few_shots: Example 1:
Problem/activity:
You’re mixing ingredients for cookies. The recipe says to combine 6 tablespoons, or ¾ stick, of butter with 1 cup of sugar. You accidentally mix in a whole stick of butter (8 tablespoons) with the cup of sugar. How can you fix this?


Mathematical concept/skill options: 
A. Analyze proportional relationships and use them to solve real-world and mathematical problems.
B. Understand ratio concepts and use ratio reasoning to solve problems.


Your response:
Thought: This problem focuses on the ratio between two ingredients. Students do not have to analyze proportional relationships in-depth, which is reserved for a higher grade level, but rather use ratios to find missing values.
Answer: B

Example 2:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Mathematical concept/skill options: 
A. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
B. Reason quantitatively and use units to solve problems.
C. Apply and extend previous understandings of numbers to the system of rational numbers.
D. Represent complex numbers and their operations on the complex plane.
E. Perform arithmetic operations with complex numbers.
F. Compute fluently with multi-digit numbers and find common factors and multiples.
G. Represent and model with vector quantities.
H. Perform operations on vectors.
I. Use properties of rational and irrational numbers.
J. Use complex numbers in polynomial identities and equations.
K. Extend the properties of exponents to rational exponents.
L. Know that there are numbers that are not rational, and approximate them by rational numbers.
M. Perform operations on matrices and use matrices in applications.
N. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.


Your response:
Thought: This problem involves understanding fractions, multiplication, and division.
Answer: A

Example 3:
Problem/activity:
Problem 1
The table below contains single equations in a single variable, single equations in two variables, and pairs of simultaneous equations in two variables. Check the appropriate box to show whether there is no solution, exactly 1 solution, or if there is an infinite number of solutions.
|                                                          | **           No solution          ** | **           Exactly 1 solution          ** | **           Infinite number of solutions          ** |
|----------------------------------------------------------|--------------------------------------|---------------------------------------------|-------------------------------------------------------|
|                       $${9=4x+7}$$                       |                                      |                                             |                                                       |
|                       $${y=3y+5}$$                       |                                      |                                             |                                                       |
|                    $${y={x\over2}}$$                     |                                      |                                             |                                                       |
|                     $${3x+6y-9=0}$$                      |                                      |                                             |                                                       |
|    $${x=y}$$                        $${1.25x=1.25y}$$    |                                      |                                             |                                                       |
| $${9.9=6x+8y}$$                        $${9x=2.5y-8.8}$$ |                                      |                                             |                                                       |
| $${11x-2y=1.5}$$                        $${11x-2y=2.5}$$ |                                      |                                             |                                                       |
Problem 2
Kim has a small container and a large container as shown.
[IMG]
It takes 16 of the small containers to fill the large container. Three small containers leave 1.95 gallons of space in the big container.
What is the size of each of the two containers?


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


Your response:
Thought: This set of problems involves solving linear equations and pairs of linear equations.
Answer: L

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_000972

6

fl_problem_001097

7

fl_problem_001392

8
options: Build a function that models a relationship between two quantities.

Build new functions from existing functions.

Interpret expressions for functions in terms of the situation they model.

Model periodic phenomena with trigonometric functions.

Prove and apply trigonometric identities.

Understand the concept of a function and use function notation.

Use functions to model relationships between quantities.

Define, evaluate, and compare functions.

Interpret functions that arise in applications in terms of the context.

Construct and compare linear and exponential models and solve problems.

Extend the domain of trigonometric functions using the unit circle.

