Connecting Benchmarks/Horizontal Alignment

• MA.912.AR.2
• MA.912.AR.3.8 
• MA.912.AR.4.3 
• MA.912.AR.5.6

 

Terms from the K-12 Glossary

• Function Notation

 

Vertical Alignment

Previous Benchmarks

• MA.6.AR.1.2

Next Benchmarks

• MA.912.F.3

 

Purpose and Instructional Strategies

In middle grades, students worked with $x$-$y$ notation and substituted values in expressions and equations. In Algebra I, students work with $x$-$y$ 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 $f$($x$) reads as “$f$ of $x$” and represents an output of a function equivalent to that of the variable $y$ in $x$-$y$ notation. 
• Instruction includes a series of functions with random inputs so that students can see the pattern that emerges (MTR.5.1). 
  • For example,

       

• Students should discover that the number in parenthesis corresponds to the input or $x$-value on the graph and the number to the right of the equal sign corresponds to the output or $y$-value. 
• Although not conventional, instruction includes using function notation flexibly. 
  • For example, function notation can been seen as $h$($x$) = 4$x$ + 7 or 4$x$ + 7 = $h$($x$). 
• 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 $x$-$y$ notation, it can lead to confusion in discussions. 
  • For example, the equations $y$ = −2$x$ + 4 and $y$ = 3$x$ + 7. Representing these functions in function notation allows mathematicians to distinguish them from each other more easily (i.e., $f$($x$) = −2$x$ + 4 and $g$($x$) = 3$x$ + 7). 
• Function notation also allows for the use of different symbols for the variables, which can add meaning to the function. 
  • For example, $h$($t$) = −16$t$² + 49$t$ + 4 could be used to represent the height, $h$, of a ball in feet over time, $t$, in seconds. 
• Function notation allows mathematicians to express the output and input of a function simultaneously. 
  • For example, $h$(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, $f$($x$) does not mean $f$ · $x$. 
• Students may need additional support in the order of operations. 
  • For example, for exponential functions, many students multiply $a$ 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 before division.

 

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 functions. 
  • When evaluating $f$($x$) = 4$x$² for $x$ = −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 Tasks

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 $V$($t$) = 9000(1 + 0.07)^$t$, where $t$ is the time in years since 1984 and $V$($t$) is the value of the painting. 
  • Part A. Create a table of values that corresponds to this function. 
  • Part B. Graph the function.

 

Instructional Items

Instructional Item 1 

• Evaluate $f$(24), when $f$($x$) = $\frac{\text{3}}{\text{2}}$$x$ + 9.  

Instructional Item 2 

• Given $f$($x$) = 2$x$⁴ − 0.24 $x$²  + 6.17$x$ − 7, find $f$(2).

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.