Helping Students Understand Measuring Using Benchmark Fractions

This summer I taught a fourth grade group in a special education extended year program. This was a great experience for me, and I was reminded how much I like teaching elementary children.

In math we worked on reinforcing basic computation and numbers and operations. However, as an integrated learning science project, we grew alfalfa sprouts, keeping a written class journal of the progress. So each day, in addition to describing the growth of the sprouts, I asked the children to measure their length. The rulers we used had only half and quarter inches on them, but initially the students would give their answer to the nearest inch. This gave me the impetus for convincing the kids they needed to use all the marks on the ruler, and not just the numbered inch marks.

So we started learning the benchmark fractions of ¼, ½, and ¾ .There was one activity that especially helped to get students to apply what they had learned. I brought in some rope (perhas it is more accurately called twine) and had each child get a partner to cut a piece that was their height. They laid down on the floor and their partner measured them against the rope and cut it for them. Then their rope was called by their name. So my rope would have been called a Marsha. I then instructed students to find and mark half by folding the rope in two equal sections. We marked the spot using masking tape. So on my rope I would have found half a Marsha. We then found ¼ and ¾ marks. In order to do this, one end of the rope must be designated as the starting point or “zero” and the other end as “one”.

Then I posed a question on the board. How many of you does it take to go from the classroom door to the gym door? I wrote each child’s name on the board, giving them a space to report their own findings. They took their own ropes and measured how many could fit down the hallway to the gym. They found it easier to do with the help of a partner. After they had written their findings on the board, I asked them why the answers were all different. Had someone done something wrong? They knew that it was because they were all different heights. I then explained that we need a common system of measurement, and that is why we use inches feet etc.

I had gone over the benchmark fractions for several days, but after we did this activity, they started to remember them and were using them in measurement with more ease. It was an active learning project that made an impression.

A System for Teaching Beginning Algebra

The school where I work uses a program called Hands-On Equations to help students understand algebra concepts. This is my second year of helping students with this program, and I see some very real benefits. It was developed by Dr. Henry Borenson, and more information about the program can be found at www.borenson.com/.

Students use manipulatives to solve algebra equations. They begin with simple ones and progress to more complicated problems. They use a picture of a balanced scale and learn the rules for keeping it balanced; these are the same rules that are used for solving equations. This system enables students younger than those usually in algebra classes to begin to solve equations. And there are smart board programs available to go with the program. In my grade book, Hands-On Equations gets a an A!

The Transformation Room: A Game to Improve Spatial Visualization

Spatial visualization, or the ability to mentally manipulate figures, is easier for some than others, but most teachers maintain that it is an ability that can be developed in students. One activity I have found and used is one of the games on www.mathplayground.com. The activity is found in the game “The X Detectives” . Upon entering the game, the little car can be driven to various locations. One location is “The Transformation Room” and that is where students can learn about reflections, translations, and rotations. A model is given to illustrate each concept, and then practice is provided. Players try to duplicate patterns in various positions by following the directions given. It is a good mental exercise and will not only reinforce the definitions of the terms reflection,translation and rotation, but will also serve to strengthen spatial visualization.

Working Backward to Improve Standardized Test Scores

One of the projects I started this year was to again look at my state’s math standards and then focus on how they are tested in the standardized test that students take every year. In my state, we refer to these standards as G.L.E.’s (Grade Level Equivalents) and the standardized test our students take is the N.E.C.A.P. (New England Common Assessment Program)

I set up a big binder with a divider and clear plastic sheet protector for each of the math GLE’s at my grade level, and also for the previous grade level which I try to review in the fall in preparation for the test. I went to the school’s library and made copies of all the math released items from the NECAP tests. There were about five years of them. I cut apart each question and identified which GLE it tested and put it in the plastic sleeve for the corresponding GLE. Doing all the items was time-consuming, but the results were worth it.

Over the years I could see which standards were tested more heavily, and this was helpful. But more than that, I now have a question bank for each skill. When teaching a specific math skill, I can look at how questions are asked to assess that skill. I can see how different vocabulary is used, and in many cases, the question is worded differently from how I would ask it , or how our math book asks it. I can give students sample questions throughout the year, and not just at practice test time. And I can save the questions I create and add them to my collection in the binder.

