Less Than Fractions

Task 240 ... Years 4 - 10

Summary

Number tiles (1 - 9) allow students to experiment with fractions less than 1 in a non-threatening, open-ended way. Early success is guaranteed because there are 36 possible answers and the obvious one is 1/2. But the greater challenge is to add two fractions (each tile can be used only once) and still get an answer less than one.
  • How many solutions are there?
  • How do you know when you have found them all?
  

Materials

  • Nine tiles numbered from 1 to 9
  • One inequation board

Content

  • addition
  • fractions, calculations
  • fractions, equivalence
  • fractions, whole & parts
  • inequalities
  • mental arithmetic
  • multiplication, calculations / times tables
  • number line
  • numbers, triangle
  • patterns, number
Less Than Fractions

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card. 		   

Exploring questions 1 and 2 soon leads to breaking the problem into smaller parts.

  • Choosing 1 as the numerator means the denominators making a fraction less than 1 are 2 ---> 9 ... 8 solutions
  • Choosing 2 as the numerator means the denominators making a fraction less than 1 are 3 ---> 9 ... 7 solutions
  • Choosing 3 as the numerator means the denominators making a fraction less than 1 are 4 ---> 9 ... 6 solutions
  • Choosing 4 as the numerator means the denominators making a fraction less than 1 are 5 ---> 9 ... 5 solutions
  • Choosing 5 as the numerator means the denominators making a fraction less than 1 are 6 ---> 9 ... 4 solutions
  • Choosing 6 as the numerator means the denominators making a fraction less than 1 are 7 ---> 9 ... 3 solutions
  • Choosing 7 as the numerator means the denominators making a fraction less than 1 are 8 ---> 9 ... 2 solutions
  • Choosing 8 as the numerator means the only denominator making a fraction less than 1 is 9 ... 1 solution
  • Choosing 9 as the numerator means there are no tiles for the denominator which can a fraction less than 1.
Adding the sequence 8 + 7 + 6 + ... + 2 + 1 gives the total number of solutions. Can you check your answer another way?

Challenge

If students have completed the first part of the task by breaking the problem into parts, then they will probably begin the Challenge with the same strategy. For example, if the first fraction is 1/2:
  • Choosing 3 as the numerator for the second fraction means the denominators making a sum less than 1 are 7 ---> 9 ... 3 solutions
  • Choosing 4 as the numerator for the second fraction means the only denominators making a sum less than 1 is 9 ... 1 solution
  • Choosing anything else as the numerator for the second fraction means that nothing else placed as the denominator makes a some less than one. Either the fraction created is improper (such as 5/4), or equal to one half or such that the sum is larger than 1.
However the first part of the task tells us that there are 35 more possibilities to try for the first fraction. That's a lot a trying, so perhaps it's time to look for unstated information that might help cut through some of the work.

A key is that each fraction chosen to be the first one has a complement - a 'partner' that makes the total exactly equal to 1. That partner may or may not be in the set of fractions that can be made with the remaining tiles once the first fraction is chosen. For example if 1/2 is chosen as the first fraction, its complement is 1/2. The 1 and 2 can't be used again, but the equivalent of 1/2 could be made with 3/6 or 4/8 (but not with 5/10 - why?)

However we don't want a sum that exactly equals 1. We want a sum that is less than one. So, once the first fraction is chosen we want a second fraction made from remaining tiles that is less than the complement of the first fraction. Therefore our work will be cut down a little by:

  1. Choosing the first fraction (36 choices).
  2. Calculating its complement.
  3. Finding all the 'tile' fractions less than the complement.
The first two steps are easy, so the problem reduces to finding a process for deciding which of two fractions is the smaller. This is the opportunity to ask students how they would make this decision. Be prepared for some interesting answers. Three approaches that work are show below and all involve creating a 'level playing field' by using the same whole to show each fraction. The examples are based on asking
  • Which is smaller, 3/5 or 4/7?

Diagram

- Draw the same whole twice.
- Divide one into sevenths and one into fifths.
- Line them up and use a straight edge to compare.

