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Section 2.1 Representations: Graphs, Data, and Algebra through Children's Literature

Children's literature is a rich source for mathematics connections. Though the stories are intended for children, the underlying mathematics is more advanced. In this lesson, we will read a story and see how it can be represented through graphs, tables, and algebra.

Subsection 2.1.1 Video: Talking About Math

In How to Learn Math for Students, Lesson 4, watch Video 4: Talking About Math. How can you benefit by actively working in groups both in and outside of class? Think about how well you and your group members currently interact. What ideas do you have about how to help your group work together more productively? Share your ideas with your group.

Student Page 2.1.2 The Eyes Have It

Because variables are so important in our work with algebra, it is helpful to further develop our facility with their use. In this activity, you will study the relationship between the numbers of monsters and the corresponding numbers of eyes (each monster has only two eyes) through tables, graphs, English sentences, and algebraic equations.

Read Horns to Toes and In Between by Sandra Boynton. (You can find an amusing version of the book on YouTube‚ÄČ17‚ÄČ.)

How many eyes do 3 monsters have? 4 monsters? 7 monsters? All the monsters in the book? Any number of monsters (with 2 eyes each)?

1.

Record the number of eyes based on the number of people in Table 2.1.2.1.

Table 2.1.2.1. Monsters/Eyes Table for Student Page Exercise 2.1.2.1
Number of Monsters, \(M\) 1 2 3 4 5 6 \(M\)
Total Number of Eyes, \(E\)             \(E =\)

2.

Graph the data in Table 2.1.2.1 using Figure 2.1.2.2. Label the horizontal axis, Number of Monsters, and the vertical axis, Total Number of Eyes.

Figure 2.1.2.2. Blank Graph for Plotting Monsters and Eyes
(a)

Starting at 0 where the two axes meet, number the tick marks on the horizontal axis increasing by 1 each time for each dark vertical line (every other tick mark).

(b)

Choose a reasonable scale for the vertical axis. Explain why the scale is reasonable.

(c)

Describe the shape of the graph.

(d)

Why do you think the graph has the shape it does?

3.

Describe the relationship between the number of monsters and the total number of eyes the monsters have. Here is one way to phase into the use of variables:

The number of Eyes is two times the number of Monsters.

Number of Eyes = 2 √ó number of Monsters

E = 2 √ó M

E = 2M

5.

Graph number of people versus total number of the chosen body part on the axes above in Figure 2.1.2.2. What do you notice?

Student Page 2.1.3 Graphing with Desmos

Read the following instructions. Open Desmos on a computer, a smart phone, or an electronic tablet. Use the directions to plot the data from The Eyes Have It. What questions do you have? Share your graph with your group.

1. Desmos.

(a)

Google ‚ÄúDesmos.‚ÄĚ

(b)

Choose Desmos Graphing Calculator.

(c)

Use with an Internet connection on a computer or download the app onto your smart phone or tablet free of charge.

2. Plotting Points with Desmos.

(a)

Click on the add item menu, ‚Äú+‚ÄĚ (upper left). Choose table.

(b)

Enter the data from your table into Desmos.

(c)

Notice that points are being plotting as you type them.

(d)

You might not be able to see all of your points on the graph.

3. Setting the Graphing Window With Desmos.

(a)

Click on the wrench icon in the upper right corner.

(b)

For the independent variable \(x\text{,}\) enter the low and high values for your data.

(c)

Choose a reasonable step size. The step is the interval between tick marks on the graph.

(d)

Include a title for the \(x\)-axis.

(e)

Repeat for the dependent variable, \(y\text{.}\)

4. Grpahing an Equation.

(a)

On the left side of the screen, type in the equation you want to plot in the next entry line.

(b)

It will automatically appear on the graphing screen.

(c)

Troubleshoot and change the equation if it doesn't fit the data.

Student Page 2.1.4 The Eyes Have It, Extended

What household items do Americans likely have in quantity? Find statistics about the average number of something each household owns. Repeat the activity in The Eyes Have It problems 1 through 3, for the item you are interested in. Create a graph with Desmos to represent the data. Share your work on this problem with your group.

1.

How did you decide what scale to use?

2.

How did you find the equation to plot?

3.

What questions do you have?

Homework 2.1.5 Homework

2.

Revisit The Eyes Have It. Using the data in Student Page Exercise 2.1.2.4 for the Human Body Parts your group chose, create a table listing the number of body parts for 2, 4, 6, 8, and 10 people. Plot the data with Desmos. Choose appropriate scales for both \(x\) and \(y\) if the scales chosen by Desmos do not show all of the data. Find an equation to fit the data. Print out the graph and data or be prepared to share them electronically.

3.

If plotting points on a coordinate grid is unfamiliar to you, or you need to dust off the skill, return to Exercise 2.1.5.2.

(a)

Compare the graph with the data in Table 2.1.5.1. Which number is plotted along the horizontal axis? Which number is plotted along the vertical axis?

Table 2.1.5.1. Data Table for Exercise 2.1.5.3
\(x\) \(y\)
\(4\) \(-3\)
\(-2\) \(5\)
\(0\) \(-1\)
\(-5\) \(-4\)
(b)

Create a new table on Desmos using the values at right. Answer Task 2.1.5.3.a again.

(c)

Predict the location of several more points. Enter the points into the table in Task 2.1.5.3.b to see if you are correct.

4.

Movie tickets cost $8.50 each.

Table 2.1.5.2. Movie Ticket Data
Number of tickets Cost of tickets
\(0\)
\(1\) \(8.50\)
\(2\)
\(3\)
\(4\)
\(10\)
\(15\)
\(n\)
Figure 2.1.5.3. Blank Graph for Movie Tickets
(a)

Complete Table 2.1.5.2 to show the cost of the numbers of movie tickets listed if each ticket costs $8.50.

(b)

Plot the data on the graph at right. Label accurate scales for both \(x\) and \(y\) so that you use most of the graph. Write titles on each axis.

(c)

Describe the relationship between the number of tickets purchased and the cost of that number of tickets.

5.

Gasoline prices sometimes vary greatly in a day. On January 28, 2015, gasoline prices in the Muskegon, MI area ranged from $1.879 to $2.199 per gallon.

(a)

Complete Table 2.1.5.4 to find how much you would pay for the numbers of gallons of gas listed for each gasoline price. Round prices to the nearest penny (hundredth of a dollar).

Table 2.1.5.4. Table of Gas Prices
Number of Gallons
of Gas Purchased, \(g\)
Price of Gas, \(P\)
$1.879 per gallon $2.199 per gallon
1
2
3
4
(b)

Accurately plot the data. Label scales and titles for each axis.

(c)

Write equations in the last row of the table that fit the data. Tell how you know your equations are correct.

(d)

How many gallons of gasoline can you purchase for $10? Show your work for one of the gasoline prices, accurate to 2 decimal places.

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