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.
![](/sites/default/files/images/ratiosr1.png)
Links to the Number Framework
Stages 7- 8 (Advanced Multiplicative to Advanced Proportional)
Ratios and Rates
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.
![](/sites/default/files/images/ratiosr1.png)
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
1/4 or 25% of the class are absent. There are 18 students present.
How many students are in the whole class?
Highlight the fact that they were using their multiplicative thinking and looking for relationships between the numbers.
Look closely at the numbers of girls and boys on each card.
Can you group any of these cards together?
If so, can you explain why you have grouped them as you have?
Give students time to explore, discuss and group.
(a. 1:3, 2:6, 3:9 and 4:12, b. 2:3, 4:6, c. 1:2, 2:4, d. 4:4, 3:3. Two do not have an equivalent: 3:4 and 2:5)
Summarise student findings, accepting all ideas. Emphasise that students are looking at part-to-part ratios (e.g. the number of boys to the number of girls in a group of children). You might write this on the board to refer back to.
Activity 2
Ask a student to read one of the cards aloud.
Ask what we call this representation (a ratio) and ask for an explanation from the students.
Record student ideas on the class chart. Elicit key ideas:
A ratio is a relationship between two numbers of the same kind and is usually expressed as ‘a to b’ or ‘a : b’. This is saying that for every amount of one thing there is so much of another thing of the same kind.
A ratio is expressed using a colon. A ratio is expressed in the order the two amounts are presented:
For example: This shows one girl to three boys. We would write the ratio as 1:3, not as 3:1.
a. 1:3, 2:6, 3:9 and 4:12 (1:3)
On each of the cards demonstrate how the 1:3 ratio is evident in each.
In each case the number of girls is 1/3 of the number of boys.
Explore the paired equivalent cards, discuss, and have the students explain the number relationships for each pair.
b. 2:3, 4:6 (there are 2 girls for every 3 boys: the number of girls is 2/3 the number of boys)
c. 1:2, 2:4 (there is 1 girls for every 2 boys: the number of girls is 1/2 the number of boys)
d. 4:4, 3:3 (there is 1 girls for every 1 boy: the number of girls and boys is the same)
Emphasise that the ratio of 3:4 remains the same. In each case there are 3/4 as many girls as there are boys.
Have student pairs discuss and create a ratio table for 2:5 and ‘draw’ what the first three would look like using simple girl/boy ‘pictures (modeled on the cards of Copymaster 1).
Session 2
SLOs:
Activity 1
Distribute the shuffled picture cards only of Copymaster 1 (Class Ratios) to student pairs.
Point out that so far the class has explored the ratio between two parts of a group, the number of girls to the number of boys. Emphasise that this kind of ratio is known as a part to part ratio.
Activity 2
(3/7, 6/14, 9/21, 12/28, 15/35, 18/42, 21/49)
1/1, 2/2, 3/3, 4/4, 5/5, 6/6 and 7/7
Ask students what they notice about these fractions. Emphasise that they are all ways of writing 1.
3/7 x 2/2 = 6/14
3/7 x 3/3 = 9/21
3/7 x 4/4 = 12/28
3/7 x 5/5 = 15/35
3/7 x 6/6 = 18/42
3/7 x 7/7 = 21/49
Elicit from the students that all fractions are an expression of 3/7 or a 3:7 part to whole ratio.
Activity 3
They should:
1. Add a group total line and complete this.
2. Explore the fractional relationships as in Activity 2, Step 2 above between the boys and the whole group.
3. Confirm that the decimal fraction for each is the same.
4. State what percentage the boys are of the whole group.
For example:
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?
Answer:
They should include (some of) these ideas.
Millie: 2:6, MM is 2/8 (1/4) of the mix (this is the same as 4:16).
MM is 25% of the mix.
Maxwell 5:11, MM is 5/16 of the mix.
MM is 31.25% of the mix.
Maxwell’s mochaccino has a slightly stronger flavour. It has 1/16 (6.25%) more mochaccino mix.
You decide to make some mochaccinos using Millie’s recipe.
If you use 18 scoops of milk, how many scoops of MM will you use?
If you use 1 1/2 scoops of MM, how many scoops of milk will you use?
If Millie’s recipe makes a small drink for one person, what quantities will you use to make a small drink for eight people using Millie’s recipe?
You decide to make your own mix.
