PERFORMANCE TRENDS IN ALGEBRAIC REASONING: 1996 to 2011
Expanded Form of the Paper Published in the Proceedings of PME-NA 2012
Peter Kloosterman
Indiana University
[email protected]
Michael Roach
Indiana University
[email protected]
Crystal Walcott
Indiana University Purdue University
Columbus
[email protected]
Doris Mohr
University of Southern Indiana
[email protected]
This study focused on the algebra knowledge of 4th and 8th grade students as measured by the
1996 through 2011 NAEP mathematics assessments. Over that period, 149 algebra-focused
items were used allowing analysis of performance on topics such as understanding and use of
algebraic expressions, understanding of numeric patterns, and ability to use equations and
inequalities to solve problems. The items described in this paper provide a good sense of how
well students perform on algebra-related tasks and overall results show gains in algebraic
thinking between 1990 and 2005 have plateaued. A key theme that ran across both grades 4 and
8 was that item wording and format can have a substantial impact on student performance.
Other themes include the fact that student performance improves even after the time that topics
are taught and that students struggle on items that require explaining an answer.
The National Assessment of Educational Progress (NAEP) has collected data on the
performance of elementary, middle, and high school students in the United States since the late
1960s. Of the content areas that NAEP has assessed, there has been more growth in mathematics
performance than any other subject area at the 4th and 8th grade levels (Kloosterman & Walcott,
2007). Growth in mathematics performance grade 12 has been modest – there was a small but
statistically significant gain between 1990 and 2000 (Braswell et al., 2001) and another between
2005 and 2009 (NAEP Data Explorer, 2012). Major changes in the grade 12 assessment
between the 2000 and 2005 administrations made growth over this period difficult to quantify.
In addition to reporting overall results, NAEP has reported results by content strand
(number and operation; geometry; measurement; algebra; and data analysis, probability, and
statistics) since 1990. Like the overall trends in mathematics performance, the trends in
performance in the algebra strand for students in grade 8 showed substantial improvement
between 1990 and 2011. In contrast, there was substantial improvement between 1990 and 2003
at grade 4 and while the 3-point gain in scale score between the 2003 and 2007 administrations
was statistically significant, it was small compared to gains in previous years. The grade 4
algebra score did not change at all between 2007 and 2011 (NAEP Data Explorer, 2012). Given
the importance of algebra and algebraic thinking (National Mathematics Advisory Panel, 2008),
this study analyzed grade 4 and grade 8 NAEP algebra data from 1996 through 2011 to discuss
overall trends in algebra and then focused on the 2003 through 2011 data to describe gains in
students’ algebraic reasoning in the areas of (a) patterns, relations, and functions, (b) algebraic
representations, (c) variables, expressions, and operations, and (d) equations.
Background
The National Center for Education Statistics (NCES) provides reports on overall findings
after each Main NAEP mathematics assessment (e.g., Braswell et al., 2001; National Center for
Education Statistics, 2011). In addition to overall results for the nation as a whole, by state, and
by demographic subgroup, the reports provide technical information such as sampling
procedures, item development and scoring procedures, and details of statistical analyses
performed on the data. A major advantage of NAEP as opposed to state-level or college
entrance (e.g., SAT) data is that NAEP data come from a representative national sample of
students and thus conclusions drawn from these data are valid for the United States as a whole.
Although any one student completes at most 20 NAEP items, different items are completed by
different individuals so when results are pooled across students, there is information on a wide
variety of mathematics skills. The 2003 Main NAEP assessment, for example, used 182 items at
grade 4 and 189 items at grade 8. Of these, 26 of the grade 4 and 48 of the grade 8 items were in
the algebra strand. After each administration, roughly one-fourth to one-third of the items are
replaced so that there are enough items to track trends over time while allowing for updates to
keep the assessment consistent with changes in curriculum. Most retired items are released to the
public and available online (see http://nces.ed.gov/nationsreportcard/itmrlsx/). One of the
reasons that items are retired is that they no longer represent what is being taught in schools and
thus released items are not necessarily representative of the NAEP assessment as a whole1.
Since the early years of NAEP, there have been interpretive reports based on the specific
items used for the mathematics assessment and some of these reports have focused specifically
on algebra. For example, Chazen et al. (2007) reported that gains in the algebra strand were
greater than gains in any other content strand for grades 4 and 8 from the mid-1990s through
2003 but that large gaps in performance based on race/ethnicity persisted. These researchers also
found that performance on items used at both grades 4 and 8 was always higher at grade 8
although the amount of difference between grades varied substantially by item. Looking at
NAEP performance of grade 8 students, Sowder, Wearne, Martin, and Strutchens (2004)
reported on 10 algebra items that were used in 1990, 1992, 1996, and 2000. Two of those items,
focused on patterns and performance increased significantly on both. Performance also
increased significantly on three of six items involving algebraic expressions or equations, and
both items involving graphing. In a similar analysis, Kloosterman et al. (2004) found significant
gains on all five pattern and informal algebra grade 4 items used from 1990 to 2000. Looking
across content areas, D’Ambrosio, Kastberg, and Lambdin (2007) argued that factors beyond
item content affected student performance. In particular, they argued that item wording and item
format impacted student performance and that the impacts varied by demographic subgroup.
Taken as a whole, these analyses show that progress varied depending on time frame and specific
task assigned.
Conceptual Framework
The NAEP assessment has always been based on a conceptual framework outlining
content and grade or age level assessed, sampling characteristics, item format, and additional
issues such as use of calculators and complexity or difficulty of items (see National Assessment
Governing Board, 2010). This framework, which is updated periodically, affects both what and
who is assessed. For example, the current framework includes provisions for accommodations
1
Our analysis of the mathematics items released at grades 4 and 8 in recent years indicates
that they are reasonably representative of the types of items found on NAEP although not
necessarily representative of the proportion of items of each type. Because of the major changes
in the content of the grade 12 assessment after 2000, grade 12 items released prior to 2005 are
not representative of the current grade 12 assessment.
2 for students with disabilities in the sample whereas the framework used 20 years ago included
students with disabilities only when they could complete the assessment without
accommodations. For the purpose of this study, it was assumed that the NAEP frameworks and
data collection procedures were adequate for item development along with collection and
reporting of performance data. In addition, it is assumed that the data presented in this paper
represent a broad national sample of what students are able to do from the perspective that
understanding performance is essential when approaching the question of where mathematics
instruction seems to be working well and where changes might be needed.
Method
Because the sampling procedure for Main NAEP was the same from 1996 to 2011, this
study focused on all items that included algebraic concepts used during that period. Given that
previous research, including the research described earlier in this document, reported on many of
the algebra items used between 1996 and 2003, the main focus was on performance between
2003 and 2011. Note that only items that have been released can be reported verbatim. The
research team was able to view most of the non-released items and those items were included in
the analyses when it was possible to say something meaningful about an item without divulging
its exact content.
