Modeling COVID19 disruptions via network mapping of the Common Core Mathematics Standards
Abstract
This paper develops a mathematical and computational modeling approach that provides a datadriven platform to address research questions relating to student pathways in K12 education. Specifically, this paper uses scalable network modeling to create a model of the Common Core Mathematics Standards. The result is an educational map that formally represents the Standards and the relationships among them. This educational map is represented mathematically as a network model that forms the basis for computational graph analytics and visualization to identify Standards and learning pathways of interest. Using the network model, we model the disruption due to COVID19 related school closures in Spring 2020. Analysis on the network model enables identification of propagating effects of the closures on later grades and reveals pathways with potential high vulnerability. When combined with schoolspecific and/or student data, this model could provide valuable analytics support to decision makers.
Keywords
Network modeling, Ontologies, Educational mapping, Intelligent tutoring systems, Common Core
1 Introduction
In the spring of 2020, millions of students abruptly shifted to online instruction, and in some cases, no instruction, as COVID19 disrupted schools nationwide. But this disruption is not simply localized to a single semester: consider, for example, the downstream effects on a fifth grader, who needs to master adding fractions in order to perform more complicated operations in sixth and later grades. Failing to master an earlier, more fundamental learning outcome will result in difficulty mastering a learning outcome in a later grade that depends on the earlier outcome. It is critical to analyze such outcome dependencies in order to address learning gaps so that deficiencies are not propagated for years to come. To study these direct and indirect COVID19 disruptions, this paper develops a graphbased, datadriven model of learning outcomes in a mathematics curriculum.
For our analysis to be widely applicable, we will consider the Common Core Mathematics curriculum. The Common Core Mathematics curriculum is a table list of 331 learning outcomes, dubbed “Standards”, for what students should be able to achieve in each grade band. The Common Core is standardized and adopted across 43 states in public school systems (National Governors Association Center for Best Practices Council of Chief State School Officers, 2010). It therefore facilitates a useful analysis that is widely applicable to all school systems who adopt the Common Core.
One major difficulty in analyzing chains of learning outcome dependencies is that of scale: if one is considering a single learning outcome and wishes to identify all downstream learning outcomes it may impact, including in later grades, it may be possible to trace and list all such downstream outcomes manually with some effort. However, such a process poses several issues. Firstly, it is difficult to replicate with the same result. Secondly, it is a manual and laborious process, with significant chance of oversight error. Thirdly, it does not allow for advanced analysis; for instance, manual lists make it difficult to denote a strong versus weak dependency and carry that forward in analysis. With these issues arising in analyzing a single outcome, how is it possible to analyze an entire curriculum of hundreds of learning outcomes?
The literature establishes the usefulness of mapping learning outcomes in a structured form and provides clues as to which structured form to use. Because we wish to analyze relationships, it is especially useful to look at network models, alternatively also referred to graph models. Courses have been linked in a curriculum through their learning outcomes in a graphbased model (Auvinen, 2011; Miller et al., 2016; Seering et al., 2015). Learning maps comprised of linked learning outcomes and activities have been created for adaptive learning (Bargel et al., 2012; Battou et al., 2011; Collins et al., 2005; Essa, 2016). Ontologies have also been created, visually linking topics, learning resources and other curriculum data in a diagramlike presentation (Bardet et al., 2008; Yudelson et al., 2015). More recently, Willcox and Huang (2017) introduced a network modeling framework for mapping educational data to leverage the unique relationshipfirst properties of graphs. Additional work referencing this network modeling approach includes graphbased visualization tools (Chen and Xue, 2018; Ghannam and Ansari, 2020; Samaranayake, 2019), curriculum development and design tools (Kaya, 2019), and adaptive learning tools (Cavanagh et al., 2019). We build upon this body of work by modeling the Common Core Mathematics Standards as a network model. To date, there has been limited research in structuring the Common Core in a network form. We emphasize the fact the Common Core Standards are presented as a list, devoid of any relationships. This is an acknowledged limitation since Standards are interrelated, and presenting them as a list loses important relationships (Daro et al., 2012; Zimba). Zimba presents the Common Core in a visual diagram with connections amongst Standards. However, as it only presents a visual diagram without an underlying network model, it is of limited analytic use. We go further by developing a structured, datadriven network model and using it to generate replicable analyses and visualizations. We chunk Standards into finergrained statements of skills mastery, dubbed “MicroStandards”, and we draw prerequisite connections between MicroStandards. In doing so, we rely on an established body of work in using experts to identify prerequisites within a hierarchy of skills (Cotton et al., 1977; White, 1974; Gagne and Paradise, 1961; Liang et al., 2017; Wang et al., 2016). By drawing prerequisite linkages between MicroStandards (finergrained skills) rather than just Standards (coarsergrained skills), we enable greater precision in relationships between statements of skills mastery (Popham, 2006; Pardos et al., 2006; Huang and Willcox, 2021). This higher level of granularity is a crucial requirement in many use cases (McCalla and Greer, 1994; Greer and McCalla, 1989; Hobbs, 1985), such as curating reusable repositories of learning content ^{1}, designing justintime interventions to address microsized learning targets (Gagne et al., 2019), intelligent tutoring systems that serve adaptive assessments to students (Huang and Willcox, 2021), etc.
