How do I compare two fields from two tables where one field is like another but not exact?

Im stumped and so are the people helping me so Ill try to ask this a different way.

I have two tables. I need to write a query to get all of the data based on where part of one field from one table is contained in the full field of the other table.

It is a classic needle in a haystack. Except all of my haystacks are not standard.

In literal terms, table_a.small_id is part of table_b.big_id.

I know the following statement wont work because it is trying to find the literal string (the actual text) table_a.small_id as opposed to the value of the variable. But if it could treat it as a variable, thats what I need to do.

SELECT table_a.x, table_a.small_id, table_b.big_id
FROM table_b, table_a
WHERE table_b.big_id LIKE '%table_a.small_id%'

Ive created a view that pulls out all of the distinct small_id possibilities. Essentially, I need to compare every big_id against the 200 possible small_ids and return the combined data that matches for each record. How the heck do you do this in standard SQL? Im using 5.0 if that helps.

Below is the sql to create the simple test databases.
SET FOREIGN_KEY_CHECKS=0;
-- ----------------------------
-- Table structure for table_a
-- ----------------------------
CREATE TABLE `table_a` (
  `id` int(11) NOT NULL auto_increment,
  `time` int(11) default NULL,
  `x` varchar(255) default NULL,
  `y` varchar(255) default NULL,
  `small_id` varchar(255) default NULL,
  PRIMARY KEY  (`id`)
) ENGINE=InnoDB AUTO_INCREMENT=8 DEFAULT CHARSET=latin1;
 
-- ----------------------------
-- Records
-- ----------------------------
INSERT INTO `table_a` VALUES ('1', '60521', '4', '12', 'i014');
INSERT INTO `table_a` VALUES ('2', '60521', '8', '32', 'i014');
INSERT INTO `table_a` VALUES ('3', '60522', '3', '74', 'i015');
INSERT INTO `table_a` VALUES ('4', '60522', '3', '60', 'i016');
INSERT INTO `table_a` VALUES ('5', '60522', '11', '50', 'i016');
INSERT INTO `table_a` VALUES ('6', '60522', '27', '8', 'i016');
INSERT INTO `table_a` VALUES ('7', '60523', '14', '43', 'i014');
 
SET FOREIGN_KEY_CHECKS=0;
-- ----------------------------
-- Table structure for table_b
-- ----------------------------
CREATE TABLE `table_b` (
  `id` int(11) NOT NULL auto_increment,
  `time` int(11) default NULL,
  `a` varchar(255) default NULL,
  `b` varchar(255) default NULL,
  `big_id` varchar(255) default NULL,
  PRIMARY KEY  (`id`)
) ENGINE=InnoDB AUTO_INCREMENT=6 DEFAULT CHARSET=latin1;
 
-- ----------------------------
-- Records
-- ----------------------------
INSERT INTO `table_b` VALUES ('1', '60521', '6', '22', '1208_i014_65');
INSERT INTO `table_b` VALUES ('2', '60521', '7', '15', '1208_i014_65');
INSERT INTO `table_b` VALUES ('3', '60522', '11', '19', '390_i015_432');
INSERT INTO `table_b` VALUES ('4', '60522', '8', '23', '90_i016_32');
INSERT INTO `table_b` VALUES ('5', '60523', '31', '50', '523_i014_9');