Ratio in the Real World

RatioRatio inin thethe RealReal WorldWorld

Task Description The students explore which ratio produces a stronger cranberry taste between a 4:3 or 3:2 mix of cranberry juice to apple juice. The students use various strategies to answer this question and apply these to a second task that asks, ‘Which rectangle is ‘more square’ a rectangle of 35 x 39 or one of 22 x 25?’ Length of Task 60 - 80 minutes Materials  Calculator Using the Activity Introductory The teacher poses the following problem to the class.

Eva Brick makes and sells her own cranberry-apple juice. In jug A, she mixed 4 Evacranberry Brick makesflavoured and cubes sells herand own 3 apple cranberry-apple flavoured cubes juice. with In jugsome A, water.she mixed In 4 jugcranberry B, she mixed flavoured 3 cranberry cubes and 23 apple flavoured cubes inwith the some same water. amount In jug B, she mixed 3 cranberry andof 2 water.apple flavoured cubes in the same amount If you ask for a drink that has the strongerof water. cranberry taste, from which jug If you ask for ashould drink shethat pour has theyour stronger drink? Please cranberry explain. taste, from which jug should she pour your drink? Please explain.

The teacher encourages the students to work on the problem individually or talk it through with a partner. The teacher shares that there multiple possible approaches for answering this question.

Whole class: The teacher draws the students together to share their approaches and responses to the question. Some students may have commenced with changing the ratios into equivalent fractions 4/3 and 3/2 and then calculated the equivalent decimal 1.33333 and 1.5 respectively. The larger response indicates the greater level of cranberry juice to apple juice. Therefore the ratio of 3:2 is stronger.

Another possible approach is converting the ratios to fractions and finding the common denominator for the fractions. So 4/3 and 3/2 would become 8/6 and 9/6. Therefore with 6 equal parts of apple juice in each mixture there would be 8 parts and 9 parts of cranberry respectively, making the second mixture stronger.

This activity is intended for use by teachers for research purposes only, as part of the Task Types and Mathematics Learning (TTML) project at Monash University. No authority is granted for persons to use these activities beyond the scope of this project, without express permission of the TTML Project Leader. RatioRatio inin thethe RealReal WorldWorld

Main Activity Building on the strategies used in the first problem the teacher poses the following problem to the students.

Which rectangle is ‘more square’ a rectangle of 35 x 39 or one of 22 x 25? Which rectangle is ‘more square’ a rectangle of 35 x 39 or one of 22 x 25?

The students are seeking a rectangle that is closest to a 1:1 ratio.

A common response to this task is for students to convert these figures into fractions 35/39 and 22/25 and then calculate the fractions into equivalent decimals 0.89 and .088. The decimal closer to 1 is the ‘more square’ rectangle, 35 x 39 is the correct response.

Alternatively, the students may reverse the fractions to 39/35 and 25/22 then convert to decimals 1.11 and 1.13. Again the number closest to 1 is correct.

Whole class: The teacher encourages students to share any difficulties they have encountered with the task and their methods for overcoming them. A discussion of incorrect responses or misconceptions will assist in developing the students’ understanding of ratio. The teacher invites the students to share any similarities or differences they found with the two problems. Key Mathematical Concepts  Comparison of ratios through conversion to fractions and decimals. Prerequisite Knowledge  Understanding of the relationship between ratios, fractions, decimals and percentages. Links to VELS

Dimension Standard Number (Level 4) Students use decimals, ratios and percentages to find equivalent representations of common fractions (for example, 3/4 = 9/12 = 0.75 = 75% =3 : 4 = 6 : 8). Working mathematically Students use the mathematical structure of problems to (Level 4) choose strategies for solutions. They explain their reasoning and procedures and interpret solutions. They create new problems based on familiar problem structures. Working mathematically Students engage in investigations involving mathematical (Level 4) modelling. They use calculators and computers to investigate and implement algorithms, explore number facts and puzzles, generate simulations, and transform shapes and solids.

Assessment To be working at Level 4, students should be able to:  Develop an appropriate strategy for comparing ratios.  Use ratio to compare the relationship between quantities.  Use ratios to find equivalent representations of common fractions. Extension Suggestions For students who would benefit from additional challenges:  The IXL website has a range of comparison word problems that some students may wish to explore. Students select the multiple choice responses and the site provides feedback if the answer is incorrect. http://www.ixl.com/math/practice/grade-6- compare-percents-and-fractions-word-problems Word problems from this site may be extended for whole class problems. Teacher Advice and Feedback Some of the students attempted to draw both rectangles to make the comparison. It was noted that to fit the rectangles on a piece of paper students were reducing the ratio terms by the same amount and therefore not keeping these numbers in proportion to the original ratio. For example some students reduced 35:39 by 3 for each term to 32:36. This method produces rectangles of different ratios. The teacher may illustrate the impact of reducing terms in this manner by referring the students to the ratio table in the ‘Making cordial’ task and asking what the effect this would have on the cordial and ratio.

Many students found the second task challenging and did not transfer their knowledge of the strategies used in the first task to the second task without prompting from the teacher. Potential Student Difficulties In the case of the cranberry-apple juice task some students developed an appropriate strategy for approaching the task however had difficulties in interpreting their results. They were unsure whether a larger decimal or one closer to 1 was the best result. Students may benefit from creating simplified ratios of cranberry to apple juice that they know in advance gives the higher concentrate of cranberry. For example without converting the ratios 5:1 compared to 2:1 to fractions or decimals most students would understand that a 5:1 ratio gives a stronger cranberry taste. The students can reproduce the steps to the task again using these new simple ratios to assist in determining the correct response.

Students who are experiencing difficulties may be given alternative ratios to compare with two equal terms. For example the problem below offers equal parts of apple juice in each jug (3:1 and 2:1).

Eva Brick makes and sells her own cranberry-apple juice. In jug A, she mixed 3 Evacranberry Brick makesflavoured and cubes sells herand own 1 apple cranberry-apple flavoured cubes juice. with In jugsome A, water.she mixed In 3 jugcranberry B, she used flavoured 2 cranberry cubes and and 1 1 apple apple flavoured flavoured cubes cubes in with the somesame water.amount In jug B, she used 2 cranberry and of1 applewater. flavoured cubes in the same amount If you ask for a drink that has the strongerof water. cranberry taste, from which jug If you ask for ashould drink shethat pour has theyour stronger drink? Pleasecranberry explain. taste, from which jug should she pour your drink? Please explain.

This slightly more difficult question offers equal parts of cranberry juice to unequal parts of apple juice (5:3 and 5:4).

Eva Brick makes and sells her own cranberry-apple juice. In jug A, she mixed 5 Evacranberry Brick makesflavoured and cubes sells herand own 3 apple cranberry-apple flavoured cubes juice. with In jugsome A, water.she mixed In 5 jugcranberry B, she used flavoured 5 cranberry cubes and and 4 3 apple apple flavoured flavoured cubes cubes in with the somesame water.amount In jug B, she used 5 cranberry and of4 applewater. flavoured cubes in the same amount If you ask for a drink that has the strongerof water. cranberry taste, from which jug If you ask for ashould drink shethat pour has theyour stronger drink? Pleasecranberry explain. taste, from which jug should she pour your drink? Please explain.

References / Acknowledgements Thank you to the teachers and students from Lloyd Street PS, for providing valuable feedback on the use of this activity.

Student work samples

Example 1: Working at Level 3-4

When comparing the proportions of two ratios, this student appears to hold the misconception that if the difference between the two terms of each ratio is the same then the ratio is the same proportion. For example the difference between the ratio of 4:3 and 3:2 is one part in each. 4 parts subtract 3 parts is 1 part. 3 parts subtract 2 parts is also 1 part. However, this student was able to represent both ratios as an equivalent decimal correctly. The student makes a real-life observation that although you may want to drink the juice from the 3:2 jug you probably won’t be able to taste the difference between the mixtures.

Example 2: Working at Level 4

This student converted the ratios to fractions and calculated the result as a decimal. The student accurately determined that the number closest to 1 is closest to the 1:1 ratio of a square.

This activity is intended for use by teachers for research purposes only, as part of the Task Types and Mathematics Learning (TTML) project at Monash University. No authority is granted for persons to use these activities beyond the scope of this project, without express permission of the TTML Project Leader.