Back to index of justice, government, and education pages by Donald Sauter.

Education Proposal -
An "Academic Coach" in every Elementary School


In my resume, I offer my services as an "Elementary School Academic Coach", whose duty is "to raise the overall performance of an elementary school by working with all students, two at a time, in a particular grade such as third. Coaching sessions will ensure that students are "getting" the classroom material and also hammer away at basic academic material required by the standardized tests, typically born of No Child Left Behind, that the students must face. The academic coach will demonstrate the marked difference a single individual with a topnotch academic background, real-life work experience, a natural gift for connecting with students, and unencumbered by the administrative component of teaching can have on grades, standardized test scores, readiness for middle school, and general enthusiasm for learning."

That's it in a nutshell. If it doesn't ignite by itself, read on for further discussion.


For a start, understand what the academic coach does not need to be: a conventionally trained teacher. This is not a disparagement of the teaching industry, and is in no way intended to set "smart outsiders" in competition with teachers.

The academic coach will be a team member. The individualized attention given by the academic coach will work hand-in-hand with the classroom instruction to produce something even greater than the sum of its parts.

The academic coach will have a clear view of where the gaps are for each student, and what material the students in general are weak on. He will communicate these weaknesses to the teachers so that they may hit the material harder, insofar as allowed by having to keep forging ahead in the curriculum.

No fantastic claim is made that the academic coach will ensure that every student latches securely onto everything that is taught - schools cover far too much material for any hope of that - but, calling on his life and work experience, he will zero in on the highest priority material.

Another reason the academic coach would not necessarily be drawn from the ranks of trained teachers is that his duties are quite unrelated to those of a teacher. The academic coach does not run a classroom or apply discipline. He does not assign grades. His charter is only to "pull up" the student he is working with, no matter whether that student is years below or years above so-called "grade level".

Because he will not be working in a class setting, the style of instruction can and will be completely different. It can be loud and boisterous (to the extent that it doesn't disturb classes.) It can be fun. In fact, it had better be! Curmudgeons need not apply. What hope is there for fostering an enthusiasm for learning if the student finds the sessions dull and dreary? The opportunity for having a break from the classroom routine is exciting for the students and a boost to class morale.

The academic coach will familiarize himself as much as possible with the expectations of the standardized test faced by students in his school, and will keep it squarely in view as he works with students.

A measure of his job performance, then, will be how the students perform on the standardized test. The problem, of course, is that it can never be known how the students would have performed without his labors. All things considered - comparison of test scores with previous years; comparison with students in grades without the academic coach; comparison with other schools; feedback from the teachers; feedback from the students themselves - a definite picture of the value of the academic coach will emerge.

More Discussion

The academic coach works with all the students in one grade level, two at a time, for approximately half-hour sessions throughout the school day. Under ideal conditions, there could be ten sessions per day, or 20 students per day receiving individualized help. For an average-sized school, every student in the targeted grade would get individualized help once or twice a week.

I propose "two at a time" because two is the magic number, superior in every way to the sainted "one-on-one" gold standard of education. A three-way session eliminates the pressure felt by a lone tutee. Attending a session with a friend makes things even more relaxed. Three heads are simply better than two. There are more ways to "connect". There are many more little tricks the coach can use keep things rolling, and fun. The coach really can give 100 percent of himself to each of two students simultaneously - not the mathematical impossibility it sounds like! Why not three? As soon as you go beyond two, it becomes a small class with the associated control problems.

The coaching sessions use what is going on in the classroom as a jumping off point; thus, the teacher does not have to worry about a student missing something that has to be made up.

The coaching sessions are for all the students, not just the weaker ones. The claim is that pulling up the top end of a class creates a more vibrant learning environment that pulls up the whole class. Schools have been criticized for shortchanging the strongest students while devoting the bulk of the resources to middle and weaker students.

A better plan than devoting himself solely to one grade, would be to split the school year in half, working a half-year with one grade, such as third, and the other half with another grade, such as fifth. Then, every student would receive personalized attention at two different phases in his elementary school career, giving the academic coach an even greater opportunity to have a positive effect.

A good academic coach recognizes that a right answer is a starting point, not the finish line. It is not a signal to drop that lesson and jump ahead. He will take a careful look into how the student gets his answers and make sure his thought process is well-tuned and general-purpose.

An academic coach makes a point of doing everything he asks his students to do. (I believe that this is a grave omission in conventional classroom teaching. I believe a parent should be able to ask to see how the teacher completed any class or homework assignment.) This serves as a check that what he is asking for is reasonable, and it gives the students examples of good answers instead of just being told what's wrong with theirs. It brings the coach and the students together on the "same playing field", which is enjoyable for them and a great confidence builder.

Going beyond simply providing feedback to the teachers, the academic coach will have no trouble whipping up worksheets for the whole class that drill the students in problem areas. These exercises can be custom-designed in a way that no textbook could possibly have envisioned. Often, the exercises will undo confusion which the textbook itself caused. The academic coach can administer such worksheets himself, so that nothing is added to the teacher's burden.

The academic coach does not require plush quarters; three desks lined up against the wall in the hallway outside of the classroom makes for an "office" with a very relaxed atmosphere.

