few_shots: 
Example:
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?


Concept/skill: 
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Answer: 
No. Completing neither of these problems requires replacing one equation by the sum of that equation and a multiple of the other equation.

Now, does the following problem or activity enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001392_neg_0

8
standard_description: Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
problem_activity: Activity
In this activity, students compare logarithmic expressions as they play a game of “war” in groups of 2. Students receive cards containing logarithms in bases 2, 5, and 10. The numbers in the expressions are written in all forms, including decimals, fractions, whole numbers, and exponential expressions. To compare the logarithms on the cards, they may:
determine if an expression is negative, zero, or positive, without finding the exact or estimated value
estimate the value of an expression such as
\(\log_2 10\)
by using logarithms with integer values
convert decimals to fractions to help find the logarithms
As students preview the deck by putting the cards in order from least to greatest, listen for how they decide on relative values of the cards. Students should not use calculators at this point.
Launch
Arrange students in groups of 2. The first 20 cards in the blackline master show log expressions with integer values. That last 10, which are labeled “challenge,” have non-integer values. Give each group only the first 20, the entire set of 30, or differentiate by giving some groups only the first 20 and some groups all of the cards.
Before students start playing, ask them to turn all the cards face up, and work with their partner to put them in order from least to greatest. The game will go more smoothly if students have a chance to gain familiarity with the contents of the card deck and evaluate each card in a lower-stakes interaction. This step may be skipped if students are comfortable determining the value of logarithmic expressions.
Speaking: MLR8 Discussion Supports.
To help students discuss the order of the cards with their partner, provide sentence frames such as: “I know _____ because . . .”, “Why did you . . . ?”, and “How do you know . . . ?”
Design Principle(s): Optimize output (for justification); Cultivate conversation
Engagement: Provide Access by Recruiting Interest.
Begin with a small-group or whole-class demonstration of how to play the game. Check for understanding by inviting students to rephrase directions in their own words. To scaffold the activity to allow for self-correction and multiple attempts, invite students to discard any cards that they may hesitate on into a separate, designated pile. Build in a few minutes for students to look through these cards after their first attempt. Then ask students to reshuffle and make a second attempt, noticing what strategies helped them to more quickly work through the more challenging cards.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Have you played the game of war with a deck of playing cards?
Your teacher will give you and your partner a set of special cards.
Shuffle and deal the cards evenly.
Each player turns one card face up. The card with the greater value wins the round and its player captures both cards and sets them aside.
If you and your partner disagree about the value of a card, discuss until you reach an agreement.
If there is a tie, each player turns another card face up. The player whose card has the greater value captures all cards (including the cards that tie).
Play until all the cards are turned up. The player with the most cards wins.
Let the logarithm war begin!
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?
Mai uses the fact that
\(\sqrt[3]{2}\)
is close to 1.25 and that
\(2^3=8\)
to make the estimate
\(\log_2(10) \approx 3\frac{1}{3}\)
. Explain Mai’s estimate. How exact is the estimate?
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 making estimates to compare expressions. Ask them questions such as "Is the expression greater than or less than 0? How do you know? Is the expression greater than or less than 1? How do you know?" Make sure they understand that they only need to decide which is bigger; they do not need to calculate exact values.
Activity Synthesis
Much of the discussions will happen in groups, as students put the cards in order before playing the game and while playing the game. Invite groups to share any insightful or useful strategies for comparing any two cards that came up before or during the game.


aligns: no