##
* *Reflection: Developing a Conceptual Understanding
Mistakes & Peer Instruction - Section 3: Exploration

When students make a mistake I like to use it to promote deeper understanding.** This has the effect of both giving value to the mistake as well as teaching them from their level of comprehension. **In this case the mistakes are made when students notice the denominators are increasing in a pattern, but ignore how the change relates to the denominator. **The questions I ask here first acknowledge what is correct about the mistake and then help to facilitate student discussion that will guide them towards the multiplicative releationship between the numerator and denominator.**

The 'student' in the discussion below represents multiple students. After each of my questions I gave the groups a couple of minutes to discuss with each other before responding.

**Me: "So, this student is noticing that the numerator is increasing by fives, is that true?"****Student: "Yes, each number is 5 more, it's going up by five every time."****Me: "What pattern is happening in the denominator? Is something similar happening there?"****Student: "Well, it looks like it's going to go up by 6 every time, but then there's the 30 and it doesn't work anymore."****Me: "So, it's not going up by 6 anymore? How many did it go up?"****Student: "It went up 12."****Student: "12 is six times two."****Me: "Did it go up 2 sixes?"****Student: "Well, we noticed something else in the bottom row. It goes one 6, 2 sixes, 3 sixes, then skips one and goes up to 5 sixes, ten sixes, then down to 8 sixes."**

This is when I reintoduce the familiar sentence frame "For every ____ there are ____", but change it to:** "For every increase of 5 in the numerator there is an increase of ______ in the denominator"** and ask how this might help them. This should focus their attention on the relative change as the numerator and denominator are changing together.

*Questioning to promote multiplicative thinking*

*Developing a Conceptual Understanding: Questioning to promote multiplicative thinking*

# Mistakes & Peer Instruction

Lesson 13 of 14

## Objective: SWBAT use peer instruction and error analysis to determine the most efficient strategies for comparing ratios.

## Big Idea: Though there are multiple methods, certain strategies are more efficient for certain types of problems.

*49 minutes*

Many of my students are still developing their multiplicative thinking. I expect this to result in mistakes when completely missing information in data tables. Having students analyze the mistakes together and discuss corrective measures with each other aids in peer instruction. Scaling ratios up and down can be really hard for students with underdeveloped multiplicative thinking and I don't want anything getting in their way. Some of them are using really big denominators, because they are not simplifying first. They will have come to class thinking they did really hard work because they multiplied big numbers. While I don't want to diminish their sense of pride and accomplishment I also want them to become more flexible in their thinking. In this lesson we focus on multiple methods, but also critique which is most efficient which helps them to better choose mathematical tools. Student work analysis.docx

*expand content*

#### Warm up

*15 min*

I expect some students to try to correct their homework and get some last minute help from their peers instead of doing the warm up. That's okay, because this warm up is such that you can jump in at any time and still be able to follow. In this warm up students are set up to dissagree:

**Juan used a common denominator of 300 to compare 12/15 and 16/20 and says they are proportional because 12/15=240/300 and 16/20=240 300.**

**Ben says he's wrong, they are proportional because when they are simplified 12/15=4/5 and 16/20=4/5.**

**What do you think? If you think one of them is wrong, what did he do wrong? If you think they are both right, who can that be?**

This is a good way to get students to critique the arguments of others (mp3). To make sure ELL students are participating I try to highlight their contributions by using their work as a conversation starter. "Let's check out what Jose did", "Jose can you explain what you did here?" Having work to refer to is really helpful both for the ELL student and for other students who may need to help them with correct terminology. It's easier for them to offer a word when they are looking at something concrete like part of a math problem.

**The key to going over this is to let the students share all their ideas and then have them come to agreement.** For the most part I expect almost everyone to realize that both methods will work to show that the two fractions are proportional. I want them to conclude that it doesn't matter if you scale up or down (simplify) to make common denominators.** As they discuss the prompt and each others ideas I listen for ideas that one is easier than the other or that they have a preference for one method over another. **If no one brings this up I can ask the class which method they would prefer to use and why.

You could extend the warm up disagreement by asking if it is still worth simplifying even if it doesn't result in common denominators. This conversation can help students see the value in simplifying just in order to avoid doing large number multiplication and makes them more flexible in their thinking.

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#### Exploration

*30 min*

In this section I play another game of "stump the student", similar to a previous lesson (ratio assessment) in which we complete a ratio table as a whole class. I give them either the numerator or denominator and they have to figure out the corresponding number. I tell students I will give them the advantage of getting to pick the numbers for our starting ratio. I just tell them to shout out some numbers and I will write them in. This way I can really be the one to choose the numbers. I am listening for small relatively prime numbers at first. Then I put a number in the top or bottom row and ask volunteers to come write the corresponding number.

I call on volunteers to give the missing number. I know that my most struggling students may be completely stumped and I am waiting to here someone ask "how are they doing that?" **I can't let it go on too long, though, because the longer they don't ask the question the dumber they feel. **After 4 or 5 rounds, if no one has asked, I ask** "Are they just reading my mind? How in the heck are they doing that?!"** **This normalizes the confusion and makes those kids feel better about not knowing and renews their interest. **I tell them to discuss what they think is going on and then play another round or two and have someone come up and explain it to the class.

Then I put up a table with incorrect responses in it and ask what mistake they think was made and why someone might make it. When students make a mistake they have usually found a different pattern and have often been thinking additively rather than multiplicatively.

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#### White boards

*4 min*

Students work on their own individual white boards and raise them up on the count of three. That way I can see everyone's response at once, give corrective feedback where needed, and no one can opt out. I want students to practice finding common denominators "Jake's way". I give them a pair of fractions which, when simplified, still need to be scaled up into common denominators. I want the original denominators to be big enough that they won't want to multiply them in order to find a common denominator.

I may start with 15/25 and 21/30 and then 12:16 and 20:30

After they have had a chance to start, I tell them to check what common denominators their partners are using and see if they used the same one. This is a good way to get them to pay attention to which is the most efficient. I usually don't have to give a lot of corrective feedback, because they help each other.

*expand content*

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- UNIT 1: Order of operations & Number properties
- UNIT 2: Writing expressions
- UNIT 3: Equivalent Expressions
- UNIT 4: Operations with Integers
- UNIT 5: Writing and comparing ratios
- UNIT 6: Proportionality on a graph
- UNIT 7: Percent proportions
- UNIT 8: Exploring Rational Numbers
- UNIT 9: Exploring Surface Area
- UNIT 10: Exploring Area & Perimeter

- LESSON 1: Which is the blackest?
- LESSON 2: Designing the floor pattern
- LESSON 3: Breaking down the design
- LESSON 4: Part to whole ratio
- LESSON 5: The secret side of ratios
- LESSON 6: Comparing ratios
- LESSON 7: Ratio soup assessment day
- LESSON 8: Scaling up ratios
- LESSON 9: Terminology for scaling ratios
- LESSON 10: There's an ap for that!
- LESSON 11: Let's get organized!
- LESSON 12: Navigating a data table
- LESSON 13: Mistakes & Peer Instruction
- LESSON 14: Mickey Mouse Proportions