standard_description: Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
problem_activity: Optional activity
In this activity, students consider a doubling relationship where the exponent is a variable. Monitor for students who connect this activity to the lessons on exponents, or who recognize that the quantities in this relationship are changing with respect to each other in a different manner than previous examples they have seen. Have these students share during the discussion.
Digital
Launch
Give students 5-7 minutes of quiet work time, followed by a whole-class discussion.
Receptive/Expressive Language: Peer Tutors.
Pair students with their previously identified peer tutors to aid in comprehension and expression of understanding.
Student Facing
A researcher who is studying mosquito populations collects the following data:
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "br": [{}], "_values": ["day in", "the study"], "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "9550"}, "_value": "\\((d)\\)"}]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of", "mosquitoes"], "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "9559"}, "_value": "\\((n)\\)"}]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {"_value": "2"}]}, {"td": [{"_value": "2"}, {"_value": "4"}]}, {"td": [{"_value": "3"}, {"_value": "8"}]}, {"td": [{"_value": "4"}, {"_value": "16"}]}, {"td": [{"_value": "5"}, {"_value": "32"}]}]}]}]}
The researcher said that, for these five days, the number of mosquitoes,
\(n\)
, can be found with the equation
\(n = 2^d\)
where
\(d\)
is the day in the study. Explain why this equation matches the data.
Use the ordered pairs in the table to graph the relationship between number of mosquitoes and day in the study for these five days.
Describe the graph. Compare how the data, equation, and graph illustrate the relationship between the day in the study and the number of mosquitoes.
If the pattern continues, how many mosquitoes will there be on day 6?
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
Give students 5-7 minutes of quiet work time, followed by a whole-class discussion.
Student Facing
A researcher who is studying mosquito populations collects the following data:
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "br": [{}], "_values": ["day in", "the study"], "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "9550"}, "_value": "\\((d)\\)"}]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of", "mosquitoes"], "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "9559"}, "_value": "\\((n)\\)"}]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {"_value": "2"}]}, {"td": [{"_value": "2"}, {"_value": "4"}]}, {"td": [{"_value": "3"}, {"_value": "8"}]}, {"td": [{"_value": "4"}, {"_value": "16"}]}, {"td": [{"_value": "5"}, {"_value": "32"}]}]}]}]}
The researcher said that, for these five days, the number of mosquitoes,
\(n\)
, can be found with the equation
\(n = 2^d\)
where
\(d\)
is the day in the study. Explain why this equation matches the data.
Use the ordered pairs in the table to graph the relationship between number of mosquitoes and day in the study for these five days.
[IMG]
Describe the graph. Compare how the data, equation, and graph illustrate the relationship between the day in the study and the number of mosquitoes.
If the pattern continues, how many mosquitoes will there be on day 6?
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
The purpose of this discussion is to help students make connections between the table, graph, and equation that describe this situation. Ask students to share who connected this activity to the lessons on exponents, or who recognize that the quantities in this relationship are changing with respect to each other in a different manner than previous examples they have seen.
Consider asking some of these questions:
“Which of the data, equation, or graph gives you more of a feel for how the mosquito population changes with the day in the study?”
“Think back to the lesson on exponents with the dot pattern and the Genie activity. How do those situations compare with this mosquito situation? How would tables of data, graphs, and equations compare?”
“What are some similarities and differences you noticed between all the relationships you've seen in the last few lessons?”
Speaking, Listening: MLR8 Discussion Supports.
To support whole-class discussion, display sentence frames such as, “The dot pattern is similar to/ different than the mosquito situation because ________,” and “The genie situation is similar to/ different than the mosquito situation because ________.” This routine will support rich and inclusive discussion about the similarities and differences between all of the relationships students have seen in the last few lessons.
Design Principle(s): Optimize output (for comparison); Cultivate conversation


aligns: no