Lessons Learned

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					                  Avoiding Common Errors
                                          on the
             10th Grade Mathematics WASL




                        Lake Chelan School District
                                     January 2008




This document is based upon Lessons Learned From Scoring Student Work, a compilation by
the Mathematics Initiative Team of common misunderstandings in student work that were
observed by Washington educators while range finding and scoring the WASL.

Problems from this document are intended for independent study by students, for guided
practice in classrooms, and as reference for mathematics teachers.




                                        Page 1                              9/14/2012
Common Error #1
Short answer and extended response items require students to show evidence of procedures,
and/or strategies. Students lose points by:
    Failing to show/describe all mathematical decisions.
    Not writing down the expressions and operations they enter in the calculator.


The sports reporter for the local newspaper was interviewing Ivan “Ice” McMillan, the captain of
the local college basketball team, about the current season.
"I see you're leading the league again this year in free-throw percentage, Ice," said the reporter.
"Sixty-eight out of eighty attempts made is not bad at all. But it's still not the 90% or better that
you've maintained in past seasons. To make 90% with only four games left, you’ll pretty much have
to make the rest of your free throws. Do you think you can do it?”
“With a little luck, I’ll do it. But, it’s not about the stats…it’s about the wins," replied Ice.
Find the minimum number of consecutive free throws made that Ice would need in order to raise
his season percentage to at least 90%.
Explain strategies used and show all of your calculations.




                                               Page 2                                      9/14/2012
Common Error #1 Student Work Samples

Though this student has written the correct solution, they did not write down any expressions or
operations, and did not show or describe any strategies or decisions made.




This student showed evidence of procedures and showed calculations.




                                           Page 3                                  9/14/2012
Common Error #2
Students sometimes provide too much information.
       When asked to “Show work using words, numbers, and/or pictures”, students do not
          have to use all of the ways.
       Students often write a narrative of “how” they solved a problem when they have already
          shown all of their work. This narrative adds no additional information and may include a
          contradiction in the explanation.
       If there is more than one answer given by the student, scorers will not choose.


Cynthia has 400 cans of pears which need to be stacked in a supermarket display. She has seen
displays in other stores with cans stacked in the shape of a square pyramid, and would like to
create this same figure with her pears. Cynthia begins her task by making the bottom layer of the
pyramid a ten-can by ten-can square, and the second layer a nine-can by nine-can square.

Will she have enough cans to finish the pyramid so that the top layer has only one can?

If she does not have enough cans, how many more cans does she need? If she does have enough
cans, how many cans will she have left over?

Show your work using words, numbers and/or pictures.




                                           Page 4                                  9/14/2012
Common Error #2 Student Work Samples

This student gave the correct solution, but also discussed what to do with the leftover cans. There
are no incorrect details given, but adding unnecessary information risks including incorrect
information.




This student limited their response to the prompt that was given.




                                            Page 5                                 9/14/2012
Common Error #3
Missing and/or incorrect labels are a common reason students lose points. Here are some
examples of ways points are lost:
    Students write $1.80 (one dollar and eighty cents) in the following ways:
      180      1.80     $1.80¢      1.80$      $1.8
    When students are given inches in a prompt, they label their answer in feet.
    When reading data from a graph, students often ignore the unit provided in the label of an
      axis or provide an incorrect label in their answer.
    Students mislabel or leave the labels off linear, square, and/or cubic measures.
    Instead of 35 ft², students mistakenly write 35² ft.
    Using the symbol for inch (") and the symbol for foot (') with a raised 2 to mean squared is
      incorrect. Example: 25'² and 25²' are both incorrect.
    Mislabeled time units include: forgetting the morning or afternoon label, forgetting the colon
      (:), and/or using a decimal point instead of a colon.


