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Patterns and Algebra
23. A juice machine charges $0.65 for a can of juice
and accepts only nickels, dimes, or quarters. The machine
requires exact change.
You have 4 nickels, 4 dimes, and 4 quarters. Make a
table and list the different ways you can use your coins
to make up exactly $0.65.
Show your work and clearly explain your answer.
Sample Item #23
Scoring Rubric
3 points
The response shows complete understanding of the
problem's essential mathematical concepts. The
table will be complete and correct, listing all
possible combinations of nickels, dimes, and quarters
that can be used to purchase a can of juice.
One possible representation is
|
Nickels
|
Dimes
|
Quarters
|
Total
|
|
0
|
4
|
1
|
$0.65
|
|
1
|
1
|
2
|
$0.65
|
|
2
|
3
|
1
|
$0.65
|
|
3
|
0
|
2
|
$0.65
|
|
4
|
2
|
1
|
$0.65
|
A clearly labeled table can serve as an explanation.
2 points
The response shows nearly complete understanding
of the problem's essential mathematical
concepts. The majority of the table will be
completed accurately, but one or two possible
combinations may be missing.
Or, the table will be complete but will have
mathematical errors and any explanation offered
will be unclear.
1 point
The response shows limited understanding of
the problem's essential mathematical concepts.
The table will contain less than three but at
least one correct combination and/or may have
major errors.
0 points
The response shows insufficient understanding
of the problem's essential mathematical concepts.
The table will be both incomplete and inaccurate,
containing major errors. There may be no explanation
of how and why decisions were made, or the explanation
offered will be illogical, unclear, or not relevant. |
24. Mr. Jones gave each of the students in his class a
one-ounce box of raisins. When the students opened the
boxes and counted the raisins, they found different amounts.
The tally sheet below shows their results.
|
Number of Raisins
|
Tally
|
Frequency
|
|
10
|
I
|
1
|
|
11
|
II
|
2
|
|
12
|
III
|
3
|
|
13
|
IIIII
|
5
|
|
14
|
III
|
3
|
|
15
|
II
|
2
|
Construct a bar graph to represent the students' findings
on the grid in your answer booklet. Be sure to label
your graph completely.
Scoring Rubric
3 points
The response shows complete understanding of the
problem's essential mathematical concepts by correctly
representing the data given in the form of a bar
graph. Axes will be labeled and the graph will
have a title. The response may contain minor errors.
2 points
The response shows nearly complete understanding
of the problem's essential mathematical concepts.
Minor errors may be evident in the title, labeling,
or representation of the data, but the student's
understanding of the concept will be evident.
1 point
The response shows limited understanding of
the problem's essential concepts. Major errors
may be evident in the title, label, or graphing
of the data.
0 points
The response shows insufficient understanding
of the problem's essential mathematical concepts.
The graph may be partially complete, incorrectly
done due to misreading of the data, or presented
in a format other than a bar graph. |
 
25. If one diameter divides a circle into 2 parts, and
two diameters divide a circle into 4 parts, as shown in
Figures A and B above, into how many parts will 4 diameters
divide a circle?
A. 8
B. 10
C. 12
D. 16
The correct answer is A.
26. If January 1 falls on a Thursday in a certain year,
what day of the week is January 13 of that same year?
A. Tuesday
B. Wednesday
C. Thursday
D. Friday
The correct answer is A.
27. Four people can sit around a square table as shown
in Figure A. If two square tables are placed together,
as shown in Figure B, six people can sit around the table.
If six of these tables are placed together in a single
row to make one long rectangular table, how many people
would be able to sit around it?
A. 13
B. 14
C. 19
D. 24
The correct answer is B.
28. Sally is making a necklace by stringing beads in
a certain pattern. A section of the unfinished necklace
is inside the box. Based on the pattern shown,
draw or describe in detail the section of the necklace
that is inside the box.
Show your work and clearly explain your answer.
Sample Item #28
Scoring Rubric
3 points
The response accurately states the pattern of
beads on the portion of the necklace inside the
box with a clear, detailed explanation of how
this determination was made. The box should contain
4 colored beads, 1 white bead, 5 colored beads,
1 white bead and 4 colored beads, in that order.
White and colored beads alternate. Beginning with
1 of each bead, the number of colored beads increases
by 1 for each repetition. i.e. 1 white, 1 colored,
1 white, 2 colored, 1 white, 3 colored, 1 white,
4 colored, 1 white, 5 colored, and so on. By examining
the pattern it can be determined that 4 colored
beads, 1 white bead, 5 colored beads, 1 white
bead, and 4 of the next 6 colored beads are in
the box.
2 points
The response shows nearly complete understanding
of the problem's essential mathematical concepts
by correctly listing the pattern of beads inside
the box. Any errors are minor. The explanation
may be somewhat unclear, causing the reader
to make inferences, but will be essentially
accurate.
1 point
The response show limited understanding of the
problem by recognizing that a pattern exists,
but incorrectly identifying the
pattern. There will be an incomplete explanation
that will cause the reader to make inferences.
0 point
The response shows insufficient understanding
of the problem's essential mathematical concepts.
The procedures, if any, will contain major errors.
An explanation, if offered, will be unclear,
illogical, or not relevant. |
29. You have to choose between two different job offers
for the same four-day period. The first job pays $5.00
each day for the 4 days. The second job pays $1.00 for
the first day, and each day after the first, you will
be paid twice the amount you received the previous day.
. Which job will pay you the greatest amount?
. Would your answer change if you were working for more
than 4 days?
Show your work and explain your answer.
Scoring Rubric
3 points
The response shows complete understanding of the
problem's essential mathematical concepts. The
student's response will clearly detail the total
amount the student would earn in both jobs, and
state that Job 1 will pay more money for the 4-day
period, or give some other reasonable method to
determine which job pays more. However, the response
will also state that Job 2 would be the better
choice if the number of days worked was greater
than 4. The response may contain minor mathematical
errors.
2 points
The response shows nearly complete understanding
of the problem's essential mathematical concepts
by correctly computing one of the two situations
presented, and providing an accurate and detailed
explanation of which job would be the better
choice under the circumstances.
1 point
The response shows limited understanding of
the problem's essential mathematical concepts.
Computations of the total in both situations
will be incorrect, but the conclusion will be
accurate.
0 points
The response shows insufficient understanding
of the problem's essential mathematical concepts.
The procedures, if any, contain major errors
or are incomplete. No conclusion or explanation
will be offered or the explanation may be unclear,
illogical, or irrelevant. |
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