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### Overview

Students will create a map of a shopping excursion by solving systems of linear equations through graphing, substitution, or elimination. Students will then be able to determine the distance traveled through the mall using Pythagorean Theorem. Students will utilize previous grade level standards as a spiral review such as plotting coordinate pairs and using those pairs to find the distance between two points.

This activity is a result of the ALEX Resource Development Summit.

## UP:MA19.6.11

### Vocabulary

• Coordinate plane
• Coordinate values
• ordered pairs
• x axis
• y axis
• Reflection

### Knowledge

Students know:
• Strategies for creating coordinate graphs.
• Strategies for finding vertical and horizontal distance on coordinate graphs.

### Skills

Students are able to:
• Graph points corresponding to ordered pairs,
• Represent real-world and mathematical problems on a coordinate plane.
• Interpret coordinate values of points in the context of real-world/mathematical situations.
• Determine lengths of line segments on a coordinate plane when the line segment joins points with the same first coordinate (vertical distance) or the same second coordinate (horizontal distance).

### Understanding

Students understand that:
• A graph can be used to illustrate mathematical situations and relationships. These representations help in conceptualizing ideas and in solving problems,
• Distances on lines parallel to the axes on a coordinate plane are the same as the related distance on the axis (number line).

## UP:MA19.7.6

### Vocabulary

Term
• like terms
• Constant
• Factor
• Expression
• Rational coefficient
• ### Knowledge

Students know:
• how to add, subtract, multiply, and divide rational numbers.
• A(b + c) = ab + ac.
• how to find the greatest common factor of two or more terms.

### Skills

Students are able to:
• apply properties of operations as strategies to add and subtract linear expressions with rational coefficients.
• Apply properties of operations as strategies to factor linear expressions with rational coefficients.
• Apply properties of operations as strategies to expand linear expressions with rational coefficients.

### Understanding

Students understand that:
• only like terms can be combined, e.g., x + y = x + y but x + x = 2x.
• To factor an expression, one must factor out the greatest common factor.
• There are many different ways to write the same expression.

## UP:MA19.8.9

### Vocabulary

• Slope
• Rate of change
• Initial Value
• Y-intercept

### Knowledge

Students know:
• how to graph points on a coordinate plane.
• Where to graph the initial value/y-intercept.
• Understand how/why triangles are similar.
• how to interpret y=mx equations.

### Skills

Students are able to:
• create a graph of linear equations in the form y = mx + b and recognize m as the slope and b as the y-intercept.
• point out similar triangles formed between pairs of points and know that they have the same slope between any pairs of those points.
• Show that lines may share the same slope but can have different y-intercepts.
• Interpret a rate of change as the slope and the initial value as the y-intercept.

### Understanding

Students understand that:
• Slope is a graphic representation of the rate of change in linear relationships and the y-intercept is a graphic representation of an initial value in a linear relationship.
• When given an equation in the form y = mx + b it generally symbolizes that there will be lines with varying y-intercepts. even when the slope is the same.
• Use of the visual of right triangles created between points on a line to explain why the slope is a constant rate of change.

## UP:MA19.8.12

### Vocabulary

• System of linear equations
• Point of intersection
• One solution
• No solution
• Infinitely many solutions
• Parallel lines
• Slope-intercept form of a linear equation
• Standard form of a linear equation

### Knowledge

Students know:
• The properties of operations and equality and their appropriate application.
• Graphing techniques for linear equations (using points, using slope-intercept form, using technology).
• Substitution techniques for algebraically finding the solution to a system of linear equations.

### Skills

Students are able to:
• generate a table from an equation.
• Graph linear equations.
• Identify the ordered pair for the point of intersection.
• Explain the meaning of the point of intersection (or lack of intersection point) in context.
• Solve a system algebraically using substitution when both equations are written in slope-intercept form or one is written in standard form and the other in slope-intercept form.

### Understanding

Students understand that:
• any point on a line when substituted into the equation of the line, makes the equation true and therefore, the intersection point of two lines must make both equations true.
• Graphs and equations of linear relationships are different representations of the same relationships, but reveal different information useful in solving problems, and allow different solution strategies leading to the same solutions.

## UP:MA19.8.27

### Vocabulary

• Pythagorean Theorem

### Knowledge

Students know:
• The Pythagorean Theorem.
• The operations and labeling within a coordinate system.

### Skills

Students are able to:
• Solve equations involving one variable and square root.
• Represent real-world and mathematical contexts involving right triangles in a variety of formats (drawings on coordinate planes, equations).
• Justify solutions and solution paths using conceptual understandings and vocabulary related to the Pythagorean Theorem (right angle, hypotenuse).

### Understanding

Students understand that:
• The properties of right triangles can be used to solve problems.
• Theorems represent general relationships that are true for all shapes that fit certain criteria.

### Phase

During/Explore/Explain
Learning Objectives

### Learning Objectives

I can solve systems of linear equations by graphing, elimination, or substitution.

I can plot points on the coordinate plane.

I can graph a linear equation in y=mx+b form.

I can apply the Pythagorean Theorem to find the distance between two points.

### Activity Details

Students will be given a copy of the story and will create a map using the information from the story. The students will solve systems of linear equations by graphing, elimination, or substitution to create the map. This is a great project that can be used in class over the course of several days in order to allow students the opportunity to grapple with real-world mathematics. The directions for completing the activity are detailed in the Google Doc. Teachers can provide a copy of the activity to each student via Google Classroom or by providing a hard copy to each student. Students will be allowed to work in small groups, but will each create their own maps.

Assessment Strategies

### Assessment Strategies

Attached rubric is part of learning activity and activity can be used as a summative assessment or a learning activity.

This can be done in the classroom or can be assigned as an independent project to be done at home.

### Variation Tips

The story could be altered to give students ordered pairs to graph rather than solving the systems of equations.

The story could be altered so all equations are given in slope-intercept form rather than the standard form.

### Background / Preparation

Each student will need their own copy of Graphville Shopping Excursion.

Graph paper will be needed for a rough draft and final copies of the product.

Colored pencils, markers, etc. are optional and up to student/teacher discretion.