Veronica is committed to flossing her teeth once per day, and uses about 18 inches of dental floss
every time she cleans her teeth. Veronica has no hobbies. She’s curious how much floss she’ll use
in the next year, and how much it will cost her. Veronica has listed varieties of dental floss below,
and their current prices. She will most likely purchase the least expensive floss.
       A.   Jessep’s unwaxed, 20 yards for $1.19
       B.   Pearly Whites waxed, mint or cinnamon, 50 feet for $1.09
       C.   Dental String waxed, 30 yards for $1.49
       D.   Mouth Care waxed, mint, 100 feet for $0.99
       E.   Pearly Whites unwaxed, 100 feet for $ 1.39
       F.   Generic waxed, 200 feet for $1.79
       G.   Nature Spring natural flossing ribbon, 30 feet for $0.59
Determine which brand of floss is the best deal, and how much Veronica will spend on dental floss
in the next year. Show all of your work.




                                             Page 6                                   9/14/2012
Common Error #3 Student Work Samples

This student incorrectly abbreviated “yards” and often placed the dollar sign ($) after the number.




This student correctly labeled their units.




                                              Page 7                                9/14/2012
Common Error #4
Students do not use inequality symbols correctly.
    Example of incorrect use: 2 < 5 > 1
    Example of correct use: 2 < 5 and 5 > 1, or 1 < 2 < 5


Rafael is using toothpicks to build a triangle. One side of his triangle is 5 toothpicks long, and a
second side is 8 toothpicks long.

What are the shortest and longest potential perimeters of Rafael’s triangle? Write the possible
perimeters P as an inequality.

Draw and label diagrams which show the triangle with the smallest perimeter and the triangle with
the largest perimeter.




                                             Page 8                                    9/14/2012
Common Error #4 Student Work Samples

This student incorrectly used inequality symbols (see last line of explanation).




This student correctly used the inequality symbols.




                                             Page 9                                9/14/2012
Common Error #5
Students have difficulties labeling fractional parts:

   Examples of mislabels:




   The diagram below shows a square piece of land that has been divided into 4 shaded parcels,
   labeled A, B, C and D. Each boundary segment drawn within the square connects midpoints of
   two other segments. What fractional portion of the large square is each shaded region? Write
   your fraction in lowest terms.


   Region A = ______ Region B = ______                        Region C = ______ Region D = ______
    all of the black             all of the white                 all of the dark grey   all of the light grey




                                         B                           C

                                 A
                                                                      D




                                                    Page 10                                          9/14/2012
Common Error #6
Students are required to estimate when given directions to estimate. Answers determined by exact
calculations that are subsequently rounded earn no credit.



While shopping for party supplies, Cindy must stay within a budget of $110 including tax.
She has selected the supplies she hopes to buy, but she did not bring a calculator with her to the
store, and she doesn’t have time to work it out by hand.

Using the list of items below, use estimation to determine whether or not she is within her budget.
Clearly explain your process for estimating. Use 8% sales tax.

Plates                   $5.89
Cups                     $4.15
Napkins                  $1.80
Silverware               $7.75
Cake                    $29.95
Soda                 5 bottles at $1.99 ea
Decorations             $22.10
Vegetable Platter       $18.78
Ice Cream           2 cartons at $2.99 ea




                                             Page 11                                 9/14/2012
Common Error #6 Student Work Samples

This student did not round until the end of the process. Rounding at the end of a problem is not
estimation.




This student correctly applied the process of estimation for the sub-total of items.
However, the student also should have estimated the tax.




                                            Page 12                                    9/14/2012
Common Error #7
Students have trouble graphing inequalities on a number line.

   Examples of correctly graphed inequalities:


               2<p




               2≤p




Which of the following inequalities has its solutions graphed on the number line below?




       A.   – 2x > -8
       B.   – 2x < -8
       C.   – 2x > -8
       D.   – 2x < -8




Joan works no more than three hours per day at her parents’ fruit stand. Write an inequality which
describes this situation, and graph it on the number line below.


      Inequality: ________________




                                          Page 13                                 9/14/2012
Common Error #8a
Students do not understand the meaning of the “remainder” in division problems.

   Two-hundred colored pencils are to be evenly distributed among 14 students. Once every
   student has the same number of pencils, what percentage of the pencils will be left over?

        A.   16⅔%
        B.   4%
        C.   2%
        D.   0.28%

Common Error #8b
Students do not know how to represent a remainder in decimal or fraction form.
       Incorrect:             Correct:


                                                  Students write 12.3
                                                  instead of 12 ¾ on the
                                                  answer line.




