In problem that have an action (change over time), there are three values: the start amount, the amount of the change, and the resulting amount. Start unknown problems are the problems of these types (join and separate) where the start amount is unknown. This is pretty easy to understand, and if you are good at identifying join and separate problems that are result or change unknown problems, you should find it only a small step to recognizing start unknown problems.
Some typical examples:
Kylie made cookies for the school fair. She sold 25 cookies, and has 10 cookies left over. How many cookies did she have to start with?
- The action is selling cookies. She has fewer cookies at the end than at the beginning, so this is a separate problem.
- We know how many she sold (change) and how many she has left over (result). The question asks for how many she started with (start), so this is a Separate, Start Unknown problem (SSU)
Luke had some puzzles. For his birthday, he got 2 more puzzles. Now he has 7 puzzles. How many puzzles did he have before his birthday?
- The action is getting puzzles as a birthday present. He has more puzzles at the end than at the beginning, so this is a join problem.
- We know how many he got (change) and how many he has at the end (result). The question asks how many did he have before (start), so this is a Join, Start Unknown problem.
In these examples, the amounts for the start, change, and end amounts are given in order, which makes them easier to understand. The same problems would be somewhat more difficult to understand if the sentence structures were changed:
Kylie sold 25 cookies at the school fair, and has 10 cookies left over. How many cookies did she have to start with?
For his birthday, Luke got 2 puzzles. Now he has 7 puzzles. How many puzzles did he have before his birthday?
In both of these versions, the start amount is not mentioned until the question. These are somewhat more natural ways to state problems like these, so children should encounter versions like this, but it's a good idea to start with easier problem statements first, and after some experience, introduce the more sophisticated problem statements.
Solving Start Unknown problems
Start unknown problems are probably the most difficult problems to solve by direct modeling, and most children who are at the direct modeling stage of their development are not able to solve start unknown problems. To solve a start unknown problem by direct modeling, you need to do a lot of guess-and-checking, because the first amount that you would start by counting out to direct model is the unknown value. Most children become ready to solve start unknown problems when they have a set of computational strategies that let them be more flexible about their computational thinking.
Some examples of flexible strategies are:
For his birthday, Luke got 2 puzzles. Now he has 7 puzzles. How many puzzles did he have before his birthday?
I realize that the problem is an unknown +2=7. What plus 2 is 7? (Use known addition facts, or direct model to find the unknown).
I recognize that this problem is ?+2=7, and that this is equivalent to 2+?=7--solve by my favorite JCU strategy.
I recognize that this problem is ?+2=7, and that I can model this by comparing 2 and 7--solve by my favorite CDU strategy.
I recognize that the total consists of the amount before and the amount change--solve by my favorite PPW-PU strategy. This is a particularly useful way to think of the problem, and it will be revisited again in an upcoming lesson.
I can imagine the process happening backwards: Luke has 7, before his birthday he had 2 less.
Kylie sold 25 cookies at the school fair, and has 10 cookies left over. How many cookies did she have to start with?
I recognize this problem as ?-25=10. What minus 25=10? Use known number facts to find the unknown.
I recognize this problem as ?-25=10, and that I can model this by comparing the change and/or end amount to the start--model using a modified compare (CRU?) strategy.
I recognize this problem as ?-25=10. I recognize that the total is the amount before the sale, and it consists of the amount sold and the amount left over--solve by my favorite PPW-WU strategy. This is a particularly useful way to think of the problem, and it will be revisited again in an upcoming lesson.
I can imagine the process happening backwards: Kylie has 10 left over. Before the fair, she had 25 more.