Ratio and proportion
National Curriculum Objectives, Y6, Ratio and proportion • solve problems involving similar shapes where the scale factor is known or can be found • solve problems involving unequal sharing and grouping using knowledge of fractions and multiples
Fractions, ratio or both? Challenge 1:
Challenge 1 Answer: Maria and Neil share £35 in the ratio 2:5. How much more money does Neil have
Neil has £15 more than Maria.
than Maria?
Challenge 2: Challenge 3:
= £35
Oliver and Petra walk a total of 42 km each week to school. Oliver walks 24 km. What is the ratio of kilometres walked each week by the two pupils in its simplest form?
Assessment:
Stefan decides to make a cake for 8 people. He uses 4 eggs, 200 g flour,
Pupils should be able to argue that the sum of money is divided into 7 equal parts: Maria’s 2 parts and Neil’s 5 parts. They should reason that £35 divided by 7 calculates the value of one part in the ratio as £5. By multiplying the respective parts belonging to Neil and Maria by 5, pupils calculate Neil to have £25 and Maria to have £10; they check that these figures total £35. Finally, they subtract Maria’s total from Neil’s to get the answer to the question. Alternatively, they will see that the ratio shows that Neil has 3 times the value of one part more than Maria.
200 g sugar, 200 g butter and 60 g dark chocolate. The following week he makes enough cake for 18 people. How much of each ingredient does he need?
Challenge 4:
The pie chart shows the results of a football team supporters’ survey in a school. The sum of pupils who support Chelsea and Leicester City is 75%. The ratio of Chelsea to
Pupils will move towards solving this type of problem by adding the numbers in the ratio then dividing the sum (in this case £35) by the result (7). When pupils have identified the value of one part, they will multiply this by the number of parts given in the ratio. They will show that this information needs the final operation of subtraction to find the answer to the question.
Leicester supporters is 7 : 8. The sum of Manchester United and Aston Villa supporters
Challenge 2 Answer:
is 45.
The ratio of kilometres walked is 4 : 3.
Assessment:
Manchester United Aston Villa
Pupils will reason that 42 km in total are covered each week by Oliver and Petra and that they cover a fraction of 42 km each. From a concrete perspective, pupils will reason that the fraction of the total journeys that Petra makes is (42 – Oliver’s journey) out of 42.
Chelsea
24 km
Oliver
= 42 km Leicester City
Petra
?
Pupils should continue to make links to fractions and, in particular, equivalent fractions to reason that if Oliver’s 24 18 this can be simplified to 47 . In the same way, Petra’s journey represents 42 of the proportion of the journey is 42 3 total journey which reduces to 7 . As both journeys are reduced to 7 equal parts, the ratio of distance travelled in its simplest form is Oliver : Petra = 4 : 3.
a)
How many pupils said they preferred Leicester City?
b) What fraction of the survey prefer Chelsea?
Pupils will start to take an algebraic approach to this type of problem by initially seeing the problem as an addition of two fractions with identical denominators, of which one numerator is unknown. 24 42
+
n 42
=
42 42
⇒ 24 + n = 42
From this, they calculate 18 km as Petra’s part of the total journey. They continue this idea by reducing both fractions to their simplest form. Once this is completed they know that the ratio of one to the other is simply the numerator values of the simplified fraction. 24 42
2
+
18 42
=
12 21
+
9 21
=
4 7
+
3 7
⇒4:3
3