fractions, decimals and percentages year 7 booklet 1

Fractions, Decimals and Percentages

Year 7

Booklet 1

Version 0.01.006

Student name: ___________________________________ Class: ___________________________ Teacher name: _______________________________ Commencement date: ______________________

www.smoothmathematics.com

This resource comes free to you, thanks to

Your school’s crest

Sneak-peak – Draft 12/10/2021

Contents 1. Fractions as part(s) of a whole – part 1. 3 2. Fractions as part(s) of a whole – part 2. 6 3. Fractions as part(s) of a whole – part 3. 9 4. Fractions as part(s) of a whole – part 4. 13 5. Fractions as “for every”. 19 6. Part of a group - multiplication. 25

6.1 Way of thinking #1. 25

6.2 Way of thinking #2. 27

6.3 When there are left-overs! 29 Answers. 32


1. Fractions as part(s) of a whole – part 1.

Question 1

State what fraction of the whole area of each shape, is shaded.

a)

Regular octagon.

Fraction shaded =

d)

Square.

Fraction shaded =

b)

Regular pentagon.

Fraction shaded =

e)

Rectangle on square grid.

Fraction shaded =

c)

Regular hexagon.

Fraction shaded =

f)


Fraction shaded =


Question 2

State what fraction of the whole area of each shape, is shaded.

a)


Fraction shaded =

b)


Fraction shaded = c)


Fraction shaded =


Question 3

Write in the missing numerators that make each statement true.

a) !"=

#=

$!

b) !"=

%=

$&

c) "'=

$!=

"#

d) "'=

!(=

$((

e) )&=

!'=

"!

f) "*=

'!=

#"

Question 4

Write in the missing denominators that make each statement true.

a) !)= ' = &

b) $'= * = !$

c) ")= $) = "(

d) )#= "( = ")

e) "&= !$ = !*

f) #*= !' = )'



Question 1

a) Shade !"!#

of the whole rectangle (on square grid).

Notice anything?

b) Shade $!"


Notice anything?

c) Shade #"%


Notice anything?

d) Shade &"%


Notice anything?

e) Shade !""!


Notice anything?


Question 2


a) $!$&=

%=

"

b) '$!=

#=

"

c) &!(=

$(=

)

d) )!(=

'

e) $!!$=

*

f) $&'!=

!$=

*

Question 3

Write in the missing denominators that make each statement true.

a) $''!= ! = $

b) $)#(= ) = $

c) &$!= ' = !

d) $!$#= "

e) "("#= )

f) #(*)= $! = '



Question 1

The region between two circles that share the same centre (concentric circles) is called an annulus. You can see an annulus, coloured red, in Figure 1.

Figure 1: An annulus, coloured red.

Below are two concentric squares.

The region between two concentric squares does not seem to have a name, so henceforth such a region shall be named squannulus.

Your job is to colour exactly !" of the squannulus, in the most creative and

colourful way you can.

Some more squannuli are provide on the next page so you can cut loose and

colour exactly !" of the squannulus in a variety of different, colourful and

creative ways.


Question 2

Below are two concentric equilateral triangles. • The small equilateral triangle has side length of 1 unit. • The large equilateral triangle has side length of 3 units. • The corresponding sides of the small and large triangle are parallel.

The region between two concentric equilateral triangles does not seem to have a name, so henceforth such a region shall be named trianngulus.

Your first job is to colour exactly !" of the trianngulus, in the most creative

and colourful way you can. Some more triannguli are provide on the next page so you can cut loose and

colour exactly !" of the trianngulus in a variety of different, colourful and

creative ways.

Your second job is to colour exactly #" of the trianngulus in a symmetrical

manner. Make it lovely and colourful. Best one wins a prize! J Extra triannguli are provided on page 14.



Question 1

Figure 2 shows a representation of a tangram puzzle. A tangram is a square cut into various shapes.

Figure 2.

What fraction of the area of the square is occupied by the two smallest right triangles?

Question 2

Figure 3 shows two identical regular hexagons enclosed in a rectangle.

Figure 3.

What fraction of the rectangle’s area is occupied by the two regular hexagons?


Question 3

Figure 4 shows two identical regular hexagons enclosed in a rectangle.

Figure 4.

What fraction of the rectangle’s area is coloured?

Question 4

Figure 5 shows a square with each side divided into four equal segments.

Figure 5.

What fraction of the square’s area is coloured?


Question 5

Figure 6 shows four small equilateral triangles packed within one large equilateral triangle.

Figure 6.

