## Week 1: Fraction divison

Thursday, January 4:  Happy New Year, welcome to MATH 5035/7035, and GO DAWGS!!!! Please bring the class activities from section 6.4 to class.

• A notetaker is needed for this class. Please see the email on eLC and go to the links there if you are interested.
• The course syllabus and all course information is on the course homepage. Find a link to it on eLC or at my math department web page.
• Recall the two types of division word problems:
• how-many-units-in-1-group
• how-many-groups
• Fraction division from the how-many-groups perspective
• solving this type of word problem by reasoning about math drawings
• Are there general methods for solving fraction division other than "keep, change, flip"? Yes!

## Week 2: Fraction divison

GO DAWGS, WOOF WOOF WOOF!!!!!!!!

Tuesday, January 9: Please bring the class activities from section 6.4 to class.

Fraction division from the how-many-groups perspective

• How-many-groups fraction division word problems
• Solving how-many-groups fraction division word problems by reasoning about math drawings
• By reasoning about like units, we can find and explain a general way to calculate fraction division that is different from “keep, change, flip”!

Thursday, January 11:  Please bring the class activities from sections 6.4 and 6.5 to class.

• Fraction division from the how-many-groups perspective
• Equivalent calculations: turning fraction division into whole number division by making like units.
• Fraction division from the how-many-units-in-1-group perspective
• How-many-units-in-1-group fraction division word problems
• Solving how-many-units-in-1-group fraction division word problems by reasoning about math drawings
• Why can we solve fraction division numerically by multiplying by the reciprocal? (Keep, change, flip)

## Week 3: Fraction divison

Tuesday, January 16:  Please bring the class activities from section 6.5 to class.

• Fraction division from the how-many-units-in-1-group perspective
• How-many-units-in-1-group fraction division word problems
• Solving how-many-units-in-1-group fraction division word problems by reasoning about math drawings
• Why can we solve fraction division numerically by multiplying by the reciprocal? (Keep, change, flip)

Thursday, January 18:  Please bring the class activities from section 6.5 to class.

• Fraction division from the how-many-units-in-1-group perspective
• Solving how-many-units-in-1-group fraction division word problems by reasoning about math drawings
• Why can we solve fraction division numerically by multiplying by the reciprocal? (Keep, change, flip)
• Given a fraction word problem, is it a division problem, a multiplication problem, or something else? How can we tell?

## Week 4: Fraction division; Ratio and proportional relationships

Tuesday, January 23:  Please bring the class activities from section 6.5 and 7.1 to class.

• Take-home test to be handed out next time.
• We will have a homework discussion at the beginning of next time.
• Given a fraction word problem, is it a division problem, a multiplication problem, or something else? How can we tell?
• Can two mixtures have the same “quality” even though they differ in size? How can we describe all the mixtures that have the same “quality”?

Thursday, January 25:  Be prepared to present homework solutions and please bring the class activities from section 7.1 to class.

• Take-home test due next time
• Homework discussion
• Can two mixtures have the same “quality” even though they differ in size? How can we describe and organize all the mixtures that have the same “quality”?

## Week 5: Ratio and proportional relationships

Tuesday, January 30:  Please bring the class activities from section 7.1 to class.

• Take-home test due on Thursday
• How can we tell if two “mixtures” will have the same “quality” or not?
• “quality” <----> ratio
• Same “quality” <----> same ratio
• Different “quality” <----> different ratio
• How can we describe ALL the “mixtures” that have the same “quality”? In other words, how can we describe ALL the pairs of quantities that are in one fixed ratio?
• Multiple batches
• Variable parts

Thursday, February 1:  Please bring the class activities from section 7.1 to class.

Solving proportion problems by reasoning about quantities from a multiple batches perspective

• Reasoning about a double number line
• Reasoning about multiplication and division with quantities
• Why should students learn methods other than setting up a proportion equation and cross-multiplying?

