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Summer Math

In addition to playing Timed Arithmetic to keep up basic skills, I like to do some math enrichment with my children during the summer. Here are some examples of things that I have done with my children. Click on 'Summer 20XX' to see the example.

Summer 2008

I got my kids interested in the Scratch programming language and I quickly realized that having some graphing skills would enhance their ability to control the position of sprites on the page. I decided to make a large grid to make it more fun and use candy as markers that they could eat after the lesson was over.

I made a giant polar-rectangular grid by piecing large pieces of white paper together and then drawing a rectangular grid, followed by a polar grid on top of that. I drew radial lines every 15° from the origin. I drew all of my lines in pencil first and then when I had them all drawn in correctly, I went over them with permanent marker. If you need to see an example of a polar rectangular grid, check out the rectangular-polar grid generator on this site. Click on: toggle axis, Grid Type-combo, Grid Size-XL. Here is a list of more specific instructions:

  1. Tape together large sheets of white or light colored paper such as artist paper. Use a tape that you can write on or just tape the back side at first so that you can write on the front side.
  2. Use a yardstick or other large straight edge to draw a horizontal and vertical line through the middle.
  3. Pick a scale such as 5 inches per unit and make scale marks along both axis.
  4. Draw horizontal and vertical lines through your scale markings to make the rectangular grid.
  5. Have someone help you draw circles by tying a string around a pencil. Have a loop end of the string around a round object such as a marker that someone holds in the middle. Adjust the string lenght so the pencil lands on the first grid. Draw all the way around. Do this for each circle.
  6. Draw your radial lines next:
    • The 45° lines is easy to draw since it goes through the the corners of the rectangular grids grids.
    • The 30° and 60° lines are also easy to draw, since they go through the grid lines at points where the circle crosses the grid. For example, when you are on your second smallest circle, you go up to the first horizontal line to find where the radial line crosses.
    • The 15° radial line comes very close to crossing the first horizontal up from the axis at the fourth circle. It should be a little inside of that circle.
  7. I also drew a large number line with tick marks every 5 inches from the center zero mark.
After getting the grid set up, and placing it on my living room floor and unrolling the number line, I would ask each child (at the time ages 6 through 11) a math question. I taught them basics of point plotting first. Here are some example questions that I asked.
  • After writing a set of Cartesian coordinates on a 3 by 5 card, I would ask the child to place a piece of candy at that point on the grid.
  • Next, I would ask another child to place another point on the grid as above.
  • Finally, I would ask the oldest child to figure the distance between the two points. We could check answers with string.
  • Of course, I would also ask questions the other way around: Place a piece of candy on the grid and ask the child to name and write down the coordinates.
  • Eventually, we discussed the polar coordinate system and I asked similar questions in that system.
  • I also asked questions on the number line, about if you start at a particular point and you move a certain distance in the positive or negative direction, where do you land? How do you represent this with an equation?
I made up many more problems on the spot, but I can't remember them all. Just try to keep it fun, and ask questions that you think they are ready for with very little explanation. The idea is to have fun with math over the summer and maybe learn a little bit of new stuff or review stuff that they have already learned so that when they see it in a more formal setting in school, it will be familiar to them.

