FIRST STEP IN CHEMISTRY - The ATOM

 An activity to introduce the atom.

Building an Atom reinforces the concepts of Periodic Table organization, atomic number, atomic mass, proton-electron pairing, shells and subshells, order in which shells and subshells are filled, future study of valence electrons, electron-sharing bonds, ions, and isotopes.

 

This example represents NEON, element #10, with 10 protons and 10 neutrons in the nucleus, and 10 electrons in the subshells.  The first ring (1s) carries 2 electrons, the second (2s) another 2, and the third (2p) a whopping 6!!






Chemistry isn't normally listed as a discrete subject until high school.  We dabble in it at the lower grades with little activities or "experiments" (Isn't it fun to make bubbles and to see the soda pop erupt from the bottle?), and in middle school we introduce a more scholastic approach to some of the foundational concepts.  But when studied at greater depth in high school, some students get stumped at the very beginning by the Periodic Chart (with all of its imbedded detail), the structure of the atom (which seems more like fiction than fact since we never actually "see" it with our own eyes), and the confounding electron cloud. 

At Tutoring Resources, the Summer Preview in Chemistry recognizes the seemingly abstract aspect of the subject and employs ways for a variety of students to gain an intrinsic understanding by doing activities that may not be possible in a classroom of 30 kids.  This year, we've expanded the program to middle school students with great success, proving once again that younger students can rise to the challenges of higher level learning.

This blog explains an activity to explore the structure of the atom through arts and crafts.  It can be completed at home with some basic craft supplies.

Supplies you'll need:
     -- wire rings, embroidery hoops, or thick wire to simulate the subshells.  Each successive shell should be of larger diameter than the previous one.
     -- styrofoam balls, wooden balls, beads, or similar objects to represent protons, electrons, and neutrons, each component of a different color to differentiate them.
     -- paint, to achieve different colors.  Use a paintbrush, not spray paint on styrofoam.  I've had the unfortunate experience of spray paint melting styrofoam.  Although this may be an interesting reaction in a chemistry experiment, physical and chemical changes come much later in the curriculum.
     -- glue or glue gun to affix the components in place.  The younger the student, the less advisable is a hot glue gun!!  I've had strapping football players react strongly to hot glue on fingers.  I've also seen a clear plastic ornament used for the nucleus -- quite attractive and allows the protons and neutrons to move around inside.
     -- string, fishing line, or thread to tie the rings together, attach the nucleus at the center of the innermost ring, and hang the "atom" for display.
     -- a periodic chart for guidance.

Building several "atoms" has pinpointed a few tips that make the construction easier and more effective.
     §  Make the nucleus first by gluing "protons" and "neutrons" together in sort of a sphere.  Before adding the last few components, glue in the string that will be used to tie the nucleus to the shells.
     §  Tie the appropriate number of rings together using the scouting method for connecting teepee poles.  This will help to spread the rings for a more spherical presentation.
     §  I tried sawing an opening in the metal rings so the styrofoam balls could be strung on, but that was a lot of effort.  I settled on cutting a slit in the styrofoam and pressing each one onto the ring.  A squirt of glue closes the slit and keeps the balls from sliding to the bottom.
     §  When cutting a length of string for tying, longer is better.  It's easier to cut off excess thread than it is to tie a square knot with only an inch to work with.
     §  A dab of glue is an effective method of setting a knot, especially in fragile threads.

STEPS:
1.  Decide the element to be constructed.*
2.  Determine the number of protons, neutrons, and electrons needed.
3.  Assemble materials.  Paint components if necessary.
4.  Construct nucleus.
5.  Tie rings together.
6.  Tie nucleus so it hangs in the center of the innermost ring.
7.  Affix electrons to appropriate rings.
8.  Tie string as a hanger and suspend far enough from a wall to allow the "atom" to move with the breezes.

