Math // Fractions

Adding and subtracting mixed numbers

Add and subtract mixed numbers with like denominators, including by converting to improper fractions or using properties of operations

Ages
9 to 10
Type
Method
Builds on
2 topics
Leads to
1 topic
Standards
1 code

Quick check

If a recipe calls for 1 and 2/5 cups of oats and another batch needs 2 and 3/5 cups, can the learner add them together to get the total — including converting when the fifths add up past a whole?

Ask this to see whether the topic is secure.

Topic // Mastery

What mastery looks like

What a learner can do once this topic is secure.
  • Calculate 2 3/5 + 1 4/5 and express as a mixed number
  • Subtract 3 1/4 from 5 3/4
  • Solve 4 2/6 − 1 5/6 by regrouping the whole number

Topic // Map

Where this fits

What to learn first and what this topic opens up, with the reason for each link. Required links come first.
  • RequiredMust come first
  • HelpfulHelps, but not essential

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Adding and subtracting mixed numbers

Ages 9 to 10 // Method

Leads to

1 topic

Topic // Research-informed

Teaching notes

From the Marble Skill Taxonomy. These notes have not been validated in classrooms, so treat them as a starting point for your own judgement.

The approach

Add and subtract mixed numbers with like denominators (ages 9–10). Two routes are worth teaching: (a) operate on whole-number and fraction parts separately, which exposes the two signature difficulties — in addition the fraction parts can sum past a whole (2 3/5 + 1 4/5 → 3 + 7/5, regroup to 4 2/5), and in subtraction the first fraction can be smaller than the second, forcing a borrow (4 2/6 − 1 5/6: regroup 4 into 3 + 6/6); and (b) convert both to improper fractions, which sidesteps the borrow and is worth teaching as the cleaner fallback. Let the awkward subtraction motivate the improper method, keep the denominator a fixed unit throughout, and always tidy an improper fraction part back into a proper mixed number.

Common misconceptions

  • Adds across the fraction parts (3/5 + 4/5 → 7/10), applying whole-number addition to numerator and denominator
  • Subtracts smaller-from-larger in each column and never borrows (4 2/6 − 1 5/6 → 3 3/6, taking |2−5|=3)
  • Leaves an improper fraction part in the answer (3 7/5) instead of regrouping to 4 2/5
  • Miscounts when converting mixed to improper (2 3/5 → 10/5 then forgets the +3/5, or writes 5/5)

Useful representations

  • Number line: jump the whole numbers, then the fraction parts, crossing whole boundaries
  • Bar model with whole tiles plus fraction tiles; a whole tile that shatters into b unit-fractions for the borrow, or b fraction tiles that fuse into a whole
  • The regroup written as 4 = 3 + 6/6 (or the improper rewrite 26/6)
  • Improper-fraction form beside the mixed form for comparison

Teaching practices

  • Teach BOTH methods and let a subtraction that needs regrouping (4 2/6 − 1 5/6) motivate why converting to improper fractions is often cleaner
  • Make the borrow physical: exchange one whole for its equivalent in the common unit fraction (1 = 6/6) before subtracting the fraction parts
  • In addition, always check whether the fraction parts crossed a whole and regroup (7/5 → 1 2/5) so the answer is a proper mixed number
  • Estimate with the whole numbers first (2 3/5 + 1 4/5 is between 3 and 5, near 4½) to police the result
  • Hold the denominator as a fixed unit — fifths add to fifths — to block the add-across error carried over from whole-number habits

Activities to try

  1. 1

    Trade a whole for unit fractions: make the borrow and carry physical

    Why it works: Borrowing is not a mysterious digit-move: it is literally trading one whole for its equivalent in the common unit fraction (1 = 6/6), and carrying is the reverse — fusing a whole-worth of unit fractions back into one whole. The denominator stays fixed because fifths only join with fifths.

    1. Work the subtraction 4 2/6 − 1 5/6 using whole tiles and fraction tiles that are locked to sixths so you cannot change the denominator.
    2. Try to take away 5 sixths when you only have 2 — notice you are stuck. Break one whole tile into 6 sixths (the borrow), so you now have 8 sixths, and finish: 8/6 − 5/6 = 3/6, leaving 2 3/6 = 2 1/2.
    3. Work the addition 2 3/5 + 1 4/5 the same way: gather the fifths, and when 5 fifths collect, fuse them into a fresh whole tile, giving 4 2/5.
    4. Optionally model the same two problems on a number line: jump the whole numbers first, then the fraction parts, and watch a jump cross a whole boundary — the same regroup seen as a different representation.
    5. Write each exchange as a number sentence: 4 = 3 + 6/6, and 7/5 = 1 + 2/5.
  2. 2

    The improper-fraction fallback, side by side

    Why it works: Converting to improper fractions avoids the borrow entirely, so it is a reliable fallback when the subtraction needs regrouping; both methods give the same answer when done carefully.

    1. Rewrite 4 2/6 and 1 5/6 as improper fractions (26/6 and 11/6), subtract to get 15/6, then convert back: 15/6 = 2 3/6 = 2 1/2.
    2. Do the same subtraction by the separate-parts borrow method and check both answers match.
    3. When converting 2 3/5 to improper, count carefully: 2 wholes is 10/5, plus 3/5 is 13/5 — do not drop the +3/5 or stop at 5/5.
    4. Decide which method you found cleaner for the subtraction, and explain why converting to improper removed the borrow.
  3. 3

    Estimate first to police the answer

    Why it works: An estimate that starts with the whole numbers and then adjusts for the fraction parts catches value errors such as a missed borrow (3 3/6 instead of about 2 1/2) or added denominators (3 7/10 instead of about 4 1/2), because a wrong answer no longer sits near the estimate. An un-regrouped answer like 3 7/5 has the correct value (it equals 4 2/5), so an estimate cannot catch it; the separate proper-fraction check does that job.

    1. Before computing 2 3/5 + 1 4/5, estimate using the whole numbers: the sum is between 3 and 5, and since both fraction parts are large, near 4 1/2.
    2. Compute the exact answer and check it sits close to your estimate.
    3. Repeat for 4 2/6 − 1 5/6: whole-number subtraction gives about 3, but because 2/6 is less than 5/6 the fraction part pulls the answer down by about a half, so expect a result between 2 and 3, around 2 1/2 — then confirm 2 1/2 is reasonable.
    4. For each result, ask: is the fraction part proper (numerator less than denominator)? If not, regroup.

Topic // Standards

Curriculum standards

Codes for the curriculum statements this topic maps to. Each publisher owns its standard text.
Common Core State Standards for Mathematics
4.NF.3.c

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