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6-N-3-1

Page history last edited by Brenda Butz 6 years, 1 month ago

6.N.3.1  Identify and use ratios to compare quantities. Recognize that multiplicative comparison and additive comparison are different.


In a Nutshell

A ratio is a type of numerical comparison. Prior to ratios, students have been making additive comparisons to determine whether one quantity is more (or less) than another. For example: “Annie has 2 dogs.  Roger has 3 more dogs than Annie.  How many dogs does Roger have?”  Multiplicative comparisons allow quantities to be compared in a different way. For example: “Annie has 2 dogs, Roger has 3 times as many dogs as Annie.  How many dogs does Roger have?” In this example, the second set is a multiple copy (3 times) of the first set. Students should understand that 3 more than (first example) has a different meaning than 3 times (second example).  When comparing ratios, the rules of comparing fractions and equivalent fractions can apply. For instance, the ratio 20 to 40 is equivalent to the ratio 1 to 2 just as the fraction 20/40 is equivalent to ½.

Student Actions

Teacher Actions

  • Develop a deep and flexible understanding by using multiple representations of ratios in order to identify relationships as part-to-whole, part-to-part, and whole-to-part, and be able to represent them in various forms.  1:2, 1 to 2, 1 out of 2.

  • Develop problem solving strategies by using procedures developed with equivalent fractions to find equivalent ratios and to compare ratios. Students should understand equivalent ratios, like fractions, are found using multiplicative comparison.  Students would see that 3:4, 6:8, 9:12  are all equivalent fractions/ratios by multiplication. (Versus additive comparison where addition is used.) Ratio tables, using equivalent ratios, can also be used to find unknown values.  For example: “If a farmer has three hay bales for every 15 cows, how many hay bales are needed for 60 cows?”

 

  • Use multiple representations such as a number line to aid in comparing ratios. Students can plot ratios on a number line, just like with equivalent fractions.   Example: “There are 14 boys in Mrs. Hardy’s class of 20 students.  There are 10 boys in Mrs. Hill’s class of 25 students.  Which class has the greater ratio of boys to total students?”  

After students simplify both ratios and plot on a number line, they can visually see the ratio of boys in Mrs. Hardy’s class to the right of Mrs. Hill’s class.  This can help students to correctly compare this ratio.

  • Engage students in making  connections among mathematical representations of ratios in order to compare ratios including pictorial and graphical representations.  The visual model below shows that the ratio 2 to 6 found by subtracting 2 from the numerator and denominator of 4/8 are not equivalent ratios.  However, if the numerator and denominator are divided by two, the resulting ratio of 2 to 4 is equivalent to 4/8.  A table of values can also be created using equivalent ratios and these ordered pairs can be plotted on a coordinate plane.  If the graph forms a straight line through (0,0), all of the ratios from the table are equivalent. 

 

 

  • Facilitate meaningful mathematical discourse between students when discussing the differences between part-to-part, part-to-whole, and whole-to-part ratios, making sure to note that all ratios including part-to-part ratios can be written in fraction form, but a fraction is typically used to compare a part-to-whole relationship.

  • Build procedural fluency for comparing ratios by encouraging students to use a variety of representations and examples in order to understand the difference between additive and multiplicative comparisons.  For example, in a classroom with 15 boys and 10 girls, “there are 5 more boys than girls” is an additive comparison where as saying “there are 1.5 times as many boys as girls” is a multiplicative comparison.  The multiplicative comparison may be expressed as a ratio of boys to girls (3 to 2, 3:2, or 1.5 to 1).

Key Understandings

Misconceptions

  • Understand that ratios can express part-to-part, part-to-whole, or whole-to-part relationships.

  • Identify and understand ratios in various contexts and represent them in multiple ways.

  • Use reasoning about multiplication and division to determine equivalent ratios.

  • Understand that a ratio is a comparison of two quantities.

  • Not understand that two ratios can be equivalent.  Example 1:2 is the same as 2:4.

  • Not understand that the order of the ratios is important.  (Ex: Students may believe that 1:2 is the same as 2:1.)

  • Think that the numbers in a ratio are absolute. (Ex: 3 dogs for every 4 cats does not mean there are only 3 dogs.)

  • Think all ratios are part to whole like fractions and have difficulty defining part to part ratios.

  • Think they can find equivalent ratios using addition.


OKMath Framework Introduction

6th Grade Introduction

 

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