Reasoning · Grade 6-2 Mixed Operations with Fractions
Problem
A short product shows the pieces cancelling.
Multiplying fractions is 'tops times tops over bottoms times bottoms', which a Grade 5 solver already knows. Once the whole product is one fraction, cancelling a factor that appears on both sides of the bar is the same simplifying you do with 6/8.
5.NF.B.4Solve An Easier Related ProblemA three-factor warm-up shows that each numerator cancels the denominator beside it, leaving only the two ends.
Dividing by a number undoes multiplying by it, so a factor above and the same factor below vanish together.
Which factors you pair up first makes no difference to the product, so the cancelling may be done in whatever order is convenient.
Running (1) to the end gives 50 1/2.
You never multiply anything big. Seeing that each numerator is one more than its own denominator — and therefore equal to the next factor's denominator — is the entire calculation, and that is a Grade 5 fraction skill, not an algebra skill.
5.NF.B.4Look For A PatternTurning divisions into products, (2) is 10.
Rewriting every division as a multiplication is the one move that lets the whole line be reordered and cancelled at once. Dividing by a number smaller than 1 makes things bigger, so ending up at 10 rather than something tiny is exactly right.
6.NS.A.1Organize Information In More WaysThe up-and-down sum is a square, so (3) is 1/121.
Adding thirteen small numbers is easy, but noticing that the total is 7 x 7 is what makes the fraction collapse. Then it is just simplifying 7/77 to 1/11 twice — ordinary Grade 4 and 5 work.
4.OA.C.5Look For A PatternWith numerators three places later, (4) is 39.
In (1) the overlap was one step, so one number survived at each end; here the numerator is 3 bigger than the denominator, so exactly three numbers survive at each end. Counting how far the shift is tells you how many leftovers to expect before you compute anything.
5.NF.B.4Look For A PatternSplit into two chains, (5) is 50/99.
The expression looks like 196 factors jumbled together, but sorting them into two piles turns it into two copies of the problem already solved in step 2. Re-sorting a product is free — that is just the commutative property.
5.NF.B.4Identify SubproblemsReading just the ends confirms every answer.
Naming the two surviving ends before doing arithmetic makes an error impossible to hide: any stray factor is instantly visible because it should have had a partner.
5.OA.A.1Look For A PatternLine the fractions up so each top matches the next bottom — then the whole middle erases itself and only the two ends are left.