All grade 4 and 8 algebra-related mathematics items used between 2003 and 2011 were
divided into the categories of (a) patterns, relations, and functions, (b) algebraic representations,
(c) variables, expressions, and operations, and (d) equations. Tables were constructed showing
the items in each category along with performance on those items for each year they were
administered. Some items fell into multiple topic areas and thus appear in more than one table.
The tables, along with patterns and trends in the data identified by the authors based on
those tables, form the results section of this paper. Note that NAEP classifies items into one of
five primary content areas (number properties and operations, measurement, geometry, data
analysis and probability, algebra). Despite this classification, some items included in content
areas other than algebra required algebraic reasoning to complete and thus were included in the
results.
Results
Scale scores for the NAEP algebra strand for students at grades 4 and 8 between 1996
and 2011 are shown in Figure 1. As can be seen in the figure there was very strong growth in
overall algebra skill of 4th graders between 1996 and 2003 but only minimal growth since then.
Growth at grade 8 was much more consistent and, given that fewer points are needed to move a
scale score up significantly at grade 8 than grade 4 (Kloosterman & Walcott, 2007), the gain of
18 points between 1996 and 2011 at grade 8 represents substantially more growth than the gain
of 17 points at grade 4.
3 Figure 1. NAEP Algebra Scale Scores. Data from the NAEP Data Explorer (http://nces.ed.gov/
nationsreportcard/naepdata/dataset.aspx).
Patterns, Relations, and Functions
Table 1 shows the 17 grade 4 items used between 2003 and 2011 that required
understanding and use of (a) shape patterns, (b) addition and subtraction patterns, (c)
multiplication patterns, and (d) other patterns. Note that the “other” classification in this and
other tables includes non-released items on which there was not sufficient information to be
certain of placement in a specific category. For each item in the table a description of the item
and the percentage of students who responded correctly to the item each year it was administered
is provided2. Exact format of released items and the diagrams that accompanied some items can
be seen by using the year, block, and item number given after the item description in Table 1 to
identify the item in the NAEP on-line Questions Tool3
(http://nces.ed.gov/nationsreportcard/about/naeptools.asp).
Looking at Table 1, it can be seen that there were four items involving shape patterns,
nine more involving addition and subtraction patterns, one involving a multiplication pattern, and
2
When an item is released, the percent of students correctly answering each item is provided
for the year of release. NAEP coded skipped items differently than we did and also included a
factor for partial credit on some items. The percent correct shown in this paper is the percent of
students who provided a fully correct answer relative to the number who answered the question
and thus percent correct in the paper may not match percent correct shown online.
3
All items, whether they are released or not, have a unique NAEP item code number. The
codes for each item, which are useful only to individuals who have access to the secure NAEP
data, are available from the authors.
4 three others that were not classified by pattern type. Perhaps the most significant trend in Table 1
is that the percentage of students who correctly responded to the items from one administration
to the next was relatively stable. This is consistent with the overall grade 4 NAEP algebra results
between 2003 and 2011 (Figure 1). Of the 16 items used for more than one administration, the
change in percent correct was 2% or less and not statistically significant (p>.01) for 11 of these
items4. None of the other items changed by more than 5%.
Another feature of the results reported in Table 1 is the substantial variation in the
percentage of students who correctly answered items within each category. This was expected
given that NAEP items are designed so that some differentiate strong from very strong students
while others differentiate weak from very weak students. In addition to the mathematical
knowledge required to answer questions, answer format and item context also appear to impact
the proportion of students who answer correctly. Performance on item 4, for example, was quite
low. That item asked students to explain how they found their answer and while only 15%
answered correctly in 2009, another 42% either found the correct answer but failed to provide a
good rationale or made a calculation mistake that resulted in an incorrect answer. A recurrent
theme of item-level analyses of NAEP data is that students do relatively poorly when they have
to explain their thinking (Arbaugh, Brown, Lynch, & McGraw, 2004). In 2005, 69% of students
correctly found the next number in the sequence 3, 6, 5, 8, 7, 10, 9 (item 7). The pattern involved
both addition and subtraction and the item used the constructed response format so students had
to figure out the correct answer rather than look at a set of answers and find the one that seemed
to work best. In contrast, item 9 involved a fraction where the numerator and denominator were
both increasing by one. In theory, this pattern is quite simple but the fact that patterns involving
fractions are rare in the grade 3 and 4 curriculum could explain why student success (28%) was
barely higher than the chance level.
Table 2 includes the patterns, relations, and functions items used at grade 8. Given the
relative importance of these topics at grade 8 as compared to grade 4, there are more items at
grade 8 and some of the items focus on graphs and relationships between sets of numbers. Item
1 in Table 2 is the same item as number 2 in Table 1 and together these table entries show that
while 74% of 4th graders could identify a relatively straightforward pattern of shapes, 90% of 8th
graders could do that. Similarly item 3 in Table 2, which required students to find the next two
numbers in the pattern 1, 6, 4, 9, 7, 12, 10, was correctly answered by 40% of 4th graders (item 5
in Table 1) and 70% of 8th graders in 2009. The difference in performance between 4th and 8th
grade is typical of trends found on Long-Term Trend NAEP where items have been used more
often at multiple grade levels (Kloosterman, 2010). It is interesting to note that these items, like
many of those used at multiple levels on LTT NAEP, focus on content that is usually not taught
after 4th grade yet older students did significantly better.
The three most difficult pattern questions for 8th graders were numbers 2 and 4, and 6.
4
Reporting of statistical significance for changes between years results in very complex
tables and the chances of spurious results are high when multiple tests for statistical significance
are run. For these reasons, the tables do not include statistical significance for percent correct
across years on individual items. Although the size of the difference in percent correct across
years varies by item, as a general rule differences of 2% or less are usually not significant at the
.01 level and differences of 4% or more are usually statistically significant at the .001 level.
5 Items 2 and 6 required written justification and, as previously noted, students regularly have
problems with items where they have to provide justification. Item 4 tells students that there is a
constant ratio between terms in the sequence 35, 280, 2240 but even with that information, only
about 1/3 were able to calculate the next term. Given that calculators were not allowed for this
item, it is hard to know whether students did poorly because they did not know what to do to find
the next term or because they could not do the relatively complex calculations.
Very few students could identify the slope in a linear setting (item 9) or relate a linear
equation to its graph (item 11). On the other hand, almost all students could infer
straightforward information from a graph (items 13a and 13b) and over half could infer average
speed from a graph of time and distance (item 13c). The fact that only 19% could explain why
the line showing distance would end when the destination point was reached (item 13d) is yet
another example of the difficulties students have in explaining mathematical concepts in writing.
The items classified as “other” in Table 2 required a variety of skills with equations and
graphs. Performance on item 14 was the lowest of these items, likely because two pieces of
information had to be added to a chart and then a generalization had to be made for the to be
counted correct. On the positive side, 85% of students identified at least one of those. Of the
remaining items, performance was over 50% only on items 17 and 20.
