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Place value inquiry

The place value inquiry is suitable for pupils aged between 10 and 12. The prompt is a number line with a microscopic view of a section of the line. The teacher can run the inquiry as open or structured depending on the experience of the class with inquiry learning and on the pupils' levels of initiative (see levels of inquiry).

Open inquiry
The inquiry starts with students posing questions and speculating about the relationship between the two parts of the number line. Different pathways can emerge from the initial phase depending on the start number (positive or negative, whole number or decimal, tenths or hundredths, and so on) and on the intervals chosen. One point that often needs reinforcing is that the intervals on each line must be the same, but each interval on the blue line is a tenth of the interval on the black one. Pupils can decide to divide the blue line into more intervals, which might lead to the appearance of recurring decimals. This, in turn, can give rise to a discussion on degrees of accuracy (and rounding) and and on the advantage of fractions in this context.
Structured inquiry
In a structured inquiry, the teacher might pose questions and set tasks, perhaps after conducting a discussion around the meaning of the prompt. A planned lesson around place value (incorporating, for example, units, tenths and hundredths on the number lines) might be followed by tasks that involve filling in missing numbers when given two pieces of information about the black or blue line. The teacher might build up to posing challenging questions, such as: If the main number line started at 5 and finished at 19, what is the fourth number on the blue line? (An interesting discussion has arisen in a year 7 class about whether it is possible to complete the missing numbers when given one number on each of the lines). Once students have practised finding the interval on the black line by dividing the difference between the start and finish numbers by the number of intervals, then they could experiment with their own start and finish points.
Extending the inquiry
A pupil in a grade 5/6 class at the Fred Varley Public School (Ontario, Canada) extended the inquiry by introducing a third number line (see the picture posted here). This represents a valuable extension to the inquiry. If the top line represents units, other number lines could be added to introduce the concepts of tenths, hundredths, thousandths and so on.

Prompt sheet
Two tasks to support inquiry.
Template for three number lines

You can read more examples of how this inquiry has developed in the classroom on the primary section of the website. 

An alternative prompt
Billy Adamson used the place value prompt with his year 7 class, tweeting "What a day, loved my #inquiry lesson with year 7!" He supplemented the original prompt with his own:
0.7 < 0.76 < 0.764 
0.7 > 0.67 > 0.467
Billy's prompt has the potential for students to engage in exploration and reasoning. Why, for example, does the inequality sign not change when more digits are placed at the end of the decimal number on the top line, but it could change when another digit is introduced on the bottom line?

Billy Adamson is the Head of Mathematics at Thurston Community College (Bury St Edmunds, UK). You can follow Billy on twitter @Billyads_47.

Reasoning through inquiry
Saleshni Cook posted a video on twitter of one of her grade 6 pupils discussing the place value prompt. She reports that her class have been "launching into some great maths with the Inquiry Maths prompts."

Saleshni is a PYP educator in Istanbul, Turkey. You can follow her on twitter @CookSaleshni.
Making connections through inquiry
Matthew Bernstein, a teacher of a grade 5/6 class at the Fred Varley Public School (Markham, Ontario), posted the pictures above on twitter. He describes how the inquiry enabled students to make connections: 
"When starting this task, students immediately made connections to measurement and the 'step' or base 10 nature of metric units. The prompt is a useful way for students to understand this concept as it is a natural connection to place value. Students who initially struggled to represent their ideas independently on the number line needed some scaffolding, which I was able to provide by asking them to count by tens on the top number line and then fill in the bottom one. Once they saw the subdivisions it was easy for them to make connections down to ones and tenths, but could also see what a number line above might look like. Students quickly saw the connection to place value and then this was reinforced the following day when we examined a 'zoomable' number line."

Matthew posts pictures on twitter of his students' inquiries @mr_bernstein.
Excitement generated by inquiry
Dr Dawn Jepson used the place value prompt with her year 7 mixed attainment class. In only her second inquiry, she was able to generate great excitement in the classroom as students reasoned about the meaning of the prompt (see the picture above). Dr Jepson described the lesson as amazing and remarked on how proud she was of her students. Her head of department added that inquiry, in producing high levels of enthusiasm in students and teacher alike, is infectious. Dr Jepson shared the outcomes of the students' inquiry in a departmental meeting.

Dr Jepson is a teacher of mathematics at Thurston Community College (Bury St Edmunds, UK).
Questions and observations
These are the questions and obseravtions of a year 7 mixed attainment class at 
Haverstock School (Camden, UK). In a structured inquiry, the teacher used the calculation that split 11 into ten equal steps to develop a mathematical approach to the resources on this page. 