The academic coach himself has an ongoing love of learning. His breadth of knowledge will astonish impressionable young students!

Academic Coach Quick Summary

Appendix 1 - Examples of feedback provided to teachers

In 2001 I worked with 3rd-graders in Prince George's County, Maryland, with an eye towards an upcoming MSPAP test. (See my page on that insane test. Better yet, don't.) I regularly found myself saying the same thing to many of the students. I reported back to the teachers in the hopes that time could be found to revisit some fundmental material which many or all of the students were weak on.

PERIMETER and AREA: Even the best students are careless in answering perimeter and area problems, often giving the area when perimeter is asked for and vice versa. By now, every 3rd-grader has been subjected to my rant:

"Perimeter and area are two different things. Do not get them confused. Perimeter is the total distance (or length) around a figure. Area is the flat space covered by the figure. Perimeter and area are calculated differently. Perimeter and area have different units. Do not answer area when they ask for perimeter. Do not answer perimeter when they ask for area."

They need to hear this about 28 more times. Get down on your knees if necessary. Perimeter and area are sure to be on the MSPAP.

Emphasize that the perimeter of a figure is given in ordinary units of length (inches, feet, miles, km, cm, etc.) and the area is given in that same unit, but with a "square" in front of it (sq. inches, sq. feet, sq. miles, etc.)

PERIMETER: Perimeter problems show up in three different guises:

1. All the sides of a figure are labeled with lengths. This is easiest for the students. They know to add up all the lengths.

2. In the case of regular figures, such as squares and rectangles, all of the sides will not have a length shown - a square will show the length of one side, and a rectangle will show the lengths of two adjacent sides. This is a stumbling block for many students. In the case of the rectangle they will stop after adding together the 2 given lengths; in the case of the square they usually square the one given length (which is the area.)

3. The figure is presented on a grid with no lengths given. One common problem is the tendency for the students to count blocks touching the edge of the figure, which will give a wrong answer. Emphasize that the units of length (one side of the little squares) need to be counted, not the blocks themselves. Another problem is the tendency for students to try to count the little units all the way around in one fell swoop. Even if they are counting the right thing, there is a big chance of making a counting mistake along the way. I suggest a standard approach: counting up the length of each side and writing it next to the side, and then adding up all the lengths. Still, this is difficult for the students when there are small or skinny features on the figure. In the figure below, few students would count up all 8 sides; probably showing just one length for the little indentation.

    |                                |
    |                             ___|
    |                            |___ 
    |                                |

The MSPAP people are also fond of asking the students to create a figure which has a specified perimeter. This is insanely inappropriate for 3rd-graders, and I would argue has little practical application. It would probably be a waste of time pursuing this in the classroom.

AREA: The students are good at counting up the unit squares inside a figure which is shown on a grid. This is all their text book expects them to do, but the MSPAP people have thrown A = L x W at them (in the Puppy Love task.) The kids are great at saying "area equals length times width", but are very poor at applying it (as they should be in 3rd grade.) I would cross my fingers and hope the MSPAP people don't hit them with A = L x W.

NORTH, EAST, SOUTH and WEST: I have seen this on two MSPAP tasks. The students are pretty good at moving their fingers across the page in the various compass directions, but are very poor at formulating a statement such as, "The church is north of the park." There is a very definite reason for this difficulty: the first place named in the statement is not the origin - it's the destination. So the student will put his finger on the church, move it southward toward the park and say (incorrectly), "The church is south of the park." Emphasize that, in statements like this, the starting point is the last thing named in the sentence; the starting point is the thing that follows the "of" or "from". (The students are more comfortable with the word "from", as in, "The church is north from the park.")

The students are slightly better at saying, "To get from the park to the church, you have to go north." However, after slogging through this unwieldy formulation, the student generally gives a random guess at the direction at the very end.

I think it would be helpful to tie together the concept of compass direction with the geometric concept of a "ray". The kids have a good understanding of lines, line segments, and rays. A ray is like an "arrow" - a line with a starting point and shooting off forever in some direction. It would be great to get across the idea that figuring out compass directions is like moving a ray around on the map so that its starting point is at the "of" or "from" location, and the arrow points to the destination location.

There is also a problem with determining compass directions if one or both of the locations is extended. For example, a student may say, "The Sahara desert is north of the Atlantic Ocean" simply because he sees the name "Atlantic Ocean" printed way down in the southern part of the ocean. Even without such a misleading label, it was confusing for the students if one of the locations was extended. For example, students would have trouble giving the direction from a house to a river. Emphasize that when we talk about the direction from this to that, we are talking about the shortest, most direct route.

BAR GRAPHS: The presenter of the MSPAP preparation meeting (Apr 3) said that the 3rd grade generally does a bar graph. I had been wondering about that since none of the 3rd grade public-release MSPAP tasks that I have seen ask for bar graphs, and it's not taught in the math text book.

The 5th grade public-release MSPAP task "Here's The Scoop" asks for a bar graph. (Actually, it asks for a line graph, which is totally inappropriate for the sort of data being plotted.) In spite of it being a 5th grade task, I had some of the 3rd graders work from it to make bar graphs. I didn't get to every student, so it seems well worthwhile to put the students through a bar graph exercise.

