08 P5 Ratio Word Problem

Mrs Lee made some curry puffs and sardine puffs. The ratio of the number of curry puffs to the number of sardine puffs was 4 : 3. She made another 65 curry puffs and sold 27 sardine puffs. The ratio became 5 : 3. How many curry puffs did she make?

Strategy : Draw model systematically (e.g., translate each statement into the “model” form). Try to look at the models drawn and adjust accordingly. 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.

The ratio of the number of curry puffs to the number of sardine puffs was 4 : 3.

Possible questions to ask: Did she bake more curry puffs or sardine puffs? How many parts for curry puff ? How many parts for sardine puffs?

It should be quite easy to draw the model for most of the students.

She made another 65 curry puffs and sold 27 sardine puffs.

Possible questions to ask : What does “made another 65 curry puff more? Do I need to “add” or “take away” for the curry puffs ? What does sold 27 sardine puffs mean? Do I need to “add” or “take away” for the sardine puffs ?

The ratio became 5 : 3.

Possible questions to ask: Did she bake more curry puffs or sardine puffs? How many parts for curry puff ? How many parts for sardine puffs? Should I drawn another 2 bars to represent that ? Are the parts equal for both “before” and “after”?

Now, take a look at the model. What is wrong with the model ? For the curry puffs, does the models show “65 made” ? For the sardine puffs, does the models show “27 sold” ?

There is a need to adjust the model!

Now the model looks like this:

The trick is to find relationship between the red parts and the yellow parts.

For Curry Puffs

5 red parts = 4 yellow parts+ 65

1 red part = 4/5 yellow parts + 13

For Sardine Puffs

3 red parts = 3 yellow parts – 27

1 red part = 1 yellow part – 9

Now, we have 1 red part in common. Redraw the model to show the red part in terms of yellow part

From the diagram , is obvious that

1 part = 13+ 9

That is of course, not the final answer.Continue and try to find  what the question is asking for.

 

 

 

 

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.

Teaching Before/After

As all mathematics teachers know, “before/after” problems are classic problems that our students will see most of the time.

Here are some of the tips that I have tried to teach my students.

(1) How is the “before bar” and “after bar”  related? That is the most difficult part. Once the students can see the relationship, half the battle is done.

(2) Is there any implied information given in the problem ? For example, I try to get my students to see if this is a special case of before/after. Like “You give me, I give you” (where the total is constant) and “Both increased or decreased by the same amount (implied difference is constant)

(3) Is the “before bar” and “after bar” the same ? Any information given in the question that implied this

This is a very broad principles that I have when I am teaching the students the “before/after”. Of course, when I am teaching that in class, I need to unpack the questions to even more minute questions so that the students can handle. This is a skill that I am still refining. Feel free to comment if you have more tips.

03 Word Problem (Another way for before/after)

Ahmad and Davis shared a box of sweets equally.
Ahmad ate 15 sweets and Davis ate 23 sweets.
After that, Ahmad had twice as many sweets as Davis.
How many sweets were there in the packet at first?

Another way to solve this before/after problem with MaPS. Hope that this will give you some ideas on how to “tackle” such classic problem.

Ahmad and Davis shared a box of sweets equally.
Possible questions to ask : What does “equally” mean? Do they have the same amount?

 Ahmad ate 15 sweets and Davis ate 23 sweets
Again break this statement into two part. Let’s take a look at the first part “Ahmad ate 15 sweets”.
Possible questions to ask: How do I show “ate 15”? Do I take away from the “before bar” or do I add on to the “before bar”?

Ahmad ate 15 sweets and Davis ate 23 sweets
Now for the second part”  “David ate 23 sweets”.
Possible questions to ask: How do I show “ate 23”? Do I take away from the “before bar” or do I add on to the “before bar”?

 After that, Ahmad had twice as many sweets as Davis.

This is the tricky part. Guess we must be go slower at this point
Possible questions to ask : Who has more now ? How do I show “twice as much”?

The model is now almost completed. Hopefully, the students can see the “answer”. If not, we will need to prompt them with questions like “What is the value of the yellow part?”

02 Word Problem (Before/After)

John had $40 more than Ali.
After John spent half of his money, Ali had $20 more than John.
How much money does John have at first?

Here is a typical before/after word problem.Again, I shall tackle this word problem “statement by statement”.

John had $40 more than Ali.

This is a simple enough statement that most pupils can draw (I hope :-)) I will be showing the screenshot of the model (instead of video)
Possible question to ask : Who has more ? How much more?

 After John spent half of his money, Ali had $20 more than John

Break this statement into 2 part. Let’s focus on “Ali had $20 more than John”.
Possible questions to ask: Who has more now? How much more? How are the “before and after” bars of Ali related ? Did Ali spend any money?

 After John spent half of his money, Ali had $20 more than John
Now, let’s handle the first part of the statement (“After John spent half of his money”).

Possible questions to ask: Does John has less money after spending money ? How are the “before and after” bars of John related related ? 

The final step is to draw the dotted lines to make the model more clearer.  Possible questions to ask : How are the “20” and “40” related to any parts on the model?

 

 

 

01 Word Problem (Comparison involving 3 persons)

Kai Yee, Lalitha and Monica share a total of $963 among themselves. If Kai Yee has twice as much money as Lalitha; and Monica has $18 more than Kai Yee, how much money does Kai Yee have?
In a step-by-step approach, we are going to unpack each statement into the models. It is okay that we do not know the exact size of the bars at first. As we go along. we have to resize the bars. The models might not be accurate at first and it is okay to resize the models. This may help the weaker students. This point is brought up in the article at the Mathematics Educator

Kai Yee, Lalitha and Monica share a total of $963 among themselves.
Possible Questions to ask :  How many bars must we draw? How do I show $963 ?

If Kai Yee has twice as much money as Lalitha
Possible Questions to ask : Who has more , Kai Yee or Lalitha? How do I show “twice a much”? Who has 1 part? Who has 2 parts?

Monica has $18 more than Kai Yee,
Possible Questions to ask : Who has more, Monica or Kai Yee? How much more?

At this point, it seems that all the information has been presented in the Model. We have to prompt the students to look out for the “implied” information.

Possible questions to ask :”Can we solve the problem by just looking at the model? Is there any information that we can put in ? How are the bars of Monica and Lalitha related?”

That is all for now. I will leave it to you to solve the rest of the problem. Too lazy to continue:-) Hope you find this useful!

Mathematics Problem Solving Process

Most of us are proficient Mathematics problem solvers such that we are able to “see” the solution once we see the problem. Even in drawing the model, we have already “see” in our head how the final model looks like. However, this is not true for our students. They are still novice problem solvers and cannot “see” the solution. Hence, it is important to make our thinking visible and unpack our thinking process into details such that our students can follow our thought process. I tried this with my class last year and it is not a very straightforward process. I have to really make a conscious efforts to break my thinking into steps that my students can follow.

In this series of word problem solving posting, I try to model such approach. Please note that I am doing this in my free time, so do not expect perfect work :-). This is in no way endorsed by the authorities but purely just me and my crazy idea of sharing my 2 cents worth on problem solving. Hope that the fellow Mathematics educators will find them useful. My approach of solving word problem is based on the following principles:

(1) Systematic and explicit instruction (Step-by-step approach)

(2) Self-instruction/questioning (For monitoring and management of their learning)

(3) Use of visual representation (Drawing of Model)

Hope you find them useful