In this paper, we develop a network model for the Common Core Mathematics curriculum and use it to analyze COVID19 disruptions. The next section presents the theoretical network model. We then illustrate mapping the Common Core curriculum into a network structure, including the process of discretizing Common Core Standards into MicroStandards and creating prerequisite linkages. With the resulting network map, we identify vertices and pathways of interest. We then model the Spring 2020 COVID19 school closures as a shock to the system, with specific MicroStandards initially impacted. Using graph analysis, we trace the propagating effects of the initial shock to later grades. Our analysis shows farreaching consequences of COVID19 disruptions and reveals learning pathways of interest. Finally, we discuss the analytic and predictive power obtained by our Common Core network model versus that of the classic Common Core Standard list.
2 The Network Model
A network model is a set of entities and relationships arranged in a graph structure in which entities are represented as vertices, or nodes, and relationships are represented as edges between vertices. Examples of entities include: educational institutions, departments, subjects, learning modules, topics, learning outcomes, etc. Examples of relationships include: prerequisite links between any two learning outcomes, parentchild relationships that denote categorical groupings, etc.
In the network model developed in this paper, we define the notion of a MicroStandard entity. Readers familiar with the Common Core will know that the Common Core defines “Standards”, mediumgrained statements of skills mastery. Our defined MicroStandards are more finegrained statements, derived from dividing up a Standard. For instance, Figure 1 shows a Standard that has been divided up into three MicroStandards, resulting in highly specific statements of skills mastery.
We then define a hasprerequisiteof relationship that points from one MicroStandard to the next MicroStandard. This relationship represents the notion that mastering one MicroStandard is necessary in order to master the next MicroStandard. Prerequisite relationships between Standards are implied in the Common Core Standards. For instance, in order to add, “Subtract and multiple complex numbers,” it is naturally obvious that a learner must first be able to define what a complex number is. By defining these hasprerequisiteof relationships, we make relationships explicit and designate them as firstclass objects in the network model. As discussed in Cotton et al. (1977); Collins et al. (2005), the identification of prerequisites between entities is sensitive to the granularity of the entities — the coarser the statement of learning, the more dimensions for interpretation there are as to what constitutes a prerequisite. By drawing hasprerequisiterelationships between MicroStandards, we inject more granularity and precision into the model because we can narrow in exactly on why a prerequisite linkage is justified.
We define the remaining entities in our model: Cluster, Domain and Grade Level / Band. These entities correspond to how Standards are grouped in the Common Core: a Cluster is a grouping of MicroStandards, a Domain is a grouping of Clusters, and a Grade Band is a grouping of Domains. To model such a notion of grouping, we further define a hasparentof relationship pointing from the child entity to the parent group entity. Figure 2 shows a schematic of the resulting network model.
We briefly introduce several basic concepts of graph theory that we will use to analyze the Common Core curriculum network. The indegree of a vertex is the number of incoming edges; the outdegree of a vertex is the number of outgoing edges. The Common Core network model belongs to a special class of graphs called directed acyclic graphs (DAG) in which there are no cycles in the graph. For DAGs, one can compute a topological sort of the vertices such that there is no edge going from any vertex in the sorted sequence to an earlier vertex in the sequence. Within the topological sort, we can rank vertices such that the $\mathit{rank}(v)$ of a vertex $v$ is the longest path from some source vertex $u$ to $v$.
