Irrational Numbers on the Number Line

Consider the roots of the first 9 Natural Numbers (√1 to √9), how many of them produce Irrational Numbers?

List them.  Graph their approximate location on a number line.

number-line-600x271

Explain how you determined where to place them on the graph.

 

Is √10 Rational or Irrational?  Explain how you know.  Where on the number line would you graph it?

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Linear Art Project

This is a fun project that I started doing to support my Art teacher and show students that Algebra can make the most amazing things.

Step 1:

Have students pick out a picture.  This is a great sell to students since they have control over what type of picture, icon, avatar, etc that they use for the project.

This was the example I used, the logo from my college, the Bemidji State University Beaver.

IMAG0600_BURST001Step 2:

Have student create a free hand drawing of the picture.  This allows them to adjust the picture, editing lines and shading that they may not want to show.  It allows them to be creative and adapt the picture for the project.

Here was my free-hand drawing of the Beaver.

IMAG0601Step 3:

Students recreate their free hand sketch on graph paper and using only lines.  They should try to make shapes end on grid points of the paper.  This part can take a while, if students created a large enough free hand sketch, you could allow window tracing to help transfer the image to graph paper.  I have students make 2 copies of the graph image, one they can decorate and one for the next step.

Here is my colored graph paper image.

IMAG0603

Step 4:

Have students create an X&Y Axis on their picture.  I allow them to create them wherever they want, and usually they have them right in the center of their drawing.  Depending on the patience level of the students, I have them label points that create their drawing.  There are times I only have them do a set amount (20 or so) and times I have them label each line segment ending.  This is totally up to you.

My XY Axis image:

IMAG0602

Step 5:

DESMOS!  Need I say anything more.  I used to have students write the equations of the lines that create their image, but with the totally awesome program DESMOS, I now have students create their image with it.  DESMOS has helped students understand how changing the slope or intercept effects the line, and with the instant drawing of the line when they enter the equation, it allows them to visually see where their line is.  This is a great error check for students, and they accept mistakes more readily than if they are writing equations on paper.

My DESMOS image.

BSU BeaverStep 6: The Finalé

To complete this project, I have students create a collage of their sketches, and a printout of the equations from DESMOS.  I then hang these posters out in the hallway for everyone to enjoy.  This attracts students from all over the building to come check out what kind of cool activities we do in 8th grade.  I am even getting new 8th grade students asking me when we will start this project!

Human Histogram

Since it looks like I may have an alteration to my teaching assignment next year, I feel it is time to dust this off and pump it up.

 

Human Height Histogram

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I started using this activity when I was teaching 7th grade.  There is a lot of physical growth that happens during the middle school years, and this is a great way for students to look at data that is meaningful- data about themselves.

We start by making predictions about which gender has the greatest average height at the beginning of the year.  Typically in 7th grade it was the girls, and we devise a way to determine it.  Students come up with a way to determine the mean height of both groups, which usually involves measurement since students are not accurate with their actual height.  We spend the day measuring all the students in the grade, and then look for the measures of central tendency.  Students will tell me the range, median, mode and mean of each gender, and they also are curious how that stacks up for the group as a whole.  I ask them how we could display this visually and they immediately tell me to graph it.  The type of graph will vary however, many 7th graders are overly “precise” in their measurement of height since they are trying to be taller than their friend.  The number one graph of choice for my students has been a bar graph, so I allow them to try this method.  After attempting to draw a bar for each student, they decide to use a different graph.  This normally results in a scatter plot where we use different colors for boys and girls.  My 7th graders are not particularly adept with reading scatter plots yet, and will complain about the representation, saying that it is more confusing than looking at a list of numbers.  So we brainstorm for a new idea.

After students struggle with other graph types, I ask them if they would want to go back to a bar graph.  I usually get a unanimous response of “YES!”  We talk about how bar graphs display data, and I ask students if instead of graphing each individual student, if we looked at graphing heights instead.  Students think on this a bit and then get into how this could help or be more complicated.  They also start arguing about the heights that we recorded, saying that there is too big of a variation of heights to graph.  I suggest “grouping” heights together.  Students really latch onto this idea, and we brainstorm on how to group the heights.  Typically students either decide on 1″ or 1/2″ intervals.  They then get to work graphing the heights and how many occurrences there are.

One thing I do during the measurement process is take a picture of each student as they are being measured.  I then print out everyone’s picture (normally a head shot, and typically multiple ones- some on colored paper) and when we as a class believe we have a good graph, allow students to “graph themselves” on the wall outside my door.  This creates a visual of the class overall.  We then create a bimodal graph slightly below it that represents each gender.  (A sample is below, but I no longer have pictures of my student ones so this is shown generically)

Histogram_of_Height_2

Students typically do not like this because some of their faces are covered.  So we represent it in another way with a vertical histogram, females on the left side and males on the right (stole this idea from Stem and Leaf plots).  This give students a great visual of the height distribution by gender.

Double histogram hieght

This is what I want students to analyze, I even take them outside and take a picture of them in this format.  They make conjectures on why this distribution occurs, what it may have looked like in earlier grades, and what it will look like when they graduate.  I then ask students to make predictions on how much they will grow over the school year, and we create one last vertical histogram from that.  Then we wait.

At the end of the year, we go through the measuring process once again.  We determine the measures of central tendency and graph our results.  Students then compare their predictions to the actual results.  When I ask students to compare the two, I get all sorts of humorous reasons why things may or may not line up.  “I drank coffee every morning this year because I was tired all the time, Mom told me coffee stunts my growth,” has to be one of my favorite student comments.

Now that I will be teaching high school as well, this will be a fun multiple year project, I may have issues storing graphs for a few years until I measure my students again, but I think it will be a great extension to this.  When I dust these off for the seniors, I can’t wait to hear what their comments are, what the remembered about it, and what kind of estimations they now come up with for their height distribution.

Achievement for All- Ruby Payne

I will start keeping tabs on what I am reading- I seem to misplace too many good ideas.  I am also going to keep a page where these will also be posted.

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Introduction:

Important Development in the middle grades:

  1. Physical Development: puberty and body image
  2. Cognitive/Intellectual development: changes in their brain
  3. Moral Development: moral compass
  4. Psychological Development: identity and differentiation from adults
  5. Social/Emotional Development: safety and belonging

These tend to be what is at the core for any issue a middle school student faces.  Being aware of these and talking with students so they can identify them are important steps in building a reflective, stable, independent young adult.

Three interlocking factors that impact cognitive framework of students:

Thinking

These are things that influence your student, and are out of your control.  They are a student’s home conditions, what they have access to at home, and relationships with people beyond the classroom.  Our students come to us with prior experiences established, being able to understand what they are, which ones you can overcome, and others you can help with are crucial to helping your student succeed.

Resources that Build Stability:

Resources

These are characteristic that will help us determine where our students are, and what resources they have available to them.  It will allow us to provide interventions to assist with developing resources for our students.

“That which doesn’t kill you, makes you stronger”

  • Ability to survive
  • Clear understanding of concrete reality
  • Ability to defend oneself
  • Strong sense of connection to others also in survival mode
  • Ability to problem-solve and “make due”
  • 6th sense about adults who may not be safe
  • Capacity to go all day without food
  • Informal, casual approach to living
  • Ability to entertain and be entertained
  • Capacity for enjoying the basics of life

 All of our students will come to us with some of these traits.  Keep them in mind, know what reactions are prompted from Strengths built from previous experiences.  Use these strengths to make connections to your student, which will in turn create trusting relationships.  This will allow your students to learn.