You use a 2:8 ratio in your first drink. What percentage is mochaccino mix?
Your second drink mix is stronger. You use a 3:7 ratio. What percentage is milk?
Your third drink mix is really thick. It is made with 40% mochaccino mix. What ratio did you use?
Answers:
6, 4 1/2 , 16MM:48m
20%, 70%, 4:6 or 2:3
Activity 4
Odds
It’s winter in the south. Four children are predicting the likelihood of school having to close early because of the snowfall that is predicted. They state their odds as Yes will close (Y) to No, won’t close (N).
Here are their odds. Y:N
Answers:
1. Toni.
2. Arapeta.
(Toni 7:9 is a 7/16 chance of closing. Filipo 3:5 is a 3/8 or 6/16 chance of closing. Mona 6:10 is a 6/16 chance of closing. Arapeta 5:11 is a 5/16 chance of closing.)
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).
(Answer. This is so with a single combination, or where the number of cups are the same for each)
Activity 2
(Answer: This is not correct. These represent different quantities. The combined ratio must be calculated from the combined quantities.
2:6 1 mochaccino made by Millie
10:22 2 mochaccinos made by Maxwell
12:28 the combined ratios of MM and m
6:14 or 3:7 the simplified combined ratio of MM and m)
Session 5
SLOs:
Activity 1
Eg. A ratio is a relationship between two amounts of the same kind (eg. people: girls to boys, drink mix, odds) usually expressed as ‘a to b’.
Elicit and record the difference and these key ideas about rates:
A rate is a special kind of ratio: a rate compares different kinds of measures such as dollars per kilogram.
A rate involves a multiplicative comparison and uses the word per and the symbol /.
dollars per hour (pay rate), metres per second or kilometres per hour (rate of speed), heartbeats per minute (pulse rate).
Activity 2
Record this in full: 30 kilometres per hour and beside this write the abbreviation.
Write on class chart:
This is equivalent to half a kilometre per minute or 30 km/hour. (30 kilometres/60 minutes)
How long to ride 60 km?
How long to ride 40 km?
How long to ride 15 km?
How far in 30 minutes?
How far in 45 minutes?
How far in 10 minutes?
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.
Ratios (Teaching unit)
Mixing Colours from Book 7: Teaching Fractions, Decimals and Percentages
Extending Mixing Colours from Book 7: Teaching Fractions, Decimals and Percentages
Rates of Change from Book 7: Teaching Fractions, Decimals and Percentages
Ratios with Whole Numbers from Book 8: Teaching Number Sense and Algebraic Thinking
Comparing by Finding Rates from Book 8: Teaching Number Sense and Algebraic Thinking
Sharing in Ratios from Book 8: Teaching Number Sense and Algebraic Thinking
Inverse Ratios from Book 8: Teaching Number Sense and Algebraic Thinking
Ratios and Rates
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.
![](/sites/default/files/images/ratiosr1.png)
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
1/4 or 25% of the class are absent. There are 18 students present.
How many students are in the whole class?
Highlight the fact that they were using their multiplicative thinking and looking for relationships between the numbers.
Look closely at the numbers of girls and boys on each card.
Can you group any of these cards together?
If so, can you explain why you have grouped them as you have?
Give students time to explore, discuss and group.
(a. 1:3, 2:6, 3:9 and 4:12, b. 2:3, 4:6, c. 1:2, 2:4, d. 4:4, 3:3. Two do not have an equivalent: 3:4 and 2:5)
Summarise student findings, accepting all ideas. Emphasise that students are looking at part-to-part ratios (e.g. the number of boys to the number of girls in a group of children). You might write this on the board to refer back to.
Activity 2
Ask a student to read one of the cards aloud.
Ask what we call this representation (a ratio) and ask for an explanation from the students.
Record student ideas on the class chart. Elicit key ideas:
A ratio is a relationship between two numbers of the same kind and is usually expressed as ‘a to b’ or ‘a : b’. This is saying that for every amount of one thing there is so much of another thing of the same kind.
A ratio is expressed using a colon. A ratio is expressed in the order the two amounts are presented:
For example: This shows one girl to three boys. We would write the ratio as 1:3, not as 3:1.
a. 1:3, 2:6, 3:9 and 4:12 (1:3)
On each of the cards demonstrate how the 1:3 ratio is evident in each.
In each case the number of girls is 1/3 of the number of boys.
