Understanding Rational Numbers and Proportions

In this lesson, students use real-world models to develop an understanding of fractions, decimals, unit rates, proportions, and problem solving. The three activities in this investigation center on situations involving rational numbers and proportions that students encounter at a bakery. These activities involve several important concepts of rational numbers and proportions, including partitioning a unit into equal parts, the quotient interpretation of fractions, the area model of fractions, determining fractional parts of a unit not cut into equal-sized pieces, equivalence, unit prices, and multiplication of fractions.

Standards & Objectives

Academic standards
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.NS.A.1
Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction...
CCSS.Math.Content.6.NS.B.3
Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
CCSS.Math.Content.6.NS.C.5
Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature...
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.7
Understand ordering and absolute value of rational numbers.
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,...
GLE 0606.2.1
Understand and explain the procedures for multiplication and division of fractions, mixed numbers, and decimals.
GLE 0606.2.2
Solve multi-step mathematical, contextual and verbal problems using fractions, mixed numbers, and decimals.
GLE 0606.2.3
Understand and use ratios, rates and percents.
GLE 0606.2.4
Understand and convert between fraction, decimal, and percent forms of rational numbers.
GLE 0606.2.5
Develop meaning for integers; represent and compare quantities with integers.
GLE 0606.3.6
Understand and use the Cartesian coordinate system.
SPI 0606.2.1
Solve problems involving the multiplication and division of fractions.
SPI 0606.2.2
Solve problems involving the addition, subtraction, multiplication, and division of mixed numbers.
SPI 0606.2.3
Solve problems involving the addition, subtraction, multiplication, and division of decimals.
SPI 0606.2.4
Solve multi-step arithmetic problems using fractions, mixed numbers, and decimals.
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 0606.3.9
Graph ordered pairs of integers in all four quadrants of the Cartesian coordinate system.
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.NS.A.1
Interpret and compute quotients of fractions, and solve contextual problems involving division of fractions by fractions (e.g., using visual fraction models...
TSS.Math.6.NS.B.3
Fluently add, subtract, multiply, and divide multi-digit decimals using a standard algorithm for each operation.
TSS.Math.6.NS.C.5
Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below...
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.7
Understand ordering and absolute value of rational numbers.
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.
 
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:

  • Represent parts of a whole using an area interpretation of fractions.
  • Determine the fractional part of a whole when parts are not cut into equal-sized pieces.
  • Develop an understanding of the quotient interpretation of fractions.
  • Find the unit cost of items that are part of a set.
  • Determine the relationship among parts of a whole that are unequal-sized pieces.
  • Express fractional parts of a whole as decimal equivalents.

Lesson Variations

Blooms taxonomy level: 
Understanding
Extension suggestions: 

Extensions:

  • Students find the cost of various-sized pieces, given that 1/8 of a cake costs $1.59 and a whole cake costs $12.72. The following table is a sample. (Tables may include other fractional parts and need not be limited to eighths, fourths, and halves.) Students may wish to use calculators with fraction capability to help them find the various prices. Using calculators may help students focus on the reasonableness of their solutions rather than on the calculations.
  • How many cookies would each person get if...
  • three people shared twenty cookies? [20/3, or 6 and 2/3 cookies for each person]
  • eight people shared twenty cookies? [20/8, or 2 and 1/2 cookies for each person]
  • x people shared twenty cookies? [20/x cookies for each person]
  • What is a rule for finding the number of cookies each person will get if a people share b cookies? 

Helpful Hints

Materials:

  • Making Four Pieces Overhead 
  • Cakes Cut Into Eighths Activity Sheet 
  • Cakes Cut Into Fourths Activity Sheet 
  • Scissors
  • Calculators (optional)

 

References

Contributors: