ESM410 Assignment 1: Problem Pictures Task - Creating open-ended questions

Student Name: Kate Borninkhof

Student Number: 212072605

Campus: Burwood

PLAGIARISM AND COLLUSION Plagiarism occurs when a student passes off as the student’s own work, or copies without acknowledgement as to its authorship, the work of any other person. Collusion occurs when a student obtains the agreement of another person for a fraudulent purpose with the intent of obtaining an advantage in submitting an assignment or other work. Work submitted may be reproduced and/or communicated for the purpose of detecting plagiarism and collusion.

DECLARATION I certify that the attached work is entirely my own (or where submitted to meet the requirements of an approved group assignment is the work of the group), except where material quoted or paraphrased is acknowledged in the text. I also certify that it has not been submitted for assessment in any other unit or course.

SIGNED: K.Borninkhof DATE: 24/08/2015

An assignment will not be accepted for assessment if the declaration appearing above has not been signed by the author.

YOU ARE ADVISED TO RETAIN A COPY OF YOUR WORK UNTIL THE ORIGINAL HAS BEEN ASSESSED AND RETURNED TO YOU.

Assessor’s Comments: Your comments and grade will be recorded on the essay itself. Please ensure your name appears at the top right hand side of each page of your essay.

Checklist

All points must be ticked that they are completed before submission.

Requirements checklist: Tick completed

The rationale addressed the rationale prompts in the assignment description. √

The rationale included relevant citations/references – which are stated. √

Created 3 quality problem picture photos. √

The photos MUST be original photos taken by yourself. √

Location of photos are stated, e.g. Taken at Deakin foreshore. √

Developed an original question for each photo with an accompanying enabling and extending prompt.

√

If your photo has numbers that you are referring to in the problem, the numbers MUST be clearly visible to be able to read in the photo.

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Open-ended questions are creative and engaging. √

Matched each problem with the appropriate mathematical content, year, definition and code from the Australian Curriculum: Mathematics

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Each question is accompanied by three possible correct responses. √

Cross-curriculum links are made to each photo. √

Reflecting on the trialling of the questions with an appropriately aged child or children. √

The trialling reflection included relevant citations/references – which are stated. √

There is evidence of reference to problem-picture unit materials. √

Problem pictures were collated into a word document using the assignment template. √

File size of the word document is under 4mb. √

Assignment is uploaded to the Cloud Deakin dropbox. √

In order to pass this assignment you must have fulfilled all aspects of the checklist.

Rationale for the use of problem pictures in the classroom

Within a mathematics classroom, effective learning strategies need to be employed by the teacher for their students to experience the highest level of meaningful learning. Effective questioning techniques, which are known as open-ended questions, are particular questions that can be solved or explained in a variety of ways and that have the potential to expose students’ understandings and misconceptions. (Sanchez, 2013)

One of the aims of The Victorian Curriculum and Assessment Authority (2015) within Mathematics is to “develop an increasingly sophisticated understanding of mathematical concepts and fluency with processes, and are able to pose and solve problems and reason in Number and Algebra, Measurement and Geometry, and Statistics and Probability.” This aim can be fulfilled through using open-ended questions, as the students will be able to make stronger connections between school mathematics and the real world by considering why and how mathematical ideas are powerful. (Ainley, 2012) Mathematics is a domain that requires the teaching to be engaging and relevant to form meaningful learning and high-level questions asked by the teacher are likely to improve students' comprehension. (Shilo, 2015)

Open-ended problem pictures are photographs of an object, scene or activity that are accompanied by one or more open-ended mathematical word problems based on the context of the photo. (Bragg & Nicol, 2011) The purpose of these problem pictures is to give students the opportunity to explore varied strategic approaches and encourages them to think flexibly about mathematics. (Bragg & Nicol, 2011) Children learn to solve problems by using previously observed strategies or by gaining insights from visual displays such as symbols and pictures. (Zhe, 2003) Considering that children are already developing and employing these strategies, encouraging the use of them in a mathematics classroom is a truly effective way of helping to build students problem solving skills, critical thinking and comprehension.

Open-ended problem pictures are as beneficial for teachers as they are for students, as they give opportunities for educators to develop their critical mathematical thinking as well. (Bragg & Nicol, 2011) As teachers are the ones that generally design the questions, it allows them to build on their professional development and help give a clearer understand of how their students are learning.

