Number lines, factors and powers of ten
Compare number representations, use divisibility, and explain what calculations with irrational numbers can establish.
What you will tackle
- Locate and compare signed rational numbers
- Use decimal place value and prime factorisation
- Find GCF, LCM and parity
- Use scientific notation and irrational-number properties
Order and distance
Each decimal position has ten times the value of the position to its right. Compare fractions using a positive common denominator or cross-products. On a number line, greater values lie farther right; distance between a and b is |a − b|, even when both are negative.
Factors, multiples and representations
Prime factorisation expresses an integer as a product of primes. A greatest common factor uses the minimum shared prime powers; a least common multiple uses the maximum powers. Even integers have factor 2; a sum of two odd integers is even. Scientific notation a × 10ⁿ uses 1 ≤ |a| < 10 for nonzero values.
Rational and irrational
A rational number is a ratio of integers with a nonzero denominator. For a nonnegative integer n, √n is rational exactly when n is a perfect square. π is irrational. Adding a rational number to an irrational number is irrational; multiplying an irrational by a nonzero rational is irrational. Two irrational numbers can have a rational sum or product, so their labels alone do not decide the result.
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
Place −3/2 and 1/4 on the displayed line and find their distance. Compare 5/6 with 7/9. In 0.047, what is the value of the digit 4? Find the GCF and LCM of 18 and 24.
Show a starting hint
Keep fraction denominators positive before comparing.
Compare with the explained answer
- The points are −1.5 and 0.25, separated by 1.75; 5/6 = 15/18 exceeds 7/9 = 14/18.
- The 4 represents four hundredths, or 0.04.
- 18 = 2·3² and 24 = 2³·3; minimum powers give GCF 6 and maximum powers give LCM 72.
Answer / response criteria: Distance 7/4; 5/6 > 7/9; 0.04; GCF 6, LCM 72.
Independent practice
Independent 1
Order −2/3, −0.7 and 1/5. Write 0.00072 in scientific notation and calculate (3 × 10⁵)(2 × 10⁻³). Classify √49, √50 and π as rational or irrational.
Compare with the explained answer
- −0.7 < −2/3 < 1/5 because −2/3 is approximately −0.667.
- 0.00072 = 7.2 × 10⁻⁴; multiply coefficients and add powers to obtain 6 × 10² = 600.
- √49 = 7 is rational; 50 is not a perfect square, so √50 is irrational; π is irrational.
Answer / response criteria: Order as stated; 7.2 × 10⁻⁴; 600; rational, irrational, irrational.
Application practice
Application 1
Two lights flash every 18 and 24 seconds and flash together now. When do they next coincide? Explain whether the sum of two odd whole-number ticket counts can be odd. Give irrational numbers with a rational product and explain whether 3 + √2 is rational.
Compare with the explained answer
- Coincidence requires a common multiple, with least positive value 72 seconds.
- Writing the counts as 2a + 1 and 2b + 1 gives 2(a + b + 1), which is even.
- √2 × √2 = 2 is a rational product; 3 + √2 is irrational, since subtracting rational 3 from an assumed rational sum would make √2 rational.
Answer / response criteria: 72 seconds; the sum is even; √2 and √2 give product 2; 3 + √2 is irrational.
What can you explain now?
Two lights flash every 18 and 24 seconds and flash together now. When do they next coincide? Explain whether the sum of two odd whole-number ticket counts can be odd. Give irrational numbers with a rational product and explain whether 3 + √2 is rational.
Compare with the explanation
72 seconds; the sum is even; √2 and √2 give product 2; 3 + √2 is irrational.
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
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Source and review status
Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.
Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.
Official source specification / guide