Some of my Favourite Measurement Puzzles

In previous posts, I have shown how we can get students to understand various measurement formulas by developing these formulas with them. Once students understand these, it is important (in the context of a Balanced Numeracy approach) that students be given opportunities to practise using these formulas with straightforward, closed questions (e.g. find the area and circumference of a circle with diameter 10 cm). At the same time, it is also important that students are given opportunities to non-standard, open problems. This is where we can help students develop their problem solving skills and their perseverance. Over the past year or so, I've collected many great puzzles from the Math community on Twitter. Below, you will find some of my favourite measurement puzzles, mainly suitable for intermediate students. Try them yourself and then see if you can get each solution in a different, more beautiful way.  

Answers are in the back of the book.

Having Fun With Math Vines (or the Grapes of Math!)

Spatial Reasoning is such an important component of Math yet it is often overlooked. Too often, Math teaching rushes to abstract symbolic representations of certain concepts at the expense of pictorial representations. Recently, I've been working with my fellow Math consultants to make some Math Vines. These seem to have struck a positive chord with teachers and students alike in allowing them to truly appreciate these concepts. It is worth remembering that many math concepts (e.g. the Pythagorean Rule, completing the square) were first discovered geometrically and only later algebracised. If a picture is worth a thousand words, how much is a Vine worth? 
Over the next few months we are hoping to build a bank of these. They are fun to make (get your students to make some!) though sometimes I do find the six second time limit a bit restricting. Do let me know what you think of these.


This next one we might use as a Minds-On when learning about the area of a parallelogram:

When students are asked to complete the square, do they know what this actually means?

Using snap cubes can help students develop a much deeper understanding of slope and y-intercept.

And this is my first Vine: the difference between two squares!

You can follow our progress on our Vine account, Math Vine a Day.

Clocking On

A few years ago I was in the school office when a Grade 12 student (whom I knew as being bright) came in and asked what time her exam began. I told her 12:30 and she replied "What time is it now?" I pointed to the analogue clock behind me (it was 11:45). She paused, then confessed, "I don't know how to tell the time".
I have heard the argument that kids these days don't need to tell the time using analogue clocks as everything is now digital but I would wholeheartedly disagree with this. In fact, I believe that if students don't learn how to read an analogue clock, then they are at a disadvantage mathematically. Being able to read analogue clocks allows students to use, develop and practice so many math skills:

Estimating ("It's about two o'clock", "It's about ten to two.")

Skip Counting ("It's (5, 10, 15) 20 to three.")

Non-standard skip-counting ("It's (5, 10, 15, 20, 21, 22) 23 minutes past four.")

Fractions ("I can see two quarter hours are the same as a half hour.")

Decomposition ("15 minutes to is the same as 45 minutes past.")

Angles (Since angles measure turn, we can see the hands of the clock continually forming angles. When is the time an acute angle? An obtuse angle?)

All of this is fantastic for developing number sense and my view is that you cannot get anywhere close to this only by reading the digits off a digital clock.

I also feel though that analogue clocks (with their built in spatial nature) are far superior than digital clocks in getting students to understand what time is. They help students see the cyclical nature of time more clearly (something which will be important when they are working with elapsed time.)

The best sort of clocks for students to learn with are geared such that the hour hand moves with the minute hand. They might cost more than cardboard versions but the learning is much better. I also look for clocks that have clear numerals on the face: I often tell parents that they should buy analogue watches for their kids that have all the numbers 1 to 12 and not Roman numerals. I also recommend the Feel Clock app (available at itunes) because it has a very clear display and animates the background to emphasise the difference between a.m. and p.m.  

If students have access to these then they can also attempt problems like this one:

I can see a clock. The two hands are nearly touching but not on top of each other. What could the time be?

I tried this in a Grade 4 class recently. Initially it was interesting seeing how many kids tried to draw their solutions. This does beg the question: why do we get students to draw clocks? It is a tricky thing to do accurately (especially for times other than the o'clocks and half-pasts) and it is not a skill needed in real life. In the time it takes for a child to draw a given time on a piece of paper, I reckon another child could show at least four different times on Feel Clock. Anyway, one student realised that they could use the analogue clocks and before long, all groups were doing so: it allowed them to tackle the problem far more effectively.

Notice how the students have recorded their times digitally: they do need to know how digital times work as this is how time is often shown. In fact, this group saw a pattern in their first three solutions which they used to predict their next six.

Extensions of his question could be:

What times are the hands at right angles?