Looking at Table 2 as a whole, one gets the sense that some but certainly not all 8th
graders can understand and explain relationships between two variables in different settings and
formats. The items in the table range from the purely visual (item 1) to the connection between
symbols and graphs (item 11), to the purely graphical (item 20). Taken together, they give an
excellent sense of how well 8th-grade students do on tasks involving patterns and functions.
Algebraic Representations
Tables 3 and 4 show 4th and 8th grade performance on items involving algebraic
representation. As was the case in Tables 1 and 2, there was little change over time at grade 4
(Table 3) while performance was stable or increasing over time on most items at grade 8 (Table
4). Over 70% of 4th graders correctly answered item 1 in Table 3 which required multiple steps
(find the next three numbers on a number line with marks at 485, 490, 495, and then use a
calculator to add them), showing that a majority of 4th graders both understand number lines and
can add 3-digit numbers with a calculator. Item 3 required interpretation of a number line where
increments were tenths rather than whole numbers and a little more than half of students were
successful. Item 7 involved a speedometer that was marked in both miles per hour and
kilometers per hour. Less than two-thirds of students correctly realized the speedometer was
showing movement about 5 mph below the speed limit – the fact that 8th graders do not drive and
thus are not used to reading a speedometer showing both miles and kilometers may explain the
relatively low performance in relation to the minimal thinking required for this item. Item 9
simply required writing “go 4 blocks north and 3 blocks east,” and that likely explains why
performance was high (69%) for an item where students had to provide an answer of more than
one word or number. Item 12 required reading a graph and then doing a very straightforward
extrapolation to find a point not on the graph. Seventy-three percent of 4th graders could read the
graph to find out how long it took to answer 3 problems and 54% were able to also complete the
extrapolation.
6 Table 4 includes a wide range of items and, as was the case with Tables 1 to 3,
performance varied substantially depending on the content, context, and item format. Items 1
and 4 were the only items that were also given at grade 4. A correct response for item 1 in Table
4 (and item 2 in Table 3) involved writing a whole number (3) and fractional quantities (3 ½, 3
¾) on a number line. Eighth-graders did substantially better than 4th graders (77% vs. 45% in
2009). Item four in both Tables 3 and 4 was the number line where increments were tenths
rather than whole numbers previously mentioned and performance on that item was also
substantially better at grade 8 than at grade 4 (89% vs. 44% in 2005). Items 8 and 9 required
identifying the coordinates of a point and, taken together, suggest that 70% or more of eighthgraders have this skill.
Item 12 involved interpolating the x intercept of a curve that crossed the x axis between 1
and 2. Performance on this item was a bit less than performance on the extrapolation item given
to 4th graders and mentioned above (item 12 in Table 3). The 4th grade item required writing in
an answer while the 8th grade was multiple choice so the likely reason that performance on the 8th
grade item was lower was that it required estimating the intercept and the use of decimals
whereas the extrapolated point on the 4th grade item was a whole number value on the grid
provided. Performance on item 21 in Table 4 improved from 55% to 65% between 2005 and
2009. This non-released item focused on finding the equation of a line and the substantial
improvement on this item indicates that student performance on finding the equation of a line is
improving in some contexts.
Variables, Expressions, and Operations
Tables 5 and 6 focus on understanding and use of variables, expressions, and operations
at grades 4 and 8, respectively. Performance on item 1 in Table 5, which involved an expression
using subtraction (32-N), was somewhat higher than performance on item 2, which involved the
expression N x 7 (82% vs. 62% in 2005). Item 3 also involved multiplication and performance
on this item (33% in 2007, 35% in 2011) was dramatically lower than item 2. One possible
explanation for this discrepancy is that the wording in item 3 says “total number of pencils” and
for some 4th graders, the word total may have prompted them to add. Support for this
explanation comes from the fact that in 2009, 36% chose the addition distractor (18+p) for the
item. Regardless, the difference in performance on items 2 and 3 indicates that relatively minor
wording changes can make a large difference in performance, especially on items such as
variable expressions that are just being introduced in the curriculum at 4th grade.
None of the variables, expressions, and operations items used at grade 8 (Table 6) were
used at grade 4 and performance on the grade 8 items was relatively stable except for two nonreleased items (6 and 13) where there was significant improvement. Item 16 required
understanding of the term square root so the improved performance there indicates more middle
school students are seeing and using square roots. The relatively strong performance on item 1
(73% in 2007) shows that by the time they are in 8th grade, many students can interpret
expressions written with multiple variables (m x p in this case). The strong performance on item
9 (80% in 2007) shows that most 8th graders understand what it means to substitute a value into
an expression. Less than half know that 6(x+6) is 6x+36 (item 4). Item 5 was by far the most
challenging for 8th graders as only 1% answered correctly in 2011. The item showed a sign
saying that the first CD is $12 and additional CDs were $6 (including tax) and students had to
write an expression for the cost of buying n CDs. A common mistake was failing to account for
7 the first CD in the expression and thus writing 12 + 6n rather than 12 + 6(n-1) or 6 + 6n.
Looking at items 1, 4, and 5 as a group indicates that many 8th graders can write simple variable
expressions (item 1) but far fewer know more than basic rules for combining variable
expressions (item 4) and very few can write linear expressions where the constant is not obvious
(item 5).
Equations and Inequalities
Given the limited use of equations at grade 4, only 5 items involving equations and no
items involving inequalities appeared on the grade 4 NAEP between 2003 and 2011 (Table 7).
Performance was strong on items 2, 3 and 4, which all required recognizing a reasonable
substitution for a variable to make an equation true. Almost all 4th graders would be able to
figure out that 29-8=21 so it is likely the item format – use of the box to represent a variable –
caused 30% to answer item 1 incorrectly. Similarly, there is nothing very complex about item 5
(see Figure 2) so the fact that less than half answered correct is likely due to lack of
understanding the diagrams used to present the problem.
Figure 2. Item 5 from Table 7
As was the case with variables and expressions, there were many more items involving
equations at grade 8 than grade 4. Items 4 and 5 in Table 8 were related in that they both
involved problems that could be solved by writing and solving simple equations (x + 3x = 152 in
item 4, x + 2x = 18 in item 5). The similarity in performance on the two items (47% and 52% in
2007) may be misleading. One could argue that a higher proportion of students correctly
answered item 5 because they only had to identify the equation needed to solve the problem.
