standard_description: Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
problem_activity: Narrative
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.
MLR8 Discussion Supports.
Display sentence frames to support partner discussion: “First, I _____ because . . .” and “I noticed _____ so I . . . .”
Advances: Speaking, Writing, Conversing, Representing
Action and Expression: Develop Expression and Communication.
Synthesis: Develop fluency with connecting rules, tables, and a coordinate grid to the same pattern. Provide access to blank or partially completed tables.
Supports accessibility for: Conceptual Processing, Attention
Launch
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.”
Activity
2 minutes: independent time
5 minutes: partner work time
Student Facing
Partner A
Rule 1: Start at 0. Keep adding 8.
Rule 2: Start at 0. Keep adding 2.
Use the rules to complete the table.
{"table": [{"_attributes": {"border": "1", "style": "width:100%"}, "tbody": [{"tr": [{"th": [{"_attributes": {"style": "text-align:center; width:14%"}}, {"_attributes": {"style": "text-align:center; width:14%"}, "_value": "A"}, {"_attributes": {"style": "text-align:center; width:14%"}, "_value": "B"}, {"_attributes": {"style": "text-align:center; width:14%"}, "_value": "C"}, {"_attributes": {"style": "text-align:center; width:14%"}, "_value": "D"}, {"_attributes": {"style": "text-align:center; width:14%"}, "_value": "E"}, {"_attributes": {"style": "text-align:center; width:14%"}, "_value": "F"}]}, {"th": [{"_attributes": {"style": "text-align:center"}, "_value": "rule 1"}], "td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"th": [{"_attributes": {"style": "text-align:center"}, "_value": "rule 2"}], "td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}]}]}]}
Which table column represents the point on the coordinate grid? Label the point with the appropriate letter.
[IMG]
Plot and label the rest of the points.
Partner B
Rule 1: Start at 0. Keep adding 10.
Rule 2: Start at 0. Keep adding 40.
Use the rules to complete the table.
{"table": [{"_attributes": {"border": "1", "style": "width:100%"}, "tbody": [{"tr": [{"th": [{"_attributes": {"style": "text-align:center; width:14%"}}, {"_attributes": {"style": "text-align:center; width:14%"}, "_value": "A"}, {"_attributes": {"style": "text-align:center; width:14%"}, "_value": "B"}, {"_attributes": {"style": "text-align:center; width:14%"}, "_value": "C"}, {"_attributes": {"style": "text-align:center; width:14%"}, "_value": "D"}, {"_attributes": {"style": "text-align:center; width:14%"}, "_value": "E"}, {"_attributes": {"style": "text-align:center; width:14%"}, "_value": "F"}]}, {"th": [{"_attributes": {"style": "text-align:center"}, "_value": "rule 1"}], "td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"th": [{"_attributes": {"style": "text-align:center"}, "_value": "rule 2"}], "td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}]}]}]}
Which table column represents the point on the coordinate grid? Label the point with the appropriate letter.
[IMG]
Plot and label the rest of the points.
Student Response
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Activity Synthesis
“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.)


aligns: no