Word Expressions

Calculate each of the following expressions.

Six hundred and forty eight more than three thousand and one. Solution Video

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Align the place columns correctly...

Align the place columns correctly... $\begin{align} 64\!&8 \\ + \ 3,00\!&1 \\ \end{align}$ = 3649

Thirty thousand three hundred fifty two less than fifty thousand five hundred. Solution

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Align the place columns correctly...

Align the place columns correctly... $\begin{align} 50,50\!&0 \\ - \ 30,35\!&2 \\ \end{align}$ = 20,148

Vertical Place Value Columns with Addition and Subtraction

Practice your addition and subtraction any way you like using mental math, or without the use of a calculator. Make sure to show your work.

Subtract: 700 - 7 Solution Video

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$\begin{align} 70\!&0 \\ - \ \!&7 \\ \end{align}$ ... = 693

Subtract: 560 - 80 Solution Video

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Subract 60 from 560 and 80... $\begin{align} & 560 - 80 \\ \\ & = (560 - 60) - (80 - 60) \\ \\ & = 500 - 20 \\ \\ \end{align}$ Borrow from the 100's column, 5-- becomes 4--... $\begin{align} & 500 - 20 \\ \\ & = 480 \\ \\ \end{align}$

Add: 4,000 + 500 + 60 + 8 Solution Video

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$\begin{align} 4,00\!&0 \\ 50\!&0 \\ 6\!&0 \\ + \ \!&8 \\ \end{align}$ ... = 4,568

Subtract: 4,568 - 3,000 - 500 - 70 - 3 Solution Video

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Step-by-step (for each place value)... $\begin{align} 4,56\!&8 \\ - \ 3,00\!&0 \\ \end{align}$ = 1,568 $\begin{align} 1,56\!&8 \\ - \ 50\!&0 \\ \end{align}$ = 1,068 $\begin{align} 1,06\!&8 \\ - \ 7\!&0 \\ \end{align}$ = 998 $\begin{align} 99\!&8 \\ - \ \!&3 \\ \end{align}$ = 995

(Yes you could just do 4,568 - 3,573...)

Ordering Whole Numbers with Place Values

Indicate whether the following statement is true or false: Solution Video 589,301 > 589,199

1. True
2. False

True. Start comparing the place values (ones, tens, hundreds...) from the LEFT-HAND side... (in the hundreds column) 3 is bigger than 9...

Ordering Decimals

Order the following decimals from greatest to least going from top to bottom. Solution

0.999

0.90

0.009

1.005

0.88

1.001

0.0095

Compare the place values: ones, tenths, hundredths, thousandths...

1.005, 1.001, 0.999, 0.90, 0.88, 0.0095, 0.009

Can also be written as

1.005 > 1.001 > 0.999 > 0.90 > 0.88 > 0.0095 > 0.009

(comparing the last two, to see more easily, you can change to: 0.0095 > 0.0090)

Representing Numbers up to 1,000,000

Which of the following is the correct word number for the following? Solution 220,012

1. Twenty two thousand and twelve
2. Two hundred thousand twenty and twelve
3. Twenty two hundred thousand and twelve
4. Two hundred twenty thousand and twelve

Two hundred twenty thousand and twelve

Which of the following is the correct phrase for the following number? Solution 502,401

1. Five hundred and two thousand four hundred and one
2. Five hundred thousand two thousand four hundred and one
3. Five hundred and two thousand four hundred and one
4. Five hundred thousand two four hundred and one

Five hundred and two thousand four hundred and one

Which of the following is the correct number? Solution Eighty-two thousand four hundred and forty-four

1. 82,444
2. 820, 444
3. 802,444
4. 8,244

82,444

Decimals and Place Values

Which of the following is two hundredths less than 0.234 Solution Video

1. 0.034
2. 0.214
3. 0.232
4. None of the above

Comparing Decimals

In a 300 meter race, Julie finished in 45.42 seconds. Christine finished in 44.51 seconds, Alice finished in 42.08 seconds, and Sarah finished in 42.24 seconds.

Who came in last? Solution

1. Julie
2. Christine
3. Alice
4. Sarah

Julie had the longest time, so she came in last.

How much time was between the first and last place? Solution

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seconds

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Alice finished first: 42.08 seconds
Julie finished last: 45.42 seconds

= 45.42 - 42.08
= 3.34

Comparing Decimals

Compare the following by filling in the blanks.

0.5 ____ 0.409 Solution

1. >
2. <
3. =
4. None of the above

0.5 > 0.409

62.37 ____ 62.370 Solution

1. >
2. <
3. =
4. None of the above

62.37 = 62.370

21.89 ____ 21.95 Solution

1. >
2. <
3. =
4. None of the above

21.89 < 21.95

Converting Decimals and Fractions

Convert 0.75 to a fraction (remember to reduce fully). Solution

1. $\dfrac{7}{5}$
2. $\dfrac{3}{4}$
3. $\dfrac{75}{10}$
4. $\dfrac{3}{2}$

$\dfrac{3}{4}$

Converting Decimals and Fractions

Convert $\dfrac{250}{100}$ to a decimal. Solution

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2.5

Fractions

David has $100. He spends $\dfrac{3}{4}$ of it on a soccer ball. How much does he have left? Solution

$

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25

Fraction and Percent Word Problems

Jon wins $100. He gives 50% to his mother, and from what is left over after giving money to his mother, he gives $\dfrac{2}{5}$ to charity. How much does Jon give to charity? Solution

$

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20 [steps not shown here]

Comparing Fractions

Compare the following by filling in the blank (without using a calculator).

