Module 5

Box Plots

Questions for my students…

1. Based on the median in both box plots, who throws away more pounds of trash?  My students throw away more trash per month.
2. Would your answer to number 1 change if we had the same amount of students as the German class?   It would be hard to answer this without data to look at. If we added more data from other students in the school, I think their answers will be similar to our class and would have minimal effect on the box plot.
3. Do you feel this is a fair comparison?  Even thought the number of students in each of the classes was substantial different I do not feel having the same amount students would not effect the numbers.  

Common Core Standards 

Write down two “first impressions” you have about the standards

The first thing I noticed is how well everything is laid out.  It is easy to read and to follow.

  

How do the concepts progress through the grades?  How do the concepts change and increase in rigor and complexity for the students?

There is what feels to me a natural progression through the grades.  For example In Kindergarten, we are introducing the topic, sorting objects and counting them.  In first, we start to organize, represent and interpret categorical data.  Second grade we and in numerical data and different ways to graph the data in bar graphs and line plots.  They solve simple questions about the data.  The standards continue in this way increasing in complexity and building upon each other in each grade.

Now look at both the Common Core and NCTM standards to respond to the following questions:

Does the Common Core Standards align with what NCTM states students should be able to know and do within the different grade level bands?  (Note that NCTM is structured in grade level bands versus individual grade levels.) Give examples of which standards align as well as examples of what is missing from the Common Core but is emphasized in the NCTM standards and vice versa.

Even thought the wording is somewhat different I do feel like they do somewhat align.  In grade, K-3 the verbiage is very similar.  However, in 3-5 grades I feel like the NCTM is more detailed where the common core just gives a few examples. Looking at 6^th grade this is the most detailed from either source. The verbiage on this page is very similar.

Would You Rather?

We were asked to pick a lesson and to examine the materials to determine which mathematical ideas it could raise for students and to consider how you might highlight those ideas.  The data activity I chose to focus on is Representations of Eagle and Whale Data.  Students were asked the question, “Would you rather be a whale of an eagle?” In this activity, students make their own representations of the results of the eagles and whales survey.

When using this activity, what mathematical ideas would you want your students to work through?  I would want my students to work through organizing, representing and interpreting the data collected.  Organize it into categories and then make a representation of the data by using cubes, stick-on notes, drawings; or any other material I have in my classroom.

How would you work to bring that mathematics out?  After everyone is done with his or her representations, I would have a class discussion.  On the whiteboard, I would make a graph and plot the class data as the students make suggestions to me.  I will ask probing questions to get them to ponder the decisions they have made.

How would you modify the lesson to make it more accessible or more challenging for your students?  By letting my students work with partners and select their own medium to display their data I am making the activity very accessible.  To make it more challenging I might add another category such as students who could not make up their mind.  On the other hand, I could have them show me another way to display the data.

What questions might you ask the students as you watch them work?  I would ask questions like,

·         Are you making sure your representations are clear so that others can read your findings?

·         Does your chart show an accurate count of the numbers of eagles and whales?

·         What does this data tell us?  

·         What animal seems to be the most popular?

What might you learn about their understanding by listening to them or by observing them? 

 By listening to them, I will be able to hear who is interpreting the concept correctly and who still needs more help.  I can ask probing questions to assess just where their inconsistencies are. Observation is a great way to see your students in action. I will use this time to ask questions and make comments on the projects they are working on.  One little question or comment can get them thinking critically about the task.

How do the concepts taught in this lesson align to the Common Core Standards?

CCSS.Math.Content.1.MD.C.4 Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Questions to ponder Miss Em….

I found this week’s module a little less intimidating, how did you do with this week’s math?

I have yet to observe a lesson on collecting data in any classroom I have been in, have you?

Do you feel it is better to let students work in groups or in pairs in activities such as theses?

Module 4

Variations about the mean

Surprising this Annenberg assignment was quite easy to understand and I went through the problem quickly.  The one part that I did struggle with was the very last set questions f - h.

     

f. What would happen to the MAD of a data set if you doubled every number in it?

g. What would happen to the variance of a data set if you doubled every number in it?

h. What would happen to the standard deviation of a data set if you doubled every number in                    it?

I wish they had provided an interactive area where you could plot out the change.  It was hard to wrap my head around these three.  Em, did you figure these out?

Generating meaning together

I really enjoyed this article.  I liked how the teacher did not just come out and tell her students what the terms range, mode, median and mean meant.  Instead, she let them investigate.  She let them search the data for their own answers.  By doing this, I believe she allowed her students to develop a deeper understanding of each concept and her students walked away feeling accomplished.

From what I remember about school, I was never given the opportunity to decipher a problem on my own.  The teacher demonstrated the correct procedure and we were expected to follow what was show to us.  Correct answers were stressed instead of understanding the concept.
I feel like this is a pattern that happened way too many times. What do you think Em?

Working with the Mean – Activity

In this activity, we were asked to use our interlocking cubes to help demonstrate the problem and our solution.  Once I figured out the first problem with the cubes, I was able to see the answer more clearly on the line plot. There are seven bags of peanuts.  Five of the bags have the following numbers of peanuts in them: 5, 7, 8, 9, and 12.  Make a representation of the five bags by using cubes.  Make another representation of the five bags on a line plot.