A class full of students has been told that the height of a shelf is 75”. After being asked to convert
75” into a different unit of measurement, Michelle believes an acceptable answer to be 6’25”.
Javier completes the conversion and obtains a measure of 6’3”. Maurice asserts that the height is
6¼ feet. Veronica writes down 2.3 yards. Who is correct? If any students made errors in their
conversions, describe the error and explain how to fix it.




                                             Page 14                                   9/14/2012
Common Error #8 Student Work Samples

This student incorrectly converted 6.25 feet into 6’25”. In addition, the student implied that
6’25” can be rounded to 6’3”.




This student correctly converted remainders using fractions, decimals and percents.




                                            Page 15                                   9/14/2012
Common Error #9
Students do not show understanding of the differences among perimeter, area, and/or volume.


The diagrams below show the front, top, and side views of an object, as well as an isometric view of
the object. This object is made of five cubes: four in the back row stacked in the shape of a square,
and one in the front. The volume of the figure is 5 un3, the surface area is 20 un2, and the perimeter
of its base is 8 units.




Below are the side, top, and front views of a different object.




Determine the volume, surface area, and perimeter of the base of this object.




        Volume = _______ Surface Area = ________ Perimeter of Base = ________


                                             Page 16                                 9/14/2012
Common Error #10
Students use 100 minutes for one hour instead of 60 minutes for one hour.



As a school project, you ask classmates to keep track of the number of hours they slept the night
before, and then turn in their results. Unfortunately, you were not very specific, and they reported
their times in many different formats. Here is the data:

7 hours and 40 minutes,
8.5 hours,
  1
6 hours,
  4
7.8 hours,
8:06:00,
5 h 20 m,
7.75 hr,
  5
8 hours
  6

   Convert each of these times into a single format of your choice, and then report the mean of the
   data in that same format.




                                Mean = _________________




                                            Page 17                                 9/14/2012
Common Error #10 Student Work Samples

This student incorrectly converted 7 hours and 40 minutes into 7.4, 8:06:00 into 8.06,
5 h 20 min into 5.2, and 8 5/6 into 8.5.




This student correctly converted between hours and minutes.




                                           Page 18                                  9/14/2012
Common Error #11
Students need a better understanding of converting within customary and metric systems.

       Examples:      2 ½ feet = 30 inches
                     1500 m =1.5 km




The doors of an elevator close at a rate of 1¾ feet per second. Which of the following is not an
equivalent speed?

        A. 21 inches per second
        B. 7 yards per second
              12
        C. 75 feet per minute
        D. 35 yards per minute




Tiko is responsible for organizing a 5K run, meaning the race will be a distance of five kilometers.
He would like to have three small water stations evenly distributed between the starting line and
finish line of the race, and one large water station at the finish line. How far apart will the water
stations be (in meters)?

        A.   1250 meters
        B.   1000 meters
        C.   12,500 meters
        D.   10,000 meters




                                             Page 19                                  9/14/2012
Common Error #12
Students do not understand how to convert between square units (for example, they mistakenly
divide square inches by 12 instead of 144 to get square feet).


The high school gym has dimensions 114 feet by 120 feet. A company has agreed to refinish the
surface of the gym floor at a cost of $20 per square yard for the first 1000 square yards, and $15 per
square yard for each additional square yard. Give the cost of resurfacing the floor. Show your
work!




                             Total Cost = ______________




                                            Page 20                                  9/14/2012
Common Error #12 Student Work Samples


This student divided by three to convert square feet into square yards. The student should have
divided by nine (one yd2 = 9 ft2).




This student correctly converted linear and square units.




                                           Page 21                                9/14/2012
Common Error #13
Students are having trouble determining whether to use the diameter or radius in a formula, and
students do not distinguish between radius and diameter at all.




A horse corral is to be built in the shape of a circle using metal fencing. Jorge would like the
distance across the corral to be 60 feet (C = 2r and A = r2).


About how much fencing will be needed to enclose the corral?

        A.   120 feet
        B.   189 feet
        C.   377 feet
        D.   2830 feet


Jorge needs to cover the corral with a fabric to prevent weeds from growing. One roll of weed-
block at the local hardware store covers 200 square feet. How many rolls of weed block will he
need to purchase?