$\quad \dfrac{3}{6} \ \text{____} \ \dfrac{2}{4}$ Solution

1. >
2. <
3. =
4. None of the above

Make a common denominator in order to compare. Use the lowest common multiple (LCM) of 6 & 4: 12... $\begin{align} \dfrac{3}{6} & \text{____} \dfrac{2}{4} \\ \\ \dfrac{3 × 2}{6 × 2} & \text{____} \dfrac{2 × 3}{4 × 3} \\ \\ \dfrac{6}{12} & = \dfrac{6}{12} \\ \\ \end{align}$

$\quad \dfrac{16}{25} \ \text{____} \ \dfrac{7}{10}$ Solution Video

1. >
2. <
3. =
4. None of the above

$\dfrac{16}{25} \ < \ \dfrac{7}{10}$

$\quad \dfrac{2}{3} \ \text{____} \ \dfrac{4}{5}$ Solution Video

1. >
2. <
3. =
4. None of the above

Make a common denominator (with the lowest common multiple, LCM)
Multiples of 3: 3, 6, 9, 12, 15, ...
Multiples of 5: 5, 10, 15, ... $\begin{align} & \dfrac{2}{3} \\ \\ & = \dfrac{2 \color{DeepSkyBlue}{× 5}}{3 \color{DeepSkyBlue}{× 5}} \\ \\ & = \dfrac{10}{15} \\ \\ \end{align}$ $\begin{align} & \dfrac{4}{5} \\ \\ & = \dfrac{4 \color{Orange}{× 3}}{5 \color{Orange}{× 3}} \\ \\ & = \dfrac{12}{15} \\ \\ \end{align}$
Therefore $\dfrac{2}{3} < \dfrac{4}{5}$

Prime and Composite Numbers

Which of the following is not a prime number? Solution

1. 23
2. 22
3. 11
4. 41

Prime numbers can only be divided by one and themselves. Composite numbers can be factored into several prime numbers.

Adding, Subtracting, Multiplying, and Dividing Fractions

Simplify the following fraction expressions without the use of a calculator. Reduce your answer fully.

$\dfrac{1}{4} + \dfrac{3}{5}$ Solution

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Make the denominator with the lowest common multiple (LCM)...
4 → 8 → 12 → 16 → $\bbox[PowderBlue, 3pt]{20}$
5 → 10 → 15 → $\bbox[PowderBlue, 3pt]{20}$ $\begin{align} & = \dfrac{1}{4} + \dfrac{3}{5} \\ \\ & = \dfrac{1}{4} × 1 \ + \ \dfrac{3}{5} × 1 \\ \\ & = \dfrac{1}{4} × \bbox[Bisque, 3pt]{\dfrac{5}{5}} \ + \ \dfrac{3}{5} × \bbox[Bisque, 3pt]{\dfrac{4}{4}} \\ \\ & = \dfrac{1 × 5}{4 × 5} + \dfrac{3 × 4}{5 × 4} \\ \\ & = \dfrac{5}{\bbox[PowderBlue, 3pt]{20}} + \dfrac{12}{\bbox[PowderBlue, 3pt]{20}} \\ \\ & = \dfrac{5 + 12}{20} \\ \\ & = \dfrac{17}{20} \\ \\ \end{align}$

$\dfrac{2}{3} + \dfrac{1}{2} - \dfrac{1}{4}$ Solution

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Make the denominator with the lowest common multiple (LCM)...
3 → 6 → 9 → $\bbox[PowderBlue, 3pt]{12}$
2 → 4 → 6 → 8 → 10 → $\bbox[PowderBlue, 3pt]{12}$
4 → 8 → $\bbox[PowderBlue, 3pt]{12}$ $\begin{align} & = \dfrac{2}{3} + \dfrac{1}{2} - \dfrac{1}{4} \\ \\ & = \dfrac{2}{3} × \bbox[Bisque, 3pt]{\dfrac{4}{4}} + \dfrac{1}{2} × \bbox[Bisque, 3pt]{\dfrac{6}{6}} - \dfrac{1}{4} × \bbox[Bisque, 3pt]{\dfrac{3}{3}} \\ \\ & = \dfrac{2 × 4}{3 × 4} + \dfrac{1 × 6}{2 × 6} - \dfrac{1 × 3}{4 × 3} \\ \\ & = \dfrac{8}{12} + \dfrac{6}{12} - \dfrac{3}{12} \\ \\ & = \dfrac{8 + 6 - 3}{12} \\ \\ & = \dfrac{11}{12} \\ \\ \end{align}$

$2 × \dfrac{3}{4}$ Solution

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You don't need a common denominator for multiplying fractions, look it's easy! $\begin{align} & = 2 × \dfrac{3}{4} \\ \\ & = \dfrac{2}{1} × \dfrac{3}{4} \\ \\ & = \dfrac{2 × 3}{1 × 4} \\ \\ & = \dfrac{6}{4} \\ \\ & = \dfrac{3}{2} \\ \\ \end{align}$

$\dfrac{1}{2} × \dfrac{2}{3} × \dfrac{1}{4}$ Solution

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With multiplication of fractions you can cancel something once on the top and bottom... $\begin{align} & = \dfrac{1}{2} × \dfrac{2}{3} × \dfrac{1}{4} \\ \\ & = \dfrac{1 × 2 × 1}{2 × 3 × 4} \\ \\ & = \dfrac{1 × \cancel{2} × 1}{\cancel{2} × 3 × 4} \\ \\ & = \dfrac{1}{12} \\ \\ \end{align}$

$\dfrac{1}{2} ÷ \dfrac{1}{3}$ Solution

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With division of fractions... multiply by the reciprocal of the fraction after the division sign ÷... $\begin{align} & = \dfrac{1}{2} ÷ \dfrac{1}{3} \\ \\ & = \dfrac{1}{2} × \dfrac{3}{1} \\ \\ & = \dfrac{1 × 3}{2 × 1} \\ \\ & = \dfrac{3}{2} \\ \\ \end{align}$

$\dfrac{3}{25} ÷ \dfrac{1}{5}$ Solution

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With division of fractions... multiply by the reciprocal of the fraction after the division sign ÷ ... And cancel across division... $\begin{align} & = \dfrac{3}{25} ÷ \dfrac{1}{5} \\ \\ & = \dfrac{3}{25} × \dfrac{5}{1} \\ \\ & = \dfrac{3 × 5}{25 × 1} \\ \\ & = \dfrac{3 × \cancelto{1}{5}}{\cancelto{5}{25} × 1} \\ \\ & = \dfrac{3 × 1}{5 × 1} \\ \\ & = \dfrac{3}{5} \\ \\ \end{align}$

More Fractions with Order of Operations (PEMDAS/BEDMAS)

Simplify the following fraction expressions without the use of a calculator. Reduce your answer fully.

