# Four Bead Mix

### Task 136 ... Years 7 - 10

#### Summary

Ratio is the focus of this task and the context is a real-life one. It is possible to purchase packs of mixed beads. Students explore different ratios given a fixed total of beads and hopefully discover that some ratios are 'nice', or 'work' for the total, but others don't. What is special about the ones that work? The task also encourages students to work the other way. That is, given a fixed ratio, what totals of beads could result.

#### Materials

• Four bags of beads
Note: Some versions of this task use four beans
• One extra container for 'packing beads'

#### Content

• ratio
• addition and multiplication facts
• counting patterns
• factors
• fractions #### Iceberg

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

One way students might approach the first question is to count out a set of 3 As, 4 Bs, 5Cs, 6 Ds. Then count out another and another and so on until the total of 180 is reached. Doing this will almost force them to record something like:

 Number Bunch Total 1 3 + 4 + 5 + 6 18 2 3 + 4 + 5 + 6 36 3 3 + 4 + 5 + 6 54 4 3 + 4 + 5 + 6 72 ... 3 + 4 + 5 + 6 ...

Hopefully students won't have to complete the whole table before realising that it will take 10 bunches to make the total of 180 beads; also, that in doing so, there must be 10 lots of 3 for Bead A, 10 lots of 4 for Bead B, 10 lots of 5 for Bead C and 10 lots of 6 for Bead D. That implies the answer to Question 1(a) is 30:40:50:60. Similarly:

• 1(b) is 20:60:80:20 to make 180 because it takes 20 bunches of 1 + 3 + 4 + 1 to make the 180.
• 1(c) is 20:30:60:70 to make 180 because it takes 10 bunches of 2 + 3 + 6 + 7 to make the 180.
1(d) hopefully helps students realise that they can't make up any ratio and expect to reach a total of 180 beads. For example, changing 1(a) to the ratio 3:4:5:7 would give a total of 19 in the first bunch, 38 in the second and so on. So the total could never reach 180. (Students should feel free to check the counting patterns using the constant function of their calculator if necessary.) The data for those ratios that do and don't work should lead students to see that the ratio numbers must sum to a number which divides exactly into 180 - a factor of 180.

So, although it is true that Bead-The-Best could put any number of each of the four beads in the pack to make the total of 180, if there were no ratio involved, it is also true that as soon as a ratio is stated, the numbers involved must sum to a factor of 180.

• How many numbers are there that are factors of 180?
• How do you know that you have found them all?
• Can all of the factors be used to build ratios of beads?
For just one factor that does work, students might be asked to explore all the ratios of Beads A, B, C D that could be made. For example 1(a) and 1(c) are two ratios that total 18. Are they the only ones?

Question 2 builds the patterns in reverse.

• If the ratio is 1:3:5:2 the total in the bag can be 11, 22, 33, ... or any multiple of 11.
• If the ratio is 3:5:3:4 the total in the bag can be 15, 30, 45, ... or any multiple of 15.
• If the ratio is 3:4:6:6 the total in the bag can be 19, 38, 57, ... or any multiple of 19.
Extensions
The task is about packing a whole bag of beads, so it is appropriate in each case to also ask the students to record the fraction of each bead in the bag. In 1(a) for example:
Bead A = 3/18, Bead B = 4/18, Bead C = 5/18, Bead D = 6/18.

However, it may also be appropriate to choose one of the Bead groups as the whole, for example Bead B. Then in 1(a) there are:
3/4 times as many Bead As, 11/4 times as many Bead Cs and 11/2 times as many Bead Ds.

#### 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.

Coloured counters or cubes could substitute for beads to turn this task into a whole class investigation. Consider preparing four bags in four colours for each pair, with each bag containing 20 counters of one colour. If resources won't stretch this far, prepare the same for groups of four. To begin you might like to explore the ideas behind the task using the students. You will need squares of coloured paper from the art room for students to hold. The colours should match the colours of your counters or cubes.

Invite 12 students to come to the front and choose any colour. Record the ratio of students with each colour. Repeat with two or three more sets of 12 students. Check that students know what the recordings mean, then begin with a challenge such as:

Now I am just going to tell you the ratio. I want you to work out the total number of students who could be standing at the front. You can use your cubes to help you.
Continue from here using the explanations above.

At this stage, Four Bead Mix does not have a matching lesson on Maths300. However, Calculating Changes members could combine this investigation with Poly Plug, Proportion & Percent.

#### 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.

The Four Bead Mix task is an integral part of:

• MWA Number & Computation Years 7 & 8 