In these lessons, we will learn some math problem solving strategies for example, Verbal Model (or Logical Reasoning), Algebraic Model, Block Model (or Singapore Math), Guess & Check Model and Find a Pattern Model.

**Related Pages**

Solving Word Problems Using Block Models

Heuristic Approach to Problem-Solving

Algebra Lessons

The strategies used in solving word problems:

- What do you know?
- What do you need to know?
- Draw a diagram/picture

**Solution Strategies**

Label Variables

Verbal Model or Logical Reasoning

Algebraic Model - Translate Verbal Model to Algebraic Model

Solve and Check.

Step 1: Identify (What is being asked?)

Step 2: Strategize

Step 3: Write the equation(s)

Step 4: Answer the question

Step 5: Check

Using the guess and check problem solving strategy to help solve math word problems.

**Example:**

Jamie spent $40 for an outfit. She paid for the items using $10, $5 and $1 bills.
If she gave the clerk 10 bills in all, how many of each bill did she use?

- Identify - What is the question?
- Plan - What strategy will I use to solve the problem?
- Solve - Carry out your plan.
- Verify - Does my answer make sense?

**Example:**

Marcus ran a lemonade stand for 5 days. On the first day, he made $5. Every day after that he made $2
more than the previous day. How much money did Marcus made in all after 5 days?

In this lesson, we will look at some intermediate examples of Find a Pattern method of problem-solving strategy.

**Example:**

The figure shows a series of rectangles where each rectangle is bounded by 10 dots.

a) How many dots are required for 7 rectangles?

b) If the figure has 73 dots, how many rectangles would there be?

**Solution:**

Rectangles | Pattern | Total dots |

1 | 10 | 10 |

2 | 10 + 7 | 17 |

3 | 10 + 14 | 24 |

4 | 10 + 21 | 31 |

5 | 10 + 28 | 38 |

6 | 10 + 35 | 45 |

7 | 10 + 42 | 52 |

8 | 10 + 49 | 59 |

9 | 10 + 56 | 66 |

10 | 10 + 63 | 73 |

a) The number of dots required for 7 rectangles is 52.

b) If the figure has 73 dots, there would be 10 rectangles.

**Example:**

Each triangle in the figure below has 3 dots. Study the pattern and find the number of dots for 7
layers of triangles.

**Solution:**

Layers | Pattern | Total dots |

1 | 3 | 3 |

2 | 3 + 3 | 6 |

3 | 3 + 3 + 4 | 10 |

4 | 3 + 3 + 4 + 5 | 15 |

5 | 3 + 3 + 4 + 5 + 6 | 21 |

6 | 3 + 3 + 4 + 5 + 6 + 7 | 28 |

7 | 3 + 3 + 4 + 5 + 6 + 7 + 8 | 36 |

The number of dots for 7 layers of triangles is 36.

**Example:**

The table below shows numbers placed into groups I, II, III, IV, V and VI. In which groups would the
following numbers belong?

a) 25

b) 46

c) 269

I | 1 | 7 | 13 | 19 | 25 |

II | 2 | 8 | 14 | 20 | 26 |

III | 3 | 9 | 15 | 21 | 27 |

IV | 4 | 10 | 16 | 22 | |

V | 5 | 11 | 17 | 23 | |

VI | 6 | 12 | 18 | 24 |

**Solution:**

The pattern is: The remainder when the number is divided by 6 determines the group.

a) 25 ÷ 6 = 4 remainder 1 (Group I)

b) 46 ÷ 6 = 7 remainder 4 (Group IV)

c) 269 ÷ 6 = 44 remainder 5 (Group V)

**Example:**

The following figures were formed using matchsticks.

a) Based on the above series of figures, complete the table below.

Number of squares | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

Number of triangles | 4 | 6 | 8 | 10 | ||||

Number of matchsticks | 12 | 19 | 26 | 33 |

b) How many triangles are there if the figure in the series has 9 squares?

c) How many matchsticks would be used in the figure in the series with 11 squares?

**Solution:**

a)

Number of squares | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

Number of triangles | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 |

Number of matchsticks | 12 | 19 | 26 | 33 | 40 | 47 | 54 | 61 |

b) The pattern is +2 for each additional square.

18 + 2 = 20

If the figure in the series has 9 squares, there would be 20 triangles.

c) The pattern is + 7 for each additional square

61 + (3 x 7) = 82

If the figure in the series has 11 squares, there would be 82 matchsticks.

**Example:**

Seven ex-schoolmates had a gathering. Each one of them shook hands with all others once. How many
handshakes were there?

**Solution:**

A | B | C | D | E | F | G | |

A | |||||||

B | ● | ||||||

C | ● | ● | |||||

D | ● | ● | ● | ||||

E | ● | ● | ● | ● | |||

F | ● | ● | ● | ● | ● | ||

G | ● | ● | ● | ● | ● | ● | |

HS | 6 | 5 | 4 | 3 | 2 | 1 |

Total = 6 + 5 + 4 + 3 + 2 + 1 = 21 handshakes.

The following video shows more examples of using problem solving strategies and models.

Question 1: Approximate your average speed given some information

Question 2: The table shows the number of seats in each of the first four rows in an auditorium. The remaining ten rows follow the same pattern. Find the number of seats in the last row.

Question 3: You are hanging three pictures in the wall of your home that is 16 feet wide. The width of your pictures are 2, 3 and 4 feet. You want space between your pictures to be the same and the space to the left and right to be 6 inches more than between the pictures. How would you place the pictures?

Try the free Mathway calculator and
problem solver below to practice various math topics. Try the given examples, or type in your own
problem and check your answer with the step-by-step explanations.

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