Reasoning · Grade 4-2 Adding and Subtracting Fractions
Problem
For A, sum minus difference halved gives 7.
Trimming away the part that makes the bars unequal turns an awkward 'sum and difference' question into a plain sharing question, and sharing 14 between two equal bars is something a fourth grader can do in one step.
4.OA.A.3Draw A DiagramTrimming away the part that makes the two bars unequal leaves two equal bars, so the rest can simply be shared in half.
The longer bar is the shorter bar together with the difference, so removing the difference from the total removes it from the longer bar alone.
Once that difference is gone the two bars are a visible statement of equality, and equal bars share a total evenly.
Only 7 and 13 fit, so no other pair works.
The differences in the list shrink by 2 every time, so each difference appears exactly once - that is why the bar picture could only ever give one answer.
4.NBT.B.4Make A Systematic ListA is proper, so it is 7/13.
Thinking of 7/13 as 7 of the 13 equal pieces of one whole makes it obvious it is less than a whole; flipping it to 13/7 would be more than a whole, so only one of the two placements can be called proper.
3.NF.A.1Guess And CheckThe same trick gives B the pair 13 and 25.
The exact same picture works with different numbers, which is the sign that this is a method rather than a lucky guess.
4.OA.A.3Draw A DiagramB is improper, so it is 25/13.
25/13 means 25 pieces when only 13 pieces make a whole, so there is more than one whole - exactly what improper is supposed to mean.
3.NF.A.1Guess And CheckSame denominator, so the sum is 32/13.
When the pieces are the same size, adding fractions is no harder than adding 7 apples and 25 apples - the denominator just names the piece and never changes.
4.NF.B.3Draw A DiagramAs a mixed number that is 2 6/13.
Pulling out whole groups of 13 pieces is the same move as trading 10 ones for a ten in place value - it just says the same amount in a friendlier way.
4.NF.B.3Draw A DiagramWhen two numbers' sum and difference are both given, chop off the difference and halve what is left - then let the words proper or improper tell you which number rides on top.