Mathematics
Grade 10
15 min
Multiply 2-digit numbers by 3-digit numbers
Multiply 2-digit numbers by 3-digit numbers
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Introduction & Learning Objectives
Learning Objectives
Accurately multiply any 2-digit number by any 3-digit number to support coordinate geometry calculations.
Apply the standard multiplication algorithm to find values like the square of a radius (r²) in the circle equation.
Verify if a point (x, y) lies on a circle by performing precise multiplication of values derived from its coordinates.
Use estimation to check the reasonableness of products calculated in circle-related problems.
Calculate the area of a circle (A = πr²) when r² is determined by a 2-digit by 3-digit product.
Deconstruct complex circle proofs and problems to identify where multi-digit multiplication is a required computational step for accuracy.
Ever had a perfect geometric proof fall apart because of a tiny multiplication error? 🤯 Let�...
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Key Concepts & Vocabulary
TermDefinitionExample
Standard Form of a CircleThe equation representing all points (x, y) on a circle with a center at (h, k) and a radius of length r.The equation (x - 2)² + (y - 3)² = 25 represents a circle centered at (2, 3) with a radius of 5.
Radius Squared (r²)The value of the radius multiplied by itself. This value appears on the right side of the standard circle equation and is crucial for many calculations.In the equation (x - 2)² + (y - 3)² = 25, the value of r² is 25.
ProductThe result obtained after multiplying two or more numbers.The product of 112 and 45 is 5040.
Partial ProductsThe intermediate results obtained in the standard multiplication algorithm when multiplying one number by each digit of the other number.In 235 × 24, the partial products are 235 × 4 (which is 940)...
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Core Formulas
Equation of a Circle
(x - h)^2 + (y - k)^2 = r^2
This is the fundamental formula for a circle. Verifying a point or finding the radius often requires squaring numbers, which is a form of multiplication. If coordinates are large, this can lead to 2-digit by 3-digit multiplication.
Area of a Circle
A = \pi r^2
To find the area, you must multiply pi (often approximated as 3.14) by the square of the radius. If r² is a 3-digit number, this becomes a 3-digit by 3-digit multiplication, but if you are given r² as a product of two numbers (e.g., from a geometric property), you may need to perform 2x3 multiplication first.
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Challenging
In a proof to show a point lies on a circle, a critical calculation is to find the value of (y-k)², which is determined to be the product of 115 and 75. A separate part of the proof states that r² is 8725. Does the calculation of (y-k)² support the proof that the point is on the circle?
A.Yes, because 115 x 75 equals 8725.
B.No, because 115 x 75 is greater than 8725.
C.No, because 115 x 75 is less than 8725.
D.It's impossible to tell without knowing (x-h)².
Challenging
The equation of a circle is (x-1)² + (y-2)² = 15750. This r² value was calculated from the product of a 2-digit integer and 126. What was the 2-digit integer?
A.115
B.120
C.135
D.125
Challenging
An engineer claims a new manufacturing process results in a circular gear with an area of exactly 10140π square mm. The design specifications state that r² must be the product of 180 and 54. Is the engineer's claim about the area correct?
A.Yes, the calculation is correct.
B.No, the correct area is 9620π.
C.No, the correct area is 10720π.
D.No, the correct area is 9720π.
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