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:
{
  "explanation": "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:
{
  "explanation": "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.
{"table": [{"_attributes": {"align": "center", "class": ["table", "table-bordered"], "style": "table-layout:fixed; width:70%"}, "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "No solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Exactly 1 solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Infinite number of solutions"}]}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9=4x+7}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y=3y+5}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y={x\\over2}}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${3x+6y-9=0}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${x=y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${1.25x=1.25y}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9.9=6x+8y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9x=2.5y-8.8}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=1.5}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=2.5}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {}]}]}]}]}
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:
{
  "explanation": "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: Apply geometric concepts in modeling situations.

Apply trigonometry to general triangles.

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

Define trigonometric ratios and solve problems involving right triangles.

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

Prove theorems involving similarity.

Find arc lengths and areas of sectors of circles.

Analyze, compare, create, and compose shapes.

Understand similarity in terms of similarity transformations.

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

Understand and apply the Pythagorean Theorem.

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

Understand and apply theorems about circles.

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

Use coordinates to prove simple geometric theorems algebraically.

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

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

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

Understand congruence in terms of rigid motions.

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

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

Experiment with transformations in the plane.

Reason with shapes and their attributes.

Prove geometric theorems.

Make geometric constructions.

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

Explain volume formulas and use them to solve problems.
problem_activity: Activity
In this activity, students begin to build their own table of trigonometric ratios (ratios of side lengths in right triangles, by angle measure of one acute angle). For now, call this the right triangle table so students remember these ratios only apply to right triangles. Building these ratios from the similar triangles they are based on helps students build connections between similar triangles and trigonometry.
Measuring several examples of triangles with the same angle measurements is essential in both the digital and paper versions of this activity. In the paper version, it provides more data points to get a more accurate ratio for the given angle. In both versions, it helps students understand that all right triangles with their angle as one of the angles will be similar and thus have the same ratios.
In the next activity and the cool-down of this lesson, students will look for patterns in the right triangle table, and also apply the ratios in the right triangle table to find unknown angle measures given a ratio of side lengths. Do not use the word trigonometry or any of the names of the ratios yet.
Digital
Launch
Provide access to devices that can run GeoGebra. Students will be working with the applet at
ggbm.at/bfxeb6cr
.
Arrange students in groups of 2-4. Invite students to create a spreadsheet to organize their data and perform calculations. Students can access a GeoGebra spreadsheet in the math tool kit or at
geogebra.org/spreadsheet
. For students needing extra support, consider providing this spreadsheet for them to use:
ggbm.at/jvvqrngb
.
Assign each group a pair of complementary angles. If students did the optional lesson choose to skip 30 and 60 or to give them to a group that needs extra practice.
Explain to students that the computer measurement is precise to at least the ten-thousandths place, but the thousandths place is sufficient for our needs.
Representation: Internalize Comprehension.
Use color coding and annotations to highlight connections between representations in a problem. For example color code the side adjacent to angle
\(A\)
, the side opposite angle
\(A\)
, and the hypotenuse.
Supports accessibility for: Visual-spatial processing.
Student Facing
Your teacher will give you some angles.
Use the applet to build 4 different right triangles for each of your angles.
Record the side lengths of each of the triangles.
Compute these 3 quotients for the acute angles in each triangle:
The length of the leg adjacent to your angle divided by the length of the hypotenuse