What fraction of the large equilateral triangle is coloured?


Figure 2.

Figure 3.

Figure 4.

Figure 5.

Figure 6.


5. Fractions as “for every”.

Question 1

State what fraction of the total number of dots in each part are blue.

a)

Fraction of all dots that are blue =

e)


b)


f)


c)


g)


d)


h)



Question 2

Colour the fraction of the total number of dots requested in each part.

a) Colour '( of all these dots.

e) Colour "' of all these dots.

b) Colour "' of all these dots.

f) Colour '$ of all these dots.

c) Colour "' of all these dots.

g) Colour '& of all these dots.

d) Colour "' of all these dots.

h) Colour "' of all these dots.


Question 3


a) "*=

!$=

!&

b) !"=

$!=

!'

c) "'=

!'=

"!

d) ")=

!(=

"(

e) '%=

!*=

')

f) )#=

$&=

!'


Question 4

Colour the fraction of the total number of dots requested in each part.

a) Colour )* of all these dots.

e) Colour #!%

of all these dots.

b) Colour )* of all these dots.

f) Colour #!%

of all these dots, in a

different way than you did in part e).

c) Colour )# of all these dots.

g) Colour )!#

of all these dots.

d) Colour )# of all these dots, in a

different way than you did in part c).

h) Colour !#("

of all these dots.


Question 5


a) #%=

"

b) #&=

'

c) &$(=

)

d) #$&=

"

d) $&*!=

"#=

'


6. Part of a group - multiplication.

6.1 Way of thinking #1 2 × 10 = 20. We can think about 2 × 10 as,

2 groups of 10 objects,

and can represent it visually as an array; something like this,

or

Instead of wanting 2 groups of 10 suppose we wanted,

"& of one group of 20 objects.

One way to think about this is,

we need 2 for every 5 of the 20 objects,

which can look like this,

So, we can write, "& × 20 = 8

One way to calculate "& × 20 (without a picture) is to:

• first figure out how many groups of 5 are in 20,

20 ÷ 5 = 4

• then determine how many objects in total for 4 lots of 2,

4 × 2 = 8


To calculate "' × 12,

"' × 12

= 2 × 4 (since 12 ÷ 3 = 4) = 8

To calculate &( × 42,

&( × 42

= 5 × 6 (since 42 ÷ 7 = 6) = 30

Question 1

Calculate each of the following.

𝑎) 34 × 12 e) "' × 27

𝑏) 27 × 42 f) &# × 56

c) "& × 30 g)

(!!

× 77

d) '& × 45 h)

(# × 72


6.2 Way of thinking #2

A different (but equal) way to think about "& × 20 is as,

"& of every one of the 20 objects.

Or in other words 20 lots of "& (of a whole/object), which could be pictured

like this,

In total we have forty-fifths, or $%&

, which equals 8 whole objects.

Convince yourself all of the red pieces fit exactly into 8 objects.

To calculate "' × 12 we could then do,

"' × 12

= "$'

= 8

To calculate &( × 42,

&( × 42

= "!%(

= 30


Question 1

Calculate each of the following, using the method on the previous page.

a) "' × 15 d)

&) × 12

b) '& × 40 e)

&) × 24

c) "' × 45 f)

"( × 14


6.3 When there are left-overs!

Suppose we need to calculate "& of one group of 23 objects.

We do not have a whole number of groups of 5. So, the “2 for every 5” way of thinking is a bit tough. (Can be done though.)

Let’s think about is as "& of every one of the 23 objects. Which could look

like this:

How many whole objects? The next diagram shows the start of moving the red two-fifths around.

Convince yourself that there are

9 whole objects and !& of a whole object that will be red.

So, "& × 23 = 9 +

!&

9 + !& is usually shortened to 9

!& (which is called a mixed number).


To calculate "& × 23 we could then do,

"& × 23

= $)&

(forty-six fifths)

Now calculate how many 5s are in 46 (to fill up the wholes) 46 ÷ 5 = 9 remainder 1. The 1 is, in fact, one-fifth (of a whole of 5). So,

"& × 23

= $)&

= 9 15

To calculate "' × 17,

"' × 17

= '&'

= 11 23

To calculate '& × 16,

'& × 16

= $#&

= 9 35


Question 1


a) "' × 16 d)

&) × 11

b) '$ × 10 e)

"( × 17

c) "' × 19 f)

'& × 42

Question 2


a) $( × 15 c)

"' × 28

b) '# × 21 d)

&) × 25


Answers To come.

fractions, decimals and percentages year 7 booklet 1

Documents