Integrating Life

Outcome N7.1

Demonstrate an understanding of division through the development and application of divisibility strategies for 2, 3, 4, 5, 6, 8, 9, and 10, and through an analysis of division involving zero.

Technology-Enhanced PBL – Miley’s Doghouse

 Indicators –

 Investigate division by 2, 3, 4, 5, 6, 8, 9, or 10 and generalize strategies for determining divisibility by those numbers.

• What does it mean for a number to be even?  Have students use base-ten blocks or other tangible counters to support their ideas.
• Once we have shown a number to be divisible by 2 and by 3, what else do we know?  Is it also divisible by 6? By 12?
• Examine the patterns for divisibility by 5 and by 10.  How do these patterns relate to divisibility by 2 and by 4?  Or 4 and 8?
• What is the relationship between the terms divisibility and factors?
• What are some multiplication “tricks” that students have been taught in the past?  For example, finger math, patterns for factors of nine, multiples of ten, skip counting,…  How can these be used to help determine divisibility?

1. Apply strategies for determining divisibility to sort a set of numbers in Venn or Carroll diagrams.

• Venn Diagram: Interactive site – http://nlvm.usu.edu/en/nav/frames_asid_153_g_2_t_1.html?open=instructions&from=topic_t_1.html                                          
•  Observe patterns and generate theorems.

* SMARTboard friendly!

• Carroll Diagram –

                                                  i.      Organise numbers according to categories

                                                ii.      Interactive resource – http://www.wmnet.org.uk/wmnet/custom/files_uploaded/uploaded_resources/850/carrollv4.swf

 * SMARTboard friendly!

 c. Determine or validate the factors of a number by applying strategies for divisibility.

• Summarize findings as a class then apply to Venn Diagram or Carroll Diagram activities.
• Use strategies to complete chart to elicit the definitions of prime and composite numbers:

• Ask why the number 1 is neither prime nor composite? 
• Observe numbers that have an even number of factors and those that have an odd number of factors.  What is special about all the numbers with an odd number of factors? (perfect squares)
• Interactive resources – http://www.shodor.org/interactivate/activities/FactorizeTwo/ and http://www.shodor.org/interactivate/activities/Factorize/
• Factor Trees – http://nlvm.usu.edu/en/nav/frames_asid_202_g_3_t_1.html?from=topic_t_1.html

*SMARTboard Friendly

 d. Explain the result of dividing a quantity of zero by a non-zero quantity.

• Verify results of zero divisors and zero dividends using a calculator.
• Discuss the relationship between terms factors and divisibility.  What does it mean for a number to have a factor of zero?  Are there any numbers that are divisible by zero?  What is  ?
• Use repeated addition (groups) of tangible counting objects like base-ten blocks to model multiplication and division (discuss concept of inverse operations).

3 groups of 4 is 12

12 divided into 3 groups contains 4 blocks in each group

12 divided into groups of 3 contains 4 groups

Can we organise 12 base-ten blocks into zero groups?  Can we organise 12 base-ten blocks into groups containing zero blocks in each one?

e. Explain (by generalizing patterns, analogies, and mathematical reasoning) why division of non-zero quantities by zero is not defined.

For more advanced students, present the “proof” that 2 is equal to 1.  Note that dividing by (A-B) is the equivalent of dividing by zero as it was already established that A = B.

Let A=B.
Multiply both sides by A: A²=AB
Subtract B²: A²-B²=AB-B²
Factor: (A+B)(A-B)=B(A-B)
Divide by A-B: A+B=B
Replace A with B (recall A=B): 2B=B
Divide by B: 2=1.

There’s a proof that 2=1

In the slide titled, “Beginning with the end in mind with learners”, Anne Davies explains that making the learning destination clear to learners helps them engage and take more responsibility for their learning.  Davies continues to provide concrete examples of how to make learning goals known to students in an easily-understandable manner.  She posits that by providing students with student samples and by discussing evidence of learning, teachers show students what success will look like.

Davies provides possible “wrong-turns” that teachers may take in their attempt to teach with the end in mind.  She warns that if the language used is not student-friendly, samples are either not available or are all exemplary, then students will not be able to visualise the path to success.  Davies also lists the detriments of using only summative assignments, test-results or evaluation of content-knowledge to evaluate student learning and of generalising student success standards and applying them to all students.

