Problem-Solving Strategy: Draw a Picture - Second Grade Mathematics
One of the greatest secrets of great mathematicians, scientists, and engineers is that they rarely try to solve complicated problems entirely in their heads: they draw pictures! Drawing a picture transforms words and numbers into visual scenes that you can inspect, count, and manipulate. In mathematics, drawing a picture does not mean creating elaborate art with shading and colors; it means drawing simple, clean math sketches—such as circles, tally marks, number lines, and boxes—that represent the mathematical relationships in a story. In this chapter, you will learn how to turn word problems into effective math sketches and use them to solve complex scenarios.
Math Sketches vs. Art Drawings
A math sketch has one single purpose: to make mathematical relationships clear and easy to count.
Art Drawing (Takes too long!):
Drawing individual feathers on birds, leaves on trees, eyelashes on characters...
Math Sketch (Fast and Efficient!):
Draw circles (O), dots (.), boxes ([ ]), or tallies (|).
Organize items in rows of 5 or 10, just like a ten-frame!
The Five-Group Organization Rule
Whenever you draw items in a math sketch, organize them in rows of 5. If a problem involves 13 apples, do not draw 13 scattered circles across your paper where you might lose count. Draw: Row 1: O O O O O (5) Row 2: O O O O O (5) Row 3: O O O (3) With five-group organization, your brain can instantly see 10 + 3 = 13 without counting one by one!
Visualizing Tricky Spatial Word Problems
Some word problems are almost impossible to solve without a sketch because they involve physical arrangements.
The Fence Post Problem:
Story: A gardener builds a straight fence that is 20 feet long. She places a wooden post every 5 feet, including a post at the very beginning and a post at the very end. How many wooden posts does she need?
Many people guess: 20 divided by 5 is 4 posts!
Let us DRAW A PICTURE:
Post 1 Post 2 Post 3 Post 4 Post 5
| | | | |
+--- 5 ft ---+--- 5 ft ---+--- 5 ft ---+--- 5 ft ---+
0 ft 5 ft 10 ft 15 ft 20 ft
Count the posts: 1, 2, 3, 4, 5 posts!
The correct answer is 5 posts, NOT 4!
Why the Drawing Saved the Day
Notice why the mathematical sketch was essential: there are 4 spaces between the posts, but 5 posts in total because a post was required at the very start (0 feet)! Without a sketch, almost everyone falls into the trap of forgetting the starting post.
Using Pictures to Solve Equal-Sharing Problems
Sketches are equally brilliant when sharing items fairly among friends.
Sharing 15 Strawberries Fairly Among 3 Children:
Step 1: Draw 3 plates (large circles):
Plate A: ( ) Plate B: ( ) Plate C: ( )
Step 2: Deal 15 strawberries (dots) one by one onto the plates:
Plate A: ( . . . . . ) Plate B: ( . . . . . ) Plate C: ( . . . . . )
Count the dots on each plate: Exactly 5 dots per plate!
Answer: Each child receives 5 strawberries.
Dealing Out Items
Dealing out dots into circles mirrors dealing playing cards: one for you, one for you, one for me, until all items are distributed. The picture shows immediately whether each group is equal and how many items belong in each share.
Practice Exercises
Exercise 1: Draw or describe a math sketch using five-group rows to solve: "A baker packed 17 blueberry tarts into a box. 8 tarts were eaten. How many tarts were left in the box?"
Exercise 2: Solve this line-up problem by drawing a picture: "In a cafeteria lunch line, Tyler is 4th from the front. Maya is standing directly behind Tyler. There are 3 children standing behind Maya. How many total children are in the lunch line?"
Exercise 3: Solve this sharing problem by drawing a picture: "Four friends share 16 colored markers equally. Draw 4 boxes for the friends and distribute 16 dots equally. How many markers does each friend receive?"
Exercise 4: Explain why organizing circles or dots into neat rows of five is more helpful in a math sketch than scattering them randomly across the page.
Solutions and Explanations
Solution to Exercise 1: Draw 17 circles arranged in neat rows: top row of 5, second row of 5, third row of 5, fourth row of 2 (total 17). To show 8 tarts eaten, cross out 8 circles: cross out the bottom row of 2, the third row of 5, and 1 circle from the second row (2 + 5 + 1 = 8). Count the uncrossed circles: 4 in the second row plus 5 in the top row = 9. There are 9 tarts left in the box.
Solution to Exercise 2: Sketch the line of children from front to back: Position 1 [Child], Position 2 [Child], Position 3 [Child], Position 4 [Tyler], Position 5 [Maya], Position 6 [Child], Position 7 [Child], Position 8 [Child]. Counting the sketch from front to back reveals 8 children in total standing in the lunch line.
Solution to Exercise 3: Draw 4 large boxes labeled Friend 1, Friend 2, Friend 3, and Friend 4. Distribute 16 dots one by one across the 4 boxes until all 16 dots are placed. Each box contains exactly 4 dots. Each friend receives 4 markers.
Solution to Exercise 4: When dots are scattered randomly across a page, your brain must count each dot individually, which is slow and makes it easy to accidentally recount the same dot or skip one. Organizing dots into structured rows of five mirrors a ten-frame, allowing you to instantly subitize quantities (such as recognizing two full rows of 5 as 10) and verify counts at a glance.