Mathematics Meaning of Fractions, Decimals, Ratios, and Percents

Sixth Grade: Mathematics Model Lesson for Unit #2: Meaning of Fractions, Decimals, Ratios, and Percents Overarching Question:

How can reasoning with ratios and rates help solve real world and mathematical problems? Previous Unit:

Factors and Multiples

This Unit:

Questions to Focus Assessment and Instruction: 1. How are ratios and decimals alike? How are they different? 2. How can you determine if two ratios are equivalent?

Key Concepts: fraction

Next Unit:

Meaning of Fractions, Decimals, Ratios, and Percents

percent decimal

This document is the property of MAISA.

Two-Dimensional Geometry

Intellectual Processes (Standards for Mathematical Practice): Model with mathematics: Solve problems involving comparisons of amounts and measurements. equivalence of fractions and ratios ratio Page 1 of 5 7/28/11

Mathematics Meaning of Fractions, Decimals, Ratios, and Percents

Lesson Abstract This lesson is an application of equivalent fractions. Having a concept of equivalent fractions is important for simplifying fractions. The number-sense of recognizing equivalent fractions is useful when students study slope and proportions. Pairs (or groups of) students use a cup of beans to find ratios to express the number of marked beans in the cup compared to the total number of beans in the cup. Theoretically, each sampling ratio should be essentially the same. The decimal representation of each ratio confirms that the ratios are, indeed, approximately equivalent. Common Core State Standards Ratios and Proportional Relationships (6.RP)________________________________________ Understand ratio concepts and use ratio reasoning to solve problems. 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 in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.” 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. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” (Expectations for unit rates in this grade are limited to non-complex fractions.) 3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. a. Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. The Number System (6.NS)_______________________________________________________ Apply and extend previous understandings of numbers to the system of rational numbers. 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 points on the line and in the plane with negative number coordinates. *d. Develop an understanding of fractions, decimals, and percents and the relationships between and among the concepts and their representations. *e. Understand and use equivalent fractions. *f. Compare and order fractions.

This document is the property of MAISA.

Page 2 of 5 7/28/11

Mathematics Meaning of Fractions, Decimals, Ratios, and Percents

*g. Model situations involving fractions, decimals and percents and use physical models (fractions strips, number line, partition, grid area, percent bar models) and drawings to help reason about the situations. * Indicates items added to CCSS Instructional Resources: Paper cups and two colors of dried beans (similar in size), Sequence of Lesson Activities Lesson Title: Bean Counting and Ratios http://illuminations.nctm.org/LessonDetail.aspx?id=L722 Selecting and Setting up a Mathematical Task: 





By the end of this lesson what do you want your students to understand, know, and be able to do?

•

Understand and use ratios to represent quantitative relationships.

•

Convert ratios to decimals and compare the results.

•

Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

•

Use ratio and rate reasoning to solve real-world and mathematical problems by reasoning about tables of equivalent ratios or decimal equivalents. In this case ratios of the dark colored beans to light colored beans in the sample will be used to set up ratios from six different samples. The number of dark and –light colored beans will be given to the students.

•

Students have studied equivalent fractions, have the ability to interpret a fraction as division of the numerator by the denominator, and can compare two decimals up to the thousandths place. Now they will compare ratios from samples of a bean population by use of equivalent fractions and decimal representations of these fractions.

•

If I have the ratio 3/4, what are some ratios that are equivalent? Ask students to explain how they were able to arrive at the answer they have given.

•

How can the ratio 3/5 be changed into a decimal with three decimal places? 3 ? You may see students use equivalent fractions to change = . Other 5 1000 students may use division 5 3.000 to generate the decimal equivalent.

•

Prepare a cup of 40 beans in which 20 of the beans are a dark color and the other 20 are a light color. It is important that both sets of beans are similar in size. This will be used as the demonstration set. If I scoop out 14 of the beans, how many would I expect to be dark? How could this result be written as a ratio? How can you compare this ratio with the ratio representing the dark colored beans to all of the beans in the cup? Have students work individually to respond and then share their answers with another student. If students all used equivalent fractions, then discuss other ways to represent this relationship (i.e. decimals). If some students already used decimals,

In what ways does the task build on student’s previous knowledge?

