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.


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.


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.


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: Reason quantitatively and use units to solve problems.

Extend the properties of exponents to rational exponents.

Perform operations on matrices and use matrices in applications.

Represent complex numbers and their operations on the complex plane.

Use complex numbers in polynomial identities and equations.

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

Perform operations on vectors.

Use properties of rational and irrational numbers.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

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.

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

Represent and model with vector quantities.

Perform arithmetic operations with complex numbers.
problem_activity: Optional activity
Here students continue to use diagrams to represent sums of decimals, but they also transition to writing addition calculations vertically. They think about the alignment of the digits in vertical calculations to help ensure that correct values are combined. This activity is optional, so students have the option to spend more time subtracting decimals in the next activity.
As in the previous activity, consider having students use physical base-ten blocks (if available), a paper version of the base-ten figures (from the blackline master), or this digital applet
ggbm.at/FXEZD466
, as alternatives to drawing diagrams.
Digital
Launch
Remind students that the term “sum” means the result of an addition. Refer to the image from the previous activity that shows how several squares and rectangles were used to represent base-ten units. Tell students to use the same representations in this activity and to keep in mind that in the process of bundling they find more sums of decimals.
Arrange students in groups of 2. Give students 10 minutes of quiet work time, but encourage them to briefly discuss their responses with their partner after completing the second question and before continuing with the rest. Follow with a whole-class discussion.
Classes using the digital activities have an interactive applet with virtual blocks. In this activity, students must redefine the value of each block to represent the place values in each problem. 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 (the arrow).
[IMG]
Action and Expression: Develop Expression and Communication.
Use virtual or concrete manipulatives to connect symbols to concrete objects or values. Encourage students to begin with physical representations before drawing a diagram. Provide access to physical or virtual base-ten blocks to support drawing diagrams.
Supports accessibility for: Conceptual processing
Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to give students an opportunity to refine their explanation about which calculation is correct for
\(0.2 + 0.05\)
. At the appropriate time, give students time to meet with 2–3 partners to share their response. Display prompts students can use to provide feedback such as, “How did the alignment of the decimals change the sum?”, “Why is aligning decimal places important here?” and "A detail (or word) you could add is _____, because . . . ." As students discuss, listen to how they talk about combining units, aligning digits of the same place values, and whether it is necessary to add zeros. Give students 1–2 minutes to revise their writing based on their conversations.​
Design Principle(s): Optimize output (for justification)
Student Facing
Andre and Jada drew base-ten diagrams to represent
\(0.007 + 0.004\)
. Andre drew 11 small rectangles. Jada drew only two figures: a square and a small rectangle.
[IMG]
If both students represented the sum correctly, what value does each small rectangle represent? What value does each square represent?
Draw or describe a diagram that could represent the sum
\(0.008 + 0.07\)
.
Here are two calculations of
\(0.2 + 0.05\)
. Which is correct? Explain why one is correct and the other is incorrect.
[IMG]
Compute each sum. If you get stuck, consider drawing base-ten diagrams to help you.
[IMG]
\(0.209 + 0.01\)
\(10.2 + 1.1456\)
The applet has tools that create each of the base-ten blocks. This time you need to decide the value of each block before you begin.
Select a Block tool, and then click on the screen to place it.
Click on the Move tool (the arrow) when you are done choosing blocks.
Subtract by deleting with the delete tool (the trash can), not by crossing out.
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
Remind students that the term “sum” means the result of an addition. Refer to the image from the previous activity that shows how several squares and rectangles were used to represent base-ten units. Tell students to use the same representations in this activity and to keep in mind that in the process of bundling they find more sums of decimals.
Arrange students in groups of 2. Give students 10 minutes of quiet work time, but encourage them to briefly discuss their responses with their partner after completing the second question and before continuing with the rest. Follow with a whole-class discussion.
[IMG]
Classes using the digital activities have an interactive applet with virtual blocks. In this activity, students must redefine the value of each block to represent the place values in each problem. 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: Develop Expression and Communication.
Use virtual or concrete manipulatives to connect symbols to concrete objects or values. Encourage students to begin with physical representations before drawing a diagram. Provide access to physical or virtual base-ten blocks to support drawing diagrams.
Supports accessibility for: Conceptual processing
Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to give students an opportunity to refine their explanation about which calculation is correct for
\(0.2 + 0.05\)
. At the appropriate time, give students time to meet with 2–3 partners to share their response. Display prompts students can use to provide feedback such as, “How did the alignment of the decimals change the sum?”, “Why is aligning decimal places important here?” and "A detail (or word) you could add is _____, because . . . ." As students discuss, listen to how they talk about combining units, aligning digits of the same place values, and whether it is necessary to add zeros. Give students 1–2 minutes to revise their writing based on their conversations.​
Design Principle(s): Optimize output (for justification)
Student Facing
Andre and Jada drew base-ten diagrams to represent
\(0.007 + 0.004\)
. Andre drew 11 small rectangles. Jada drew only two figures: a square and a small rectangle.
[IMG]
If both students represented the sum correctly, what value does each small rectangle represent? What value does each square represent?
Draw or describe a diagram that could represent the sum
\(0.008 + 0.07\)
.
Here are two calculations of
\(0.2 + 0.05\)
. Which is correct? Explain why one is correct and the other is incorrect.
[IMG]
Compute each sum. If you get stuck, consider drawing base-ten diagrams to help you.
[IMG]
\(0.209 + 0.01\)
\(10.2 + 1.1456\)
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
The goal of this discussion is to help students understand that vertical calculation is an efficient way to find the sums of decimals. Discuss:
“When finding
\(0.008 + 0.07\)
, why do we not combine the 8 thousandths and 7 hundredths to make 15?” (Hundredths and thousandths are different units. If each hundredth is unbundled into 10 thousandths, we can add 70 thousandths and 8 thousandths to get 78 thousandths).
“How do we use representations of base-ten numbers to add effectively and efficiently?” (Make sure to put together tenths with tenths, hundredths with hundredths, etc. Also, make sure that if a large square represents
\(\frac{1}{10}\)
for one summand, it also represents
\(\frac{1}{10}\)
for the other.)
“When adding numbers without using base-ten diagrams or other representations, what can we do to help add them correctly?” (Pay close attention to place value so we combine only like units. It is helpful to line up the digits of the numbers so that numerals that represent the same place value are placed directly on top of one another.)
Additionally, consider using color coding to help students visualize the place-value structure, as shown here.
[IMG]


correct_option_index: 5