I’ve blogged about the first two ratio and rate standards for 6th grade in “Ratios vs.Rates.” These standards are conceptual understanding standards. In 6.RP.A.3, we have a mixed standard that is both an application standard and procedural skills and fluency standard.
CCSS.MATH.CONTENT.6.RP.A.3
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors.
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.
This standard gives examples of ratio reasoning: tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. The only forms of ratio reasoning required in future standards are tables and equations. The standards do not explicitly address tape diagrams and number lines. So should you teach them? Maybe. Tape diagrams and number lines help explain ratios. Many students need various models to gain understanding. Are they required? They shouldn’t be, but it depends on how the assessment designers interpret the standard or if they just based the new assessment on older assessments without regard to the CCSS. If you are allowed to review your assessments, teach what matches the assessment. If you cannot review your assessments, teach what works for your students and hope that the assessment follows CCSS the same way you interpret the standards.
Each person brought a dozen cookies. There were 48 chocolate chip cookies and half as many oatmeal scotchies. How many people brought oatmeal scotchies?
A student might draw a tape diagram to help answer the question.
The standard has four sub-standards. One sub-standard (3.B) is an application standard; two are procedural skills and fluency standards. 3.C is both.
CCSS.MATH.CONTENT.6.RP.A.3.A
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors.
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.
6.RP.A.3.A has the students working with tables and with the coordinate plane. The coordinate plane is introduced in Math.5.G.A.1 and 2. The students should know what the coordinate plane is and how to plot coordinates. In 6th-grade, they need to use a table to make ordered pairs and plot the points. Addition and multiplication tables are introduced in 3rd grade, 3.AO.9. In 4.MD.1, the students create conversion tables. 4.NF.A.2 involves comparing fractions. So, the 6th-grade skills are just a new application for established skills. The students will apply conceptual understanding of ratios to the old skills.
CCSS.MATH.CONTENT.6.RP.A.3.B
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors.
Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
3.B is the application standard. It focuses on unit rates. The students will have experience with equivalent ratios using a table and equivalent fractions from 3.NF.3.B and 4.NF.3. The CCSS states that for 6th grade, the expectations for unit rates in this grade are limited to non-complex fractions. Based on this expectation, the example given would have to be converted to minutes before determining the unit rate. 7:4= 7/4:1 vs. 420:4 = 105:1 Then we get the equivalent ratio of 105/1 = 2100/x. So the students will need to learn how to tell if they need to convert to a different measure before determining the unit rate. The students will need strong factoring skills to determine if a conversion is needed.
The standards states the students hould “solve unit rates including those involving unit pricing and constant speed.” You must specifically address theses two typeof unit rate problems and you should also include other types of problems.
| Unit Pricing | Constant Speed | Other Problems |
| Bear bought 3 bottles of honey for $6. How much did each bottle cost? | The train moves at a constant rate of 136 miles per hour. How long would it take to travel 100 miles? | It takes 3 gallons of paint for the room. What is the unit rate per wall assuming the room in square? (Conversion needed.) |
| Each shirt costs $12. How much would it cost to buy 12 shirts? | The bike club wants to bike 27 miles in 3 hours or less. What is the minimum rate they need to keep to meet the goal? | The recipe calls for 8 eggs for 4 servings. How many eggs would be needed for 6 servings? |
CCSS.MATH.CONTENT.6.RP.A.3.C
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors.
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.
6th-grade is the first time the students learn about percents. Finding a ratio per 100 isn’t much different than finding equivalent ratios. The students are supposed to look at percent as x%/100 = 3/4. They are not looking at percent as a decimal. 4.NF.6 has the students using fractions with denominators of 10 or 100 to create decimals; some students may connect percents and decimals but it is not required. If you taught the added concepts of part-to-part and part-to-whole when you taught the concept of ratios, it might make understanding percents easier. After the students gain skills creating percents, they get to apply the skills. The students are given the part and percent and must determine the whole.
Sometimes I wonder if I read the standards too literally. 6.RP.A.3.C says, “find a percent of a quantity as a rate per 100.” This is straightforward; the students take an initial ratio then convert it to an equivalent ratio per 100. It is one skill. We do not need to teach all the associated skills yet. The standard continues with “solve problems involving finding the whole, given a part and the percent.” Again, it is particular. The students should solve problems requiring one specific algorithm, nothing more. 7.RP.A.3 has students solving a range of percent problems, but the sixth-grade standard is particular about the type of problem the students must solve. Do I teach exactly what the standard says? Am I just being too persnickety?
What to do? Do I teach exactly what is in the standards? Teach what I always taught? Teach to the test? Just do what the textbook says?
I have debated what teaching to the standards mean with many teachers and administrators. Some say that the standards are the floor, not the ceiling, implying that you should teach beyond the standards. Others say tojust teach the materials because the materials are “data driven.” Yet the students are supposed to be proficient in all the grade level standards in time for the standardized tests. We need to teach as much of the standards as possible before the end of the year (2/3s into the year) test. Is there time to “add” to the standards in the rush for the test. (Yes, I am letting my attitude show a little.) Well, it is up to you, the programs you use to teach, expectations of the next grades teachers, and your administration. I usually taught above the standard but learned to skip the lessons in my textbooks that went outside the standard – sometimes even skipping a whole chapter to teach the actual standard, not the publisher’s interpretation of what should be taught. But to come to grips with this conundrum, I am studying standards’ based instruction. It is pretty cool, but I will talk about that in another blog. Back to the standards.
It is easy to find percent problems, but problems’s that specifically meet the standard (part to whole) are harder to find. Some examples:
- Matt is 8 and 50% his sister’s age. How old is his sister?
- Twii bought a bicycle using a 15% off coupon. She payed $170. What was the original cost of the bike?
- After 35 miles, Sam said she had 40% more to go. How long is the whole trip?
Using the mix of problems normally found in 6th grade worksheets should be fine for most students. Since the standard is specific, you can have struggling students skip all the problems that are not specific to the standard and save you all a lot of frustration.
CCSS.MATH.CONTENT.6.RP.A.3.D
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors.-grade
Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
In third grade, the students form simple equivalent fractions using visual or physical models. Fourth grade had the students understand multiplying by a/a to find an equivalent fraction and then apply that concept. So the students have the skills to convert ratios. Measuring to specific units is taught in grade 2. Fourth grade students create number pairs of measurement conversions (4.MD.A.1). The students now need to write the comparative ratios: 1:12 for feet to inches, 3:1 for yards to feet… This is a great time to help the students to create reference pages. Tables, charts, and diagrams are helpful. Cootie Catchers are a great way to help the students memorize measurement conversions.
6.RP.A.3 Is a complex standard. It is also one of the most foundational standards for future math classes. Ratios are part of probability, slope, scale, vectors, trig… The skills the students learned about fractions and the coordinate grid in previous grades get taken a little further and applied a little differently. You can teach the ratios unit fairly quickly; however, time needs to be spent keeping the skills fresh. Adding occasional ratios review is essential. But 6th-grade has many essential concepts that are foundational. That is why I am starting with the 6th-grade standards.