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: Understand congruence and similarity using physical models, transparencies, or geometry software.

Define trigonometric ratios and solve problems involving right triangles.

Visualize relationships between two-dimensional and three-dimensional objects.

Apply trigonometry to general triangles.

Translate between the geometric description and the equation for a conic section.

Understand similarity in terms of similarity transformations.

Understand and apply the Pythagorean Theorem.

Use coordinates to prove simple geometric theorems algebraically.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Experiment with transformations in the plane.

Analyze, compare, create, and compose shapes.

Draw, construct, and describe geometrical figures and describe the relationships between them.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Apply geometric concepts in modeling situations.

Prove theorems involving similarity.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Find arc lengths and areas of sectors of circles.

Solve real-world and mathematical problems involving area, surface area, and volume.

Explain volume formulas and use them to solve problems.

Classify two-dimensional figures into categories based on their properties.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Understand and apply theorems about circles.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Understand congruence in terms of rigid motions.

Prove geometric theorems.

Make geometric constructions.

Reason with shapes and their attributes.
problem_activity: Activity
This activity introduces students to graphic scales. Students interpret them, use them to find actual measurements (heights of tall buildings), and express them non-graphically. In earlier lessons, students used markings on an index card or sheet of paper to measure a drawing and create scaled copies. Here, they use non-standard measuring tools again to solve problems (MP5). Students make markings on an unmarked straightedge to measure scaled lengths on a drawing, and then use the given scale to determine actual lengths. This measuring strategy builds on students’ work with measurement in grades 1 and 2 (i.e., placing multiple copies of a shorter object end to end and expressing lengths in terms of the number of objects).
As students work, encourage them to be as precise as possible in making their marks and in estimating lengths that are less than 1 scale-segment long. Most distances students are measuring here are not line segments but rather distances between a point (the tip of a building) and a line (the ground).
Monitor the different ways students reason about scaled and actual distances. Here are two likely approaches for finding the difference between the Burj Khalifa and the Eiffel Tower (the second question):
Find the actual height of each tower and then find their difference.
Find the difference in scaled heights, and then use the scale to find the difference in actual heights.
Select students who use these approaches to share during the whole-class discussion in this sequence.
Launch
Display the scale drawing of the structures. Before beginning the work of the task, students may be interested in or eager to share the locations of the structures. Consider taking a few minutes to elicit what they know, or display a world map showing the locations of the structures.
Ask students what the segment labeled with “0 m” and “100 m” might mean. Some students are likely to say that it also conveys a scale. Verify that a scale can indeed be communicated graphically; an actual distance is not represented by a numerical measurement, but rather, by the length of the segment.
Provide access to index cards or sheets of paper students can use to measure. Tell students to check their answers to the first question with a partner. Tell them to discuss as necessary until they reach an agreement before proceeding to work on the rest of the problems. Give students 4–5 minutes of quiet work time and partner discussion.
Student Facing
Here is a scale drawing of some of the world’s tallest structures.
[IMG]
About how tall is the actual Willis Tower? About how tall is the actual Great Pyramid? Be prepared to explain your reasoning.
About how much taller is the Burj Khalifa than the Eiffel Tower? Explain or show your reasoning.
Measure the line segment that shows the scale to the nearest tenth of a centimeter. Express the scale of the drawing using numbers and words.
Student Response
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Student Facing
Are you ready for more?
The tallest mountain in the United States, Mount Denali in Alaska, is about 6,190 m tall. If this mountain were shown on the scale drawing, how would its height compare to the heights of the structures? Explain or show your reasoning.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students may not measure heights of the buildings at a right angle from the ground line. Remind students that heights are to be measured perpendicular to the ground or base line.
If needed, demonstrate how to use the edge of a sheet of paper or an index card to measure lengths on a scale drawing with a graphic scale.
Activity Synthesis
Poll the class for their answers to the first question. Ask what might be some sources of discrepancies. The main issue here is measurement error, but there are also different methods to make the measurements and estimate the heights of the buildings. Two methods to estimate the heights of the buildings (and their limitations) are:
Estimate how many times “taller” each building is compared to the line segment giving the scale. Then multiply this number by 100 m. The accuracy here is not very good unless, for example, the height of the building is very close to being a whole number times the length of the scale.
Measure the segment giving the scale (in centimeters, for example), and then express the scale using centimeters (for example, 0.7 cm represents 100 m). Then measure each building and use the scale to find the actual height. Estimating and rounding will be necessary when measuring the scale and when measuring the buildings.
Ask previously selected and sequenced students to highlight the different approaches for comparing the heights of the Burj Khalifa and the Eiffel Tower. The main difference between the two strategies is the order of arithmetic. Taking the difference of the actual building heights means multiplying scaled heights first and then subtracting. Taking the difference of the scaled heights and then applying the scale factor means subtracting scaled heights first and then multiplying by the scale factor.
If time permits, discuss the scale drawing of the towers more broadly. Consider asking:
Besides height information, what other information about the towers does the drawing show? (Widths of the buildings and their overall shapes.)
What information does it
not
show? (The depth of each building, any projections or protrusions, shapes of different parts of the buildings, etc.)
How is this scale drawing the same as that of the basketball court? (They both show information on a single plane and are drawn using a scale.) How are they different? (The basketball court is a flat surface, like the drawing of the court. The drawing of the towers is a side view or front view; the actual objects represented are not actually flat objects.)
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. For each response or observation that is shared, ask students to restate and/or revoice what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This will provide more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making


correct_option_index: 11