Thinking Critically: Problem-Solving Strategies

Problem-Solving Strategies in Mathematics

• When presented with mathematical word problems, organizing into problem-solving teams and employing systematic strategies allows for efficient solutions.

• A classic problem type involves finding the specific counts of two distinct entities (such as two types of animals) given the total count of entities and the total count of a shared attribute (such as feet).

Strategy 1: Guess and Check

• The guess and check strategy involves making an initial guess at the solution, testing that guess against the problem's conditions, and using the result to refine subsequent guesses.

• If a guess produces a value that is too large or too small, the guess is systematically adjusted to yield a smaller or larger result as required.

• Problem Context:

  • Total animals = $37$ (implied by individual animal combinations summing to $37$: $20 + 17 = 37$, $30 + 7 = 37$, $25 + 12 = 37$).

  • Target total feet = $98$ feet.

  • Chickens have $2$ feet each.

  • Pigs have $4$ feet each.

• Step 1: Mary's Initial Guess:

  • Mary guesses $20$ chickens and $17$ pigs.

• Step 2: Joe's Evaluation of Mary's Guess:

  • Chicken feet calculation: $2 \times 20 = 40$ feet.

  • Pig feet calculation: $4 \times 17 = 68$ feet.

  • Total feet calculation: $40 + 68 = 108$ feet.

  • Evaluation: $108$ feet is too high compared to the target of $98$ feet ("That's too many feet").

• Step 3: Sue's Adjustment:

  • Sue reasons that increasing the number of chickens (which have fewer feet) will reduce the overall total foot count.

  • Sue guesses $30$ chickens and $7$ pigs.

  • Chicken feet calculation: $2 \times 30 = 60$ feet.

  • Pig feet calculation: $4 \times 7 = 28$ feet.

  • Total feet calculation: $60 + 28 = 88$ feet.

  • Evaluation: $88$ feet is too low compared to the target of $98$ feet ("That should give us fewer feet").

• Step 4: Carlos's Proportional Refinement:

  • Carlos analyzes the errors in both previous guesses relative to the target of $98$ feet:

  • Mary's guess yielded $108$ feet, which is $10$ feet too many ($108 - 98 = 10$).

  • Sue's guess yielded $88$ feet, which is $10$ feet too few ($98 - 88 = 10$).

  • Since $108$ is $10$ too much and $88$ is $10$ too few, the exact target lies directly in the middle.

  • Carlos guesses $25$ chickens, which is halfway between Mary's guess of $20$ chickens and Sue's guess of $30$ chickens.

  • Evaluating Carlos's guess with $25$ chickens and $12$ pigs:

  • Chicken feet: $2 \times 25 = 50$

  • Pig feet: $4 \times 12 = 48$

  • Total feet: $50 + 48 = 98$ feet (exact match).

Strategy 2: Making a Table to Look for Patterns

• Making a table to record systematic attempts is a powerful refinement of the guess and check method.

• By listing sequential changes, key mathematical patterns emerge that make finding the exact solution straightforward without relying on random guessing.

• Nandita and Ann's Table Structure:

• Step-by-Step Breakdown of Table Rows:

  • Row 1: $20$ chickens, $17$ pigs.

  • Chicken feet: $2 \times 20 = 40$

  • Pig feet: $4 \times 17 = 68$

  • Total feet: $40 + 68 = 108$

  • Row 2: $21$ chickens, $16$ pigs.

  • Chicken feet: $2 \times 21 = 42$

  • Pig feet: $4 \times 16 = 64$

  • Total feet: $42 + 64 = 106$

  • Row 3: $22$ chickens, $15$ pigs.

  • Chicken feet: $2 \times 22 = 44$

  • Pig feet: $4 \times 15 = 60$

  • Total feet: $44 + 60 = 104$

• Pattern Analysis & Table Shortcut:

  • Each time the number of chickens increases by $1$ (and the number of pigs decreases by $1$):

  • Chicken feet increase by $2$

  • Pig feet decrease by $4$

  • Net total feet decrease by $2$ (since $-4 + 2 = -2$)

  • To drop from $108$ feet down to the required $98$ feet, a total decrease of $10$ feet is needed ($108 - 98 = 10$).

  • Since each row decreases the total feet count by $2$, the number of additional rows required is:     $\frac{10}{2} = 5$

  • Adding $5$ more chickens to the initial $20$ yields $20 + 5 = 25$ chickens and $17 - 5 = 12$ pigs.

Strategy 3: Drawing a Picture / Visual Representation

• Visual representation allows problem solvers to model constraints visually.

• Mike proposes drawing $37$ circles to represent the heads of the $37$ animals.

• Feet are then drawn onto the circles systematically to fulfill the constraint of $98$ total feet:

  • Draw $2$ legs on each of the $37$ circles, utilizing $2 \times 37 = 74$ feet.

  • Calculate remaining feet to distribute: $98 - 74 = 24$ remaining feet.

  • Distribute the remaining $24$ feet in pairs of $2$ to turn circles into pigs: $\frac{24}{2} = 12$ pigs.

  • The remaining circles with only $2$ legs represent chickens: $37 - 12 = 25$ chickens.