The purpose of this unit is to synthesise students’ fraction and decimal place value knowledge, thus enabling them to work with rates and ratios with understanding and competence.
Implicit within students’ understanding of fractions and percentages is relational thinking, or the ability to see and use possibilities of variation between numbers in an equation. Just as fractions and percentages express part-whole relationships, a ratio expresses a relationship between two numbers of the same kind. It is usually expressed as ‘a to b’ or ‘a : b’, meaning that for every amount of one thing, there is so much more of another thing of the same kind. Ratios can express a part-to-part relationship, such as the number of boys to the number of girls in a group of children or a part-to-whole relationship, such as the number of boys to the number of children in the whole group.
It is not a significant conceptual shift for students to work with ratios, as their proportional nature is familiar, and the students’ ability to understand and work easily with multiplicative relationships is fundamental to any proportional thinking. This thinking has been consolidated in their fractional and percentage work, so the shift in their work with ratios is rather in the contexts in which ratios are most often used, and the interpretation of these contexts. However, students do encounter for the first time, the colon notation used in the expression of a ratio.
Students need to understand that, as with fractions, ratios can be reduced by common factors of the quantities, to make the simplest form. For example, a ratio of 20:60 can be represented in its simplest form as a ratio of 1:3. This can also be expressed as a fraction. The first amount is one third of the second amount. However, in this 1:3 part-to-part ratio, we can also see that there are 4 parts altogether (1 + 3) so the relationship of the first quantity (1) to the whole (4) can be expressed as 1/4. Students may not have recognised before that every fraction is in fact a ratio. It is important to model well with equipment both the part-to-part and part-to-whole relationships that make up ratios.
Rates
The key idea that students need to develop is that a ratio compares two amounts of the same kind of thing (eg. people: girls to boys, drink mix, odds) whilst a rate is a special kind of ratio that compares different kinds of measures such as dollars per kilogram. A distinguishing feature of a rate is that it uses the word per and the symbol /. Students therefore need to be able to co-ordinate pairs of numbers, and be competent in working with fractions, decimals and percentages, to calculate the multiplicative relationship between the number pairs and to make comparisons of rates.
A rate is a very important kind of ratio because many practical tasks in our lives involve some kind of rate: speed (kilometres per hour), remuneration (dollars paid per hour of work), health (pulse, heartbeats per minute), or the price of something (cost per unit bought), to name a few.
Constant rates are ordered pairs that result in a straight line when plotted on a number plane. The slope of the line is the unit rate, and the relationship between the numbers on the line does not change. This co-linear representation of a constant rate captures visually this multiplicative relationship ‘in action’.
Proportional thinking underpins one’s ability to make sense of and use easily ratios and rates in our daily lives. These ideas are presented in five sessions however, as they include complex concepts that are fundamental to a student’s success with ratios and rates, these sessions can be extended over a longer period of time.
Note: One very well known ratio is the golden ratio, also known as the golden mean. The golden ratio is a special number approximately equal to 1.618. It appears many times in geometry, art, architecture and other areas. Your students may well enjoy investigating exploring this remarkable ratio that has fascinated people for centuries.
Links to the Number Framework
Stages 7- 8 (Advanced Multiplicative to Advanced Proportional)
The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:
The contexts for this unit involve collections of discrete objects (e.g. people, scoops of mochaccino mix). These contexts might be supplemented by, or adapted to, better reflect students' interests, cultural backgrounds, or to make connections to learning from other curriculum areas. Examples might include ratios in cooking, ratios of positions in sports teams or genders in a class, or in the dimensions of human faces. Consider how you can utilise these ratio and rate problems as a way to make connections between mathematics and your students' 'real-world' contexts.
Te reo Māori kupu such as ōwehenga (ratio), hautanga (fraction), whakarea (multiplication, multiply), and pāpātanga (rate) could be introduced in this unit and used throughout other mathematical learning.
Session 1
SLOs:
Activity 1
Activity 2
Girls | 3 | 6 | 9 | 12 | 15 | 18 | 21 |
Boys | 4 | 8 | 12 | 16 | 20 | 24 | 28 |
Session 2
SLOs:
Activity 1
Activity 2
Girls | 3 | 6 | 9 | 12 | 15 | 18 | 21 |
Boys | 4 | 8 | 12 | 16 | 20 | 24 | 28 |
Total in group/class | 7 | 14 | 21 | 28 | 35 | 42 | 49 |
Activity 3
Girls | 2 | 4 | 6 | 8 | 10 | 12 | 14 |
Boys | 5 | 10 | 15 | 20 | 25 | 30 | 35 |
Total in group/class | 7 | 14 | 21 | 28 | 35 | 42 | 49 |
The fraction of boys in each group: 5/7, 10/14, 15/21, 20/28, 25/35, 30/42, 35/49
These can be explored this way:
5/7 x 1/1 = 5/7
5/7 x 2/2 = 10/14,
5/7 x 3/3 = 15/21,
5/7 x 4/4 = 20/28,
5/7 x 5/5 = 25/35,
5/7 x 6/6 = 30/42,
5/7 x 7/7 = 35/49
Activity 4
Session 3
SLOs:
Activity 1
Activity 3
Mochaccino Mix
Millie and Maxwell are creating their own homemade ‘mochaccinos’ with scoops of mochaccino mix (MM) and scoops of milk (m).
Which cup has the stronger mochaccino flavour?
How do you know?
2:6 (MM:m) | 5:11 (MM:m) |
Millie | Maxwell |
Millie's | Maxwell's |
Activity 4
3:5 | 7:9 | 5:11 | 6:10 |
Filipo | Toni | Arapeta | Mona |
Activity 5
Session 4
SLOs:
Activity 1
Mochaccino Mix
Millie and Maxwell are creating their own homemade ‘mochaccinos’ with scoops of mochaccino mix (MM) and scoops of milk (m).
2:6 (MM:m) | 5:11 (MM:m) |
Millie | Maxwell |
Activity 2
Session 5
SLOs:
Activity 1
Activity 2
Distance (km) | Time (minutes) |
30 | 60 |
10 |
Activity 2
Dear families and whānau,
We have been working with rates and ratios in class. Your child would like to tell you about what they have been learning and invite you to solve with them and discuss these problems that they have worked on in class.
Perhaps you could also discuss and explore rates that you know and use in your own lives.
We trust that you find this both challenging and enjoyable.
We hope that you learn together and enjoy the mathematics.
Printed from https://nzmaths.co.nz/resource/ratios-and-rates at 2:21am on the 26th April 2024