Day 35: Equivalent Ratios

6th Grade Math Standards: 6th Grade Math Standards: 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: Determine if two ratios measuring the same units are equivalent

Quote of the Day: “It is a new school year and things seem to be going pretty well. Suddenly some popular kids start teasing you and calling you names. At first you brush it off - these things happen. But it continues. Every day they follow you, they taunt you, they make fun of what you’re wearing, they make fun of what you look like, they tell you you’re a loser - in front of everybody. Every day.” The fixed mindset took the incident more personally. ‘I would think I was a nobody. I would think I was stupid and weird and a misfit.’ Then they’d say they wanted violent revenge, saying they’d explode with rage at them, punch their faces in or run them over. They strongly agreed with the statement, ‘My number one goal would be to get revenge.’

The growth mindset students saw it as a psychological problem of the bullies, a way for bullies to gain status or charge their self-esteem. ‘I’d think that the reason he/she is bothering me is probably that he/she has problems at home or at school with grades...I would want to forgive them eventually. My number one goal would be to help them become better people.’ Now individual children can’t usually stop the bullies, especially when the bullies attract a group of supporters, but the school can.” - Carol Dweck from Mindset

Agenda:

1. Jumpstart on Gatorade calories vs. Coke calories
2. Review the homework specifically the point between (2,1) and (2,4)
3. Do the following problem: A 24 pack of water at Store A costs $8 and a 32 pack of water at Store B costs $10. Which is the better buy and why? 
4. Notes on equivalent ratios
5. Start the homework on equivalent ratios

The Assessment: I told students that their ticket to start the homework was me checking off the first homework problem. When seven students stood up simultaneously to be checked, I realized I was going to have a room of chaos shortly with students waiting to be checked and not waiting to do math. Thus, after checking the first correct answer, I gave the student feedback on the fact that she had made the numerator have the same number through multiplication and had her go around and check others. I often use this peer mentor method, and today it proved useful.

The other assessment I had was with the problem described in step three of the agenda. Where I noticed they were lacking was in emphasizing how important units were. Several students said a store was better because the value 3 is cheaper than 3.20, but the unit was really 3 bottles and 3.20 bottles. In effect the idea of getting 3 bottles per dollar was of course worse than getting 3.20 bottles per dollar. It was interesting to see one student solve this problem trying to use the least common multiple too.

Homework: Page 63-65 odds only on equivalent ratios. I especially like the question that asks students to find if $28 in 4 weeks is equivalent to $49 in 49 days.

My Glass Half-Full Take: The problem that I took the picture of motivates students. They enjoy the challenge. And I don't just mean the students that enjoy math. Everyone is enjoying it. It's when I do notes that I start to lose them. Speaking of notes...

One Thing to Do Differently: I came up with the conclusion by the third time I'd done notes that to tell if two rates are equivalent simply multiply or divide a rate or two rates until either the numerator or denominator is the same. In my last class I made the students repeat. "To see if two ratios are equivalent." To see if two ratios are equivalent. "Make the two numerators." Make the two numerators. "Or." Or. "Denominators." Denominators. "The same." The same. It proved to be the most effective way I did it all day, although there was still a good deal of confusion. Yet, when the students had no direction from the notes, they weren't without a clue as referenced by the pictures above (although all of the pictures do have their flaws).

Link of the Day: Tape diagrams have been a problem for parents to understand, so I made a video.