        A.   1 roll
        B.   2 rolls
        C.   15 rolls
        D.   57 rolls




                                             Page 22                                   9/14/2012
Common Error #14
Students need to understand the difference between expressions and equations and a correct way
to represent expressions and equations using variables.

         Examples of expressions:
         4  3,   t  2,   20t   t  tickets
         Examples of equations:
         4  3  12,   c  20t   c  cost, t  tickets



In the year 2008, Hugh posed the following “Magician’s Trick” to one of his friends:

1.   Pick the number of days per week that you would like to eat out at a restaurant (1-7).
2.   Multiply this number by 2.
3.   Add 5.
4.   Multiply by 50.
5.   If you have already had your birthday this year add 1758. If you haven't, add 1757.
6.   Now subtract the four-digit year that you were born.

You should now have a three-digit number.
The first digit was your original number (days eating at a restaurant this week).
The second two digits are your current age (assuming the current year is 2008).

Assign a variable to the number you chose in Step 1, write an algebraic expression,
and use the expression to explain how the trick works.

Also, this particular sequence only works for the year 2008. How would you alter the
sequence to make it work for 2009?




                                                         Page 23                       9/14/2012
Common Error #14 Student Work Samples

This student wrote an equation rather than an expression, and is missing a pair of parenthesis in
the equation: [(x2) + 5]50.




This student correctly wrote an expression. However, notice that the student wrote a “run-on
equation” on the top line while exploring the trick (72  14+5  1950, etc.).




                                           Page 24                                 9/14/2012
Common Error #15
Students do not understand how to define variables. They give the value of the variable instead of
what the variable means. In addition, students need practice writing expressions and equations
that represent a situation. They can solve a problem, but do not write an equation or expression
that represents what they have done.


One hot summer day, Zach and Jorge were sitting on their front porch. Zach noticed a rather large
flock of crows had settled on nearby telephone lines. He was impressed by how many birds were
in the flock, so he decided to count them. As he finished counting, a truck passed by and scared
away some of the crows. In fact, exactly 10% of the birds flew away. A few moments later some of
the crows returned to the lines. Jorge observed, "Now the number of crows that have returned is
10% of the number that did not fly away when the truck passed."

The next day, Zach again noticed crows sitting on the power lines. "This is quite a coincidence.
Now there are 10% more sitting there than started out yesterday," Zach commented. At that very
moment, a loud clap of thunder was heard, scaring away some of the crows. Jorge said, "It's
amazing, really amazing. Just 10% of those crows were scared enough to leave."

On which day were more birds left sitting on the lines, day one or day two? How do you know?
Use algebraic expressions or equations to demonstrate your choice, not guess and check. Make
sure to define your variables.




                                          Page 25                                 9/14/2012
Common Error #15 Student Work Samples

This student did not use any variables.




This student defined variables, though they should have defined x more specifically as the
beginning number of birds. This student correctly wrote and simplified algebraic expressions.




                                          Page 26                                 9/14/2012
Common Error #15, continued…
Students do not understand how to define variables. They give the value of the variable instead of
what the variable means. In addition, students need practice writing expressions and equations
that represent a situation. They can solve a problem, but do not write an equation or expression
that represents what they have done.




The diagram above illustrates the Distributive Property: a(a + b) = a2 + ab.




What does this diagram illustrate? Make sure you give "both sides" of the equation, as
shown in the illustration above.




                                           Page 27                                 9/14/2012
Common Error #15 continued Student Work Samples

This student did not correctly write algebraic equations.




This student correctly converted the geometric diagram into an algebraic equation.




                                            Page 28                                  9/14/2012
Common Error #16
Students need practice writing equations with two variables. As always, students should define
any variables they use.



Chelan’s Hungry Goat Pizza Parlor serves students 6” mini-pizzas at lunch.

Two pizzas and two large drinks total $13 before tax. Define variables and write an equation for
this situation.




Four pizzas and three large drinks total $24 before tax. Using the variables you defined above,
write a second equation for this situation.




What is the cost of one mini-pizza and one large drink? Show all work.




