standard_description: Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.
problem_activity: Activity
Positive and negative numbers are often used to represent
changes
in a quantity. An increase in the quantity is positive, and a decrease in the quantity is negative. In this activity, students see an example of this convention and are asked to make sense of it in the given context.
Launch
Arrange students in groups of 2. Give students 30 seconds of quiet work time followed by 1 minute of partner discussion for the first two problems. Briefly, ensure everyone agrees on the interpretation of positive and negative numbers in this context, and then invite students to finish the rest of the questions individually. Follow with whole-class discussion.
Representation: Develop Language and Symbols.
Use virtual or concrete manipulatives to connect symbols to concrete objects or values. For example, demonstrate “change” by adding or taking away phones in a whole-class discussion.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Speaking, Representing: MLR8 Discussion Supports.
To support small-group discussion, provide sentence frames such as: “The change between Monday and Tuesday is ___ because....”. Some students may see the change in the column labeled “change” while others may note that on Monday, there were 18 phones, and on Tuesday, there are only 16. This will help students use the table representation to reason about the inventory.
Design Principle(s): Support sense-making
Student Facing
A store tracks the number of cell phones it has in stock and how many phones it sells.
The table shows the inventory for one phone model at the beginning of each day last week. The inventory changes when they sell phones or get shipments of phones into the store.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}}, {"_attributes": {"scope": "col"}, "_value": "inventory"}, {"_attributes": {"scope": "col"}, "_value": "change"}]}]}], "tbody": [{"tr": [{"th": [{"_value": "Monday"}], "td": [{"_value": "18"}, {"_value": "-2"}]}, {"th": [{"_value": "Tuesday"}], "td": [{"_value": "16"}, {"_value": "-5"}]}, {"th": [{"_value": "Wednesday"}], "td": [{"_value": "11"}, {"_value": "-7"}]}, {"th": [{"_value": "Thursday"}], "td": [{"_value": "4"}, {"_value": "-6"}]}, {"th": [{"_value": "Friday"}], "td": [{"_value": "-2"}, {"_value": "20"}]}]}]}]}
What do you think it means when the change is positive? Negative?
What do you think it means when the inventory is positive? Negative?
Based on the information in the table, what do you think the inventory will be at on Saturday morning? Explain your reasoning.
What is the difference between the greatest inventory and the least inventory?
Student Response
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Activity Synthesis
Tell students that we often use positive and negative to represent changes in a quantity. Typically, an increase in the quantity is positive, and a decrease in the quantity is negative.
Ask students what they answered for the second question and record their responses. Highlight one or two that describe the situation clearly.
Ask a few students to share their answer to the third question, and discuss any differences. Then discuss the answer to the last question.


aligns: no