12					x
11					x
10					x
9				x	x
8			x	x	x
7		x	x	x	x
6		x	x	x	x
5	x	x	x	x	x
4	x	x	x	x	x
3	x	x	x	x	x
2	x	x	x	x	x
1	x	x	x	x	x
	1	2	3	4	5	6	7

Now use your representations to figure out how many peanuts could be in the other two bags so that the mean number of peanuts in all seven bags is 8.  Try to figure this out without adding up the peanuts in the five bags.  Find at least two different sets of numbers for the two bags that will work to solve this problem.

12					x		X
11					x		X
10					x		X
9				x	x		X
8	?	?	x	x	x		X
7	?	x	x	x	x		X
6	?	x	x	x	x		X
5	x	x	x	x	x		X
4	x	x	x	x	x		X
3	x	x	x	x	x	x	X
2	x	x	x	x	x	x	X
1	x	x	x	x	x	x	X
	1	2	3	4	5	6	7

The mean tells me if all things were equal, I would have 8 peanuts in each bag for a total of 56 peanuts all together, fair share value.  From there I looked at the bags that had more that 8, which was 4 and 5.  I counted the overage, which was 5, and subtracted that from 8, which is 3 and placed that in the 6^th bag.  The I counted the missing peanuts in bags 1 and 2, which were 4.  Added that number to 8 and got 12.  You could use any combination that equals up to 15 in the last two bags.

  

What is the least number of peanuts there could be in one of the additional bags?  What is the greatest number? 

  The least number there could be is 1 and the greatest number would be 14.

12					x
11					x
10					x
9				x	x
8			x	x	x
7	?	x	x	x	x		x
6	?	x	x	x	x		x
5	x	x	x	x	x		x
4	x	x	x	x	x		x
3	x	x	x	x	x		x
2	x	x	x	x	x		x
1	x	x	x	x	x	x	x
	1	2	3	4	5	6	7

 What could be in the two bags if the mean number of peanuts was 7?

 I approached this problem the same way.  Counted the overage in bags 3,4,5 which was 8. ( that is one over my mean) I then counted the missing peanuts in bag 1, which is 2, added that to 7, which gave me 9.  Remembering that I was over 7 in my initial count of overages by 1 and the 1 that I placed in bag 6 I subtracted them from the 9, which left me 7 in bag 7.

What if the mean number was 10?

12					x
11					x		x
10	?	?	?	?	x		xx
9	?	?	?	x	x		xx
8	?	?	x	x	x	x	xx
7	?	x	x	x	x	x	xx
6	?	x	x	x	x	x	xx
5	x	x	x	x	x	x	xx
4	x	x	x	x	x	x	xx
3	x	x	x	x	x	x	xx
2	x	x	x	x	x	x	xx
1	x	x	x	x	x	x	xx
	1	2	3	4	5	6	7

                               

Again, the approach was the same.  I counted the overage in bag 5, which was 2.  Subtracted that number from 10, which gave me 8.  I put that number in bag 6.  I then counted the missing peanuts in bag 1, 2, 3, and 4, which was 11.  Add 11 to the 10.  And put that in bag 7.

This was fairly easy for me to do but sometimes when things seem easy its because I missed the concept all together. Did you get similar outcomes?

Video - How Much Taller

While watching this video I noticed that the students were confusing two separate concepts average or mean and mode. I did not like the way the teacher cut every child off while they were explaining their thinking and gave her interpretation of what they were going to say or trying to say.  I do not know what she was trying to do, help them or guide them to the wrong answers. 

Em, did this bother you at all? What do you think she was trying to do? Do you think it helped their understanding or hurt it?

In my opinion, the children did not understand the definition of mean.  Jason’s answer of “53 because that’s the average” At first I was impressed that he knew to look for the average until me gave his explanation of what the average was “most common height.”  Jason knew he wanted to find the average but gave the definition of finding the mode. 

It is hard to tell what the teacher was teaching in this clip or what her intentions were for her students to take away.  Nevertheless, from this clip alone I feel she only confirmed in her students misconceptions about what the mean means and how to find it.

Choose at least two questions:

Annual salary is often a touchy subject for teachers whose low pay and high workloads are axiomatic.  Search the virtual archives of a newspaper in an area where you would like to teach.  Look for data about averages and entry-level salaries as well as information about pay scales and increases.  Evaluate the data.  What does it tell you?  What doesn’t it tell you?

Interesting data…

Teacher w/ bachelor’s degree with 0-2 year experience $2,566.67 a month.  I will not be eligible for a raise until my 6th year working and then the raise is minuscule $35.oo extra a month. Looks like I will be eligible for a raise every year after that.  I thought about trying to be an assistant for one to two years just to become more comfortable with being in the school system but a teacher’s aid makes $1,935.51 -$3,087.64 a month.  I am sure this depends on experience, time in position and schooling. I am sure I would be hired at the lowest pay. 

North Carolina Public Schools

Find examples of averages in a daily newspaper, from the sports page, or any page. Then describe what these averages “mean” – their significance, implications within the context of the story, and so forth.

I found a story related to the first question I answered in this weeks State Port Pilot entitled

School officials say new-teacher pay plan a good first step towards proper salaries

This article tells about the possibilities of a pay increase for beginning teachers from $30,800 to $35,000 over the next two years. Here is an excerpt from the article:

“According to data collected by the National Association of Educators, the national average starting salary is $36,141.”  This means that of all the starting salaries in all the states the average pay is $36,141. We will still be underpaid based on average but we would no longer be the least paid teachers in America. What is doesn't say is will this pay increase travel up the line or will teachers that have been working for years still be underpaid for their qualifications. Nothing is official as of yet so I wouldn’t hold my breath!