$\dfrac { 11 }{ 4 } \ - \ 1$ Solution

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$\begin{align} & = \dfrac { 11 }{ 4 } \ - \ \color{RoyalBlue}{1} \\ \\ & = \dfrac { 11 }{ 4 } \ - \ \color{RoyalBlue}{\dfrac { 4 }{ 4 }} \\ \\ & = \dfrac { 11 \ - \ 4 }{ 4 } \\ \\ & = \dfrac { 7 }{ 4 } \\ \\ \end{align}$

$\left( \dfrac { 2 }{ 3 } +\dfrac { 1 }{ 2 } \right) \times \dfrac { 2 }{ 5 }$ Solution

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$\begin{align} & = \left( \dfrac { 2 }{ 3 } + \dfrac { 1 }{ 2 } \right) \ \times \ \dfrac { 2 }{ 5 } \\ \\ & = \left( \dfrac { 4 }{ 6 } + \dfrac { 3 }{ 6 } \right) \ \times \ \dfrac { 2 }{ 5 } \\ \\ & = \left( \dfrac { 4 \ + \ 3 }{ 6 } \right) \ \times \ \dfrac { 2 }{ 5 } \\ \\ & = \dfrac { 7 }{ 6 } \ \times \ \dfrac { 2 }{ 5 } \\ \\ & = \dfrac { 7 \ \times \ 2 }{ 6 \ \times \ 5 } \\ \\ & = \dfrac { 14 }{ 30 } \\ \\ & = \dfrac { 7 }{ 15 } \\ \\ \end{align}$

$3 \ \div \ \dfrac { 6 }{ 11 }$ Solution

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$\begin{align} & = 3 \ \color{WildStrawberry}{\div} \ \color{RoyalBlue}{\dfrac { 6 }{ 11 }} \\ \\ & = 3 \ \color{WildStrawberry}{\times} \ \color{RoyalBlue}{\dfrac { 11 }{ 6 }} \\ \\ & = \dfrac { 3 }{ 1 } \ \times \ \dfrac { 11 }{ 6 } \\ \\ & = \dfrac { 3 \ \times \ 11 }{ 1 \ \times \ 6 } \\ \\ & = \dfrac { 33 }{ 6 } \\ \\ & = \dfrac { 33 \ \color{Magenta}{\div 3} }{ 6 \ \color{Magenta}{\div 3} } \\ \\ & = \dfrac { 11 }{ 2 } \\ \\ \end{align}$

$\left( \dfrac { 2 }{ 9 } \ \div \ \dfrac { 4 }{ 3 } \right) \ \div \ \dfrac { 1 }{ 5 }$ Solution

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$\begin{align} & = \left( \dfrac { 2 }{ 9 } \ \times \ \dfrac { 3 }{ 4 } \right) \ \div \ \dfrac { 1 }{ 5 } \\ \\ & = \left( \dfrac { 2 \ \times \ 3 }{ 9 \ \times \ 4 } \right) \ \div \ \dfrac { 1 }{ 5 } \\ \\ & = \left( \dfrac { 6 }{ 36 } \right) \ \div \ \dfrac { 1 }{ 5 } \\ \\ & = \left( \dfrac { 6 \ \color{ForestGreen}{\div \ 6} }{ 36 \ \color{ForestGreen}{\div \ 6} } \right) \ \div \ \dfrac { 1 }{ 5 } \\ \\ & = \left( \dfrac { 1 }{ 6 } \right) \ \color{Red}{\div} \ \dfrac { 1 }{ 5 } \\ \\ & = \dfrac { 1 }{ 6 } \ \color{Red}{\times} \ \dfrac { 5 }{ 1 } \\ \\ & = \dfrac { 1 \ \times \ 5 }{ 6 \ \times \ 1 } \\ \\ & = \dfrac { 5 }{ 6 } \\ \\ \end{align}$

...FRACTIONS HOMEWORK (GR6)

Simplify the following fraction expressions without the use of a calculator. Reduce your answer fully.

$\dfrac { 17 }{ 8 } \ - \ \dfrac { 2 }{ 1 }$

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$\dfrac { 9 }{ 4 } \ - \ 1$

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$\dfrac { 32 }{ 11 } \ + \ 3$

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$\dfrac { 23 }{ 7 } \ - \ 2$

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$\dfrac { 15 }{ 9 } \ - \ 1$

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$\dfrac { 24 }{ 5 } \ - \ 4$

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$5 \ - \ \dfrac { 11 }{ 6 }$

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$\dfrac { 5 }{ 12 } \ \div \ \dfrac { 4 }{ 3 }$

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$\dfrac { 8 }{ 25 } \ \div \ \dfrac { 9 }{ 5 }$

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$\left( 4 \ \div \ \dfrac { 2 }{ 3 } \right) \ \div \ \dfrac { 24 }{ 9 }$

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Adding Fractions Word Problems

David walks $\dfrac{3}{4}$km home from a soccer game, then walks $\dfrac{2}{5}$km to the corner store to buy a treat, then he walks $\dfrac{1}{1}$km to his friends house. How far has David travelled in total? Solution

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To add fractions, make a common denominator... the lowest common multiple (LCM)

In this case the LCM is 20... $\begin{align} & = \dfrac{3}{4} + \dfrac{2}{5} + \dfrac{1}{1} \\ \\ & = \dfrac{3 × 5}{4 × 5} + \dfrac{2 × 4}{5 × 4} + \dfrac{1 × 20}{1 × 20} \\ \\ & = \dfrac{15}{20} + \dfrac{8}{20} + \dfrac{20}{20} \\ \\ & = \dfrac{15 + 8 + 20}{20} \\ \\ & = \dfrac{43}{20} \\ \\ \end{align}$

Mixed Fractions - Basics

True or false? Solution $5\left(\dfrac{2}{5}\right) = 5\dfrac{2}{5}$

1. True
2. False

Careful, look at each one closely,

$\begin{align} & 5\left(\dfrac{2}{5}\right) \\ \\ & = \dfrac{5(2)}{5} \\ \\ & = \dfrac{10}{5} \\ \\ & = 2 \end{align}$ $\begin{align} & 5\dfrac{2}{5} \\ \\ & = 5 + \dfrac{2}{5} \\ \\ & = \dfrac{25}{5} + \dfrac{2}{5} \\ \\ & = \dfrac{25 + 2}{5} \\ \\ & = \dfrac{27}{5} \end{align}$

Adding & Subtracting Mixed Fractions

Given the following 3 different examples of ways to solve, use whatever method you prefer, and enter your answer as a fully reduced, mixed fraction.