What questions will you ask to help students access their prior knowledge?

Launch: 

How will you introduce students to the activity so as to provide access to all students while maintaining the cognitive demands of the task?

This document is the property of MAISA.

Page 3 of 5 7/28/11

Mathematics Meaning of Fractions, Decimals, Ratios, and Percents have them share their ideas with the rest of the class. •

Using the same demonstration cup of beans, have students consider how many beans will be dark if I scoop out a sample of 27 beans. Since this will not be an obvious answer (13.5), it allows students to consider whether a decimal answer makes sense given the context.

•

For the class activity, prepare a cup with 45 beans for each pair of students. Select 30 dark colored beans and 15 light colored beans. Activity Sheet A is provided to record the results of six samples the students will take. All students can do the same ratio of dark beans to the whole sample. Check to be sure students know that all beans must be returned to the cup and the cup shaken before a new sample is selected.

•



What will be heard that indicates that the students understood what the task is asking them to do?

Students will be writing ratios of dark beansin sample and comparing them to

total # of beans the known ratio in the cups given. The comparisons should be made by using either equivalent fractions or decimals. A calculator may be used to support these computations.

Supporting Student’s Exploration of the Task: 







What questions will be asked to focus students’ thinking on the key mathematics ideas?

What questions will be asked to assess student’s understanding of key mathematics ideas?

What questions will be asked to encourage all students to share their thinking with others or to assess their understanding of their peer’s ideas?

How will you extend the task to provide additional challenge?

•

What connections would you expect to see in the ratios of dark beans to total beans for the different samples that you select? How will these ratios be related to the dark beans to the total beans in your population (cupful) of beans?

•

How does the average ratio of the decimals for all six trials compare to the decimal you calculated for the whole cup? Why do you think this relationship exisits?

•

How are ratios similar to and different from fractions?

•

What method did you use to compare your ratios? Is there any other way to compare the ratios?

•

How are the strategies you used similar to or different from your past work with fractions?

•

How can you use the information you know about the original cup of beans and it’s dark beansin sample ratio to compute the missing values in questions

total # of beans 2 and 3 on the worksheet? •

Provide a variety of different cups with simpler or more complicated ratios of dark beans : total beans geared to the levels of your students.

•

If students have already studied percents, have them compare the percentage of each sample that is dark, explaining how to find the percent in each case.

•

Give students a cup of beans, some dark and some light and tell them the total number of beans. Ask students to come up with a plan to determine how many of the beans in the cup are marked.

This document is the property of MAISA.

Page 4 of 5 7/28/11

Mathematics Meaning of Fractions, Decimals, Ratios, and Percents

Sharing and Discussing the Task: 

What specific questions will be asked so that all students will: o

o



Make sense of the mathematical ideas that you wanted them to learn? Expand on, debate, and question the solutions being shared?

o

Make connections between the different strategies that are presented?

o

Look for patterns?

o

Begin to form generalizations?

•

How close did each of the decimals you calculated come to the actual ratio? What seems to be a reasonable range of values?

•

Given the ratio of dark beans to total beans as a decimal, how could you find the ratio of dark beans to total beans?

•

Given the ratio (written as a decimal) of dark beans to total beans, can you find the total number of beans in the cup? What additional information would you need to do this calculation?

•

Given a pair of ratios, how can you tell if they are equivalent?

What will be seen or heard that indicates all students understand the mathematical ideas you intended them to learn?

Formative Assessment:

Ask students to predict outcomes for these examples: 1. In this class there are 28 students. One half of them are female. How many of them do you expect to be female? If 9 students are called to the office, how many are likely to be girls? [4 to 5] 2. A florist used 25 flowers to make a bouquet. Five of the flowers are daffodils. If 10 flowers are removed from the bunch at random, how many of them do you expect to be daffodils? 3. Write another example of your own.

This document is the property of MAISA.

Page 5 of 5 7/28/11