Day 31: Unit Rates and Tape Diagrams

6th Grade Math Standards: 6.RP.1 Understand the concept of a ratio and use ratio language to describe the relationship between two quantities. For example, "The ratio of wings to beaks at the birdhouse in the zoo was 2:1, because for every 2 wings there was one beak." "For every vote candidate A received, candidate C received nearly 3 votes."

6.RP.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 ¾ cup of flour for each cup of sugar.” “We paid $75 for 
15 hamburgers, which is a rate of $5 per hamburger.”

6.RP.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.
b. 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?
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.
d. Use ratio reasoning to convert measurement units; manipulate and transform units
appropriately when multiplying or dividing quantities

The Learning Objective: Find the unit rate of a product to determine the value of that product for any given number of units. Find an equivalent ratio given an original ratio and the sum value of the two units.

Quote of the Day: "I believe the most important thing in any endeavor is hope. You cannot believe it is hopeless because if you do, it is." - Dr. Henry Friendman

Agenda:

1. Upon entering the room the students simplified 4 ratios that had a greatest common factor bigger than 1.
2. We reviewed the homework particularly the question that I had liked yesterday about 6 students having a baseball glove in a class of 24. What was the ratio of students that had a glove to the students that did not have a glove? 
3. We reviewed the exit ticket from yesterday. In one class in particular, more than 70% of students made a mistake on at least one of the two problems. We discussed circling what the ratio is asking for since most students ignored the word total. 
4. I gave the students this picture and asked them to consider a price that they would think is too high to pay for the apple juice, a price that is too low, and a price that seems just right. It was amazing. 
5. We took notes on unit rate
6. We revisited the 6-pack turned 5-pack issue described in step four and found the unit rate of that to make sense of what was a fair price.
7. In the second part of class, I gave students the following prompt: The Patriots are going to outscore the Jets tonight by a ratio of 3:2. If there are 85 total points, what is the final score? 
8. We took notes on tape diagrams. I was shocked by how much they enjoyed it. I think they enjoyed it so much though because of the struggle they had before starting the unit. 
9. The students answered the question described in Step 7
10. The students did the tape diagram homework
11. The students did select unit rate problems for homework on a different sheet

The Assessment: There were a few:

1. Last night's homework. Every students but about ten (out of sixty-five) answered the baseball glove question wrong.
2. I cold called students to help me answer the jumpstart questions. 
3. I went around the room on the Dan Meyer Apple Juice pre-assessment to see what students were writing. They were really thinking today on this one. Many students wrote a too high of $3.29 and other students were discovering the unit cost and multiplying by five. I was particularly pleased with who was putting $3.29 for a too high answer because these students are showing growth in the early part of the year.
4. On the pre-assessment to tape diagrams, one out of sixty-five students could tell me the score was going to be 51 to 34. After taking notes on three problems, more than half the class was able to do this. The students that were left confused were helped by their peers. 
5. I did start to look at homework answers for tonight as well. 

Homework: Tape diagram practice as well as the unit rate worksheet (not scanned here) and of course the Weekly Quiz which is due tomorrow

My Glass Half-Full Take: I loved throwing the challenging problem at the students before we did the notes today. It was such a helpful way to get them to buy into what we were doing.

One Thing to Do Differently: As we were doing the notes on unit rate in the first part of class one of my students said "Mr. Schruender I'm bored." I put my ego aside because quite frankly I was too. The problem with the five cans came to life partially because it was a picture but mainly because it was a problem worth solving. The bored student was one of the students who was actually finding the unit rate and multiplying by five to find a price that seemed fair. In his defense my notes were far more basic than this type of problem. I was putting him in fourth grade when he was ready for grade 6 and a half. The notes could have been differentiated. There was one problem that gave the students ounces and prices. I know now that if I had asked those students who tested out of the can problem that they could have been asked to determine two different unit rates in the same problem ($1.36 per ounce of ice cream and 0.72 ounces of ice cream per $1 - that sort of thing).

As a a side note, I'm really glad that student said he was bored. I actually came up with the Patriots vs. Jets problem on the spot as a way to say, "ok class I dare you to get this without me actually teaching you how to do a thing." The kids relished the challenge. And I relished the fact that the kids were relishing the challenge.

Link of the Day: Seth Godin discusses why some people just aren't good at math. I agree with everything he writes except for the fact that he thinks standardized tests don't measure anything. Although I think their value is inflated in our society, they can inform us of a student's fluency. I also think they can do a lot more of bad than good when we spend 100 percent of our time preparing students that never do well on these tests to prepare to take yet another one. If it's broke for these students, shouldn't we fix it?