                                           Page 29                                 9/14/2012
Common Error #16 Student Work Samples




This student correctly wrote equations and solved a system, but did not define the variables.




                                           Page 30                                 9/14/2012
Common Error #16 Student Work Samples, continued




This student correctly defined variables and wrote equations with two variables.




                                           Page 31                                 9/14/2012
Common Error #17
Students need more practice plotting numbers on a number line and plotting points on coordinate
grids. Students are not placing or marking points on number lines (see drawing below) and
students often reverse ordered pairs.
                                            No point marked on
                                            number line to indicate
                                            location.




                             0


Complete the following steps.
1. Plot the following four points:       A(6, 3)           B(2, -5)   C(-4, -4)   D(-8, 2)
2. Connect the points in alphabetical order: A to B, B to C, C to D, and join D to A. This
forms quadrilateral ABCD.

3. Find the midpoint of each segment of this quadrilateral.

4. Label the midpoints in this manner:     AB has midpoint Q, BC has midpoint is R, CD has
midpoint S, and DA has midpoint T.

5. Connect the four midpoints in alphabetical order forming quadrilateral QRST.


Quadrilateral ABCD was an ordinary quadrilateral, but QRST is a special one. What is
QRST? Show how you know that QRST is this special quadrilateral.




                                           Page 32                                  9/14/2012
Common Error #17 Student Work Samples

This student did not correctly graph the ordered pairs.




This student correctly plotted the given ordered pairs and found the midpoints of the segments.




                                            Page 33                                9/14/2012
Common Error #18
Students do not correctly draw polygons, both regular and irregular.



A homework problem in Tao’s geometry book asks him to draw an isosceles triangle in which at
least one of the angles is 40. One of Tao’s friends claims that there are two answers to this
question, because there are two different triangles which can be drawn that meet the criteria. Draw
both triangles and label the angle measures.




                                           Page 34                                 9/14/2012
Common Error #18 Student Work Samples


The student correctly drew the first triangle, but did not make the second triangle isosceles.




Both triangles are correctly drawn, marked and labeled.




                                            Page 35                                  9/14/2012
Common Error #19
Students do not draw a reflected figure the same distance from the line of reflection as the original
figure, as well as preserve the shape and size of the original figure.



Draw a horizontal or vertical line of your choosing on the coordinate plane below, and draw the
reflection of the triangle over your line.




Draw a non-vertical or non-horizontal line of your choosing in the space below, and draw the
reflection of the figure over your line.




                                            Page 36                                  9/14/2012
Common Error #19 Student Work Samples



Incorrect reflection.




Correct reflection.




                                  Page 37   9/14/2012
Common Error #20
Students need practice sorting figures using more than one attribute (example: four-sided figures
with exactly one line of symmetry). When sorting figures with specific attributes, students
mistakenly assume that there must be an equal number of figures for each attribute. Give students
practice situations where the sorting does not result in equal numbers of figures in each category.


Sort each of the geometric figures below into three groups based on the attributes of each figure.
Describe the criteria you are assigning each group. When deciding on your criteria, you may not
have a figure which fits into two different groups.




               Group I                            Group II                       Group III
   Criteria:                        Criteria:                        Criteria:




               Figures:                           Figures:                       Figures:




                                                Page 38                                9/14/2012
Common Error #20 Student Work Samples




This student did not create three mutually exclusive groups. All of the items in Group II may also
be put in Group III.




                                           Page 39                                  9/14/2012
 Common Error #20 Student Work Samples, continued




This student correctly sorted the figures into groups.




                                            Page 40      9/14/2012
Common Error #21
Students need to understand measures of central tendency: mean, median, and mode.


Wes and his six children are playing in a carnival “house of mirrors”. The weights of the seven
individuals (in pounds) are: 34, 45, 60, 51, 45, 210 and 64. Compute the mean, median and mode
of these weights.

A carnival manager asks, “What is the typical weight of someone who uses the house of mirrors?”
Decide which of your statistics (mean, median or mode) best answers the manager’s question.
Explain why you choice best describes the data.




                                          Page 41                                9/14/2012
Common Error #21 Student Work Samples

This student correctly computed the mean and median, but confused the mode with the range. In
addition, the student did not determine which measure best described the data.