* To make the exercise more challenging, I prepared a few styrofoam balls in three different colors.  The students needed to count how many of each color were available and then select an element that would need no more components than they had at their disposal.  This approach reinforced the idea of protons and electrons being equal and the value of an element's atomic number.

Did the activity work?  Well, from the social media posts, the enthusiasm of requests to "make another one," and the fluency with which the kids can now speak the names of  the first 10 elements, I think atom building will be a feature of the Chemistry Preview for many years to come.

Atom display: (L to R) Neon, Helium, Lithium, Hydrogen, Boron





EXTENSIONS:  The concepts introduced here can be expanded to introduce Periodic Table BLOCKS, electron configuration, elements with higher atomic numbers, isotopes, and ions.

An anecdote from the experience of a seventh grader:  In working on the valance shell, the student dropped one of the electrons and questioned what had happened to the element.  This became a learning opportunity to introduce ions.

QUICK STEPS TO COMPLETE THE SQUARE

This article is going to be a “quickie” because I believe Completing the Square should be just that....quick, easy.  So let’s dispose of all the middle steps and just move swiftly from the General Quadratic Equation to the Vertex Form.

We’ll use a sample equation.......12X^2 + 8X - 10 = Y

1.  Start with the pattern
     of the Vertex Form.................................A(x + h)^2 + K = Y

2.  From the original equation,
     think of the X-variable
     terms as a separate entity......(12X^2 + 8X) - 10 = Y

3.  Factor out the leading
     coefficient, so that the
     coefficient on the X^2
     is just 1..............................12(X^2 + 8/12 X) - 10 = Y

     I’d reduce that improper
     fraction...............................12(X^2 + 2/3 X) - 10 = Y

4.  Fill in “A” in the
     Vertex Form..........................................12(X + h)^2 + K = Y

5.  Fill in the “h” value
     of the Vertex Form
     with 1/2 the new
     coefficient on X in
     Step 3..............................................12(X + 2/6)^2 + K = Y

     I like smaller numbers, so I’m reducing
     that pesky improper fraction................12(X + 1/3)^2 + K = Y

6.  Think for a second about what we’ve
     done.  By creating the (X + h) binomial
     and squaring it, we’ve actually added
     more to the equation.  (FOIL it through
     if you need proof.)  So we need to
     remove it again.

     Square “h”, multiply it by “A”,
     and subtract it from the
     constant in the General
     Form equation...................12(X + 1/3)^2 + (- 10 - 12/9) = Y

     I feel I’m doing more reducing that any
     real work here.....................12(X + 1/3)^2 + (- 10 - 4/3) = Y

7.  A little arithmetic and,
     TAH-DAH, I’m ready
     to solve for X and label
     the vertex on a graph......................12(X + 1/3)^2 - 34/3 = Y


Many texts add the step of finding the perfect square trinomial before factoring it into the binomial squared.  I think it’s just a way to insure that you subtract out the extraneous constant value but I prefer "the elegant solution."

ALGEBRA II: QUADRATIC EQUATIONS AND THE PARABOLA

EVERYTHING YOU NEED TO KNOW ABOUT

QUADRATIC EQUATIONS AND THE PARABOLA
 IN THE ALGEBRA II CURRICULUM

Quadratic equations can be expressed in several forms, each emphasizing a different aspect of graphs.  In Algebra I we learned the General Form and practiced factoring, substituting various values of X or Y, the Quadratic Formula, and (maybe) Completing the Square.  In advanced Algebra courses, we manipulate the General Form to openly express some of the familiar coordinates (like the vertex) and to examine in greater depth some of the properties of the Parabola.

While what follows are simple explanations of a Parabola that opens up, a few transformations will describe graphs which open down, right, or left.  If more detail on transformations is needed for your class, use the comments section to request examples.

