##
* *Reflection: Developing a Conceptual Understanding
Which is blackest the sequel - Section 2: Warm up

The first attempt my students make to express their ideas is not always the easiest to understand. At first I wasn't sure if it was worth the valuable time it would take to try to understand their thinking. It is really tempting to just move on to the next kid until you hear the ideas or words you're listening for. I took a risk, knowing it would take more of our precious classtime and found that it was well worth it.

When explaining why graph C was the blackest my kids said things like:

- "the line goes more across"
- "the line goes more over to the right"
- " the graph for C stretches farther"

I had no idea what they were talking about until I asked for some clarification and elaboration:

**"what do you mean by that?"****"show me on the graph what you mean?"****"why does that mean it is blacker?"****"why does that make sense?"**

Only then did I understand that they were describing that the line for "C" was closer to the axis representing the black tiles, was more horizontal, and was moving more in the black direction which meant that the black tiles were increasing faster than the white tiles, making the pattern blacker.

When a student suggested that his points (3/12, 8/12, 36/12) were easier to compare, the above conversation helped my students understand why common denominators were helpful. I asked the class to decide why his points were easier and they said "it's easiest to see where the points on the graph are at the same level". I continued probing with similar questions as above. When I asked **"what does he mean by 'at the same level'?"** someone asked to come to the projector and highlight the level where the white tiles on the y-axis were equal to 12. I then could follow up with **"what's happening in terms of black and white tiles at this level?" **

- "
**why is it easier to compare**when the points are at the same level like this?" (it's easier to see that graph C has a greater number of black tiles for the same number of white tiles) - "
**are there other places on the graph where this will be true?"** - "
**how could we predict**other levels where it would be easy to compare?" (wherever the number of white tiles/denominator is the same)

**Being able to use their explainations in conjunction with the context and the graph helped us make much more meaningful mathematical connections.** I think it was through these connections that my students finally gained a better understanding of comparing fractions with common denominators.

*Developing a Conceptual Understanding: Making student thinking visible*

# Which is blackest the sequel

Lesson 8 of 10

## Objective: SWBAT use a graph to find common denominators.

## Big Idea: Common denominators are found when "points" on a graph are at the same "level" in relation to the y-axis.

*54 minutes*

This lesson morphed as I was going over the warm up. I started with the graph because I wanted them to notice that the ratios representing the different tile designs were easier to compare when the number of white tiles is equal for both. But as students were explaining where on the graph the ratios were easiest to compare they came up with this idea of the "same level". Exploring their thinking made me realize that this was a really nice model for ELL students and also really helpful for visual learners. It also allowed them to "see" the multiplication on the graph. It is so important to really listen to the ways students express their ideas, especially ELL students who may not be using the vocabulary you are listening for. If I had not asked this student to come up and show us what he meant we would have missed it entirely.

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#### Warm up

*20 min*

In this warm up finding ratios on a graph.docx I show three proportional graphs that they are familiar with and ask which one shows the "blackest" floor design. Students have many ways of seeing and explaining this and it is important that the teacher really listens, because it's easy to make the wrong assumption about what they mean. This is especially true for ELL students who may not be using the vocabulary we are listening for. At first I just want them to figure it out without numbers, just by what they see in the structure of the graph.

Most students can see that graph C shows the "blackest" design. They may say "because it goes farther". Encourage them to explain what they mean by "farther" by asking them in what direction (to the right) which is the direction in which black tiles are increasing. They may say it is more horizontal or flat. Ask them where the graph would be if the design used only black tiles and no white tiles (on the x-axis) to emphasize the closer to that axis the "blacker" the design.

As soon as students mention the ratios (coordinates at the points) shift the conversation to the numbers. It is important first to establish the fact that graph C is the blackest and have a graphic way to prove it so there is no disagreement when comparing the ratios. The most obvious observation they may start with is that there are more black tiles than white tiles on line C. At this point have them label each point on the graphs. Tell them we are going to try to ignore the clues on the graph and just look at the numbers.

Ask them which two ratios (one from B and one from C) make the comparison the most obvious (which would provide the most convincing evidence). (2:3 and 9:3) Ask them why. Answers may vary. "they are at the same level on the graph"(see highlighting) "the white tiles are equal at these points" "the denominators are the same". Model this on the graph so they can see what the common denominators mean in the ratios. Have them look for the same pair of ratios on graphs A & B. They may also point them out for A and C.

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#### Exploration

*25 min*

When students have identified the points 3:12 and 8:12 on graphs A and B I ask them how we could have predicted the "level" at which it would have been easy to compare 1:4 and 2:3. How could they have predicted the 12. Students will try to explain it graphically at first. Then take the graph away. Write just the ratios and ask how they could have predicted the denominators would "level" at 12 just by looking at the numbers. It shouldn't take long for students to suggest multiplying the two denominators. "Will that make a common denominator every time?" (yes)

Students now work in their math family groups on a series of scenarios in which they figure out which is blackest and are given only the ratios. Warm up which is blackest ratios.docx

The first one shows ratios with common denominators. When explaining their choice they may only mention black tiles. If they do, I show them a figure they saw in a previous lesson (Which is blackest?) Which is blackest handout.docx and remind them they can't just talk about the black tiles. There is something really important that must be true for the white tiles. This encourages them to point out that "for the same number of white tiles (denominator) one has more black tiles".

When they get to the last scenario, if they don't try common denominators ask them what made the other scenarios easier (common denominators) and ask if they could scale these ratios up to the same "level" so they have common denominators. This is when a discussion of common multiples might be appropriate. Many of my students have a limited mastery of their multiplication facts, so showing them how to list multiples in order to find common denominators may be helpful. I also point out that multiplying the two denominators always results in a common multiple. It's easiest to show this in a multiplication chart and it fits into their idea of "levels".

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#### White boards

*9 min*

I don't want students to be bogged down by their insufficient mastery of multiplication facts, so, those that feel they need it can use their multiplication table from their daily planner. They can completely understand the process of scaling up ratios and finding common denominators but can get the whole problem wrong if they mess up the multiplication. Unfortunately, this level of frustration can make them think they really don't understand and block their learning.

Students do their work on individual white boards, but can work collaboratively with their math family groups. They hold up their white boards at the same time so I can give corrective feedback if necessary and no one can opt out.

I give a pair of fractions and ask them to compare which shows the blackest floor design or the greatest amount. I start with fractions that both need to be changed. Although some may think it is easier to find common denominators if only one needs to be changed I have found that the one multiple that is often missed is the first one. Also, it requires more reasoning to determine that only one needs to change. I also like to start out with fraction that match the fraction circles I have access to in case they try to tell me they are equal.

2/3 and 3/4 (they already have experience with these denominators 1/2 and 2/3

If they needed my fraction circles for scaffolding I try a couple more (1/2 and 1/3, 2/5 and 1/3).

If they didn't need my fraction circles I move on to 1/2 and 3/4, 2/3 and 5/6. I expect them to come up with different ways as some students use the lowest common denominator and others do not. We finish by explaining how they can both represent the same thing and we can make the same determination about which one is more.

#### Resources

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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: Patterns in the graph
- LESSON 2: What is it trying to tell us?
- LESSON 3: Keep it in proportion
- LESSON 4: Recognizing proportional relationships in a graph
- LESSON 5: Are they proportional?
- LESSON 6: Writing arguments
- LESSON 7: Clarify & Correct arguments
- LESSON 8: Which is blackest the sequel
- LESSON 9: Scaling up, scaling down, scaling all around
- LESSON 10: Ratio assessment day