One team of Year 6 teachers was interested in developing a Guided Inquiry for exploring positive and negative integers with their students. They wanted the inquiry to involve investigating everyday situations that use integers and to generate an authentic need for locating and representing these numbers on a number line (ACMNA125). What better way to engage students with learning about integers in Year 6, than with their inquiry, What is the best game that you can create to model positive and negative integers? If you would like to try this inquiry in your own classroom then please see the four phases of the 4D framework (Allmond, Wells & Makar, 2010) outlined below:
In all phases of the inquiry, Checkpoints can be used to share interim ideas and challenges. You can download the Authentic Problems Teachers’ Guide from the re(Solve): Maths by Inquiry website to find more information on this Guided Inquiry process. The Year 6 teachers who designed this unit were also interested in using the information about their students’ learning, generated through the inquiry, as summative assessment information. This assessment information would contribute to their current assessment schedule. The team here at Inquiry Maths Pedagogy in Action (IMPACT) worked with the Year 6 teachers to develop possible formative and summative assessment opportunities that could complement the Guided Inquiry, What is the best game that you can create to model positive and negative integers? (Now available from the Members section of this website). Part of this required the students to apply ideas gained from playing games with integers to an unfamiliar context to demonstrate transfer to cartesian planes. If you wish to use Our Marking Guide and suggested summative assessment questions in your own classroom, then you will need to ensure you do not explore games involving locating and representing positive and negative integers on a cartesian plane, such as in Battleships, prior to the assessment. Our Marking Guide  available from the Members section of this website If you decide to give this inquiry a try then we hope your class enjoys finding out What is the best game that you can create to model positive and negative numbers? You can use Our Marking Guide as well as the suggestions for assessment to generate assessable information about your students. We hope you find this useful and welcome your feedback (Contact).
What mathematical problem solving could students participate in when engaged in the Guided Inquiry, Record Breaking: Are athletes getting better over time? The openended nature of this question lends itself to students devising multiple solution pathways as students consider the authenticity of the context. The ambiguity of the word ‘athletes’ means an answer depends upon whether students focus on women, men or children. Does getting ‘better’ mean faster, jumping higher/further, lifting more? Described in detail in the Record Breaking inquiry unit (Thinking Through Mathematics, Book 3, unit 8), adaptations for conducting the inquiry in different year levels – and alignment with the Australian Curriculum in each of these year levels – can be found below on this Research Page of the IMPACT website. An article exploring this inquiry has recently been published in the Australian Primary Mathematics Classroom journal (Muir & Wells, 2019) and includes further illustrations of the mathematics in action in an Australian Year 5/6 classroom. These illustrations include different data displays typical of the work produced by students and exchanges made by students that include conclusions made in the Defend phase. From the article: Student's refined question: Are athletes getting better at jumping?
The Guided Inquiry approach provided students in Year 5/6 the opportunity to engage in authentic mathematical problem solving that required understanding of data representations, and fluency with interpretation, beyond simplistic representations. The reasoning by students (see the above example) required explanation of their analysis and evaluation of authentic data (about athletes) to justify conclusions reached in the Defend phase. From Muir, T. & Wells, J. (2019). Are athletes getting better over time? Australian Primary Mathematics Classroom, 24(3), 1520.
Favourite Books: Does our class have a favourite type of book? is an inquiry that was published in Book 1 of the Thinking through Mathematics series. While the inquiry was originally designed for students aged 68 years of age, we have now aligned the mathematical intent of the inquiry with the Australian Curriculum: Mathematics. We think this inquiry would best suit exploration in a Year Three classroom but would also be good with Year Two or Year Four  the adaptations are provided below. Any childrens' book can be used to introduce the 'Maths Investigator' concept introduced in this unit. This unit provides an opportunity for children to share their love of reading. It also enables reluctant readers to discover the types of books their peers are reading and hopefully encourage them to read some of the suggested books. The nature of the data collection offers opportunities for a variety of data representations to be trialled and evaluated. Year Three Work together with the students to formulate question(s) to investigate from books you read together. Guide students to plan their data collection process and support students to choose their own ways to represent their data.