Summer 2009

This summer I decided to focus on counting techniques. We will probably do some review with the grids later in the summer. My college students had troubles with counting problems, so I thought I would introduce the idea to my kids, but focus more on listing all possible results, so that when they get older, and they learn the theory behind how to calculate the number of ways to do things, they already have a good foundation. When I teach this to my college students again, I think I will do something similar and introduce listing early in the semester so when we are ready to talk about theory, they have the foundation they need to do so. When I want to give my kids a treat, I use M&M's™ for their manipulatives. Otherwise, I can do things like have them rearrange dolls or each other in a line. Again, I want the focus to be fun learning. Here are some questions that I have thought of so far, and I will probably come up with more as the summer goes on. (The questions and comments that are in the parenthesis, will be too difficult for most kids just learning these concepts. These questions are meant to be for the adults that are giving the questions to the kids.)
  1. How many ways can your rearrange 2, 3, or 4 different items in a line? (How many ways can you arrange any n items in a line?)
  2. Suppose you have 5 different colors of candy and you plan to eat 3 of them all at once. How many ways can you do this?
  3. Suppose you have 5 different colors of candy and you plan to eat 3 of them, one right after another. How many ways can you do this?
  4. Suppose you have 7 candies where each one is a different color. Three are in one bowl and 4 are in a different bowl. How many ways can you select one candy from each bowl?
  5. How many distinguishable subsets of candy do you have if you have 2 pieces of type A, 1 piece of type B, and 3 pieces of type C? (This problem was inspired by calculating the number of factors of a number from its prime factors. Do you see why this is essentially the same problem?)
  6. Suppose you are going distribute subsets of 8 pieces of candy among 3 kids, but since you are using them as prizes for eight rounds of a game the distribution is not even. How many ways can the 8 candies be distributed? (This problem is inspired by the number of terms from the expansion of (a + b + c)8. Do you see why it is the same problem?)
  7. Suppose you have 5 different pieces of candy and your parent decides that you had a good enough meal so you can have them all if you wish. You decide that you might want to save some for later. How many ways can select some of the candies for consumption right now? Include in your listing the choice of all candies are eaten now and the choice that you decide to save them all for later. (This is the same as asking for the number of subsets from a set of 5 items.)
  8. You have two different kinds of candy. You have 3 pieces of M&M's™, where each one is a different color. You have two different kinds of truffles. You are allowing yourself to eat exactly 3 pieces of candy and you want to have at least one truffle. How many ways can you do this? (This is inspired by the problem of how many ways can you field a coed team where at least one player must be female and you have a limited number of males and females to choose from.)
  9. How many distinguishable ways can you put the candy in a row if you have 2 pieces of type A, 1 piece of type B, and 2 pieces of type C?
Remember, that for young kids (below high school age) the focus should be on listing. Older students and adults can start their focus on listing and then try to find patterns to calculate answers without listing. Older kids can do similar problems with bigger numbers. An example of listing, for number one would be:
RGYB, RGBY, RYGB, RYBG, etc.
where R stands for red, G stands for green, Y stands for yellow, and B stands for blue.

Summer 2010

I had one child that was skipping 6th grade math. She and I were both a bit nervous about that, so I downloaded the sixth grade curriculum and made up problems to go with it so that she could do at least a sampling of sixth grade problems before moving on to seventh grade. My youngest daughter needed to practice her math facts for speed, so we just worked on that. My oldest daughter needed a break to prevent burn out, so I gave her one. However, with all 3 kids, when I run across an interesting problem, I can usually ask it and they all show interest in solving problems, so even though we didn't do as much formal enrichment, this summer, my kids are still used to being asked questions and figuring out answers and more importantly they seem to enjoy it somewhat. Of course there are times when they are not in the mood to think, so I try not to push too hard.

Summer 2011

I would like to plug some holes in my kids education and ward off problems that I see come up in college students as a result of not a strong enough background. Here is a working list:
  • Study set ideas for both discrete and continuous sets including intersection and union. For continuous sets, also introduce how to represent sets on a number line and with interval notation.
  • Properties of Numbers including commutative, associative, and distributive
  • Using properties of numbers to justify steps in a proof or in an algebraic manipulation
  • Complex Fractions: simplifying fractions inside of fractions
  • choosing scales on a graphing grid and making sure that the scale is commensurate, especially when a fraction is used as the scale
  • unit analysis for application problems, especially those that involve a lot of conversions

Coming Soon: I plan to add some more counting problems, focusing on the basics: "The Fundamental Theorem of Counting" and "The Addition Principle". I also want to put in some problems that deal with graphing principles, visualizing fractions, and working with decimals. I have run across several interesting problem types while subbing for a Math for Future Educator's course.

Last Update: February 27, 2011