TAKS Tutorial by maclaren1

VIEWS: 25 PAGES: 72

									 TAKS Tutorial
Algebra Objectives 1 – 4
                  Part 2
Remember…
This tutorial and all the ones presented
   previously can be found on Mrs. Nelson’s
   website.
There is a link to her site on the Spring High
   website.
If you missed Objectives 1 – 4 part 1, you
   definitely want to go to the website to go
   through the lesson.
Today, we have a great deal to
cover! Topics include:
   Simplifying expressions
   Setting up & solving linear equations
   Setting up & solving linear systems
   Setting up & solving linear inequalities
   Writing linear equations from information
   Slope and intercepts
Let’s start with expressions.
This problem was on the 2006 test.
                                  This expression
                                  certainly can be
                                  simplified the
                                  ―traditional‖ way:


                                    - 21x  12x  6x - 13  24x
                                          2                       2
                   Distribute :
                                              3x  18 x  13
                                                2
                   Combine like terms :
The only problem with the
“traditional” way is that some of
you do not correctly distribute the
negative/subtraction signs!
                         So let’s look at an
                         alternate way to do
                         this problem with a
                         calculator.
                         It takes longer, but if
                         you are someone
                         who makes simple
                         mistakes…this is
                         for you!
The only variable in the expression is
x. Store a value into your calculator
for x—any number except 1, 0, or -1.
Select and press your number, press STO,
 press X,T,Θ,n , and ENTER

                           Now, enter your
                           expression, just as you
                           see it, into your
                           calculator…
                           -3x(7x-4)+6x-(13-24x^2)
                           Be sure to double check
                           that you typed the
                           expression correctly!
Now, press 2ND MATH to get TEST
                                     We are looking for equal
                                     expressions so select =

                                         You should see the = sign
                                      Now, press ENTER. If this
                                         appear after your
                                      answer choice is correct and
                                          two expressions are equal,
                                      theexpression
                                      you will see a ―1‖ as an
                                         (NO, do not press
Now, oneanswer! We is the the answer. If these two
    Looks like option F
    correct
             by one, enter were          ENTER—the calculator
firstlucky the first time. just as it expressions are not equal in
      answer choice You                  will not simplify this
appears. Double check to
      might check the other           value, you will see ―0‖ as an
                                         expression for you!)
     options to see that you
make sureget “0”.
              you typed it in         answer.
correctly.
That was a lot to grasp in one
problem. Let’s try another one and
practice with the calculator. (2003)
                         Type in the (2 option
                         Get TEST first answer
                         Type variable used is
                        Try again withnd Math)B.
                          The in the expression
                         exactly Remember,
                         choice. as it appears.
                        You do NOTahave to ―1‖
                          ―n‖. Select
                         and select = number
                         Double for ―n‖ and is
                         if it useeverything! Press
                          to is check
                        retypeequal; ―0‖ if itstore
                         not.
                          nd ENTER
                         correctness!
                        2it in the calculator.
That was a lot to grasp in one
problem. Let’s try another one and
practice with the calculator. (2003)
                       That againthe answertoB.
                              like
                       Lookswasn’t the correct go
                        Use the backspace
                        Try         with option
                                 option D. Don’t nd
                       must beequal sign. Either
                        to the NOT have to
                       answer choice, either. 2
                        You do Check
                       assume!the first it out!
                        delete everything! the
                                         answer
                       Enter, backspace toPress
                        retype
                       =, nd ENTER option it.
                        2 and type in
                        choice or type over C.
We will now move on to solving equations…
This problem was on the 2006 test




                                   We are looking
                                   for C when
                                   they have
                                   given us F.
We will now move on to solving equations…
This problem was on the 2006 test

                                       We can solve
                                       this equation
                                       the ―traditional‖
                                       way—using the
                                       ―undo‖ process.


