Sensim Math · Depth 한국어

2-2 · Multiplication Facts

Bound an inequality at equality

3.OA.A.43.OA.C.7 · take · grade 3

Archetype: Pin Down a Number from Digit and Range Conditions · step in a 9-type progression

▶ Practice — 12 problems

Find every number from 11 to 99 that can go in the □\square to make the statement true.

6×8<9×□6\times8<9\times\square

Show solution

Understand

We need every digit from 1 to 9 that can fill the box so that 9 times the box is larger than 6 times 8.

Givens
  • The inequality is 6 x 8 < 9 x box.
  • The box must be a whole number from 1 to 9.
Unknowns
  • All values from 1 to 9 that make 9 x box greater than 48.
Constraints
  • The box is a single whole number between 1 and 9.
  • The relationship must be strictly greater than (not equal).

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

First compute the fixed left side, then test the box values in order from small to large to find where 9 x box first passes 48; listing the candidates keeps every value from 1 to 9 accounted for.

Execute

#6 Guess and Check 3.OA.C.7
Work out 6 times 8 so the inequality becomes a comparison with a single number.
6×8=486 \times 8 = 48
6 times 8 is a basic multiplication fact, so the left side is just the number 48.
#6 Guess and Check 3.OA.A.4
Test multiples of 9 in order: 9x5 = 45 is not greater than 48, but 9x6 = 54 is greater than 48. So the box must be 6 or larger.
9×5=45, 9×6=549 \times 5 = 45,\ 9 \times 6 = 54
Checking the nine-times facts shows exactly when the right side becomes bigger than 48.
#2 Make a Systematic List 3.OA.A.4
Every box value 6 and above (and at most 9) makes the right side larger than 48, so collect them.
□∈{6,7,8,9}\square \in \{6, 7, 8, 9\}
Since bigger box values only make 9 x box larger, all of 6, 7, 8, 9 keep the inequality true.
Answer: 6, 7, 8, 9

Review

Checking the smallest answer: 9x6 = 54 > 48, true; and the value just below, 9x5 = 45, is not greater than 48, so 6 is correctly the cutoff and 6, 7, 8, 9 all work.

You could find where the two sides are equal: 48 divided by 9 is between 5 and 6, so the box must be more than 5, giving 6, 7, 8, 9.

Standards · min grade 3

  • 3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding which box values keep 9 x box greater than 48.
  • 3.OA.C.7 Fluently multiply and divide within 100 — Computing 6 x 8 and the nine-times facts used in the comparison.
💡 Turn one side into a number, then test the times-table facts -- this is just Grade 3 multiplication you already know!