References for the rationale:

Ainley, 2012 cited in: Day, L 2014, 'Purposeful Statistical Investigations', Australian Primary Mathematics Classroom, 19, 3, pp. 20-26

Bragg, L, & Nicol, C 2011, ‘Seeing mathematics through a new lens’, Australian Mathematics Teacher, 67, 3, pp. 3-9,

Sanchez, W.B, 2013, ‘Open-ended questions and the process standards', Mathematics Teacher, 107, 3, pp. 206-21

Shilo, G 2015, 'Formulating Good Open-Ended Questions in Assessment', Educational Research Quarterly, 38, 4, pp. 3-30

Victorian Curriculum and Assessment Authority, 2015. AusVELS Mathematics: Rationale and Aims. Retrieved from <http://ausvels.vcaa.vic.edu.au/Mathematics/Overview/Rationale-and-Aims>

Zhe, C 2003, 'Worth One Thousand Words: Children's Use of Pictures in Analogical Problem Solving', Journal Of Cognition & Development, 4, 4, pp. 415-434

Problem Picture 1

Location: Backyard garden in Lilydale

Problem Picture 1 - Questions

Grade level: Grade Six

Question 1This table has a perimeter of 10m (width: 1.5m, length: 3.5m). Find three different tables settings that have a perimeter of 10m and draw them. Explain your answers.

Answers to Question 1

AusVELS - Measurement and GeometryContent strand/s, year, definition and code Using units of measurement. Level Six. “Solve problems involving the comparison of lengths and areas using appropriate units” (ACMMG137)

Enabling PromptMy table has a length of 7m and a width of 3m, with a perimeter of 20m. It is in the shape of a rectangle. Come up with three different rectangle shaped tables and their widths and lengths that will equal a perimeter of 40m.

Answers to Enabling Prompt

AusVELS Content strand/s, year, definition and code Using units of measurement. Level Five. “Calculate the perimeter and area of rectangles using familiar metric units” (ACMMG109)

Justification for change to the original questionIn this question, I changed the format to find three tables, focusing on them being rectangles rather than just equalling a 40m perimeter. Changing it to this, changes the direction of the question to be more about using their times tables to come up with different widths and lengths to all equal a 40m perimeter. The challenge that remains is making sure either the lengths or widths are longer than the other.

Extending PromptIf one person takes up a space of 1m on the table, create three different tables with an area of 30m2 and find the number of people that can fit on it.

Answers to Extending Prompt

AusVELSContent strand/s, year, definition and code Using units of measurement. Level six. “Solve problems involving the comparison of lengths and areas using appropriate units” (ACMMG137)

Justification for change to the original questionTo make this question slightly harder, although still keeping in line with a level six AusVELS standard, I incorporated area as well as perimeter and having to find how many people could fit. To continue extending this question, you could change the amount of space one person takes up so that it doesn’t equal the same as the perimeter and they actually have to spend more time thinking about it and working it out.

Cross-Curriculum LinksCreate an imaginary text using the photograph as your basis for the setting of your story.

AusVELS - Cross-curriculum English, Writing (Literacy), Level 6, definition and code Plan, draft and publish imaginative, informative and persuasive texts, choosing and experimenting with text structures, language features, images and digital resources appropriate to purpose and audience (ACELY1714)

Report of Trialling Problem Picture 1 Child’s pseudonym, age and grade level: “Sally”, Age 12, Grade 6

Original Question: This table has a perimeter of 10m (width: 1.5m, length: 3.5m). Find three different tables settings that have a perimeter of 10m and draw them.

Child’s response to the question:

Reflection on child’s response:The question presented to my student was to find three different table settings with a perimeter of 10m and to display this information. The question was not read out to the student as she is at level six and was able to read it to herself. Sally approached this question really enthusiastically and exactly as I’d intended her to. She initially sketched out the worded question example of a table being 3.5m long and 1.5m wide. When I asked her why she had done this, she explained that it helped her to understand better how she had to answer the question.

