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###### Week 1: The base-ten system; reasoning with fractions

Tuesday, August 14: Welcome to MATH 5020/7020!

• How many toothpicks are in the bag? How can we organize them to see how many there are?

Thursday, August 16:  If you have your book, please bring the class activities from section 2.2 to class.

• Follow up on base-ten structure from last time:

• A digit in a number tells you how many of a specific base-ten unit there are.

• The value of a place’s unit is ten times the value of the place to the right.

• This structure is consistent across whole numbers and decimals.

• What are fractions?

• What are limitations to defining the fraction A/B as "A out of B"?

• The Common Core definition of fraction

• Reasoning with the Common Core definition of fraction to solve problems

###### Week 2: Reasoning with Fractions

Tuesday, August 21: Please bring the class activities from section 2.2 to class.

• Reasoning with the Common Core definition of fraction to solve problems (A/B means the amount formed by A parts, each of size 1/B of the unit amount/whole).

• What are similar ideas in the Common Core defintion of fractions and base ten?

• Why is it important to attend to a fraction's unit amount (whole)?

• How can we think of a fractions as numbers just like whole numbers?

Thursday, August 23:  Please bring the class activities from section 2.2 to class.

• How can we think of a fractions as numbers just like whole numbers?

• Describing quantities with numbers: a measurement sense of number

• Why is it important to attend to a fraction's unit amount (whole)?

###### Week 3: Reasoning about and with partitioning and equivalent fractions

Tuesday, August 28: Please bring the class activities from sections 2.2 and 2.3 to class.

• How can we identify unit amounts for fractions and whole numbers in situations? From a measurement perspective, numbers are the result of measurement questions of the form “How many/much of this unit amount does it take to make this quantity?” For example, if a situation involves 2/3 of a gallon, then 2/3 is the answer to “how much of 1 gallon does it take to make this amount?” so the unit amount for the number 2/3 is “1 gallon”. Another way to think about why “1 gallon” is the unit amount: the quantity is 2/3 of the 1 gallon.

• Reasoning about quantities in math drawings to solve fraction problems.

• Equivalent fractions

• Every fraction is equal to infinitely many others

• Given a fraction, how can we find other fractions that are equal to it, and why does that method work?

• Equivalent fractions can be useful in problem solving!

Thursday, August 30:  Please bring the class activities from section 2.3 to class.

• Instead of "unit amount" we can also say "referent unit" or "referent quantity" or just "referent" so you don't confuse it with "unit fraction" (which means 1/2, 1/3, 1/4, etc). Are any of those terms better? For example, the referent for 5 in "5 kilometers" is "1 kilometer"; the referent for 3/4 in "3/4 of a cup of flour" is "1 cup of flour."

• Equivalent fractions

• Using math drawings to explain why a fraction A/B is equal to (A•N)/(B•N).

• Equivalent fractions can be useful in problem solving!

• Making common partitions

• Reasoning about common multiples to make common partitions

• Caution: sometimes common partitions are NOT achieved by common denominators!

###### Week 4: Reasoning about and with equivalent fractions

Tuesday, September 4: Please bring the class activities from section 2.3 to class.

• Equivalent fractions and making common partitions

• Equivalent fractions and common partitions can be useful in problem solving!​​

• Caution: sometimes common partitions are NOT achieved by common denominators!

Thursday, September 6: Please bring the class activities from section 2.3 to class.

• Solving problems by reasoning about equivalent fractions and common partitions.

•  Caution: sometimes common partitions are NOT achieved by common denominators!

###### Week 5: Reasoning to compare fractions; percent

Tuesday, September 11:  Please bring the class activities from section 2.4 to class.

• What are ways we can compare fractions?

• Standard methods for comparing fractions that work in all cases.

• Reasoning for comparing fractions that is efficient in some special cases.

• Comparing fractions by relating them to benchmark numbers.

Thursday, September 13:  Please bring the class activities from section 2.5 to class.

• Reasoning about percentages using tables and math drawings.

###### Week 6: Percent; Why we add and subtract fractions the way we do; fraction addition and subtraction word problems

Tuesday, September 18: Please bring the class activities from sections 2.5 and 3.4 to class.

• Solving percent problems by reasoning about tables and math drawings.

• Why do we add and subtract fractions with like denominators by keeping the same denominator?

• Why our (Common Core) definition of fraction is more helpful for adding and subtracting fractions than an “A out of B” view of fractions.

• In addition and subtraction equations A + B = C or A – B = C, each number A, B, C in the equation must have the same referent (i.e., must refer to the same unit amount).

Thursday, September 20: Please bring the class activities from section 3.4 to class.

• Fraction addition and subtraction word problems.

• In addition and subtraction equations A + B = C or A – B = C, each number A, B, C in the equation must have the same referent (i.e., must refer to the same unit amount).

• Distinguishing fraction addition and subtraction word problems from other word problems and why it’s so important to pay close attention to the unit amount/referent that each fraction is of.

• How equations model word problems – your thoughts.

• What is multiplication? Your thoughts.

• Our class definition of multiplication.

###### Week 7:  Multiplication: what is it?

Tuesday, September 25:  Please bring the class activities from section 4.1 to class.

• Discussion of test 1

• What is multiplication? Your thoughts.

• Our class definition of multiplication.

Thursday, September 27:  Work on the group quiz.

###### Week 8:  Multiplication: reasoning about and with properties

Tuesday, October 2:  Please bring the class activities from sections 4.1 to class.

• We will be using our class definition of multiplication page 1page 2 to see how multiplication is a coherent concept that applies across many different kinds of quantitative situations (word problems) that involve whole numbers, fractions, or decimals.

