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# Friday, May 17

Final projects. Please have a board for your game ready, as well as written instructions for how to play it. You will set up your games at the beginning of class on Friday, and then everyone will make rounds around the room playing all the different games. Make sure to play every game at least once, and to make a record of the outcomes since you will need those for the reflections. I will leave paper next to each game board so you can write down the results of playing the game on it. Make sure that your writing is understandable to the creators of the game. While you are playing, think about the probabilities of playing each game. When you are done, if there is time, we will discuss each game and the probabilities associated with it.

# Wednesday, May 15

Class work: Measures of center, comparing sets of data

• Start with looking at Illuminations for 5-10 minutes. I will let you play with the two graphing tools, Data Grapher and Advanced Data Grapher. You can find them here.
• Do an activity with box plots. Handout will be given in class.
• Short lecture on mean, median, mode. How is each one found in a set of data? When is each used? The activity I will give you may help answer the second question.

# Friday, May 10

Class work: Graphing data

Sections covered: 30.1, 30.2.

• We will go over how to make graphs in Excel, and then work on the problems from the packet I gave you last time. Note that we will be working in Excel, but you should also know how to do this by hand, as your students will probably need to know. We will discuss histograms, and note that in middle school and high school mathematics, it is too complicated to have bins of different widths, and we will therefore assume that all bins have the same width. We will review the difference between bar graphs and histograms.
• I will show you graphing capabilities of Illuminations.
• Here are the lecture notes.
• When we are done with this, we will work on the middle school worksheets. Do what you can. If you are not familiar with a topic, just make a note of it, and we will review it on Monday.

# Wednesday, May 8

Class work: Surveys, sampling, and visual representations of data

Sections covered: 29.1-30.2

Review questions for today:

1. Which square roots between 1 and 25 are you able to construct on paper?
2. (7 from 25.1) Suppose that a rope fits exactly around the equator. Then 40 feet more are added to the rope, and this longer rope is raised uniformly all around the equator. Which of the following could “walk” under the rope: a bacterium, an ant, a mouse, a 3-year-old child, or an adult human? You do not need to know the radius of the Earth, but if you insist on using it, it is approximately 4000 miles). Here is an explanation of this problem!

Note that these are not necessarily the types of problems that will be on the test, but I think they are fun and they help you review the concepts you will need (like the Pythagorean theorem and circumference of a circle).

New material:

• I want to talk about the ice cream problem briefly, and to discuss why the number of cones when order doesn’t matter is half of the number when the order does matter. Note that the number of ice cream cones when repetition is not allowed but order matters is $8\cdot 7$, but the number of cones when order does not matter is $8 \cdot 7/2$ because each combination is repeated twice. This generalizes to the lottery extra credit worksheet.
• Next, we will talk about surveys, and finish up the notes from last time.
• Finally, we will talk about different types of data and different ways to represent them. We also looked at some interesting graph. Look up happiness index to see how it’s calculated.
• We will work on some practice worksheets next time. Then next Monday we will talk about the measures of center, on Wednesday we will do some more practice with statistics, and on Friday we will review and have a game day.

# Final review topics

Here are the topics that you should expect on the final.

