OK, so your problem at this time is not with tools, but with joining two or three pieces of wood so that you can reinforce the end(s) of a paddle.

Let's start with working with the tools you have. it would also be nice to know the approximate thickness of the pieces of wood you wish to join, and how thick the final product is going to be. For example, if you are working with 1 by 6 stock, which is 3/4 inch thick, and you plan to thin the blade down to 3/8th inch thick, or thinner, then we should know this before we start.

Also, is your wood flat, or have you already done some shaping and rounding to the wood? If it is flat we can machine it directly. If it has been rounded then we need to clamp it to something flat so it can be cut or shaped.

You can cut a tongue and groove joint with a saw.

If you are working with a table saw you'll cut a groove in the ash tip and a tongue on the cedar. The reason for this is that your ash piece will probably be only a few inches wide, and would need a slot in the side of the part. With a narrow part this is rather easy, but if you try to use a table saw to cut a slot in the end of a (probably) 6 foot long cedar board you'll have to stand the thing on end. If the top end of it doesn't hit your ceiling it will certainly be hard to keep tight against the fence on the saw.

Let's assume your wood is flat and not yet shaped or rounded. Let's also assume you will be thinning the paddle to about 3/8 inch, and finally lets assume you have some scrap pieces of wood.

Normally, when joining two pieces of 3/4 inch thick wood I'd like to cut away about half the wood from each piece. With a tongue and groove joint that would give me a 3/8 inch tongue sitting in a 3/8 groove. When I cut that 3/8th inch groove in a 3/4 inch piece of wood, the wood on either side of the groove is only 3/16 th inch thick (a bit less than 1/4 inch) thick. Other people like to figure on splitting the joint into thirds. With 3/4 inch thick stock they would make all parts of the joint 1/4 inch thick. For most situations the differences in these typse of joints would be minimal.

In your case, though, we have to recalculate things. remember that we assumed you would be thinning your blade ends to 3/8th inch. If you make a joint using the "usual" calculations, your tongue would be 3/8th inch thick, and you'ld end up removing ALL the wood on either side, totally eliminating the groove! So, that just won't work.

In your case, you would want a groove about 1/8th inch wide. That would make the tongue about 1/8th inch wide, too. You would want these centered on the thickness of the wood, and later, when you thin down the blade ends, the joint should be centered still.

Since most tablesaw blades make a slot, or kerf, that is 1/8th inch or slightly thinner, you can start by running a slot that is up to 3/16ths deep into the center of your pieces of ash. Measure carefully, and run several scraps to ensure you get the right depth. If the blade is too narrow, you'll need to make two passes. Make the first cut a tiny bit offcenter. Then rotate the wood and run it through with the opposite side agaist the fence. The further your blade is from the center the wider your final groove will be.

Once you have the grooves cut in the ash tips you can work on the tongues. Since the grooves are 3/16th inch deep and about 1/8th inch wide you'll need to set your tablesaw so the blade is not quite 5/16th of an inch above the table. Here is how I figure that. Your wood is 3/4 inch thick. If i measured it in eighths that would be 6/8ths of an inch. I need to leave a tongue that is 1/8th inch, so I need to remove 5/8ths of wood, total. Since I want that tongue centered I'll remove half of the waste ( half of 5/8 is 5/16) from each side.

I'll mark a piece of scrap wood 3/16 th of an inch from the end, set it next to the blade while the saw is OFF, place a 1 to 2 inch wide block at the end of the board, and then lock my fence so it is just touching the added block. that leaves my fence away from the blade by the distance of the block, plus about 1/8inch. I'll place the sliding crosscut guide in its groove on the saw table top, move the spacer and cedar board back so they are not near the blade, and turn the saw on. I use the spacer against the fence to set the end of the board at the proper distance, remove it, and slide the board past the sawblade. this will remove most of the wood. A second pass with the board about 1/8th inch away from the spacer should take care of any remaining wood on that side of the tongue. Now i flip the board over and do the other side, first with the spacer next to the fence, and then with some additional space to remove the extra wood. Flip the boards end for end and repeat the process on the other end.

