Demonstration 1 of 4
Factor pairs as rectangles
How many different rectangles can 36 tiles make, and what happens with 13 tiles or 1?
Each rectangle uses every tile with no gaps. Searching candidates in order, the pairs start to repeat after the square pair, so the list is complete there.
\[36 ÷ 4 = 9\]
A factor pair is two positive whole numbers whose product is the number. Each pair is one rectangle of tiles. A prime has exactly two factors, 1 and itself.
Predict first. How many rectangles will 13 tiles make?
Choose an example
Constructed example: the chapter's 36 square tiles, plus 6, the prime 13, and 1.
Calculated values
- Factor pairs
- 1 x 36, 2 x 18, 3 x 12, 4 x 9, 6 x 6
- Positive factors
- 1, 2, 3, 4, 6, 9, 12, 18, 36
- Number of factors
- 9
- Type
- composite (more than two factors)
Divide by candidates in order and record the partner: 36 / 1 = 36; 36 / 2 = 18; 36 / 3 = 12; 36 / 4 = 9; 36 / 6 = 6. The search stops once the partners would reverse. It has 9 factors, more than two, so 36 is composite.
Use the idea
To list factors, divide by 1, 2, 3 and so on, write each partner, and stop when the partners turn around.
Where the conclusion applies
Positive whole numbers and positive factors only. 1 is neither prime nor composite because it has one factor; 0 is outside the definitions entirely.
Check your understanding: List the factor pairs of 24. Is 24 prime or composite?
Chapter 8 source: section "Factors, multiples, primes, and boundaries". Demonstration C08-D01.
Demonstration 2 of 4
Different trees, same primes
If you start 360 with a different split, do you end with different primes?
The tree keeps splitting composite numbers until only primes remain. The branches change with the first split, but the final inventory never does.
\[360 = 2 × 2 × 2 × 3 × 3 × 5\]
\[360 = 2^3 × 3^2 × 5\]
Each branch splits a number into two factors. A shaded endpoint is prime and cannot split further. The exponent counts how many times a prime repeats.
Predict first. Starting with 360 = 45 x 8, how many 2s will appear at the endpoints?
Choose an example
Constructed example: the chapter's factor trees for 360 that start with 36 x 10 and 45 x 8, plus its 18 x 20 arrangement.
Calculated values
- First split
- 360 = 36 x 10
- Endpoints in tree order
- 2, 2, 3, 3, 2, 5
- Prime inventory
- 2 x 2 x 2 x 3 x 3 x 5
- With exponents
- 2^3 x 3^2 x 5
First split: 36 x 10 = 360. The endpoints are 2, 2, 3, 3, 2, 5. Sorted: 2 x 2 x 2 x 3 x 3 x 5. Check: 2 x 2 x 2 = 8, 3 x 3 = 9, and 8 x 9 x 5 = 72 x 5 = 360. Three 2s, two 3s and one 5, whichever split comes first.
Use the idea
Start a factor tree with whatever split you see first. Then multiply the endpoints back together to check the inventory.
Where the conclusion applies
A positive whole number greater than 1. The order of factors does not matter. Stopping at a composite endpoint such as 4 leaves the factorization unfinished.
Check your understanding: Using 84 = 2 x 42, what is the prime factorization of 84?
Chapter 8 source: section "Opening 360". Demonstration C08-D02.
Demonstration 3 of 4
A negative base or a minus outside
Why is (-3)^2 equal to 9 while -3^2 is -9?
Each bar is the product after one more factor. A negative base flips the sign with every factor, so even counts end positive. A minus outside waits until the power is done.
\[(-3)^2=(-3)(-3)=9\]
\[-(3\times3)=-9\]
\[(-2)^3=(-2)(-2)(-2)=4(-2)=-8\]
The base is what gets repeated; the exponent counts the copies. Parentheses around -3 make -3 the base. Without them, only 3 is the base and the minus is applied last.
Predict first. With base 3 and exponent 3, will (-3)^3 and -3^3 agree?
Choose an example
Constructed example: the chapter's (-3)^2, -3^2 and (-2)^3, with the matching cases filled in.
Calculated values
- Expression
- (-2)^3
- Base
- -2
- Value
- -8
- The other form
- -2^3 = -8
(-2)^3 = (-2) x (-2) x (-2) = -8. The base is the negative number -2, so every factor is negative. -2^3 is also -8: with an odd number of factors the values agree, but for different reasons.
Use the idea
Before calculating a power, circle the complete base. Keep any minus sign outside the circle separate until the end.
Where the conclusion applies
Positive whole numbers 2 and 3, used as negative bases or with a minus outside, and positive whole-number exponents. Powers are done before the outside minus, as in the chapter's order of operations.
Check your understanding: What are (-2)^4 and -2^4?
Chapter 8 source: section "Signs and exponents have different jobs". Demonstration C08-D03.
Demonstration 4 of 4
A square root reverses a square
Which nonnegative number squares to 144, and how do its prime factors show it?
A perfect square splits its prime factors into matching pairs. Taking one from each pair gives the side of the square. An unpaired prime leaves tiles over.
\[√144 = 12\]
\[144 = 2^4 × 3^2 = (2^2 × 3)^2 = 12^2\]
The root symbol names the principal square root: the nonnegative number whose square is the given number. Paired prime factors split evenly into two equal halves.
Predict first. Will 50 tiles make a complete square?
Choose an example
Constructed example: the chapter's 144 = 12 x 12, plus 36, 81 and the non-square 50.
Calculated values
- Prime factors
- 2^4 x 3^2
- Unpaired primes
- none
- Principal square root
- 12
- Squaring check
- 12 x 12 = 144
144 = 2^4 x 3^2. Every prime factor is paired, and one from each pair gives 2 x 2 x 3 = 12. Check: 12 x 12 = 144. The number -12 also squares to 144, but the root symbol names only the nonnegative one.
Use the idea
To check a proposed square root, square it and confirm the root is not negative.
Where the conclusion applies
Whole numbers and perfect squares for whole-number roots. When solving x^2 = 144 both 12 and -12 work, but the root symbol by itself names only 12.
Check your understanding: What is the principal square root of 196, and what are its paired prime factors?
Chapter 8 source: section "A square root reverses a square". Demonstration C08-D04.