While none of the students made perfect bar graphs, it seems that they have a pretty good idea of them. One of their biggest stumbling blocks is surely the fault of their math text book. There are dozens of graphs in the math book, and they all are presented in nice, neat boxes. This is very misleading. This is not what graphs are about. Graphs have a pair of axes, not four sides. Many of the students start their graph by drawing a box of arbitrary size. It then becomes nearly impossible to mark off the increments on the x- and y-axes so that they fit sensibly within the box. Emphasize that a graph starts with the two axes. Mark off the increments in even steps from zero. If you go past the end of an axis, just extend it.

                    |                            |              |
                    |                            |              |
  Start with this:  |                 NOT this:  |              |
                    |                            |              |
                    |                            |              |
                    |_______________             |______________|

DICTIONARIES: I also heard the presenter say that the students may look up words that they don't know in a dictionary (since they can't ask the tester.) I had put some students through dictionary word searches, and found all of them to be very poor at it, although, to be fair, I don't know how good a 3rd-grader is expected to be. Some turn pages one at a time - and not necessarily in the right direction. Some spend loads of time looking up and down the wrong page for a given word. Worse yet, some take forever to find the word even when they are on the right page. Obviously, this is a very important skill to have, whether or not it can be improved enough before the MSPAP to help lift even a single student over the "satisfactory" bar. (By the way, I hope this lack of skill in flipping pages to zero in on a target is not a side effect of all our modern-day poking and punching at computers.)

WRITTEN ANSWERS: Generally speaking, the 3rd-grade students are very weak in forming complete, self-contained, single-sentence answers to questions of all types. This is a very important, basic skill. It would seem to me that there's not much hope for satisfactory paragraphs and letters and essays on the MSPAP if they can't nail this.

Arranging the exact, same words from the question into a statement format is more than just a convenience or matter of style; it helps to ensure a good, sensible answer. For example, it would prevent a student from answering how much money was collected when the question was, "How many items were sold?"

In fact, the students are familiar with the concept, and I have learned from them that it is called "restating". Still, they are very poor at it. Emphasize how easy this is: just put the key words of the question together in a statement and "fill in" the answer. Emphasize that the answer should be self-contained; it should make complete sense to someone who didn't hear the question. Pronouns should be avoided. Explain that "My answer is 15 apples" doesn't cut it, even though it is a sentence.

I think they should be drilled in this, and that such tests would be much more valuable than spelling tests, say. I kept it fun for the kids by encouraging imaginative, fanciful answers - anything goes, as long as the question is clearly and fully answered. Here are some samples:

    Q: What does the farmer do in the morning?
    A: In the morning the farmer milks the cows. 
    A: In the morning the farmer flies to the moon to sell his cheese.
    Q: How many apples are in the bag?
    A: There are 8 apples in the bag.
    A: There aren't any apples in the bag, just 12 baby dinosaurs.  
    Q: Where was Billy hiding?  
    A: Billy was hiding under his bed.  
    A: Billy was hiding in a red and blue egg carton.  

POWER WORDS: I don't know about the 3rd-graders but, speaking for myself, I think the list of "power writing transition words" is overwhelming. For the students, I would pare it back drastically, and beg them to work these words in wherever they can:

    For example
    For instance
    In conclusion

UNITS: Beg the students to remember to show units when giving answers to math problems. (Reminder: perimeter is in units of length; area is in square units.)

CONTINENTS: Knowledge of the continents has been necessary on at least two public-release MSPAP tasks. I've drilled a lot of the students in the continents, but there is evidence it hasn't stuck. I tell them that continents are the biggest "chunks" of land on our planet. I point out that Europe and Asia are considered to be separate continents, even though they're attached. I try to get across the distinction between countries - ours, for example - and continents, and point out where some of the most famous countries are.

NORTH, EAST, SOUTH, WEST, revisited: I've had some good results by scribbling a quick compass rose on a piece of scrap paper, tearing it out, and moving it around the the map. It might be worth getting the students to do this when answering NESW questions. Slide the compass rose to the starting location (the one that follows the "of" or "from"), and then just look at which arrow is pointing toward the destination.

CLIMATE: When working with the students on the "Deserts" task, none of them knew what "climate" means.

PROFIT and LOSS: When I was working with the students on the "Lemonade Stand" task, none of them had a firm grasp of "profit" and "loss".

CHARTS: In the "Lemonade Stand" task, most students were at a loss when asked to "fill in the chart below". Charts are a very common way of presenting information, and it seems likely that they'll pop up on the MSPAP. I think they need more work in pulling information out of simple, basic charts.

SYMMETRY: Ms. Lawrence has said that symmetry is always on the MSPAP. I haven't seen it worked into any of the public-release MSPAP tasks, but my work with the students in symmetry indicates they have a very good handle on it. This surprises me, actually, because of 1) the scariness of the word itself, 2) the inherent trickiness, and 3) the abysmal presentation in the math text book. The math text confuses 3-dimensional objects with their 2-dimensional representations. The text authors themselves never seem sure whether or not something is symmetrical.

