options: Understand the connections between proportional relationships, lines, and linear equations.

Represent and solve problems involving addition and subtraction.

Work with addition and subtraction equations.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Solve equations and inequalities in one variable.

Work with radicals and integer exponents.

Interpret the structure of expressions.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Represent and analyze quantitative relationships between dependent and independent variables.

Solve systems of equations.

Perform arithmetic operations on polynomials.

Use properties of operations to generate equivalent expressions.

Understand the relationship between zeros and factors of polynomials.

Use the four operations with whole numbers to solve problems.

Rewrite rational expressions.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Write and interpret numerical expressions.

Analyze patterns and relationships.

Gain familiarity with factors and multiples.

Generate and analyze patterns.

Work with equal groups of objects to gain foundations for multiplication.

Represent and solve equations and inequalities graphically.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Multiply and divide within 100.

Represent and solve problems involving multiplication and division.

Understand and apply properties of operations and the relationship between addition and subtraction.

Create equations that describe numbers or relationships

Use polynomial identities to solve problems.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Reason about and solve one-variable equations and inequalities.

Add and subtract within 20.

Write expressions in equivalent forms to solve problems.

Understand solving equations as a process of reasoning and explain the reasoning.

Understand properties of multiplication and the relationship between multiplication and division.
problem_activity: Activity
Having just seen an example of the meaning of
\(a\)
and
\(b\)
in an exponential expression
\(a \boldcdot b^x\)
, students now focus on interpreting these numbers using graphs. They graph the equations from the previous task, noticing that
\(a\)
is the vertical intercept of the graph while the number
\(b\)
determines how quickly the graph grows (since in these cases
\(b>1\)
). A larger value of
\(b\)
corresponds to a more rapid rate of growth for the bacteria population. The axes for the graphs have been labeled here, but in future activities, students will have to think strategically about how to label the axes to most effectively plot the points.
Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Display the equations from the previous activity for all to see (if they are not already visible).
Conversing: MLR8 Discussion Supports.
Use this routine to help students describe the meaning of variables in exponential expressions. Arrange students in groups of 2. Invite Partner A to share their observations for the first graph and Partner B to share their observations for the second graph. Provide sentence frames such as “I can connect the equation to the graph by….” and “I notice that….” Invite the listener to press for additional details referring to features of the graphs, such as the vertical intercept, specific points, and the steepness of the curve. This will help students justify how features of the graph can be used to identify matching equations.
Design Principle(s): Support sense-making; Cultivate conversation
Student Facing
Refer back to your work in the table of the previous task. Use that information and the given coordinate planes to graph the following:
a. Graph
\((t,n)\)
when
\(t\)
is 0, 1, 2, 3, and 4.
[IMG]
b. Graph
\((t,p)\)
when
\(t\)
is 0, 1, 2, 3, and 4. (If you get stuck, you can create a table.)
[IMG]
On the graph of
\(n\)
, where can you see each number that appears in the equation?
On the graph of
\(p\)
, where can you see each number that appears in the equation?
Student Response
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for free access to Student Response.
Anticipated Misconceptions
Students may have trouble graphing the points, particularly finding the appropriate vertical (
\(n\)
or
\(p\)
) values. Ask them to find the coordinates of the grid points on the vertical axis and use that to estimate the vertical position of their points.
When calculating values by hand, many students may mistakenly write an expression like
\(100 \boldcdot 3^2\)
as
\(300^2\)
. Remind them that the expression
\(100 \boldcdot 3^2\)
means
\(100 \boldcdot 3 \boldcdot 3\)
.
Activity Synthesis
Make sure students recognize two key takeaways from this activity:
The vertical intercept of the graph is the initial bacteria population size. It is the size of the population when first measured, or when
\(t\)
, the number of hours since measurement, is 0.
The growth factor of the populations is represented by how quickly the graph increases.


correct_option_index: 26