A Qualitative Problem
Which of these yields an answer greater than 1? Circle those that are greater than 1.
1024/999
68/72
4753/3412
4753/5000
These problems are not for teaching, they are DIAGNOSTIC. If your student can't do them easily, there is something wrong with your student's concept of division. These problems do not teach that concept. You can tell your student the mindless method for solving these problems, but that isn't really math or teaching.
Qualitative problems are problems where you don't have to calculate a number as the answer. They are simple, if you know the required concept. Usually, they are difficult and require a calculator if you don't know the concept. Once, some educators talked a college physics professor into putting some qualitative problems on his test. Up to then, his problems required students to calculate the correct answer, which they could do by memorizing a formula. The professor's reservation was that the problems were too easy. The students did horribly on the qualitative problems, and the professor was shocked.
However, the only reason these qualitative problems worked is that teachers have not taught their students a mindless method of solving these problems. There of course is an easy mindless method for the above.
Basic Division
If you have six apples and three children, can you divide the apples evenly among the children so that each child gets the same number of apples?
A1. 4 apples and 2 children
A2. 8 applies and 2 children
A3. 9 apples and 3 children.
Base 10
B1. 6 boxes of apples, all containing the same number of apples. 3 children.
B2. 6 boxes of apples and 3 more apples. 3 children.
B3. 6 boxes, 12 extra apples, 3 children.
B4. 7 boxes of apples, 2 extra apples, 3 children. If it matters, the boxes all contain 10 apples.
B5. 5 boxes of apples, 4 extra apples, 3 children.
B6. 9 crates of apples, all cotaining the same number of apples. 3 children.
B7. 9 crates, 6 boxes, and 3 more apples. 3 children.
B8. 4 crates of apples, 1 box, and 4 extra apples. The crates contain exactly 10 boxes. The boxes still contain 10 apples.
B9. 5 crates of apples, 5 boxes, and 5 extra apples.
-----Any number of fillers in here ----
Variations. There are two ways to do these problems. The easiest is to first split up the crates evenly, then open the leftover crate and split up the boxes evenly, then open the leftover boxes and split up the remaining applies. The other method is to figure out the total number of apples and divide by 3. It is important to see the first way (which would be the easiest in real life also). So a student probably shown be shown that method, if the student is not using it. Or maybe the student can be led to this method by asking the student to be practical. If the student gets the answer, say, 132, does the student really open all the crates and boxes and hand out the apples individually? Or does the student put them back in the crates and boxes. (That's inefficient, so a student might get tired of writing that out.)
But once the student is using it, a variation is to use both methods. Of course (?), the two methods are the same, which is the whole point -- this is designed to make sense of the procedure of long division.
Students can also check their answer. This might help them see the correspondence between division and multiplication. Or do a multiplication and then the corresponding division problem.
Helping Kiki
Perhaps obviously, this webpage was created before I had the idea of doing addition, subtraction and multiplication. The following problems follow the Aldus Lookridge story line of the other pages.
Kiki is supposed to visit two families on Monday. They each get the same amount. But she lost the piece of paper saying how much each family gets. Aldus put in her car 2 crates, 4 boxes, and 2 loose apples. Can you tell her what each family gets?
Kiki is supposed to visit two families on Tuesday. They each get the same amount. Again, she lost the piece of paper saying how much each family gets. Aldus put in her car 3 boxes. Can you tell her what each family gets?
Kiki is supposed to visit two families on Wednesday. They each get the same amount. Again, she lost the piece of paper saying how much each family gets. Aldus put in her car 5 boxes and 8 loose apples. Can you tell her what each family gets?
Kiki is supposed to visit three families on Wednesday. They each get the same amount. Again, she lost the piece of paper saying how much each family gets. Aldus put in her car 7 boxes and 2 loose apples. Can you tell her what each family gets?
Kiki is supposed to visit three families on Thursday. They each get the same amount. Again, she lost the piece of paper saying how much each family gets. Aldus put in her car 3 crates, 4 boxes and 8 loose apples. Can you tell her what each family gets?
Kiki is supposed to visit three families. They each get the same amount. Again, she lost the piece of paper saying how much each family gets. Aldus put in her car 4 crates, 7 boxes, and 7 loose apples. Can you tell her what each family gets?
Kiki is supposed to visit four families. They each get the same amount. Again, she lost the piece of paper saying how much each family gets. Aldus put in her car 7 crates, 5 boxes and 2 loose apples. Can you tell her what each family gets?
Variation: Other Bases
You can do other bases, or just other combinations -- if the student likes the calculation, you could do, for example, 4 crates, 5 boxes, and 5 apples, 3 children, where the crates all have 8 boxes, the boxes all have 7 apples.
Fractions
1 apple, 2 children.
3 apples, 2 chilren
61 apples, 2 children
1 apple, 3 children. Ground rule -- the student does not have to know how to divide an apple equally into three parts. The student can simple say that it has been done.
2 applies, 3 children.
5 apples, 7 children.
Half of an apple, 2 children. (Ground rule for teachers -- half of a half seems like an acceptable answer to me.)