$5\,\dfrac{2}{5} - 3\,\dfrac{7}{8}$ Solution

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Just method #1 is shown here... $\begin{align} & = 5 - 3 + \left(\dfrac{2}{5} - \dfrac{7}{8}\right) \\ \\ & = 2 + \left(\dfrac{2}{5} - \dfrac{7}{8}\right) \\ \\ & = 1 + \left(1 + \dfrac{2}{5} - \dfrac{7}{8}\right) \\ \\ & = 1 + \left(\dfrac{40}{40} + \dfrac{16}{40} - \dfrac{35}{40}\right) \\ \\ & = 1 + \dfrac{40 + 16 - 35}{40} \\ \\ & = 1 + \dfrac{21}{40} \\ \\ & = 1\,\dfrac{21}{40} \\ \\ \end{align}$

The Commutative Property

The following expressions are equal 5 ÷ 6 × 18 = 5 × 18 ÷ 6

1. True
2. False

Solve the following, without the use of a calculator Solution 3 ÷ 11 × 55

1. $\dfrac{165}{11}$
2. 15
3. $\dfrac{3}{5}$
4. Not possible

$\begin{align} &= 3 ÷ 11 × 55 \\ &= 3 × 55 ÷ 11 \\ &= 3 × (55 ÷ 11) \\ &= 3 × (5) \\ &= 15 \\ \end{align}$

The following expression equals 3100 Solution 3100 ÷ 999 × 999

1. True
2. False

3100 ÷ 999 × 999

= 3100 × 999 ÷ 999

= 3100 × 1

= 3100

The Distributive Property

The following is correct... Solution $\begin{align} &= (45 + 75) ÷ 15 \\ &= 45 ÷ 15 + 75 ÷ 15 \\ &= (45 ÷ 15) + (75 ÷ 15) \\ &= (3) + (5) \\ &= 8 \\ \end{align}$

1. True
2. False

Yup, this is how the 'distributive property' works.

Try calculating the following using the distributive property, without the use of a calculator Solution (5.5) × 8

1. 11
2. 22
3. 33
4. 44

$\begin{align} &= (5.5) × 8 \\ &= (5 + 0.5) × 8 \\ &= 5 × 8 + 0.5 × 8 \\ &= (5 × 8) + (0.5 × 8) \\ &= 40 + 4 \\ &= 44 \\ \end{align}$

Adding Decimals

Add the following decimals without the use of a calculator...

   10,240.15 + 29.1 Solution Video

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Make sure to line up all the decimals overtop of eachother... $\begin{align} 10240\!\!&.15 \\ + \ \ 29\!\!&.1 \\ \end{align}$ = 10269.25

   6.99 + 10.01 + 1.3 + 5.22 + 0.05 + 0.25 Solution Video

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Make sure to line up all the decimals overtop of eachother... $\begin{align} 6\!\!&.99 \\ 10\!\!&.01 \\ 1\!\!&.30 \\ 5\!\!&.22 \\ 0\!\!&.05 \\ + \ \ 0\!\!&.25 \\ \end{align}$ = 23.82

   2.640 + 34.3 + 0.06 Solution $\begin{align} 2\!\!&.640 \\ 34\!\!&.3 \\ + \ 0\!\!&.06 \\ \end{align}$

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Make sure to line up all the decimals overtop of eachother (shown above)

You could use placeholder zeros: $\begin{align} 2\!\!&.640 \\ 34\!\!&.300 \\ + \ \ 0\!\!&.060 \\ \end{align}$ ... = 37

Subtracting Decimals

Subtract: 391,246.123 - 123.4 Solution Video

1. 289,123.332
2. 391,122.723
3. 390,100.231
4. 392,218.649

Make sure to line up all the decimals overtop of eachother $\begin{align} 391,246\!\!&.123 \\ + \quad \ \ \ 123\!\!&.4 \\ \end{align}$ You could use placeholder zeros: $\begin{align} 391,246\!\!&.123 \\ + \quad \ \ \ 123\!\!&.400 \\ \end{align}$ = 391, 122.723

Multiplying by Factors of 10

Find and state the pattern in the following.

Solution $\begin{align} 5025 ÷ 100 & = 50.25 \\ \\ 5025 ÷ 10 & = 502.5 \\ \\ 5025 ÷ 1 & = 5025. \\ \\ 5025 ÷ 0.1 & = 50250. \\ \\ 5025 ÷ 0.01 & = 502500. \\ \\ \end{align}$

As you divide by a smaller factor of ten each time, the decimal place moves one place to the right, and the resulting number gets bigger.

Solution $\begin{align} 235 × 100 & = 23500 \\ \\ 235 × 10 & = 2350 \\ \\ 235 × 1 & = 235 \\ \\ 235 × 0.1 & = 23.5 \\ \\ 235 × 0.01 & = 2.35 \\ \\ \end{align}$

As you multiply by a smaller factor of ten each time, the decimal place moves one place to the left, and the resulting number gets smaller.

Multiplying and Dividing by Factors of 10

Match each operation below with the other operation that does the same thing. Solution

× 0.1

÷ 0.01

÷ 0.1

× 100

÷ 10

× 10

× 0.1 = ÷ 10

÷ 0.01 = × 100

For example:
55 × 0.1 = 5.5
55 ÷ 10 = 5.5

33 ÷ 0.01 = 3300
33 × 100 = 3300

Starting with the number below, drag and drop each operation into the box with the correct answer. Solution 1525

× 10

÷ 0.01

× 0.1

÷ 100

= 15.25

= 152.5

= 15250

= 152500

As you multiply by a smaller factor of ten each time, the decimal place moves one place to the left, and the resulting number gets smaller.

