EduGAINs



OverviewIn this activity students examine the first differences in a table of values to determine if a relationship is linear or non-linear.Important Mathematical Ideas A linear relationship has a constant first difference.A positive constant first difference indicates an increasing linear relationship when the differences of the independent variable are positive.A negative constant first difference indicates a decreasing linear relationship when differences of the independent variable are positive. Prior KnowledgeCharacteristics of a good graph (appropriate title, labeled axes with units, uniform scale, points plotted correctly, independent variable on the horizontal axes, and dependent variable on the vertical axes).Plotting and interpreting points on a scatter plot.Identifying trends in scatter plots (e.g., correlation, line or curve of best fit).Subtracting integers. Common MisconceptionsConfusing dependent and independent variables.Calculating first differences to test whether a relationship is linear without the independent variable increasing by a constant.Subtracting the second entry for the dependent variable in a table from the first entry in the table instead of vice versa.Curriculum NotesIn Unit 5 students will make connections between the value of the first differences and rate of change of a linear rmation to Support/ Enhance/ Extend LearningStudents are asked to keep a journal for each unit in the course. It should contain notes of important mathematical ideas with examples and new vocabulary.ePortfolio may be used for these journal entries.Students can make individual choices whether this is a paper or digital personal resource.Consider a variety of formats as alternatives to journal entries (e.g., student note, pair/share, group discussion, exit card, poster, electronic posting).Develop a Word Wall and continue it throughout the unit as new vocabulary and terms arise that require clarification.Minds OnTask 1: Multiple ChoiceFive-question multiple choice activity.Review of:what a linear relationship isconstant increases from a table of valuesmatching graphs to tablesAnswers are provided.Students can work in pairs in order to discuss their reasoning.A discussion may be lead which makes connections about:appropriate labelling of the axesthe independent and dependent variablesthe table of valuesActionTask 2: First DifferencesStudents will watch and make notes on:First Differences Ex1 video first differences are calculated in a table of values where the dependent variable increases by a constant valueFirst Differences Ex 2 videofirst differences are calculated in a table of values where the dependent variable decreases by a constant valueSome students may benefit from a review of subtracting integers:Gap Closing Intermediate/Senior Module 3 IntegersStudent BookFacilitator’s GuideIt is important that students realize:the independent variable must be increasing by the same incrementthe first difference of each dependent variable (value) is calculated by subtracting the value of the previous dependent variable Some students may find it easier to start at the bottom of the table and work up.Journal Prompts and Sample ResponsesMake notes in your journal as you watch the two videos about first differences.You find the first differences in a table of values by finding the difference in consecutive values for the dependent variable when the values for the independent variable change consistently (i.e., by the same amount).The independent variable must be increasing by the same amount.You find the first difference between values of the dependent variable by subtracting the previous value from each.To find first differences determine by how much the dependent value is increasing or decreasing, also called the change in the dependent variable.Find the first differences for the table Temperature vs. Time from the video First Differences Ex2.The value of the first differences is -3.Task 3: Perimeter TaskStudents will:explore shipping costs by calculating the perimeter of tiles with natural number side lengths between 1 cm and 5 cmmodel the data in a table and a scatter plotcalculate the first differencesmake a connection between the first differences and a linear relationshipcheck their answers with the provided sample solutionStudents can use the online Linear Graphing Tool to construct the graph for each task and then screen capture it onto the Task assignment page.Consider organizing as a Think/Pair/Share or sharing results through a Gallery Walk.Task 4: Jody's Pay TaskStudents will:explore Jody's pay based on the number of hours she worksmodel the data in a table and a scatter plotcalculate the first differencesmake a connection between the first differences and a linear relationshipcheck their answers with the provided sample solutionStudents can use the online Linear Graphing Tool to construct the graph for each task and then screen capture it onto the Task assignment pageConsider organizing as a Think/Pair/Share or sharing results through a Gallery WalkTask 5: Area TaskStudents will:explore possible shipping costs by calculating the area of tiles with natural number side lengths between 1 cm and 5 cmmodel the data in a table and a scatter plotcalculate the first differencesmake a connection between the first differences and a non-linear relationship which looks like a curvecheck their answers with the provided sample solutionStudents can use the online Linear Graphing Tool to construct the graph for each task and then screen capture it onto the Task assignment pageConsider organizing as a Think/Pair/Share or sharing results through a Gallery WalkConsolidationTask 6: Discussion Prompt and Notes Reflect on the three tasks (Perimeter Task, Jody's Pay Task and Area Task) and describe connections between the values of the first differences, the shape of the graph and the type of relationship. Justify your answers.When there was a constant first difference, the graph showed a linear relationship. This happened in the Perimeter Task and Jody's Pay task.When there wasn't a constant first difference, the graph showed a non-linear relationship. This happened in the Area Task. There was a curve of best fit on the scatter plot but not a line of best fit.Task 7: Journal Prompt and Sample ResponseDescribe what you have learned about First Differences.You find the first differences in a table of values by finding the difference in consecutive values for the dependent variable when the values for the independent variable are increasing by the same amount.If the first differences are equal then the relationship is linear. If the first differences are not the same, then the relationship is non-linear.Task 8: Practice Exercise: Deep Sea Diver's TaskStudents will:use first differences to determine what type of relationship (linear or non-linear) there is from a given set of data (depth of diver (m) vs. time (min))confirm their finding by graphing the datacheck their answers with the sample solution providedTask requires skill with subtraction of negative integers or the ability to reason about the change in depth with negative numbers and requires the ability to graph in Quadrant 4.Task 9: Practice Exercise: Hot Air Ballooning TaskStudents will:use first differences to determine what type of relationship (linear or non-linear) there is from a given set of data (height of hot air balloon (m) vs. time (min))confirm their finding by graphing the datacheck their answers with the sample solution providedTask 10: Assignment 2 Sunflower Experiment Missing Data Posted with unit.Sample solution posted in Teacher Guide on the vLE.Task 11: Student ReflectionStudents are asked to reflect on their understanding of this topic.These reflections can be used as assessment for learning to help determine next steps for individual students. ................
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