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?


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_001392

8
options: Use complex numbers in polynomial identities and equations.

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Apply and extend previous understandings of numbers to the system of rational numbers.

Extend the properties of exponents to rational exponents.

Compute fluently with multi-digit numbers and find common factors and multiples.

Perform operations on matrices and use matrices in applications.

Represent complex numbers and their operations on the complex plane.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Use properties of rational and irrational numbers.

Perform arithmetic operations with complex numbers.

Represent and model with vector quantities.

Reason quantitatively and use units to solve problems.

Perform operations on vectors.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
problem_activity: Optional activity
In this activity, students use symbols and diagrams to find a sum that requires regrouping of base-ten units. Use this activity to give students more explicit instruction on how to bundle smaller units into a larger one and additional practice on using addition algorithm to add decimals.
Again, consider having physical base-ten blocks, a paper version of the base-ten figures, or this digital applet
ggbm.at/n9yaWPQj
available as alternatives to diagram drawing, or to more concretely illustrate the idea of bundling and unbundling.
Digital
Launch
Arrange students in groups of 2. Give groups 4–5 minutes to discuss and answer the first question, and then 7–8 minutes of quiet time to complete the remaining questions. Provide access to base-ten representations, if needed.
[IMG]
Classes using the digital activities have an interactive applet with virtual blocks. Note that the applet can only represent ones, tenths, and hundredths, so the pieces are large enough to manipulate. To use the bundling and unbundling features, the pieces must be aligned on the light blue grids. To bring a piece into the workspace, select one of the green tool icons and then click on the workspace. To move it, you must click on the Move tool
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organization and problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention
Student Facing
Here are two ways to calculate the value of
\(0.26 + 0.07\)
. In the diagram, each rectangle represents 0.1 and each square represents 0.01.
[IMG]
Use what you know about base-ten units and addition of base-ten numbers to explain:
Why ten squares can be “bundled” into a rectangle.
How this “bundling” is reflected in the computation.
The applet has tools that create each of the base-ten blocks. Select a Block tool, and then click on the screen to place it.
[IMG]
One
[IMG]
Tenth
[IMG]
Hundredth
Click on the Move tool when you are done choosing blocks.
[IMG]
Find the value of
\(0.38 + 0.69\)
by drawing a diagram. Can you find the sum without bundling? Would it be useful to bundle some pieces? Explain your reasoning.
Calculate
\(0.38 + 0.69\)
. Check your calculation against your diagram in the previous question.
Find each sum. The larger square represents 1, the rectangle represents 0.1, and the smaller square represents 0.01.
[IMG]
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Arrange students in groups of 2. Give groups 4–5 minutes to discuss and answer the first question, and then 7–8 minutes of quiet time to complete the remaining questions. Provide access to base-ten representations, if needed.
[IMG]
Classes using the digital activities have an interactive applet with virtual blocks. Note that the applet can only represent ones, tenths, and hundredths, so the pieces are large enough to manipulate. To use the bundling and unbundling features, the pieces must be aligned on the light blue grids. To bring a piece into the workspace, select one of the green tool icons and then click on the workspace. To move it, you must click on the Move tool
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organization and problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention
Student Facing
Here are two ways to calculate the value of
\(0.26 + 0.07\)
. In the diagram, each rectangle represents 0.1 and each square represents 0.01.
[IMG]
Use what you know about base-ten units and addition to explain:
Why ten squares can be “bundled” into a rectangle.
How this “bundling” is represented in the vertical calculation.
Find the value of
\(0.38 + 0.69\)
by drawing a diagram. Can you find the sum without bundling? Would it be useful to bundle some pieces? Explain your reasoning.
Calculate
\(0.38 + 0.69\)
. Check your calculation against your diagram in the previous question.
Find each sum. The larger square represents 1.
[IMG]
[IMG]
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?
A distant, magical land uses jewels for their bartering system. The jewels are valued and ranked in order of their rarity. Each jewel is worth 3 times the jewel immediately below it in the ranking. The ranking is red, orange, yellow, green, blue, indigo, and violet. So a red jewel is worth 3 orange jewels, a green jewel is worth 3 blue jewels, and so on.
If you had 500 violet jewels and wanted to trade so that you carried as few jewels as possible, which jewels would you have?
Suppose you have 1 orange jewel, 2 yellow jewels, and 1 indigo jewel. If you’re given 2 green jewels and 1 yellow jewels, what is the fewest number of jewels that could represent the value of the jewels you have?
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
If students have difficulty drawing the diagrams to represent bundling, it might be helpful for them to work with actual base-ten blocks, paper printouts of base-ten blocks, or digital base-ten blocks so that they can physically trade 10 hundredths for 1 tenth or 10 tenths for 1 whole.
Activity Synthesis
Focus the whole-class debriefing on the idea of choosing appropriate tools to solve a problem, which is an important part of doing mathematics (MP5). Highlight how drawings can effectively help us understand what is happening when we add base-ten numbers before moving on to a more generalized method. Discuss:
“In which place(s) did bundling happen when adding 0.38 and 0.69?” (In the hundredth and tenth places.) “Why?” (There is a total of 17 hundredths, and 10 hundredths can be bundled to make 1 tenth. This 1 tenth is added to the 3 tenths and 6 tenths, which makes 10 tenths. Ten tenths can be bundled into 1 one.)
“How can the bundling process be represented in vertical calculations?” (We can show that the 8 hundredths and 9 hundredths make 1 tenth and 7 hundredths by recording 7 hundredths and writing a 1 above the 3 tenths in 0.38.)
“Which method of calculating is more efficient?” (It depends on the complexity and size of the numbers. The drawings become hard when there are lots of digits or when the digits are large. The algorithm works well in all cases, but it is more abstract and requires that all bundling be recorded in the right places.)
Listening, Speaking, Representing: MLR7 Compare and Connect.
Use this routine when students discuss different ways to calculate and represent decimal sums. Display three different representations for calculating the sum of 6.03 and 0.098 (using base-ten blocks, diagrams, and vertically). Ask students to identify where the bundling occurred and how it is shown in each representation. Draw students’ attention to how the action of bundling is represented. For instance, in the base-ten blocks the pieces are physically joined or traded, in the diagram the hundredths are circled, and in the vertical calculation the “1” is notated. Emphasize the mathematical language used to make sense of the different ways to represent bundling. These exchanges strengthen students' mathematical language use and reasoning.
Design Principle(s): Support sense-making; Maximize meta-awareness


correct_option_index: 4