However, the numbers in item 5 were small enough that guess and check was an appropriate
strategy. It is possible that if students had been asked to solve problem 5, use of guess and check
would have made the percentage of students answering correctly higher. The numbers in item 4
were large enough that, even though there were only 5 answer choices to test, guess and check
8 was a relatively difficult strategy compared to writing and solving an equation. Sixteen percent
of students selected 38 as the correct answer to item 4, which suggests that these individuals
found the number of hot dogs sold by solving an equation or by guess and check but failed to
remember that they were being asked for the number of hot dogs that Carmen rather than Shawn
sold. When those who incorrectly selected 38 are pooled with those who correct solved the
problem, it suggests that the number of individuals who can solve linear combination problems is
higher than performance on either item 4 or 5 indicates. The supposition that some students
prefer informal methods of solving linear combination problems is supported by looking at item
18 which is solvable using the equation 2.5x + 1(5-x) = 8 or 1x + 2.5(5-x) = 8. Performance on
this constructed response item was quite good (76% in 2007) and the need to use decimals when
writing and solving equations for this item strongly suggests that many if not most successful
students used a guess and check or other informal solution method. The fact that students were
asked for two pieces of information (number of newly released movies and number of classics)
probably improved performance because students had to think about what the number they
calculated represented.
Discussion
The goal of this study was to use performance on NAEP items to determine the strengths and
weaknesses of American students when it comes to algebraic reasoning. The large number of
algebraic reasoning items used by NAEP since 2003, and the large proportion of those that were
released, makes NAEP an excellent sense of student performance on a wide range of algebraic
tasks. The fact that performance on most items has been relatively stable in recent years,
especially at grade 4, indicates that using items released several years ago as indicators of current
student performance is justified. Even during the years 1990 through 2003 when performance in
the area of algebra was increasing substantially there was relatively modest gain on many NAEP
algebra items (Chazen et al. 2007; Kloosterman et al., 2004, Sowder et al., 2004) so the minimal
gains on items between 2003 and 2011 are not that surprising. Although released items are
available online, this study went far beyond what is available online by (a) categorizing items
with algebra content by sub-topic, (b) documenting trends in performance over time on items, (c)
including data on secure items, and (d) identifying issues and themes about what students know
and can do with respect to algebra. It is the last of these points – identifying issues and themes –
that will be the focus of this section.
What Makes an Item Difficult?
One thing that is clear from the tables in this paper is that exact item content, format, and
context make a difference. Item 1 in Table 7, where 4th graders were asked to find the number
that goes in the box in the equation □ – 8 = 21, is a good example of how unusual formats or
symbolism can make it appear students know less than they do. Although item 3 in Table 7 was
multiple choice rather than constructed response, we know from that item that 90% of 4th graders
realize that if n + 4 = 12, then n = 8 and thus it is likely that had n or x been used rather the □ in
item 1, performance on the item would have been higher than 70%. A previously noted example
of how wording can affect performance involved items 2 and 3 from Table 5. On those items
students had to select multiplication expressions N x 7 and 18 x p and the nearly 30% difference
in performance can perhaps be explained by the wording of the two items. In item 3, the use of
the word “total” may have led 4th graders to incorrectly assume addition was the appropriate
operation.
9 Item 2 in Table 2 provides an example of how context may affect performance. The
item, shown in Figure 3, shows a pattern of the perimeter of hexagons placed side by side and
asks students to find the perimeter of the 25th figure and explain how they found that perimeter.
In 2007, 19% of students correctly solved the problem and provided an explanation and another
19% found the answer but did not accurately describe their process. Students did not need to use
the figure to solve the problem, which leads to the question of whether performance would have
been lower or higher had the figure been omitted and students simply given the sequence 6, 10,
14, 18, … and asked to find the 25th term. This would have meant less reading and interpreting
of the figure (see D’Ambrosio et al, 2007 for a discussion of the impact of reading on NAEP
items) and thus saved students time. On the other hand, the diagram of the first four figures may
have helped visualize the need to add 4 for each new hexagon and thus removing the diagram
may have made the problem harder. A related policy question is whether simply providing the
sequence would have been more in line with the algebraic reasoning that the NAEP framework
says this item should be measuring. Only half of the students who got the right answer also
explained their process to receive full credit. As noted earlier in this paper and in Arbaugh et al.,
(2004), students often do poorly when they have to explain their reasoning.
Figure 3. Item 2 from Table 2
What do we want elementary and middle school students need to know about algebraic
reasoning?
For the purposes of this study, we have defined algebraic reasoning as what is measured
by current and recently released NAEP items. With the introduction of the Common Core State
Standards (www.corestandards.org), the most important aspects of algebraic reasoning are likely
to become what is specified in the standards. For each of the grades K through 5, the CCSS
specify standards for “operations and algebraic thinking.” At grades 3 and 4 the standards focus
on representing and solving multi-step word problems with all four operations, understanding
factors and multiples, and identifying and explaining number and shape patterns. With respect to
10 these standards, NAEP data tell us, for example, that most 4th graders have at least a minimal
understanding of variable expressions as long as those expressions are used in standard formats
and contexts (Tables 5 and 7). They can also identify straightforward numeric and geometric
patterns as long as they do not have to describe those patterns (Table 1). Multi-step word
problems are in the number and operation strand of NAEP and thus not described in this paper
and while there are a few NAEP items dealing with factors and multiples, those are also
considered number and operation rather than algebra. The eight Common Core Standards for
mathematical practice (making sense of problems, abstract reasoning, constructing arguments,
modeling, using tools, precision, use of structure, regularity in reasoning) apply to all grade
levels and, while those standards are open to interpretation, the poor performance on any items
requiring justification suggests ability to construct arguments, interpret a model, and explain
reasoning are generally poor at both grades 4 and 8.
At grades 7 and 8, the CCSS do not specify algebraic reasoning standards, focusing instead
on ratios and proportional relationships, expressions and equations, solving problems using
equations and formulas, and defining and evaluating functions. Although there are more algebra
items on the 8th-grade than the 4th-grade NAEP assessment, the proportion of items in the algebra
strand that map to CCSS algebra skills seems smaller. Moreover, the data from those items
suggest that it will be difficult to meet those standards in the short term. For example, only a
third of 8th graders correctly answered item 4 in Table 2 where they had to calculate a ratio to
find the next term of a sequence. Just over one third correctly answered a question involving
rates (item 15, Table 2) and performance on the item requiring identification of the graph of y = – 2x + 1 (item 11, Table 2) was near the chance level. Although 72% could read a non-linear
graph (item 26, Table 4), only 48% could find change in y given change in x for a linear function
(item 22, Table 4). In short, if we define algebraic reasoning as what is outlined by the CCSS,
8th-grade performance is acceptable in some areas but certainly not all.
Do we need to continually repeat ideas in the curriculum?
Although LTT NAEP includes more items that are used at multiple levels then Main
NAEP, the evidence from the items used in Main NAEP is parallel to the trend found in LTT
(Kloosterman, 2010) in that performance improves as students get older. Items 1 and 3 in Table
2, items 1, 4, and 10 in Table 4, and item 1 in Table 8 were also used at grade 4. Performance of
8th graders was substantially higher than performance of 4th graders on all items even though
items 1 (fractions) and 4 (decimals) in Table 4 were the only ones used at both levels that
involved content that would normally be taught in middle school. One could argue that the
improvement on items where content was not taught suggests that students eventually figure out
that content even though it is not taught so more repetition is not needed. On the other hand, the
relatively poor performance at any level on items that required more complex reasoning suggests
that at least the process of solving complex problems needs continued emphasis.