Year Two The Favourite Books inquiry can be used to introduce younger students to the idea of sorting data into categories to identify categorical variables (using Venn diagrams and Carroll diagrams). Students could sort books by genre or topic for example.
If you are interested in trying this inquiry in your classroom, Favourite Books: Does our class have a favourite type of book?, download the inquiry unit from the Members section of this site for more detail. For more alignment information, including information about the Proficiencies, download the alignment document here.
We hope your students enjoy finding out which type of book is their class favourite.
Students:
Year Six Students determine the number of pools that might run in a competition, by drawing on their knowledge of triangular numbers. A handball tournament can be one way to identify triangular numbers and represent them using a real life context.
Year Five Students decide on an appropriate inquiry question that could be answered if the class conducted a tournament. As a class, have students decide how pool participants will be determined (random, seeded, etc.). Pool members can construct a workable draw and data collection sheet which includes match results and durations.
The full version of Round Robin: Who is the best handball player in our class? is available in the Members section of this site. For further information regarding alignment with the Australian Curriculum: Mathematics, including how the inquiry supports student development in each of the Proficiencies, please download the alignment document we have created.
We hope you enjoy running a handball tournament in your classroom. Record breaking is an inquiry unit you will find on the Resources page of this website. The inquiry can also be found in Book 3 of the Thinking through mathematics series, for students aged 1013 years. An excellent way to mathematically consider records broken at the Olympic, Commonwealth, Paralympic, Pacific or Youth Olympics (depending on which is most relevant to your location and the year), in this inquiry students explore the notion that athletic ability has continued to improve over time. This is a popular classroom topic and such an inquiry could take place in many different year levels. The beauty of inquiry pedagogy is the ability to open tasks up so students of various abilities can successfully participate  low floor, high ceiling tasks. Here we consideration, with a little imagination, ways in which you might adapt this unit for your own year level. Foundation Year Consider whether students can jump further from a standing jump or a frog jump. Direct comparison to determine which is longer. Each student jumps. Records which jump was further (using markers to enable comparison) and then yes/no questions are asked to determine the most common response for the class.
Year One Similar to above, but the students measure their jumps using informal objects. This provides an opportunity to discuss the need for uniform objects (imagine if you wanted to compare each other’s jumps). Count and record the jumps. How could we record the class data? What does the data mean? What would the data look like for other classes (inference).
Year Two Consider whether students get better at jumping over time. Have the student record a jump. Practise jumping for a short period each day and then record the jump distance and weekly intervals. After three jumps (say three consecutive Mondays), students compare their jump data (you could use lengths of string/wool – blue for first jump, red for second, green for third etc). Did students jump further with practice? How can they record this data? What inferences can they make? Measure the string lengths with informal objects, how much further/less did the student jump from one jump to the next? Show your working (evidence).
Year Three As Year 2, however the measurement are now able to be made in centimetres.
Year Four As at Yr 3, with the additional connection between metric measures (metres and centimetres) and decimal place value notation. ie 123 cm is 1m 23cm. NB measurement should not be used to introduce decimal notation but only introduced once decimal PV is in place. With this age group, consider the jumping events as these use length to two decimal places only (cm). Using timed events involves students with Base 60 and, if using hundredths (eg running or swimming) or thousandths of seconds (eg kayaking) – this can be quite difficult.
Year Five Intended year level of document. Be very careful of using events as cautioned in Year 4 notes).
Year Six As Year 5 with the additional connection between metric measures (metres and centimetres) and decimal place value notation. ie 123 cm is 1m 23cm is 1.23m. Opportunities to extend the maths for this age would include: average time, proportional reasoning (Is the 200m run in twice the time of the 100m etc).
At all levels where students are constructing data representations (graphs, tables, tallies etc) – there are multiple opportunities to compare these representations and discuss the relative merits of, for example, a stem and leaf plot with a line graph.
We hope you are able to adapt the inquiry, Record Breaking: Are athletes getting better over time? , to your own year level. 
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