| 
  • If you are citizen of an European Union member nation, you may not use this service unless you are at least 16 years old.

  • You already know Dokkio is an AI-powered assistant to organize & manage your digital files & messages. Very soon, Dokkio will support Outlook as well as One Drive. Check it out today!

View
 

Pre-Algebra Unit 2: Linear Equations and Functions (redirected from Pre-Algebra Unit 3: Linear Equations and Functions)

Page history last edited by Brigit Minden 10 months, 1 week ago Saved with comment

  

Pre-Algebra Unit 2: Linear Equations and Functions

Unit Driving Question

How can we use graphs and other representations to gain knowledge of real-world situations? 

 

Essential Questions

  1. How can we identify relationships in the world around us?

  2. What does the rate of change communicate?

  3. How can you represent linear relationships?

  4. How can you identify a situation as linear or nonlinear?

  5. In what real world situations would you use scatterplots, and why? 

 

Big Ideas

  1. A function is a relationship between an independent and dependent variable.
  2. Rate of change describes how one quantity changes in respect to another.
  3. Multiple representations can be used to express and analyze linear relationships.
  4. Functions can be identified as linear if they can be expressed in slope intercept or graphed in a straight line.
  5. Data can be displayed and interpreted using scatterplots. 

  

Technology Resources

 

Launch Task

1 Lesson 

 

Big Ideas for Development Lessons

5-6 Weeks (approximately 1 week per big idea)

Big Idea 1: A function is a relationship between an independent and dependent variable.

OAS-M: PA.A.1.1

Key Resources 

 

  1. Inputs and Outputs (OpenUp): This is the first of two lessons introducing students to the idea of a function as a rule that assigns to each allowable input exactly one output. In the classroom activities, students take turns guessing each other's rules from input-output pairs. Then they use input-output diagrams with a given rule to fill out a table with inputs and associated outputs.  A table alone does not give enough information to determine a rule if it applies to inputs that are not listed in the table, so we have to know the rule explicitly to be able to determine exactly how to find the output for each input.

  2. Introduction to Functions (OpenUp): Students learn the term function for a rule that produces a single output for a given input. They also start to connect function language to language they learned in earlier grades about independent and dependent variables.

  3. Equations for Functions (OpenUp): In this lesson students transition from input-output diagrams and descriptions of rules to equations. This lesson also introduces the use of independent and dependent variables in the context of functions. For an equation that relates two quantities, it is sometimes possible to write either of the variables as a function of the other.

  4. Commuting Times (Desmos Activity): This activity illustrates the relationship between a dataset (which is usually not a function) and a model of the data (which,in Algebra, is a function).

 

Computer Science Integration Activities-For Bootstrap Trained Teachers:

 

Big Idea Probe

  • Unit 3: Big Idea 1 Probe: Create two tables, one that creates a function and one that does not.  Allow students to describe what characteristics create a function. 

 

Evidence of Understanding 

 

Recognize that a function is a relationship between an independent variable and dependent variable.

  • The value of the independent variable determines the value of the dependent variable.

 

 

Big Idea 2: Rate of change describes how one quantity changes in respect to another.

OAS-M: PA.A.2.3

Key Resources

 