• Properties of polygons (which one have congruent diagonals, which ones have perpendicular diagonals, etc.); hierarchy of quadrilaterals (a square is a rectangle, etc.); regular polygons; angle measures in polygons. Practice problems: 2, 3, 6, 15 from Section 17.1; 2-4 from Section 17.2; 2, 3 from Section 17.3.
• Symmetry may come up, probably as part of a different problem. Practice problems: 4 from Section 18.1.
• Circles may also come up in context of measurement, but you should know some basic properties of circles. Practice problems: 1 from Section 21.1.
• Polyhedra, spheres, cones, and cylinders will probably also come up in terms of measurement, but you should know their properties. Practice problems:  5, 6 from Supplementary exercises after Section 16.2; 5, 16 from Section 16.3; 1, 4, 9 from Section 21.2.
• Ideas of measurement, having some benchmarks for different units, knowing some basic conversions. Practice problems: 10 from Section 23.1; 6-8, 27, 31, 33 from Section 23.2.
• The meaning of area and perimeter; formulas for the area and perimeter of polygons (triangles, quadrilaterals, and polygons that can be broken up into triangles) and circles. Practice problems: 3, 4, 7, 11, 12, 17 from Section 24.1; 2, 3, 6-8, 14, 18, 19 from Section 25.1.
• Similarity: properties of similar figures and finding missing lengths and angles given pairs of similar figures. Practice problems: 2, 9-14 from Section 20.1; 2 from Section 20.2.
• Pythagorean theorem: using the theorem in different contexts, and being able to outline a proof of the theorem. Practice problems: 9, 11, 14, 28 from Section 26.1.
• Surface area and volume of prisms, pyramids, cylinders, and spheres; volume of cones; knowing how to apply the formulas in problems. Practice problems: 1, 11-13 from Section 24.2; 2, 13, 14, 18, 20, 21 from Section 25.2; 2, 3 from Supplementary exercises after Section 25.2.
• Probability: problems similar to ones we did in class and on homework, mostly those containing probability trees or tables. Practice problems: 12, 13, 14, 18, 19 from 27.2; 3-9 from 28.1; 1-10 from 28.2; 7 from 28.3.
• Statistical graphs: analyze and create pie charts, bar graphs, histograms, line graphs, and boxplots, and know when each is applicable. Practice problems: 2, 3, 7, 8, 9 from 30.1; 3 from 30.2; 3-10 from 30.3.
• Mean, median, mode: computing each, knowing the difference between the three, knowing situations in which each of the three is appropriate,  knowing how each changes when data changes. Practice problems: 1, 3, 4, 5, 8, 10, 12, 17, 18 from 30.4.

# Monday, May 6

Class work: Introduction to statistics

Sections covered: Chapter 29

2.MD.10. Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems1 using information presented in a bar graph.

3.MD.3. Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.

4.MD.4. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.

5.MD.2. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.

6.SP.1. Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.

6.SP.4. Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

7.SP.1. Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

7.SP.2. Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.

Activities: We will follow these lecture notes closely. All information about what we are doing is contained in them. We will follow chapter 29 from the book closely, and also look at some Connected Math activities.

Here is an Illuminations lesson that you can look at:


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# How loud or flashy are verbal and somatic spells? [duplicate]

I was trying to cast prestidigitation on a guard behind his back and my DM said that he would definitely know I did it because the verbal part of the spell would be in a loud voice. He also said that the somatic part would be flashy and with grand movements of the hands. How loud does the verbal need to be and how flashy are the gestures?

# It is up to the DM, but they are noticeable.

In 5e nowhere does it define anything specifically defining the volume of spells or how grand gestures have to be. These kinds of details are left entirely up to the DM to decide.

Verbal (V)

Most spells require the chanting of mystic words.

[...]

Somatic (S)

Spellcasting gestures might include a forceful gesticulation or an intricate set of gestures. [...]

This is all the detail we get from the rules. It is worth noting that the rules do even say that there is a potential range of how obvious the somatic component might be, so it may even depend on the spell. In the end, your DM will have to decide how they want it to work in their world.

Jeremy Crawford has shed some light on the intent for verbal components though (in an old unofficial tweet):

The verbal component of a spell must be audible to work. How loud is audible? That's up to the DM.

One thing is sure though: both the verbal and somatic components are definitely noticeable however. We know this because there is at least one class feature, a Sorcerer's Subtle Magic metamagic, whose only purpose is to make them not noticable:

When you cast a spell, you can spend 1 sorcery point to cast it without any somatic or verbal components.

# Flashy/loud enough to always be noticed

The reason I say this is because there is a specific mechanic option that removes those components from notice: the Sorcerer's Subtle Spell Metamagic:

When you cast a spell, you can spend 1 sorcery point to cast it without any somatic or verbal components.

Simply allowing any caster to 'hide' their components completely minimizes this class feature.

As long as the components are heard/seen, then the DM/player can decide together an appropriate volume/flashiness. The volume/movement isn't mechanically important here - it's just that the somatic/verbal is always noticed.