Next, Anne Davies provides information on the benefits of involving learners as partners in their learning by co-constructing criteria.  She explains that this practice can make learning more explicit for students and can familiarise them with the language of assessment.  Once again, she follows this advice with possible “wrong-turns” saying that if teachers control the rubric creation or brainstorming session too much, students will not benefit fully.  She also adds that students may not be ready to construct criteria in terms of their content knowledge or ability and that the criteria are being established too early in the process.  After suggesting that some rubrics may not support struggling learners, Davies provides an example for discussion.

The “wrong-turns” provided after Davies explains the benefit of involving learners in their assessment using self-assessment include fall back on those identified in the criteria creation process.  Davies explains that self-assessment may be troublesome if the students are not familiar with the language of assessment, if their criteria differs from that of their teacher, or if the audience for their self-assessment findings is either non-existent or not valued by the student.

A third way to involve learners in the assessment process is to provide students with specific, descriptive feedback that supports improvement and learning.  Davies warns that if there isn’t enough feedback, if it isn’t timely, continuous, specific enough or is merely evaluative, then student learning does not benefit to the fullest capacity.

Davies sites brain research, asserting that having students set learning goals in addition to constructing criteria and self-assessing can improve their learning.  However, she explains that differing opinions on quality levels in relation to products, processes or evidence of knowledge can hinder the benefit of setting learning goals as can student misunderstanding of what successful goal completion will look like.  Also, if the students did not set their own goals and take ownership of them or if they have set goals that are unattainable, then they will not reap the full benefits.

Once learners have been made aware of the desired learning outcomes and have acted as partners in their learning, they should be required to collect key pieces of evidence of their learning. The evidence they collect should be in relation to the desired learning outcomes, criteria, and goals.  Students should be asked to reflect on their chosen evidence and be provided with both a structure and a time to present their collection.  “wrong-turns” exist when the collected evidence does not match intended outcomes, when it is limited in scope or when it is too complicated.  Students must be given an audience that consists of more than their teacher or their report-card and they should articulate the rationale for their choices.  Davies suggests that student-parent-teacher conferences provide opportunity for beneficial rewards but warns that they must be focused on learning and improvement.

 http://www.annedavies.com/PDF/We%27reNomads-Davies2009%5B1%5D.pdf

By Murray Guest

http://www.annedavies.com/PDF/16B_Murray_Sask_math%203.pdf

Murray Guest begins his article on AFL in Saskatchewan math classes with an assumption that math teachers are already using AFL strategies and that AFL is tied to student improvement.  He continues to provide an overview of what AFL means and reinforces the fact that AFL is research-based.  After differentiating between assessment and evaluation, Guest provides specific examples of how math teachers have practiced AFL for years.  He then offers three methods, based on his own classroom practice, of making the AFL process more transparent for the students.  By removing marks from quizzes and homework, Guest focuses his students on the feedback for improvement without evaluating work that is done during the practice stage.  He also incorporates student self-assessment into his AFL repertoire.

Shifting his focus on the benefits AFL has for the teacher, Guest exposes areas where math teachers have traditionally used student assessment to guide their own practice.  He expands on this by providing ideas for adjustment that can help the teacher benefit even more from the AFL process while making it even more transparent and useful to the student.  Guest posits that by sharing learning outcomes with students at the beginning of the class and providing alternatives to the traditional pencil-and-paper tests, students are in a better position to learn and to provide evidence of their learning.

N6.1 Demonstrate understanding of place value including:

• greater than one million
• less than one thousandth
• solving situational questions using technology.

a. Explain, concretely, pictorially, or orally, how numbers larger than one million found in mass media and other contexts are related to one million by referencing place value and/or extending concrete or pictorial representations.
What are the students interested in? If they are interested in sports, have them research salaries or costs of race horses. If they are interested in health, they could research how many litres of pop are consumed each year in the world.
Use authentic sources such as Tobacco article or Energy_Revenue_Deficit and have children organize numbers found within in place-value charts.
Contrast the way the numbers in above article are written with the way the numbers in Tomato Article are written. Why?
Use visualizations such as Planet Distances to examine large values. Have students use authentic data like http://www.allsands.com/science/planetsdistanc_aju_gn.htm (convert to metric) to create scale representations.

b. Change the representation of numbers larger than one million given in decimal and word form to place value form (e.g., $1.8 billion would be changed to $1 800 000 000) and vice versa.
Have students create popular media using authentic, large-number data. For example, they could write a newspaper article using planet distances, internet hits, or whatever data they researched in step a.
c. Explain, concretely, pictorially, or orally, how numbers smaller than one thousandth found in mass media and other contexts are related to one thousandth by referencing place value and/or extending concrete or pictorial representations.
To bridge from large to small, use visualization like Cells Alive to see what the common rhinovirus looks like when magnified one million times.

