Open-ended Question about Triangle Areas

Open-ended Question about Triangle Areas

This question is quite vague – a useful feature in my opinion! Describing the area of the triangle as “slightly smaller” means that there are many possible correct areas. Students are not intimidated and willing to try to make a triangle.

The task is also able to be differentiated to many levels. At the most basic, students can draw accurate triangles and count squares and parts of squares. Students who know how to find areas of right-angled triangles using the formula (1/2)bh can draw these. To find the final side length requires Pythagoras, though the teacher can decide not to request this. More advanced students can draw triangles and find their areas using the formula (1/2)ab sinC. Finding the third side length of these triangles requires the cosine rule.

This is great starter or ender to lessons. It can be used to consolidate or review area concepts.

Do you like vague questions?

Leadership Styles

I have been reading today about instructional leadership and transformational leadership. Here are the definitions I have uncovered so far.

Instructional leadership focuses on how the school leader engages with teaching and learning. A strong, directive leader becomes a culture builder in a school by communicating a mission. The leader talks over and over about the mission and it is embedded into classroom practice and policies. The leader also takes an active role in managing staff and the curriculum. They work directly on teaching and learning issues. They are highly visible and have high expectations. The instructional leader protects teaching time and promotes professional development.

Transformational leadership describes a process by which a leader increases support for common goals. They seek to improve staff and themselves. There is a collaborative culture in which all are encouraged to participate and grow.

According to the article I have been reading by Viviane Robinson (2007), instructional leadership has a much greater positive impact on student outcomes than transformational leadership.

I am aware that I am an emerging leader; I am still developing my style of leadership. I can see the impact that I could have as an instructional leader. It seems to me that I may be helped by developing some charisma, though! (Is that even possible?) I have worked under a headteacher who was a dynamic instructional leader; he was also so demanding that many staff felt burnt out. I think sometimes I tend to the collaborative structures of transformational leadership because I am conflict-averse. Seeking others’ opinions seems like a safe way to proceed since then I cannot displease anyone.

I have more to say and I need more time to think and write. But I need to get back to the business of planning lessons.

I think that planning lessons is my prime task as a teacher. One article I read today (also Viviane Robinson) said that educational leaders should focus more on leading teaching and learning. This gives me lots to (hopefully) think about and digest soon.

Swap and Spot the Error

Here is a strategy I use to get students thinking. My year 11s have learned some trigonometry recently and everyone is working at their own pace. Some are still doing basic trig and others are doing complex multistep problems with bearings or in 3D. I used this slide (revealing one sentence at a time) to run a starter activity.

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First I encouraged them to make a problem that would challenge their partner (who is the person sitting next to them). Then their partner was told to answer the question, but make a mistake on purpose. Then a third person checks over the work and tries to spot the error.

Each student is engaged in this starter and no one sits passively. (I created it after reading Kagan’s Cooperative Learning book, which states that as many students as possibly should be active during an activity.) The problems the students made up were more interesting and many were more challenging than textbook or worksheet questions. And each student was exposed to three problems: the one they created, the one they solved, and the one they checked. My year 11s enjoyed the social aspect, too.

“Swap and Spot the Error” is a task that you can use for many topics. Try it and let me know how it goes.

Equations of Lines, and Other Coordinate Geometry

This is just a simple worksheet with twelve questions about finding the equations of straight lines and about gradients and the distance between two points. It’s laid out as twelve questions and I asked my students to work through them in any order.

A few of my students really like to cut out the questions individually and paste them in, writing their answers below. It helps them keep their work neat. I like anything that makes a worksheet more interesting for them! Even just giving them the choice of what order to do the questions in seems to make them feel more resilient.

The worksheet is available here in pdf format.

Mean, Median, and Mode: Sorting Cards

I have a set of cards with statements like these:

Some of the statements are always true, others are never or sometimes true. I give each pair of students a set of these cards and a laminated board (below). The students have to discuss and sort the statements.

Next, I ask students for feedback and I will often lead a whole class discussion highlighting some of the common misconceptions. This activity always leads to good discussion, both between the pairs and as a class. Here are pdf documents: the cards and the board. If you try it, please leave a comment telling me how it goes.

True or False Sorting Cards for Arithmetic Sequences and Series

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I have noticed that my students get a bit confused sometimes with arithmetic sequences. They think that the statement above is true: the sixth term in a sequence can be found by doubling the third term. So I made up a set of true/false sorting cards that highlighted some common misconceptions. There are seven statements like the one above that I ask students to sort and then we discuss them as a class.

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I made these cards with Tarsia; a pdf is available here.

Links Picnic #3

Links Picnic is an opportunity for us to share things from around the web that help and inspire maths teachers. Here are my picks. Please add yours in the comments section. You can leave a link to your own Links Picnic!

—Maths and Engineering fonts

—Answers to the question, “Why do I have to study algebra?”

–Seven posters answering the question, “When will I ever need maths?” I really like that they link to other school subjects, not careers to which students can’t relate.

–A collection of infographics, from deep to ditzy.

–The huge catalog of activity cards from SMILE is available from the UK’s National STEM Centre.

Equations of Straight Lines: Easy Mini Whiteboard Activity

My year 8 students did this task today as part of their lesson. The mini whiteboards (MWBs) have squares on one side, so they can use these for drawing axes.

I love using the MWBs because students feel less inhibited about making mistakes. Also, it gets them working faster; I seem to have a few students who want to draw perfect axes every time, and this can take up to five minutes!

This task is self-differentiating, since students choose the lines they want to draw. But one partner usually tries to outsmart the other, so most students get to try drawing (or guessing) the lines they find “harder”. Even though students can choose to stick to easy examples, they are encouraged not to by the structure of the task.

A pdf of this activity is available. Let me know if you try it!

Do you use mini whiteboards in class?

A Pythagoras’ Theorem Open Question


An open question like this one:

  • is easy to pose,
  • is easy to understand,
  • can lead to lots of practice (if that’s your goal for the students),
  • can be kept simple or extended to suit lots of students,
  • can lead to an interesting discussion,
  • can lead to a generalisation.

I used this with a year 11 class that are revising for their Foundation IGCSE exam. I asked students to come up to the board and sketch some triangles that fit the criteria. Some triangles had 10 cm as the length of one of the shorter sides. Others had 10 cm as the length of the hypotenuse. Some students had chosen to draw isosceles triangles; other students drew scalene triangles.

This task can be made more simple by asking students to draw accurate triangles and measure the sides. Then they can extend this to checking using Pythagoras’ theorem.

On the other hand, this task can be extended by asking students to generalise what they have found. If 10 cm is length of the hypotenuse, what can be said about the other two sides? If the triangle is isosceles, there is only one answer. But if it is scalene, perhaps students will call the length of the second side x. Then they can come up with a formula for the third side in terms of x. And how does this change if 10 cm is not the length of the hypotenuse but of one of the other sides?

Thinking Systematically

One of my goals at the moment is to encourage systematic thinking. My students should learn to think mathematically, and systemisation is an important part of that. Tomorrow’s year 10 starter will be there two little questions. They are from the ATM book Eight Days a Week.

My students are very algebra-focused, so these questions are good because they cannot resort to a formula!