3 Mapping the Common Core
The Common Core Mathematics Area comprises 331 Standards across ten grade bands from Kindergarten through High School. Standards are mediumgrained statements of skills mastery. From Kindergarten through Grade 8, Standards are grouped into Domains. In the High School grade band, Standards are grouped under Clusters, and Clusters are further grouped by Domains. As an example, Table 1 illustrates a set of Standards in the “Vector & Matrix Quantities” Cluster, further nested under the “Number & Quantity” Domain in the High School grade band.
Domain: Vector and Matrix Quantities 
Cluster: Represent and model with vector quantities 
A.1 Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., $v$, $v$, $v$, $v$). 
A.2 Solve problems involving velocity and other quantities that can be represented by vectors. 
A.3 Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. 
Cluster: Perform operations on vectors. 
B.4.A Add vectors endtoend, componentwise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes 
B.4.A Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum 
⋮ 

To create MicroStandards, we divide a Standard into finergrained statements of skills mastery. To do this, we determine whether a Standard contains multiple discrete skills. In the interests of preserving fidelity, this determination was largely based on grammatical clues, such as periods, semicolons separating independent clauses, numbered points, etc. In all cases, we attempted to preserve the original wording of a Standard and did not introduce new meaning when splitting it into discrete statements. For instance, in Figure 1, Standard A.1 has two complete sentences with one independent clause. We split this Standard to create three distinct MicroStandards with original wording: “Represent vector quantities as having both magnitude and direction” is a distinct skill from being able to “Represent vector quantities by directed line segments,” which is yet distinct from “Use appropriate symbols for vectors and their magnitudes.” The figure illustrates a Standard broken into three MicroStandards. Dividing up Standards in this way results in finergrained entities that drive more powerful analytics and precise analysis.
The next step in creating the network model is to draw prerequisite relationships between MicroStandards. Focusing on one grade band at a time, we review the MicroStandards within the given grade band. We determine whether a given MicroStandard is a prerequisite to another MicroStandard via a topdown decomposition with subject matter experts established in literature (Cotton et al., 1977; White, 1974; Gagne and Paradise, 1961). These subject matter experts are active researchers in the field of education and mathematics. We first identify (within a grade band) a candidate set of the most synthesizing skills — that is, the skills that build upon the most prior skill. For each MicroStandard in the candidate set, we then identify the immediate Microoutcomes within that grade band that are necessary for learning the synthesizing MicroStandard. We thus create the prerequisite relationships between the target synthesizing MicroStandard and the prerequisite MicroStandards. Next, we take the previouslyidentified prerequisite MicroStandards and in turn identify their prerequisites. Note that we draw only direct prerequisite relationships: that is, if MicroStandard A requires MicroStandard B, and MicroStandard B requires C, we draw a relationship between A and B, and a relationship between B and C, but we do not draw a relationship between A and C. This level by level decomposition is a breadthfirst traversal and gives us a tentative version of the partial dependency tree. Because this initial version was formed by one subject matter expert, we check the reasonableness of the dependencies by polling at least two other subject matter experts. Any revisions are agreed upon in consensus. In this way, we progress through all the grade bands, constructing the intragrade prerequisite relationships.
After the intragrade prerequisite relationships are constructed, we step through the grades again to draw intergrade prerequisite relationships. Starting from the most downstream grade band (i.e., the High School grade band), we identify the most fundamental MicroStandards in a given Cluster or Domain, i.e., the MicroStandards that do not have any intragrade prerequisites. We then identify any prerequisites in the previous grade band; if none can be found in the immediate preceding grade band, we step back to the next preceding grade band and begin the search again. After every grade band iteration, we again check for consensus amongst experts in the updated linkages. In this way, we step through all the grade bands and construct intergrade prerequisite relationships.
Table 2 shows the total number of mapped entities and relationships for the Common Core. Figure 3 shows a zoomedin visualization of the resulting network map of MicroStandards grouped within several Clusters and two Domains in the High School grade band.
