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: Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = �� ÷ 3, 6 × 6 = ?.
problem_activity: Narrative
The purpose of this activity is for students to learn about
prime numbers
and
composite numbers
. Students are given a set of cards with rectangles on them. They sort the rectangles by area and then attempt to draw an additional rectangle for each category. They notice that some areas can be represented by more than one rectangle and some areas can only be represented by one rectangle.
During the synthesis, highlight that the side lengths of each rectangle represent one factor pair (each pair of side lengths should be used only once), and that the area of each rectangle represents a multiple of each side length. Students learn that a number with only one factor pair—1 and the number itself—is a prime number, and a number with more than one factor pair is a composite number.
Here is an image of the cards for reference.
[IMG]
MLR8 Discussion Supports
. Invite students to take turns selecting a rectangle, and explaining how they should sort it to their partner. Display the following sentence frames: “This rectangle belongs with _____ , because . . .” Encourage students to challenge each other when they disagree.
Advances: Conversing, Representing
Engagement: Develop Effort and Persistence.
Chunk this task into manageable parts to support organizational skills in problem solving. Some students may benefit from explicit guidance for how to begin. For example, before sorting, students can find the area of each rectangle.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing, Organization
Required Materials
Materials to Gather
Grid paper
Inch tiles
Materials to Copy
Card Sort: Area
Required Preparation
Create a set of cards from the blackline master for each group of 2.
Launch
Groups of 2
Give each group a set of cards from the blackline master.
“Sort the cards into categories in any way that makes sense to you.”
2 minutes: partner work time
Ask students to share ways in which they sorted.
Activity
“If you did not already, sort the rectangles by their area.”
3–5 minutes: partner work time
Ask students to check their work with another group to make sure the cards in each category match.
“Now, create at least one rectangle to add to each category in your card sort.”
3–5 minutes: partner work time
Observe the rectangles students add to each category. Monitor for students who notice that no new rectangles could be drawn for the area of 7 square units.
Student Facing
Your teacher will give you a set of cards to sort.
Sort the cards by area. Record your sorting results. Be prepared to explain your choices.
For each group of sorted cards, think of at least one more rectangle. Name its length and width. Be prepared to explain your reasoning.
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Select 2–3 students to share the rectangles they added to each category.
“Why were you able to create more rectangles for some areas and not others?” (Some of the numbers had more factor pairs. For some numbers, there was only one possible factor pair.)
Revoice student reasoning. “Only one rectangle can be made for the area of 7. Numbers like this are called
prime numbers
. Prime numbers have only one factor pair: 1 and itself.”
“Numbers like 15 that have more than one factor pair are called
composite numbers
.”
“What other composite numbers did you work with? How do you know they are composite?” (Twenty-four is a composite number because I can make 2 rows of 12 or 4 rows of 6. Eighteen is composite because it has factor pairs of 2 and 9 and 3 and 6.)


aligns: no