options: Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.
problem_activity: Warm-up
The goal of this opening activity is to activate, through a familiar context, what students know about functions from middle school.
Students first encounter a relationship in which two quantities—the number of bagels bought and price—do not form a function. They see that for some numbers of bagels bought, there are multiple possible prices. The relationship between the number of bagels bought and the
best
price, however, do form a function, because there is only one possible best price for each number of bagels.
Students contrast the two relationships by reasoning about possible prices, completing a table of values, and observing the graphs of the relationships.
Launch
Arrange students in groups of 4. Give students a few minutes of quiet time to think about the first question, and then a couple of minutes to share their thinking with their group. Pause for a class discussion.
Invite students to share an explanation as to how each person in the situation could be right. If possible, record and display students’ reasoning for all to see. After the reasoning behind each price is shared, direct students to the table in the second question. Ask students to write down “best price” in the header of the second column and then complete the table.
Student Facing
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A customer at a bagel shop is buying 13 bagels. The shopkeeper says, “That would be $16.25.”
Jada, Priya, and Han, who are in the shop, all think it is a mistake.
Jada says to her friends, “Shouldn’t the total be \$13.25?”
Priya says, “I think it should be \$13.00.”
Han says, “No, I think it should be \$11.25.”
Explain how the shopkeeper, Jada, Priya, and Han could all be right.
Your teacher will give you instructions for completing the table.
{"table": [{"_attributes": {"border": "1", "style": "width:100%"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col", "style": "text-align:center; width:50%"}, "br": [{}], "_values": ["number", "of\u00a0bagels"]}, {"_attributes": {"scope": "col", "style": "text-align:center; width:50%"}}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {}]}, {"td": [{"_value": "2"}, {}]}, {"td": [{"_value": "3"}, {}]}, {"td": [{"_value": "4"}, {}]}, {"td": [{"_value": "5"}, {}]}, {"td": [{"_value": "6"}, {}]}, {"td": [{"_value": "7"}, {}]}, {"td": [{"_value": "8"}, {}]}, {"td": [{"_value": "9"}, {}]}, {"td": [{"_value": "10"}, {}]}, {"td": [{"_value": "11"}, {}]}, {"td": [{"_value": "12"}, {}]}, {"td": [{"_value": "13"}, {}]}]}]}]}
Student Response
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Activity Synthesis
Consider displaying a table to summarize the different possibilities for calculating the price of 13 bagels (or the prices for 6 or more bagels), as well as the table showing the best price for each number of bagels bought.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number", "of bagels"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["shopkeeper\u2019s", "price"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["Jada\u2019s", "price"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["Priya\u2019s", "price"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["Han\u2019s", "price"]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {"_value": "1.25"}, {}, {}, {}]}, {"td": [{"_value": "2"}, {"_value": "2.50"}, {}, {}, {}]}, {"td": [{"_value": "3"}, {"_value": "3.75"}, {}, {}, {}]}, {"td": [{"_value": "4"}, {"_value": "5.00"}, {}, {}, {}]}, {"td": [{"_value": "5"}, {"_value": "6.25"}, {}, {}, {}]}, {"td": [{"_value": "6"}, {"_value": "7.50"}, {"_value": "6.00"}, {}, {}]}, {"td": [{"_value": "7"}, {"_value": "8.75"}, {"_value": "7.25"}, {}, {}]}, {"td": [{"_value": "8"}, {"_value": "10.00"}, {"_value": "8.50"}, {}, {}]}, {"td": [{"_value": "9"}, {"_value": "11.25"}, {"_value": "9.75"}, {"_value": "8.00"}, {}]}, {"td": [{"_value": "10"}, {"_value": "12.50"}, {"_value": "11.00"}, {"_value": "9.25"}, {}]}, {"td": [{"_value": "11"}, {"_value": "13.75"}, {"_value": "12.25"}, {"_value": "10.50"}, {}]}, {"td": [{"_value": "12"}, {"_value": "15.00"}, {"_value": "12.00"}, {"_value": "11.75"}, {"_value": "10.00"}]}, {"td": [{"_value": "13"}, {"_value": "16.25"}, {"_value": "13.25"}, {"_value": "13.00"}, {"_value": "11.25"}]}]}]}]}
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number", "of bagels"]}, {"_attributes": {"scope": "col"}, "_value": "best price"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {"_value": "1.25"}]}, {"td": [{"_value": "2"}, {"_value": "2.50"}]}, {"td": [{"_value": "3"}, {"_value": "3.75"}]}, {"td": [{"_value": "4"}, {"_value": "5.00"}]}, {"td": [{"_value": "5"}, {"_value": "6.25"}]}, {"td": [{"_value": "6"}, {"_value": "6.00"}]}, {"td": [{"_value": "7"}, {"_value": "7.25"}]}, {"td": [{"_value": "8"}, {"_value": "8.50"}]}, {"td": [{"_value": "9"}, {"_value": "8.00"}]}, {"td": [{"_value": "10"}, {"_value": "9.25"}]}, {"td": [{"_value": "11"}, {"_value": "10.50"}]}, {"td": [{"_value": "12"}, {"_value": "10.00"}]}, {"td": [{"_value": "13"}, {"_value": "11.25"}]}]}]}]}
Ask students,
“’Number of bagels’ and ’price’ do not form a
function
, but ’number of bagels’ and ’best price’ do form a function. Why is this? What do you recall about functions?”
“If we graph the relationship between ’number of bagels’ and ’price,’ what do you think the graph would look like?”
“If we graph the relationship between ’number of bagels’ and ’best price,’ what would the graph look like?”
After students make some predictions, display the two graphs for all to see. In the first graph, solid blue dots represent the shopkeeper’s price, open green circles Jada’s price, red squares Priya’s price, and yellow triangles Han’s price. In the second graph, each X represents the best price for each number of bagel.
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Because there are multiple possible prices when the number of bagels is 6 or greater, price is not a function of the number of bagels bought.​​​​​​
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Because there is only one best price for a particular number of bagel, best price is a function of the number of bagels bought.
Emphasize that a function assigns one output to each input. Clarify that the word “function” in mathematics has a very specific meaning that does not necessarily agree with how “function” is used in everyday life (for instance, as in the sentence “The function of a bridge is to connect two sides of a river.”).


correct_option_index: 4