few_shots: Example:
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
$$-\frac{1}{4}$$


Standard options: 
A. Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
B. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
C. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.


Your response, in json format:
{
  "explanation": "This problem involves understanding fractions and division to divide rational numbers, such as 1, -4, 4, and -1. This problem also involves an understanding of multiplication: if p and q are integers, then -(p/q) = (-p)/q = p/(-q).",
  "answer": "B"
}

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_001097

7
options: Use square root and cube root symbols to represent solutions to equations of the form x^(2) = p and x^(3) = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that Ö2 is irrational.

Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3^(2) × 3^(–5) = 3^(–3) = 1/3^(3) = 1/27.

Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.
problem_activity: Activity
In this activity, students continue to develop their understanding of square roots. Students first find the areas of three squares, estimate the side lengths using tracing paper, and then write the exact side lengths. Then they make a table of side-area pairs. They then graph the ordered pairs from the table and use the graph to estimate the values of squares with non-integer side lengths, such as the ones they drew in the previous activity.
Digital
Launch
Arrange students in groups of 2. Provide access to calculators. Remind students about the meaning and use of square root notation. Have students work together on the first problem and check each other’s work. Have them make their graphs independently and then check with their partners that they look the same. Follow with a whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts. Check in with students after the first 2-3 minutes of work time. Check to make sure students have attended to all parts of the original figures.
Supports accessibility for: Organization; Attention
Student Facing
Find the area of each square and estimate the side lengths using your geometry toolkit. Then write the exact lengths for the sides of each square.
[IMG]
Complete the tables with the missing side lengths and areas.
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}, {"_attributes": {"scope": "col"}, "_value": "side length,", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "833"}, "_value": "\\(s\\)"}]}], "td": [{"_value": "0.5"}, {}, {"_value": "1.5"}, {}, {"_value": "2.5"}, {}, {"_value": "3.5"}, {}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "row 1"}, {"_attributes": {"scope": "col"}, "_value": "area,", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "81"}, "_value": "\\(a\\)"}]}], "td": [{}, {"_value": "1"}, {}, {"_value": "4"}, {}, {"_value": "9"}, {}, {"_value": "16"}]}]}]}]}
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}, {"_attributes": {"scope": "col"}, "_value": "side length,", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "833"}, "_value": "\\(s\\)"}]}], "td": [{"_value": "4.5"}, {}, {"_value": "5.5"}, {}, {"_value": "6.5"}, {}, {"_value": "7.5"}, {}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "row 1"}, {"_attributes": {"scope": "col"}, "_value": "area,", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "81"}, "_value": "\\(a\\)"}]}], "td": [{}, {"_value": "25"}, {}, {"_value": "36"}, {}, {"_value": "49"}, {}, {"_value": "64"}]}]}]}]}
Plot the points,
\((s, a)\)
, on the coordinate plane shown here.
Use this graph to estimate the side lengths of the squares in the first question. How do your estimates from the graph compare to the estimates you made initially using your geometry toolkit?
Use the graph to approximate
\(\sqrt{45}\)
.
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. Provide access to calculators. Remind students about the meaning and use of square root notation. Have students work together on the first problem and check each other’s work. Have them make their graphs independently and then check with their partners that they look the same. Follow with a whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts. Check in with students after the first 2-3 minutes of work time. Check to make sure students have attended to all parts of the original figures.
Supports accessibility for: Organization; Attention
Student Facing
Find the area of each square and estimate the side lengths using your geometry toolkit. Then write the exact lengths for the sides of each square.
[IMG]
Complete the tables with the missing side lengths and areas.
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "side length,", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "833"}, "_value": "\\(s\\)"}]}], "td": [{"_value": "0.5"}, {}, {"_value": "1.5"}, {}, {"_value": "2.5"}, {}, {"_value": "3.5"}, {}]}, {"th": [{"_attributes": {"scope": "col"}, "_value": "area,", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "81"}, "_value": "\\(a\\)"}]}], "td": [{}, {"_value": "1"}, {}, {"_value": "4"}, {}, {"_value": "9"}, {}, {"_value": "16"}]}]}]}]}
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "side length,", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "833"}, "_value": "\\(s\\)"}]}], "td": [{"_value": "4.5"}, {}, {"_value": "5.5"}, {}, {"_value": "6.5"}, {}, {"_value": "7.5"}, {}]}, {"th": [{"_attributes": {"scope": "col"}, "_value": "area,", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "81"}, "_value": "\\(a\\)"}]}], "td": [{}, {"_value": "25"}, {}, {"_value": "36"}, {}, {"_value": "49"}, {}, {"_value": "64"}]}]}]}]}
Plot the points,
\((s, a)\)
, on the coordinate plane shown here.
[IMG]
Use this graph to estimate the side lengths of the squares in the first question. How do your estimates from the graph compare to the estimates you made initially using your geometry toolkit?
Use the graph to approximate
\(\sqrt{45}\)
.
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
Invite students to share their graphs from the second problem and display the graph for all to see. Ask: “What relationship does the graph display?” (The relationship between the side length and area of a square.) Select 2–3 students to share their answers to the last problem, including how they used the graph to estimate
\(s\)
for the tilted square they drew earlier.
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. After a student shares their response to the last problem, ask students to restate and/or revoice what they heard using 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.
Design Principle(s): Support sense-making


correct_option_index: 0