Mixed numbers in 4th grade

Liam's work shows $\frac{7}{8} - \frac{3}{8} = \frac{4}{4}$. The numerator subtraction $7 - 3 = 4$ is correct, but the denominator must stay $8$, not change to $4$. The best description is that Liam changed the denominator.

1. 1Look at Liam's work shown below.
2. 2Check the subtraction. When the denominators are the same, subtract the numerators and keep the denominator unchanged. The correct result would be $\frac{7}{8} - \frac{3}{8} = \frac{7 - 3}{8} = \frac{4}{8}$. But Liam wrote $\frac{4}{4}$ — the denominator changed from $8$ to $4$ instead of staying at $8$.
3. 3For reference, the correct difference $\frac{4}{8}$ is not yet in simplest form. The numbers $4$ and $8$ share a common factor of $4$, so $\frac{4}{8} = \frac{1}{2}$. However, the main issue in Liam's work is not simplification — it is that the denominator changed.
4. 4Liam's denominator changed from $8$ to $4$, which is not correct. When subtracting fractions with the same denominator, the denominator must stay the same. So the best description is that Liam changed the denominator.

Common mistake. A common mistake is checking only the numerator and not noticing the denominator changed. Even if $7 - 3 = 4$ is correct, the denominator must remain $8$, giving $\frac{4}{8}$, not $\frac{4}{4}$.

The idea underneath. When subtracting like-denominator fractions, two checks matter: the numerator subtraction must be correct, AND the final answer must be in simplest form.

The model uses the wrong jump length. The student jumped $3$ ticks ($\frac{3}{5}$) instead of $2$ ticks ($\frac{2}{5}$). A correct jump should land at $\frac{2}{5}$, not $\frac{1}{5}$.

1. 1The expression is $\frac{4}{5} - \frac{2}{5}$. A correct number-line model should start at $\frac{4}{5}$, jump to the left (subtraction), and have a jump length of $\frac{2}{5}$.
2. 2Check the start point. The jump begins at $\frac{4}{5}$, which matches the first fraction in the expression. The start point is correct.
3. 3Check the direction. The jump arc goes to the left, which is the correct direction for subtraction. The direction is correct.
4. 4Check the jump length. The jump goes from $\frac{4}{5}$ to $\frac{1}{5}$. Counting the tick marks gives a distance of $\frac{3}{5}$, but to subtract $\frac{2}{5}$, the jump should only be $\frac{2}{5}$ long.
5. 5The error is the jump length. The student jumped $3$ ticks ($\frac{3}{5}$) instead of $2$ ticks ($\frac{2}{5}$), so the diagram lands at $\frac{1}{5}$ instead of the correct answer $\frac{2}{5}$.

Common mistake. A common mistake is to confuse the jump length with the landing point. The jump length should match the second fraction in the expression ($\frac{2}{5}$), not the answer. To subtract $\frac{2}{5}$, count back exactly $2$ ticks from $\frac{4}{5}$.

The idea underneath. On a number-line model, the start point is the first fraction, the jump direction matches the operation (right for $+$, left for $-$), and the jump length matches the second fraction. The landing point is the answer to the expression — it is not the same as the jump length.

$\frac{3}{10} - \frac{1}{10} = \frac{2}{10}$, which simplifies to $\frac{1}{5}$.

1. 1Subtract the numerators. The denominator stays the same.
2. 2Now simplify $\frac{2}{10}$ by dividing both the top and the bottom by $2$.
3. 3The simplified difference is $\frac{1}{5}$.

Common mistake. A common mistake is stopping at $\frac{2}{10}$ without simplifying. After subtracting, always check whether the numerator and denominator share a factor — here both $2$ and $10$ can be divided by $2$, giving $\frac{1}{5}$ in simplest form.

The idea underneath. When fractions share a denominator, you only subtract the numerators — the denominator stays the same. The work is not done until you check whether the result can be written in simpler form.

The sum $\frac{5}{8} + \frac{3}{8} = \frac{8}{8}$, and Liam's model shows all $8$ parts shaded. The model correctly shows the sum.

1. 1First, find the sum. When adding fractions with the same denominator, add only the numerators and keep the denominator the same.
2. 2So Liam should shade $8$ parts of the rectangle. Now look at his model and count the shaded parts.
3. 3Liam shaded all $8$ parts. The model shows $\frac{8}{8}$, which matches the sum $\frac{8}{8}$. The shaded count is correct.
4. 4The rectangle has $8$ equal parts, matching the denominator $8$. Both the partition count and shaded count are correct. The best description is: 'The model correctly shows the sum.'

Common mistake. A common mistake is assuming that a fully shaded bar must be wrong. Always compute $5 + 3 = 8$ first — when the sum equals the denominator, all parts should be shaded, and that is exactly what Liam did.

The idea underneath. When adding fractions with the same denominator, only the numerators add — the denominator stays the same. To check a model, compute the sum first, then count the shaded parts to verify they match the new numerator.