What’s 192 plus 192?

My son, “Daddy, what’s 192 plus 192?”

Me, “What’s 200 plus 200?”

My son, “400.”

Me, “What’s 192 plus 192 then?”

My son, “I don’t know.”

Me, “Try breaking it down like this. 100 plus 100 plus 90 plus 90 plus 2 plus 2. Does that help?”

My son, “Hrmm. Okay, it’s 284. No, I mean 384. Do you want to know why I want to know?”

Me, “Absolutely. Why do you want to know what 192 plus 192 is?”

My son, “I started with 1 plus 1, and now I’m at 8192, and I want to know what 8192 plus 8192 is.”

Me, “Do you mean you added 1 plus 1 to get 2, and then 2 plus 2 to get 4, and so on?”

My son, “Yes.”

 

What makes this exchange especially interesting to me is that I remember attempting to do exactly the same calculations my son is doing, at about the same age.

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Problem solving with exponents

This post by Michael Pershan has an excellent way to give students opportunities to play around with exponents and problem solve with them.

The basic idea is, give students a bunch of dots in an array, and ask them to find ways to change the picture so that it represents a power.

A related, and more generic, question is:

How do I change this diagram/image/equation/formula so that it represents <something else>?

 

Open-ended problem solving: Make the largest possible number

Use the numbers 2 and 3 only once (but any combination of mathematical symbols you know), make the largest possible number you can. Make the smallest possible number you can. Make a number as close to zero as you can.

Bonus, use any numbers you like, and then look for patterns between the numbers. In general, which collection of operations will produce the largest number for any two given initial numbers?

Thanks to Michael Pershan for the idea.

Lesson planning: Homeschooling my son

My wife and I are homeschooling my son, and so I work with him on his mathematics. For the first part of the year, our experiences have been fairly informal, but I feel like he and I will benefit from more structure, and so I’m planning on writing lessons and keeping track of the resources we have used, as well as reflecting on those lessons to help me plan what’s next. My son’s number sense and understanding of number operations is fairly strong. Our work here will continue informally, but for at least the next month, I’m going to focus on geometry with him.

The Common Core cluster standard in grade 2 geometry is:

2.G.A. Reason with shapes and their attributes

And the specific standards in this cluster are:

2.G.A.1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces (Sizes are compared directly or visually, not compared by measuring.). Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Example task from Illustrative Mathematics:
http://www.illustrativemathematics.org/illustrations/1506
Example MARS task: Don’s Shapes

2.G.A.2. Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Example task from Illustrative Mathematics:
http://www.illustrativemathematics.org/illustrations/827
Example MARS task: Half and Half

2.G.A.3. Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Example task from Illustrative Mathematics:
http://www.illustrativemathematics.org/illustrations/826
Example MARS task: Half and Half

My guess is that my son would find all of these tasks fairly easy to do, but I plan on spending our first few days together doing these tasks, and seeing if I can surface my son’s understandings and misunderstandings related to these standards. In terms of a more open-ended project, I suspect that the creation of mathematical art with polygons may be appropriate. For example, my son and I recently created as many hexagons as we could using his pattern blocks (We made one mistake. Can you find it?).

Hexagons

Open-ended problem solving: Fermi problems

A Fermi problem is an estimation problem, where the objective is to answer a question that requires a number of assumptions be made, and for which the final answer can usually be best given as a range of possible values.

Some examples of Fermi problems are:

Could the forests of North America serve as the lungs of the world?
How many phone books are delivered in NYC?
How many piano tuners are there in the United States?
How many different civilizations like ours exist in our galaxy?

 

An activity that students could do would be to brainstorm their own Fermi problems, and then choose one of the problems to attempt to solve in a small group.

Open ended project: Investigate facial recognition algorithms

Here is a sample outline for what a project about facial recognition algorithms might look like.

  1. Investigation phase:

    What is automated facial recognition? How does it work? What does a computer do to look at two different photos of someone and determine if they are the same person? What algorithms are in use? How successful are these algorithms? Where is facial recognition used?

  2. Problem posing:

    What factors affect the success of automated facial recognition? How efficient are these algorithms? What are the problems with these algorithms?

  3. Experiment
  4. Share results

 

Open-ended problem: How likely is it a quadratic equation will factor?

This question comes from Professor Wright via Twitter:

When Professor Wright means factor, he means, be written in the form (dx + e)(fx + g) with d, e, f, and g all being rational numbers.

Here are some questions to extend this investigation:

How often does ax2+bx+c factor with rational values of a, b, and c? Does the range of numbers we choose to look at matter? What if we allow d, e, f, and g to be irrational numbers? How often does this factor if a is 1? What if a is 2? Is there a relationship between the value of a, and the likelihood the quadratic function factors? Can you write a computer program to test out your hypothesis?