Is it realistic to expect students to explain their reasoning?
In this paper, the only items on which students had to explain thinking to get full credit
were item 4 in Table 1 (dot patterns) and items 13c and 13d in Table 2 (Josh’s car trip).
Percentage correct on these questions was 15, 54, and 19 respectively. All that was required for
the item where 54% answered correctly was stating how the distance formula was used. The
other questions required more than application of a formula and the performance reflects that.
The dot pattern item was used three times and performance dropped over that period. Based on
11 these three items, it is hard to answer the question of whether students can be expected to explain
their reasoning. The data are limited and one could argue that many students have never been
expected to explain their thinking and thus more focus on explanation might improve that ability.
All that can be said for sure is that getting all students to explain their reasoning will be
challenging.
How do we prepare students for high stakes standardized assessments?
NAEP data are potentially useful in both developing high stakes assessments and
preparing students for such assessments because the data provide insight into what a national
sample of students are likely to do on a wide variety of items. At the very least, performance on
the □ – 8 = 21 item (item 1, Table 7) and the balance scale item (Figure 2) shows that students
need to be aware of any unusual notation or representation, that might be used in an assessment.
NAEP has a wide range of items from patterns and paths to graphs and word problems and in
that sense is a good sample of what students may see on other assessments. As Lambdin and
Morge (2006) suggest, released items can be used to help teachers get a sense of how their
students compare to state and national samples and whether there seem to be item contexts or
formats that are a particular challenge to individual students. The organization of items by topic
in this paper makes selection of released items easier for teachers looking for assessment items
on specific algebraic topics.
In closing, we note that while good research usually raises as many questions as it
answers, the NAEP data described here give the best sense we are likely to get of the algebraic
thinking skills of 4th and 8th grade students in the United States. Although there are algebraic
reasoning skills that are not covered by NAEP, there are items relating to most of the topics we
expect students to master. With respect to performance, the picture is much better than it was 20
years ago – Kloosterman and Walcott (2007) argue that the mathematics gains at grades 4 and 8
over the last 20 years are equivalent to 2 grade levels. That being said, we have challenges ahead
when we consider current performance in light of the NCTM Principles and Standards and the
Common Core State Standards.
References
Arbaugh, F., Brown, C., Lynch, K., McGraw, R. (2004). Students’ ability to construct responses (1992-2000):
Findings from short and extended-constructed response items. In P. Kloosterman & F. K. Lester, Jr. (Eds.).
Results and interpretations of the 1990 through 2000 mathematics assessments of the National Assessment of
Educational Progress (pp. 337-362). Reston, VA: National Council of Teachers of Mathematics.
Braswell, J. S., Lutkus, A. D., Grigg, W. S., Santapau, S. L., Tay-Lim, B., & Johnson, M. (2001). The nation’s
report card: Mathematics 2000. Washington, DC, National Center for Education Statistics (Report No. NCES
2001-517).
Chazan, D., Leavy, A. M., Birky, G., Clark, K., Lueke, M., McCoy, W., & Nyamekye, F. (2007). What NAEP can
(and cannot) tell us about performance in algebra. In P. Kloosterman & F. K. Lester, Jr. (Eds.). Results and
interpretations of the 2003 mathematics assessment of the National Assessment of Educational Progress (pp.
169-190). Reston, VA: National Council of Teachers of Mathematics.
D’Ambrosio, B. S., Kastberg, S. E., & Lambdin, D. V. (2007). Designed to differentiate: What is NAEP
measuring? In P. Kloosterman & F. K. Lester, Jr. (Eds.). Results and interpretations of the 2003 mathematics
assessment of the National Assessment of Educational Progress (pp. 289-309). Reston, VA: National Council
of Teachers of Mathematics.
Kloosterman, P. (2010, November). How much do mathematics skills improve with age? Findings from LTT
NAEP. Paper presented at the annual meeting of the School Science and Mathematics Association. Fort Myers,
FL.
12 Kloosterman, P. & Walcott, C (2007). The 2003 mathematics NAEP: Overall results. In P. Kloosterman & F. K.
Lester, Jr. (Eds.). Results and interpretations of the 2003 mathematics assessment of the National Assessment of
Educational Progress (pp. 23-42). Reston, VA: National Council of Teachers of Mathematics.
Kloosterman, P., Warfield, J., Wearne, D., Koc, Y., Martin, W. G. & Strutchens, M. (2004). Knowledge of
mathematics and perceptions of learning mathematics of fourth-grade students. In P. Kloosterman & F. K.
Lester, Jr. (Eds.). Results and interpretations of the 1990 through 2000 mathematics assessments of the National
Assessment of Educational Progress (pp. 71-103). Reston, VA: National Council of Teachers of Mathematics.
Lambdin, D. V., & Morge, S. P. (2006). Investigating student understanding by examining NAEP test items and
student performance data. In C. A. Brown & L. V. Clark (Eds.), Learning from NAEP: Professional
development materials for teachers of mathematics (pp. 41-50). Reston, VA: National Council of Teachers of
Mathematics.
NAEP Data Explorer (2012). http://nces.ed.gov/nationsreportcard/naepdata/
National Assessment Governing Board (2010). Mathematics framework for the 2011 National Assessment of
Educational Progress. Washington, DC: U.S. Department of Education.
National Center for Education Statistics (2011). The nation’s report card: Mathematics 2011. Washington, DC:
Institute for Education Sciences, U.S. Department of Education.
National Mathematics Advisory Panel (2008). Foundations for success: Final report of the National Mathematics
Advisory Panel. Washington, DC: U.S. Department of Education.
Sowder, J. T., Wearne, D., Martin, W. G. & Strutchens, M. (2004). What do 8th-grade students know about
mathematics? In P. Kloosterman & F. K. Lester, Jr. (Eds.). Results and interpretations of the 1990 through
2000 mathematics assessments of the National Assessment of Educational Progress (pp. 105-144). Reston, VA:
National Council of Teachers of Mathematics.
This paper is based upon work supported by the National Science Foundation under the REESE Program,
grant number 1008438. Opinions, findings, conclusions and recommendations expressed in the paper are
those of the authors and do not necessarily reflect the views of the National Science Foundation.