If one rhinovirus is 1/1000 000 of the size of a pin head, how big is it in terms of a square mm?
d. Explain how the pattern of the place value system (e.g., the repetition of ones, tens, and hundreds), makes it possible to read and write numerals for numbers of any magnitude.
Use Place Value Tents (template example ) to help students break up numbers and represent them in expanded form.

e. Solve situational questions involving operations on quantities larger than one million or smaller than one thousandth (with the use of technology).
f. Estimate the solution to a situational question, without the use of technology, involving operations on quantities larger than one million or smaller than one thousandth and explain the strategies used to determine the estimate.

Web resources – http://www.emints.org/ethemes/resources/S00001120.shtml

Betty Hollas is an engaging speaker who presents useful, applicable activities to complement differentiated instruction. Some of my favourite examples include:

1. Show, Don’t Tell
This is a general vocabulary activity which would work very well with math terms.

Give the students the words to learn and their definitions (on a handout or projected). Working in small groups, students are to DRAW a representation of their term then come up with a GESTURE that represents the term.

Taking turns, the group members present their picture and gesture both before others are allowed to guess at the term being illustrated. Once it has been correctly identified, ALL students repeat the gesture together.

Imagine the ease with which you could review these terms – show the picture and ask for the gesture, gesture and ask for the word,…

2. R.A.F.T.
This activity has been well-used in ELA classrooms for quite a while. But let’s think about applying it to the Math classroom.

R – Role of the writer –

A – Audience for the writer –

F – Format of the writing –

T – Topic of the writing –

• Have a circle write to his polygon friends about how lucky he is to have no straight sides.
• Have an improper fraction write to the textbook company complaining that his name is unfair.
• Have the variable x write to a psychiatrist to get help with his identity crisis

3. Cubing
This activity consists of rolling a die with different choices/activities on each of its six faces. This activity can be made much more fun using the SMART Notebook software’s interactive cubes.

4. We Have, Who Has?

I have a triangle, who has two sets of parallel sides?
I have a parallelogram, who has a set of points equidistant from a central point?
I have a circle, who has three sides?

I have 12, who has the square root of 400?
I have 1 ½, who has ¾ + ½ ?

DI – Is not a YES or NO proposition. We do not have a choice about whether or not we should differentiate – we must!

DI – Is not a program, a resource, a new bandwagon. It is a way of thinking.

DI – Strengthens shift from the teacher-centred “I taught it – they should have learned it” focus on how hard we are teaching to the more student-centred focus on how well students are learning.

DI – can help students who need differentiation because of varied interests, abilities or readiness levels.

THE MORE WAYS WE TEACH, THE MORE STUDENTS WE REACH!

We can all definitely do something with this!

http://www.youtube.com/watch?v=YtjD3mRrVT4

What does PBL look like with technology integration? 

Here is a problem involving permutations and combinations that was designed for a grade 12 (Math B30) audience.  I have used it with grade 9 and grade 11 students as well. The feedback?  Why can’t we do more like this!

A good way to encourage collaboration is to display the problem on the SMARTboard and let the class work through it as a large group.

Last week I attended an SPDU session on Inquiry Learning. It was interesting, applicable, but really nothing new. What defines the difference between inquiry, PBL, situated learning,… Or are they all simply slightly different applications of constructivism?

In ETEC 512 we created a concept map of learning theories and in ETEC 530 we created one just on constructivism.  Reviewing the important concepts and how I linked them together and organised them, I realize that the term INQUIRY LEARNING is nowhere to be found.

So what is Inquiry Learning anyway?  And how is it different?

In my opinion, Inquiry Learning is the practice of asking students questions instead of giving them information.  Feedback tends to turn the assessment back onto the student, asking them to evaluate their performance and suggest areas for improvement.

I think that I have taught using the inquiry approach for many years.  I just didn’t know what it was called!

Math lends itself very well to inquiry processes.  Experienced math teachers can develop progressions of questions that will enable students to apply possessed knowledge, construct new knowledge, and transfer knowledge to new and different problems.

Here is a sequence that exemplifies my understanding of inquiry learning:

2/7 + 3/7

2/x + 3/x

2/7 + 3/14

2/x + 3/xy

2/7 + 3/5

2/x + 3/y