Equivalent fractions: change part size while preserving value

Where we are on the map

The meaning of a fraction is secure; now we connect several notations for one value.

Begin with a familiar image

Split each part of a half into two again. The shaded length stays fixed, although its name becomes 2/4.

Precise meaning

Multiplying or dividing numerator and denominator by the same nonzero number preserves value: a/b = ak/bk.

The lesson’s support signal

meaning → model → calculation → check → explain in your own words

×k on top and bottom → smaller parts → more selected parts → same point

Worked example

14/21 = 2/3 because both terms divide by 7. Check backward: 2/3 × 7/7 = 14/21.

Example with a fading prompt

Change the numbers in the example and repeat the solution without copying its steps. Estimate first, calculate exactly, and check against the original condition.

Recall without a prompt

1. Define the main idea in one sentence.
2. Rebuild the support signal from memory.
3. Solve the example with different numbers and explain why every step is valid.

Find and correct the mistake

Changing 1/2 to 2/3 by adding 1 to both terms is wrong: it changes the size of the parts and moves the point.

Transfer to a new setting

Apply the same relationship to something you can measure yourself. Name the quantities, units, and the boundary within which the model is valid. Check the answer by a second representation or an inverse operation.

Exercise

Required. Prove that 14/21 and 10/15 are equal by simplifying, cross-products, and a number-line model.

With your own data. Create one everyday example with the same structure and show it in words, a formula, and a check.

Optional. Seven days later, change the numbers and repeat without the support map.

Open perspective

Common-sized parts let us compare unequal fractions on one scale.

Course contents