Mathematics
Grade 10
15 min
Properties of multiplication
Properties of multiplication
What you'll learn
- Identify and name the commutative, associative, and distributive properties of multiplication when given examples with numbers.
- Solve multiplication problems using the associative property to regroup factors and simplify calculations, achieving at least 80% accuracy on a worksheet.
- Explain how the distributive property can be used to break down a multiplication problem into smaller, easier problems, demonstrating understanding through verbal explanations or written solutions to word problems.
- Apply the distributive property to solve multiplication problems with a two-digit number multiplied by a one-digit number with 75% accuracy.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Apply the distributive property to expand the standard form of a circle's equation into its general form.
Identify the use of the multiplicative property of zero when calculating the x- and y-intercepts of a circle.
Recognize the role of multiplicative factors when completing the square to find a circle's center and radius.
Analyze the effect of scalar multiplication on a circle's radius and its corresponding equation.
Use the commutative and associative properties to simplify and rearrange algebraic expressions derived from circle equations.
Prove that two general form equations are equivalent by identifying a common multiplicative constant.
Ever wonder how a simple rule like `a(b+c) = ab + ac` is the key to describing the perfect shape of...
2
Key Concepts & Vocabulary
TermDefinitionExample
Distributive PropertyThe property that states multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products.To expand `(x - 5)^2`, we write it as `(x - 5)(x - 5)`. Using the distributive property, this becomes `x(x - 5) - 5(x - 5) = x^2 - 5x - 5x + 25 = x^2 - 10x + 25`.
Scalar MultiplicationThe multiplication of a geometric object's parameters (like a circle's radius) by a single number (a scalar). In the context of circles, this results in a dilation.If a circle with radius `r = 3` is dilated by a scalar of 2, the new radius is `r' = 2 * 3 = 6`.
Multiplicative Property of ZeroThe property that states the product of any number and zero is zero.To find the y-intercept of a circle, we set `x = 0` in its e...
3
Core Formulas
Standard to General Form Expansion
(x - h)^2 + (y - k)^2 = r^2 \rightarrow x^2 + y^2 - 2hx - 2ky + (h^2 + k^2 - r^2) = 0
This conversion is achieved by applying the distributive property to the squared binomials `(x-h)^2` and `(y-k)^2` and then rearranging the terms. It's the primary application of the distributive property in this chapter.
Scalar Dilation of a Circle
A circle x^2 + y^2 = r^2 dilated by a scalar factor k > 0 becomes x^2 + y^2 = (kr)^2.
When a circle centered at the origin is dilated, the new radius is the original radius multiplied by the scalar factor. Note that the term on the right side of the equation is multiplied by `k^2`.
Multiplicative Equivalence of General Form
Ax^2 + Ay^2 + Bx + Cy + D = 0 \iff k(Ax^2 + Ay^2 + Bx + Cy + D) = 0
For...
4 more steps in this tutorial
Sign up free to access the complete tutorial with worked examples and practice.
Sign Up Free to ContinueSample Practice Questions
Challenging
The equation k(x - 2)^2 + k(y + 1)^2 = 16 is equivalent to the general form equation 3x^2 + 3y^2 - 12x + 6y - 3 = 0. What must be the value of the scalar k?
A.4
B.3
C.1/3
D.16
Challenging
A student attempts to dilate the circle x^2 + y^2 = r^2 by a scalar k, but incorrectly writes (kx)^2 + (ky)^2 = r^2. This equation simplifies to x^2 + y^2 = (r/k)^2. This error effectively applies a dilation by what multiplicative factor?
A.1/k
B.k^2
C.-k
D.1/k^2
Challenging
To prove that A(x-h)^2 + A(y-k)^2 = Ar^2 is equivalent to the simplified general form for any non-zero A, one must first apply the distributive property to expand the binomials. What is the essential subsequent multiplicative step to complete the proof?
A.Multiply the equation by A
B.Multiply the equation by -1
C.Multiply the equation by 1/A
D.Multiply the equation by r^2
Want to practice and check your answers?
Sign up to access all questions with instant feedback, explanations, and progress tracking.
Start Practicing FreeMore from Circles in the coordinate plane
Mathematics for other grades
Frequently asked questions
What grade level is "Properties of multiplication"?
Properties of multiplication is a Grade 10 Mathematics lesson on ExcelOS.
What will I learn in Properties of multiplication?
You'll be able to: Identify and name the commutative, associative, and distributive properties of multiplication when given examples with numbers; Solve multiplication problems using the associative property to regroup factors and simplify….
Is "Properties of multiplication" free to practice?
Yes. You can read the tutorial preview for free, and signing up for a free ExcelOS account unlocks the full tutorial and all practice questions with instant feedback.
How many practice questions are included with Properties of multiplication?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.