# Well-structured problems

## Strategy 1: Well-structured problems

Problem solving begins in kindergarten (or earlier) with simple, well-structured problems that connect to a child’s everyday understanding of how things work. Typical early problems involve joining or separating quantities, and they are structured so that children can easily model the action in a problem, using counters.
• Jennie has 3 shells. Her brother gives her 5 more shells. Now how many shells does Jennie have? (joining 3 shells and 5 shells)
• Pete has 6 cookies. He eats 3 of them. How many cookies does Pete have then? (separating 3 cookies from 6 cookies)
• 8 birds are sitting on a tree. Some more fly up to the tree. Now there are 12 birds in the tree. How many flew up? (joining, where the change is unknown)

The types of well-structured problems get progressively more complicated, and the solutions that children develop to solve them get progressively more sophisticated, leading eventually to fluency and problem solving competence. Other types include part-whole, comparison, rate, array, and grouping problems.
(See Carpenter et. al., 1999)

Algebra and geometry problems can also be structured to follow a pattern, making the solution approach more apparent.
Algebra examples can be analyzed using a bar model:

1) You’re driving on a vacation. You drive at 50 mph for 7 hours. How far have you driven?

Counting out 7 groups of 50: d = 50 · 7 This procedure generates the formula d = r · t

2) You drive 300 miles in 6 hours. How fast were you driving, on average? (how many miles do you go in each hour)

Dividing 300 into 6 groups: r = 300/6
What formula would you create from this example? Can students see how this relates to d = r · t?

3) How long does it take you to drive 400 miles at 50 mph?

Counting how many 50’s in 400 (measurement division). 400 / 50 = ___
How does the formula relate to this calculation?

Geometry problems can be modeled with simple drawings:

1) A rectangular table is 6 feet long by 2 feet wide. What is the area of its surface? (Optional discussion question: Would a 15 square foot table cloth fit over it?)
6 · 2 = x

2) A rectangular table is 6 feet long and has an area of 18 square feet. How wide is it?
6 · x = 18

3) A rectangular table is 2.5 feet wide and has an area of 10 square feet. How long is it?
x · 2.5 = 10

Whenever you introduce new types of real-life problems, structure them carefully to allow students to build on what they already know.