Obviously using some scrap and making several test cuts is important to good alignment. You don't want the tongue to be be cut too thin. Your sawcuts will be rough, and you'll clean them up with a sanding block -- nothing more than a piece of 80 grit sandpaper wrapped around a wood block. If the tongue is a bit thick, go over it a few more times with that sanding block until you get a snug fit. Apply glue, tap the pieces together and clamp as best you can until the glue sets.

This gives you a square edged. tongue and groove.

For more strength you might want to modify this and make a fat "V" shaped groove, and a matching tongue.

To do this, simply tilt the blade on the tablesaw by 5 to 10 degrees. You'll need a few more tests to find where to set the fence so you get your cuts to line up in the middle of the board, but this is really pretty quick. Ideally the first cut looks like this: / and when you rotate the board and run the wood through a second time you get two cuts which look like this: /\.

You can make this groove deeper than a square cut tongue -- considerably deeper -- and this will give you a much stronger joint. If the groove goes 1/2 inch deep, it is possible that the bottom of the groove will be 1/8th inch wide, but the opening will be over 3/8ths wide. No problem. This tapering joint will be the full thickness of your thinned paddle end, and have significant glued surface and be very strong. When you go to cut the tongue on your cedar, you'll remove a thinner amount for a longer distance. You want the tongue you cut to be as wide as the widest part of the groove it is to match, and as deep. You cut a square tongue as described above, then use a utility knife, pocket knife or chisel to remove some of the excess from the end. Don't take off too much, just taper it a bit. Take one of the scraps you cut when seting up the cuts for the groove, and line it with a piece of sandpaper. Just fold a sharp crease in the sandpaper, with the grit side on the inside of the fold, and drop that fold into the groove in your test piece. Slide that along the roughly cut tongue, and the soft cedar will rapidly take on the exact shape of the groove you've cut. It may take 5 minutes of hand sanding, but you only have to do two ends, so I doubt you'll get bored.

Again, glue and clamp and away you go.

If you are careful to keep the saw perfectly straight, you can cut these tongue and grooves with a handsaw. The best type to use is one called a backsaw, which is what is commonly used in a miterbox. Lay your board flat on a workbench and cut with the saw horizontal. This way you can insert thin pieces of plywood or wedges under the board to raise it ot the proper height for cutting the groove. The saw blade is supported on a narrow piece of wood or plywood so it is parallel to the top of the workbench. A bit of wax on that board will help the blade slide freely. Clamp down the ash or cedar part, and press teh saw down flat on the waxed board it is running on as you gently feed the teeth into the end of the board you want to cut. The set-up is tedious but you only have to make two very shallow grooves, so take your time. When you cut the tongues, leave them a bit big and sand them down to fit. If the board you are sliding your saw on is tilted a bit you can cut those fat "V" shaped grooves and tongues, too.

Now, here is another thought: Why bother with tongue and groove joints at all?
Yo can use a scarf joint -- basically a long diagonal cut -- to add those ash tips to your cedar blades. Becasue of the angle that the wood is cut at you'll have a tip that is about 2 inches wide on the face of the blade, and maybe 6 inches wide on the back. If the cedar is dark the contrast with the lighter ash can be dramatic and rather pretty. The diagonal joint is plenty strong, self centering (as you'll be removing the excess and overhang when you thin the blades), and simple.

You can make this with just a plane, or there are a few ways to cut away most of the wood with a backsaw, tablesaw, router, etc. You can clean up the joint surfaces with a sanding block if you don't have a plane. It just takes a bit longer. sahpe the ash surface first. Then, when the cedar part is close to fitting, sandwich a sheet of sandpaper betweent he ash and cedar parts with the grit facing the cedar. As you move the ash and sandpaper the cedar gets shaped to the exact profile of the ash, givig a neat fit.

Hope this helps

PGJ