13 Table 1: Performance on Grade 4 Items Involving Understanding and Use of Patterns, Relations, and
Functions
Item Description
Shape patterns
1. In the pattern shown above, which of the following would go into
the blank space? [2007-M9 #17]
2. Which of the figures below should be the fourth figure in the
pattern shown above? [2005-M4 #3]
3. Kiara set her beads on a table to make a repeating pattern. Some
of the beads rolled off the table. Here is what was left. Which of
the following should Kiara use to replace the missing beads in the
pattern? [2009-M10 #8]
4. A pattern of dots is shown above. How many dots would be in the
6th picture? (1 dot, 3 dots, 6 dots, 10 dots, _ , _ ) Explain how
you found your answer. [2009-M10 #13]
Addition and subtraction patterns
5. Write the next two numbers in the number pattern. 1 6 4 9 7
12 10 __ __ Write the rule that you used to find the two
numbers you wrote. [2009-M5 #12]
6. The numbers in the pattern 14, 26, 38, __, __ are increasing by
12. Which of these numbers is part of the pattern? [2007-M11 #3]
7. In the number pattern (3, 6, 5, 8, 7, 10, 9, ?), what number comes
next? [2005-M12 #6]
8. Which rule describes the pattern shown in the table? [2005-M12
#14]
9. Addition pattern; extend pattern, numerator and denominator
each increasing by one
10. The table shows how the "In" numbers are related to the "Out"
numbers. When 38 goes in, what number comes out? [2007-M7
#12]
11. The table shows the number of edges for each prism. What is the
number of edges for a prism if the bottom face has 7 sides?
[2011-M12 #14]
12. Continue pattern using shapes to identify given shape and term-term number larger than number of shapes given
13. Sam folds a piece of paper in half once. There are 2 sections. Sam
folds the paper in half again. There are 4 sections. Sam folds the
paper in half again. There are 8 sections. Sam folds the paper in
half two more times. Which list shows the number of sections
there are each time Sam folds the paper? [2011-M9 #14]
Multiplication patterns
14. Every 30 minutes Dr. Kim recorded the number of bacteria in a
test tube. Which best describes what happened to the number of
bacteria every 30 minutes? [2011-M8 #14]
Other
15. Identify number in pattern
16. A number pattern is given, use rule for this pattern to make
another beginning with a different number
17. Identify rule for function table
14 Percent Correct
2005 2007 2009
Item Type
2003
MC
45
47
MC
74
74
46
MC
70
72
72
SCR
20
15
15
40
SCR
35
37
40
MC
52
55
57
SCR
65
69
MC
21
24
MC
28
28
MC
20
20
28
MC
28
29
MC
65
64
MC
MC
34
35
MC
ECR
45
46
MC
2011
29
24
23
35
34
34
18
20
53
Table 2: Performance on Grade 8 Items Involving Understanding and Use of Patterns, Relations, and
Functions
Item Description
Patterns
1. Which of the figures below should be the fourth figure in the
pattern shown above? [2005-M4 #3]
2. Each figure in the pattern below is made of hexagons that
measure 1 centimeter on each side. Show how you found your
answer. If the pattern of adding one hexagon to each figure is
continued, what will be the perimeter of the 25th figure in the
pattern? [2007-M7 #14]
3. Write the next two numbers in the number pattern. 1 6 4 9 7
12 10 __ __. Write the rule that you used to find the two
numbers you wrote. [2009-M5 #11]
4. In the sequence below, the ratio of each term to the term
immediately following it is constant. What is the next term of this
sequence after 2240? 35, 280, 2240, __ [2009-M10 #9]
5. 1, 9, 25, 48, 81, ... The same rule is applied to each number in the
pattern above. What is the 6th number in the pattern? [2005-M12
#13]
6. Extend and generalize a number pattern
7. According to the pattern suggested by the four examples above,
how many consecutive odd integers are required to give a sum of
144? [2005-M3 #12]
Graphs of functions
8. From the starting point on the grid below, a beetle moved in the
following way. It moved 1 block up and then 2 blocks over, and
then continued to repeat this pattern. Draw lines to show the path
the beetle took to reach the right side of the grid. [2005-M4 #10]
9. Identify slope from verbal description
10. Find change in y given x for line equation
11. Which of the following is the graph of the line with equation
y = –2x + 1? [2007-M11 #11]
12. Which of the following is an equation of a line that passes
through the point (0, 5) and has a negative slope? [2011-M12#7]
13. The linear graph below describes Josh’s car trip from his
grandmother’s home directly to his home. [2011-M8 #15]
(a) Based on this graph, what is the distance from Josh’s
grandmother’s home to his home?
(b) Based on this graph, how long did it take Josh to make the
trip?
(c) What was Josh’s average speed for the trip? Explain how you
found your answer.
(d) Explain why the graph ends at the x-axis.
(a) through (d) all correct
Other
14. Sarah has a part-time job at Better Burgers restaurant and is paid
$5.50 for each hour she works. She has made the chart below to
reflect her earnings but needs your help to complete it. (a) Fill in
the missing entries in the chart. (b) If Sarah works h hours, then,
in term of h, how much will she earn? [2007-M9 #10]
15. An airplane climbs at a rate of 66.8 feet per minute. It descends at
twice the rate that it climbs. Assuming it descends at a constant
rate, how many feet will the airplane descend in 30 minutes?
15 Item Type
2003
MC
90
SCR
SCR
67
SCR
Percent Correct
2005 2007 2009
90
18
19
68
70
70
32
32
33
12
14
18
47
25
18
48
26
29
MC
60
61
ECR
MC
40
40
SCR
54
54
MC
MC
MC
22
2011
23
MC
31
ECR
90
ECR
94
ECR
54
ECR
19
ECR
12
SCR
24
27
27
MC
38
37
37
11
[2007-M11 #19]
16. In the equation y = 4x, if the value of x is increased by 2, what is
the effect on the value of y? [2005-M3 #10]
17. Which of the following equations represents the relationship
between x and y shown in the table above? [2005-M12 #17]
18. Tom went to the grocery store. The graph below shows Tom's
distance from home during his trip. Tom stopped twice to rest on
his trip to the store. What is the total amount of time that he spent
resting? [2009-M10 #10]
19. The number of gallons of water, y, in a tank after x hours may be
modeled by the linear equation y = 800 – 50x. Which of the
following statements about the tank is true? [2011-M12 #15]
20. For 2 minutes, Casey runs at a constant speed. Then she gradually
increases her speed. Which of the following graphs could show
how her speed changed over time? [2011-M9 #3]
16 MC
33
34
MC
51
54
MC
MC
MC
41
42
44
46
48
48
69
70
Table 3: Performance on Grade 4 Items Involving Understanding and Use of Algebraic Representations
Item Description
Number lines
1. On the number line above, what is the sum of the numbers to
which the arrows X, Y, and Z point? [2011-M8 #8]
2. Jorge left some numbers off the number line below. Fill in the
numbers that should go in A, B, and C. [2009-M5 #15]
3. On the number line above, what number would be located at
point P? [2005-M4 #5]
4. On the number line, what number does P represent? [2009-M10
#7]
5. Fill in the four missing numbers on the number line above. [2005M12 #18]
6. Choose correct time line based on before and after statements
7. The speedometer shows how fast Dale is driving. If the speed
limit is 55 miles per hour (mph), which of the following is true?
