Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
Purpose and Instructional Strategies
In middle grades, students worked with
notation and substituted values in expressions and
equations. In Algebra I, students work with
notation and function notation throughout
instruction of linear, quadratic, exponential and absolute value functions. In later courses,
students will continue to use function notation with other function types and perform operations
that combine functions, including compositions of functions.
- Instruction leads students to understand that () reads as “ of ” and represents an
output of a function equivalent to that of the variable in - notation.
- Instruction includes a series of functions with random inputs so that students can see the
pattern that emerges (MTR.5.1).
- Students should discover that the number in parenthesis corresponds to the input or -value on the graph and the number to the right of the equal sign corresponds to the
output or -value.
- Although not conventional, instruction includes using function notation flexibly.
- For example, function notation can been seen as () = 4 + 7 or 4 + 7 = ().
- Instruction leads students to consider the practicality that function notation presents to
mathematicians. In several contexts, multiple functions can exist that we want to consider
simultaneously. If each of these functions is written in - notation, it can lead to
confusion in discussions.
- For example, the equations = −2 + 4 and = 3 + 7. Representing these
functions in function notation allows mathematicians to distinguish them from
each other more easily (i.e., () = −2 + 4 and () = 3 + 7).
- Function notation also allows for the use of different symbols for the variables, which can
add meaning to the function.
- For example, () = −162 + 49 + 4 could be used to represent the height, ,
of a ball in feet over time, , in seconds.
- Function notation allows mathematicians to express the output and input of a function
- For example, (3) = 7 would represent a ball 7 feet in the air after 3 seconds of
elapsed time. This is equivalent to the ordered pair (3, 7) but with the added
benefit of knowing which function it is associated with.
Common Misconceptions or Errors
- Throughout students’ prior experience, two variables written next to one another indicate
they are being multiplied. That changes in function notation and will likely cause
confusion for some of your students. Continue to discuss the meaning of function
notation with these students until they become comfortable with the understanding. In
other words, () does not mean · .
- Students may need additional support in the order of operations.
- For example, for exponential functions, many students multiply by the growth
factor and then raise the product to the value of the exponent.
- For example, students may think that the multiplication is always performed
Strategies to Support Tiered Instruction
- Instruction is provided to determine the order of operations required once a given input is
placed into the function for evaluation. Students may need additional support determining
the correct order of operations to perform.
- Teacher models using parentheses to help organize order of operations when evaluating
- When evaluating () = 42 for = −1, teacher can model the use of
parentheses by writing the expression 4(−1)2 rather than without using
parentheses writing 4 ⋅ −12 . This will help students visualize the operations.
- Teacher provides instruction for identifying the operations in various functions as they
relate to the order of operations using a graphic organizer.
Instructional Task 1 (MTR.3.1)
- The original value of a painting is $9,000 and the value increases by 7% each year. The value
of the painting can be described by the function () = 9000(1 + 0.07), where is the
time in years since 1984 and () is the value of the painting.
- Part A. Create a table of values that corresponds to this function.
- Part B. Graph the function.
Instructional Item 1
- Evaluate (24), when () = + 9.
Instructional Item 2
- Given () = 24 − 0.24 2 + 6.17 − 7, find (2).
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.