I gave Sally roughly five minutes to come up with her answers and she didn’t ask for any help or assistance during this time. After finishing three new designs with a 10m perimeter, I asked her to go through her answers and explain them to me. She started off by telling me what perimeter was (length + length + width + width) and that she had to find two numbers that she could times by 2 that would equal 10. So in light of her thinking in this way, she would come up with equations (see above) that would give her a perimeter of 10. “It is important to include an instruction such as the following to the end of each open-ended problem photo: explain or illustrate your response. This prevents students from simply responding in an ad hoc manner with little consideration of the complexity of the problem.” (Bragg & Nicol, 2011)

In her third attempt, it is seen that she started with 1.5 x 1.5, which equalled 3, but she quickly realised that that was the example question so she modified it to be 2 x 2 instead. It is obvious that Sally has strengths in her four operations and even though being an ESL student, she was able to easily comprehend the question and proceed by answering correctly.

The AusVELS component of this question: “Solve problems involving the comparison of lengths and areas using appropriate units” (2015), was directly addressed in the question as it involved comparing and finding lengths to suit a number of different problems. Area was referred to in my extending prompt.

If I were to modify this question, disregarding my enabling and extending prompts, I would make it slightly more difficult as it seemed to be not much of a challenge for Sally. I could have trialled the question on more students to gain a better idea of how it was approached by varied development levels but due to circumstances and timing, I was only able to test it on one child.

References for reflection on the trial of question 1:

Bragg, L, & Nicol, C 2011, ‘Seeing mathematics through a new lens’, Australian Mathematics Teacher, 67, 3, pp. 3-9,

Victorian Curriculum and Assessment Authority, 2015. AusVELS: Mathematics. Retrieved from <http://ausvels.vcaa.vic.edu.au/Mathematics/Curriculum/F-10>

Problem Picture 2

Location: Kitchen at home

Problem Picture 2 - Questions

Grade level: Grade Six

Question 2Create a worded problem using the photograph of the fruit in the bowl and come up with different ways that probabilities can be used. (There are 14 pieces of fruit in the bowl: 2 lemons, 3 pears and 8 oranges.)

Answers to Question 21. Jennifer’s mum asked her if she would like a piece of fruit. Without asking which type she’d prefer, what is

the probability that she would be given an orange? Fraction: 8/14. = 4/72. John only eats pears. Out of all of 14 pieces of fruit in the bowl, what is the percentage that he can eat?

(3/14 = (0.214 x 100)) = 21%3. What is the decimal representation of pears and lemons in the bowl? (5/14) = 0.36

AusVELS – Statistics and ProbabilityContent strand/s, year, definition and code Chance. Level Six. “Describe probabilities using fractions, decimals and percentages” (ACMSP144)

Enabling PromptDetermine the probability of picking each type of fruit out of the bowl using fractions and simplify as much as possible. Put in order from most likely to least likely, which fruit you would randomly pick.

Answers to Enabling Prompt1. Picking a lemon out of the bowl = 2/14 = 1/72. Picking a pear out of the bowl = 3/143. Picking an orange out of the bowl = 8/14 = 4/7

Most likely = orange (4/7). Second most likely = pear (3/14). Least likely = lemon (1/7)

AusVELS Content strand/s, year, definition and code Chance. Level Five. “List outcomes of chance experiments involving equally likely outcomes and represent probabilities of those outcomes using fractions” (ACMSP116)

Justification for change to the original questionI changed the original question to just finding out fractions for the fruit rather than finding out fractions, decimals and percentages. However, I thought that just doing fractions was a bit too easy and more of a closed question so I added the extra part of ranking the fruit from most likely to least likely to choose which makes it slightly harder considering they have to deal with mixed fractions.

Extending PromptUsing the picture of the fruit bowl as an example, create your own worded problem and a solution for something (using ideas within the classroom) using probabilities.

Answers to Extending Prompt1. Q) In my pencil case of 20 pencils, I have 3 red pencils. What is the chance that I would randomly pick out a

red pencil? A) 3/20 = 0.15 = 15%

2. Q) If there are 24 students in my class and one quarter of them are not at school today, how many students are there in the classroom?

A) 1/4 of 24 = 6. 6 students are away so 18 students are at school.3. Q) My friends teacher said that 90% of the students in their class passed the maths quiz that they did

yesterday. In a class of 30 students, how many students passed?A) 10% of 30 = 3.