  1. Meet Slope (OpenUp): In this lesson, students use what they have learned about similar triangles to show that the slope between any two points on a line in a coordinate plane will always be the same. (Once this is accomplished, we have earned the right to refer to the slope of a line.) The slope of a line is defined as the quotient of the lengths of the vertical side and the horizontal side of a right triangle whose hypotenuse lies on the line. Then, students find the slopes of given lines and draw lines with given slopes. 
  2. An Equation of a Line (Open Up): In this lesson, students find an equation of a line using what they know about slope. The purpose is not to refine and catalog every possible form of an equation, but to establish that knowing a point on the line and its slope (or equivalently, knowing two points on the line) enables us to write an equation that represents a relationship between the coordinates of a generic point (x,y) on the line, and that the slope can often be seen as a parameter in the equation. 
  3. Introduction to Linear Relationships (OpenUp): In this lesson, the focus is proportionality vs. linear relationships and rate of change. The meaning of the vertical intercept of the graph  may come up in the various activities, and the term vertical intercept itself is introduced in the last activity. The discussion of slope and vertical intercept will continue in the next lessons. 
  4. Slopes Can Be Negative (OpenUp): In this lesson students get their first glimpse of lines that visually slope downhill. After reflecting initially on commonalities and differences of lines that slope in different directions, students explore a situation where one quantity decreases at a constant rate in relation to a second quantity. They interpret a graph of the situation and reason that it makes sense for the slope to be negative by contextualizing back to the situation. The scenario is then extended to consider a quantity that does not change with respect to another, and students realize that a flat graph has a slope of zero. 
  5. Equations of All Kinds (OpenUp): In this lesson, students continue their study of lines with negative slopes and consider how to write equations for horizontal and vertical lines. They first explore how a line with negative slope and non-zero intercept can be thought of as a translation of a line with negative slope and intercept (0,0).
  6. Polygraph, Lines Part 2 (Desmos Activity): This activity follows up on Polygraph: Lines, using the discussions (and students' informal language) in that activity to develop academic vocabulary related to the graphs of linear functions. 
  7. Stations Activity- Slope and Slope-Intercept Form: Students will work in collaborative groups to complete station activities that provide opportunities for students to develop concepts and skills related to coordinate graphing.

 

Big Idea Probe

 

Evidence of Understanding 

 

Know that the slope of a linear function equals the rate of change.

  • Zero slope is a horizontal line.

  • Undefined slope is a vertical line.

  • Positive slope is an increasing line when read from left to right.

  • Negative slope is a decreasing line when read from left to right.

 

Big Idea 3: Multiple representations can be used to express and analyze linear relationships.

OAS-M: PA.A.2.1, PA.A.2.2, PA.A.2.3

Key Resources 

 

  1. Solutions of Linear Equations in Two Variables (OpenUp):  In this lesson, students extend their work with linear equations by exploring what it means to be a solution of an equation, and the relationship between the equation, its solutions, and its graph. 
  2. Solutions to Linear Equations in Context (OpenUp):  This lesson continues the exploration of linear equations in two variables, their solutions, and their graphs. Students now consider real world situations that can be represented by linear equations and find solutions within the context presented. They also consider whether their graphs will go on indefinitely as they think about solutions that make sense in the given situation. 
  3. Comparing Lines and Linear Equations(OER Commons): In this lesson, students will interpret speed as the slope of a linear graph and also translate between the equation of a line and its graphical representation.
  4. Stations Activity- Scatter Plots: Students will work in collaborative groups to complete station activities that provide opportunities for students to develop concepts and skills related to analyzing data using appropriate graphs.

 

 

Big Idea Probe

  • Unit 3: Big Idea 3 Probe: This activity asks students to notice and use properties of linear functions to make groups of three. Different properties will lead to different groupings by different students. Later we ask students to make conjectures about different groupings – why might another student have grouped the cards in a particular way?  

 

Evidence of Understanding 

 

Represent linear functions in multiple ways.

  • Represent linear functions with tables.
  • Represent linear functions with verbal descriptions.
  • Represent linear functions with symbols.
  • Represent linear functions with graphs. 

 

Translate from one representation to another.

 

Identify, describe, and analyze linear relationships between two variables.

 

Identify the graphical properties, slope, and intercepts of linear functions.  

 

 

Big Idea 4: Functions can be identified as linear if they can be expressed in slope intercept or graphed in a straight line.

