# Constant Dimensions

Students will measure the length and width of a rectangle using both standard and non-standard units of measure. In addition to providing measurement practice, this lesson allows students to discover that the ratio of length to width of a rectangle is constant, regardless of the units. For many middle school students, this discovery is surprising. This activity involves the applicaton of several skills such as, plotting points, creating a table, measuring, ordered pairs, and coordinate plane.

### Standards & Objectives

Academic standards
CCSS.Math.Content.6.EE.A.2
Write, read, and evaluate expressions in which letters stand for numbers.
CCSS.Math.Content.6.EE.B.7
Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all...
CCSS.Math.Content.6.EE.B.8
Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that...
CCSS.Math.Content.6.EE.C.9
Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one...
CCSS.Math.Content.6.G.A.2
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge...
CCSS.Math.Content.6.G.A.3
Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same...
CCSS.Math.Content.6.G.A.4
Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures....
CCSS.Math.Content.6.NS.C.6
Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to...
CCSS.Math.Content.6.NS.C.8
Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and...
CCSS.Math.Content.6.RP.A.1
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, "The ratio of wings...
CCSS.Math.Content.6.RP.A.2
Understand the concept of a unit rate a/b associated with a ratio a:b with b != 0, and use rate language in the context of a ratio relationship....
CCSS.Math.Content.6.RP.A.3
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams,...
CCSS.Math.Content.7.RP.A.1
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different...
CCSS.Math.Content.7.RP.A.2
Recognize and represent proportional relationships between quantities.
CCSS.Math.Content.7.RP.A.3
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and...
CCSS.Math.Content.8.EE.B.5
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships...
CCSS.Math.Content.8.EE.B.6
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive...
CCSS.Math.Content.8.F.A.3
Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For...
CCSS.Math.Content.8.SP.A.3
Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For...
GLE 0606.1.1
Use mathematical language, symbols, and definitions while developing mathematical reasoning.
GLE 0606.1.3
Develop independent reasoning to communicate mathematical ideas and derive algorithms and/or formulas.
GLE 0606.2.3
Understand and use ratios, rates and percents.
GLE 0606.3.2
Interpret and represent algebraic relationships with variables in expressions, simple equations and inequalities.
GLE 0606.3.4
Use expressions, equations and formulas to solve problems.
GLE 0606.3.5
Use multiple representations including symbolic algebra to model and/or solve contextual problems that involve linear relationships.
GLE 0606.3.6
Understand and use the Cartesian coordinate system.
GLE 0606.4.4
Develop and use formulas for surface area and volume of 3-dimensional figures.
GLE 0706.2.3
Develop an understanding of and apply proportionality.
GLE 0706.2.4
Use ratios, rates and percents to solve single- and multi-step problems in various contexts.
GLE 0706.3.5
Understand and graph proportional relationships.
GLE 0706.3.6
Conceptualize the meanings of slope using various interpretations, representations, and contexts.
GLE 0706.3.7
Use mathematical models involving linear equations to analyze real-world phenomena.
GLE 0706.4.2
Apply proportionality to converting among different units of measurements to solve problems involving rates such as motion at a constant speed.
GLE 0706.4.4
Understand and use ratios, derived quantities, and indirect measurements.
GLE 0806.3.4
Translate among verbal, tabular, graphical and algebraic representations of linear functions.
GLE 0806.5.2
Select, create, and use appropriate graphical representations of data (including scatterplots with lines of best fit) to make and test conjectures.
SPI 0606.2.6
Solve problems involving ratios, rates and percents.
SPI 0606.2.7
Locate positive rational numbers on the number line.
SPI 0606.2.8
Locate integers on the number line.
SPI 0606.3.1
Represent on a number line the solution of a linear inequality.
SPI 0606.3.3
Write equations that correspond to given situations or represent a given mathematical relationship.
SPI 0606.3.5
Translate between verbal expressions/sentences and algebraic expressions/equations.
SPI 0606.3.9
Graph ordered pairs of integers in all four quadrants of the Cartesian coordinate system.
SPI 0706.1.3
Recognize whether information given in a table, graph, or formula suggests a directly proportional, linear, inversely proportional, or other nonlinear relationship.
SPI 0706.2.6
Express the ratio between two quantities as a percent, and a percent as a ratio or fraction.
SPI 0706.2.7
Use ratios and proportions to solve problems.
SPI 0706.3.1
Evaluate algebraic expressions involving rational values for coefficients and/or variables.
SPI 0706.3.4
Interpret the slope of a line as a unit rate given the graph of a proportional relationship.
SPI 0706.3.5
Represent proportional relationships with equations, tables and graphs.
SPI 0806.1.1
Solve problems involving rate/time/distance (i.e., d = rt).
SPI 0806.1.3
Calculates rates involving cost per unit to determine the best buy.
SPI 0806.3.4
Translate between various representations of a linear function.
SPI 0806.3.7
Identify, compare and contrast functions as linear or nonlinear.
SPI 0806.4.2
Apply the Pythagorean theorem to find distances between points in the coordinate plane to measure lengths and analyze polygons and polyhedra.
SPI 0806.4.4
Convert between and within the U.S. Customary System and the metric system.
SPI 0806.5.3
Generalize the relationship between two sets of data using scatterplots and lines of best fit.
TSS.Math.6.EE.A.2
Write, read, and evaluate expressions in which variables stand for numbers.
TSS.Math.6.EE.B.7
Solve real-world and mathematical problems by writing and solving onestep equations of the form x + p = q and px = q for cases in which p, q, and x are all...
TSS.Math.6.EE.B.8
Interpret and write an inequality of the form x > c or x
TSS.Math.6.EE.C.9
Use variables to represent two quantities in a real-world problem that change in relationship to one another.
TSS.Math.6.G.A.2
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and...
TSS.Math.6.G.A.3
Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side that joins two vertices (vertical or...
TSS.Math.6.G.A.4
Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these...
TSS.Math.6.NS.C.6
Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent...
TSS.Math.6.NS.C.8
Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value...
TSS.Math.6.RP.A.1
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, the ratio of wings to beaks...
TSS.Math.6.RP.A.2
Understand the concept of a unit rate a/b associated with a ratio a:b with b != 0. Use rate language in the context of a ratio relationship.
TSS.Math.6.RP.A.3
Use ratio and rate reasoning to solve real-world and mathematical problems.
TSS.Math.7.RP.A.1
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units. For...
TSS.Math.7.RP.A.2
Recognize and represent proportional relationships between quantities.
TSS.Math.7.RP.A.3
Use proportional relationships to solve multi-step ratio and percent problems.
TSS.Math.8.EE.B.5
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in...
TSS.Math.8.EE.B.6
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; know and derive...
TSS.Math.8.F.A.3
Know and interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.
TSS.Math.8.SP.A.3
Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a...

