Try...Check....Revise (Math 1512)

I am infamous for the saying "If at first you don't succeed, try...and then try again!"  I have worked with children in childcare settings for 12 of the last 13 years and my goal was to always make the best big people out of the little people I blessed to share time with.  Part of achieving this goal was to make them independent thinkers.  While it often seems easier to just "do it for them," in the long run kids do not benefit from this.  Children need to learn that with success comes failure and many times there will be several failures before finally succeeding.  The most basic situation I can think of is completing puzzles with toddlers.  Little kids LOVE to dump out puzzles, and then when the pieces do not easily go back into place chances are they will walk away.  I loved to sit with my little ones and "coach" them through reassembling the puzzle.  "Turn this piece...oops! , try again, ok...now try this piece here, wow...good job! "  Now, how does any of this pertain to our math lesson???
Try...Check....Revise!
*Use the given facts to try and check one answer. Then revise and try again until you solve the problem.  "If at first you don't succeed, try...and then try again!"  

-Solving Linear Equations (equations where the value of y is known, they can be associated with linear functions)  Try-Check-Revise is relatively efficient when the numbers in the equation are integers.  The equation 12x + 35 = 131 can be solved using try-check-revise.

To begin, use number sense to estimate the solution.  For this particular problem one may estimate the solution to be 10.  Substitute 10 for x,

12(10) + 35= 155, since 155 is greater than 131, the actual solution must be less than 10. "Try again!"  

After a couple more attempts the solution proves to be 8.... 12(8) + 35 = 96 + 35 = 131 

A spreadsheet can also be used to solve linear equations, and it provides a high-speed application of the try-check-revise strategy.  This approach also starts by using number sense and estimation to think of a reasonable range for the solution.  This independent thinking gives students the opportunity to "attempt" to solve the equation based off of their number sense and then be willing to continue problem solving until the correct solution is found.  This was a fun web site that I found dealing with linear equations....lots of opportunity to Try-Check-Revise! Cool Math- Linear Equations

 

Fractions, put to use! (Math 1510)

My oldest daughter is 10 and in so many ways takes after her mother, which at times is quite scary.  We share a love of cooking.  I thought at first this common interest would be wonderful, but so far it has been quite a patience tester for this momma.  I have been cooking for about 25 years.  I am the oldest of 9 siblings and my mother absolutely despises the kitchen.  As soon as I was old enough to follow her directions, I was responsible for cooking.  It did not take long and I was creating my own "masterpieces" to share at the family table.  I have to remind myself often (like every 7 1/2 seconds! ) that my daughter is LEARNING.  It is going to take time for her to do things quickly and without the kitchen looking like a tornado went through it.  To get into the fractions lesson here, the other day I decided that we were going to make my grandmothers Lumberjack Molasses Cookies.  My children and I love these cookies.  My grandmother has been healing in a nursing home for almost 7 months now and we are planning a visit to southern Minnesota soon to visit, so I knew she would appreciate having this treat brought to her.  On a side note, these cookies freeze very well!  We started by measuring out the amount of molasses that I had in the cupboard and Aubrie (my daughter) said, "just a little more than 1 1/2 cups."  Great....this means we could make 1 1/2 bathes of these cookies.  Knowing this is what I wanted to do, I gave her the recipe and asked her to figure out the amounts for the rest of the ingredients to make 1 1/2 batches.

*The original recipe reads: 
2c. white sugar                      1tsp. salt
1c. shortening                        2tsp. cinnamon
1c. molasses                          2tsp. ginger
1c. sour cream                      1/2tsp. cloves
2tsp. baking soda                  6c. flour
2 beaten eggs
Mix as for a cake, refrigerate over night, roll out, cut, bake- 350° for 12-15 minutes
Lumberjack Frosting:
2 1/4 cups Powered Sugar, add 2 unbeaten egg whites and whip for 3 minutes.  While beating add 1/4 tsp. Cream of tarter and 1tsp. vanilla.  Beat for 3 or 4 more minutes.  Frost cooled cookies.

At first my daughter's solution to figuring out measurements was to only fill the measuring cups and spoons that she had used, 1/2 full this time.  While this is a creative solution, I explained that simply guessing what seems to be 1/2 full would not be exact enough and her cookies would probably not work out.  So we sat down together and started multiplying our measurements by 1/2 so that we would know exactly how much we needed of each ingredient.