Depending on the fractions being compared, this process needs a reasonable amount of time and accurate measurement and consideration of scale. If the drawing is too small the decision won't be conclusive. Further if it was necessary to know by how much one was smaller than the other, there is no real answer except 'that much' and pointing.

However, choosing not only a common whole, but a way to divide it into parts that accommodate both fractions:

also answers which is smaller and measures how much smaller. 		Common Denominator

Adding the 35ths whole to the diagram in the previous column is the visual/measurement equivalent of choosing a denominator against which both of the fractions can be compared - a common denominator.

The common denominator could be chosen by trial and improve... - Let's try tenths. Will that work?
- Nah! It only works for fifths.
- Well what about trying fifteenths?
- ...
(of course there is an assumption in this conversation that the students have seen the point of only using multiples of one or other of the denominators.)

...or the common denominator can be found be using the denominators to calculate the parts the whole has to be divided into to represent the middle whole in the diagram. Then using equivalent fractions and the identity property of multiplication:

  • 3/5 x 7/7 = 21/35
  • 4/7 x 5/5 = 20/35
and now we know which one is smaller. 		Darren's Method

Darren discovered this method in Grade 5 while doing exercises on comparing fractions.

Multiply on the cross and

write the answers above the numbers

The one with the smaller 'hat' is the smaller fraction.

Probably Darren didn't know why this algorithm works. Do you?
So, over to you:

  • How many solutions are there?
  • How do you know when you have found them all?

We look forward to posting solutions from your students here.

Extension

  • What happens if we investigate greater than fractions?

Note: This investigation has been included in Maths At Home. In this form it has fresh context and purpose and, in some cases, additional resources. Maths At Home activity plans encourage independent investigation through guided 'homework', or, for the teacher, can be an outline of a class investigation.

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works. 		   

It is very easy to get a class started with this investigation. A pair of students only has to fold and tear a piece of paper into 9 parts (there's a little discussion about thirds hovering in just that request if you want it) and number them one to nine. Then use another piece of paper to draw up a pair of boxes (in the middle of a landscape page) large enough to fit their number tiles. Draw your own boxes on the board as you explain.

It's easy to state the initial problem too:

Now put your tiles in the boxes to make fractions. When you make a fraction less than one, come and write it on our board.
Highlight your interest in less than fractions by including the inequality in the first part of the task and invite students to include it on their paper too. You might comment that the shape of the 'less than' sign is easy to remember because it is like a capital L turned a little.

As the data grows there will be opportunity for discussion about how we know a fraction is bigger than, equal to, or less than 1. A number line could be introduced to mark approximately where some of the less than fractions lie. The underlying intent of the conversation is to illustrate that everyone of us knows something about fractions, together we know more about fractions and by asking and investigating questions we can help each other learn even more about them. The first of those questions is:

How many less than fractions are there ... and how do we know when we have found them all?

Follow this through and when the 36 possibilities have been discovered and recorded introduce the second challenge.

Now let's see if we can use out knowledge to try something a little more challenging. Can we find any fractions that make this true...?
Turn the inequation on the board into the second one on the card by including fraction boxes to the left of the middle and the plus sign. Invite the students to do the same on their paper. The discussion above (and the students' suggestions) will guide the rest of the lesson. As suggested above, we would love to see your class's work investigating the number of solutions to this challenge.

At this stage, Less Than Fractions does not have a matching lesson on Maths300. However Maths300 does contain several lessons which explore fraction concepts and develop skill in operating with fractions. Those lessons are:

  • Lesson 33, Fraction Estimation
  • Lesson 72, Fraction Magic Square
  • Lesson 77, Rectangle Fractions
  • Lesson 84, Number Charts
  • Lesson 144, Rod Mats
  • Lesson 182, Fractions To Decimals
  • Lesson 191, Fractions & Fractions Charts
Most of these lessons include software support and each piece of software offers several levels of difficulty to allow for the range of students in your classroom.

Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner. 		   

Less Than Fractions is not in any MWA kit. However it can be used to enrich the Number & Computation kit at Years 5/6 and the Number & Computation kit at Years 9/10.

Green Line
Follow this link to Task Centre Home page.