This student correctly computed the mean, median and mode, and correctly choose a measure
(the median) which best represented the data.




                                         Page 42                               9/14/2012
Common Error #22
Students do not make lists of all possible outcomes.



Mick and Tasha are playing a board game. In order to win, Tasha needs to roll a sum of at least ten
using her pair of six-sided dice. Mick argues that this role is not likely, because the probability of
her getting a sum of ten is less than 25%. Tasha agrees that she is not likely to win, but claims that
the probability is higher than 25%, because there are eleven possible sums (2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12) and three of them are 10 or higher. Who is correct? Explain how you know.




                                             Page 43                                  9/14/2012
Common Error #22 Student Work Samples


This student did not list all possible outcomes. For example, for the sum of three, the student
should have listed {2,1} and {1,2}.




                                           Page 44                                  9/14/2012
Common Error #22 Student Work Samples, continued


This student listed all possible outcomes and correctly computed the probability.




                                           Page 45                                  9/14/2012
Common Error #23
Students need more experience using geometric probability.



A game at a local arcade features a ramp that a ball is rolled down. At the end of the ramp, the ball
is launched into the target shown below. If the ball lands in the largest circle, the player earns 100
points. However, if the ball lands in any of the smaller circles, the player earns 200, 300, 400 or
500 points. The large circle has a diameter of two feet. The small circles each have a diameter of
4”. What is the probability that a ball which randomly lands within the target will earn at least 400
points?




                                               500


                                               400


                                 100
                                               300


                                               200




                                            Page 46                                  9/14/2012
Common Error #23 Student Work Samples


This student did not use areas to compute probabilities.




This student correctly computed areas in order to find probabilities.




                                            Page 47                     9/14/2012
Common Error #24
Students have difficulty determining probabilities of dependent events.

A lottery game based on ping-pong balls drawn from a machine contains fifty balls numbered 1
through 50 and four colored balls (red, blue, yellow, green). When a lottery ticket is purchased, the
player must choose a color and a number. Each week, the machine chooses one colored ball and
one numbered ball. The ticket owner is a winner if their ticket matches both balls chosen by the
machine.

What is the probability that the machine will choose RED-48?

        A.   o.5%
        B.   1.85%
        C.   8.0%
        D.   27%


If a player purchases 10 tickets, what is their likelihood of winning?

             1
        A.
            10
            10
        B.
            54
              1
        C.
            200
             1
        D.
            20

If each ticket costs $5, and the winning payout is $500, do you think this is a fair game?
Explain why or why not.




                                             Page 48                                 9/14/2012
Common Error #24 Student Work Samples



This student did not use compound probabilities to determine the likelihood of winning, and
therefore could not accurately explain why the game is not a fair game.




This student correctly used compound probability to compute the likelihood of winning, and was
able to determine the fairness of the game.




                                           Page 49                                 9/14/2012
Common Error #25
When students are using the “guess and check” method, they need to identify this as their method
or show at least two different guesses and checks to earn strategy points. When students are
using “making a list” as a strategy, they should have at least 3 iterations to get credit for the
strategy.


Humberto has been given the puzzle at the right. He must
fill in each of the shapes with an integer so that the given              Total
totals are valid. From this diagram he was able to write the                6
following equations:

      +=7
      +=5                                                    Total                  Total
      +=6                                                      5                      7

Find the number that goes in each shape. Show the method
you used for finding each number.




                                           Page 50                                9/14/2012
Common Error #25 Student Work Samples

This student identified guess-and-check as their strategy, but only showed one guess.




This student identified guess-and-check as their strategy, and showed multiple guesses.




                                          Page 51                                 9/14/2012
Common Error #26
When asked to write questions, students have difficulty writing questions that can be answered
from the given information.
    Example: When students write, “The cost of a milkshake and a donut = ____”, they do not
   receive credit for writing a question. This is not a question, this is “fill-in-the-blank”. They should
   write, “What is the cost of a milkshake and donut?”
    An expression or an equation is not a question. Example: “x + y + 2. Find x and y.”




                       ●                      ●                              ●
                   ●       ●              ●       ●                      ●       ●
                                      ●       ●       ●              ●       ●       ●
                                                                 ●       ●       ●       ●


Write a question that may be asked of a geometry student who has been shown this pattern, and
give the answer to the question.