This document lists the questions and observations about the prompt from two other year 7 mixed attainment classes at Haverstock School. The teachers of the classes were Amy Flood and Ollie Rutherford. Ollie reports that, "There were so many amazing points from students and overall the inquiry went really well!"
Using students’ questions to structure inquiry
Matt Carvel used the second half of a one hour lesson to introduce the prompt to his year 7 mixed attainment class. Students asked questions and made observations:
  • Is it to do with measuring? 
  • Are they centimetres and millimetres? 
  • Is the top line the same but bigger?
  • What’s in between the gaps on the bottom line?
  • There are ten sections on top.
  • There are 10 little pieces in the line above.
  • Between 1 and 2, there is 1.1 to 1.9 points in between.
  • The units could be in between. 
For the second lesson Matt chose to focus on the four statements in italics above. He introduced the lesson with this on the board:
Matt created a task that focused on the two numbers at the start and end of the blue line on the top number line. The six tasks involved an increasing level of mathematical complexity. (Students had previously been taught how to choose the task with an appropriate level of challenge.)
As the lesson developed, students began to make up their own questions. One chose 32.8 as the start number and 328 for the end. On the bottom line he went up in intervals of 32.8, but realised that he would not end at 328. When he presented his work to the class, another student, who had devised a method of dividing the difference between the start and end numbers by 10, offered a way of correcting the calculations. Matt called the second student up and the two discussed a solution in front of the class so all students would benefit. Matt reports that “it was the best moment of my teaching career so far.”

Matt Carvel
is a newly qualified mathematics teacher at Longhill High School, Brighton (UK). You can follow Matt on twitter @MrCarvel.
Structure and independence in the same inquiry
These are the questions and observations of a year 7 mixed attainment class at Haverstock School in Camden (London, UK). Each pair of students went on to consider the six regulatory cards before sharing their choice with the rest of the class. As a whole, the class selected the following four cards:
  • Decide on the aim of the inquiry: The majority of students had a clear understanding of the number lines. However, those that picked this card required a further discussion on what they could do with the number line to inquire further.
  • Change the prompt: Students who selected this card wanted to experiment by using different increments on the top line to see how this changes the bottom line. They proceeded independently from the teacher.
  • Practise a procedure: The teacher structured the inquiry by handing out one of two tasks depending on the prior attainment of each student.
  • Inquire with another student: Students who chose this card, it turned out, were expressing a desire to be given direction. The teacher gave them one of the same two tasks as above.
The teacher responded flexibly to the choice of cards to structure the inquiry for those students who required guidance, while allowing others to develop their inquiry independently. At the end of the lesson, students fed back on their reasoning during the tasks or on their own findings.

You can follow Haverstock mathematics department on twitter @havamaths.
Open inquiry with year 7   
Nichola Sowinska contacted inquiry maths after using the place value prompt with her year 7 class: "Some really great thinking came out of it." The students posed the following questions (see the photograph for a sample):
  • If the line was between two and three, what would the intervals represent?
  • Is the black line showing units or tenths? This will affect the blue line.
  • Is it positive or negative?
  • If it was between five and six, what would the blue 'bits' be?
  • If either line was longer, would it affect any of the other numbers?
  • How many decimal places could the blue line show?
  • Is there a specific number of blue lines (intervals) between two black lines?
  • If there were more numbers on the black line, would it affect the blue line?
  • What would the decimals be as fractions or percentages?
  • Does there have to be a specific number of blue lines between black lines?
  • Is there a formula to work out the next term?
This set of rich questions shows a class in the process of trying to construct an understanding of the number line based on their existing knowledge of place value. Multiple inquiry pathways could develop around the relationship between the black and blue lines. Once the ends of the black line are defined numerically, then the intervals in both lines are fixed. In their questioning, students have begun to speculate about what section of the number line is represented in the diagram (for example, between two and three or five and six). How would intervals on the blue line change if the intervals on the black line represented tenths or negative numbers?
As the class was new to inquiry, Nichola decided to keep the students together: "We went on to look at how many different ways we could split up the line." The class finished the inquiry by presenting diagrams supporting their ideas on mini-whiteboards. In the space of one lesson, the students had been involved in questioning, discussing, reasoning and presenting their ideas.

Nichola Sowinska 
is a maths teacher in Peterborough (UK). She is studying for a Masters degree at the Institute of Education, London. You can follow Nichola on twitter @NixxSunshine.
Structuring the inquiry  
Caitriona Martin
also tried out the prompt with a year 7 class. She invited students to pose a question or make a comment, but then she structured the rest of the inquiry: "It’s the first time we’ve done an inquiry lesson properly together so I kept it fairly closed and made the students all look into the same thing. Our key question was “What if it didn’t go up in ones?” They started trying out different intervals on the scale. Some made it go up in twos, some in fives, some in tens, some in tenths. It was interesting to see how they tackled working out the 10 sub-divisions in the lower scale from there."

In the second lesson, the class continued with the structured inquiry. Caitriona evaluates the learning that occurred through the inquiry: "In the style that we did it, it was a useful activity to see how the students made sense of 10 smaller steps being equal to one big step. I think it honed their reasoning and logic skills." She also discussed the structure of the inquiry: "As an inquiry, it would be interesting to see where other classes take it, whether they could be more creative or abstract with it as it felt rather closed, but that would definitely have been influenced by the style with which I conducted the inquiry."

Caitriona Martin
is second-in-charge of the maths department at St. Andrew's School, Leatherhead (UK). She has been teaching through inquiry since qualifying as a teacher. You can follow her on twitter @MrsMartinMaths.