VertexMath · MTH1W · Free worksheet

Number Sense Review

Try each problem first. You don’t need to finish everything.

Core path ~55 min Full sheet ~1 h 45 min

Recommended: K1–K5, then one A and one T problem

K = Knowledge & Understanding · A = Application · T = Thinking · Boss = Challenge

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Quick reference

Number sets: natural $\{1,2,3,\dots\}\subset$ whole $\{0,1,2,\dots\}\subset$ integers $\subset$ rationals. Simplify first: $\frac{12}{4}=3$ is a natural number.
BEDMAS: brackets, exponents, then $\times$ and $\div$ left to right, then $+$ and $-$ left to right. $-3^2=-9$ , but $(-3)^2=9$ .
Exponent laws (same base): $a^m\cdot a^n=a^{m+n}$ , $\dfrac{a^m}{a^n}=a^{m-n}$ , $(a^m)^n=a^{mn}$ , $a^0=1$ , $a^{-n}=\dfrac{1}{a^n}$ .
Percent change $=\dfrac{\text{new}-\text{original}}{\text{original}}\times100\%$ . Always divide by the original value.
Successive percents multiply: $30\%$ off then $20\%$ off is $\times\,0.7\times0.8$ , not $50\%$ off.

Show your work on separate paper. Show your work in the dot-grid space.

Core Skills (K)

Number Sets · Integers · Fractions · Exponent Laws · Scientific Notation

K1 Sort and Order

Use these six numbers: $-\frac{14}{2},\quad 0.\overline{4},\quad -0.6,\quad 0,\quad \frac{9}{3},\quad -\frac{2}{3}$

a)
List the numbers that are integers.
b)
List the numbers that are whole numbers, and the numbers that are natural numbers.
c)
Order all six numbers from least to greatest.

Stuck? Peek at a hint

Simplify each fraction and write each number as a decimal before you classify or order it.

K2 Integers & BEDMAS

Evaluate.

a)
$-15-(-9)+4$
b)
$-|3-11|+(-2)(-3)(-5)$
c)
$-3^2-4(2-6)\div8$

Stuck? Peek at a hint

Rewrite subtracting a negative as adding a positive. In $-3^2$ , the exponent applies to $3$ only.

K3 Fractions & Mixed Numbers

Evaluate. Give each answer in lowest terms, as a mixed number when it is greater than 1.

a)
$3\frac{1}{6}-1\frac{3}{4}$
b)
$2\frac{1}{4}\times1\frac{1}{3}$
c)
$\dfrac{7}{8}-\dfrac{1}{2}\div\dfrac{4}{3}$

Stuck? Peek at a hint

Convert each mixed number to an improper fraction first. In (c), divide before you subtract.

K4 Exponent Laws

a)
Evaluate $-2^4+(-3)^2+7^0$ .
b)
Simplify $\dfrac{m^{10}\cdot m^{2}}{m^{4}}$ .
c)
Simplify $\dfrac{(2y^{4})^{3}}{4y^{10}}$ .

Stuck? Peek at a hint

Same base: add exponents when multiplying, subtract when dividing. A power of a product raises every factor, numbers included.

K5 Scientific Notation

a)
Write $0.000\,608$ and $5\,040\,000$ in scientific notation.

The table gives approximate masses. Order the four objects from lightest to heaviest.

$\text{Object}$	Grain of rice	Grain of salt	Mosquito	Sesame seed
$\text{Mass (kg)}$	$2.5\times10^{-5}$	$6.0\times10^{-8}$	$2.5\times10^{-6}$	$3.5\times10^{-6}$

c)
Evaluate $\dfrac{(6\times10^{5})(4\times10^{-2})}{8\times10^{-3}}$ . Give the answer in scientific notation.

Stuck? Peek at a hint

In (b), look at the powers of 10 first. In (c), work with the first factors and the powers of 10 separately.

Real World (A)

Unit Prices · Sales Tax & Discounts · Very Small Masses

A1 Best Buy on Oats

A store in Thunder Bay sells oats in three bags, S, M and L. The graph shows the mass $m$ of each bag, in kilograms, and its cost $C$ , in dollars.

a)
Find the unit price of each bag, in dollars per kilogram.
b)
Which bag is the best buy? How much do you save on 15 kg of oats by buying it instead of bag L?
c)
An opened bag stays fresh for 4 weeks. Sam eats 0.5 kg of oats a week and throws out stale oats. Which bag gives Sam the lowest cost per week?

Stuck? Peek at a hint

A unit price is the cost divided by the mass: read both from each point.

A2 Boxing Day Jacket

A winter jacket is priced at $185. On Boxing Day it is $30\%$ off, and an extra $20\%$ is taken off the sale price at the checkout. The $13\%$ HST is added after both discounts.

a)
Find the price before tax. What single percent discount does this equal?
b)
Find the total with HST, to the nearest cent.
c)
You pay in cash. Canada no longer has a penny, so cash totals are rounded to the nearest 5 cents. How much do you pay?