With the resulting network map, we can analyze the curriculum for MicroStandards of interest. Table 3 shows some example graph analytics. Across all grade bands, the vertex with the highest indegree is that of MicroStandard 4.NBT.1 Recognize that in a multidigit whole number, a digit in one place represents ten times what it represents in the place to its right. That is, MicroStandard 4.NBT.1 has the highest number of adjacent followon MicroStandards in our network model of the Common Core. There are five vertices that tie for the highest outdegree (i.e., they are the MicroStandards that have the highest number of direct prerequisite MicroStandards in our network model). Table 3 lists these as MicroStandards 1.0A.6, 2.0A.2, 3.OA.7, 3.OA.9, and GCO.4 in grades 1, 2, 3, 3, and High School, respectively. This kind of analysis provides insight into the elements of the curriculum that have the potential for causing or experiencing large disruption.
Metric  MicroStandard  Grade 
Highest indegree  4.NBT.1 Recognize that in a multidigit whole number, a digit in one place represents ten times what it represents in the place to its right.  4 
Highest outdegree  1.0A.6 Add and subtract within 20; 2.0A.2 Fluently add and subtract within 20; 3.OA.7 Fluently multiply & divide within 100; 3.OA.9 Identify arithmetic patterns; GCO.4 Develop definitions of rotations, reflections, and translations  1; 2; 3; 3; 
Highest incoming rank  9 (17 vertices) 

Highest outgoing rank  9 (6 vertices) 

Finally, we conduct a topological sort of the entire Common Core Mathematics curriculum to look at learning pathways of interest. Of particular interest are learning pathways that are especially long, since these pathways may be highly vulnerable to disruption. These pathways can be found by tracing the vertices with the highest rank in both the incoming and outgoing directions. A total of 17 vertices tie for the highest outgoing rank of nine. For example, GGPE.3 Derive the equation of an ellipse given the foci in High School has a prerequisite path length of nine; Figure 4 visualizes this path. Note that in our visualization, arrows point from a more fundamental MicroStandard to a downstream one, since it is more intuitive to visualize learning flow in this direction. This is in contrast to the underlying mathematical model depicted in Figure 2, where the directed hasprerequisiteof edge in the graph points from the downstream MicroStandard to its prerequisite. Six vertices tie for the highest incoming rank of nine. For example, 2.MD.6 Represent whole numbers as lengths on a number line in Grade 2 leads to a downstream path of length nine, across four grade bands. This branching pathway is visualized in Figure 5.
4 Example Application: COVID Disruption in Massachusetts
The resulting network map represents a structured view of how learners move through the Common Core Mathematics curriculum. With this network model, we can follow learning paths, assign probabilities or weights to the edges between vertices, and replicate our analyses. As one application example, we analyze the disruptions caused by school closures on March 15, 2020 in Massachusetts. From March 15 to the end of the school year, schools were either entirely closed or had adopted online learning in Massachusetts. In our example analysis, we consider any MicroStandard scheduled to be taught during this time to have been disrupted.
For every MicroStandard that was directly impacted during this time, we assign the vertex a boolean attribute of directly_impacted = true and color that vertex red for visual illustration. For each MicroStandard that was directly impacted, we follow incident incoming edges of type hasprerequisiteof to arrive at other vertices of type MicroStandard that depend on the impacted MicroStandard. Formally, we conduct a breadthfirst search to discover the MicroStandards in order of ascending immediacy: the immediate neighbors of the initial vertex are the next MicroStandards to be disrupted; the neighbors of these next MicroStandards are further next in line, and so forth. We assign these downstream vertices a boolean attribute of indirectly_impacted=true and color them yellow. We note that our modeling approach is not limited to boolean attributes as used here; vertices can be attached different types of values such as continuous probability values, categorical values, discrete values, etc.
In one analysis, we analyze the downstream impact to sixth graders. Using the sixth grade syllabus of Cambridge Public Schools (Cambridge Public Schools, 2015), we estimated there was a total of 27 MicroStandards scheduled to be taught during the period of school closures. To show some examples of pathway analyses: Figure 6 illustrates a path of a single directlyimpacted MicroStandard, 6.G.4, colored red, located at the top of the figure. This MicroStandard leads to 6.G.4, another directlyimpacted MicroStandard, which leads to 7.G.6, a downstreamimpacted MicroStandard in the seventh grade. In this simple example, we observe how one directlyimpacted MicroStandard in the sixth grade leads to a downstream disruption of one MicroStandard in the seventh grade.