How many times a day do the hands form a straight line?

Have You Checked Your Work?

"Have you checked your work?"
I reckon this is one of the most common questions that teachers ask their students. And it is usually more of a rhetorical question as we ask it when we know that students have made a glaring error or have come up with an answer that makes no sense.
And usually the students look briefly at the work and say "Yep!" as if to say "Yep, it's still there!"
Checking your work is a vital piece of Polya's four step approach to problem solving. In many ways, I think it should feedback into the first three parts (Understand the Question, Make a Plan, Carry Out the Plan). My sense of things is that this maybe the weakest link for a lot of students.
I have been wondering why this is and now I'm thinking that maybe it is the question itself ("Have you checked your work?") which is causing the problem. Maybe there are better prompts to get the students to reflect on what they have written.
As a case in point, I was in a Grade 9 class recently and they were working in groups on this problem: try it yourself before you read on!!

What was interesting is that many students stopped after they found an answer. They didn't think to consider of there was more than one answer. It was almost as if they stopped out of habit: I have an answer, so now I'll just wait for further instructions.

Biting my tongue, I managed to avoid saying "Have you checked your work?" Instead I asked "Which two sides are equal?" They pointed to the 3x-4 and the 5x-8. I then asked "Are these the only possibilities?" 

Immediately, this had a much better effect than "Have you checked your work?" The students realised that the 3x-4 could be the same as x+6, or the 5x-8 could be the same as the x+6. There were some comments along the lines "They don't look the same!" but they were reminded that they were told that the diagram is not to scale!

But even if they had considered each of the three possible isosceles triangles, and had done the algebra correctly, there was one geometric error that kept on coming up and was not seen by the students: notice the 2cm, 2cm, 8cm triangle:

"How do you know this is an isosceles triangle?"

"Because two sides are the same."

"OK...how do you know it is a triangle?"
"Errr...because it has three sides that are joined together, (said in that 'D'uh' tone!).

"OK...if you can draw me that triangle, then I'll give you twenty dollars!"

Sometimes when we ask students to check their work, they might just check the algebra (which in this case was correct) but not realise that what this leads to is an impossible shape. Which is why I really like this question as it will force students to appreciate the triangle inequality: any two sides of a triangle must add up to more than the third side.
But to get students into this habit of thoroughly seeing if their solutions make total sense, we must provide them with rich questions that force them to look for different cases, or to examine the validity of their solutions.
If, however, we only provide them with questions where there is only one answer (and probably just one way to get this), then they won't get into the habit of effectively reflecting on their work. Thus, we cannot expect them to be complete problem solvers.

                           *                                   *                                    *
The problem above is adapted from the University of Waterloo's CEMC's Problem of the Week. 

Using Pattern Blocks to Reason and Prove

As part of one of our Ministry's Numeracy initiatives, we tried out the following task in a Grade 6 classroom. 

Find the size of each of the angles in the pattern blocks shown. Protractors are not allowed!

Our Minds-On discussion showed that the students had a good grasp of what acute, obtuse and right angles were. At the same time there was some disagreement as to what a polygon was: it was only after a fair bit of using counter-examples to their definitions that they agreed that polygons are closed 2-D figures made up of entirely straight lines (which begs the question, are pattern blocks polygons?). 
When the students began the task proper (and after they asked me a dozen times if I really meant they couldn't use their protractors!) it was interesting to see their strategies.
Some just estimated angles (below). It was a quick fix to show these students that some of their answers were contradictory and that they needed exact answers instead. 

Some recalled that there are 180° in a triangle so this meant that each angle in the equilateral triangle was 60° and progressed from there. Others put compared three equilateral triangles to two right angles (or squares) as shown:

Others benefited from marking the angles they were trying to find actually on the pattern block as this helps them see the amount of 'turn' between the two sides (check out this post for some ideas on how angles measure turn):

As much as possible, I am learning to hang back and not jump in and show students what to do: when I do this, I realise that it is me doing the maths, not them; it is me reasoning and proving, not them. There certainly are some awkward pauses when I do this but this is most likely because the students are mulling over what they could do.
And sure enough, they could solve the problem without my help:

And for fun, they could verify their solutions in more than one way:

So when it comes to problem solving, I really like the philosophy from Singapore's Ministry of Education:

Teach less, learn more.