## Week 6: Ratio and proportional relationships

Tuesday, February 6:  Please bring the class acitivities from section 7.2 to class.

• Brief test discussion
• Solving proportion problems by reasoning about quantities from a multiple batches perspective
• Using multiplication to express solutions to proportion problems when taking a multiple batches perspective:
• Multiply with an “original batch”
• Multiply with a “unit-rate batch”

Thursday, February 8:  Please bring the class activities from section 7.2 to class.

• Solving proportion problems by reasoning about quantities from a variable-parts perspective.
• Using multiplication to express solutions to proportion problems when taking a variable-parts perspective

## Week 7: Solving proportion problems by reasoning about multiplication and division with quantities;

Tuesday, February 13:  Please bring the class activities from section 7.2 to class.

• Solving proportion problems by reasoning about quantities from the variable-parts and multiple-batches perspectives.
• Using multiplication to express solutions to proportion problems.
• Rates: unit rates (at the base-unit level) and multiplicative comparisons that compare total amounts.

Thursday, February 15:  Please bring the class activities from sections 7.2 and 7.3  to class.

• Solving proportion problems by reasoning about quantities from the variable-parts and multiple-batches perspectives.
• Using multiplication to express solutions to proportion problems.
• Given a ratio A to B, and two quantities in that ratio, there are two ways to interpret the values of the ratio A/B and B/A as rates: (1) as unit rates that tell us the number of base units of one quantity per 1 base unit of the other quantity and (2) as multipliers that compare total amounts of the two quantities.
• Equations for lines
• What is slope and what does it tell us?
• Why do lines have the kind of equations they do?

## Week 8: Developing equations for proportional relationships and lines

Tuesday, February 20: Please bring the class activities from section 7.4 to class.

• Preparation for take-home test on Thursday
• Why lines have the kind of equations that they do
• What the slope of a line tells us
• Developing equations to relate quantities that vary together in a proportional relationship using a variable parts perspective

See geogebra sketch: https://ggbm.at/Wf9ACCtg

• Developing equations for lines using a variable parts perspective

See geogebra sketch: https://ggbm.at/B8NtGvmy

Thursday, February 22:

## Week 9: Developing equations for proportional relationships and lines; Inversely proportional relationships

Tuesday, February 27:  Please bring the class activities from sections 7.4 and 7.5 to class.

• Developing equations for lines using a variable parts perspective

See geogebra sketches: https://ggbm.at/Wf9ACCtg  https://ggbm.at/B8NtGvmy

• Inversely proportional relationships

Thursday, March 1:  Please bring the class activities from section 7.5 to class.

• Proportional relationships versus other relationships
• When is a relationship between quantities proportional? When the quantities remain in a fixed ratio.
• What kind of equations characterize proportional relationships?
• What kind of graphs characterize proportional relationships?
• Inversely proportional relationships

## Week 10: Inversely proportional relationships

Tuesday, March 6: Please bring the class activities from section 7.5 to class.

• What topics should we prioritize in the rest of the course?
• Some relationships are not proportional.
• Inversely proportional relationships are one type of relationship that we can describe using multiplication. They share characteristic equations and graphs.

Thursday, March 8: Please bring the class activities from section 7.5 to class.

• Plans for what topics to discuss after spring break.
• In a situation involving quantities, how can we determine if a relationship is proportional or not?
• Characteristics of proportional vs inversely proportional relationships:
• Proportional relationships: constant quotient (the constant of proportionality) – a fixed rate,  Y = m•X so Y/X = m (a constant).
• Inversely proportional relationships: constant product – such as a fixed amount of work (e.g., fixed amount of person-hours to do one fixed job), Y • X = c (a constant)
• Can we use “linear interpolation” in inversely proportional situations?