As you divide by a smaller factor of ten each time, the decimal place moves one place to the right, and the resulting number gets bigger.

Multiply Whole Numbers

Solve, without the use of a calculator.

25 * 200 = Solution

1. 5,000
2. 500
3. 5,500
4. 50,000

First take care of the integers (25)(2) = 50 Then add the zeros at the end of 2 0 0... 5 0 0 0

Multiply Decimal Numbers by 10, 100, 1000, and 10000

Solve, without the use of a calculator.

5.2513 × 100 = Solution

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Move the decimal by the number of zeros in the number, 1 0 0, so by 2 zeros. Think: move the decimal to the right to make the number bigger, because you are multiplying by one hundred... = 525.13

2.3 × 1000 =

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Dividing Numbers by 10, 100, 1000, and 10000

Solve, without the use of a calculator.

345,000 ÷ 1,000 =

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15,000 ÷ 50

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Multiply Whole Numbers by 0.1, 0.01, and 0.001

Solve, without the use of a calculator.

25 × 0.1 = Solution

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Count the number of places, behind the decimal, 0.1 There is one place behind the decimal, 25. Move the decimals in the question by the same amount, 2.5

123456 × 0.001 = Solution

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Count the number of places, behind the decimal, 0.001 There are three places behind the decimal, 123456. Move the decimals in the question by the same amount, 123.456

200 × 0.001 = Solution

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Count the number of places, behind the decimal, 0.001 There are three places behind the decimal, 200. Move the decimals in the question by the same amount, 0.200

Multiply Decimal Numbers by Whole Numbers

The rules for multiplying decimals are:

1. Count the total number of decimal places in the numbers (your magic number).
2. Multiply normally, as if without decimals
3. Once you have the final product, put the decimal at the furthest position to the right
4. Move the decimal to the left by the total number of decimal places (your magic number).

Multiply the following without the use of a calculator. Solution $\begin{align} 30\!\!&.31 \\ × \ 5\!\!&.0 \\ \end{align}$

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Count the total number of decimal places in 30.31 and 5.00 = 2 + 2 = 4 decimal places.
$\begin{align} 30\!\!&.31 \\ × \ 5\!\!&.00 \\ \end{align}$
$\begin{align} 0000 \!\!& \quad \\ 0000\emptyset \!\!& \\ 15155\emptyset\emptyset \!\!& . \\ \end{align}$ Move the decimal 4 places left.
= 151.550̸0̸
= 151.55

Multiplying Decimals

Remember, the rules for multiplying decimals are:

1. Count the total number of decimal places in the numbers (your magic number).
2. Multiply normally, as if without decimals
3. Once you have the final product, put the decimal at the furthest position to the right
4. Move the decimal to the left by the total number of decimal places (your magic number).

$\begin{align} 7\!\!&.8 \\ × \ 3\!\!&.4 \\ \end{align}$ Solution

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Count the total number of decimal places
= 1 + 1 = 2 decimal places. $\begin{align} 7\!\!&.8 \\ × \ 3\!\!&.4 \\ \end{align}$
$\begin{align} 312 & \\ \quad 234\emptyset & \\ \end{align}$
$\begin{align} 2652 \!\!& . \\ \end{align}$ Move the decimal, by the same places as the question, 2 places left.
= 26.52

$\begin{align} 100\!\!&.06 \\ × \ 0\!\!&.4 \\ \end{align}$

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Count the total number of decimal places
= 2 + 1 = 3 decimal places. $\begin{align} & = \\ \\ & = \\ \\ & = \\ \\ & = \\ \\ & = \\ \\ & = \\ \\ \end{align}$

$\begin{align} 3\!\!&.45 \\ × \ 0\!\!&.32 \\ \end{align}$

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Count the total number of decimal places
= 2 + 2 = 4 decimal places. $\begin{align} & = \\ \\ & = \\ \\ & = \\ \\ & = \\ \\ & = \\ \\ & = \\ \\ \end{align}$

$\begin{align} 0.123& \\ × \ 250& \\ \end{align}$

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Count the total number of decimal places
= 3 decimal places. $\begin{align} & = \\ \\ & = \\ \\ & = \\ \\ & = \\ \\ & = \\ \\ & = \\ \\ \end{align}$

$\begin{align} 1050& \\ × \ 1.002& \\ \end{align}$

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Count the total number of decimal places
= 3 decimal places. $\begin{align} & = \\ \\ & = \\ \\ & = \\ \\ & = \\ \\ & = \\ \\ & = \\ \\ \end{align}$

$12.3 × 0.0025$

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Count the total number of decimal places
= 1 + 4 = 5 decimal places. $\begin{align} & = \\ \\ & = \\ \\ & = \\ \\ & = \\ \\ & = \\ \\ & = \\ \\ \end{align}$

Divide Whole Numbers by 0.1, 0.01, and 0.001

Dividing whole numbers by decimal factors of ten (0.1, 0.01, and 0.001...) will make the number in your answer bigger.

The decimal place will move 3 times in the expression below. Solution 21.5 ÷ 0.001

1. True
2. False

0 . 0 0 1

÷ means move the decimal to the right.
Set up the number by adding more zeros on the end. See how this does not change the number yet, it's still twenty-one point five. $\begin{align} & = 21.50000 \\ \\ \end{align}$ Then move the decimal place over in the correct direction, and the correct amount of times. $\begin{align} & = 21500.00 \\ \\ & = 21500 \\ \\ \end{align}$ That's it!
The number is 21,500

The following is correct. Solution 11.5 ÷ 0.001 = 11500

1. True
2. False

÷ 0 . 0 0 1 means move the decimal three times, to the right...

11.5 ÷ 0.001 = 11500

The following is correct. Solution 22.3 ÷ 0.01 = 22300

1. True
2. False

22.3 ÷ 0.01 = 2230

÷ 0 . 0 1 means move the decimal twice, to the right.

What does the following equal? Solution 432.1098 ÷ 0.00001

1. 432109.8
2. 4,321,098
3. 43,210,980
4. 0.004321098

Move the decimal place to the right, making the number bigger.