Alignment of this item to academic standards is based on recommendations from content creators, resource curators, and visitors to this website. It is the responsibility of each educator to verify that the materials are appropriate for your content area, aligned to current academic standards, and will be beneficial to your specific students.

Learning objectives:

Learning Objectives:

Students will:

• Critique various units of measure based on their appropriateness for this particular activity.
• Use a linear graph to model, analyze and make predictions.
• Draw conclusions about the relationship of two dimensions based on collected data.
Essential and guiding questions:

Questions for Students

• Although more pennies were used than M&M’s when measuring the width, did the size of the width actually change?
• Take a look at your six points. Do they appear randomly or does there appear to be a pattern?
• If someone used gumballs to measure the length and width, and their ordered pair were placed at (22, 10), would we suspect that they made a good measurement? What if the ordered pair had the coordinates (16, 10.5)? What is your reasoning?
• What unit of measure could be used that would have produced a point very close to the origin? Why?
• What remained constant even when the units of measurement changed?
• What algebraic rule is associated with the ordered pairs? Write an equation that shows how the two dimensions are related for this rectangle. Can this rule be written in the form length : width = __ : __ ?
• If the length of the rectangle is 13 wooches, determine the width of the rectangle in wooches using (1) your line of best fit and (2) your algebraic rule.

### Lesson Variations

Blooms taxonomy level:
Understanding
Extension suggestions:

Extensions:

• Using the data that students collected for their rectangle, graphing calculators can be used to perform a linear regression and determine the line of best fit.
• You can provide other rectangles and ask students if the ratio is the same for every rectangle. Though it’s not the same for every rectangle, students should see that rectangles with the same ratio have the same "shape," leading to the concept of similarity.

### Helpful Hints

Materials:

• Rectangle Measure Activity Sheet
• Rulers (both inches and centimeters)
• Alternate units of measure (pennies, paper clips, M&M’s, beads, width of index finger, width of pencil)
• Graphing calculator (optional)

Contributors: