Mixing Paint and Other Problems

Purpose

In this unit we look at ratio in the context of objects, physical space, geometric quantities and mixable things such as paint.

Achievement Objectives
NA4-4: Apply simple linear proportions, including ordering fractions.
Specific Learning Outcomes
  • Given a ratio, find relative numbers of objects, lengths of sides.
  • Given a ratio, find the relevant fractions.
  • Given fractions making a whole, find the relevant ratio.
  • Find ratios between three objects.
Description of Mathematics

Ratios, proportion, fractions are all related items that come up in both real life and mathematical situations. A ratio is a relationship between measures. For example, green paint might be mixed in a ratio of 1:3 meaning one measure of blue paint is combined with three measures of yellow paint. If the ratio is applied consistently then different quantities of paint can be made of the same colour. A four litre mixture will contain one litre of blue and three litres of yellow but a 1 litre mixture will contain 250mL of blue and 750mL of yellow.

Ratio is a particularly valuable concept in the context of scale drawings. Here the ratio of the drawing to the actual object gives the idea of the relative size of the drawing to the object.

In mathematics, ratios and proportions are fundamental to trigonometry, coordinate geometry and calculus.

The unit This is to That, Number, Level 4, gives an introduction about ratios and relates them to proportion and fractions. You may want to provide those experiences prior to undertaking this unit.  

Opportunities for Adaptation and Differentiation

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:

  • Beginning with simple ratios, e.g., 1:2, 1:3, and 2:3, and using objects like connecting cubes to model the ratios.
  • Using symbols for ratios and fractions alongside physical and diagrammatic models.
  • Asking predictive questions to encourage students to think beyond what is visible, e.g., If you make three copies of this ratio, 2:3, what ratio do you have in total?
  • Developing students’ ability to find common factors when learning basic multiplication and division facts, e.g., Simplify the ratio 20:25.
  • Progressing to working with the symbols alone, with the option of folding back to materials.

The contexts for this unit are mixing substances and scaling to make models. The unit uses contexts like scale models, mixing fruit drink, and making fertiliser for the garden. These contexts might be supplemented by others that are of special interest to your students. Examples might include ratios in cooking, ratios of positions in sports teams or genders in a class, or in the dimensions of human faces.

Required Resource Materials
  • Copymasters One, Two, Three, Four, Five and Six
  • Geometric instruments: ruler, compasses, protractor, a tape measure several metres long
Activity

Session 1

In this session, students are reintroduced to ratios.

  1. Start with a whole class quick fire quiz about ratios in which students work in pairs or threes. At the end of each question, process students’ answers and decide collectively on the correct answer. Suggested questions include (use the names of people in the class):
    • I have twice as many eggs as you. You have 6 eggs. How many do I have?
    • Jane has three times as many dogs as Bill. Bill has 4 dogs. How many dogs does Jane have?
    • Bob has five times as many brazil nuts as Jenny. Jenny has four brazil nuts. How many brazil nuts does Bob have?
    • I have twice as many eggs as you. I have 10 eggs. How many do you have?
    • Jane has three times as many toy cars as Bill. Jane has 15 toy cars. How many cars does Bill have?
    • Bob has five times as many strawberries as Jenny. Bob has 30 strawberries. How many strawberries does Jenny have?
    • I have twice as many eggs as you. What is the ratio of the number of my eggs to the number of your eggs?
    • Jane has three times as many toy cars as Bill. What is the ratio of the number of Jane’s toy cars to the number of Bill’s cars?
    • Bob has five times as many strawberries as Jenny. What is the ratio of the number of Bob’s strawberries to the number of Jenny’s strawberries?
    • If the ratio of my eggs to your eggs is 2:1, what fraction of our eggs do you have? What fraction do I have?
    • If the ratio of Jane’s toy cars to Bill’s is 3:1, what fraction of their cars does Jane own? What fraction does Bill own?
    • If the ratio of Bob’s strawberries to Jenny’s is 5:1, what fraction of the strawberries does Jenny have? What fraction does Bob have?
  2. Discuss the meaning of ratios like 2:3 and 1:5. For example, 2:3 means two units of one measure for every three units of the other measure. In ratios the units are the same. If the units are different that is called a rate, such as two windows painted for every three hours or two kilometres in three hours (snail’s pace).
  3. Break the class up into groups with no more than four in a group. Give each group Copymaster 1. Ask students to make up their own 20 questions along the lines of the questions in 1 above. Ask students to record their own questions on the Copymaster, and their answers on the back..
  4. Spend time sharing the problems. Students from different groups could be paired up to share their favourite problems. Each group might share their favourite problem with the class.

Session 2

In this session, we consolidate ideas relating to ratio.

  1. Divide the session up into three parts and get the class to attempt the problems in Copymaster 2Copymaster 3, and Copymaster 4 in each part. Students might work individually or in groups.

Session 3

In this session students draw diagrams about ratios.