   

    GENERAL EQUATION FORM            AX^2 + BX + C = Y

    VERTEX FORM                                  a (X-h)^2 -k = Y

    STANDARD FORM                             X^2 = 4PY
                                                               (X-h)^2 = 4P (Y-k)

-------------------------------------------------------------------------

GENERAL FORM

        AX^2 + BX + C = Y

    This is the form that is used in systems of equations and matrices.

    Coefficients from the General Form are also used in the quadratic formula to find X intercepts.

    These X values give the intercepts of the X axis -- the roots.
  
    The X value of the Vertex is the midpoint between the two roots.  Substitute to find the Y coordinate of the Vertex.

    The axis of symmetry is .... X = the X value of the vertex.

    The Y intercept is C.

Graphing from the General Form

    Factoring or solving for the root is generally the first step in graphing from the General Form.  And, of course, the y-intercept is expressly given as the constant.

    Finding the vertex takes a little more calculation.


                            X^2 - 10X + 21 = Y

X = 7, 3

Vertex X = 5
Vertex Y = -4

Y intercept = 21

 

VERTEX FORM

        a (x-h)^2 - k = y

    In this form, (h,k) is the vertex of the parabola; hence the name “vertex form.” 

    To move from the General Form to the Vertex Form, you will need to complete the square.  (See the blog, Quick Steps to Completing the Square, 7-22-13)

    To find the roots (solutions), solve for X by isolating the (x-h)^2, taking the square root of both sides, and isolating  X.

    The y intercept is calculated by substituting 0 (zero) for X.
  

Graphing from the Vertex Form

    As the name implies, the vertex is the first step in graphing from this form.  Calculations are needed to find the axis intercepts.

a (x-h)^2 -k = y

Vertex = (5, -4)

X intercepts = (7,0) and (3,0)

Y intercept = 21

Did you notice that the same 4 points are used to graph from both the General and Vertex Forms?  If a fifth point is required for class, use the symmetry principle to find the point directly across from the y-intercept and the same distance from the axis of symmetry but on the other side.

 

STANDARD FORM

        X^2 = 4PY

        (x-h)^2 = 4P (y-k)

    This form emphasizes the Focus and Directrix of the graph.  (h,k) is the vertex and P is the distance from the vertex to the Focus point along the axis of symmetry (X = Vertex X) and the distance from the Vertex to the Directrix.  The Directrix is the horizontal line, Y = Vertex Y - P.

    To solve for the roots, substitute 0 (zero) for Y.

    To solve for the Y intercept, substitute 0 (zero) for X.

Graphing from the Standard Form

    Details represented in this form are especially useful in physics.  The focus is the point to which any ray striking the ‘”cup” of the parabola is reflected.

(X - 5)^2 = 1(Y + 4)

     4P = 1

       P = 1/4

Vertex = (5, -4)

Focus = (5, -15/4)

Directrix, Y = -17/4

X intercepts = (3,0), (7,0)

Y intercept = (0,21)

Do you notice anything “special” about the Standard Form compared with the Vertex Form?  They are actually the same, but the Standard Form specifically mentions the 4P value.  A little algebra and you’ll see that  4P = 1/ a.  All we’ve really done is isolate the squared binomial by moving everything else to the other side of the equation.

Any of these equations can describe a Parabola opening down by changing P to the Arithmetic Inverse or a Parabola opening left or right by exchanging X with Y and vice versa. 

Relation of the Focus, Directrix, and Latus Rectum*

    Any point on the Parabola is equidistant from the Focus and the Directrix; that’s the definition of the Parabola, and the distance is explicitely expressed in the Standard Form as P.  The Latus Rectum is the length of a line perpendicular to the Axis of Symmetry, through the focus, and intersecting the curve of the Parabola.  It is also directly expressed in the Standard Form as 4P.

    *  Please excuse a little silliness in the middle of all this serious math.  I work with Middle Schoolers quite a bit and, although many schools exclude it from the curriculum, mention of the “Latus RECTUM” always brings a giggle, something that is too often missing in the math classroom.