Analyze functions using different representations.
problem_activity: Optional activity
This activity, designed for an extra class period, gives students an opportunity to gather and analyze data for bounce heights for multiple balls. Each ball should be sufficiently bouncy to allow measurement of at least 4 bounces. Good examples include: tennis balls, basketballs, super balls, golf balls, and soccer balls. It will also be important to find a surface that is hard, flat, and level. Any padding will dampen the bounces, and any slant or irregularity on the surface will affect the direction of the bounce. A tiled or concrete floor, or a flat and paved surface outdoors should work. Students will need measuring tapes and may need some practice gathering the data.
Notice how students record the bounce heights. Recording these heights to the nearest inch or centimeter will already be challenging and anything beyond that is too much precision. This activity is a good opportunity to choose a degree of precision appropriate to the context and the measuring device used (MP6).
As in the previous task, monitor for how students process their data and decide on an appropriate factor to quantify the bounciness of each ball. Students should now be comfortable with the fact that the data is not exactly exponential but may still choose different ways for deciding on an appropriate exponential decay factor.
Here is a typical rebound factor for several types of balls:
Tennis ball: 0.5 to 0.6
Super ball: 0.9
Basketball: about 0.6
Making graphing or spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Arrange students in groups of 3 or 4. Give each group a measuring tape and a ball. Explain to students that their job is to determine the rebound factors of several balls by gathering data on their rebound heights. They then need to use mathematics to model the relationship between the number of bounces and the height of a bounced ball.
Students should drop them all from the same height. Consider letting students realize this on their own.
Engagement: Provide Access by Recruiting Interest.
Leverage choice around perceived challenge. Invite students to select one or two balls to bounce, instead of 3. Chunking this task into more manageable parts may also benefit students who benefit from additional processing time.
Supports accessibility for: Organization; Attention; Social-emotional skills
Student Facing
Your teacher will give your group three different kinds of balls.
Your goal is to measure the rebound heights, model the relationship between the number of bounces and the heights, and compare the bounciness of the balls.
[IMG]
Complete the table. Make sure to note which ball goes with which column.
| ** *             n,            * **           number of bounces | * **             a,            ** *           height for ball 1 (cm) | ** *             b,            * **           height for ball 2 (cm) | ** *             c,            * **           height for ball 3 (cm) |
|-----------------------------------------------------------------|----------------------------------------------------------------------|----------------------------------------------------------------------|----------------------------------------------------------------------|
|                                0                                |                                                                      |                                                                      |                                                                      |
|                                1                                |                                                                      |                                                                      |                                                                      |
|                                2                                |                                                                      |                                                                      |                                                                      |
|                                3                                |                                                                      |                                                                      |                                                                      |
|                                4                                |                                                                      |                                                                      |                                                                      |
Which one appears to be the bounciest? Which one appears to be the least bouncy? Explain your reasoning.
For each one, write an equation expressing the bounce height in terms of the bounce number
\(n\)
.
Explain how the equations could tell us which one is the most bouncy.
If the bounciest one were dropped from a height of 300 cm, what equation would model its bounce height
\(h\)
?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
If Ball 1 were dropped from a point that is twice as high, would its bounciness be greater, less, or the same? Explain your reasoning.
Ball 4 is half as bouncy as the least bouncy ball. What equation would describe its height
\(h\)
in terms of the number of bounces
\(n\)
?
Ball 5 was dropped from a height of 150 centimeters. It bounced up very slightly once or twice and then began rolling. How would you describe its rebound factor? Explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students may struggle to measure the heights of the bounces. Consider allowing phones or other technology that can record a video of the bounces so that it can be replayed in slow motion.
Activity Synthesis
Depending on available time, you may choose to have groups of students prepare a presentation for sharing their findings or simply discuss the the data and findings as a whole class.
As in the previous task, highlight different methods for estimating the rebound factor (taking the quotient of two successive values, taking an average of successive quotients, or making a general estimate of successive quotients). Also highlight the inherent inaccuracy of bounce height measurements which in turn influence how accurately we should report the successive quotients (MP6). Probably no more than one significant decimal digit should be used.
Focus the discussion on the meaning of the rebound factors. Ask questions such as:
“Does a larger factor means that the ball is more bouncy or less bouncy?”
“Does a larger factor mean that the heights are decreasing more quickly or more slowly?”
Emphasize the fact that in a situation modeled by a function
\(f\)
given by
\(f(n) = a \boldcdot b^n\)
, the number
\(b\)
expresses the growth factor. In situations where
\(0<b<1\)
, a
larger
value of
\(b\)
means that the function decays more slowly (because more of the quantity remains). In terms of comparing the bounciness of the balls, the larger the successive quotients in the table columns, the bouncier the ball is. While all of the balls eventually will lie still on the ground, a larger successive quotient (or rebound factor) means that the bounce heights decrease more slowly.
Writing, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to help students improve their writing by providing them with multiple opportunities to clarify their explanations through conversation within their group. Give students time to share their response to the question, “Explain how the equations could tell us which one is the most bouncy.” Provide listeners with prompts for feedback that will help their partner add detail to strengthen and clarify their ideas. For example, students can ask their partner, “How did you use the data in the table?” or “Can you explain the rebound factor in your equations?” Next, provide students with 3–4 minutes to revise their initial draft based on feedback from their group. This will help students produce a written generalization for how they can estimate the rebound factor and use successive quotients in a real-world exponential situation.
Design Principle(s): Optimize output (for explanation)


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