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
The purpose of the task is for students to compare signed numbers in a real-world context and then use inequality signs accurately with negative numbers (MP2). The context should help students understand “less than” or “greater than” language. Students evaluate and critique another's reasoning (MP3).
Launch
Allow students 5–6 minutes quiet work time followed by whole-class discussion.
Representation: Internalize Comprehension.
Activate or supply background knowledge. Some students may benefit from access to partially completed or blank number lines. Consider preparing blank number lines with just the tick marks to get students started.
Supports accessibility for: Visual-spatial processing; Organization
Student Facing
Here are the low temperatures, in degrees Celsius, for a week in Anchorage, Alaska.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "_value": "day"}, {"_attributes": {"scope": "col"}, "_value": "Mon"}, {"_attributes": {"scope": "col"}, "_value": "Tues"}, {"_attributes": {"scope": "col"}, "_value": "Weds"}, {"_attributes": {"scope": "col"}, "_value": "Thurs"}, {"_attributes": {"scope": "col"}, "_value": "Fri"}, {"_attributes": {"scope": "col"}, "_value": "Sat"}, {"_attributes": {"scope": "col"}, "_value": "Sun"}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "_value": "temperature"}], "td": [{"_value": "5"}, {"_value": "-1"}, {"_value": "-5.5"}, {"_value": "-2"}, {"_value": "3"}, {"_value": "4"}, {"_value": "0"}]}]}]}]}
Plot the temperatures on a number line. Which day of the week had the lowest low temperature?
The lowest temperature ever recorded in the United States was -62 degrees Celsius, in Prospect Creek Camp, Alaska. The average temperature on Mars is about -55 degrees Celsius.
Which is warmer, the coldest temperature recorded in the USA, or the average temperature on Mars? Explain how you know.
Write an inequality to show your answer.
On a winter day the low temperature in Anchorage, Alaska, was -21 degrees Celsius and the low temperature in Minneapolis, Minnesota, was -14 degrees Celsius.
Jada said, “I know that 14 is less than 21, so -14 is also less than -21. This means that it was colder in Minneapolis than in Anchorage.”
Do you agree? Explain your reasoning.
Student Response
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Student Facing
Are you ready for more?
Another temperature scale frequently used in science is the
Kelvin scale
. In this scale, 0 is the lowest possible temperature of anything in the universe, and it is -273.15 degrees in the Celsius scale. Each
\(1\text{ K}\)
is the same as
\(1^\circ\text{C}\)
, so
\(10\text{ K}\)
is the same as
\(\text-263.15^\circ\text{C}\)
.
Water boils at
\(100^\circ \text{C}\)
. What is this temperature in
\(\text{K}\)
?
Ammonia boils at
\(\text-35.5^\circ \text{C}\)
. What is the boiling point of ammonia in
\(\text{K}\)
?
Explain why only positive numbers (and 0) are needed to record temperature in
\(\text{K}\)
.
Student Response
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Anticipated Misconceptions
Some students may have difficulty comparing numbers on the negative side of the number line. Have students plot the numbers on a number line in order to sequence them from least to greatest. Make it evident that temperatures become warmer (i.e., greater) as they move from left to right on the number line.
Activity Synthesis
Ask students to explain how they could tell which of the two negative numbers was greater in problem 2. After gathering 1 or 2 responses, display the following image for all to see:
[IMG]
Explain to students that as we go to the right on the number line, we can think of the temperature as getting hotter, and as we go to the left, we can think of the temperature as getting colder. Numbers don’t always describe temperature, though. So we use the word “greater” to describe a number that is farther to the right, and “less” to describe numbers that are farther to the left. For example, we would write
\(6 > \text-50\)
and say “6 is greater than -50 because it is farther to the right on the number line.” Equivalently, we could write
\(\text-50 < 6\)
and say “-50 is less than 6 because -50 is farther to the left on the number line.”
Ask students to make up their own list of negative numbers and plot on a number line. Then ask them to write several inequality statements with their numbers. As time allows, have students share one inequality statement with the class to see if all agree that it is true. If time allows, have students swap their list with a partner to plot and check their partner’s number lines and inequality statements.
Speaking: MLR7 Compare and Connect.
Use this routine to support students’ understanding of both the language and the symbolic notation of inequalities. Ask students to consider what is the same and what is different between the following sentences: “6 degrees is warmer than -50 degrees,” “6 is greater than -50,” and “
\(6 > \text-50\)
.” This helps students make the connection between the language of the context, the written inequality statement and the symbolic inequality statement.
Design Principle(s): Optimize output (for comparison); Maximize meta-awareness


aligns: no