Move the decimal 5 times: 0 . 0 0 0 0 1

Dividing Decimals

Solve the questions below using the following example as a guide.

26.64 ÷ 1.2 Solution

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Convert the decimals first: move both 2 spaces... $\begin{align} & = 1.2 \big) \!\!\! \overline{\ \ 26.64} \\ & = 12 \big) \!\!\! \overline{\ \ 266.4} \\ \end{align}$ Then, long division... (make sure to place the decimal directly above) $\begin{align} 22.2 & \\ 12 \big) \!\!\! \overline{\ \ 266.4} & \\ \underline{24} \ | \ | & \\ 2 6 \ | & \\ \underline{\ 2 4 \ } | & \\ 2 \, 4 & \\ \underline{\ 2 \, 4} & \\ 0 & \\ \end{align}$

4.263 ÷ 0.21 Solution

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Convert the decimals first: move both 2 spaces... $\begin{align} & = 0.21 \big) \!\!\! \overline{\ \ 4.263} \\ & = 21 \big) \!\!\! \overline{\ \ 426.3} \\ \end{align}$ Then, long division... (make sure to place the decimal directly above) $\begin{align} 20.3 & \\ 21 \big) \!\!\! \overline{\ \ 426.3} & \\ \underline{4 2} \ | \ \ | & \\ 0 \, 6 \ \ | & \\ \underline{\ \ 0 \ } \ | & \\ 6 \, 3 & \\ \underline{\ 6 \, 3} & \\ 0 & \\ \end{align}$

2.5235 ÷ 0.0035 Solution

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Convert the decimals first: move both 4 spaces... $\begin{align} & = 0.0035 \big) \!\!\! \overline{\ \ 2.5235} \\ & = 35 \big) \!\!\! \overline{\ \ 25235} \\ \end{align}$ Then, long division... $\begin{align} 721 & \\ 35 \big) \!\!\! \overline{\ \ 25235} & \\ \underline{245} \ | \ | & \\ 7 3 \ | & \\ \underline{\ 7 0 \ } | & \\ 3 \, 5 & \\ \underline{\ 3 \, 5} & \\ 0 & \\ \end{align}$

Estimation with Addition & Subtraction

Use estimation and mental math to determine the most accurate answer. Some examples of estimation according to current curriculum guidelines:

55 + 105 + 37 Solution

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The curriculum says 105 should round to 100, the nearest largest place value. $\begin{align} & = 60 + 100 + 40 \\ \\ & = 200 \end{align}$

999 - 113 - 25 Solution

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$\begin{align} & = 1000 - 100 - 30 \\ \\ & = 870 \end{align}$

1025 - 175 - 63 Solution

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$\begin{align} & = 1000 - 200 - 60 \\ \\ & = 740 \end{align}$

4535 - 2590 Solution

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$\begin{align} & = 4535 - 2590 \\ \\ & = 5000 - 3000 \\ \\ & = 2000 \end{align}$

3567 - 2631 Solution

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$\begin{align} & = 3567 - 2631 \\ \\ & = 4000 - 3000 \\ \\ & = 1000 \end{align}$

128,120 + 282,123 Solution

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$\begin{align} & = 128,120 + 282,123 \\ \\ & = 100,000 + 300,000 \\ \\ & = 400,000 \end{align}$

Estimation with Multiplication & Division

A student collected 223 pop bottles for a recycling program at the school. If the pop bottles were collected over 4 days, then on average how many were collected each day? Solution

1. 30
2. 45
3. 50
4. 55

A little bit more than 200 ÷ 4 = 50

...which is a bit more than 50

...which is ≈ 55

Unit Rates and Dividing Decimals by Whole Numbers

A calculator costs $8.50 at The Bookey School Supply Store. If another store, The Penney Store, is selling 5 calculators for $46.25 what is the price for one calculator at The Penney Store, and which store has the better deal? Solution

1. $9.25, The Penney Store
2. $8.25, The Bookey Store
3. $9.35, The Penney Store
4. $9.25, The Bookey Store

$\begin{align} 9.25 & \\ 5 \big) \!\!\! \overline{\ \ 46.25} & \\ \underline{45\,} \ | \ | & \\ 1 2 \ | & \\ \underline{\ 1 0 \ } | & \\ 2 \, 5 & \\ \underline{\ 2 \, 5} & \\ 0 & \\ \end{align}$

Order of Operations (PEMDAS, BEDMAS)

Simplify the following, without the use of a calculator.

Solution 6 x (4 ÷ 2) + 8 + (10 + 4) – 2 × 6

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Use PEMDAS/BEDMAS...

Solution 3 × 4 ÷ 3 × 4 × (64 ÷ 16)

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Use PEMDAS/BEDMAS...

Solution Video (4 × 2) + 3² - 4 ÷ 2

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Use PEMDAS/BEDMAS... $\begin{align} & = (4 × 2) + 3^2 - 4 ÷ 2 \\ \\ & = (8) + 3^2 - 4 ÷ 2 \\ \\ & = 8 + 9 - 4 ÷ 2 \\ \\ & = 8 + 9 - 2 \\ \\ & = 15 \\ \\ \end{align}$

Ratios

A bubble gum container holds 24 blue and 32 red gumballs. What is the ratio of red to blue?

1. 24 to 32
2. 32 to 24
3. 3 to 4
4. 4 to 3

Fill in the missing number below to make the ratios equal: 2 to 5 = ___ to 20

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Converting Decimals, Fractions, and Percents

Fill in the blank.

Solution

Fraction	Percent
$\dfrac{2}{4}$	50 %
$\dfrac{3}{4}$	75 %
$\dfrac{2}{5}$
$\dfrac{1}{10}$	10%

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%

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Fraction	Percent
$\dfrac{2}{4}$	50 %
$\dfrac{3}{4}$	75 %
$\dfrac{2}{5}$	40 %
$\dfrac{1}{10}$	10%

Solution

Decimal	Percent
0.3	30%
0.25	25%
	5%
0.01	1%

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Decimal	Percent
0.3	30%
0.25	25%
0.05	5%
0.01	1%

Multiplying Positive and Negative Numbers

Which of the following is incorrect? Solution Video

1. -1 × -1 = -1
2. -1 × +1 = -1
3. +1 × -1 = -1
4. +1 × +1 = +1

A double negative (-1 × -1) reverses the negative (-) sign into a positive sign...