For what it's worth, if the students are to get any more work in symmetry, I have found that they get a kick out of completing patterns that make surprise words out of symmetrical letters. The letters with a horizontal line of symmetry are:

    B C D H K X    E I O

The letters with a vertical line of symmetry are:

    H M T V W X Y    A O U I

From the first group you can make words like BOOK, BOX, HEX, DECK, HIKED and CODE with the top or bottom half of each letter missing. From the second list you can make words like

    M   M   W   W   T   T                              
    A   O   I   O   U   A                                
    M   U   T   W   M   X                              
    A   T   H       M   I                              
        H           Y                                 

with the right or left half of each letter missing.

I also found a good demonstration of symmetry that the kids got a kick out of. Fold a piece of paper. Use the fold as a line of symmetry and draw half of a figure with very dark, heavy pencil lines. Close up the paper over the half-figure and rub firmly over it. Open it up and you get the complete, symmetric figure.

READING SCALES: I know there was a scale reading question on the benchmark test, and there is a thermometer scale (a cruelly misleading one) in the "Hot And Cold" MSPAP task. This is an important skill, and I'm not so sure students ever get lessons devoted specifically to it. There are about 25 graphs in their math text book, and the various scales use increments of 1, 2, 5, 10, 20 and 200. Thus, one often needs to read between the printed numbers on the scale. I've seen many instances of students counting each tick mark as 1 when it really represented a step of 2, for instance. I've attached a page with a variety of printed scales. I've used this with some of the 3rd-graders, and my experience indicates they could all use a lot of practice. I tried to get them to see that there is a familiar tick mark pattern for each of the three cases, where the smallest ticks divide the bigger (labeled) steps into 2 parts, 5 parts, or 10 parts - and those three cases are about all you're ever likely to meet.

MEASURING WITH A RULER: The big, bad exception to the above, of course, is a ruler scale, which isn't decimal, but continually divides by 2. I've only worked with a few students on making real measurements with a ruler, but indications are that they could use practice. Not only are they not careful about placing the starting point of the ruler, but they are weak at reading the half- and quarter-inches, etc. I think this activity might fit in well with their current work in fractions.

MONEY: Money shows up in lots of the MSPAP tasks, so it's worth being comfortable with it. They seem pretty good at adding and subtracting money, but it might be worthwhile to revisit this, especially relating the subtraction problems to the notion of profit and loss.

I also think a useful exercise for them is to make up specified amounts of money less than $100 using just the most familiar bills and coins: $20, $10, $5, $1, quarters, nickels, dimes and pennies - and using the least number of pieces of money. That's a very practical problem, and since it's the hardest way to do it, any other way should be easy in comparison. I also like this problem for it's purely arithmetical aspects. There's a whole symphony of counting by 25s, 20s, 10s, 5s and 1s going on in your head, and when you see the next count will "go over", you have to downshift to the next smaller piece of money. I suggest having the students put their answers in a format something like the following (where q = quarter, d = dime, n = nickel and p = penny):

        3 x $20 
        4 x $ 1 
        2 q
        1 d
        1 n
        2 p

I wouldn't accept "wise guy" answers like, 74 ones and 67 pennies.

With a lot of the students, I went a step further and had them figure out in their heads what the change from a hundred dollar bill would be if that was all they had to pay with.

REVISIT THE TEXT!: I don't think we've gotten it through to all the students yet how simple it is to dig answers out of a reading passage. Many of them still ponder the question with their eyes turned heavenward, and while this often gives rise to a most wonderful answer, it's not likely to be what the tester wanted. (Here's a gem from just today. Q. When is it incorrect to fly the flag? A. It will be incorrect to fly the flag if you do not have enough string.) C'mon kids! When you're answering test questions, just turn your brains off! (Whoops, I didn't mean it like that.)

ROUNDING: The 3rd-graders are pretty good at rounding 2-digit numbers to the nearest ten. They can round off 23 and 66, for example. Still, none of them have grasped rounding completely and can apply it to the general case of rounding a number of any size to any specified position. Most of them will break down when you add another place or two and ask for rounding to the nearest 100 or 1000, say. They are almost sure to choke on 0s and 9s (for example, rounding 7070 to the nearest 100; or rounding 596 to the nearest 10.) Most of them cannot round 74 to the nearest 100; or round 381 to the nearest 1 (not a trick question.)

I don't think the "recipe" for rounding is too hard for any 3rd-grader.

1. Think about what place you're rounding off to: to the nearest one, the nearest ten, the nearest hundred, the nearest thousand, etc.
2. If you're rounding off to the nearest ten (hundred, thousand, etc.) focus on the tens (hundreds, thousands, etc.) place.
3. The number in that place will either stay right where it is, or will go up one if the digit just to the right of it is 5 or greater.
3a. If the number 9 has to go up one, just put a 0 in that place and carry a 1 to the next higher place, as in regular addition.
4. Put zeros in all the places lower than the place you're rounding off to.
5. Look at your answer and confirm that it is, in fact, the nearest multiple of ten (hundred, thousand, etc.) to the original number.

In reality, of course, rounding is much easier than all these words make it sound. The weakness in rounding indicates a fundamentally weak "number sense" in the students and I suspect more time should be spent in first and second grade drilling the students in counting by ones, tens and hundreds. Students need to know numbers before they can go on to any more advanced math work.

