The standards require students to build deep conceptual understanding of ratios and rate, which they can then use to solve problems. Over time, they should be able to use a variety of representations and methods to explain their thinking, rather than relying on a single method. Tape diagrams also known as bar models or strip diagrams are thin rectangles resembling pieces of tape that can be divided into sections to represent parts of a problem.
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In the elementary grades, a tape diagram with two sections can represent a simple addition or subtraction problem; later on, a diagram with several equal-sized sections can represent a multiplication or division problem. They also help to build conceptual understanding of equivalent ratios, as in this task.
Shanni and Mel are using a ribbon to decorate a project in their art class. Draw a tape diagram to represent this ratio. Grade 6, Module 1, Lesson 3 Available from engageny. When used early in the unit on ratios, this task can help students understand the nature of equivalent ratios. Since each model has the same number of units, students can see that equivalent ratios all have the same unit rate. This is one of the defining features of equivalent ratios. Tape diagrams are also useful aids for problem-solving.
The process of constructing a tape diagram from a problem requires students to read the problem, determine the quantities involved, and determine the relationships among those quantities. At first, students might find it tedious to create a diagram and then find a solution to the problem, but tape diagrams have two advantages.
First, they help students slow down and think about each problem before they try just anything. Second, they allow students to decode very tricky problems that are difficult to solve using only an arithmetic method. Take this problem, for example. The Business Direct Hotel caters to people who travel for different types of business trips. On Saturday night there is not a lot of business travel, so the ratio of the number of occupied rooms to the number of unoccupied rooms is 2: 5.
However, on Sunday night the ratio of the number of occupied rooms to the number of unoccupied rooms is 6: 1 due to the number of business people attending a large conference in the area. If the Business Direct Hotel has occupied rooms on Sunday night, how many unoccupied rooms does it have on Saturday night? Grade 6, Module 1, Lesson 6 Available from engageny. Without drawing the diagrams first, the relationships between the two ratios and and the given quantity occupied rooms on Sunday might be tough to discern.
But once students take the time to construct both tape diagrams, the situations becomes much more clear.
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Finding the solution just requires a few simple calculations, as shown here:. In addition to being a useful instructional tool, ratio tables are also useful for solving problems. This task is just one example. A mixture of concrete is made up of sand and cement in a ratio of How many cubic feet of each are needed to make cubic feet of concrete mix? While students might represent this problem in a number of ways, a ratio table might be the fastest and simplest: They can easily record the facts given in the problem, as well as the quantities they need to find.
Asking them to explain how to complete these intermediate rows might give them a clue about how to get started. Total Mixture. Just as the name implies, double number lines are diagrams containing two parallel number lines labeled with different units. In particular, double number lines are handy where percents are involved, allowing students to make sense of problems beyond those that just require finding percent of a number.
The task below relates ounces of soda and grams of sugar, allowing students to solve based on a ratio of 20 grams of sugar to every 6 ounces of soda. A school cafeteria has a restriction on the amount of sugary drinks available to students. Drinks may not have more than of sugar. Based on this restriction, what is the largest size cola in ounces the cafeteria can offer to students? My estimate is between 6 and 12 oz but closer to 6 ounces. I need to find of 6 and add it to 6.
A 7 oz cola is the largest size that the school cafeteria can offer to students. Grade 6, Module 1, Lesson 12 Available from engageny. Notice how the double number line allows students to see where the answer falls in a range of equivalent ratios. The calculations shown are only one possible solution; students could also use unit rate reasoning, for example, to find the amount of soda per gram of sugar, and then multiply to find the amount of soda containing 25 grams of sugar.
If given a choice of methods, students will probably find yet more ways to solve. After students have had plenty of exposure to conceptual representations like ratio tables and tape diagrams, they may be ready to solve certain problems by writing equations. Notice, however, how closely these equations resemble portions of the table that students used to approach the problem at first.
Once the concept of ratio has been introduced, students should get plenty of practice solving a variety of ratio problems. A balanced instructional unit will include problems that involve finding equivalent ratios and also problems that require students to work with ratio tables and graph ratio relationships on the coordinate plane. A This example involves both: S tudents need to complete a ratio table based on a description of a situation, and then use their entries in the table to create a graph. The distance between Yonkers and Morgantown is miles.
The total trip will take 8 hours. Dinner service starts once the train is miles away from Yonkers. What is the minimum time the players will have to wait before they can have their meal? The minimum time is 5 hours. Grade 6, Module 1, Lesson 14 Available from engageny. Constantly considering how the parts of a representation relate to the original situation is a good habit of mathematical thinking to encourage.
In addition to using equivalent ratios to solve problems, students should also be able to use unit rate thinking to solve problems. A runner ran 20 miles in minutes. If she runs at that speed,. If this task is used in the course of a lesson and students are allowed to solve in several ways, it might be interesting to discuss the contrasts among the methods they used with the class. How are they similar and different, in terms of the operations involved? Which one is the most efficient?
But if parts c and d were moved first, they might be more inclined to use a unit rate solution. One remarkable aspect of the ratio standards is that percents and measurement conversions , which in the past were often taught as isolated skills, are now treated as ratio situations. This is good news, as students are now able to approach them using the same type of reasoning and tools as with other ratio problems.
This task is a fairly straightforward percent problem. How much did she pay for the vase? One of the sample solutions shows a percent table really, a type of ratio table that illustrates the relationship between ratios and percents. Double number lines like those shown in the section above could also help students find the correct answer. Similar strategies can be employed to solve problems involving measurement conversions.
You can read the full text of the Standards for Mathematical Practice here. The idea is that students should cultivate the same habits of mind in increasingly sophisticated ways over the years. In other words, the Practices help students get the content. The table below contains a few examples of how the Mathematical Practices might help students understand and work with ratios in Grade 6.
Opportunities for Mathematical Practices:. Teacher actions:. When students are allowed to notice and explain this concept, and then to apply it in a variety of problem contexts, they look for and make use of structure. For an example, see Exercises of this introductory lesson plan on percents. Ask them to explain how this approach is similar and different to using a unit rate.
Students can model with mathematics MP. Show students a variety of ways to model ratio situations such as those presented above: tape diagrams, ratio tables, double number lines, and equations , and during the course of the year, give them interesting problem contexts that require these models.
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Have students explain how their representations model the situation at hand, and why they selected a certain type of representation for a given situation. Students can reason abstractly and quantitatively MP. For example, a ratio of implies equivalent ratios of , , etc. As a simple exploration activity, you might have students count intentionally constructed sets of objects—like checkers—where there is a consistent ratio of black checkers to red.
To start, students are simply counting quantifying what they observe. Then you can guide students to look for patterns between the numbers of black and red checkers in each group. After examining several groups—perhaps one group per table of students—students should come to see the underlying structure of a black checkers to b red checkers. Finally, you can abstract this structure using ratio language e.