5.A.1.1 Describe the process of finding quotients involving multi-digit dividends using models, place value, properties, and the relationship of division to multiplication.

BIG IDEA 1 BENCHMARKSMA.5.A.1.2 Estimate quotients or calculate them mentally depending on the context and numbers involved

BIG IDEA 1 BENCHMARKSMA.5.A.1.3 Interpret solutions to division situations including those with remainders depending on the context of the problem.

MA.5.A.1.4 Divide multi-digit whole numbers fluently, including solving real-world problems, demonstrating understanding of the standard algorithm and checking the reasonableness of results.

BIG IDEA 1 BENCHMARKS

• Depth of understanding involves the ability to work with numbers flexibly and easily, not the ability to perform the same procedure over and over again.

• Children must be able to make sense of the algorithm, explore informal strategies before being introduced to more formal algorithm, use a variety of invented strategies, able to explain procedure.

Juli K. Dixon; Transforming Teaching:From Dissonance to Depth; NCSM 42nd Annual Conference, San Diego, CA

Cognitive Dissonance

Cognitive dissonance is a psychological phenomenon which refers to the discomfort felt at a discrepancy between what you already know or believe, and new information or interpretation. It therefore occurs when there is a need to accommodate new ideas, and it may be necessary for it to develop so that we become "open" to them.

Juli K. Dixon; Transforming Teaching:From Dissonance to Depth; NCSM 42nd Annual Conference, San Diego, CA

Two basic types of problem types in division

Measurement: You have a group of objects and you remove subgroups of a certain size repeatedly. The basic question is—how many subgroups can you remove?Example: You have 15 lightning bugs and you put three in each jar. How many jars will you need?

2nd problem type for division

Partitioning: You have a group of objects and you share them equally into a number of subgroups. How many go in each subgroup?Example: 20 kids are going to the circus. 5 cars can take them there. How many kids can ride in each car?

What comes to mind when you see this?

Division should start at the concrete level using place value blocks.

• Repeated subtraction—goes along with measurement interpretation of division

• Distributive—goes along with partitioning; our traditional algorithm

• You pass repeated subtraction on the way to the distributive, but you can also stop there. It’s simpler but not as efficient, but for certain kids, it might be more appropriate to stop with this algorithm.

2 algorithms for division2 algorithms for division

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How kids polish their work with repeated subtraction. They learn to subtract more than

1 group at a time. • For the problem 50 ÷ 3, they might do:• 50• -30 10 (ten 3’s subtracted.)• 20• -15 5 (5 more 3’s subtracted.)• 5• -3 1 (1 more 3 subtracted)• 2 Answer: 16 with r = 2

128 ÷ 22 238 ÷ 31 1243 ÷ 41 954 ÷ 4

Try these yourself—use repeated Try these yourself—use repeated subtraction. Record!!subtraction. Record!!

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• Easier to understand• Not perfect but fewer mistakes than in long

division—no estimation as in 331 ÷ 46• Tied in with measurement interpretation of

division in Grade 3 Big Idea 1.• Not as efficient—takes a lot of paper

Key points to Key points to Repeated Subtraction?Repeated Subtraction?

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• 63 gumdrops for a class of 20 students. How many does each student get?

• 155 people waiting in line to ride the monorail. Each ride holds 50 people. How many rides are necessary for everyone to get a ride?

• I have four cups of flour. I use all of the flour to make 3 batches of cookies. How much flour does each batch take?

Making sense of remaindersMaking sense of remainders