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B. Number · Number Sense: Scientific Notation, Real Numbers, Square Roots and Percents

Number Sense: Scientific Notation, Real Numbers, Square Roots and Percents: Unit Review

A1 B1.1 B1.2 B1.3 B1.4About 40 minutes
Learning goals
  • I can write, compare and use very large and very small numbers in scientific notation.
  • I can classify, compare and order rational and irrational numbers.
  • I can estimate and calculate square roots in real situations.
  • I can use fractions, decimals and percents, including percents over 100% and under 1%, to solve problems.

Words to know

scientific notation
A number from 1 up to (not including) 10, times a power of 10: 4.5 × 10⁶, 3 × 10⁻⁴.
rational number
A number that can be written as a fraction of integers. Its decimal terminates or repeats.
irrational number
A number whose decimal never ends and never repeats, like √2 or π.
real numbers
All rational and irrational numbers together.
square root
A number that, multiplied by itself, gives the original number: √0.49 = 0.7.
percent
A number out of 100. 225% = 2.25 and 0.5% = 0.005.

Let's review!

In this unit you wrote very large and very small numbers in scientific notation, compared them and saw how they are used in everyday life. You met the real number system and irrational numbers like √2 and π, and compared and ordered real numbers. You estimated square roots and used them for decimals, fractions, areas and circles. Finally, you moved flexibly between fractions, decimals and percents, including percents over 100% and under 1%, to solve problems.

This review has a few questions from every lesson. Try each one, then press Show answer to check. If a part feels tricky, go back to that lesson for more practice.

Number 10 or more58 000 000point hops 7 places LEFT= 5.8 × 10⁷positive exponentNumber less than 10.000 09point hops 5 places RIGHT= 9 × 10⁻⁵negative exponent
The one rule for scientific notation, both ways. Big numbers: the decimal point hops left and the exponent is positive. Numbers less than 1: it hops right and the exponent is negative.

Scientific Notation for Very Large Numbers

1

Write 58 000 000 in scientific notation.

2

Write 6.04 × 10⁵ in standard form.

3

Fix 36 × 10⁴ so it is in scientific notation.

Very Small Numbers and Negative Exponents

1

Write 0.000 09 in scientific notation.

2

Write 4.2 × 10⁻³ in standard form.

3

What is 10⁻⁴ as a decimal?

Comparing and Using Very Large and Very Small Numbers

1

Which is greater: 7 × 10⁻⁴ or 2 × 10⁻³?

2

Order from least to greatest: 3.1 × 10⁶, 9 × 10⁵, 3.01 × 10⁶

3

How many times as large is 6 × 10⁸ as 3 × 10⁴?

The Real Number System

1

Classify −√25.

2

Classify √12.

3

Show that 2.75 is rational.

Irrational Numbers: Decimals That Never Repeat

1

Rational or irrational: 0.363 636…?

2

Rational or irrational: √40?

3

Is 22/7 equal to π?

Comparing and Ordering Real Numbers

1

Order from least to greatest: √8, 2.8, 2 5/6

2

Which is greater: −√3 or −1.7?

3

Find a rational number between √5 and √6.

−2−101234−√3 ≈ −1.734/11 = 0.36…√2 ≈ 1.412.75π ≈ 3.14√12 ≈ 3.46rationalirrational
Real numbers from this review on one number line. Rational and irrational numbers sit side by side; together they fill the whole line.

Estimating Square Roots

1

Between which two whole numbers is √70?

2

Estimate √40 to one decimal place.

3

A square has an area of 90 m². About how long is each side?

Square Roots of Decimals and Fractions, and Square Roots in Real Life

1

Find √0.25.

2

Find √(36/49).

3

A circle has an area of 78.5 cm². Using π ≈ 3.14, find its radius.

Moving Flexibly Between Fractions, Decimals and Percents

1

Write 3 1/5 as a decimal and a percent.

2

Write 0.2% as a decimal and a fraction.

3

Write 1/3 as a percent.

Solving Problems with Percents Over 100% and Under 1%

1

Find 175% of 40.

2

Find 0.3% of 2 000.

3

A game costs $60 before tax. What is the total with 13% HST?

Choosing the Best Form: Problems with Fractions, Decimals and Percents

1

Which is cheaper on a $90 item: 1/3 off or 30% off?

2

Of 200 students, 3/5 play an instrument and 25% of those play piano. How many play piano?

3

Is 20% off, then 10% off the sale price the same as 30% off? Explain.

Just for fun

Riddle: I never end, I never repeat, and I show up every time you go around a circle. Who am I?

π! (Not to be confused with pie, which also ends up going around... your stomach.)

How to use this review

  • Work through one section at a time and write your steps, not just the answers.
  • Put a check mark beside each question you got right on your own. Put a star beside any you needed help with.
  • If you starred two or more questions in a section, go back to that lesson and redo its "More practice" questions.
  • Finish with the quiz. For every quiz mistake, read the explanation and find the lesson it came from.
  • Mistakes are useful: they show exactly what to practise next.
Check your understanding

Quiz

1.

What is 4 500 000 in scientific notation?

2.

What is 0.000 3 in scientific notation?

3.

Which number is greatest?

4.

Which number is irrational?

5.

Which statement is true?

6.

Which list is in order from least to greatest?

7.

Which is the best estimate of √45?

8.

What is √0.36?

9.

What is 0.5% of 600?

10.

Which is equal to 2 1/4?

For parents and teachers

Use mistakes to choose what to revisit. Each review section matches one lesson.

  • Counts zeros instead of decimal places, or writes coefficients like 36 × 10⁴: revisit Scientific Notation for Very Large Numbers.
  • Thinks 10⁻⁴ is negative, or moves the decimal point the wrong way: revisit Very Small Numbers and Negative Exponents. Redo the paper-thickness activity.
  • Compares coefficients before exponents, or mixes up metric prefixes: revisit Comparing and Using Very Large and Very Small Numbers.
  • Forgets that a number belongs to several sets, or thinks every root is irrational: revisit The Real Number System. Redo the sorting mat.
  • Thinks repeating decimals are irrational, or that π = 3.14: revisit Irrational Numbers: Decimals That Never Repeat.
  • Orders negative numbers backwards, or doesn't use enough decimal places: revisit Comparing and Ordering Real Numbers.
  • Halves instead of taking a square root, or checks guesses by doubling: revisit Estimating Square Roots.
  • Writes √0.09 = 0.03, or takes the root before dividing by π: revisit Square Roots of Decimals and Fractions, and Square Roots in Real Life.
  • Writes 0.5% as 0.5, or 2.4 as 24%: revisit Moving Flexibly Between Fractions, Decimals and Percents.
  • Doesn't believe percents can exceed 100%, or uses the wrong whole: revisit Solving Problems with Percents Over 100% and Under 1%.
  • Adds percents from different wholes, or uses the original whole in a second step: revisit Choosing the Best Form: Problems with Fractions, Decimals and Percents.

A few mistakes are normal. Praise the strategy your child explains and the habit of checking, not only the right answer.