Big Idea 1 : BIG IDEA 1 (Archived)



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Analyze and represent linear functions, and solve linear equations and systems of linear equations.

General Information

Number: MA.8.A.1
Title: BIG IDEA 1
Type: Big Idea
Subject: X-Mathematics (former standards - 2008) - Archived
Grade: 8
Body of Knowledge: Algebra

Related Benchmarks

This cluster includes the following benchmarks
Code Description
MA.8.A.1.1: Create and interpret tables, graphs, and models to represent, analyze, and solve problems related to linear equations, including analysis of domain, range, and the difference between discrete and continuous data.
MA.8.A.1.2: Interpret the slope and the x- and y-intercepts when graphing a linear equation for a real-world problem.
MA.8.A.1.3: Use tables, graphs, and models to represent, analyze, and solve real-world problems related to systems of linear equations.
MA.8.A.1.4: Identify the solution to a system of linear equations using graphs.
MA.8.A.1.5: Translate among verbal, tabular, graphical, and algebraic representations of linear functions.
MA.8.A.1.6: Compare the graphs of linear and non-linear functions for real-world situations.


Related Access Points

This cluster includes the following access points.

Independent

Access Point Number Access Point Title
MA.8.A.1.In.a: Use information from physical models, diagrams, tables, and graphs to solve addition, subtraction, multiplication, and division number sentences (equations) based on real-world problems.
MA.8.A.1.In.b: Identify the relationship between two sets of related data, such as ordered number pairs in a table.
MA.8.A.1.In.c: Translate problem situations into number sentences (equations) involving addition and subtraction of two-digit numbers and multiplication and division facts using information from physical and visual models, tables, and pictographs.

Supported

Access Point Number Access Point Title
MA.8.A.1.Su.a: Use information from physical models, diagrams, tables, and pictographs to solve number sentences (equations) involving addition and subtraction with one-digit and two-digit numbers.
MA.8.A.1.Su.b: Describe the relationship (1 more or 1 less) between two sets of related numbers.
MA.8.A.1.Su.c: Translate real-world situations into number sentences (equations) involving addition and subtraction using information from physical and visual models, tables, and pictographs.

Participatory

Access Point Number Access Point Title
MA.8.A.1.Pa.a: Solve simple real-world problems involving quantities using language, such as number names, more, less, same, larger, smaller, and none.
MA.8.A.1.Pa.b: Solve simple problems involving joining or separating sets of objects or pictures to 8.
MA.8.A.1.Pa.c: Distinguish between the position of two objects, such as first and next.


Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

Educational Software / Tool

Name Description
Free Graph Paper: A variety of graph paper types for printing, including Cartesian, polar, engineering, isometric, logarithmic, hexagonal, probability, and Smith chart.

Formative Assessment

Name Description
Solving Real-Life Problems: Baseball Jerseys: This lesson unit is intended to help you assess how well students are able to:
  • Interpret a situation and represent the variables mathematically.
  • Select appropriate mathematical methods to use.
  • Explore the effects of systematically varying the constraints.
  • Interpret and evaluate the data generated and identify the break-even point, checking it for confirmation.
  • Communicate their reasoning clearly.

Lesson Plans

Name Description
Interpreting Distance-Time Graphs: This lesson takes a formative assessment approach to assessing whether students can interpret distance-time graphs. Whole-class discussion, group work, and individual activities help students interpret distance-time graphs related to real-world scenarios.
Discovering Kepler's Law for the Periods of Planets:

Students listen to a video that describes Kepler's determination that planetary orbits are elliptical and then will use data for the solar distance and periods of several of the planets in the solar system, then investigate several hypotheses to determine which is supported by the data.

The Linear Function Connection:

The students will compare two linear functions that have been represented in different ways (equation, table, graph, verbal description). They will be able to find and compare the rate of change, or slope, of the function from any of the representations.