Our multiplication of fractions began:
2c. sugar.... “easy mom, 1/2 of that is 1c”.....ok, no fraction needed :) (3c.)
1c. shortening..... “half of that is 1/2c.” (1 1/2c.)
1c. molasses....... “half of that is 1/2c.” (1 1/2c.)
1c. sour cream..... “half of that is also 1/2c.” (1 1/2c.)
2tsp. soda..... “half of that would be 1tsp.” (3tsp.)
2 beaten eggs.... “so, we need 1 more egg” (3 eggs)
1tsp salt........ “plus another 1/2 tsp” (1 1/2tsp.)
2tsp cinnamon..... “and another 1tsp.” (3 tsp.)
2tsp ginger..... “plus 1tsp.” (3tsp.)
1/2 tsp cloves..... “half of a half”.....ok, lets figure this one out.
  *We divided using the common denominator method.  We lucked out that our denominators were alike...  1/2 ÷ 1/2= 1/4....ok, so we need 1/2 tsp + 1/4 tsp,
finding a common denominator and then adding is the plan this time too!
1/2 + 1/4= 1/2 becomes 2/4 + 1/4= 3/4 , we need (3/4tsp.)
6c. flour.... “half of 6 is 3, 6+3=9” (9cups)

Frosting:
2 1/4c. Powered sugar....”well half of 2 is 1” and “half of 1/4....hmmmmm?”
again we will multiply using common denominators.  1/4 x1/2= 1x1=1, 4x2=8, = 1/8, so we need 2 1/4c. plus 1 1/8 cup....with the complexity of adding these two fractions we just left it at this, she added 3c+1/4c+ 1/8c. of powered sugar to the bowl (3c+1/4c+1/8c)
2 unbeaten egg whites “plus one more white”
1/4 tsp. Cream of Tarter “ugh, another 1/4? well half of 1/4 is 1/8”  * she did the same thing this time that she did with the powered sugar, (1/4tsp + 1/8tsp)
1tsp. vanilla “and another 1/2” (1 1/2 tsp.)

This entire process of converting took us a while, but I feel like she truly understood the method behind properly figuring out the halves. 

The cookies were then mixed and refrigerated, the next day baked, frosted and enjoyed them!! And....the half-batch we made is stored in the freezer for our special visit with Grandma next week!
 
                                              

My feet or your feet?? (Math 1512)

 

                                                                                                                                                         

                                                                   In the house and on the street,

How many, many feet you meet.

Up in the air feet

Over a chair feet

More and more feet

Twenty-four feet.

Here come more and more..................................and more feet!

Left foot. Right foot.

Feet. Feet. Feet.

Oh how many feet you meet.

Measurement involves a comparison of an attribute of an item or situation with a unit that has the same attribute.  Lengths are compared to units of length, areas to units of area, time to units of time and so on.  Before anything can be measured meaningfully, it is necessary to understand the attribute to be      measured. (definition from:http://www.teachervision.fen.com/measurement/printable/56860.html?detoured=1)

Some basic measurement terminology:

*attribute- the aspects that can be measured

*length- the amount of distance between two points or objects

*area- the amount of space that needs to be covered

*volume- the amount of space that needs to be filled

*mass- how heavy or light something is

*temperature- how warm or cold something is

*time- how long it takes for something to happen

Choosing the type of Measure and Unit:

-distance between two towns- length

-a person's height- length

-carpet needed to cover the floor- area

-amount of land owned by a farmer- area

-amount of water to fill a swimming pool- volume

-amount of liquid medication for a child- volume

-amount of weight in an elevator- mass

-how warm it is outside- temperature

 

To measure something, one must perform three steps:

1. Decide on the attribute to be measured.

2. Select a unit that has that attribute.

3. Compare the units, by filling, covering, matching, or some other method with the attribute of the object being measured. 

In order to consistently measure something we must use a standard unit of measurement, that is, measures must mean the same thing to everybody.  Two standard measurement systems that have been developed and agreed upon are the English system an metric system.  

This website gives great information and illustrations on these two systems.....Enjoy!

*English System

*Metric System  

Teaching measurement is a very complicated lesson.  I spent quite a bit of time looking through this site created by IXL.  The lessons are divided up age-appropriately and I am amazed as to the amount of information that is taught at even the youngest of ages.  This is the section specified for first grade.
*Measurement- First Grade.....with resources like this available I am excited to have a classroom full of students roaming the classroom with rulers and thermometers in hand!