               9
        104  C  32   Subtract 32
               5
               9
         72  C        Multiply by 5
               5
        360  9C       Divide by 9
          40  C
Alternate method

                       We can solve
                       this equation by
                       using the table
                       feature of the
                       graphing
                       calculator.
                   Enter the equation.
                   Go to the table.
                   Scroll down the
                   table until you find
                   104 in the y-column
Alternate method
                      You we could use
                        Or need to be
                        the to see in the
                      able graph and
                      window features of
                        CALC where the
                      two lines
                        the graphing
                      intersect. That
                        calculator
                      place looks way
                    Enter the equation
                      off to the right.
                    in y1 and 104 in y2.
                   Adjust the window
                     Adjust the try the
                   again. Let’s window
                     — at 50.
                   xmaxYou need ymax to
                     be higher than 104
                     Graph
Alternate method
                    Press 2nd TRACE
                    so that you get
                    CALC. Now,
                    select Intersect.

                   Move the cursor to
                   be close to the point
                   of intersection.
                   Enter again for the
                   second curve? And
                   guess?
This problem was NOT multiple
choice. You have to bubble in
your answer correctly!
Be careful!!! After
  going through all      4   0
  that work to get the
  correct answer, you
  don’t want the
  problem to be
  scored as wrong
  because you didn’t
  bubble in the
  answer properly!
This problem was on the 2003 test.
Let’s approach this one differently,
since it is multiple choice.



           We could transform the equation so we can
               use the calculator, but too many people
                    Option F is not it.
               mess that up—especially with the
                  Neither is option G
               subtraction sign!
                  Option H is the one!
          Since we were given answer choices for y, we
          will use the calculator to substitute and
          simplify until we find the y-value that gives us
          18 for an answer. We already know x is 3.
Here’s an inequality problem from the 2003
test where they ask you to graph, but they
don’t look at the graph! We’ll solve this
problem the same as the last one.

In other words, don’t be fooled by how
  complicated a problem looks!
Study the problem. See what you know that
  can be used to make the problem easy to do!
Let me again mention—the state
graders are not going to look in
the test booklets for your graph!
We are NOT going to use the grid
to do this problem!
                      We are going to substitute these
                      coordinates into the expression.
                      We want an answer that is less
                      than 12!


Option D gives
us 12. We want
an answer that     A is more than 12
is less than 12.   And so is B

Sure enough, C      Looks like C is the answer,
is the answer.      but let’s check to be sure.
TAKS will not always give you an
expression, equation or inequality.
You may just have to read a word problem and
 come up with one of your own.

The next several problems fit in this category.
2006


       First, you want to find the profit on ONE
       tool set.
       If it sells for $19.95 and is made for
       $4.37, the profit is the difference in the
       two prices.
       19.95 – 4.37
       Eliminate B & C
       Now, the profit is made on every tool set
       sold, which is s. We would multiply s to the
       profit (19.95 – 4.37) on each tool set.
2006



                                  Let’s reason. A total of
                                  80 backpacks were
                                  sold. That means, if 1
                                  $35 backpack was sold,
                                  79 of the $50 backpacks
                                  were sold.
                                  If 2 $35 were sold, then
 In each case, I subtracted the   78 $50 were sold.
 number of $35 backpacks, x,      If 10 $35 were sold,
 from the total of 80.            then 70 $50 were sold.
 That is 80 – x .
2006



               Since x is the number of
               $35 backpacks, I would
               need to multiply 35 and
               x together to get the
               amount of money made
               by those purchases.

       Since (80 – x) is the number of
       $50 backpacks sold, I would
       need to multiply 50 and (80 – x).
2006
       ―Total‖ implies
       addition. Adding b
       and c should be 220.
       b + c = 220
2006
       As for the
       money…Every
       brownie sold for $.75
       so $.75 needs to be
       multiplied to b, the
       number of brownies.

                  Please
                  note that
                  $.50, the
                  cost of
                  each
                  cookie is
                  multiplied
                  to c.
In some cases, you will be expected
to use the information given in a
problem situation to set up the
procedure and to solve for the
requested information.

How you do the problem is up to you and will
 not be checked. That you come up with the
 correct answer is the important piece this
 time.
2006        Perimeter is the sum of the lengths of
            the 3 sides: 5x + 10


                                             Anytime a
                                             figure is used,
                                             DRAW &
                                             LABEL it in the
                   2x + 5       2x + 5       test booklet!
                                            We have no
                                            idea how long
                                            the base is so
5x + 10 = 95
                            x               that is ―x‖.
  5x    = 85
                                            Each leg is 5
       x = 17                               more than
                                            double the
                                            base: 2x + 5
That is not the only way you can do
this problem. You can just use the
answers and figure it out.