SPRING BREAK: week of March 12

## Week 11: Number theory (GCF and LCM)

Tuesday, March 20:  Please bring the class activities from section 8.5 to class

• What topics would you like to discuss for the rest of the course?
• Factors and multiples. Greatest common factor (GCF) and least common multiple (LCM).
• Three methods for finding the GCF and LCM of two counting numbers:
• based on what GCF and LCM mean
• by factoring into products of prime numbers
• using the “slide method”

Thursday, March 22: Please bring the class activities from sections 8.5 and 12.9 (Pythagorean theorem) to class.

• Three methods for finding the GCF and LCM of two counting numbers:
• based on what GCF and LCM mean
• by factoring into products of prime numbers
• using the “slide method”
• Seeing how square roots arise as side lengths of “tilted squares”.

## Week 12: Square roots and the Pythagorean theorem

Tuesday, March 27:  Please bring the class activities from sections 8.5 and 12.9 (Pythagorean theorem) to class.

• Do we want to think through why the “slide method” works one more time?
• Seeing how square roots arise as side lengths of “tilted squares”.
• Why is the Pythagorean theorem true? We can prove it by reasoning about areas.

Thursday, March 29:  Please bring the class activities from 12.9 (Pythagorean theorem) to class.

• Why is the Pythagorean theorem true? We can prove it by reasoning about areas.
• Distance formula: how does it come from the Pythagorean theorem?
• Problem solving with the Pythagorean theorem.

## Week 13: Statistics

Tuesday, April 3:  Please bring the class activities from Sections 15.1 and 15.3 to class.

•  Statistical problem solving
• formulate a question
• collect data
• analyze data
• interpret results
• Statistical questions versus other kinds of questions
• statistical questions anticipate variability
• Measures of center of numerical data: mean, median, mode
• The mean as “leveling out” numerical data

Thursday, April 5

Please bring the class activities from Section 15.3 to class.

• Measures of center of numerical data: mean, median, mode
• These measures can be different. Depending on the purpose, one might be a better choice than the others.
• The mean as “leveling out” numerical data
• Why do we calculate the mean by adding all the numbers and dividing by how many numbers there are? Why does that give us a reasonable single-number summary of the data?
• The mean as “balance point” of a dot plot or histogram
• When a dot plot or histogram has a “tail” the “tail” pulls the mean toward it.

## Week 14: Statistics

Tuesday, April 10:  Please bring the class activities from Section 15.4 to class.

• Please hand in the research study materials if you were participating in the study.
• No class on the last day, Tuesday April 24 – extra time to study for the final!
• Distributions of random samples
• Distributions of random samples have a characteristic shape
• How is the mean of a distribution of random samples related to the population the samples come from?
• How is the distribution of random samples of size 10 different from the distribution of random samples of size 20?
• We can describe how variable a set of numerical data is using measures of variability:
• The range
• The interquartile range (we often use that together with the median)
• The Mean Absolute Deviation (we can use that with the mean)

Thursday, April 12:  Please bring the class activities from Section 15.4 to class.

• Shall we schedule a review session for Friday April 27? The final exam is Tuesday May 1.
• Distributions of random samples
• How is the mean of a distribution of random samples related to the population the samples come from?
• How is the distribution of random samples of size 10 different from the distribution of random samples of size 20?
• We can describe how variable a set of numerical data is using measures of variability:
• The range
• The interquartile range (we often use that together with the median)
• The Mean Absolute Deviation (we can use that with the mean)
• Box and whisker plots

## Week 15: Statistics

Tuesday, April 17:  Please bring the class activities from section 15.1 to class.

Using random samples to draw inferences about a population.

• A random sample tends to be representative of the full population.
• How can we understand (informally) that the relationship between a random sample and the full population is (approximately) proportional?
• Reasoning about multiplication and division to draw an inference about a population from a random sample.

Thursday, April 19:  Please bring the class activities from section 15.1 to class.

• We will have a review session of Friday, April 27, 10 - 11:30 am  in our usual classroom, 229 Aderhold (feel free to come and go as you like). I will also return your tests at that time. If you need someone to pick your test up for you, please email me.
• Reasoning about multiplication and division to draw an inference about a population from a random sample.