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Unit 3: Unit Rates and Percentages


In the previous unit, students began to develop an understanding of ratios and rates. They started to describe situations using terms such as “ratio,” “rate,” “equivalent ratios,” “per,” “constant speed,” and “constant rate” (MP6). They understood specific instances of the idea that 
a:b is equivalent to every other ratio of the form sa:sb, where s is a positive number. They learned that “at this rate” or “at the same rate” signals a situation that is characterized by equivalent ratios. Although the usefulness of ratios of the form a/b:1 and 1:b/a was highlighted, the term “unit rate” was not introduced.

In this unit, students find the two values ab and ba that are associated with the ratio a:b, and interpret them as rates per 1. For example, if a person walks 13 meters in 10 seconds at a constant rate, that means they walked at a speed of 13/10 meters per 1 second and a pace of 10/13 seconds per 1 meter.

Students learn that one of the two values (a/b or b/a) may be more useful than the other in reasoning about a given situation. They find and use rates per 1 to solve problems set in contexts (MP2), attending to units and specifying units in their answers. For example, given item amounts and their costs, which is the better deal? Or given distances and times, which object is moving faster? Measurement conversions provide other opportunities to use rates.

Students observe that if two ratios a:b and c:d are equivalent, then a/b = c/d. The values a/b and c/d are called unit rates because they can be interpreted in the context from which they arose as rates per unit. Students note that in a table of equivalent ratios, the entries in one column are produced by multiplying a unit rate by the corresponding entries in the other column. Students learn that “percent” means “per 100” and indicates a rate. Just as a unit rate can be interpreted in context as a rate per 1, a percentage can be interpreted in the context from which it arose as a rate per 100. For example, suppose a beverage is made by mixing 1 cup of juice with 9 cups of water. The percentage of juice in 20 cups of the beverage is 2 cups and 10 percent of the beverage is juice. Interpreting the 10 as a rate: “there are 10 cups of juice per 100 cups of beverage” or, more generally, “there are 10 units of juice per 100 units of beverage.” The percentage—and the rate—indicate equivalent ratios of juice to beverage, e.g., 2 cups to 20 cups and 10 cups to 100 cups.

In this unit, tables and double number line diagrams are intended to help students connect percentages with equivalent ratios, and reinforce an understanding of percentages as rates per 100. Students should internalize the meaning of important benchmark percentages, for example, they should connect “75% of a number” with “3/4 times a number” and “0.75 times a number.” Note that 75% (“seventy-five per hundred”) does not represent a fraction or decimal (which are numbers), but that “75% of a number” is calculated as a fraction of or a decimal times the number.

Work done in grades 4 and 5 supports learning about the concept of a percentage. In grade 5, students understand why multiplying a given number by a fraction less than 1 results in a product that is less than the original number, and why multiplying a given number by a fraction greater than 1 results in a product that is greater than the original number. This understanding of multiplication as scaling comes into play as students interpret, for example,

  • 35% of 2 cups of juice as 35/1002 cups of juice.
  • 250% of 2 cups of juice as 250/1002 cups of juice.


Mathematics Learning Targets for Unit 3

I can ...

I can see that thinking about "how much for 1" is useful for solving different types of problems.  

When I know a measurement in one unit, I can decide whether it takes more or less of a different unit to measure the same quantity. 

I know that when we measure things in two different units, the pairs of measurements are equivalent ratios. 

I can convert measurements from one unit to another, using double number lines, tables, or by thinking about "how much for 1."

I understand that if two ratios have the same rate per 1, they are equivalent ratios. 

When measurements are expressed in different units, I can decide who is traveling faster or which term is the better deal by comparing "how much for 1" of the same unit. 

I can choose which unit rate to use based on how I plan to solve the problem. 

When I have a ratio, I can calque its two unit rates and explain what each of them means in the situation. 

I can give an example of two equivalent ratios and show that they have the same unit rates. 

I can multiply or divide by the unit rate to calculate missing values in a table of equivalent ratios. 

I can solve more complicated problems about constant speed situations. 

I can choose how to use unit rates to solve problems. 

I can explain the meaning of percentages using dollars and cents as an example.

I can create a double number line with percentages on one line and dollar amounts on the other line. 

I can use double number line diagrams to solve different problems like "What is 40% of 60% or "60% is 40% of what number?"

I can use tape diagrams to solve different problems like "What is 40% of 60%? or "60% is 40% of what number?"

When I read or 
hear that something is 10%, 25%, 50%, or 75% of an amount, I know what fraction of that amount they are referring to. 

I can choose and create diagrams to help me solve problems about percentages. 

I can solve different problems like "What is 40% of 60%?" by dividing and multiplying. 

I can solve different problems like "60 is what percentage of 40?" by dividing and multiplying. 

                                                                                                        Image result for math cartoon                 

Open Up Resources

Students are allowed to access resources from the units using the links below.

Student:  https://im.openupresources.org/6/students/index.html
Families:  https://im.openupresources.org/6/families/index.html

Dear Parents or Guardians,

I am ecstatic to inform you that after school tutorial for my class, 6th Grade Mathematics will be held on every Wednesday of the week. During these tutorial sessions, students are given a chance to gain much needed enrichment of the course as well as an opportunity to complete assignments that students may have missed due to excused absences. The hour(s) for tutorial is from 3:30 p.m. to 5:30 p.m.

 

All students must be picked up from the school campus by someone who has been identified by school records as an authorized person to sign – out/pick up students. If your student walks to and from school, he/she will be released after tutorial under the same terms as they would at the completion of the school day. If for any reason, there are changes in how a student is to be picked up, the parent or guardian must provide notice in writing twenty – four hours before the schedules tutorial session. This request will be verified and communicated to the school administration.     

 

Additionally, even though you are signing below, the administration staff requires a note to be given to Mrs. Miller verifying that your child will be staying after school on the designated day.

 

 

Thank you,

 

  

Ms. G

Email: cgranville@macon.k12.ga.us

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