Appendix 2 -
Custom-made worksheets addressing observed weaknesses in 3rd-grade students

Sample 1 - compass directions.

This exercise comes with a basic map of the U.S. showing very clearly all the states and their names.

                 COMPASS DIRECTIONS: North, East, South, West

REMEMBER: the place after the "of" or "from" is the starting point of 
the direction!  Imagine the compass rose shifted to the starting point.

Section 1.  Simple!  These states are right above or below or beside each 
other.  Fill in the blanks with the correct compass direction.  Use the 
whole word, or just the abbreviation:

  N (north)      E (east)        S (south)       W (west)           

 1. To get from Utah to Colorado, you have to go _____________.

 2. Colorado is _____________ from Utah.

 3. To get from California to Oregon, you have to go _____________.

 4. Oregon is _____________ of California.  

 5. To get from Alabama to Mississippi, you have to go _____________.

 6. Mississippi is _____________ of Alabama.

 7. To get from Kentucky to Tennessee, you have to go _____________.

 8. Tennessee is _____________ of Kentucky.

 9. Pennsylvania is _____________ from New York. 

10. North Dakota is _____________ of South Dakota.

11. South Dakota is _____________ of North Dakota. 

12. Georgia is _____________ from Florida.

Section 2.  Still simple!  These states are straight above or below or 
across from each other, but might not be touching.  

13. Nevada is _____________ of Colorado.

14. Ohio is _____________ from Georgia. 

15. Iowa is _____________ of Louisiana.

16. New Mexico is _____________ of Montana. 

17. New Mexico is _____________ of Wyoming.

18. Maryland is _____________ of California.

19. Idaho is _____________ from Minnesota.

20. New York is _____________ of Maryland.  

21. Virginia is _____________ of Maryland. 

22. Maryland is _____________ of Pennsylvania. 

23. Canada is _____________ of Mexico.  (These are countries!)

Section 3.  A little trickier!  These states are not straight up or down 
or across from each other.  Use the compass points that are in between 
N, E, S and W: 

  NE (northeast)    SE (southeast)    SW (southwest)    NW (northwest)

24. Utah is _________________ of New Mexico.

25. Ohio is _________________ of Arkansas.

26. Nebraska is _________________ of Montana. 

27. South Carolina is _________________ from Illinois. 

28. Idaho is _________________ from Texas.

29. Kansas is _________________ of Wisconsin. 

Section 4.  Fill in the blank with the name of a state.  For each one, 
there are several good answers to choose from.  

30. Kansas is south of _______________________.

31. Missouri is east of _______________________.

32. Washington state is west of _______________________.

33. Indiana is north of _______________________.

34. New Mexico is southeast from _______________________.

35. Maine is northeast of _______________________.

36. Nebraska is northwest from _______________________.

37. Arizona is southwest of _______________________.

Sample 2 - rounding.

Weakness in rounding shows a fundamentally weak understanding of numbers.

ROUNDING is the easiest thing on earth.  You're already great at rounding
to the nearest 10.  Rounding to any other place is the exact same idea!  

HOW TO ROUND: Look at the place you're rounding to.  That digit will either 
stay the same, or "round" up by 1 if the digit next to it (to the right) is 
5 or bigger.  Then put zeros in all the lower places.  

If you have to round a 9 up to the next number, just put a 0 in that place and 
carry a 1 to the next higher place, as you do in ordinary addition.  Zero is 
treated just like any other number - it either stays at 0, or rounds up to 1. 
If there is no digit in a place, pretend a 0 is there.  For example, to 
round 62 to the nearest 100, think of 62 as 062.  (The answer is 100, right?)  

Fill in all the blanks.  (Do the "10" column first, if you want.)

Round  to the nearest:      10              100             1000

  23        -->        ____________     ____________     ____________     

  59        -->        ____________     ____________     ____________     

 701        -->        ____________     ____________     ____________     

   2        -->        ____________     ____________     ____________     

  98        -->        ____________     ____________     ____________     

1234        -->        ____________     ____________     ____________     

 102        -->        ____________     ____________     ____________

 999        -->        ____________     ____________     ____________     

2468        -->        ____________     ____________     ____________     

 909        -->        ____________     ____________     ____________     

8000        -->        ____________     ____________     ____________     

 555        -->        ____________     ____________     ____________     
7033        -->        ____________     ____________     ____________     

 359        -->        ____________     ____________     ____________     

 894        -->        ____________     ____________     ____________     

2950        -->        ____________     ____________     ____________     

 500        -->        ____________     ____________     ____________     

6414        -->        ____________     ____________     ____________     

  45        -->        ____________     ____________     ____________     
5050        -->        ____________     ____________     ____________     

   8        -->        ____________     ____________     ____________     

3579        -->        ____________     ____________     ____________      

Sample 3 - halves.

An exercise to drill the kids in dividing even numbers in half. (I used real division signs in the worksheet, rather than the ascii approximations you see here.)

                          Halves are easy!

You can divide something into any number of parts, of course, but a very 
common thing in real life is to divide something in half.  This means the 
same thing as dividing by 2, or multiplying by 1/2.  

