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Grade 9 · Polynomial products

Calibration table: Explain a binomial area product

Calibration table investigation: explain a binomial area product. Change the quantities, follow the motion, and explain the result using the displayed mathematical relationship.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Calibration table: Explain a binomial area product. Square x²: 9. Two strips x(a+b): 24. Corner ab: 15. Whole product: 48One rectangle, four area productsx² = 9ax = 15bx = 9ab = 15Sides: (3+5) and (3+3)Sum of four areas = 48
x, a, and b are positive in this area model; signed algebra requires additional interpretation. The picture is fitted with one common scale for both dimensions. All area terms are in square units.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Square x²
9
Two strips x(a+b)
24
Corner ab
15
Whole product
48

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From experiment to screen.

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Understand what you are seeing

The idea behind the motion.

Partitioning a rectangle expresses one product as the sum of four areas. The two cross terms combine into x(a+b), giving the coefficient of x. Reversing that comparison explains factoring a monic quadratic when its positive constants have the required sum and product. This investigation begins with added length a = 5; added width b = 3; shared length x = 3. Treat the named setting as a constructed classroom model, then explain which relationships would still hold if its numbers changed.

A relationship to keep

(x+a)(x+b) = x²+(a+b)x+ab

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Represent the given quantities

    Read each full side as x plus its added length rather than multiplying only the added pieces.

  2. STEP 2

    Follow the changing model

    Playback reveals the square, the two strips, and the corner rectangle in order.

  3. STEP 3

    Check and explain the relationship

    Add the four areas and compare with multiplying the full sides. The agreement illustrates the distributive identity.

Your turn to explain

Make a prediction. Test your reasoning.

Calibration table: Evaluate (x+5)(x+3) at x=3, and identify its four area parts.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

48 = 9 + 15 + 9 + 15. The coefficient of x is 8.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 9 · Algebra

Multiply every pair of binomial terms

Use distribution to multiply two binomials and collect the result.