Primary+Round+2+June+2013

= Primary Round 2 June 2013 = 1 out of every 3 Canadian households has a dog. How many dogs would you predict for the students in your class? Source: Marian Small - Past Presentations and resources []

Questions To Consider
As you view the student thinking about the problem consider 2-3 of the following questions to focus on. Please __**join this wiki**__ to share your thinking to the posted questions or a sample of a student solution on a division page. When you request to __**Join Now**__ (top left of menu bar) please identify your board, role and grades you teach, or are responsible for.

Solution A: []
What does the student know/understand? Pat: This student understands the relationship or the correspondence between the number of dogs and the number of Canadian households. This student seems to be thinking about quanities and how they relate but demonstrates additive thinking only.

What strategies does the student use to solve the problem? What additional strategies should the student consider? Pat: This student used pictures and symbols, to create a repeating pattern showing how she divided 21 students in her class into equal groups of three. The people are represented by a picture symbol, [stick figure] and the dog by the capital letter, "D". The student demonstrates an understanding that one person out of three has a dog by writing the capital, "D" on one of the stick figures in each group of three stick figures.

What probing questions would you ask the student? Pat: This student did not predict as the question asked. I would ask the student what the word predict means. I wonder if this student has had many experiences predicting or if the student misunderstood the question. I think that generally the word "predict" is interpreted by teachers as "figure out and tell in order to make a reasoned prediction". Predicting and estimating seem to fall into the same category for both teachers and students. Although this means they don't predict strictly speaking, they are definitely thinking as mathematicians and are using their reasoning to make sense.

What scaffolding questions would you ask the student? Pat: I would ask the student when checking to make sure if all 21 students were accounted if there was a quicker way to check. I would be looking to see if the student might suggest counting the number of groups and multiplying by three, [7 groups x 3 people in each group] or [ 7 groups x __ = 21]

I would ask the student why she wrote 21 divided by 7 = and then erased the algorithm. What was she thinking?

What feedback would you give the student? You might consider focusing on the mathematical processes or the student’s next steps. Pat: I would tell the student that I liked the way she accounted for all 21 of the students in her class by making groups of 3 students and 1 dog. I would ask if there is a different way to show her thinking. [manipulatives, T table, graph] I thought it was interesting the way this student began to write the division sentence, indicating that s/he understood that this is a division situation (and perhaps her/his drawing indicates fair share thinking), but that s/he intuitively understood that it wasn't 21 divided by 7, but her/his understanding was sufficiently fragile that s/he could not reason it all the way through.

What are your next steps for instruction? Pat: Take the student from repeated addition to multiplication. I think I would revisit that partial understanding, carefully building her/him up for what s/he did see, and talk through how all of the numbers are related to build multiplicative thinking and to build the understanding of the fact family. I would also discuss how long it took to draw all of those stick figures; perhaps a gallery walk or congress/bansho would reveal other ways of keeping track that mathematicians might use that don't take so long to make.

Solution B: []
What does the student know/understand? Pat: This student understands the relationship or the correspondence between the number of dogs and the number of Canadian households. This student seems to be thinking about quanities and how they relate but demonstrates additive thinking only.

What strategies does the student use to solve the problem? What additional strategies should the student consider? Pat: This student used logical reasoning and pictures. This student only made 20 sticks even after she says she needed to add a stick for the 21st person. I would ask her to check her pictures to make sure she has shown exactly what she says she has shown. I might suggest using a number line or maniupulatives such as unifix cubes. I like your idea of using the cubes - we could show the person with a dog with a different colour of cube - perhaps this could be introduced by saying something like, "I was wondering what it might look like if we used unifix cubes... would you try that to see what it might look like? I will come back in a few minutes and we can talk about it."

What probing questions would you ask the student? Pat: This student did not predict as the question asked. She solved the problem and then made her predicition. I would ask the student what the word predict means.I wonder if this student has had many experiences predicting or if the student misunderstood the question.


 * I would ask the student how her solution would have changed if there were only 20 people in her class and not 21**. I love this question! I was wondering what the student was going to do! That would be a great way to get at the idea of predictions and how mathematicians use 'almost facts' to make predictions.

What scaffolding questions would you ask the student? Pat: I would ask the student to show me the steps she would take when checking her answer.

What feedback would you give the student? You might consider focusing on the mathematical processes or the student’s next steps. Pat: I would tell the student I understood why she drew sticks and gave a "D" to every third person. I'd ask the student to solve this problem using a different strategy? What are your next steps for instruction? Pat: I would ask, "If you skip counted to find the number of dogs for the students in your class would you get the same answer? What other operation might give us the same answer?"
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