                                                  Page 52                                    9/14/2012
Common Error #26 Student Work Samples



This student did not ask a question, and gave a mystery answer.




This student wrote an appropriate question, and gave a correct answer.




                                          Page 53                        9/14/2012
Common Error #27
Students need practice drawing conclusions and giving quantitative support for their conclusions.
    Valid conclusions must be based on the data or describe the data.
    Support must use the specific data and/or information specific from the item.



               September       October      November      December        January         February
MedTech           $15           $18           $20           $24             $27             $29
GSA, Inc          $40           $32           $28           $27             $26             $26
FD Harris         $22           $27           $29           $28             $23             $16

The chart above shows the value of three corporate stocks over the past few months. Write two
mathematical conclusions that can be reached by analyzing this data.




Make two predictions based on trends shown in the table. Explain how your predictions are based
on the data.




                                           Page 54                                  9/14/2012
Common Error #27 Student Work Samples



This student did not make conclusions that could be drawn from the data given.




This student also made predictions that do not rely on the given data.




                                           Page 55                               9/14/2012
Common Error #27 Student Work Samples


This student used the given data to make specific quantitative conclusions, though these
conclusions did not require much analysis.




This student uses quantitative evidence to back up their predictions, though the patterns and
evidence cited may be faulty in nature.




                                           Page 56                                 9/14/2012
Common Error #27, continued…
When students are asked to write questions or make conclusions that can be obtained from given
data, they should use the information to make calculations, rather than just restate the information
that was given.
                                                   Number of Burglaries in Smallville, WA (2004-2006)

                            16


                            14


                            12
     Number of burglaries




                            10
                                                                                                                                  2004
                            8                                                                                                     2005
                                                                                                                                  2006

                            6


                            4


                            2


                            0
                             Jan.   Feb.   March     April   May    June      July     Aug.   Sept.   Oct.   Nov.   Dec.
                                                                   Month of the year

The line graph above shows the number of burglaries that occurred in Smallville over a three year
period.
● Write two questions that may be asked of a student analyzing this graph.
● Use the data to write two new mathematical facts that can be obtained from the data in this chart.
  The facts may not be the answers to your two questions.
  Example: “December had the fewest number of burglaries - a total of nine.”




                                                                        Page 57                                            9/14/2012
Common Error #27 Student Work Samples



This student asked questions which required the reader to restate data on the graph, but did not
require any analysis on the part of the reader. The student’s first fact/conclusion did include some
analysis, as data from multiple years is accessed. The student’s second fact/conclusion simply
restated data on the graph.




This student’s questions required analysis (though basic) on the part of the reader, rather than
regurgitating data off of the graph. In addition, the facts cited came from analysis, rather than
simply restating data on the graph.




                                            Page 58                                  9/14/2012
Common Error #28
Students need practice describing a plan to gather mathematical information, including where and
how the information can be found. When a student is asked to plan a survey, the plan must
include:
     WHO will be surveyed,
     HOW the participants will be selected to ensure a representative sample, and
     WHAT questions will be asked to ensure they are specific, unbiased, and focus on the
    desired information.


In 2007, the City of Chelan changed a four lane highway (two lanes eastbound, two lanes
westbound and no turn lane) into a two lane highway (one lane eastbound, one lane westbound
and a center turn lane). The city leadership would like to find out what the residents think of this
change. As City Manager, you have been assigned the task of polling the residents.

Devise a plan for determining the popularity and effectiveness of the new road system. In your
plan, include questions to be asked, and a plan for deciding who will be asked the questions.




                                            Page 59                                  9/14/2012
Common Error #28 Student Work Samples



The questions asked by this student are non-specific, and the plan to gather information is brief
and vague.




This student gave a specific plan for who will be surveyed, and how the sample will span multiple
age groups (though we don’t know whether these numbers are representative of the target
population).




                                           Page 60                                  9/14/2012
Common Error #28 Student Work Samples, continued


This student asked specific questions, and gave a specific plan for collecting the data.




