Ratio WODB

Which one doesn’t belong?  Can you find more than one?

WODB

What is 3/5 Follow Up

One great thing about blogs, you can reflect on what you have done in class, and remind you when you haven’t done a follow up you intended to do.

 

So, I needed to go back to What is 3/5.  My students had made some good connections during that week and I wanted to see how much of it was retained.  I went with a basic warmup today.

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I decided on this because I really want students to expand their thinking past one basic shape for fraction representation.  As I expected however, the first two diagrams students drew were circles and rectangles.  I was pleasantly surprised when one student drew a pentagon.

IMAG0531

At first I was worried when I was walking around the room, typical comments on why their figure was 3/5 included “3 out of 5 are shaded.”  I am thinking, WHAT? We covered that fully just a month ago, how could they backtrack again?  My worries were unfounded when for the last explanation students showed me they were indeed thinking of EQUAL parts, not just 3 parts out of 5.  In the first responses it is easier to “explain” that way without fully relating everything.  When I asked students to revise it so that I would know exactly what they were thinking, they came up with what a few students already had (as you can see in the student work photo):

3 out of 5 equal pieces are shaded

I am proud of them for remembering the concept and how to explain it so I could understand their thinking, and told them so.

 

Multiplying Fractions via Area

Nicora has a great post on multiplying fractions, check it out here.  This is something similar that I end up reviewing with my 8th graders.  She was wondering about what I have found so here it is….

I start with AREA, Chris(@trianglemancsd) may call it an array, whatever works.  I started approaching multiplication this way since they are pounded with Area problems in 7th grade.  The concept and procedure is ingrained and ready to be built upon.

I start with a multiplication problem I am pretty confident all of my students will know, such as 3×5.  Students will quickly tell me it is 15.  I ask how they can represent that math question and answer, and I get a variety of methods.  I will get at least one student who shows me an area problem.  I pick that example and we talk about area and what it means, how to represent it and I ask them to draw a diagram on graph paper showing it.  This is what I get:

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This is exactly what I want to see, even though I kinda manipulated the students into showing me a representation I want to build upon.  I then ask them to show me representations of 3 and 5 using the graph paper grid, and then I create the following construction to connect to their previous representation of  3×5.

IMAG0485

 

This is not a big stretch from what they already do and know, so I have them practice it a bit and come up with new problems in order to try and stump their friends or even me.  Students try and get tricky, and at least one will introduce the idea of fractions into the mix.  Once again, they are leading their learning where I would like them to go.

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Students make the grid naturally like they did for 3×5.  When asked why, I have had some who say “because we did it for the other problem.”  A lot of mathematical thinking there, although some will pick up on it and say it is a representation of the area of 3×5.  I will ask for other ideas, and students will tell me that the reason we have a grid of 15 is because it is the LCM of the denominators.  Surprisingly, the shading of the grid has never been an issue with any of the students I have worked with.  Typical student talk is “you can only shade on the first row, but only three of those.”  When I ask why, students have said that it is the area model of  1×3.  A few students will reply “because it’s a double grid Mr. Anderson, you have a grid of unshaded squares and a grid of shaded squares.”  Some students have even suggested that I separate the drawings of shaded and unshaded squares- creating area models of the numerator and denominator.  This really relates to the old rule of “multiply numerators and multiply denominators”, and students made that connection on their own.

The thing that I have to be careful of with this is that they use two fractions that are less than one.  As long as they do, this works out very well for them.  Students do try to do things like 3x(1/5) and get confused on the outcome, which would also look like 3/15 or 1/5.  When I have student discussion about this, I always have one student who brings up that they can change the representation of 3 to 3/1.  They then draw their separate area models for the numerator and denominator to find their answer of 3/5.

 

 

Shoebacca -3Act Math

ACT 1:

Show them this clip and this picture:

Shoebacca_The+Big+Shoe_3

What questions do you have after seeing this Video and Picture?

Ask students to write down their questions, I normally ask students to find at least 3.  When I observe that most students have questions written, I ask them to share those questions with their neighbor.  I then throw up a Microsoft Word document and start typing down questions students supply.  Students from my classroom came up with all sorts of different questions, some we can easily answer and others that we can’t.  I am looking for a key question or questions to start this lesson.  If students do not ask one of these questions, I tell them that I hope I can answer most of the questions provided, but that I need them to consider one of these questions first.

  • What is the shoe size?  Is it like a 52?
  • How big of a foot would you need to wear that?
  • How big/tall would a person be to wear this shoe?

Any of these type of questions will lead students down the inquiry I hope to explore with them.

 

ACT 2:

This part can take a few different paths, depending on your focus for the day (you could use this idea with Proportions, Scatter Plots and Statistics).  I started this part by asking what type of mathematical operation or ideas they would need to find an answer.  The most common ideas students came up with are:

  • guess/ estimation
  • fractions
  • proportions
  • equations
  • graphs

These are all excellent ideas and I place them on the board as reminders to where we want to go throughout the investigation.  Depending on what focus I use this for, students normally fall into 3 strategies for finding our answer.

  1. Proportions: Most students want to make a proportion relating themselves to Shoebacca.  This is a great strategy to begin with, but since I teach 8th grade students, I want them to push themselves.  This strategy would work well with our 7th grade standards.
  2. Statistics: One thing that comes up when we start exploring Shoebacca is the fact that we can’t accurately predict how big someone’s foot depending on their height.  Students immediately think of mean (or average) and start taking down everyone’s foot size and height (this is also great review of measurement and conversion- students this past year put up their height in feet, inches and I asked for inches only, and for some classes, cm).  Then the next step students wanted to take was strategy 1, they made a ratio of their class average and made a proportion.  Once again, a great way to tackle this problem, but I am relentless- I want them to really explore different options of solving this problem and how those strategies relate and compare.
  3. Graphs: When I first designed this problem, this was naturally MY first thought.  One thing all great teachers do is anticipate how students will approach a problem and then tie it into the method they are presenting.  I anticipated the first two approaches and hope my students will make connections between them and this one.  One of the reasons I like looking at graphs is it allows students to use software to enter data, find trends, and display it graphically.  Students take measurements of their class, place it in a spreadsheet and create a graph.  They can then look at lines of best fit (before you automatically create it with the program) and make a conjecture based on their data.

Some other considerations:  The length of the trailer which will determine how big the shoe length is, can be varied.  I normally go with what the students discover- many will do searches on trailer sizes.  The most common lengths I get in class are 12′, 14′,16′, 18′ and 20′.  This variation of the trailer size could be a extension of the problem.

Here are the statistics for this year: 8th grade shoe 13-14

 

ACT 3:

These are the results my students got this year:

  1. Proportion: varies with individual   65″/10″ = ?/240″ so ? = 1560″ or 130′
  2. Statistics: 66.8″/9.7″ = ?/240″ so ? = 1652.8″ or 137′ 9″
  3. Graph: students drew a line of best fit and got 1080 or 90′

Some great discussions resulted in the difference in these estimates for height.