standard_description: Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., p^(2)). For example, by truncating the decimal expansion of Ö2, show that Ö2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
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


aligns: no