Lattice Multiplication (Math 1510)

Lattice Multiplication-
There has not been very many times during this class that I felt I had bragging rights, but after mastering lattice multiplication I was like a 10 year old who just bought the coolest new iPod!
We were actually camping with a group of friends, and I was the super cool friend that was in the camper working on homework.  I had frequent visitors requesting my presence at the campfire, but knowing the chapter deadline was quickly approaching I was determined to accomplish some school work.  When I had the ahhhhhaaaa! moment with Lattice Multiplication I emerged and challenged the others.  Keep in mind this was a group of construction workers, police officers, social workers, business owners....all of whom work with numbers on a relatively frequent basis and are only in their early 30’s.  I handed them all a piece of paper and a pen, and we began.....yes, there were a lot of laughs and about 15 minutes later they had all successfully completed their first Lattice Multiplication!  Of course there was a deep conversation as to why this method would be taught, and “what is wrong with learning multiplication the way we did?”  I explained the importance of teaching in various methods in order to accommodate various learning styles.  I have since worked with my older two children on this method and a few of my sisters, I enjoy exposing others to something new.



This information was found at: Cool Math 4 Kids

Lattice Multiplication   A fun and easy way to multiply bigger numbers (page 1 of 4)

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This is a really cool method for multiplying bigger numbers.  It's a lot easier than the regular way and it's kind of fun too. Before I show you the whole thing, I need to show you how to do some smaller stuff first... We're going to multiply easy numbers from our times tables, but we're going to put our answers in special boxes. Here's the box...  It's got a spot for tens and a spot for ones: When we look at a number, remember... So, we'll put the 1 (the tens digit) in the top spot and the 4 (the ones digit) in the bottom spot

What do we do when our answer is just a single digit? We'll just be doing this multiplication box over and over again...  Then, we'll do a little easy addition and we'll get our answer! OK, let's start doing one: Set it up like this: First, we do   Then, we do Then, we do Then, the last one: Now, just add down the diagonal stripes...  Start at the bottom and work your way up the stripes: We get our answer by reading down the left side and across the bottom.  (Just ignore the first zero!)   This YouTube Video also gives great step by step directions on Lattice Multiplication: While some times it is scary to try something "out of the normal," I think this method will be very helpful for many students...and Fun too!!

Graphically speaking....Math 1512

While I was doing the lesson on graphing in our text book, I thought a lot about how elementary students could put the various types of graphs to use.  Last school year one of my daughters was in Kindergarten.  As a parent I was amazed as to the amount of graphing she did.  Most of the graphs were either tally mark style or bar graphs, but none the less very impressive.  I was impressed that she was able to properly gather, record and display the information in an effective manner.  For many students, graphing allows them to have the important visual reinforcement they need to properly understand the results of the information they were working with.   Graphing also gives students the opportunity not only to use their mathematical skills while computing the results, but also communication skills while gathering information and also writing and composition skills while displaying the
information.                                       

*Simply stated, graphing is the display of data!

Our text explains that, every data display "tells a story."  Students are great story tellers, so why not take advantage of this and see just how creativity and knowledge can combine?

*A graphic display of data should always have certain characteristics.  Specifically, a graphic display should have a title explaining what the data represents and labels for the units or categories used in the display.  The display itself should include scales that show the units or categories used in the display. 

 

The display of data that is done in a visual form can involve these various types of graphs:

*Frequency tables: various data categories of interest are listed in a table

*Tally marks: are placed in each category to indicate the frequency with which observed data items fall into a given category

*Dot plots: a horizontal number line where dots are placed above the corresponding scale marking

*Stem and leaf plot: a vertical display where the larger #'s 10's, 100's etc...digit is the stem and the units digits are the leaf

*Pictograph: A graph commonly used in newspapers, a small icon or figure is used to represent the data values

*Histograph: similar to a stem and leaf plot, but without the individual leaves

*Bar graph: used when categories represent discrete groups, used to show a variation over time or to compare items

*Circle graph or Pie graph: often used to compare parts of the total in a data display, a circular region sub-divided into a number of pie shaped sectors.

*Line graph: used to describe either discrete or continuous events, constructed by plotting the ordered pairs of data on a rectangular coordinate grid.

*Scatterplots: displays two-variable data that includes a selected number of ordered pairs

 

These two websites contain some great examples of the various types of graphs:

Beacon Learning Center 

Types of Graphs 

 

 

A MUST TRY!!!!  This site is aimed towards kids and lets you create various types of graphs.  After all data is input and your graph is created there is an option to save and even print....a super fun classroom resource!