## Noticing vs understanding

However, just because they notice you've cast a spell doesn't mean they know what you've cast. Xanathar's contains some optional rules that cover identifying spells.

• I was not asking if he would notice the spell. I was asking how loud the verbal is and how flashy the somatic. – Eternallord66 Feb 13 at 14:53
• @Eternallord66 Gotcha - adjusted my headline. – NautArch Feb 13 at 14:54

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## Integers !!!

Hey guys ,
was wondering if you guys know of any lecture notes regarding integers ?
i would like to further my knowledge in this field ... cheers

 Recognitions: Gold Member Science Advisor Staff Emeritus Are you serious? There are thousands of different "fields" involving integers. You will have to narrow down the search! What, exactly, are you looking for? Number theory problems such as solve Diophantine equations? Analysis questions such the the definition of integers and basic properties?
 Recognitions: Homework Help Science Advisor While you're at it, HallsofIvy, how about some notes on real numbers too?

## Integers !!!

I guess to start with I would like to look at the basics such as the proof of : If S ⊂ ℤ , then there is a natural number g such that S={ gn : n ∈ ℤ }

And also i came across a proposition somewhere in some maths books ( but it's only a small part ) Saying that :
- If A and B are subgroups of ℤ , then so is their intersection A ∩ B and so is the set
{m+n : m ∈ A , n ∈ B }
they followed on by saying this which i don't really get the picture
- A ∩ B contains every subgroup contained in both A and B
- A + B is contained in every subgroup containing both A and B

Recognitions:
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 Quote by garyljc I guess to start with I would like to look at the basics such as the proof of : If S ⊂ ℤ , then there is a natural number g such that S={ gn : n ∈ ℤ } And also i came across a proposition somewhere in some maths books ( but it's only a small part ) Saying that : - If A and B are subgroups of ℤ , then so is their intersection A ∩ B and so is the set {m+n : m ∈ A , n ∈ B } they followed on by saying this which i don't really get the picture - A ∩ B contains every subgroup contained in both A and B - A + B is contained in every subgroup containing both A and B someone please help me out =) thanks
You are using non-standard fonts that do not display on my web-reader. Please try LaTex or just stating the problem in words.

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Homework Help
 Quote by garyljc I guess to start with I would like to look at the basics such as the proof of : If $S \subset \mathbf{Z}$ , then there is a natural number g such that $S = \{ gn : n \in \mathbf{Z} \}$.
Am I misreading this, or is this simply not true? For example, take S = {1, 2, 3}, this is finite, while any set of the form $\{ g n : n \in \mathbf{Z} \}$ is necessarily {0} or countable infinite.

 Quote by garyljc And also i came across a proposition somewhere in some maths books ( but it's only a small part ) Saying that : - If A and B are subgroups of $\mathbf{Z}$ , then so is their intersection A ∩ B and so is the set $\{m+n : m \in A , n \in B \}$ they followed on by saying this which i don't really get the picture - $A \cap B$ contains every subgroup contained in both A and B - A + B is contained in every subgroup containing both A and B
You can also just check the group axioms and see that it is true.

 this is what i copied exactly from the book ... but anyways , do you know of any sites that has such notes or reading material that could help me ?
 by the way i did make a mistake If S ⊂ ℤ is a subgroup , then there is a natural number g such that S={ gn : n ∈ ℤ } this should be the right one halls , where could i get latex ?
 Recognitions: Gold Member Science Advisor Staff Emeritus You don't have to "get it" at all, it's part of this forum. Start with [ tex ] (without the spaces) and end with [ /tex ] and use LaTex syntax in between. Here's an example: $$e^x= \sum_{n=0}^\infty \frac{x^n}{n!}$$ Click on that to see the code. More on LaTex syntax can be found here: http://physicsforums.com/showthread.php?t=8997

 Quote by garyljc but anyways , do you know of any sites that has such notes or reading material that could help me ?
Check here.

 thanks thanks .... halls ... i'll get it done right away =)
 Recognitions: Gold Member Science Advisor Staff Emeritus [quote]I guess to start with I would like to look at the basics such as the proof of : If , then there is a natural number g such that .