Problem
Match denominators or numerators to compare
A fraction 4/□ sits between 2/5 and 8/9. I need the biggest whole number that can fill the box and still keep both inequalities true.
Fractions
Your answer
How to solve
Strategy Solve an Easier Related Problem — The middle fraction already has numerator 4, so instead of finding a common denominator I make the numerators the same. Rewriting 2/5 and 8/9 so they each have a numerator of 4 turns a messy three-way comparison into an easy one: with equal numerators, the fraction with the smaller denominator is larger. Then I guess-and-check the boundary to be sure I pick the largest box.
1STEP 1
Give the outer fractions the numerator 4
Rewrite each outer fraction with numerator 4: 2/5 doubles to 4/10 and 8/9 halves to 4/4.5.
2/5=(2 × 2)/(5 × 2)=4/10, 8/9=(8 ÷ 2)/(9 ÷ 2)=4/4.5
Making all three numerators equal to 4 is the same trick as finding a common denominator, just easier here because the box already starts with a 4 on top.
4.NF.A.1Solve An Easier Related Problem2STEP 2
Compare fractions that share a numerator
With equal tops a bigger bottom is a smaller fraction, so the box sits between 4.5 and 10.
4/10 ＜ 4/□ ＜ 4/4.5 → 4.5 ＜ □ ＜ 10
Cutting a 4-piece bar into more equal parts makes each part smaller, so a larger denominator gives a smaller fraction — the box has to be smaller than 10 but bigger than 4.5.
3.NF.A.3Solve An Easier Related ProblemWhen two fractions have the same numerator, the one with the bigger denominator is the smaller fraction.
Why?
A bigger denominator cuts the same whole into more pieces, and cutting a fixed whole into more parts makes each part smaller.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Taking the same number of the smaller pieces therefore gives less than taking that many of the bigger pieces.
Why?
Each piece on one side pairs with one piece on the other, and in every pair the piece from the more finely cut whole is the smaller one.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
3STEP 3
Pick and check the largest natural number
The largest whole number strictly inside is 9; 10 fails because 4/10 is just 2/5 again.
2/5=0.4 ＜ 4/9≈ 0.444 ＜ 8/9≈ 0.889
Testing the boundary value 9 confirms it works while 10 just barely fails, so 9 is the largest box that keeps 4/□ inside the range.
4.NF.A.2Guess And CheckAnswer
9
4/9 ≈ 0.444 really does land between 0.4 and 0.889, so the answer sits inside the range. Using 10 would give 4/10 = 2/5, which is equal (not greater), so 9 being the largest makes sense.
Takeaway
Make the tops match instead of the bottoms — then the fraction with the bigger bottom is the smaller one.
- Give the outer fractions the numerator 4
- Compare fractions that share a numerator
- Pick and check the largest natural number
Where next?
Another one like thissuggested
Step back and solidifysuggested