Explore the paired equivalent cards, discuss, and have the students explain the number relationships for each pair.
b. 2:3, 4:6 (there are 2 girls for every 3 boys: the number of girls is 2/3 the number of boys)
c. 1:2, 2:4 (there is 1 girls for every 2 boys: the number of girls is 1/2 the number of boys)
d. 4:4, 3:3 (there is 1 girls for every 1 boy: the number of girls and boys is the same)
Emphasise that the ratio of 3:4 remains the same. In each case there are 3/4 as many girls as there are boys.
Have student pairs discuss and create a ratio table for 2:5 and ‘draw’ what the first three would look like using simple girl/boy ‘pictures (modeled on the cards of Copymaster 1).
Session 2
SLOs:
Activity 1
Distribute the shuffled picture cards only of Copymaster 1 (Class Ratios) to student pairs.
Point out that so far the class has explored the ratio between two parts of a group, the number of girls to the number of boys. Emphasise that this kind of ratio is known as a part to part ratio.
Activity 2
(3/7, 6/14, 9/21, 12/28, 15/35, 18/42, 21/49)
1/1, 2/2, 3/3, 4/4, 5/5, 6/6 and 7/7
Ask students what they notice about these fractions. Emphasise that they are all ways of writing 1.
3/7 x 2/2 = 6/14
3/7 x 3/3 = 9/21
3/7 x 4/4 = 12/28
3/7 x 5/5 = 15/35
3/7 x 6/6 = 18/42
3/7 x 7/7 = 21/49
Elicit from the students that all fractions are an expression of 3/7 or a 3:7 part to whole ratio.
Activity 3
They should:
1. Add a group total line and complete this.
2. Explore the fractional relationships as in Activity 2, Step 2 above between the boys and the whole group.
3. Confirm that the decimal fraction for each is the same.
4. State what percentage the boys are of the whole group.
For example:
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?
Answer:
They should include (some of) these ideas.
Millie: 2:6, MM is 2/8 (1/4) of the mix (this is the same as 4:16).
MM is 25% of the mix.
Maxwell 5:11, MM is 5/16 of the mix.
MM is 31.25% of the mix.
Maxwell’s mochaccino has a slightly stronger flavour. It has 1/16 (6.25%) more mochaccino mix.
You decide to make some mochaccinos using Millie’s recipe.
If you use 18 scoops of milk, how many scoops of MM will you use?
If you use 1 1/2 scoops of MM, how many scoops of milk will you use?
If Millie’s recipe makes a small drink for one person, what quantities will you use to make a small drink for eight people using Millie’s recipe?
You decide to make your own mix.
You use a 2:8 ratio in your first drink. What percentage is mochaccino mix?
Your second drink mix is stronger. You use a 3:7 ratio. What percentage is milk?
Your third drink mix is really thick. It is made with 40% mochaccino mix. What ratio did you use?
Answers:
6, 4 1/2 , 16MM:48m
20%, 70%, 4:6 or 2:3
Activity 4
Odds
It’s winter in the south. Four children are predicting the likelihood of school having to close early because of the snowfall that is predicted. They state their odds as Yes will close (Y) to No, won’t close (N).
Here are their odds. Y:N
Answers:
1. Toni.
2. Arapeta.
(Toni 7:9 is a 7/16 chance of closing. Filipo 3:5 is a 3/8 or 6/16 chance of closing. Mona 6:10 is a 6/16 chance of closing. Arapeta 5:11 is a 5/16 chance of closing.)
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).
(Answer. This is so with a single combination, or where the number of cups are the same for each)
Activity 2
(Answer: This is not correct. These represent different quantities. The combined ratio must be calculated from the combined quantities.
2:6 1 mochaccino made by Millie
10:22 2 mochaccinos made by Maxwell
12:28 the combined ratios of MM and m
6:14 or 3:7 the simplified combined ratio of MM and m)
Session 5
SLOs:
Activity 1
Eg. A ratio is a relationship between two amounts of the same kind (eg. people: girls to boys, drink mix, odds) usually expressed as ‘a to b’.
Elicit and record the difference and these key ideas about rates:
A rate is a special kind of ratio: a rate compares different kinds of measures such as dollars per kilogram.
A rate involves a multiplicative comparison and uses the word per and the symbol /.
dollars per hour (pay rate), metres per second or kilometres per hour (rate of speed), heartbeats per minute (pulse rate).