All of these number sentences say the exact same thing: Half of 6 is 3.  

      .              3       6                         1
    6 - 2 = 3      ____     --- = 3     6 / 2 = 3     --- x 6 = 3
      .           2| 6       2                         2

Or, you can think of it the other way: two halves equal a whole.  In this 
example, 3 is the half, so:   3 + 3 = 6    or    2 x 3 = 6

Any even number can be divided in half without a remainder.  

Try to do these in your head:

Half of    2 is _____.     Half of   20 is _____.     Half of  200 is _____.   

Half of   12 is _____.     Half of   14 is _____.     Half of   16 is _____.   

Half of    6 is _____.     Half of   60 is _____.     Half of  600 is _____.   

Half of    8 is _____.     Half of   18 is _____.     Half of   28 is _____.   

Half of 6000 is _____.     Half of 8000 is _____.     Half of 2000 is _____.   

Half of   10 is _____.     Half of  100 is _____.     Half of 1000 is _____.   

Half of    4 is _____.     Half of   44 is _____.     Half of  444 is _____.   

Half of    2 is _____.     Half of   24 is _____.     Half of  248 is _____.   

Half of   20 is _____.     Half of   22 is _____.     Half of   24 is _____.   

Half of   26 is _____.     Half of   28 is _____.     Half of   30 is _____.   

Half of   32 is _____.     Half of   34 is _____.     Half of   36 is _____.   

Half of   38 is _____.     Half of   40 is _____.     Half of   50 is _____.   

Half of   10 is _____.     Half of  100 is _____.     Half of  102 is _____.   

Half of  104 is _____.     Half of  106 is _____.     Half of  108 is _____.   

Half of  120 is _____.     Half of  130 is _____.     Half of  140 is _____.   

Half of  160 is _____.     Half of  180 is _____.     Half of  200 is _____.   

Half of  222 is _____.     Half of  444 is _____.     Half of  666 is _____.   

Sample 4 - comfort with zero.

Third-graders can do this one, for the most part, but speed and accuracy are the important things. Since accuracy is more important than speed, it was gratifying to see signs of erasures on almost every worksheet, even the perfect ones, showing that the student was mentally double-checking his answers.

                     THE WORLD'S EASIEST MATH TEST

      2      0      6      0      1      0      2      0      4      0
    + 0    x 4    + 0    + 5    x 0    x 6    + 0    x 7    + 0    + 8
   ----   ----   ----   ----   ----   ----   ----   ----   ----   ----

      0      5      0      9      0      7      0      8      0      1
    x 3    + 0    + 9    x 0    x 3    + 0    x 2    + 0    x 6    x 0
   ----   ----   ----   ----   ----   ----   ----   ----   ----   ----

      9      0      7      0      2      0      5      0      6      0      
    x 0    + 9    + 0    x 3    x 0    + 1    x 0    + 3    x 0    + 8
   ----   ----   ----   ----   ----   ----   ----   ----   ----   ----

      0      8      0      7      0      1      0      2      0      4
    + 4    x 0    x 5    x 0    + 6    x 0    + 2    x 0    + 5    x 0
   ----   ----   ----   ----   ----   ----   ----   ----   ----   ----

      6      0      4      0      9      0      7      0      8      0      
    + 0    x 4    x 0    + 9    x 0    + 3    + 0    x 1    + 0    + 3
   ----   ----   ----   ----   ----   ----   ----   ----   ----   ----

      0      8      0      5      0      6      0      7      0      1
    + 2    x 0    + 5    + 0    x 7    + 0    + 1    + 0    x 6    + 0 
   ----   ----   ----   ----   ----   ----   ----   ----   ----   ----

Sample 5 - powers of ten.

This one was actually whipped up for 5th-graders who were doing conversions between mm, cm, dm, m, etc. (This group of students also needed familiarization with those units of measure, and basic work in making measurements. I made a worksheet to address that, as well.) In any case, everything here should be doable by the end of 3rd grade. I'm not happy with the crowded, scary-looking layout.

                Multiplying and Dividing by "Powers of Ten"

Numbers like 10, 100, 1000, etc. are called "Powers of 10".  To MULTIPLY 
counting numbers by a power of ten, just ADD ZEROS.  

3 x 10 = _______    9 x 10 = _______    10 x 6 = _______    10 x 13 = _______  

2 x 100 = _______    7 x 1000 = _______     100 x 19 = _______   

10 x 130 = _______   10000 x 43 = _______   26834 x 10000000 = ______________

To DIVIDE numbers that end in zero by a Power of 10, just ELIMINATE ZEROS.  

30/10 = _______    600/10 = _______    340/10 = _______    77000/100 = _______

77000/1000 = _______      3000000/1000 = _______      3000000/10 = _______   

12300/10 = _______   404000/1000 = _______   500/100 = _______   10/10 = _____

To MULTIPLY a number that has a decimal point by a Power of 10, just move 
the decimal point to the RIGHT.  