            Let’s look at option A. If the base is 17,
            then based on what the problem says, the
            legs are each 2(17) + 5 = 39
            The perimeter means to add the two
            legs and the base:
            17 + 39 + 39 = 95

             95 is the perimeter we want, so A is
             the answer.
2006   This part means $.11 times ―x‖




                                 ―x‖ is the number
                                 of minutes Lisa
                                 can talk. Notice: all
                                 graphs have a
                                 start at zero. That
                                 is because Lisa
                                 cannot talk for less
                                 than 0 minutes,
                                 which is no talking
                                 at all.
                              The connection fee is
                              charged, no matter
                              what.
2006   This part means $.11 times ―x‖




                                 ―Not more than‖
                                 means we have
                                 an inequality—this
                                 value is the upper
                                 limit. Lisa wants to
                                 spend $5 or less.
                               .50 + .11x ≤ 5
                                    .11x ≤ 4.50
                                        x ≤ 40.9090…
Again, on this problem we could use
the answer choices to work toward
the solution.

                       All of the graphs have 0
                       as a lower limit of
                       minutes. That makes
                       the 0 a ―non-issue‖.
                        We need to look at the
                        upper limit of the graphs
                        and see which one
                        keeps us at $5 or less.
                       Let’s start with option D
                       since it has the largest
                       upper limit.
Again, on this problem we could use
the answer choices to work toward
the solution.

                       If minutes is 50, then
                       .50 + .11(50) = 6. $6 is
                       over Lisa’s limit.

                       If minutes is 45, then
                       .50 + .11(45) = 5.45 This
X                      value is also over Lisa’s
                       limit

X                    If minutes is 40, then
                     .50 + .11(40) = 4.9 $4.90 is
                     under Lisa’s $5 limit, so
                     option B is the correct one.
2003                                   Anna must get the
                                       same amount of
                                       money, m, for
                                       selling the plates
                                       as it takes to
                                       make the plates
                                       before she can
                    ―x‖ represents the number of plates
                                       make any profit.
                    Anna must sell

                    Her costs: m = 750 + 10x
                    Her revenue: m = 25x
As we discussed earlier, you can solve by graphing the
two equations and finding the intersection. Or you can use
the table and find where the two y values are the same.
Or you can use substitution and solve algebraically.
2003




       750 + 10x = 25x
       750    = 15x
       50 = x
2003




       And, of course, you always have the option of
       working with the answer choices to see which
       one gives you equal values for her costs and
       how much she makes selling the plates.




          20 plates is a NO
          50 plates looks NO
          30 plates is also like
          a winner.
Now, it is time to move on to
the mechanics of lines.
We need to talk about
 Slope
 Y-intercepts
 X-intercepts
 Equations of lines through points


If we do not get everything covered before the
   end of the session, please visit Mrs. Nelson’s
   website to complete the power point lesson!
About slope…
                    The first thing you
                    need to note is
                    that the slope is

               X    negative. That
                    means the line
                    should be going
                    downward.
                    Eliminate the
                    ones going up.
                   Now, look for the

   X               point (4, -3) on the
                   remaining 2 graphs.
                   The point must be
                   ON the line.
Rate of change is slope


                 Graph the line and see what it
                 looks like.




          The line is horizontal. It is ―running‖ but it
          is NOT ―rising‖. It has 0 rate of change
          because the y-coordinates of every point
          on the line are exactly the same.
You want to remember:
   Horizontal lines have 0 slope
   Vertical lines have undefined slope
   Diagonal lines have some fraction for a slope
   If the diagonal line goes up as it goes right, the
    slope is positive
   If the diagonal line goes down as it goes right, the
    slope is negative
   The steeper the line (the closer it is to the y-axis),
    the bigger the slope
   The flatter the line (the closer it is to the x-axis), the
    closer the slope is to zero (which would be
    horizontal).
 2006

                              You are expected to know that
                              the x-intercept is where the
                              graph ―touches‖ the x-axis.
                              You are also expected to know
                              that the y-coordinate is 0 for an
                              x-intercept.
Option 1: put 0 in   2/3(0) = -6x + 12
for y and find x.
                        0 = -6x + 12
                        6x = 12
                         x=2
 2006

                                 You are expected to know that
                                 the x-intercept is where the
                                 graph ―touches‖ the x-axis.