• -1 × -1 = +1
• -1 × +1 = -1
• +1 × -1 = -1
• +1 × +1 = +1

Time

Which of the following is the longest time when you set something in a microwave and enter a time (in minutes:seconds)? Solution

1. 1:00
2. 0:90
3. 1:05
4. 0:30

0:90 or 90 seconds is actually longer than 1:00 or 1 minute, which is only 60 seconds.

Appropriate Metric Units: Millimeter, Centimeter, Decimeter, Meter, Decameter, Kilometer

Which of the following units of measurement would be most appropriate for measuring the height of your refrigerator?

1. Millimeter
2. Centimeter
3. Meter
4. Kilometer

Converting Metric Units of Distance

Match the units of meters to the equivalent units of kilometers given that 1000 m = 1 km. Solution

990 m

9,000 m

99 m

0.99 m

9 km

0.099 km

0.99 km

0.0099 km

Make sure the number of kilometers is always a factor of 1,000 lower than the number of meters.

9 km = 9,000 m

0.9 km = 900 m

(The 'km' moves down one place value, so the 'm' moves down one place value as well)

Converting Metric Units of Mass

Fill in the blank, given that 1000 g = 1 kg

Grams (g)	Kilograms (kg)
220	0.22
	0.5
12,500	12.5

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grams

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Converting Metric Units of Volume

Fill in the blank, given that 1000 mL = 1 L

Milliliters (mL)	Liters (L)
4	0.004
34
2344	2.344

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L

Hint Unavailable

Area of Triangles

Two triangles below have the same area. The triangle on the left has a base of 8 cm, and a height of 8 cm. Determine the dimension of base of the triangle on the right. Solution

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cm

Use the formula for the area of a triangle and compare the bases and heights

You know the area of a triangle is: $Area = \dfrac{1}{2} × base × height$ The area of triangle 1: $\begin{align} Area & = \dfrac{1}{2} × base × height \\ \\ & = \dfrac{1}{2} × 8 × 8 \\ \\ & = 4 × 8 \\ \\ & = 32 \\ \\ \end{align}$ The area of triangle 2: $\begin{align} Area & = \dfrac{1}{2} × base × height \\ \\ & = \dfrac{1}{2} × base × 4 \\ \\ & = 2 \\ \\ \end{align}$ Since the areas are equal... $\begin{align} 32 & = 2 × base \\ \\ base & = 16 \\ \\ \end{align}$

Area of Triangles

What new dimensions would be twice the area of the dimensions of the triangle below? Solution base = 4cm, height = 6cm

1. base = 8cm, height = 12cm
2. base = 4cm, height = 8cm
3. base = 6cm, height = 6cm
4. base = 8cm, height = 6cm

You know the area of a triangle is: $Area = \dfrac{1}{2} × base × height$ Use guess & check to find the answer choice that has twice the area... eventually check D) $\begin{align} Area & = \dfrac{1}{2} × base × height \\ \\ & = \dfrac{1}{2} × 8 × 6 \\ \\ & = 4 × 6 \\ \\ & = 24\ cm^2 \\ \\ \end{align}$

Area of Parallelogram

Parallelograms are quadrilaterals that have opposite sides that are parallel and equal in length.

Calculate the area of the following parallelogram (diagram not to scale). Solution

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cm²

Hint Unavailable

Chop off the end at the dotted line and place it in the nook on the right-hand side to make a rectangle with the dimensions: width = 10
height = 6 Then calculate the area of this rectangle, $\begin{align} Area & = (width) × (height) \\ \\ & = (10) × (6) \\ \\ & = 60\ cm^2 \\ \\ \end{align}$

The area of a different parallelogram is 48 cm². If the height is 8 cm, calculate the width of one of its sides. Solution

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cm

Hint Unavailable

You have the area and height in the formula: $\begin{align} Area & = (width) × (height) \\ \\ 48 & = (width) × (8) \\ \\ width & = 6\ cm \\ \\ \end{align}$

Area of Compound Shape

Calculate the area of the following shape using compound shapes. Solution

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cm²

Hint Unavailable

See that this shape is made of two triangles plus one rectangle. The triangles and rectangle have: height = 5 cm Calculate the base lengths of the triangles. $\begin{align} base & = \dfrac{12 - 8}{2} \\ \\ & = 2\ cm \\ \\ \end{align}$ Now that you have all the dimensions, calculate the area of each shape $\begin{align} Area_{triangle\ 1} & = \dfrac{1}{2} × b × h \\ \\ & = \dfrac{1}{2} 2 × 5 \\ \\ & = 1 × 5 \\ \\ & = 5\ cm^2 \\ \\ \end{align}$ $\begin{align} Area_{triangle\ rectangle} & = w × h \\ \\ & = 8 × 5 \\ \\ & = 40\ cm^2 \\ \\ \end{align}$ $\begin{align} Area_{triangle\ 1} & = \dfrac{1}{2} × b × h \\ \\ & = \dfrac{1}{2} 2 × 5 \\ \\ & = 1 × 5 \\ \\ & = 5\ cm^2 \\ \\ \end{align}$ Add the area of each shape together to get the total area. $\begin{align} Area_{total} & = 5\ cm^2 + 40\ cm^2 + 5\ cm^2 \\ \\ & = 50\ cm^2 \\ \\ \end{align}$

Conversions Between m² and cm²

The diagram below shows the same size squares, one is in units of meters, the other is in units of centimeters.

Calculate the area of each square. Solution

Area of a square is the length times width. The length is the same as the width in a square. First: $\begin{align} Area & = (length)(width) \\ \\ & = (1\ m)(1\ m) \\ \\ & = 1\ m^2 \\ \\ \end{align}$ Second: $\begin{align} Area & = (length)(width) \\ \\ & = (100\ cm)(100\ cm) \\ \\ & = 10,000\ cm^2 \\ \\ \end{align}$

True or false? Solution 1 m² = 10,000 cm²

1. True
2. False

This is the correct conversion.