                                            Page 61                                  9/14/2012
Common Error #29
Students have trouble making comparison in mathematics.
    Students should discuss how things are alike and/or different.
    Students need to correctly use and apply more, most, always, never, and almost.
    Students should use factual attributes, not opinions.
          Example of incorrect use of a comparison word: “Don has more apples.”
          A better comparison: “Don has more apples than Juan.”
             “More” is a comparative word and needs how and what is being compared.
          Example of opinion: “These are pretty. These are not.”
         Example of insufficient comparison: “The first building has squares.”
         A better comparison: “These are squares. The other shapes are not squares because the
         side lengths are not equal.”

         Example of insufficient comparison: “The snakes are longer.”
         A better comparison: “The snakes are longer than the worms.”

Figure A and Figure B are equilateral polygons with sides of length 10 mm. Compare attributes of
these figures, considering such things as perimeter, area, and how the figures may be classified.




                               Figure A
                                                          Figure B




                                          Page 62                                 9/14/2012
Common Error #29 Student Work Samples

This student correctly stated that the areas of the figures are different, but did not specify which
figure had the larger area.




This student made several correct and specific comparisons.




                                             Page 63                                   9/14/2012
Common Error #30
Students need more practice interpreting and describing graphs.


Over the period of 1980 to 2005, surveys have been conducted asking respondents, "Which
attribute would be most important to you in your choice of your next vehicle?" The share of
respondents answering that fuel economy (miles per gallon) was the most important attribute is
shown in the bars on the graph below. The fuel price (adjusted to 2004 dollars) is represented by
the line on the graph.




Analyze the information in this graph and make two logical interpretations or conclusions based on
the combined details of this graph. Clearly explain how you arrived at each statement.




                                          Page 64                                 9/14/2012
Common Error #30 Student Work Samples

This student made multiple statements which demonstrate that the student did not understand how
to interpret and gain meaning from the graph.




This student demonstrated an understanding of the connection between gasoline price and the
desire for fuel economy. However, the student only made one logical interpretation. Students
should address the years 2004-2005, which are a break in the pattern of the graph.




                                         Page 65                                9/14/2012
General Tips:

 Students need to answer the question that is being asked!!


 Students are using words or a mixture of words and symbols when asked to write an
  equation using variables.      Incorrect: $ = (5)(hours)       Correct: C = 5h


 When students provide more than one answer, scorers will not choose which one is
  correct. When students write over an answer, they are not making their answers clear
  enough to score. Students need to cross out answers they do not want scored.


 Students should cross out work rather than erase. Students can earn points for work that is
  crossed out if it ends up being correct and supports their answer. When students erase
  work, they show no evidence of a strategy or procedure (no partial credit).


 Remind students to check for the reasonableness of their answers. They are submitting
  answers that defy reality (example: cost of burger = $550).


 Students need to use the variable given in the prompt when a defined variable is provided.


 Students need to use a ruler or straight edge when drawing figures.


 “Edges” are used with three-dimensional objects. “Sides” are used with two-dimensional
  figures.


 Students should write the number and its unit/label when extracting data from charts or
  graphs.


 Students are using the terms “equal” and “even” incorrectly to mean the same thing.
  Example of incorrect use: Paul mowed an even number of acres on both days.


 Students are using the word “it” without providing information to whom or what is being
  referred. Students should avoid the use of “it.” Students should clarify all pronouns.




                                        Page 66                                  9/14/2012
 When plotting points on a coordinate grid, students are showing the tracking lines that help
  them locate the points. These “extra” lines are confusing students when it is time to
  connect the points, and also make scoring the students’ figures difficult. In addition, the
  extra lines may cause students to lose points, as they force the scorer to choose intent.
                                5
                                4
                                3
                                2
                                1

                                0    1   2   3   4     5



 Students are using run-on equations to support work. Run-on equations give false
  information and do not earn points for supporting work.

   “A football team gained 10 yards, gained 17 yards, and then lost 3 yards.”

   Example of a run-on equation:
      10 + 17 = 27 – 3 = 24

   Part of the run-on equation gives false information:
      10 + 17 ≠ 27 – 3

   Remedy: students should use only one equal sign per line of work.




                                             Page 67                                9/14/2012

				
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