Patterns are Predictable

I made the point in previous posts (The Study of Patterns and Next-Near-Far-Any ) that algebraic reasoning is based on our ability to notice patterns and generalise from them. As our students become more proficient at spotting patterns we must get them to realise that patterns are predictable: if you have spotted the pattern rule correctly, then you will be able to predict any term with 100% accuracy. 
This differs from the predictions we make in, say, handling data, where our predictions (even though they might be well thought out) aren’t guaranteed to hold. 
The Next-Near-Far-Any continuum is also useful in helping our students make predictions. First get them to predict what might come next. Then predict a ‘near’ term such as the tenth (this might be done continuing the pattern one at a time). From junior grades onwards, students might then be asked to predict a ‘far’ term such as the hundredth. As it is now impractical to continue the pattern one at a time, students will now begin to generalise patterns between the position number and the term number. This is more easily done when we use visual patterns. From intermediate grades onwards, students might be asked to predict any term: this is true generalising. In other words, for position number n, what is the term number in terms of n?
A great source of pattern prompts can be found at Fawn Nguyen's Visual Patterns site. These type of questions are 'Here are the first three terms: figure out what happens next.' They can be relatively straightforward with colour being used as a clue to how the pattern grows:

to more complex such as this one which asks 'How many visible spots in this view are there?' (The first term has 33 visible dots):

I love these types of questions but I like to balance them questions where students are given the chance to make their own patterns: I am always blown away by how creative they can be. Last week I was in a Grade 5/6 class and we asked the students to make their own patterns. They then had to look at someone else's pattern and predict the tenth and hundredth terms. Here is a selection of what we saw:

Notice how some of these combine growing patterns with repeating patterns. For example, to predict the hundredth term of the first pattern above, you would need to notice that the vertical line increases by one each time and that the colours rotate in a cycle of four. Since every fourth term is a blue vertical line with a single green square, we can predict that the hundredth term will be 100 vertical blue squares attached to a single green square. It was great seeing the students make and justify their predictions.
A variation on this is where you give a single term and ask the students to fill in the pattern from this single piece of information. This one (from Marian Small) is my favourite and I've used it with students from grades 4 to 11.

This is the fifth term. Make the first four terms. 
Students always come up with a variety of patterns: some increasing; some decreasing; some linear; some non-linear. I like to plot these patterns on a single graph and ask students to explain why these plots cross at the same point.



And if you need further evidence of the creativity of students, a colleague gave the dice problem above to his computing class and before long, one of his students had written the following program (in Turing, I believe) to solve the problem. It is, I hope you agree, a thing of beauty.

function dice (term : int) : int
    var dotnum : int := 0
    dotnum += 10 %Tops of yellow and red dice *
    dotnum += term * 15 %Sides of yellow and red dice *
    dotnum += 5 * term ** 2 %Tops of white dice *
    dotnum += 3 * term ** 2 %Sides of white dice *
    dotnum += 6 * term ** 2 %Sides of white dice
    dotnum -= 6 * term %Covered sides of white dice
    result dotnum
end dice

Bar Models 2

Following on from my last post, here are a couple of further examples of bar models being used.
The first is in a Grade 6 class and we wanted to specifically give them a multiplicative comparison problem. These problems tend to be more difficult for students compared to equal groups problems; I think this may be because students have a hard time visualising what is happening. Bar models can certainly help here.
Our question was:

Mr. Jacobs eats some candies. Mrs. Delaney eats four times as many candies as him. Together, they eat a total of 260 candies. How many candies did each of them each?

A typical response is shown below.

By using bars to represent how many candies Mr. Jacobs has and how many candies Mrs. Delaney has, the calculations needed to be performed are clearer to see: the combined five bars must represent the total 260 candies. Thus one bar must be 260÷5=52.

In another example (this time from a Grade 2/3 class) the student uses a bar model to visualise this part-part-whole problem:

A school has 83 water bottles. If 29 are filled, how many are not filled?

A typical response is shown below:

What I like about this, is that the student drew a bar model first and used this to help decide what number sentence could be written. Once this was done, the student decided to use an empty number line to calculate the difference. So whilst this might be thought of as a subtraction problem (when the unknown isolated, you get 83-29) it can be solved by addition. The beautiful thing about bar models is that they allow students to visualise both ways.
I'm not sure if students will become experts at using bar models to solve problems unless we as teachers model them effectively and consistently. In many ways a whole school approach is necessary. In primary, teachers should model how to use bar models to solve additive thinking problems. In junior grades, teachers need to model how to use bar models to solve multiplicative thinking problems. In intermediate grades, teachers need to model how to use bar models to solve proportional reasoning problems.

The pay off will be massive.