Unit 3: Lesson 1: Understanding Proportional Relationships

Unit 3: Lesson 1: Understanding Proportional Relationships

Learning Goals

• Comprehend that for the equation of a proportional relationship given by

   $y\ =\ kx,\ k$  represents the constant of proportionality.

• Create graphs and equations of proportional relationships in context, including an appropriate scale.

• Interpret diagrams or graphs of proportional relationships in context.

I Can Statements

• I can graph a proportional relationship from a story.

• I can use the constant of proportionality to compare the pace of different animals.

Open Ended

What do you notice? What do you wonder?

What You Noticed Possible Responses

• The second set of axes are not labeled.

• If the first graph is about speed, then f is twice as fast as g.

• Graph g is something going a speed of 2 centimeters every second.

• Graph f is something going a pace of about 0.25 seconds per 1 centimeter.

What You May Wonder Possible Responses

• What do the two points mean?

• Why does one graph show two lines while the other only has one?

• What does g and f represent?

• 
  

Ladybug and Ant Race Task

A ladybug and ant move at constant speeds. The diagrams with tick marks show their positions at different times. Each tick mark represents 1 centimeter.

Ladybug and Ant Race Task

• Q1: Lines u and v also show the positions of the two bugs. Which line shows the ladybug’s movement? Which line shows the ant’s movement? Explain your reasoning.

• Q2: How long does it take the ladybug to travel 12 cm? The ant?

Ladybug and Ant Race Task

• Q3: Scale the vertical and horizontal axes by labeling each grid line with a number. You will need to use the time and distance information shown in the tick-mark diagrams.

• Q4: Mark and label the point on line u and the point on line v that represent the time and position of each bug after traveling 1 cm.

  
  

Possible Responses

• Q1: Ladybug: line u ; Ant: line v

• Q2: Ladybug: 6 seconds, ant: 4 seconds

Ladybug and Ant Race Task Continued.

• Refer to the tick-mark diagrams and graph in the earlier activity when needed.

• Q1:Imagine a bug that is moving twice as fast as the ladybug. On each tick-mark diagram, mark the position of this bug.

• Q2:Plot this bug’s positions on the coordinate axes with lines  and , and connect them with a line.

• Q3:Write an equation for each of the three lines.

Possible Responses

• Q1:

• 
  

• 
  

• 
  

Possible Responses

Q2: See graph. Line n represents a bug moving double the ladybug’s distance in the same amount of time.

Q3: Possible Responses

•  $Equation\ for\ Ladybug\ y=\frac{1}{2\ }x$  

  
  

•  $Equation\ for\ Ant\ y=\ \frac{1}{3}x$  

•  $Equation\ for\ new\ bug\ y=\ \frac{1}{4}x$  ; (twice as fast as ladybug).

• Where  represents the distance traveled and  represents elapsed time.

Cool Down

• 1.On the same axes, draw a line for a third turtle that is going half as fast as the turtle described by line g.

• 2.Explain how your line shows that the turtle is going half as fast.

Cool Down Possible Responses

• 1.A line through (0,0), (1,1), (2,2) , etc.

• 2.Looking at the values for 2 seconds, turtle g moves 4 cm and the third turtle moves only 2 cm. This third turtle covers half the distance in the same amount of time.