Sensim Math · Depth 한국어

3-2 · Multiplication

Estimate a product to bound an unknown

3.NBT.A.33.OA.C.7 · take · grade 3

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

▶ Practice — 11 problems

Find every one-digit number that can go in the □\square.

□×95>20×30\square \times 95 > 20 \times 30

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Understand

Find every one-digit number that can replace the square so that (square) times 95 is greater than 20 times 30.

Givens
  • The inequality is (square) x 95 > 20 x 30.
  • The square must be a one-digit number.
Unknowns
  • All one-digit numbers that make the inequality true.
Constraints
  • The square is a single digit (0 through 9).

Plan

#6 Guess and Check · also uses: #8 Analyze the Units

First compute the fixed right side (20 x 30 = 600). Then estimate: each step up in the square adds about 95. Test the boundary digit to find where the product first passes 600, then list every digit at or above it.

Execute

#8 Analyze the Units 3.NBT.A.3
Multiply 20 by 30 to get the number the left side must beat.
20×30=60020 \times 30 = 600
Multiplying two multiples of 10 is 2 x 3 with two zeros tacked on.
#6 Guess and Check 3.OA.C.7
Since 95 is close to 100, (square) x 95 is roughly (square) hundreds. Six hundred would need about 6, so test near there: 6 x 95 = 570 (not over 600) and 7 x 95 = 665 (over 600).
6×95=570<600,7×95=665>6006 \times 95 = 570 < 600,\quad 7 \times 95 = 665 > 600
Estimating 95 as about 100 quickly shows which digit is the turning point.
#6 Guess and Check 3.OA.C.7
From 7 up, the product only grows, so 7, 8, and 9 all satisfy the inequality (8 x 95 = 760, 9 x 95 = 855). Digits 6 and below give 570 or less, which is not over 600.
7×95=665, 8×95=760, 9×95=8557 \times 95 = 665,\ 8 \times 95 = 760,\ 9 \times 95 = 855
Once a digit works, every larger digit works too, since the product keeps increasing.
Answer: 7, 8, and 9

Review

The boundary 6 x 95 = 570 is just under 600 and 7 x 95 = 665 is just over, so 7 being the smallest that works makes sense; all one-digit values are 7, 8, 9.

Divide instead: 600 / 95 is a little over 6.3, so the square must be at least 7. The one-digit numbers 7, 8, 9 follow directly.

Standards · min grade 3

  • 3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Computing 20 x 30 and estimating with multiples of 10.
  • 3.OA.C.7 Fluently multiply and divide within 100 — Testing one-digit multiples of 95 against 600.
💡 Treat 95 as almost 100 and you can guess the boundary -- Grade 3 estimating points right to 7, 8, 9!