  1. Try to get hold of some scale models. Talk about the items and the way that ratio/scale is used for that object. Model cars or aeroplanes are a good example as the scale is usually displayed on the box. This box shows a scale of 1:43 which means a 1cm length on the model represents a 43cm length on the actual truck.
    truck
  2. Show students a metre ruler. Get different students to answer the following questions
    How long would a stick be it the ratio of the ruler to the stick was 2:1; 5:1; 10:1; 1:2; 1:5; 1:10? For example, a 2:1 ratio would make the stick 50cm since every two centimetres on the ruler is represented by 1cm on the stick.
  3. You might create a length of 25cm out of a long straw.
    What is the length ratio of the metre ruler and the straw?
    How might we figure that out?
    Noting that the straw fits into the ruler four times shows that four units on the rule match with one unit on the straw. The ratio of ruler to straw is 4:1.
  4. Get students to work in pairs on Copymaster 5. The straw task will support students to understand what is required. Physical lengths made from 1 metre-long straws (available at dollar shops) will be helpful to some students. For example, compare strips that are 40cm and 100cm long.
    How many times does 40cm fit into 100cm? (2 ½ times)
    Each 10cm on the short straw is matched by how many centimetres on the metre straw? (25cm by finding quarters)
    Is there a simpler way to record the ratio 10:25? (2:5 using a common factor of five)
  5. You might progress to using ratios to compare areas. Ratios are used in standard paper sizes such as A3, A4, and A5. Four A5 sheets fit into an A3 sheet so the ratio of area is 1:4.
    paper
    Area is a more complex attribute than length so stay with rectangles and areas that map tidily into one another.
  6. Cut an A4 sheet in half, length-wise, to make an A5 sheet (coloured paper is good). Locate the A5 sheet in the corner of the A3 sheet.
    What is the ratio of areas for these two sheets of paper?
    Since the A5 sized sheet fits four times into the A3 sheet, the ratio of areas is 1:4 (meaning 1 cm2 of the A5 corresponds to 4cm of the A3 sheet.
  7. Ask students to create two rectangles in the area ratio of 1:2. That will mean the small sheet is half the area of the large sheet.
  8. Progress to 1:5 then 2:3 to see if students can transfer the ratio concept.

Session 4

In this session we draw a physical space to scale.

  1. Use maps with various scales. You might print off maps of parts of the local area using the Zoom function on Google Maps. Discuss the scale. Use questions like:
    I want to walk from the school gate to the town library. How far will it be? How long will it talk to walk that distance? How do you know?
    I want to ride from Taupō to Wairakei. How far is that? How long will the ride take?
  2. Talk about drawing the school campus to scale (or use the classroom or some other physical space). Have a discussion with the whole class
    What would be a useful scale to use?
    How would you decide that? 
    (size of paper; size of campus)
  3. Google maps will provide a very accurate bird’s eye view of the school. Use a printed map to create a treasure hunt. Each group needs a metre ruler, trundle wheel, or tape measure and some treasure to hide (Crunchie Bars are good).
  4. Let the groups go outside and hide their treasure. They must use the measuring instruments to accurately mark the location of their treasure on the map. Locations should be some distance from buildings.
  5. When the groups return get them to exchange maps and try to find the treasure as marked on the map.
  6. After the treasure is found gather the class.
    How is the scale on the map an example of a ratio?
    How did you find the location of the treasure?
    How was the scale useful?

Session 5

In this session students consider equivalent ratios in the context of mixing paint and other things.

  1. Pose this problem:
    Henry mixes orange juice to water in a 1 litre jug in the ratio 3:1.
    Reina mixes 9 litres of orange juice with 3 litres of water in a big container.
    Which mixture will taste the most strongly of orange?
  2. Model the situation with orange and white cubes or draw a diagram. You might start with a model of Reina’s mixture, using each cube to represent 1 litre of liquid.
    cubes
  3. Discuss the fractions in the mixture, 9/12 orange and 3/12 water.
    Is there a simpler way to record these fractions?
    Students might notice that 9/12 = ¾ and 3/12 = ¼.
    Show how the fractions can be found by re-unitising the cubes into parts of 3 litres.
    cubes
    Try to show Henry’s mixture now. What does that look like?
  4. If students are consistent with the model, they will use one cube to represent 1 litre and sub-divide that cube into four parts to show 3:1
    cubes
    What fractions are in this ratio?
    Students might notice the ¾ orange and ¼ water are also present.
    What do the fractions tell us about taste?
    You may need to mix different amounts of orange juice and water in the same ratio to convince students that taste is conserved if ratio is kept constant.
  5. Present similar problems, such as:
    Koru mixes liquid seaweed to water in 5 litre container. He uses a ratio of 3:7.
    Israel mixes 150 mL of liquid seaweed with 350mL of water to fertilise his tomato plants.
    Which mixture will have the strongest smell of seaweed?
    Model the problems using cubes or diagrams.
  6. Let students work on the problems in Copymaster 6. Use collaborative grouping so students can share ideas. Roam the room as students work. Look for:
    • Do they know how to record ratios and explain their meaning?
    • Do they recognise when ratios are the same, such as 1:3 and 100:300?
    • Do they understand standard units of measurement and how those unit connect, e.g., 1000ml = 1L?
    • Do they recognise that when ratios are not equal, that signifies a difference in a characteristic or attribute, like colour, taste or strength?
  7. Gather the class and discuss answers, highlighting the points above.
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Level Four