Stuck? Peek at a hint

After $30\%$ off you pay $70\%$ of the price, so multiply by $0.70$ .

A3 Bacteria in a Dish

A lab dish holds a colony of $4.0\times10^{6}$ bacteria. Each bacterium has a mass of about $2.5\times10^{-15}$ kg. Give each answer in scientific notation.

a)
Find the total mass of the colony, in kilograms.
b)
How many of these bacteria have a total mass of 1 g? ( $1\text{ g}=10^{-3}$ kg)
c)
A model says that, if the colony kept doubling every 20 minutes for 2 days, it would hold $8.9\times10^{49}$ bacteria. Find their mass and how many times Earth's mass ( $6.0\times10^{24}$ kg) it is. What does this tell you about the model?

Stuck? Peek at a hint

Multiply the first factors, add the exponents, then rewrite so the first factor is between 1 and 10.

Think It Through (T)

Error Analysis · Counterexamples

T1 Spot the Flaw

Gas went from $1.25/L to $1.50/L. A student claims:
"That is a $20\%$ increase. If the price drops back to $1.25/L, that will be a $20\%$ decrease."

a)
Check each of the two claims. Give the correct percent for any claim that is wrong.
b)
Both changes are $0.25/L. Explain why their percents are different.
c)
A price rises by $60\%$ . By what percent must the new price fall to return to the original price?

Stuck? Peek at a hint

Divide each change by the price before that change.

T2 Always True?

Decide whether each claim is always true. If it is, show why. If it is not, give a counterexample.

a)
For every positive whole number $n$ , $3^n+3^n=6^n$ .
b)
For every number $x$ , $(x^2)^3=(x^3)^2$ .
c)
A power with a negative exponent is always a negative number.

Stuck? Peek at a hint

Test each claim with small numbers, and try more than one value before you decide.

Boss Level

Optional challenge

Mixing Juice

Bottle A holds juice and water in the ratio $1:3$ . Bottle B holds juice and water in the ratio $3:5$ .

a)
1 L from A is mixed with 1 L from B. Find the ratio of juice to water in the mix.
b)
How many litres from B must be mixed with 2 L from A to get juice and water in the ratio $1:2$ ?
c)
Can any mix of A and B have juice and water in the ratio $1:1$ ? Explain.

Stuck? Peek at a hint

Find what fraction of each bottle is juice: a $1:3$ ratio splits the bottle into 4 equal parts.

Answer key

Final answers only. Try first, then check.

Core Skills

a) $-\frac{14}{2}=-7$ , $0$ , $\frac{9}{3}=3$
b) Whole: $0$ and $\frac{9}{3}$ . Natural: $\frac{9}{3}$ only.
c) $-\frac{14}{2},\ -\frac{2}{3},\ -0.6,\ 0,\ 0.\overline{4},\ \frac{9}{3}$

a) $-2$
b) $-38$
c) $-7$

a) $1\frac{5}{12}$
b) $3$
c) $\frac{1}{2}$

a) $-6$
b) $m^{8}$
c) $2y^{2}$

a) $6.08\times10^{-4}$ and $5.04\times10^{6}$
b) Grain of salt, mosquito, sesame seed, grain of rice
c) $3\times10^{6}$

Real World

a) S: $6.00/kg; M: $5.00/kg; L: $5.40/kg
b) Bag M; five M bags cost $75 and three L bags cost $81, so you save $6.
c) Bag S: $3.00 per week. An M bag goes stale with 1 kg left, so it costs $15 per 4 weeks, or $3.75 per week (L: $6.75 per week).

a) $103.60; a $44\%$ discount (not $50\%$ )
b) $117.07
c) $117.05

a) $1.0\times10^{-8}$ kg
b) $4.0\times10^{11}$ bacteria
c) About $2.2\times10^{35}$ kg, about $3.7\times10^{10}$ times Earth's mass. Impossible: food and space run out long before, so the colony cannot keep doubling.

Think It Through

a) The $20\%$ increase is correct. The drop is a $16\frac{2}{3}\%\approx16.7\%$ decrease, not $20\%$ .
b) Each percent is measured from the price before the change: $0.25 out of $1.25 for the increase, $0.25 out of $1.50 for the decrease.
c) $37.5\%$

a) Not always: it works for $n=1$ , but $n=2$ gives $3^2+3^2=18$ and $6^2=36$ .
b) Always true: both equal $x^{6}$ .
c) False: e.g. $2^{-1}=\frac{1}{2}>0$ .

Boss Level

Boss

a) $5:11$
b) $4$ L
c) No: A is $\frac{1}{4}$ juice and B is $\frac{3}{8}$ juice, so any mix is between $\frac{1}{4}$ and $\frac{3}{8}$ juice, always less than $\frac{1}{2}$ .

Free cheat sheet

Number Sense Cheat Sheet

The whole unit on one page: every rule, a worked example for each idea, and the traps that cost marks.

Unit Summary

One printable page with every rule.

↓ Unit Summary PDF

Build your own

The same page with blanks: fill it in from memory, then check.