3 x 9 = 27.27 students passed the test.

AusVELSContent strand/s, year, definition and code Chance. Level Seven. “Assign probabilities to the outcomes of events and determine probabilities for events” (ACMSP168)

Justification for change to the original questionThis extending prompt basically used the same idea as the original question but instead gave the responsibility of thinking up the questions to the students and having them figure out their own worded questions as well as finding out the answers to them as well to make sure they worked.

Cross-Curriculum LinksResearch and create a poster about the benefits of eating one of the fruits in the picture.

AusVELS - Cross-curriculum Health and Physical Education, Learning focus, level six, definition and code They investigate different food-selection models such as the Healthy Eating Pyramid and the Australian Guide to Healthy Eating and their characteristics

Report of Trialling Problem Picture 2 Child’s pseudonym, age and grade level: “Sally”, Age 12, Grade six

Original Question: Create a worded problem using the photograph of the fruit in the bowl and come up with different ways that probabilities can be used. (There are 14 pieces of fruit in the bowl: 2 lemons, 3 pears and 8 oranges.)

Child’s response to the question:

Reflection on child’s response:The question presented to my student was for her to create a worded problem that would relate to the photograph of the bowl of fruit and would incorporate the use of probabilities. Referring to her work sample shown above, it seemed as though she might not have completely understood the task as she repeated the information that I’d given her in her answer. Additionally, she did not use probabilities, but rather just found the single number answer. I believe this was due to my wording of the question. I had struggled a bit to come up with a simple and comprehensible way for a grade six to understand what I was asking but it was difficult.

After Sally had finished answering the question, I needed to prompt her to continue so that she was able to show me that she could use probabilities (hence the further questioning of ‘decimal, fraction, percentage’). She struggled to determine the fraction (8 oranges out of 14 pieces of fruit, 8/14) but that might have been due to my wording again. Once I had guided her to the answer, she clicked and knew straight away which showed me that she did know what

to do, it just wasn’t clear on my behalf. She also wasn’t sure how to come to the decimal number from the fraction so I guided her in that as well, however she knew straight away how to turn the decimal into the percentage.

The AusVELS intent relating to my question was “Describe probabilities using fractions, decimals and percentages” (ACMSP144) which I think is definitely addressed in my question; however, the execution of it was lost in translation in the wording of my question. If I were to modify it, I would need to write it with clearer instructions, or as seen in my extending prompt, (using the picture of the fruit bowl as an example, create your own worded problem and a solution for something using ideas within the classroom, using probabilities.) giving the student the chance to create their own problem with less restriction and more relevance. Using the fruit bowl is fairly relevant to students, but a lot of the time, math context is not relevant or of immediate interest to a majority of children so using items around the classroom that they see every day is much more significant and beneficial to use. (Sparrow, 2008)

References for reflection on the trial of question 2:

Sparrow, L. (2008), Real and relevant mathematics: Is it realistic in the classroom?. Australian Primary Mathematics Classroom, 13(2), 4-8

Victorian Curriculum and Assessment Authority, 2015. AusVELS: Mathematics. Retrieved from <http://ausvels.vcaa.vic.edu.au/Mathematics/Curriculum/F-10>

Problem Picture 3 Location: Catalogue from home

Problem Picture 3 - Questions

Grade level: Grade Six

Question 3Find out the new prices of each of the items of clothing when discounted at 10%, 25% and 50%.

Answers to Question 31. Roll neck jumper - $29.99.

10% = 29.99 – 2.99 = $27.00 25% = 29.99 – 7.49 = $22.50 50% = 29.99 – 14.99 = $15.00

2. Zip Coated Jeans - $59.99 10% = 59.99 – 5.99 = $54.00 25% = 59.99 – 14.75 = $45.24 50% = 59.99 – 29.99 = $30

3. Zip Coat -$109.99 10% = 109.99 – 10.90 = $99.09 25% = 109.99 – 27.49 = $82.50 50% = 109.99 – 54.99 = $55.00

AusVELS - Number and AlgebraContent strand/s, year, definition and code Money and Financial Mathematics. Level Six. “Investigate and calculate percentage discounts of 10%, 25% and 50% on sale items, with and without digital technologies” (ACMNA132)

Enabling PromptWork out the GST on each item of clothing. (GST = 10%)

Answers to Enabling Prompt1. Roll neck jumper - $29.99.