In another more complex example: Figure 7 traces the downstream path of a single directlyimpacted MicroStandard, 6.NS.8, colored in red, located at the top of the figure. 6.NS.8 has three immediate downstream MicroStandards: 6.NS.8, 6.G.8, and 7.G.4. While both 6.NS.8 and 6.G.3 were scheduled to be taught during school closures and are thus directly impacted, 7.G.4 was not scheduled to be taught during that time. 7.G.4 is in fact a MicroStandard taught in the seventh grade. 7.G.4 leads to another MicroStandard in the seventh grade, 7.G.6, which in turn leads to an eighth grade MicroStandard 8.G.8. 8.G.8 has five immediate downstream MicroStandards: GGPE.1, GGPE.3, GGPE.3, GGPE.2 and GGPE.7. These five MicroStandards are all located in the High School grade band and they lead to even more downstream MicroStandards. In this example, we observe that a single MicroStandard impacted 17 downstream MicroStandards spanning three grade bands. Our sixth grade analysis showed that from an initial 27 MicroStandards, there resulted a total of 37 downstream impacted MicroStandards, spanning a total of four grade bands. Note that because the High School grade band is counted as a single grade band, more than four grades are likely to have been impacted. All disrupted outcomes in this example are listed in Table 4.
5 Discussion
In mapping the Common Core Mathematics Standards, our process of chunking Standards and identifying linkages between the resulting MicroStandards requires some level of subjective input. In chunking the Standards, we attempted to preserve the original wording as closely as possible and used grammatical hints such as periods, independent clauses, etc. to divide up Standards. This process of dividing up Standards not only achieves improved uniformity with respect to grain size across MicroStandards, but also enables more precise relationships between MicroStandards to be drawn. Even with a panel of subject matter experts, there is unlikely to be complete agreement on all prerequisite relationships; the results presented here based on our own modeling of the relationships are intended to be illustrative. Even if the modeling approach highlights points of disagreement and/or multiple potential prerequisite paths, this in itself could be a useful outcome. Further revision of linkages between MicroStandards is an ongoing and future undertaking. We note that because we leveraged network models in which relationships are firstclass objects, it is a straightforward task to rerun analyses after entities and relationships are edited.
In drawing relationships between MicroStandards, we acknowledge that there may be missing or extraneous linkages. This is an issue that will be present for any model representing a complex dataset. However, the power of our graph model approach is such that irrelevant links can be surfaced and discarded, and missing links can be revealed when one layers in student activity data. For example, with the incorporation of student activity into the graph model, we can observe which linkages are indeed relevant or missing, and prune and add as needed. In addition, we have simplified linkages to boolean values — either an edge exists or it does not. It is straightforward to expand the model so that edges admit numerical weights to indicate the strength of the relationship between two MicroStandards (although assigning these weights will again require subjective expert input). For instance, the numerical strength of a relationship can be a result of a panel vote of experts or even an algorithmicallyderived value from application of machine learning. In our particular COVID19 application case, assigning edge weights will lead to nonboolean determinations for whether downstream MicroStandards are impacted and is an area of future work.
The mapped network form of the Common Core Mathematics curriculum yields important insights not obtainable with its classic list form. Vertices with high indegrees are important since they represent MicroStandards upon which many other MicroStandards rely. Disruption to achieving high indegree MicroStandards will lead to many failures downstream. Vertices with high outdegrees represent the MicroStandards most sensitive to disruption, as they rely on a great amount of prerequisite mastery. Also of interest are long paths: when MicroStandards require the learner to retrieve knowledge from a long time ago, there may be greater chance of failure. Long learning paths indicate that additional support may be needed, such as justintime interventions. For instance, Essa (2016) proposes an adaptive learning framework with granular learning objects that serve to surface justintime actionable insights and feedback. These observations are important for curriculum design under normal circumstances, but become critical in a crisis situation such as COVID19 when learning is widely disrupted. In this paper, we have chosen a particular grade and state to introduce the initial COVID19 shock. We emphasize that our datadriven network model enables rapid and scalable analyses under different inputs, such as choosing an earlier grade.
The graph analysis conducted in this paper is illustrative and does not represent the full capability of the network model, nor its significance for curriculum design and adaptive learning applications. There is much scope for further analysis. For instance, graph partition analysis can be useful for discovering and designing parallel tracks of study. A learner model can be superimposed over the base network map to track how individual learners progress through the curriculum. While other studies have visualized the Common Core form with linkages (Zimba), to our knowledge, this is the first study to formally construct a network model of the Common Core and unlock graphbased analysis techniques.