OAS-M: PA.A.1.3, PA.A.1.2, PA.A.2.4, PA.A.2.5

Key Resources

 

  1. Understanding Relationships (OpenUp): In all three lessons students are asked to interact with different representations of proportional relationships, pull information from different representations to answer context related questions, and work fluidly with finding different representations given one piece of information.  This lesson starts with a more concrete representation of moving objects and has students verify and create information from a given diagram. 
  2. Proportionality and Representations (OpenUp): In this lesson, students match different representations of proportional relationships, create arguments to support why those representations are matches, and point out the constant of proportionality in each representation. 
  3. Translating to y=mx+b (OpenUp): This lesson introduces the idea that any line in the plane can be considered a vertical translation of a line through the origin. In the previous lesson, the terms in the expression are more likely to be arranged b+mx because the situation involves a starting amount and then adding on a multiple. In this lesson, mx+b is more likely because the situation involves starting with a relationship that includes (0,0) and shifting up or down. 

 

Bootstrap Computer Science Integration Activity

 

Big Idea Probe

  • Unit 3: Big Idea 4 Probe: In this activity, students practice finding equations of lines in order to land a plane on a runway. Most of the challenges are well-suited to slope-intercept form, but depending on the goals of an individual class or student they are easily adapted to other forms of linear equations. 

 

Evidence of Understanding 

 

Identify a function as linear.

  • It can be expressed in the form y=mx+b.
  • Its graph is a straight line.

 

Use linear functions to represent and explain real-world and mathematical situations.

 

Predict the effect of a graph on a linear function.

  • Changing the slope effects the graph.
  • Changing the y-intercept effects the graph.

 

Solve problems involving linear functions.

  • Interpret the results in the original context.

 

Identify a function as proportional.

  • Y-intercept must be zero.

    • The line crosses through the origin (0,0).

 

 

Big Idea 5: Data can be displayed and interpreted using scatterplots.

OAS-M: PA.D.1.3

Key Resources

 

  1. What a Point on a Scatterplot Means (OpenUp): Students interpret points in a scatter plot in terms of the context, and add points to a scatter plot given information about an individual in the population. They compare individuals represented by different points and informally discuss trends in the data.
  2. Fitting a Line to Data (OpenUp): In the previous lesson, students focused primarily on the details of a scatter plot. In this lesson, their focus becomes more holistic, and they begin to see a set of data points as a single thing that can be analyzed, not just a bunch of disconnected points. For the first time, they see that sometimes we can model the relationship between two variables with a line, although they continue to analyze the connections between the scatter plot and the line by comparing individual points. 
  3. Describing Trends in Scatterplots (OpenUp): In this lesson, students are introduced the terms positive association and negative association. They use fitted lines to help them understand this language and tie it back to their work in an earlier unit on linear relationships. 
  4. Analyzing Bivariate Data (OpenUp): In this lesson, students bring everything they have studied in the unit so far to analyze and interpret bivariate data in context. They create a scatter plot, identify outliers, fit a line, and determine and interpret the slope of the line. They compare actual and predicted values. They reflect on what they have learned about modeling bivariate data. 
  5. Scatter Plot Capture (Desmos): In this activity, students use observations about scatterplot relationships to make predictions about future points in the plot. In particular, students focus on linear vs. nonlinear association, strong vs. weak association, and increasing vs. decreasing plots. 
  6. Stations Activity- Data and Relationships: In this activity, students will complete stations activities to analyze scatter plots and data sets.

 

Computer Science Integration Activities-For Bootstrap Trained Teachers: 

  1. Bootstrap Lesson: What Influences Temperature: (Note: This lesson is for Bootstrap-trained teachers and also focuses on PA.A.2.2) 

 

Big Idea Probe

 

Evidence of Understanding 

 

Collect, display, and interpret data using scatterplots.

 

Use the shape of a scatterplot to interpret the data.

  • Informally estimate a line of best fit.

  • Make statements about the average rate of change.

  • Determine the correlation or relationship of the data.

    • Positive correlation is increasing from left to right.

    • Negative correlation is decreasing from left to right.

    • No correlation or unpredictable correlation shows no pattern.

 

Use the shape of the scatterplot to make predictions about values not in the original data set.

 

Use appropriate titles, labels, and units when constructing scatterplots.

 

 

Unit Closure

1 Week (includes time for probes, re-engagement, and assessment)  (H4)

 

 

Comments (0)

You don't have permission to comment on this page.