Mathematical Connections for Benchmarking
 Math to self -- recalling an experience that relates to the problem
Math to real life—recalling or imagining a situation where you may require math
similar to the problem. Students may recall working on similar problems
For primary grades these are often the only type of connections we see, as these
young children haven’t experienced enough math to make connections within
mathematics.
 Math to other subjects—for instance, statistics in social studies, or graphing
in science
Connections within the math:
 Noticing patterns or relationships
 Noticing how certain math concepts are related (like multiplication to
repeated addition, or money to percents, percents to fractions or ratios,
probability to fractions)
 Connecting symbolic notation to pictures or diagrams, like a fraction
symbol to a fraction diagram, an area model, a graph to an equation (this is
also a representation and a communication)
 Providing a key to diagrams to help the reader understand representation
(this is also a representation and a communication)
 Finding alternative strategies to solve (this may also be reasoning and
proof)
 Connecting appropriate vocabulary to diagrams and algorithms (also
communication)
 Providing and identifying a proof (also reasoning and proof)
 Creating multiple representations (connecting a table to an equation or
graph, a fraction of a whole model to fraction of a set, drawing
manipulatives and providing written descriptions)
Extending a solution
 A true extension of a solution is when a student recognizes a mathematical
relationship, pattern, or rule that could be applied to efficiently solve the
problem and solve for any unknown. For example, two years ago there
was a problem about roofing. There was a linear relationship between the
number of shingles needed and the number of houses to be done. Many
students drew a linear graph, or a table of values where they identified a
problem. Expert problem solvers wrote the rule for the linear function, and
could then use that rule (equation) or graph to determine the number of
shingles needed for any number of houses.
 Problem posing: Many students “extend” their solution by creating a new
problem that deals with similar math. This is one type of connection but
maybe not the strongest mathematical connection, and can be contrived,
though the way the rubric is worded we did feel it was valid and
encouraged our students to do it. We recognize this as a way of making a
mathematical connection.
How many, what type, and what level of connections are appropriate depends on
the grade level and problem. Certainly problems that require sophisticated
mathematical vocabulary or diagrams and algorithms allow students to showcase
the mathematical connections they are making.
Please refer to the exemplars website and examine the criteria and rational in the
rubrics attached to the summative assessment tasks.
http://www.exemplars.com/education-materials/free-samples