This approach has certainly been working backward; I started with the test and worked to match a skill to it rather than the other way around. But I have gained some good information from it.

Better Than Luck

There is an activity I like to do with my classes when they are in the probability unit. I start by holding up my brown paper bag and telling them that there are colored tiles in the bag. I pull out one tile of each color to show them as I say there are red tiles, blue tiles, and green tiles. And there are twenty tiles total in the bag. Can they now predict how many of each color are in the bag? At this point, some try, but I point out they are just making random guesses. I do encourage them to write down their guess to see how lucky they will be.

Next, I go around the room letting each student pull out one tile from the bag without looking. They must put it back in the bag once we have seen what color it is. Another student keeps a tally on the board. We do this several times to get some data. Then we proceed to turn the data into a fraction of how many picks were of each color. The fraction is then turned into a decimal, and then a percent. Next, we use the percent to make our (educated) guess about how many of the tiles are red, blue and green. For example, if 25% of our picks were blue, we would guess that there are 5 blue tiles in the bag because 5 is 25% of 20.

After figuring out all of our guesses, known as experimental probability, and discussing our final hypothesis and the process used to make it, we are ready for the “big reveal”. We examine how close we came and most always discover that statistical data based upon probability is better that just plain old luck!

Exploring Careers

I wrote about this website last year, but didn't get my students using it as much as I have been doing with them lately. The website is:
www.xpmath.com
There is a variety of information on there including a "math arcade" where students can play games that help them with specific skills.

But this week I asked my students to do a little research into careers using this website. I typed up a worksheet that has two parts. The first part asks them to list a career or occupation and then tell how many math skills are used by someone in that field. They then have to list some of the math skills listed for that job. This is accomplished by clicking on "math jobs". The second half of the worksheet has the students list a math skill and how many careers in the database require that math skill. They then list a few of the careers. A simple click on one of the math skills gives this information.

The worksheets make an impressive display on my wall and students can see not only that math is important in everyday life, but that being proficient at math skills will give them more careeer options.

One Hundred Eighty Degrees

There is an activity I like to do to help kids remember that the number of degrees in a straight line and the sum of the angles in a triangle are the same number. It is a good visual demonstration to drive the point home.

On a piece of paper draw a straight line using a ruler. This can also be done on an overehead transparency if you are demonstrating it for a whole class. Have several triangles cut out of construction paper. I ususally have one that is scalene, one that is isosceles, and one that is equilateral. Tear off the three angles of the triangle and arrange them with their points together on the straight line. I do this with each of the three different triangles.

Some years, I have told the class that I would give ten dollars to anyone that could come up with a triangle that had angles that didn't fit together on a straight line. But I am sure that I will never have to pay up, because it works for every triangle every time!

ETA Cuisenaire Daily Math Reinforcers

I've been using this resource for nearly ten years, and it has been a tremendous way to improve student math skills as well as review and pre-teach concepts that need to be met. These work well as a warmup for math classes as well as a tool for remedial groups. There are three sheets to each reproducible page. Each daily sheet has a mental math section, a word problem, and a skill review section. The newest edition of the manual also includes concept sheets and section quizzes. The series has a book for each of grades 1-8.

The mental math section has three questions which I just read to the class. I usually read them twice and ask the students to answer the questions by thinking about it, but not by writing anything down except the answer. I've seen a number of students improve tremendously in the area of mental math as the year progresses. When we go over the questions, we share how we thought about the problem to get the answer. Then we realize that there is often more than one way to solve the problem. Sometimes we have the opportunity to correct faulty reasoning or misconceptions.

The word problem section requires students to list the steps to solving the problem in a format that fits the acronymn, TIPS. (Thought, Information, Plan, Solution)Occasionally, problems are given that do not have enough information to solve. Students know this and if there is not enough information, they write NEI and then tell what information they would need to solve the problem.

When I was teaching regular ed middle school math, I started many of my classes with this activity as a warm-up. I handed the slips out as students came in the door, and then just went to the front of the room and started by saying "Number 1" which was the mental math section. When I did this, students settled right down. I allowed about eight minutes for the work. Now as a remedial teacher, I often use these daily warm-ups with my students. In both situations, I go over the slips inviting students to give answers and model solutions.

The ADD acronymn is on every sheet for Arithmetic Daily Developed. This series has truly helped my students become more proficient in math.