10% = 29.99 GST= $2.99

1. Zip Coated Jeans - $59.99 10% = 59.99 GST= $5.99

2. Zip Coat -$109.99 10% = 109.99 GST= $10.90

AusVELS Content strand/s, year, definition and code Money and Financial Mathematics. Level Five. “Create simple financial plans” (ACMNA106) Elaboration: “identifying the GST component of invoices and receipts”

Justification for change to the original questionThe modification for this question just involved reducing the amount of work required in the original question. However, instead of the question just being, “Find 10% of the clothing price” I linked it to GST as stated in the AusVELS so that it had more meaning and relevance.

Extending PromptMassive clothing sale! Everything must go!

Items of clothing originally marked as $24.50, $38.99 and $96.75 are all being reduced by 10%, 25% and then 50% respectively each week until they’re sold. Find the new prices of each item of clothing at each percentage discount.

Answers to Extending Prompt $24.50

10%= 24.50 - 2.45 = $22.0525%= 24.50 – 6.125 = $18.3550%= 24.50 – 12.25 = $12.25

$38.99 10%= 38.99 – 3.89 = $35.1025%= 38.99 – 9.75 = $29.2550%= 38.99 – 19.49 = $19.50

$96.7510%= 96.75 – 9.67 = $87.0825%= 96.75- 24.18 = $72.5750%= 96.75 – 48.37 = $48.38

AusVELSContent strand/s, year, definition and code Money and Financial Mathematics. Level Six. “Investigate and calculate percentage discounts of 10%, 25% and 50% on sale items, with and without digital technologies” (ACMNA132)

Justification for change to the original questionThe difference I made to the original question was just using more complex numbers to increase the difficulty.

Cross-Curriculum LinksReferring to the middle image on the catalogue, create a design brief that will fulfil the requirements of the final design.

AusVELS - Cross-curriculum Cross-curriculum area, Content strand/s, year, definition and code Design, Creativity and Technology, Level 6, “They contribute to the development of design briefs that include some limitations and specifications by posing questions about and identifying situations, problems, needs and opportunities for the creation of useful products and simple systems.”

Report of Trialling Problem Picture 3 Child’s pseudonym, age and grade level: “Sally”, age 12, grade six.

Original Question: Find out the new prices of each of the items of clothing when discounted at 10%, 25% and 50%.

Child’s response to the question:

Reflection on child’s response:The question posed to Sally was to find out the new prices of each of the items of clothing when discounted at 10%, 25% and 50%. She understood easily what she needed to do and proceeded to complete the activity with no assistance or clarification.

This was clearly the easiest of the questions out of all three for her to answer and the one she seemed most confident with. I allowed her to finish all of the questions before asking her to explain her working out of the answers to me. As seen in the trialling of Question 2 (create a worded problem using probabilities) she had no issue converting her decimal to percentage, which obviously helped her in this situation as prices are in decimal format. Sally’s method of working out her answers involved eliminating zeros from the question, which helped her work out that 10% of $30 was $3. From there she simply added as many increments of 10% or halved it to work out 5% and she was able to find all of her percentage discounts.

I think it’s incredibly important to be able to change easily between fraction, decimal and percentage as they’re commonly linked in every day life. “Flexibility in moving from one representation to another is useful in deepening students’ understanding of rational numbers and helping them think flexibly about these numbers” (NCTM, 2000, as cited in Whitin & Whitin, 2012)

The AusVELS link for this question stated: “Investigate and calculate percentage discounts of 10%, 25% and 50% on sale items, with and without digital technologies” (ACMNA132) which is addressed quite explicitly in my original question as it asks specifically to find those three percentage discounts.

References for reflection on the trial of question 3:

Whitin, D, & Whitin, P 2012, 'Making Sense of Fractions and Percentages', Teaching Children Mathematics, 18, 8, pp. 490-496

Victorian Curriculum and Assessment Authority, 2015. AusVELS: Mathematics. Retrieved from <http://ausvels.vcaa.vic.edu.au/Mathematics/Curriculum/F-10>