Describing and Graphing Situations

15 min

Narrative

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 of 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 Task

<p>Trays of bagels at a bagel shop.</p>

<p>Fresh bagels sign. Prices listed. 1 bagel, 1 dollar 25 cents. 6 bagels, 6 dollars. 9 bagels, 8 dollars. 12 bagels, 10 dollars.</p>

Your teacher will give you instructions for completing the table.

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.

number of bagels BESTPRICE\phantom{BEST PRICE}
1
2
3
4
5
6
7
8
9
10
11
12
13

Sample Response

Sample response:

  • The shopkeeper multiplied the price of 1 bagel by 13, which gives $16.25.
  • Jada multiplied the price of 6 bagels by 2 to get the price of 12 bagels, which is $12.00, and then added the price of 1 bagel, which is $1.25. Her total is $13.25.
  • Priya started with the price of 9 bagels, which is $8.00, and added the price 4 individual bagels, which is 4(1.25)4(1.25), or $5.00. Her total is $13.00.
  • Han started with the price of 12 bagels, which is $10.00, and added the price of 1 bagel, which is $1.25. His total is $11.25.
number
of bagels		 best price
1 1.25
2 2.50
3 3.75
4 5.00
5 6.25
6 6.00
7 7.25
8 8.50
9 8.00
10 9.25
11 10.50
12 10.00
13 11.25
Activity Synthesis (Teacher Notes)

Consider displaying a table to summarize the different possibilities for calculating the price of 13 bagels (or the prices for 6 or more bagels) and a table showing the best price for each number of bagels bought.

number
of bagels		 shopkeeper’s
price		 Jada’s
price		 Priya’s
price		 Han’s
price
1 1.25
2 2.50
3 3.75
4 5.00
5 6.25
6 7.50 6.00
7 8.75 7.25
8 10.00 8.50
9 11.25 9.75 8.00
10 12.50 11.00 9.25
11 13.75 12.25 10.50
12 15.00 12.00 11.75 10.00
13 16.25 13.25 13.00 11.25
number
of bagels		 best price
1 1.25
2 2.50
3 3.75
4 5.00
5 6.25
6 6.00
7 7.25
8 8.50
9 8.00
10 9.25
11 10.50
12 10.00
13 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 represent Jada’s price, red squares represent Priya’s price, and yellow triangles represent Han’s price. In the second graph, each X represents the best price for each number of bagels.

<p>Discrete graph of bagel prices. number of bagels and price in dollars.</p>
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.​​​​​​

<p>Discrete function graph. number of bagels and best price in dollars.</p>
Because there is only one best price for a particular number of bagels, 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”).

Standards
Building On
  • 8.F.1·Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
  • 8.F.A.1·<p>Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. <span>Function notation is not required in Grade 8.</span></p>
Addressing
  • F-IF.1·Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
  • F-IF.1·Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
  • F-IF.1·Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
  • HSF-IF.A.1·Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If <span class="math">\(f\)</span> is a function and <span class="math">\(x\)</span> is an element of its domain, then <span class="math">\(f(x)\)</span> denotes the output of <span class="math">\(f\)</span> corresponding to the input <span class="math">\(x\)</span>. The graph of <span class="math">\(f\)</span> is the graph of the equation <span class="math">\(y = f(x)\)</span>.

10 min

10 min