## 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
- What are your current thoughts about
- 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?

- When is a relationship between quantities
- 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 can be answered (or addressed) by collecting data
- 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.
- Brief discussion of the Mean Absolute Deviation. Youtube video about the MAD: https://m.youtube.com/watch?v=UBh48VErmZg

## Week 16: Review

**Tuesday, April 24**: No class. Review for the final exam.

**Friday, April 27**: Review session 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.