Create A Graph 

 

Estimation....Math 1510

Just Guess, Guess-timate, Close Enough!  These are all common terms associated with Estimation.  To estimate means to approximately calculate.  Common examples of estimations can be costs of items, measures, quantities and values.  The skill of estimation can be very useful in everyday life.  Probably the most used estimation of cost is done while grocery shopping.  Before the days of credit cards it was essential to know that if you went into the store with $50.00, you would have enough money to purchase all the items you placed into your cart.  I remember "keeping track" for my mom as we paced up and down the aisles many times.  Of course in an effort to avoid the embarrassing event at the check out of not having enough money, I was told to round-up when estimating.
I found this neat website that has all sorts of ideas for teaching children various estimation skills at the grocery store.  Math at the Grocery Store

Computational estimation is a process for finding a number reasonably close to the exact answer for a calculation.  There are four computational estimation techniques: rounding, substitution of compatible numbers, front-end estimation, and clustering. 
*Rounding- the process of replacing a number or numbers in a calculation with the closest multiple of 10,100, 1000 etc... 

How to round numbers

Make the numbers that end in 1 through 4 into the next lower number that ends in 0. For example 74 rounded to the nearest ten would be 70.
Numbers that end in a digit of 5 or more should be rounded up to the next even ten. The number 88 rounded to the nearest ten would be 90.

*Substitution of compatible numbers- involves replacing some or all of the numbers in a computation with numbers that are easy to compute mentally

Compatible Number Strategy
Compatible numbers are number pairs that go together to make “friendly” numbers.  That is, numbers that are easy to work with.  To add 78 + 25 for example you might add 75 + 25 to make 100 and then add 3 to make 103.

*Front end estimation- involves calculating with the leftmost, or front-end, digit of each number as if the remaining digits were all zeros

Front-End Estimation
This strategy involves adding from the left and then grouping the numbers in order to adjust the estimate.  For example 5239 + 2667 might be calculated in the following way: Seven thousand (5000 + 2000), eight hundred (600 +200) – no, make that 900 (39 and 67 is about another hundred).  That’s about 7900

*Clustering- involves looking for the number about which the addends cluster and then multiplying by the number of addends

Clustering in Estimation
Clustering involves grouping addends and determining the average.  For example, when estimating 53 + 47 + 48 + 58 +52, notice that the addends cluster around 50.  The estimate would be 250 (5 x 50)

As you can tell when it comes to estimation there are several options to help guess-timate. The approximate answer you are looking for can easily be found as long as the situation allows for "close enough" to be good enough!

Math 1512 **I'm seeing stars**

Star Polygons!  Wow...this is quite the procedure to figure out, but it is so neat to see the end result.
*Utilizing a circle, equally spaced points on the circle are connected in an order specific to each problem and completed the star.

For example: n=the # of equally spaced points on the circle
                      d=the dth point to which the segments are drawn

Points ABCDE are spread equally on a circle. For the five points, every two points are connected by a line.  This is denoted {n} ={5 }.   
                                    d       2

                        

   * I found this example on Hyper Flight-Geometric Construction, and thought it would be a great way to put clock templates to use in the classroom for a completely different purpose than telling time!

A five pointed star just for kids. If you can tell time you can sketch this and other stars by hand. Oh, use the template at first.

The illustration above is not copyrighted.

Draw a star, any perfect star. Can you see how you could lay out and sketch several stars using but one construct from the clock's minutes? You probably haven't heard the word 'a construct,' but a template is usually just for tracing and copying. Can you join the points by skipping some? Odd and even number of points makes a big difference. 

After mastering the skill of equally spacing the points on the circle and connecting the segments, the lesson goes into greater detail on calculating the dent angles and point angles.

*the point angles would be best described as the star point angles

*the dent angles would be best described as the angle formed by the bent line between two points that form the point angles

As best described by our text book, (O'Daffer,2008,p694) star shaped polygons have n congruent point angles with measure a, and n congruent dent angles B such that B= (360) + a 

                                                                                                              n

A six pointed, start shaped polygon with point angle 30° is denoted by 630°.  For this measurement the dent angle can be calculated by: (360) + 30°=90°  

                                              6

Constructing the star shaped polygon can be done as follows:

     a. Plot 6 equally spaced points on the circle

     b. Connect two of the points A and B, and construct 45° angles at those two points, which then   produces a 90° dent angle.

     c. Using a compass the other dent angle points can be drawn

I had several circles drawn on many pieces of paper at first when I was starting my drawings and then realized that I could utilize the same circle several times by borrowing my kids marker set to simply draw each star in a different color.