5.1 x 10 = _______          66.31 x 10 = _______          4.301 x 10 = _______

66.31 x 100 = _______      3.1416 x 100 = _______     77.7777 x 1000 = _______

1.2345678 x 10000 = ____________       1.2345678 x 10000000 = ____________   

1.2345678 x 10 = ____________          10.2345678 x 1000 = ____________   

To DIVIDE a number that has a decimal point by a Power of 10, just move 
the decimal point to the LEFT.  

2.5/10 = _______   22.5/10 = _______   30.45/100 = _______   61.5/10 = _______

5432.1/1000 = _______     666777.88/10000 = __________      9.33/10 = _______

30.08/10 = _______         30.08/100 = _______         300.08/100 = _______

REMEMBER:  If a counting number (1, 2, 3, 4, 5...) doesn't show a decimal 
point, it's really there after the one's place.  For example,  

       5 = 5.     20 = 20.    36 = 36.     100 = 100.     701 = 701.

46/10 = _______   504/100 = _______   76760/1000 = _______   499/1000 = _______

876/100 = _______    876/10 = _______    876/1000 = _______   22/100 = _______

IF YOU RUN OUT OF DIGITS when moving the decimal point, just SUPPLY ZEROS.  

3.1 x 100 = _______      6.2 x 1000 = _______      14.3 x 100 = _______   

4.234 x 100000 = _________   718.1 x 1000 = ________   55.43 x 1000 = ________

3/100 = _______    9/1000 = _______     99/1000 = _______    16/100 = ________

6.6/100 = _______     1.23/100 = ________    88.2/10000 = ________   

1.9/1000 = _______      67.4 x 100 = _______      67.4/1000 = _______  

Sample 6 - lines, line segments and rays.

This one was also for 5th-graders. The Addison Wesley math textbook expected the students to become familiar with the geometric figures without ever actually drawing them. A big problem with the text was that the figures were always presented so small that the students had a hard time grasping the distinction between the figures and the symbols used to name the figures.

In the actual worksheet I use the proper symbol for "angle", as opposed to the angle bracket "<" used below. Also, the line and ray symbols have the proper neat, unbroken appearance.

                  Lines, Rays, and Angles and Their Names

Tell whether each of these geometric figures is a line, ray, or angle.  
Then use a ruler to actually draw them below.  (I ran out of letters 
so I named three points #, $, and *.)

  MV  _______________

  JS  _______________

< BWN  _______________

  #F  _______________

  ZE  _______________

  YD  _______________

< QH*  _______________

  LU  _______________

  KT  _______________

  $G  _______________

  IR  _______________

  XC  _______________

             .      .      .      .      .      .      .      .
             A      B      C      D      E      F      G      H
             .                                                .
             *                                                I
             .                                                .
             $                                                J
             .                                                .
             #                                                K
             .                                                .
             Z                                                L
             .                                                .
             Y                                                M
             .                                                .
             X                                                N
             .      .      .      .      .      .      .      .
             W      V      U      T      S      R      Q      P

Sample 7 - basic additions.

Nothing ground-breaking with this one, or at least I hope there isn't. (Should I have to make something like this up???) Many of my 3rd-graders were still weak in the single-digit additions - presumably because they weren't drilled in it regularly in 1st and 2nd grade.

One thing I emphasize is, no matter which order the numbers (addends) are given in, think of the big number (addend) first. (Two exceptions may be 5+7 and 5+8 where the brain quickly sees 5+5+2 and 5+5+3, respectively.)

                           All The Additions

2+6=_______    5+5=_______    6+1=_______    8+8=_______    3+8=_______

0+7=_______    8+3=_______    4+4=_______    8+0=_______    3+2=_______

4+9=_______    5+1=_______    7+0=_______    9+5=_______    1+1=_______ 

0+9=_______    6+8=_______    1+5=_______    7+7=_______    2+3=_______    

7+3=_______    4+7=_______    9+0=_______    3+6=_______    1+8=_______    

0+5=_______    4+1=_______    8+6=_______    6+6=_______    2+2=_______    

1+0=_______    8+1=_______    2+9=_______    5+7=_______    5+2=_______    

9+7=_______    0+0=_______    4+3=_______    6+4=_______    7+6=_______    

9+2=_______    3+5=_______    2+2=_______    8+4=_______    9+9=_______    

3+1=_______    5+3=_______    4+0=_______    1+7=_______    0+1=_______    

7+5=_______    6+2=_______    5+6=_______    6+0=_______    2+7=_______    

7+9=_______    6+7=_______    5+0=_______    0+2=_______    3+3=_______    

0+6=_______    2+4=_______    7+1=_______    1+3=_______    4+6=_______    

9+3=_______    0+4=_______    4+2=_______    2+1=_______    9+6=_______    

4+5=_______    3+0=_______    8+2=_______    6+3=_______    7+4=_______    

2+0=_______    1+4=_______    8+5=_______    1+6=_______    8+7=_______    

9+8=_______    0+3=_______    3+7=_______    5+4=_______    6+9=_______    

3+9=_______    7+2=_______    5+9=_______    0+8=_______    7+8=_______    

8+9=_______    3+4=_______    2+8=_______    1+9=_______    4+8=_______    

1+2=_______    9+1=_______    6+5=_______    9+4=_______    5+8=_______    

Sample 8 - multiplication from easiest to hardest.

