What should you look at before calculating?
FNEGE says calculators are not allowed. Mental math is therefore useful, but you do not have to keep everything in your head. A line of scratch work may make a transformation safer than a long chain of memorized operations. The methods below are our teaching recommendations, not official instructions.
Look for a structure first. A number close to 100 may be easier to break down. A product followed by division may be simplified. A familiar fraction can sometimes replace a percentage. If nothing simple stands out, written calculation is still a valid method. A shortcut should reduce the number of steps while keeping the work easy to check, not force you to learn a new trick for every number.
Before calculating the exact result, set an order of magnitude. Multiplying about 50 by about 20 should give about 1,000, not 100 or 10,000. This estimate may not be enough to choose an answer, but it immediately catches a misplaced decimal point or zero.
How can you turn a product into two simple calculations?
Both breakdowns work in this example. Compare how much calculation each requires before choosing which one to reuse.
Calculez 48 × 25 sans calculatrice.
Choose an answer.
The solution, step by step
1,200.
The method in action
- First method: 25 = 100 ÷ 4. So 48 × 25 = 48 ÷ 4 × 100 = 12 × 100 = 1,200.
- Second method: 48 = 50 − 2. So 48 × 25 = 1,250 − 50 = 1,200. This also works, but requires knowing 50 × 25.
- Check: 48 is slightly less than 50, so the result should be slightly less than 1,250. The answer fits that benchmark.
Why simplify before multiplying?
A large product may just be a complicated way to write a small quotient. Keep the parentheses in your scratch work so you can see exactly what you are dividing.
Calculez (84 × 35) ÷ 21.
Choose an answer.
The solution, step by step
140.
The method in action
- Since 84 ÷ 21 = 4, the expression becomes 4 × 35 = 140. There is no need to calculate 84 × 35.
- The direct method gives 2,940 ÷ 21 = 140. It confirms the answer but uses larger intermediate numbers.
- An independent estimate gives about 80 × 35 ÷ 20 = 140. This rough check is consistent; simplification provides the exact proof.
The pitfall
Simplifying across addition changes the result
In (84 + 35) ÷ 21, you cannot replace only 84 ÷ 21 with 4 and then write 4 + 35. The divisor applies to the entire sum. You must write 84 ÷ 21 + 35 ÷ 21, or add first and then divide. The earlier simplification worked because it was a product.
Another limitation: an estimate is not an exact equality. If the choices are 139, 140, and 141, “around 140” does not distinguish them. Keep the exact value when the choices are close.
What should you record after a mental math exercise?
For each calculation, note the transformation you used, the number of steps, and any mistake. Then apply the same method to different numbers. If you get a wrong answer quickly, slow down when transforming the expression rather than when multiplying.
Alternate exercises where a shortcut is obvious with those where it is not. The goal is to recognize an opportunity, not force a breakdown every time. End each set by explaining one solution in a line or two: a method you can justify is easier to check under time pressure.
Key takeaways
- Look for a structure before calculating.
- Simplify factors in a product, but do not cancel across addition.
- An estimate checks the result; it does not always replace an exact calculation.