6 Conclusion
We present a datadriven graphbased approach for modeling the Common Core Mathematics curriculum. Our main result is that the network structure makes possible scalable analysis in tracing relationships and effects in learning paths in the Common Core Math Standards. Using COVID19 school closures in spring 2020 as an initial shock, we trace the propagating effects in the network starting in sixth grade reaching through high school. Because our approach includes first discretizing the Common Core Standards into more finegrained statements of skills mastery, we are able to identify with a higher level of precision which MicroStandards will experience disruption. We have not validated our predictions against student assessment data given ongoing COVID19 conditions, but our main result reveals vulnerable learning pathways to investigate. Validation constitutes an important area for future research. Finally, we note that in the process of validation there must be necessary revisions, and an important advantage of our network modeling approach is that our graph structure enables easy revision of vertices and edges.
Data access
We make the mapped network dataset publicly available via API access at the MIT Mapping Lab (https://mapping.mit.edu).
No. 
Outcome 
Impact Type  Grade 
1. 
[6.EE.2c] Evaluate expressions at specific values of their variables. 
Directlyimpacted  Grade 6 
2. 
[6.EE.2c] Perform arithmetic operations, including those involving whole number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). 
Directlyimpacted  Grade 6 
3. 
[6.EE.5] Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? 
Directlyimpacted  Grade 6 
4. 
[6.EE.5] Use substitution to determine whether a given number in a specified set makes an equation or inequality true. 
Directlyimpacted  Grade 6 
5. 
[6.EE.7] Solve realworld and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. 
Directlyimpacted  Grade 6 
6. 
[6.EE.8] Write an inequality of the form x > c or x < c to represent a constraint or condition in a realworld or mathematical problem. 
Directlyimpacted  Grade 6 
7. 
[6.EE.8] Recognize that inequalities of the form x > c or x < c have infinitely many solutions 
Directlyimpacted  Grade 6 
8. 
[6.EE.8] Represent solutions of inequalities x > c or x < c on number line diagrams. 
Directlyimpacted  Grade 6 
9. 
[6.EE.9] Use variables to represent two quantities in a realworld problem that change in relationship to one another 
Directlyimpacted  Grade 6 
10. 
[6.EE.9] Write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. 
Directlyimpacted  Grade 6 
11. 
[6.EE.9] Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. 
Directlyimpacted  Grade 6 
12. 
[6.G.1] Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes 
Directlyimpacted  Grade 6 
13. 
[6.G.1] Apply techniques that find the area of polygons by composing into rectangles or decomposing into triangles in the context of solving realworld and mathematical problems. 
Directlyimpacted  Grade 6 
14. 
[6.G.2] Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths 
Directlyimpacted  Grade 6 
15. 
[6.G.2] Show that the volume of a right rectangular prism with fractional edge lengths is the same as would be found by multiplying the edge lengths of the prism. 
Directlyimpacted  Grade 6 
16. 
[6.G.2] Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving realworld and mathematical problems. 
Directlyimpacted  Grade 6 
17. 
[6.G.3] Draw polygons in the coordinate plane given coordinates for the vertices 
Directlyimpacted  Grade 6 
18. 
[6.G.3] Use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate 
Directlyimpacted  Grade 6 
19. 
[6.G.3] Apply techniques of drawing on the coordinate plane and finding side lengths in the context of solving realworld and mathematical problems. 
Directlyimpacted  Grade 6 
20. 
[6.G.4] Represent threedimensional figures using nets made up of rectangles and triangles 
Directlyimpacted  Grade 6 
21. 
[6.G.4] Use the nets made up of rectangles and triangles to find the surface area of these figures. 
Directlyimpacted  Grade 6 
22. 
[6.G.4] Apply techniques using nets made up of rectangles and triangles in the context of solving realworld and mathematical problems. 
Directlyimpacted  Grade 6 
23. 
[6.NS.8] Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. 
Directlyimpacted  Grade 6 
24. 
[6.NS.8] Use coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. 
Directlyimpacted  Grade 6 
25. 
[6.SP.1] Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. 
Directlyimpacted  Grade 6 
26. 
[6.SP.2] Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. 
Directlyimpacted  Grade 6 
27. 
[6.SP.3] Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. 
Downstream impacted  Grade 6 
28. 
