Question

do you understand? do you know how? 1. essential question what is slope? 2. reasoning how is the slope related to a unit rate? 3. look for relationships why is the slope between any two points on a straight line always the same? 4. what is the slope of the line? cost of grapes 5. the scale of a model airplane is shown in the graph. a. find the slope of the line using (\frac{y_2 - y_1}{x_2 - x_1}). b. what does the slope mean in the problem situation? model airplane

Explanation:

1. What is slope?

• # Brief Explanations:
• Slope measures the steepness of a line. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. Mathematically, for two points \((x_1,y_1)\) and \((x_2,y_2)\) on a line, slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
• # Answer:
• Slope is the ratio of the vertical - change to the horizontal change between two points on a line.

1. How is the slope related to a unit rate?

• # Brief Explanations:
• A unit rate is a rate with a denominator of 1. In the context of a line representing a proportional relationship, the slope is equal to the unit rate. For example, if a line represents the cost \(y\) of \(x\) items, the slope of the line gives the cost per one item (the unit rate).
• # Answer:
• In a proportional relationship, the slope is equal to the unit rate.

1. Why is the slope between any two points on a straight line always the same?

• # Brief Explanations:
• A straight line has a constant steepness. Since slope measures steepness, and the steepness doesn't change along a straight - line, the ratio of the vertical change to the horizontal change (the slope) between any two points on the line will be identical.
• # Answer:
• Because a straight line has a constant steepness.

1. What is the slope of the line (Cost of Grapes graph)?

• # Explanation:
• ## Step1: Identify two points on the line
• Let's take the points \((2,6)\) and \((4,10)\) from the graph.
• ## Step2: Apply the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\)
• Here \(x_1 = 2,y_1=6,x_2 = 4,y_2 = 10\). Then \(m=\frac{10 - 6}{4 - 2}=\frac{4}{2}=2\).
• # Answer:
• 2

1. a. Find the slope of the line (Model Airplane graph) using \(\frac{y_2 - y_1}{x_2 - x_1}\)

• # Explanation:
• ## Step1: Identify the two given points
• The points are \((3,5)\) and \((6,10)\).
• ## Step2: Use the slope formula
• \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{10 - 5}{6 - 3}=\frac{5}{3}\).
• # Answer:
• \(\frac{5}{3}\)

1. b. What does the slope mean in the problem situation?

• # Brief Explanations:
• In the context of the model airplane graph where \(x\) is in centimeters and \(y\) is in feet, the slope \(\frac{5}{3}\) means that for every 3 - centimeter increase in the measurement on the model (in the \(x\) - direction), the actual length of the airplane (in feet, \(y\) - direction) increases by 5 feet.
• # Answer:
• For every 3 - centimeter increase in the model's measurement, the actual length of the airplane increases by 5 feet.

Answer:

1. What is slope?

• # Brief Explanations:
• Slope measures the steepness of a line. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. Mathematically, for two points \((x_1,y_1)\) and \((x_2,y_2)\) on a line, slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
• # Answer:
• Slope is the ratio of the vertical - change to the horizontal change between two points on a line.

1. How is the slope related to a unit rate?

• # Brief Explanations:
• A unit rate is a rate with a denominator of 1. In the context of a line representing a proportional relationship, the slope is equal to the unit rate. For example, if a line represents the cost \(y\) of \(x\) items, the slope of the line gives the cost per one item (the unit rate).
• # Answer:
• In a proportional relationship, the slope is equal to the unit rate.

1. Why is the slope between any two points on a straight line always the same?

• # Brief Explanations:
• A straight line has a constant steepness. Since slope measures steepness, and the steepness doesn't change along a straight - line, the ratio of the vertical change to the horizontal change (the slope) between any two points on the line will be identical.
• # Answer:
• Because a straight line has a constant steepness.

1. What is the slope of the line (Cost of Grapes graph)?

• # Explanation:
• ## Step1: Identify two points on the line
• Let's take the points \((2,6)\) and \((4,10)\) from the graph.
• ## Step2: Apply the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\)
• Here \(x_1 = 2,y_1=6,x_2 = 4,y_2 = 10\). Then \(m=\frac{10 - 6}{4 - 2}=\frac{4}{2}=2\).
• # Answer:
• 2

1. a. Find the slope of the line (Model Airplane graph) using \(\frac{y_2 - y_1}{x_2 - x_1}\)

• # Explanation:
• ## Step1: Identify the two given points
• The points are \((3,5)\) and \((6,10)\).
• ## Step2: Use the slope formula
• \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{10 - 5}{6 - 3}=\frac{5}{3}\).
• # Answer:
• \(\frac{5}{3}\)

1. b. What does the slope mean in the problem situation?

• # Brief Explanations:
• In the context of the model airplane graph where \(x\) is in centimeters and \(y\) is in feet, the slope \(\frac{5}{3}\) means that for every 3 - centimeter increase in the measurement on the model (in the \(x\) - direction), the actual length of the airplane (in feet, \(y\) - direction) increases by 5 feet.
• # Answer:
• For every 3 - centimeter increase in the model's measurement, the actual length of the airplane increases by 5 feet.