                            This equation is not in the correct
                            form for our calculator. So, divide by
                            2/3—Not you! Tell the calculator to
Option 2: graph the         do it. Just be sure to use
line and see where          parentheses!
it crosses the x-
axis.
                  y1= (-6x+12)/(2/3)
                    x = 2 at this point
2003

       Find the point
       (5, -1) on the grid.
       You need to use
       the equation
       x – 3y = 6 to find
       the y-intercept.

       There are a few
       ways to do that!

                    Remember, the state graders are
                    NOT looking at your graphs in
                    your booklet. This grid is for you!
2003

       The x-coordinate of the y-intercept is
       always zero. Substitute 0 in for x and find y.
       0 – 3y = 6       -3y = 6      y = -2
       You could transform the equation x – 3y = 6
       into slope-intercept form to find the y-intercept.
       -3y = 6 – x       y = -2 + 1/3 x
         You could put the equation into a form that
         the calculator will graph and find the y-
         intercept visually.
                                               x = -2
         -3y = 6 – x    y = (6 – x)/-3          here
2003

       Eliminate A and D

   X   since the b-value,
       the y-intercept, is
       NOT -2.
       Make a point
       where the y-
       intercept is -2.

       Now, count the
       slope from (0,-2)      up 1    right 5

   X   to (5,-1).            The slope is 1/5.
  2006




  Use your formula chart
  and draw the line
  between R and S.
You can see that the y-
intercept cannot be 4.5, so
eliminate those two choices.
                                up 12  3
 Now, count the slope                   1.5
 between R and S.              right 8 2
2004                There are
                    numerous ways to
                    do this problem.

        You could type each answer choice
        in your calculator, go to the table,
        and look for BOTH points.




       BOTH points are here—this is it!
 2004                               Option 2



                       You are given graph paper at the
                       end of the math section. Make
                       yourself a set of axes and graph the
                       two points.           The y-intercept is
                                                   at 3.5
                                                  7/2 = 3.5




Draw the line between them.
Count the slope:
down 2  2  1
           
right 4   4   2
 2004                                       Option 3



                           You could use the slope formula on
                           the formula chart to calculate the
                           slope. Then, you could use slope-
                           intercept form or point-slope form,
                           which is also on the formula chart to
                           calculate the y-intercept.
                  y  mx  b
                      1
                  2     (3)  b use the slope and (3,2) as(x, y)
4-2     2    1        2
                4   3  2b get rid of the fraction by multiplyin g everything by 2
-1 - 3  4   2    7  2b         add 3 to both sides
                  7
                    b          divide both sides by 2
                  2
Now, let’s talk “linear systems” 2003




          X                   X         We – 3y needthe given
                                            Neither 0
                                         2x just = of the y by
                                            equations is in
                                        itself. Using parentheses
                                             -3y = 0 – 2x
                                            calculator the
                                        will allow us form. We
                                        opportunity – 2x)/-3 into
                                               y = (0 for the
                                            need to get them
                                            that form ―calculate‖
                                        calculator tofirst.
                                         put this function in y1
                                        instead of us.

      X                                    That does is too
                                          The slope in FNOT
                                           mean compared to
                                          steep as they have to
                                          this graph.
                                           be in slope-intercept
                                           form.
The line going through (0, 0) must go
upward.
Now, let’s make sure…



         X                 X    x + 2y = -7
                                    2y = -7 – x
                                    y = (-7 – x)/2
                                put this function in y2

      X
Sure enough, we have a match.
Now, let’s make sure…



      X               X         You were not asked,
                                but let’s find the point
                                of intersection for the
                                two lines—the solution
                                to the system.


    X
           Yes, this point is
                (-3, -2)
 2006




Count the slope, follow the
pattern, and extend each
line.

The point of intersection,
the solution, is (10,2).
Practice Problems

								
To top