This handout will not be too useful by itself to a 3rd-grader, but I gave it out so they could follow along with my pep talk on how easy multiplication really is. The point is, if you grant me that multiplying by 0, 1, 2, 3, 5 and 9 is easy, and that the squares are easy, then there are only six other single-digit multiplications that need to be nailed. Just six! (Admittedly most students need extra work with the "big 3's" - 21, 24, and 27.)

                       Multiplication is not so bad!

0 times anything is 0.  

1 times anything is the number itself.

Multiplying by 2 just gives the even numbers:  2 4 6 8 10 12 14 16 18 20.

Here are the multiples of 3:  3 6 9 12 15 18 21 24 27 30.

The "5 times" are a cinch.  
    5 times an even number ends in 0:     10     20     30     40     50
    5 times an odd number ends in 5:   5     15      25     35     45

You can figure out the "9 times" easily.  Always start by thinking, 
9 times anything has to be less than 10 times the same thing.  

For example, here's 9 x 7 .
Step 1. The answer has to be in the 60s, (because it must be less than 10 x 7)
Step 2. The two digits have to add up to 9.  In this case 6 + 3 = 9.  
        So the answer is 63.

Get familiar with the "squares" (when a number is multiplied by itself.) 
    Number:  1  2  3  4  5  6  7  8  9  
    Square:  1  4  9 16 25 36 49 64 81

And here are the only other products you need to know!  

    24 = 4 x 6    (Twice as big as 2 x 6, if you get stuck.)

    28 = 4 x 7    (Twice as big as 2 x 7, if you get stuck.)

    32 = 4 x 8

    42 = 6 x 7

    48 = 6 x 8    (It rhymes!  Six times eight is forty-eight.)

    56 = 7 x 8    (Notice the nice pattern: 5 6 7 8)

Sample 9 - familiarity with the multiplication table answers.

This pep talk makes the same point, but from the opposite direction - there are really very few answers to the single-digit multiplications. Get familiar with them, and then it's easy to match the right answer to the problem at hand. More fundamentally, it's important to think of the two multipliers and the product as a multiplication "fact family." Then you've got division automatically licked.

                       Multiplication is not so bad!

The multiplication table looks big and scary, but there are not so 
many different answers to memorize.

There are NO answers in the 90s.

There is only ONE answer in the 80s.
  81 = 9 x 9

There is only ONE answer in the 70s. 
  72 = 9 x 8

There are only TWO answers in the 60s.
  63 = 9 x 7
  64 = 8 x 8  (a "square")

There are only TWO answers in the 50s.
  54 = 9 x 6
  56 = 7 x 8

There are only FOUR answers in the 40s.
  42 = 6 x 7
  45 = 9 x 5  (easy!)
  48 = 6 x 8
  49 = 7 x 7  (a "square")

There are only THREE answers in the 30s.  
  32 = 4 x 8
  35 = 5 x 7  (easy!)
  36 = 6 x 6  (a "square")
  36 = 9 x 4

There are FIVE answers in the 20s, but all easy.
  21 = 3 x 7
  24 = 3 x 8
  24 = 4 x 6
  25 = 5 x 5  (an easy "square"!)
  27 = 9 x 3
  28 = 4 x 7

The answers in the teens are all EASY, since they are mostly just 
multiples of 2 or 3.

  12 = 2 x 6
  12 = 3 x 4
  14 = 2 x 7
  15 = 5 x 3  
  16 = 2 x 8
  16 = 4 x 4  (a "square")
  18 = 2 x 9
  18 = 3 x 6

Sample 10 - multiplication table answers shown in the 100 chart.

The tables on this sheet make the same point as above - the times table is big, but there are really very few answers. The times table is no-frills, but perfectly useful to a student who still needs it. Note that in the discussions above, I've been harping on single-digit multiplications (0 through 9), but these tables go from 1 to 10. No problem.

                        The TIMES TABLE

     1     2     3     4     5     6     7     8     9    10

     2     4     6     8    10    12    14    16    18    20

     3     6     9    12    15    18    21    24    27    30

     4     8    12    16    20    24    28    32    36    40

     5    10    15    20    25    30    35    40    45    50

     6    12    18    24    30    36    42    48    54    60

     7    14    21    28    35    42    49    56    63    70

     8    16    24    32    40    48    56    64    72    80

     9    18    27    36    45    54    63    72    81    90

    10    20    30    40    50    60    70    80    90   100

               All the ANSWERS in the TIMES TABLE
                   arranged in counting order

     1     2     3     4     5    6     7     8     9    10

     .    12     .    14    15    16     .    18     .    20

    21     .     .    24    25     .    27    28     .    30

     .    32     .     .    35    36     .     .     .    40

     .    42     .     .    45     .     .    48    49    50

     .     .     .    54     .    56     .     .     .    60

     .     .    63    64     .     .     .     .     .    70

     .    72     .     .     .     .     .     .     .    80

    81     .     .     .     .     .     .     .     .    90

     .     .     .     .     .     .     .     .     .   100

Answers in italics can be gotten two ways.  For example,      

                       2 x 9 = 18

                       3 x 6 = 18


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