FocusInterpreting rate of change and y-intercept in linear relationships using football statistics and game situations.
Materials & Prep
Football-themed task sheet with tables and graphs, grid paper, colored pencils, calculators, and exit ticket. Prepare representations showing yards gained over carries and minutes elapsed versus score difference.
Opening
6 minContentIn a linear relationship, the rate of change describes how the output changes for each one-unit increase in the input. The y-intercept is the output value when the input is 0. In context, both require units and a statement tied to the situation.
Student taskAnnotate a football table showing carries and total rushing yards. Identify the input, output, rate of change, and y-intercept, then write one interpretation for each.
EvidenceAnnotated table and two context-based interpretation statements.
Representing Meaning from a Table
12 minContentFor the table with carries 0, 1, 2, 3 and rushing yards 5, 9, 13, 17, the rate of change is 4 yards per carry and the y-intercept is 5 yards. The rate of change represents additional yards per carry; the y-intercept represents the initial 5 yards when carries equal 0. A complete interpretation includes the quantity, direction, units, and condition.
Student taskComplete two football table analyses. Students calculate the rate of change, identify the y-intercept, and write interpretations that distinguish change per input unit from starting value.
EvidenceCompleted table analyses with correct calculations, units, and situation-specific explanations.
Interpreting a Graph
12 minContentOn a graph of score difference versus minutes elapsed, the slope represents the change in score difference for each minute, and the y-intercept represents the score difference at the start of the game, when 0 minutes have elapsed. A negative slope indicates that the score difference decreases as time passes. Example: points (0, 11), (2, 7), and (4, 3) have rate of change −2 points per minute and y-intercept 11 points.
Student taskRead a football graph, identify two points, determine the rate of change and y-intercept, and write explanations of what both values mean. Classify each relationship as increasing or decreasing from its graph.
EvidenceLabeled graph, slope calculation, y-intercept identification, and two complete interpretations using units and context.
Application and Error Analysis
10 minContentA numerical answer is incomplete without its contextual meaning. The statement “the slope is 3” must specify that the score difference changes by 3 points per minute, while “the intercept is 8” must specify that the score difference is 8 points when the selected input is 0. Rate of change and y-intercept may be found from either a table or a graph, but the input and output quantities must remain consistent.
Student taskAnalyze two football representations and revise an incorrect interpretation that confuses rate of change with starting value. Produce a short comparison explaining how the table and graph communicate the same two features.
EvidenceRevised interpretation, comparison response, and accurate use of input, output, units, direction, and conditions.
Exit Ticket
5 minContentA table gives minutes played as 0, 5, 10, 15 and completed passes as 2, 5, 8, 11. The rate of change is 3 completed passes per 5 minutes, equivalent to 0.6 completed passes per minute, and the y-intercept is 2 completed passes.
Student taskDetermine the rate of change and y-intercept, then explain what each means in the football situation.
EvidenceIndividual exit ticket with both values and context-specific interpretations.
DifferentiationA step-labeled table-and-graph analysis template and reduced problem set for students needing support; mixed-representation problems with non-integer rates and a requirement to compare two possible interpretations for extension.
Assessment criteriaProficient work identifies the rate of change and y-intercept accurately from tables and graphs, includes appropriate units, states the change per one input unit, identifies the output when the input is 0, and explains both values in the football situation without confusing rate of change with starting value.