07 Challenging Word Problem

Ali and John had 550 sticks of satay altogether. Ali had 50 more sticks than John. When John sold 1/3 as many sticks of statay as Ali, he was left with twice as many sticks as Ali. How many sticks did John sell?

The challenge is to “simplify” or “reduce” the model into some common parts that we can find the value of the common part. There might be different parts in the model and the trick here is to express the different parts into 1 common part that we can find. In algebraic terms, we are trying to express the multiple variables into 1 variable so that we can solve the equation. In this example, I am going to draw the model systematically. For the better students, they can skip a few steps if they can “visualize” the initial models in their hard.

Ali and John had 550 sticks of satay altogether. Ali had 50 more sticks than John. 

Most students should be able to draw this model without much prompting. Possible questions to ask : Who had more? How do I represent 550 in the model?

When John sold 1/3 as many sticks of statay as Ali, he was left with twice as many sticks as Ali.

There are two parts to this sentence. I shall tackle the 2nd part first as the 2nd part seems more easier (After all, “twice as many” does not seem as difficult as “1/3”).  Always tell the pupils to draw models that they find it more easier first. Possible questions to ask : Who had more now? How are the two “after bars” related? 

 When John sold 1/3 as many sticks of statay as Ali, he was left with twice as many sticks as Ali.

Now, for the first part. This is more complicated than the 2nd part as it involves “1/3” and “sold”. Possible questions to ask : What does the word “sold” mean? How can I show the “sold” in the after model? How can I show “1/3 sold”?

The model drawn below has captured all the information in the question. Get the students to see that it is impossible to solve for any of the parts. There is a need to “reduce” the model to some common part.  Possible questions to ask : Are all the parts in the model the same? Can I find any of the part? If not,  what must I do?

The trick is to relate the cyan parts and the blue parts together. Possible questions to ask : Is there anyway that the cyan parts and blue parts are related ? 
Some students might see the relationship. We can use MaPS to rearrange the parts so that the relationship is clearer.

Get the students to articulate the relationship

2 cyan parts = 1 blue part  + 50

Using this relationship, we can attempt to “simplify” the model. Emphasize the aim is now to convert the model into common part (i.e. I choose blue part). Get the students to articulate why they choose that as “common part”. It is okay if students choose cyan part to be the “common part”.

This is how the model looks like now:

We are still not too happy as the cyan parts stick out like a sore thumb :-) Possible questions to ask: How can I “get rid” of the cyan parts?  How can I get “2 cyan parts” as I know the relationship? Is there another way to draw the model? 

Now, get the students to redraw the model horizontally.

Now, the model looks much more “doable”. Possible questions to ask: Does the model looks more “friendly” now? Can you express the 2 cyan parts into blue parts?”

Finally, we get to the “solvable” model.

The battle is 75% won. The students just need to solve for the blue part. And make sure the final answer is what the question is asking.

 

 

06 Fraction Word Problem

4/5 of Peter’s money is twice as much as John`s money. What fraction of Peter`s money is John’s money?

This may seem a very easy question. But this is a really good test if the student has understood fraction. It is important that the students understand the concept of “in fraction, all parts are equal” very well. With that strong understanding, they can solve any problems involving fraction.

4/5 of Peter’s money is twice as much as John`s money.

Try to draw any bars to represent Peter and John. With the initial arbitrary bars, we try to resize the bars to reflect the correct relationship. Possible Questions to ask: “Who has more” ? “How do I show 4/5 ? How do I show “twice as much”?

 What fraction of Peter`s money is John’s money?

Possible Questions to ask: “Are the Peter’s parts and John’s parts equal? What does the word “fraction” mean?”

 

 

05 Ratio Word Problem (Involving 3 persons)

In my humble opinion, students usually have difficulty with mathematical word problems is  that they do not have strong basic mathematical concepts. For example, in teaching of ratio, the students must know clearly that “all parts are equal”. This fundamental principle can help them solving involving even the most difficult problem. For this word problem, I will try solving a ratio word problem

The ratio of the number of Alex`s marbles to Raju`s is 2:1 and the ratio of the number of Raju`s marbles to Jim`s is 4:5. Find the ratio of the number of Alex`s marbles to Raju`s to Jim`s.

The ratio of the number of Alex`s marbles to Raju`s is 2:1 and the ratio of the number of Raju`s marbles to Jim`s is 4:5.
Let’s “attack” the first part of the statement (The ratio of the number of Alex`s marbles to Raju`s is 2:1). Possible Questions to ask: Who has more ? What does it mean by 2:1? How do you show it on the model?

The ratio of the number of Alex`s marbles to Raju`s is 2:1 and the ratio of the number of Raju`s marbles to Jim`s is 4:5.
Now for the 2nd part (the ratio of the number of Raju`s marbles to Jim`s is 4:5). Possible Questions to ask: Who has more ? What does it mean by 4:5?

Instead of building on the model for the first part, I will just draw the 2nd part “separately” before trying to make the association with the first statement.

 Find the ratio of the number of Alex`s marbles to Raju`s to Jim`s.

Get the students to look at the models drawn and get them to see why there is a need to break the bars to “common or same parts”. After all, ratio is all about “same parts”.

Possible Questions to ask: Can I find the ratio by looking at the model ? Are the parts equal? What does it mean by ratio? If Raju has 4 parts, how many parts does Alex have?

04 Common Part

In word problems, it is common that students are faced with models as shown below (with the parts of the 2 bars not being equal). We have to teach them how to further “simplify” the bars so that we can find the answer we want. I guess most of us would have taught our students the technique of breaking the bars into common parts.

There will be a need to ask questions so that the students see why there is a need to find a common part. In this case, image the total is 72.
Possible questions to ask: Are the yellow and white parts equal ? Why can’t we say “2 yellow parts + 3 white parts = 5 parts”? How can I further simplify these bars (i.e leading on the need to find the common part and hence the need to find the  common multiple o 2 and 3) ? Why must we choose the lowest common multiple of 2 and 3?)
Do note that in the video the parts are shaded in the same colour if they are equal.

One of the feature of the MaPS is that the same parts are shaded in the same colour. Bring the students’ attention to the fact all the common parts are shaded in the same colour (i.e. pink). Hopefully, this will let the students visualize what is going on when we break the bars into common parts.

My view is that students need to be convinced on why there is a need to find the common part. Hopefully, this reasoning will be etched on their minds and can adopt this very useful technique when there is a need.