Lesson 11

The purpose of this warm-up is for students to discuss the patterns they see in points plotted on a coordinate grid, which will be useful when students graph ordered pairs consisting of corresponding terms from two patterns in a later activity. While students may notice and wonder many things about this image, the location of the points and their coordinates are the important discussion points.

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses. 

• “How can we use coordinates to describe the location of each point?” (The point \(D\) is at \((8,4)\) since its horizontal coordinate is 8 and its vertical coordinate is 4. The other points are harder to tell though the vertical coordinate of \(B\) is 4.)

The purpose of this activity is for students to generate two patterns from rules and then graph them on the coordinate grid. Students first identify a point on the coordinate grid with one of the pairs of numbers from the patterns and then plot the rest of the points. Students may notice that the points on the graph are regularly spaced. They are invited to share this and other observations in the synthesis.

• Groups of 2
• “You and your partner will each start some problems about patterns and the coordinate grid independently. After a couple minutes, work with your partner to complete the problems.”

• 2 minutes: independent time
• 5 minutes: partner work time

• “How did you decide where to place the points on the grid?” (I used the top row to decide how far over to go on the horizontal axis and the second rule to decide how far up to go on the vertical axis.)
• Invite students to share completed graphs for parts A and B.
• “How are the two graphs the same?” (There is a point at the bottom left, \((0,0)\), on each. The points are regularly spaced and go up and to the right. They all lie at the intersection of gridlines.)
• “How are the two graphs different?” (The numbers on the axes are different. The ones for partner B get bigger really quickly.)

The purpose of this activity is for students to generate numerical patterns given two rules, form ordered pairs consisting of the corresponding terms, and graph the ordered pairs on the coordinate grid. The structure of the activity is the same as the previous activity but this time the multiplicative factor relating the two rules is a fraction. Monitor for students who express the relationship (MP8) between the two patterns by saying that

• the numbers in the second pattern are double the numbers in first pattern and half more
• the numbers in the second pattern are \(2\frac{1}{2}\) times the numbers in the first pattern

• Groups of 2.

• 5 minutes: independent time
• 5 minutes: partner work time
• Monitor for students who:
  • notice the additive relationship for each rule
  • notice the multiplicative relationship between rule 1 and rule 2

If students don’t plot and label the points from the table correctly, plot points A and B and ask, “How do each of these points represent the rules?”

• Invite previously selected students to share.
• “What does point \(D\) represent in terms of the two rules? How do you know?” (When rule 1 is 6, rule 2 is 15. The coordinates are \((6,15)\) and the horizontal coordinate is rule 1 and the vertical coordinate is rule 2.)
• Display: \((10,20)\)
• “10 is a number in rule 1 and 20 is a number in rule 2. Is the point with coordinates \((10, 20)\) on your graph?” (No, the points that represent the two rules are \((10, 25)\) or \((8, 20)\). The 10 from rule 1 and the 20 from rule 2 don’t match up with each other.)