The other day a teacher asked me to create a lesson that had elements of a Number Talk involving percents.  Her students had recently been discussing proportional reasoning and the constant of proportionality. Knowing this, I decided to combine percents and proportional reasoning with Desmos.  Here’s the resulting lesson.

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Part A:  Percents and the Double Number Line

1. Draw the line  (Steps 2 – 7 are designed to be the low entry piece) 
2. Fill in the symbols (% and $)
3. Me: “Whisper to your elbow partner what I added to the number line”
4. Write in 100% and $50
5. Me:  “Discuss with your elbow partner the meaning of the 100 and 50.
6. Write in 0% and $0
7.  Me:  ” Discuss with your elbow partner the meaning of the 0 and 0.”
8. Me:  “What percent do you think I’m going to add to the double number line next?”
   1. Typically, students will say 50% – which is the percent I’m looking for.  If a student offers a different percent, I say, “Thanks, I’m going to come back to that one.” (see step 13)
9. Me:  Write the dollar value that 50% of $50 represents.
10. Call on student to give answer and strategy.
11. Repeat for steps 9 and 10 for 25%
12. Repeat step 9 for 75%.  Make sure to give ample “think time”
    1. Strategies:  Students came up with 4 different strategies for getting $37.50.  The process of discussing the different strategies was my number talk element.
       1. 75/2 = $37.50
       2. You can add 25% and 50% to get 75%, therefore I added 12.50 and 25 to get $37.50
       3. 25 • 3 = 75, therefore 12.50 • 3 = $37.50
       4. 100% – 25% = 75%, therefore 50 – 12.50 = $37.50
13. Depending on time, ask students to find 10%, 5%, 1% 16%  etc…  This is when you can address the guess from step 8

Part B:  Desmos & The Constant of Proportionality

1. Go to link:  bit.ly/PercentDesmos
2. Ask students to finish filling in the x/y chart.
3. Discuss the fact the information forms a diagonal line.
4. Ask students to add more information to the chart.  i.e. Enter in the dollar amount for a percent greater than 100.
5. If the student information is correct, then the point will fall “in line”.  If not, then the point will distort the line. (see pic)
   

   View of graph with an incorrect x/y chart entry

6. Ask the students to move the red line so it completely covers the purple line and then acknowledge the constant of proportionality (k value)
7. Group question:  Why is the constant of proportionality 2?
   1. This questioned garnered a range of responses!
      1. b/c we divided by 2   (25/2 = 12.50)
      2. b/c we multiplied by 2  (50 •  2 = 100)
      3. When we multiply the x column by 2 we get the y column
      4. When you divide the percent by 2 you get the x column
      5. 2x = y
      6. 75/37.5 = 2
      7. y/x = 2
      8. y/2 = x
   2. Testing the equations
      1. At this point, I asked students to enter 2x = y into Desmos and asked them to describe what happened.
      2. Student Responses:
         1. The line changed color,
         2. The line got longer
         3. The line now goes into infinity.
      3. This led students to understand that the equation represents the pattern in the x/y chart and that all 3 equations shared are equivalent.
      4. 
      5. When entering y/x = 2, the line changed color again.  This time green.
      6. 
      7. 1. When entering y/2 = x, the line changed color again.  This time red.
         2. 

The lesson met the teachers expectations!!  She intends to revisit this process using a different dollar amount, such as $40 in place of $50.

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• My blog post on Number Talks:  A Touch of Number Sense
• For other activities and posts involving Desmos, click here