few_shots: Example:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
[IMG]
Counters, 10 per pair of students
Decks of number cards 0-5 (with 4 of each number) or two 0-5 dice
Setup
This activity is done in pairs. The students sit side by side with 10-frame and deck of cards or dice. If they have cards, the cards are shuffled and placed face down in front of them.
Action
Player 1 flips a card or rolls the die. They place that many counters on the 10-frame. Player 2 flips a card or rolls the die. They place that many more counters on the 10-frame next to the counters already on the ten frame. The students count the total together.
A recording sheet can be added when the students have been introduced to equations and number sentences. For those students who can handle higher numbers, a deck of 0-10 cards and two 10-frames can be used.
This game can also be played alone.


Mathematical concept/skill options: 
A. Write and interpret numerical expressions.
B. Generate and analyze patterns.
C. Analyze patterns and relationships.
D. Use properties of operations to generate equivalent expressions.
E. Use the four operations with whole numbers to solve problems.
F. Work with radicals and integer exponents.
G. Solve systems of equations.
H. Represent and solve problems involving multiplication and division.
I. Perform arithmetic operations on polynomials.
J. Understand the relationship between zeros and factors of polynomials.
K. Understand properties of multiplication and the relationship between multiplication and division.
L. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
M. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
N. Use polynomial identities to solve problems.
O. Apply and extend previous understandings of arithmetic to algebraic expressions.
P. Rewrite rational expressions.
Q. Interpret the structure of expressions.
R. Add and subtract within 20.
S. Analyze and solve linear equations and pairs of simultaneous linear equations.
T. Represent and analyze quantitative relationships between dependent and independent variables.
U. Solve equations and inequalities in one variable.
V. Create equations that describe numbers or relationships
W. Reason about and solve one-variable equations and inequalities.
X. Represent and solve equations and inequalities graphically.
Y. Understand solving equations as a process of reasoning and explain the reasoning.
Z. Understand and apply properties of operations and the relationship between addition and subtraction.
AA. Gain familiarity with factors and multiples.
AB. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
AC. Work with addition and subtraction equations.
AD. Multiply and divide within 100.
AE. Work with equal groups of objects to gain foundations for multiplication.
AF. Represent and solve problems involving addition and subtraction.
AG. Write expressions in equivalent forms to solve problems.
AH. Understand the connections between proportional relationships, lines, and linear equations.


Your response:
Thought: This task teaches students to understand addition as putting together counters, or adding to already placed counters.
Answer: AB

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_task_001477

K
options: Write and interpret numerical expressions.

Reason about and solve one-variable equations and inequalities.

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

Represent and analyze quantitative relationships between dependent and independent variables.

Represent and solve problems involving addition and subtraction.

Work with radicals and integer exponents.

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

Interpret the structure of expressions.

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

Apply and extend previous understandings of arithmetic to algebraic expressions.

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

Analyze patterns and relationships.

Use polynomial identities to solve problems.

Use the four operations with whole numbers to solve problems.

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

Rewrite rational expressions.

Write expressions in equivalent forms to solve problems.

Add and subtract within 20.

Solve equations and inequalities in one variable.

Understand properties of multiplication and the relationship between multiplication and division.

Perform arithmetic operations on polynomials.

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

Use properties of operations to generate equivalent expressions.

Work with addition and subtraction equations.

Generate and analyze patterns.

Solve systems of equations.

Multiply and divide within 100.

Represent and solve equations and inequalities graphically.

Gain familiarity with factors and multiples.

Understand the relationship between zeros and factors of polynomials.

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

Represent and solve problems involving multiplication and division.

Create equations that describe numbers or relationships

Work with equal groups of objects to gain foundations for multiplication.
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
Teachers with a valid work email address can
click here to register or sign in
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: 32