The length of the leg opposite from your angle divided by the length of the hypotenuse
The length of the leg opposite from your angle divided by the length of the leg adjacent to your angle
Find the mean of each type of quotient.
What do you notice? What do you wonder?
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 3-4. Give each group a copy of the blackline master to complete.
Assign each group a pair of complementary angles. If students did the optional lesson choose to skip 30 and 60 or to give them to a group that needs extra practice.
Discuss rounding with students before they begin. Ask students whether it makes sense to measure the short side of a triangle as about 5.2 cm and the hypotenuse as pretty close to 7.1 cm and then report that the ratio of the short side to the hypotenuse is exactly 0.73239437. (No, that's too many digits to report.)
Explain to students that the rule of thumb used by scientists is that if your measurements have two digits then your calculated answer should have two digits as well. For example, if your tools measure to the tenths place, say 5.2 cm, so that your measurements have one digit you’re confident in - the 5 in the one’s place, and one digit you may have had to estimate a bit - the 2 in the tenths place, then your answer should be rounded to two digits - 0.73. In the case where the number starts with a zero we count digits starting at the first non-zero digit, so in this case we will round to the hundredths place.
As students finish their measurements, have them find the mean of each column and record that on the class Right Triangle Table.
Check students’ mean data before they enter it to ensure that it is close to the actual values:
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "angle"}, {"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math"], "data-png-file-id": "4258"}, "_value": "\\(\\div\\)"}], "_values": ["adjacent leg", "hypotenuse"]}, {"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math"], "data-png-file-id": "4258"}, "_value": "\\(\\div\\)"}], "_values": ["opposite leg", "hypotenuse"]}, {"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math"], "data-png-file-id": "4258"}, "_value": "\\(\\div\\)"}], "_values": ["opposite leg", "adjacent leg"]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "10 degrees"}, {"_value": "0.985"}, {"_value": "0.174"}, {"_value": "0.176"}]}, {"td": [{"_value": "20 degrees"}, {"_value": "0.940"}, {"_value": "0.342"}, {"_value": "0.364"}]}, {"td": [{"_value": "30 degrees"}, {"_value": "0.866"}, {"_value": "0.500"}, {"_value": "0.577"}]}, {"td": [{"_value": "40 degrees"}, {"_value": "0.766"}, {"_value": "0.643"}, {"_value": "0.839"}]}, {"td": [{"_value": "50 degrees"}, {"_value": "0.643"}, {"_value": "0.766"}, {"_value": "1.192"}]}, {"td": [{"_value": "60 degrees"}, {"_value": "0.500"}, {"_value": "0.866"}, {"_value": "1.732"}]}, {"td": [{"_value": "70 degrees"}, {"_value": "0.342"}, {"_value": "0.940"}, {"_value": "2.747"}]}, {"td": [{"_value": "80 degrees"}, {"_value": "0.174"}, {"_value": "0.985"}, {"_value": "5.671"}]}]}]}]}
This table is needed for the next activity, so it should be recorded to display at that time.
Representation: Internalize Comprehension.
Use color coding and annotations to highlight connections between representations in a problem. For example color code the side adjacent to angle
\(A\)
, the side opposite angle
\(A\)
, and the hypotenuse.
Supports accessibility for: Visual-spatial processing.
Student Facing
Your teacher will give you some angles.
Draw several right triangles using the angles you receive.
Precisely measure the side lengths of the triangles.
Complete the tables by computing 3 quotients for the acute angles in each triangle:
The length of the leg adjacent to your angle divided by the length of the hypotenuse
The length of the leg opposite from your angle divided by the length of the hypotenuse
The length of the leg opposite from your angle divided by the length of the leg adjacent to your angle
Find the mean of each column in your table.
What do you notice about your table? What do you wonder about your table?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Some students might struggle to identify opposite and adjacent legs. Prompt students to highlight the angle in question to accurately identify sides. If you point to “your angle” then the leg you are touching is adjacent and the leg you are not touching is opposite. The hypotenuse is always the longest side.
[IMG]
Activity Synthesis
Invite groups to share what they noticed and wondered about their data. (Each ratio is about the same for the angle. The ratio of the opposite side to the hypotenuse for angle
\(A\)
is the same as the ratio of the adjacent side to the hypotenuse for angle
\(B\)
.)
Speaking: MLR8 Discussion Supports.
Give students additional time to make sure that everyone in their group can explain what they noticed and wondered about the relationships between the quantities represented in the table. Invite groups to rehearse what they will say when they share with the whole class. Rehearsing provides students with additional opportunities to speak and clarify their thinking, and will improve the quality of explanations shared during the whole-class discussion.
Design Principle(s): Support sense-making; Cultivate conversation


correct_option_index: 3