Posted in Geometry

Convert a Boring Geometry Problem to Exploratory Version

The following problem (or proving activity, if you like to call it that) is a typical textbook geometry problem. It is tough and guaranteed to scare the wits out of any Year 9 student.

proving triangle

When I used the given condition to construct the figure using GeoGebra, the only thing I can move is A or B and what it does is simply to reduce or enlarge the circle. Pretty boring. So I thought of making C dynamic. The way to do this is to construct point C along the circle and then construct a perpendicular line to AB. With C moving along the circle, the once static and close task is now a dynamic, exploratory and task.

kinds of triangles

Your students will observe that for triangle ABC to be an equilateral triangle, CD must be the perpendicular bisector of AB. You can now ask them the problem: Given that CD is the perpendicular bisector of radius AB, prove that ABC is an equilateral triangle, which is what the textbook is asking them to do.

In presenting the problem the way I’ve shown above, you did not only make the problem more interesting (hopefully) and accessible to the majority of the learners (I’m sure most of them can answer the questions), you have also given learners the chance to explore the problem first and be familiar with the situation.

Note that you will be doing a disservice to your students’ geometry life if you will stop at #5 and not give them the opportunity to prove. Proving is what makes mathematics different from other disciplines. It would be a shame if they will go through life only complaining about x and not of proving as well. I’m joking but you know what I mean. You may want to check some of my favorite post about teaching geometry through problem solving: Unpacking mathematics – a geometry example and Problem Solving Involving Quadrilaterals.

Posted in Algebra

Guest Post: Supporting Older Students Who Struggle with Maths

Often as teachers we struggle to support those students for whom maths is challenging. As a teacher myself, I found many of these students experienced difficulties which continued from their primary years right through until adulthood. Working with these students is what inspired me to write ‘Real World Maths- building skills for diverse learners’ ISBN 9780987542724 – a new publication for Banksia Publishing, distributed by Admark Education. I decided to make this title available as both a print paperback book and a full colour PDF e-book so that teachers can access the information and student pages in the way which suits their classroom needs the best.

real world mathsSometimes students who struggle with maths might have additional needs such as an intellectual or learning disability, be learning English as an additional language, or have missed significant periods of their schooling due to reasons such as illness or travel. Some may have a condition known as dyscalculia, which makes learning and retaining maths and numeracy skills extremely difficult. Often these students in particular have trouble with skills such as memorizing times tables or performing calculations which work in a backwards direction, such as subtraction tasks. They also tend to have difficulty with using their working memory to keep facts or signs in their head whilst they read the rest of a question, or remember which steps to perform next in a process and which they have already done. They might experience difficulty visualizing objects such as three dimensional shapes and imagining what might happen if these objects are turned or transformed in some way. This makes answering questions related to space and transformation quite difficult to do.

As the rest of their class moves forwards, often these students continue to struggle and can becoming increasingly frustrated with their inability to retain skills and progress with their peers. This can lead to acting out in class, failure to attend, personal frustration and distress and a decrease in self esteem and confidence. As teachers, it is critical to identify which students are struggling to keep pace with their peers and the expected learning outcomes, and take steps to provide age appropriate work which is success focused and will ensure skills are able to continue to develop over time.

Real World Maths – building skills for diverse learners’ (Banksia Publishing, 2013) provides a wide range of practical, easy to use resources for upper primary and junior to middle secondary students. It is an innovative title which focuses on addressing areas of need for students who struggle with their maths skills. It includes:

o    Strategies for supporting visual and auditory working memory

o    Ideas for student goal setting and skill building in maths

o    Place value

o    Fractions, decimals and percentages

o    Working with maps, distances and directions

o    Time, dates and calendar skills

o    Managing money and building financial literacy skills

o    Working with shapes, patterns and designs

o    Collecting and using maths information through practical activities

o    Over 50 copiable student pages to use in the classroom

-Article Author: Anne Vize M.Ed (Special Education).

Feel free to email Anne Vize with any queries to banksiapublishing@gmail.com

Posted in Teaching mathematics

Math Knowledge for Teaching

The mathematics that engineers, accountants, etc and teachers of mathematics know are different. They should be. There are some engineers, accountants, chemists, etc who become very good mathematics teachers but I’m sure it is not because they have ‘math knowledge for engineering’ for example but because they were able to convert that knowledge to ‘math knowledge for teaching’.

What is math knowledge for teaching?

It includes knowledge of mathematics but on top of that according to Salman Usiskin, it should also include knowledge of:

  1. ways of explaining and representing ideas new to students;
  2. alternate definition of math concepts as well as the consequences of each of these definitions;
  3. wide range of application of mathematical ideas being taught;
  4. alternate ways of approaching problems with and without calculator and computer technology;
  5. extensions and generalizations of problems and proofs;
  6. how ideas studied in school relate to ideas students may encounter in later mathematics study; and,
  7. responses to questions that learners have about what they are learning.

appreciating teachersI don’t know why some people especially politicians think teaching is easy. Surely college preparation is not enough to learn all these. You certainly need to be a practicing teacher to even start knowing #1 and #7.  Teachers need more support in acquiring these knowledge when they are already in the field than when they are still in training.

I started this blog to contribute towards helping teachers to acquire the seven listed by Mr. Usiskin. After 250 posts, it looks like I have not even scratched the surface 🙂

More posts: teaching mathematics and levels of teaching mathematics

Posted in Algebra

A Challenging Way of Presenting Math Patterns Problems

Mathematics is the science of patterns. Part of the math skill students need to learn in mathematics is to see regularities.  The usual way of introducing pattern searching activity is by showing a sequence of figures or numbers and then asking the students to find or draw the next one.   Here’s a more exciting way of presenting math patterns problems. Bernard Murphy of MEI (Innovators of Mathematics Education) shared this with us. MEI is an independent charity committed to improving mathematics education and is also an independent UK curriculum development body.

The figure below is the third in a sequence of pattern.

visual pattern

How does the first, second, fourth, fifth figure look like?

Here are three of the patterns I produced and the questions you could ask the learners after they produced the sequence of patterns. Note that the task is open-ended. There are other patterns learners can make.

1. How many unit squares will there be in Figure 50?

linear pattern

2. How will you count the number of unit squares in Figure n in this pattern?

y = 4x + 7

3. This is my favorite pattern. How many unit squares will there be in Figure n?

y=x(x+1)+2x+1; y=(2x+1)(x+1)-x^2

Note that in all the sequences, Figure 3 looks the same. Note also that for each of these sequences, you can have several expressions depending on how you will count the squares. Of course the different algebraic expressions for a particular sequence will simplify to the same expressions.

You can use this activity to teach sequences, linear function, and quadratic function. But this is not just what makes this activity a mathematical one. To be able to see regularity is already a mathematical skill and much more of course if they can generalize them as well in algebraic form.

I am so tempted to just give you the equations but that would mean depriving you of the fun. Anyway, here are two examples on how you can think about counting the squares in Figure n: Counting Hexagons and Counting Smileys. Have fun.