Erica Burnison C. A. JACOBS INTERMEDIATE, DIXON, CA
7th Grade Math : Unit #5 - Writing and comparing ratios : Lesson #1

Which is the blackest?

Objective: SWBAT understand and explain the meaning of a ratio.
Standards: 6.RP.A.1 MP1 MP2 MP3
Subject(s): Math
60 minutes
1 Intro & Rationale - 0 minutes

This lesson introduces the concept and vocabulary of ratio within the context of designing a tile floor for the bathroom. They answer the question:

Which is blackest?

Students will engage in argumentation as a way to both make sense of the problem as well as to negotiate the meaning of ratio. I ask students to explain a phenomena before I give them the vocabulary needed to do so. I believe this helps connect the big ideas to the terminology and makes the term more relevant to student needs. It also allows them to use language that is descriptive and comprehensible to them in order to connect meaning to the concept. The exploration phase of this lesson relies on students generating ideas for further exploration, so the role of the teacher is really to listen to the small group discussions for entry points to bring to the attention of the whole group.

I use some informational text to introduce the terminology and notation because my students have traditionally been far too reliant on the teacher as the giver of information. I want them to learn to be more self reliant and turn to text or to each other. 

2 Warm up - 20 minutes

For the Which is blackest? Warm Up students are given three black and white tile patterns for a bathroom floor design.  The prompt asks, "Which is the blackest?" Students are expected to explain their reasoning in making  choice. They need to consider several factors:

  • The 3 floors are all different sizes, but have the same number of black tiles. 
  • Students often think that they are all the same because they have the same number of black tiles. 

As I watch students work on this warm up I sometimes ask, "Which one looks blackest? Why do you say that?"

As I circulate I encourage students to continue giving evidence and asking for evidence. I'll use prompts like:

  • What makes you say that?
  • Where in the figure do you see that?
  • What question could you ask her to help you understand what she means?

Students often respond with justifications like:

  1. Figure A has the fewest number of white squares, so it is the blackest.
  2. The sizes of the floors are getting bigger, but only white squares are being added.
  3. As the floors get smaller, the white tiles are being taken away. 
  4. They may explain that the black tiles are covering more of the floor in the smaller floors.
  5. Some will observe that in figure C the black and white tiles are more "even". 
  6. You may hear them say there are more black tiles than white tiles in figure A. I would point out that this is also true of figure B and ask why figure A appears blacker than figure B. All of these ideas involve comparing the white tiles to the black tiles. Once you hear these types of observations being made you can move on to introduce the idea of ratio.

Observation #1 has the potential to end the discovery in a misconception, because the students are not yet comparing the black and white tiles. The pattern observed here does not hold up in all three figures. Because of this, I am prepared to provide counterexamples. I have this other handout Which is blackest handout extended ready. It extends the pattern in Figure A so that it has more white tiles than floor B and an equal number of white tiles to floor C, but still appears the "blackest".

After allowing students to work together on this task, I will have math family groups share out with the whole class. I will ask each group, "Which Figure is blackest? Why?" I will listen carefully to their explanations and revoice explanations that I want the whole class to think about. 

The students will get there!
Adjustments to Practice

I was really nervous going into this lesson that the students would not be able to make the necessary observations and conclusions to define the idea of ratio. I want to do less of the telling and have them do more of the figuring and discovering, but I was not entirely sure that they would be able to. I found that students are much more capable than I thought and they often exceed my expectations

The most important thing for me as a teacher in this lesson was to listen to the language students were using to explain their observations and encourage them to elaborate their thinking. Because they don't initially use academic language it can be difficult to understand their meaning. Don't fear that their observations are on the wrong track! Instead, probe them for more detail. When they say things like "As the floors get smaller, the white tiles are being taken away" ask them to tell you a little more about that (or "how does that make this floor blacker?"). They may reply that when only the white tiles are taken away and none of the black tiles are, the floor ends up with more black than white tiles. 

The students may be thinking much more concretely than you do. When I first heard some of the ways in which students were explaining their ideas my heart sank, because I didn't really understand what they were thinking and how it applied. Once I probed for more, however, their thinking and its relevance became clear.

Using their own language helps them attach meaning to the quantities and will help them make sense of the new terminology. It also creates a need for more precise language which they will more readily recognize and accept when it is introduced. When students work in a group and hear all the different ways to explain the same idea, they gain a more complete nuanced understanding of the concept and the terminology.

3 Exploration - 30 minutes

After students have shared their initial observations, conclusions, and explanations I tell them that I noticed at first they were only counting the black tiles, but found that wasn't good enough to answer the question. I point out that they discovered the need to look at both black and white tiles. It sounds as if they are trying to compare the two. As I hand out the Ratio reading.docx reading I tell them they will find some vocabulary to help them do this better. I give them about 3 silent minutes to read to themselves then see if they can use the idea of ratio to compare the black and white tiles and answer the question.

The key to this entire lesson is just being a listener and let students generate the ideas and questions for the class. When you hear an idea ratio introduction student ideas to follow.docx that will shift the level of understanding have that student explain and show their thinking to the class and ask the class to pursue this idea. Be sure to leave the discussion of the disagreement or critique/verification of the suggestion to the math family groups. Circulate and encourage explanation and use of evidence! Tell them to show each other what they mean. Ask others in the group to explain the reasoning of their peers. If they start to talk directly to you tell them you are walking away so they will turn the discussion into the group. It is so important they do not see you, the teacher, as the giver or judge of information. They need to convince and be convinced by each other!

After students hear another student's idea for the first time I want them to make sense of the new information, so I may ask them to try it out for themselves or verify it.

After students have had a chance to digest the reading and use ratios to explain which figure is blackest I like to debrief the reading and ask them what they found the most helpful. They tended to like the examples and the different models like OOOOLLL for ratio.

4 Wrap up - 4 minutes

I give students a small piece of grid paper and ask that they draw a design that is the same "blackness" as floor A. I am looking to see that they are trying to use the same ratio of black and white tiles. They may have difficulty making a design as their pieces go "out of bounds". I tell them this happens all the time when tiling an actual floor and just to outline their design pieces that are not broken up to show the ratio.

If they are obviously not repeating the same ratio I refer them back to the sentence frame "for every...."