Asteroids: In this lesson, students show and discuss that the planets' orbits follows a pattern, compare sizes of asteroids relative to the size of Earth and Earth's moon, make asteroid models, research impact craters, and discuss the likelihood and effects of an asteroid impact on Earth.
Who goes faster, earns more, drives farthest?:

Given a proportional relationship, students will determine the constant of proportionality, write an equation, graph the relationship, and interpret in context.

Problem-Solving Tasks

Name Description
Smiles:

In this online problem-solving challenge, students apply algebraic reasoning to determine the "costs" of individual types of faces from sums of frowns, smiles, and neutral faces. This page provides three pictorial problems involving solving systems of equations along with tips for thinking through the problem, the solution, and other similar problems.

Drip, Drop, Drip, Drop: Students design an experiment to model a leaky faucet and determine the amount of water wasted due to the leak. Using the data they gather in a table, students graph and write an equation for a line of best fit. Students then use their derived equation to make predictions about the amount of water that would be wasted from one leak over a long period of time or the amount wasted by several leaks during a specific time period.

Teaching Idea

Name Description
Learning About Rate of Change in Linear Functions Using Interactive Graphs:

This resource features two pairs of interactive graphs to help students explore rate of change and linear relationships. "Users can drag a slider on an interactive graph to modify a rate of change (cost per minute for phone use) and learn how modifications in that rate affect the linear graph displaying accumulation (the total cost of calls). In this first part, Constant Cost per Minute, the cost per minute for phone use remains constant over time. In the second part, Changing Cost per Minute, the cost per minute for phone use changes after the first sixty minutes of calls." (from NCTM's Illuminations)

Tutorial

Name Description
Graphing Linear Functions:

This is a video which explicitly demonstrates graphing linear functions using the x- and y-intercepts.

Unit/Lesson Sequence

Name Description
Direct and Inverse Variation: "Lesson 1 of two lessons teaches students about direct variation by allowing them to explore a simulated oil spill using toilet paper tissues (to represent land) and drops of vegetable oil (to simulate a volume of oil). Lesson 2 teaches students about inverse variation by exploring the relationship between the heights of a fixed amount of water poured into cylindrical containers of different sizes as compared to the area of the containers' bases." from Insights into Algebra 1 - Annenberg Foundation.

Virtual Manipulative

Name Description
Multi Bar Graph:

This activity allows the user to graph data sets in multiple bar graphs. The color, thickness, and scale of the graph are adjustable which may produce graphs that are misleading. Users may input their own data, or use or alter pre-made data sets. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.



Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

Problem-Solving Task

Title Description
Smiles:

In this online problem-solving challenge, students apply algebraic reasoning to determine the "costs" of individual types of faces from sums of frowns, smiles, and neutral faces. This page provides three pictorial problems involving solving systems of equations along with tips for thinking through the problem, the solution, and other similar problems.

Tutorial

Title Description
Graphing Linear Functions:

This is a video which explicitly demonstrates graphing linear functions using the x- and y-intercepts.

Virtual Manipulative

Title Description
Multi Bar Graph:

This activity allows the user to graph data sets in multiple bar graphs. The color, thickness, and scale of the graph are adjustable which may produce graphs that are misleading. Users may input their own data, or use or alter pre-made data sets. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.



Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.

Problem-Solving Task

Title Description
Smiles:

In this online problem-solving challenge, students apply algebraic reasoning to determine the "costs" of individual types of faces from sums of frowns, smiles, and neutral faces. This page provides three pictorial problems involving solving systems of equations along with tips for thinking through the problem, the solution, and other similar problems.

Teaching Idea

Title Description
Learning About Rate of Change in Linear Functions Using Interactive Graphs:

This resource features two pairs of interactive graphs to help students explore rate of change and linear relationships. "Users can drag a slider on an interactive graph to modify a rate of change (cost per minute for phone use) and learn how modifications in that rate affect the linear graph displaying accumulation (the total cost of calls). In this first part, Constant Cost per Minute, the cost per minute for phone use remains constant over time. In the second part, Changing Cost per Minute, the cost per minute for phone use changes after the first sixty minutes of calls." (from NCTM's Illuminations)