[2007-M9 #7]
Coordinate grids and graphs
8. From the starting point on the grid below, a beetle moved in the
following way. It moved 1 block up and then 2 blocks over, and
then continued to repeat this pattern. Draw lines to show the path
the beetle took to reach the right side of the grid. [2005-M4 #10]
9. The map below shows how to go from the school to the park.
Complete the written directions that are started below. [2005M12 #17]
10. Locate points on grid/coordinate system
11. Joe rode his bicycle from his house to his friend’s house. He rode
1.7 miles along the path below. The path is marked every 0.5
mile. Put an X on the path to show how far Joe rode to his
friend’s house. [2011-M12 #12]
12. The graph shows the total number of minutes it took Selena to do
math problems. How many minutes did it take her to do 3
problems. Selena continues to work at the same rate. How many
problems will she do in 40 minutes? [2011-M9 #9]
Other
13. On the scale above, 2 cylinders balance 1 cube. Which of the
scales below would balance? [2011-M12 #7]
17 Percent Correct
2005 2007 2009
Item Type
2003
MC
71
72
73
72
SCR
40
43
46
45
SCR
56
57
46
47
76
64
76
63
30
63
30
MC
SCR
44
56
59
MC
MC
62
76
65
SCR
22
24
SCR
69
69
SCR
SCR
61
MC
MC
2011
71
31
54
47
45
44
Table 4: Performance on Grade 8 Items Involving Understanding and Use of Algebraic Representations
Item Description
Number line
1. Jorge left some numbers off the number line below. Fill in the
numbers that should go in A, B, and C. [2009-M5 #14]
2. Which of the graphs below is the set of all whole numbers less
than 5? [2005-M12 #10]
3. What number is represented by point A on the number line
above? [2005-M12 #12]
4. On the number line above, what number would be located at
point P? [2005-M4 #5]
5. Weather records in a city show the coldest recorded temperature
was –20° F and the hottest was 120. Which of the following
number line graphs represents the range of recorded actual
temperatures in this city? [2007-M7 #5]
Graph
6. If the points Q, R, and S shown above are three of the vertices of
rectangle QRST, which of the following are the coordinates of T
(not shown)? [2005-M3 #13]
7. The map above shows eight of the counties in a state. The largest
city in the state can be found at location B-3. In which county
could this city lie? [2005-M12 #14]
8. Which point is the solution to both equations shown on the graph
above? [2009-M10 #7]
9. The graph above shows lettered points in an (x, y) coordinate
system. Which lettered point has coordinates (-3, 0)? [2007-M11
#2]
10. From the starting point on the grid below, a beetle moved in the
following way. It moved 1 block up and then 2 blocks over, and
then continued to repeat this pattern. Draw lines to show the path
the beetle took to reach the right side of the grid. [2005-M4 #10]
11. According to the graph, between which of the following pairs of
interest rates will the increase in the number of months to pay off
a loan be greatest? [2011-M12 #9]
12. On the curve above, what is the best estimate of the value of x
when y = 0? [2005-M3 #11]
13. Interpret slope from verbal description
14. Find coordinate of collinear points
15. Group of points on a line
16. The linear graph below describes Josh’s car trip from his
grandmother’s home directly to his home. [2011-M8 #15]
(a) Based on this graph, what is the distance from Josh’s
grandmother’s home to his home?
(b) Based on this graph, how long did it take Josh to make the
trip?
(c) What was Josh’s average speed for the trip? Explain how you
found your answer.
(d) Explain why the graph ends at the x-axis.
(a) through (d) all correct
17. Which of the following is the graph of the line with equation
y = –2x + 1? [2007-M11 #11]
18. For the figure above, which of the following points would be on
the line that passes through points N and P? [2009-M10 #14]
19. Identify true statement about graph of a line
18 Percent Correct
2005 2007 2009
Item Type
2003
SCR
75
76
MC
36
37
MC
41
43
SCR
89
89
MC
59
MC
58
61
MC
85
86
MC
69
78
72
75
SCR
54
54
MC
49
MC
SCR
MC
77
59
67
MC
MC
77
2011
72
71
70
70
18
27
53
18
29
56
12
11
50
48
ECR
90
ECR
94
ECR
54
ECR
19
ECR
MC
MC
MC
12
22
23
25
50
52
55
31
11
20. Which of the following is an equation of a line that passes
through the point (0, 5) and has a negative slope? [2011-M12 #7]
21. Find equation of the line
22. Find change in y given x for linear equation
23. Identify graph of exponential growth
Equation
24. The length of a rectangle is 3 feet less than twice the width, w (in
feet). What is the length of the rectangle in terms of w? [2009M10 #8]
25. The number of gallons of water, y, in a tank after x hours may be
modeled by the linear equation y=800-50x. Which of the
following statements about the tank is true? [2011-M12 #15]
26. Mrs. Brown would like to pay off a loan in 180 months.
According to the graph, what should be the approximate percent
of the interest rate on her loan? [2011-M12 #10]
27. The admission price to a movie theater is $7.50 for each adult and
$4.75 for each child. Which of the following equations can be
used to determine T, the total admission price, in dollars, for x
adults and y children? [2011-M8 #3]
Inequality
28. Graph the solution set for 3 < x < 5 on the number line below.
[2011-M9 #11]
29. Select graph for inequality
30. Arrow on number line
Other
31. Use algebraic model to estimate height
32. Use algebraic model to make prediction
33. The map above gives the distances, in miles, between various
locations in a state park. Traveling the shortest possible total
distance along the paths shown on the map, from the visitor
center Teresa visits the cave, waterfall, and monument, but not
necessarily in that order, and then returns to the visitor center. If
she does not retrace her steps along any path and the total
distance that Teresa travels is 14.7 miles, what is the distance
between the cave and the monument? [2005-M3 #15]
19 MC
26
29
31
MC
MC
MC
55
60
47
48
65
48
48
MC
48
50
51
MC
46
48
48
SCR
75
75
72
MC
70
70
SCR
24
22
MC
MC
MC
SCR
MC
43
44
71
45
73
38
24
23
74
40
21
Table 5: Performance on Grade 4 Items Involving Understanding and Use of Variables, Expressions,
and Operations
Item Description
Variables in equations
1. Paco had 32 trading cards. He gave N trading cards to his friend.
Which expression tells how many trading cards Paco has now?
[2007-M9 #5]
2. N stands for the number of hours of sleep Ken gets each night.
Which of the following represents the number of hours of sleep
Ken gets in 1 week? [2005-M12 #12]
3. Each of the 18 students in Mr. Hall’s class has p pencils. Which
expression represents the total number of pencils that Mr. Hall’s
class has? [2011-M12 #15]
4. Give algebraic expression that represents given situation
Variables in tables
5. What number does n represent in the table? [2009-M10 #2]
6. Which rule describes the pattern shown in the table? [2005-M12
#14]
20 Percent Correct
2005 2007 2009
Item Type
MC
MC
2003
79
82
MC
61
62
MC
83
33
MC
MC
MC
34
51
21
82
24
2011
84
83
35
Table 6: Performance on Grade 8 Items Involving Understanding and Use of Variables, Expressions,
and Operations
Item Description
Expression
1. If m represents the total number of months that Jill worked and p
represents Jill's average monthly pay, which of the following
expressions represents Jill's total pay for the months she worked?