[6.SP.4] Display numerical data in plots on a number line, including dot plots, histograms, and box plots. 
Directlyimpacted  Grade 6 
29. 
[6.SP.5a] Summarize numerical data sets in relation to their context by reporting the number of observations. 
Downstream impacted  Grade 6 
30. 
[6.SP.5c] Summarize numerical data sets in relation to their context by giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation) 
Downstream impacted  Grade 6 
31. 
[6.SP.5c] Summarize numerical data sets by describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. 
Downstream impacted  Grade 6 
32. 
[6.SP.5d] Summarize numerical data sets in relation to their context by relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. 
Downstream impacted  Grade 6 
33. 
[7.G.1] Solve problems involving scale drawings of geometric figures. 
Downstream impacted  Grade 7 
34. 
[7.G.4] Use the formulas for the area and circumference of a circle to solve problems. 
Downstream impacted  Grade 7 
35. 
[7.G.6] Solve realworld and mathematical problems involving area of 2D objects 
Downstream impacted  Grade 7 
36. 
[7.G.6] Solve realworld and mathematical problems involving volume and surface area of 3D objects. 
Downstream impacted  Grade 7 
37. 
[7.SP.1] Understand that statistics can be used to gain information about a population by examining a sample of the population. 
Downstream impacted  Grade 7 
38. 
[7.SP.1] Understand that generalizations about a population from a sample are valid only if the sample is representative of that population. 
Downstream impacted  Grade 7 
39. 
[7.SP.1] Understand that random sampling tends to produce representative samples and support valid inferences. 
Downstream impacted  Grade 7 
40. 
[7.SP.2] Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. 
Downstream impacted  Grade 7 
41. 
[7.SP.2] Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. 
Downstream impacted  Grade 7 
42. 
[7.SP.3] Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. 
Downstream impacted  Grade 7 
43. 
[7.SP.4] Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. 
Downstream impacted  Grade 7 
44. 
[8.G.6] Explain a proof of the Pythagorean Theorem and its converse. 
Downstream impacted  Grade 8 
45. 
[8.G.7] Apply the Pythagorean Theorem to determine unknown side lengths in right triangles 
Downstream impacted  Grade 8 
46. 
[8.G.8] Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. 
Downstream impacted  Grade 8 
47. 
[GC.2] Identify and describe relationships among inscribed angles, radii, and chords 
Downstream impacted  High School 
48. 
[GC.3] Construct the inscribed and circumscribed circles of a triangle 
Downstream impacted  High School 
49. 
[GC.3] Prove properties of angles for a quadrilateral inscribed in a circle 
Downstream impacted  High School 
50. 
[GC.4] Construct a tangent line from a point outside a given circle to the circle. 
Downstream impacted  High School 
51. 
[GGPE.1] Derive the equation of a circle of given center and radius 
Downstream impacted  High School 
52. 
[GGPE.1] Complete the square to find the center and radius of a circle given by an equation. 
Downstream impacted  High School 
53. 
[GGPE.2] Derive the equation of a parabola given a focus and directrix. 
Downstream impacted  High School 
54. 
[GGPE.3] Derive the equation of hyperbola given the foci. 
Downstream impacted  High School 
55. 
[GGPE.3] Derive the equation of ellipse given the foci. 
Downstream impacted  High School 
56. 
[GGPE.4] Use coordinates to prove simple geometric theorems algebraically. 
Downstream impacted  High School 
57. 
[GGPE.5] Prove the slope criteria for parallel and perpendicular lines 
Downstream impacted  High School 
58. 
[GGPE.5] Use the slope criteria for parallel and perpendicular lines to solve geometric problems. 
Downstream impacted  High School 
59. 
[GGPE.6] Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 
Downstream impacted  High School 
60. 
[GGPE.7] Use coordinates to compute perimeters of polygons and areas of triangles and rectangles 
Downstream impacted  High School 
61. 
[GSRT.9] Derive the formula for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. 
Downstream impacted  High School 
62. 
[SID.1] Represent data with plots on the real number line (dot plots, histograms, and box plots). 
Downstream impacted  High School 
63. 
[SID.2] Use statistics appropriate to the shape of the data distribution to compare center and spread of two or more different data sets. 
Downstream impacted  High School 
64. 
[SID.3] Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). 
Downstream impacted  High School 