↓ Build-your-own PDF

Phone card

The essentials on one screen. Save it to your photos.

↓ Phone card (image)

Most testedAn $80 hoodie is 25% off, and 13% HST is added to the sale price. What do you pay? (card 7)

1Number sets

UseSimplify before you classify: $18/6=3$ is natural, so it is also whole, an integer and rational.

Trap $0$ is whole, an integer and rational, but not natural: natural numbers start at $1$ .

2Order and $|x|$

$|-7|=|7|=7$

Use $|x|$ is the distance from $0$ , never negative. To order, write each number as a decimal: further left is smaller.

Trap $-0.7<-0.6$ : $-0.7$ is further left, even though $0.7>0.6$ .

3Integer signs

$a-(-b)=a+b\qquad a+(-b)=a-b$

Multiply or divide non-zero integers:

even number of negatives	positive
odd number of negatives	negative

Example $(-2)(-3)(-4)=-24$ : three negatives.

Trap $-12-(-5)=-12+5=-7$ , not $-17$ .

4BEDMAS

UseBrackets (and $|\;|$ bars), Exponents, then $\times$ and $\div$ left to right, then $+$ and $-$ left to right.

Example $-2[-8-4\lvert-3\rvert^2]=-2[-8-36]=88$

Trap $-4^2=-16$ : the exponent acts on the $4$ only. $(-4)^2=16$ .

5Fractions

$\dfrac{a}{b}\times\dfrac{c}{d}=\dfrac{ac}{bd}$

$\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\times\dfrac{d}{c}$

UseAdd or subtract: rewrite over the lowest common denominator (LCD). Multiply or divide: change mixed numbers to improper fractions first ( $2$ and $1/3$ is $7/3$ ).

Trap $1/2+1/3=3/6+2/6=5/6$ , not $2/5$ : never add the denominators.

Trap $3/4-1/2\times1/3=3/4-1/6=7/12$ , not $1/12$ : multiply first.

6Ratios and rates

$\dfrac{a}{b}=\dfrac{c}{d}\;\Rightarrow\;ad=bc$

$\text{unit price}=\text{price}\div\text{quantity}$

UseScale a ratio by multiplying both terms by the same number. Compare unit prices, not package sizes.

Example$4.49 for 500 g is $8.98/kg, so $7.99 for 1 kg is cheaper.

Trap $2:3$ scaled to $10:?$ is $10:15$ (multiply by 5), not $10:11$ (add 8).

7Percents

$\text{percent change}=\dfrac{\text{new}-\text{original}}{\text{original}}\times100$

$\text{up } r\text{: }\times(1+r)\qquad\text{down } r\text{: }\times(1-r)$

Use $r$ is the percent as a decimal: 13% is $0.13$ , 0.4% is $0.004$ .

Example$80, 25% off, plus 13% HST: $80\times0.75\times1.13=67.80$ , so $67.80.

Trap25% off, then 10% off: $\times0.75\times0.9=\times0.675$ , so 32.5% off, not 35%.

8Exponent laws

$a^m\times a^n=a^{m+n}\qquad(a^m)^n=a^{mn}$

$\dfrac{a^m}{a^n}=a^{m-n}\qquad(ab)^n=a^nb^n$

$a^0=1\qquad a^{-n}=\dfrac{1}{a^n}$

UseAdd or subtract exponents only with the same base. No law works on sums: $3^2+3^2=18$ .

Trap $x^{10}\div x^2=x^8$ , not $x^5$ : subtract the exponents.

Trap $5^{-2}=1/25$ , a positive number, not $-25$ or $-1/25$ .

9Scientific notation

$a\times10^n,\quad1\le a<10$

Use $n>0$ for numbers $\ge10$ ; $n<0$ for numbers between $0$ and $1$ . Multiply the $a$ 's and add the exponents, then fix $a$ .

Example $(4\times10^5)(3\times10^{-2})=12\times10^3=1.2\times10^4$

TrapCompare the powers first: $1.1\times10^{-2}>9.9\times10^{-8}$ .

Trap $10^5\div10^{-3}=10^{5-(-3)}=10^8$ , not $10^2$ .

Which percent question?

Percent of an amount decimal × amount

Part as a percent part ÷ whole × 100

Price goes up or down change ÷ original

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Core Skills (K)

K1 Sort and Order

K2 Integers & BEDMAS

K3 Fractions & Mixed Numbers

K4 Exponent Laws

K5 Scientific Notation

Real World (A)

A1 Best Buy on Oats

A2 Boxing Day Jacket

A3 Bacteria in a Dish

Think It Through (T)

T1 Spot the Flaw

T2 Always True?

Boss Level

Mixing Juice

Answer key

Core Skills

Real World

Think It Through

Boss Level

Number Sense Cheat Sheet

Unit Summary

Build your own

Phone card

1Number sets

2Order and ∣x∣|x|∣x∣

3Integer signs

4BEDMAS

5Fractions

6Ratios and rates

7Percents

8Exponent laws

9Scientific notation

Which percent question?

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What is VertexMath?

2Order and $|x|$