[2007-M11 #6]
2. If n is any integer, which of the following expressions must be an
odd integer? [2011-M8 #8]
3. Consider each of the following expressions. In each case, does
the expression equal 2x for all values of x? (a) 2 times x, (b) x
plus x, (c) x times x. [2007-M9 #2]
(a) 2 times x (Yes or No)
(b) x plus x (Yes or No)
(c) x times x (Yes or No)
(a) through (c) all correct
4. Which of the following is equal to 6(x + 6)? [2005-M12 #3]
5. The Music Place is having a sale. Write an expression that shows
how to calculate the cost of buying n CD’s at the sale. [2011-M9
#13]
6. Expression for seats on a van
7. Simplify an algebraic expression
8. Choose statement corresponding to expression
Equation
9. If x = 2n + 1, what is the value of x when n = 10? [2007-M7 #1]
Inequality
10. If a > 0 and b < 0, which of the following must be true? [2011M12 #17]
Other
11. 3 + 15 ÷ 3 – 4 × 2 = [2003-M10 #12]
12. Determine the value of x
13. Given a whole number square root less than 10, find the original
number.
21 Item Type
2003
MC
73
Percent Correct
2005 2007 2009
72
2011
73
MC
40
40
3
1
MC
MC
SCR
31
41
30
44
32
MC
MC
MC
63
67
MC
78
80
MC
MC
MC
MC
27
68
38
46
28
52
66
58
61
64
29
Table 7: Performance on Grade 4 Items Involving Understanding and Use of Equations
Item Description
1.
2.
3.
4.
5.
□ – 8 = 21 What number should be put in the box to make the
number sentence above true? [2009-M5 #7]
What number does n represent in the table? [2009-M10 #2]
What value of n makes the number sentence n + 4 = 12 true?
[2011-M12 #3]
The weights on the scale above are balanced. Each cube weighs 3
pounds. The cylinder weighs N pounds. Which number sentence
best describes this situation? [2007-M7 #4]
On the scale above, 2 cylinders balance 1 cube. Which of the
scales below would balance? [2011-M12 #7]
22 Item Type
2003
SCR
68
Percent Correct
2005 2007 2009
2011
70
70
70
MC
MC
82
84
91
83
92
90
MC
79
45
44
MC
81
47
Table 8: Performance on Grade 8 Items Involving Understanding and Use of Equations and Inequalities
Item Description
Linear equation in one variable
1.
□ – 8 = 21 What number should be put in the box to make the
number sentence above true? [2009-M5 #6]
If 15 + 3x = 42, then x = [2007-M9 #4]
Which of the following equations is NOT equivalent to the
equation n + 18 = 23? [2011-M8 #5]
4. At the school carnival, Carmen sold 3 times as many hot dogs as
Shawn. The two of them sold 152 hot dogs altogether. How many
hot dogs did Carmen sell? [2007-M11 #15]
5. Robert has x books. Marie has twice as many books as Robert
has. Together they have 18 books. Which of the following
equations can be used to find the number of books that Robert
has? [2011-M12 #5]
6. Which of the following equations has the same solution as the
equation 2x + 6 = 32? [2009-M10 #1]
Linear equation in two variables
7. If x = 2n + 1, what is the value of x when n = 10? [2007-M7 #1]
8. Which of the following equations represents the relationship
between x and y shown in the table above? [2005-M12 #17]
9. Angela makes and sells special-occasion greeting cards. The table
above shows the relationship between the number of cards sold
and her profit. Based on the data in the table, which of the
following equations shows how the number of cards sold and
profit (in dollars) are related? [2007-M7 #15]
10. The number of gallons of water, y, in a tank after x hours may be
modeled by the linear equation y = 800 – 50x. Which of the
following statements about the tank is true? [2011-M12 #15]
11. The point (4, k) is a solution to the equation 3x + 2y = 12. What is
the value of k? [2011-M12 #12]
12. The admission price to a movie theater is $7.50 for each adult and
$4.75 for each child. Which of the following equations can be
used to determine T, the total admission price, in dollars, for x
adults and y children? [2011-M8 #3]
13. A rectangle has a width of m inches and a length of k inches. If
the perimeter of the rectangle is 1,523 inches, which of the
following equations is true? [2011-M9 #10]
14. In the equation y = 4 – x, if the value of x is increased by 2, what
is the effect on the value of y? [2005-M3 #10]
Quadratic equation
15. The formula d = 16t2 gives the distance d, in feet, that an object
has fallen t seconds after it is dropped from a bridge. A rock was
dropped from the bridge and its fall to the water took 4 seconds.
According to the formula, what is the distance from the bridge to
the water? [2007-M7 #9]
16. A reasonable prediction of the distance d in feet, that a car travels
after the driver has applied the brakes can be found by using the
formula d = 0.055r2, where r is the speed of the car in miles per
hour. If Mario is driving at 60 miles per hour and applies the
brakes, then according to the formula, how many feet will
Mario’s car travel before it stops? [2011-M8 #12]
Other
17. The temperature in degrees Celsius can be found by subtracting
2.
3.
23 Percent Correct
2005 2007 2009
Item Type
2003
MC
87
87
87
MC
MC
79
80
83
MC
47
MC
MC
51
MC
87
60
59
52
52
53
69
70
70
78
54
80
52
54
47
MC
MC
2011
47
MC
46
48
48
MC
30
32
33
MC
70
70
MC
48
49
52
50
MC
33
MC
34
48
50
MC
MC
35
35
37
18.
19.
20.
21.
22.
23.
32 from the temperature in degrees Fahrenheit and multiplying
the result by 5/9. If the temperature of a furnace is 393 degrees in
Fahrenheit. What is it in degrees in Celsius, to the nearest degree?
[2007-M9 #14]
At Jorge's local video store, "New Release" video rentals cost
$2.50 each and "Movie Classic" video rentals cost $1.00 each
(including tax). On Saturday evening, Jorge rented 5 videos and
spent a total of $8.00. How many of the 5 rentals were New
Releases and how many were Movie Classics? [2007-M11 #18]
Which of the graphs below is the set of all whole numbers less
than 5? [2005-M12 #10]
Ravi has more tapes than magazines. He has fewer tapes than
books. Which of the following lists these items from the greatest
in number to the least in number? [2005-M3 #5]
Graph the solution set for 3 < x < 5 on the number line below.
[2011-M9 #11]
Solve an algebraic inequality
Solve linear inequality
24 SCR
77
76
MC
36
37
MC
87
87
76
SCR
MC
MC
24
31
56
57
60
22