The following table shows the conversion between square units of meters and centimeters. Fill in the blanks. Solution

meters2	centimeters2
0.5 m2	_________
1 m2	10,000 cm2
2 m2	20,000 cm2
3 m2	30,000 cm2
4 m2	_________

1. 25 cm², and 9,000 cm²
2. 250 cm², and 30,000 cm²
3. 1,000 cm², and 4,000 cm²
4. 5,000 cm², and 40,000 cm²

See the pattern, increasing by the same amount each time...

Representing Angles

Which angle is greatest? Solution

1. Right
2. Obtuse
3. Acute
4. None of the above

Obtuse is greater than 90˚

Classifying Triangles by Angle

Look at the differences in the angles in these triangles...

An acute triangle can have more than one acute angle in it. Solution

1. True
2. False

True

An obtuse triangle can have more than one obtuse angle in it. Solution

1. True
2. False

Obtuse triangles only have 1 obtuse angle.

A right angle is anything less than 90˚ Solution

1. True
2. False

A right angle is exactly 90˚

A three-sided polygon that has equal angles must be an acute triangle Solution

1. True
2. False

True. And an acute triangle with all angles equal is called an equilateral.

What angle would you measure first to classify the triangle. Solution

1. ∠ A
2. ∠ B
3. ∠ C
4. It does not matter

Measure the biggest angle first, if greater than 90˚ it is obtuse...

(If equal to 90˚ it is right, and if less than 90˚ it is acute)

Classifying Triangles by Side Lengths

Look at the differences in the side lengths in these triangles, indicated by the different number of tick marks.

An isosceles triangle has all unequal side lengths. Solution

1. True
2. False

An isosceles triangle has 2 equal side lengths.

All three side lengths of an equilateral triangle are always the same in the triangle. Solution

1. True
2. False

True

Which is the scalene triangle? Solution

1. Two side lengths are equal
2. No side lengths are equal
3. All side lengths are equal
4. None of the above

None of the side lengths are equal in a scalene triangle.

Explain how the side lengths are shown as similar or different. [1] Solution

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With the use of tick marks. The same number of tick marks indicates the side lengths are the same. For example, equilateral would have three single tick marks, isosceles triangle would have two single tick marks and a double, and a scalene triangle would have on of each of: single tick mark, double tick mark, and triple tick mark.

Classifying Shapes

Which shapes are congruent? Solution

1. I
2. II
3. III
4. IV
5. all of the above

Congruent means same size and same shape.

(The rest are similar shapes, with the same shape but different size).

Polygons

Which one of the following is not a property of a polygon? Solution

1. Closed shape
2. Has to have more than 1 angle
3. All sides are straight lines
4. 2-dimensional

Polygons:

• Closed shape
• All sides are straight lines
• 2-dimensional

Polygons

What is a "regular polygon"? Provide at least one example. [3] Solution

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Any two of the following with same sides and angles: equilateral triangle, square, pentagon, hexagon, ...

Regular polygons have all side lengths and angles equal.

Polygons

Which of the following polygons is not 4-sided? Solution

1. Quadrilateral
2. Rectangle
3. Rhombus
4. Square
5. Trapezoid
6. Parallelogram
7. Kite
8. All of the above are 4-sided

All are 4-sided:

• Quadrilateral
• Rectangle
• Rhombus
• Square
• Trapezoid
• Parallelogram
• Kite

Quadrilateral Polygon Classification Exercise

Given the 8 terms, arrange into the categories or hierarchy - "what falls under what"...? Solution

• Square (all sides equal, all right angles)
• Quadrilateral (4 sides)
• Rectangle (all right angles)
• Polygon (closed, 2D, straight sides)
• Rhombus (all sides equal)
• Parallelogram (2 parallel sides)
• Kite (two different pairs of equal, adjacent sides)
• Trapezoid (at least one pair of parallel sides)

This is the general order... Google 'em!

Determine the Pattern Rule

Given the following pattern, determine the pattern rule. Solution Video 5.5, 11, 16.5, 22, 27.5, 33...

1. Add 5.5 to the term number.
2. Divide 11 by 2 and add to term number.
3. Subtract 5.5 from the term number and add 1.
4. Multiply 5.5 by the term number.

Determine the pattern rule quickly: +5.5

Pattern Rules: Term Number

What is the term number of 60 in the following pattern? Solution Video 15, 30, 45, 60, 75, 90...

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Term numbers start at 1

60 is the 4^th term in the pattern, so the term number is 4.

The 5th term in the following pattern is 54. Solution Video 9, 18, 27, ...

1. True
2. False

Determine the pattern rule first: +9. Then find the 5th term. 5th term is 45.

Patterning: Filling in Tables

Complete the following table by finding the missing numbers for the number and cost of tickets. Solution Video

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Number of Tickets	Cost
1	$18.00
3	$23.00
5	$28.00
7	$33.00
9
	$43.00

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Follow the pattern

Find the pattern rule for the number of tickets, and the pattern rule for the cost.

Number of tickets: +2
Cost: +5. The missing numbers are 11 tickets, and $38.00

Patterning

Start with 3 and add 10 to each term to get the next term. Determine the term number when the term is 53 Solution Video

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Start with 3, and then use the pattern rule of 10 up to 53... The term number is 6.

Representing Pattern Rules from Words

Make a table of values and a graph of the coordinates for the following pattern rule. Let 'x' be the term number, and let 'y' be the term. Solution Start with 2, then add one to each term and double it to get the next term.

x	y
1	2
2	__
3	__
4	__
5	__

x	y
1	2
2	6
3	14
4	30
5	62

[Graph not shown]

Patterning with Multiplication and Division

Determine the pattern rule for the following pattern, without the use of a calculator. Solution Video 3, 12, 48, 192, 768...

1. Start with three then multiply by 3 to get the next term
2. Start with three then multiply the term number by 3 to get the next term
3. Start with three then multiply by 4 to get the next term
4. Start with three then multiply the term number by 4 to get the next term

Find out what it multiplies by each time: × 4. The pattern rule is multiply by 4...