• When we are given a quantitative situation (word problem), we have to look for structure.

• We can use our class definition of multiplication to explain why multiplication applies in a quantitative situation (word problem).

Thursday, October 4: Please bring the glass activities from section 4.3 and 4.4 to class.

• To email me either use sybilla@uga.edu OR if you want to reply to an email from eLC do so INSIDE eLC otherwise it will bounce back.

• Using our class definition of multiplication to explain why a word problem is a multiplication problem.

• The commutative, associative, and distributive properties are fundamental properties that allow us to calculate flexibly and efficiently by reorganizing and breaking problems apart into simpler problems.

• Explaining where the commutative and associative properties of multiplication come from (why they are valid) by grouping items in two different ways.

• Explaining where the distributive property of multiplication over addition/subtraction comes from (why it is valid) by grouping items in two different ways.

###### Week 9:  Multiplication: how can we use properties of multiplication and where does the whole number algorithm come from?

Tuesday, October 9: Please bring the class activities from sections 4.3, 4.4, and 4.5 to class.

• Explaining where the distributive property of multiplication over addition/subtraction comes from (why it is valid) by grouping items in two different ways.

• Using properties of multiplication to:

• make multiplication calculations easier to do mentally

• facilitate learning the basic multiplication facts

Thursday, October 11: Class is cancelled due to the hurricane. Please see the assignment page for a replacement assignment due by the end of the day on Tuesday, October 16.

###### Week 10: Multiplication: whole number algorithm; how does multiplication extend to fractions?

Tuesday, October 16: Please bring the class activities from sections 4.2, 4.5, and 4.6 to class.

• Multiplying by 10 and powers of 10--why is it special in base ten?

• Using properties of multiplication to make calculations easier to do by breaking them apart.

• Developing and explaining the partial products written method and the standard whole number multiplication algorithm.

• How the standard algorithm arises from place value and properties of multiplication.

Thursday, October 18: Please bring the class activities from sections 4.6 and 5.1 to class.

• Developing and explaining the partial products written method and the standard whole number multiplication algorithm.

• How the standard algorithm arises from place value and properties of multiplication.

• How can we understand multiplication as a coherent concept that extends from whole numbers to fractions?     Geogebra sketch: https://ggbm.at/bmz9sgpx

###### Week 11: Reasoning about fraction multiplication

Tuesday, October 23: Please bring the class activities from section 5.1 to class.

• Interpreting what multiplication means when the multiplier is a whole number and the multiplicand is a fraction.

• Interpreting what multiplication means when the multiplier is a fraction and the multiplicand is a whole number.

Geogebra sketch for ounces: https://ggbm.at/ntebhz49   Our class definition of multiplication: Page 1 and Page 2

Thursday, October 25: Please bring the class activities from section 5.1 to class.

• Interpreting what multiplication means when the multiplier and the multiplicand are fractions.

• Why do we multiply fractions the way we do? What is the reasoning behind the procedure?

FALL BREAK Friday, October 26

###### Week 12: Reasoning about fraction multiplication

Tuesday, October 30: Please bring the class activities from section 5.1 to class.

• Interpreting what multiplication means when the multiplier and the multiplicand are fractions.

• Why do we multiply fractions the way we do? What is the reasoning behind the procedure?

Thursday, November 1: No class due to a project you are doing with Dr. White. We will make up this class next Thursday, 8 - 9:15 am.

###### Week 13:  Division: what is it? Connecting division with fractions

Tuesday, November 6:  Please bring the class activities from section 5.1 to class.

• Why do we multiply fractions the way we do? What is the reasoning behind the procedure?

• Distinguishing fraction multiplication word problems from other word problems.

Thursday, November 8: Two class periods today: 8 - 9:15 am and our usual 9:30 - 10:45 am (to make up the missed class on 11/1). Please bring the class activities from section 6.1 to class.

• What does division mean?

• We can view division as multiplication with an unknown factor:

• How many units in 1 group?

• How many groups?

###### Week 14: Division: explaining the connection with fractions; interpreting quotients and remainders

Tuesday, November 13: Please bring the class activities from sections 6.1 and 6.2 to class.

• Identifying division word problems as "how many units in 1 group?" or as "how many groups?" problems.

• The Fundamental Theorem of Fractions: how are division and fractions related?

• Division with remainder: Interpreting quotients and remainders in whole number division word problems.

Thursday, November 15: Please bring the class activities from sections 6.3 and 6.2 to class.

• Explaining why the standard algorithm for whole number division works in terms of dividing base-ten bundles equally among groups.

• Division with remainder: Interpreting quotients and remainders in whole number division word problems.

Activity:

1. Organize 372 toothpicks into base-ten bundles. Be sure to make each bundle of a hundred out of 10 bundles of ten.

2. Distribute those 372 toothpicks equally among 3 groups. Record how you did it (Step 1 ... Step 2 ... Step 3 ... etc).

3. If it fits, write some notation that captures or corresponds to the steps you took in part 2.

4. See if you can think of a different set of steps for distributing the toothpicks equally among the 3 groups.

THANKSGIVING BREAK: November 19 - 23

###### Week 15: Division: where does the whole number algorithm come from? Review

Tuesday, November 27: Please bring the class activities from section 6.2 to class.

• NOTE: hang on to your textbook because we will use it again next semester!

• Division with remainder: Interpreting quotients and remainders in whole number division word problems of both types:

• how many units in 1 group division

• how many groups division

Thursday, November 29: Review, based on your questions.

• Would you like to schedule a review session for sometime next Monday, Tuesday, or Wednesday?

• NOTE: hang on to your textbook because we will use it again next semester!

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