Download Grade 7 Equivalent Ratios Lesson and Resources

Assembly Line Grade 7 Ratios Clarification
Standard 6.0 Knowledge of Number Relationships and Computation/Arithmetic
Topic C. Number Computation
Indicator 3. Analyze ratios, proportions, or percents
Objective b. Determine or use ratios, unit rates, and percents in the context of a problem
Clarification of Math Discussion Terms
A RATIO is a comparison of two numbers or quantities. The ratio of 3 to 5 can be
written as 3 out of five, 3:5, or as a fraction where the first number becomes the
3
numerator and the second, the denominator: . A PROPORTION is a statement that two
5
ratios are equivalent, an equation that illustrates that the two ratios are the same
3 18
proportional amounts of a whole thing. For example, the proportion =
indicates that
4 24
if a whole thing were divided into 4 equal parts, taking 3 of the 4 parts would be the same
proportional amount as taking 18 parts out of the same thing divided into 24 equal parts.
EQUIVALENT RATIOS can be generated by using the number 1 as the identity
element for both multiplication and division. That is, any number multiplied or divided
2
2 1
× 1=
,
÷1 =
by 1 gives the identical original number : 8 × 1 = 8, 57 ÷ 1 = 57,
9
9 2
1
2 673 10,000
etc. The number 1, however, can be written in infinitely, such as ,
,
,
2
2 673 10,000
37.5 n abc
,
,
etc. Therefore, multiplying or dividing a fraction or ratio by a version of
37.5 n abc
2 7
2 × 7 14
= . This illustrates that 2 out of
1 keeps its identical proportional value: × =
3 7
3 × 7 21
3 is the same proportional part as 14 out of 21.
2
14
can also be thought of as
reduced to lowest terms by dividing by the common
3
21
14
7
are divided by 1, written as , the
factor, 7. If the numerator and denominator of
21
7
14 ÷ 7
2
result is called the reduced, or simplified form:
=
If the numerator and
21 ÷ 7
3
denominator of the reduced form have no common factors, the ratio is said to be in
lowest terms. Since there are an infinite number of fractions that are equivalent to a given
2
ratio such as , when comparing ratios, it is convenient to write the fractions in reduced,
3
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or simplified form:
6 2
= ,
9 3
8 2
= ,
12 3
10 2
= ,
15 3
22 2
= ,
33 3
17 2
= ,
51 3
42
2
=
63
3
2,000,000
2
= etc.
3,000,000
3
Classroom Example 1
7
42
Is
equal to
?
13
78
Method 1: Using multiplication
7
42
is equal to
if there is a common factor by which 7 and 13 are multiplied to get
13
78
42
:
78
7×?
42
=
,
13 × ?
78
7 × 6 42
7
42
=
, therefore
does equal .
13 × 6 78
13
78
Method 2: Using divisors
42
Starting with
, we can find divisors, or factors of 42 and 78 that would simplify, or
78
42
7 42 ÷ ?
7
42 ÷ 6
7
reduce
to
:
=
.
=
.
78
13 78 ÷ ?
13 78 ÷ 6
13
Often students will do the simplifying, or reducing to lowest terms in several steps. For
example, a student might recognize that 42 and 78 are both even numbers, making them
42 ÷ 2
21
divisible by 2. Dividing 42 and 78 by 2 would yield an equivalent ratio:
=
,
78 ÷ 2
39
however if we want the ratio in lowest, or simplest terms, as is desirable for comparison
21
purposes,
would have to be reduced again, since 21 and 39 have a common factor, 3 :
39
21 ÷ 3 7
= . Although successive divisions will give the same final reduced form, it is
39 ÷ 3 13
more efficient to simplify by dividing by the GREATEST COMMON FACTOR, as
42
when dividing by 6 rather than 2 to reduce . The fraction will be in lowest or
78
simplified form if the numerator and denominator of the fraction have no common
factors, which is to say that they are RELATIVELY PRIME.
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Method 3: Using Cross-Products
A useful characteristic of proportions is the CROSS- PRODUCT PROPERTY, which
a
c
7 42
states that if
=
, then a × d = b × c. Therefore, if
=
, then 7 × 78 = 13 ×
b
d
13 78
42.
7 × 78 = 546 and 13 × 42 = 546, so yes,
7
42
does equal .
13
78
Classroom Example 2
4
Write five ratios that are equal to .
9
Answers will vary. Some possible answers would include:
8 4
= ,
18 9
12 4
= ,
27 9
20 4
= ,
45 9
24 4
= ,
54 9
36 4
= ,
81 9
40 4
= ,
90 9
44 4
= ,
99 9
48 4
= ,
108 9
28 4
= ,
63 9
52
4
= ,
117 9
32 4
= ,
72 9
72 4
=
162 9
Students might be encouraged to check their answers by reducing or simplifying their
4
fractions to see if they do, in fact, reduce to . This would also offer the opportunity to
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compare and contrast the efficiency of dividing by the greatest common factor, such as
40
dividing by 10 when reducing , compared with successive divisions in cases when the
90
72 72 ÷ 2
36 36 ÷ 9
4
:
=
,
=
greatest common factor is not obvious, as in
162 162 ÷ 2
81 81 ÷ 9
9
The Math in the Puzzle
In the Assembly Line puzzle, players must demonstrate their knowledge of ratios and
proportion by using gear ratios to label premium
proportion of premium
cans. To achieve the correct
cans to total cans in the collection bins, players must study
visual clues. Clues include the “can counter” array of small square lights at the top of the
screen that matches the layout of the collection bin, the numerical relationship between
the
cans and the total number of cans in the destination bin, and the appearance of
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each division on the conveyor belt. The visual clues on the belt are especially useful
when the player is presented with two possible sets of gears that meet the required gear
ratio.
In the screen shot above, when the collection bin has place-holders for 3 premium
cans out of the 15 total spots reserved for all cans, the target ratio is
3
1
or . Given the
15
5
choice of left gears with 4, 5, or 6 gear teeth, and right gears with 20, 25, and 26 teeth,
players must choose between two acceptable combinations to get the required
1
ratio.
5
The markings on the belt will give the clue as to the correct gears. Note also that in all
cases, both original gears on the opening screen of the puzzle must be replaced, even if
one is being replaced by a gear with the same number of teeth.
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