Activity 2
Record this in full: 30 kilometres per hour and beside this write the abbreviation.
Write on class chart:
This is equivalent to half a kilometre per minute or 30 km/hour. (30 kilometres/60 minutes)
How long to ride 60 km?
How long to ride 40 km?
How long to ride 15 km?
How far in 30 minutes?
How far in 45 minutes?
How far in 10 minutes?
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.
Ratios (Teaching unit)
Mixing Colours from Book 7: Teaching Fractions, Decimals and Percentages
Extending Mixing Colours from Book 7: Teaching Fractions, Decimals and Percentages
Rates of Change from Book 7: Teaching Fractions, Decimals and Percentages
Ratios with Whole Numbers from Book 8: Teaching Number Sense and Algebraic Thinking
Comparing by Finding Rates from Book 8: Teaching Number Sense and Algebraic Thinking
Sharing in Ratios from Book 8: Teaching Number Sense and Algebraic Thinking
Inverse Ratios from Book 8: Teaching Number Sense and Algebraic Thinking
Chocolate choices
This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.
A PDF of the student activity is included.
Click on the image to enlarge it. Click again to close. Download PDF (1045 KB)
solve ratio problems
compare ratios
Number Framework Links
Use this activity to help students learn to express fractions as ratios and to express ratios in their simplest form by dividing by common factors (stages 6 and 7).
Counters or cubes
Question 1 involves straightforward proportional thinking and can be solved either by counting chocolates and comparing numbers or by making connections between the sizes of the boxes. In question 1a, the students might notice that if they turned box A 90 degrees, it would fit twice into box B, so 2A = B. Box B would fit neatly over the top of box C, leaving 12 chocolates uncovered at the bottom, which is the number of chocolates in box A, so C = B + A or C = 3A. Once they know the relationship between the sizes of the boxes, it’s easy for them to determine the weight
and cost of each.
Question 2 requires the students to work out Tania’s best option. This can be done by counting the strawberry hearts for each combination of boxes that will make up 1 kg. Box A gives Tania 3:9 (or 3/12), box B gives her 8:16 (or 8/24), and box C gives her 13:23 (or 13/36).
If your students have had little experience working with ratios, you will need to show them carefully how the ratio 3:9 (favourites to non-favourites) can equally well be expressed as the fraction 3/12 (favourites out of total number of chocolates in the box).
There are four combinations of boxes that will make up 1 kg: 4 of box A (4A), 2 of box A and 1 of box B (2A + B), 2 of box B (2B), or 1 of box A plus 1 of box C (A + C). The last two of these options will give Tania the same number of strawberry hearts: 16. Expressed as a ratio of favourites to non-favourites, this is 16:32 or 1:2.
Give your students time to explain their strategies. Make sure they understand that the ratios 16:32 and 1:2 are identical. They may think that the ratio 16:32 contains within itself the number of chocolates and 1:2 doesn’t. This is a misconception. A ratio never means anything without its context. To make sense of the 16:32 (or 1:2), we need to know the “story” that goes with the numbers: in this case, the strawberry hearts and the number of chocolates in the boxes.
Question 3 doesn’t introduce any new information or processes but gives students a chance to apply and consolidate the ideas they have met in question 2. They may find it useful to collate the information and answers for both questions in a table.
In question 4, the students need to work out which combination of boxes would give each of the three friends the same number of favourite chocolates. Encourage them to make use of the work they have already done. There are only four combinations to be considered, and they know from question 2 that both 2B and A + C give Tania 16 of her favourites. The only remaining task is to find which of these two combinations also gives Atama and Chloe 16 of their favourites.
Answers to Activity
1. a. 500 g and 750 g
b. Box A (250 g) should cost $4.30.
Box B (500 g) should cost $8.60.
2. a. There are two possibilities: 2B or A + C. Both would give Tania 16 strawberry hearts.
b. The ratio of favourites to non-favourites would be 16:32 = 1:2.
3. Atama would get most caramel circles (18) by choosing 2 x B. The ratio of favourites to non favourites would be 18:30 = 3:5.
Chloe would get most peppermint squares (20) by choosing 4 x A. The ratio of favourites to non favourites would be 20:28 = 5:7.
4. If they buy boxes A and C, each person will get 16 of their favourite kind.