Introduction to linear equations

Introduction to linear equations 

Forming and solving linear equations is usually a teething trouble that teachers and parents go through with the primary and middle schoollers. Here I am going to share my experience in conceptualising linear equations with my kids:

STEP 1: We all are familiarised with common balance, right? We have seen that in the Grocers as well as in our home kitchens while moms measure portions of flour for cake making. Now let us weigh an apple using the balance and let us also assume that an apple weighs exactly 10 gms. So, our balance will look like this after putting a 10 gms weight to counter the apple.

This gives us the first equation! Clearly, we use the “=” sign because the weight of the apple is exactly 10 gms.

STEP 2: Let us add 5 gms weight to one side of the balance. And then

It is no longer balanced! 15 grams on the right hand side of the balance is certainly heavier than the apple that was mere 10 grams.

STEP 3: Let us balance this then. Just an addition of a 5 gms to the left side (apple side) will do the trick.

So, we have learnt that:

We need to add the same value to BOTH side of the equation without disturbing the balance.

STEP 4: Let us do the same experiment of adding the same “thing” to both sides. Say, we add an orange to the left and right hand side. The “equation” continues to balance.

STEP 5: Let us add 5 gms again to either side. It will obviously remain balanced.

STEP 6: Can we now subtract 5 gms from either side? Looks like, we can.

So, we have learnt that:

We need to subtract the same value to BOTH side of the equation without disturbing the balance.

STEP 7: Let us go back to our apple + orange equation once again. Let us just DOUBLE the quantities on either side. The balance remains horizontal, i.e the equation remains balanced.

So, we have learnt that:

We need to multiply BOTH sides of the equation with the SAME VALUE so as not to disturb the balance.

Solving the first equation

Introduction to solving linear equations

The problem: Let us look at our apple and balance example to form an equation like the below:

Clearly, there are "known" and "unknown" values in the equation. The weights with 1 (gram) values are known because we exactly know how heavy each of them is. The weight of the apple is unknown and denoted by "A" here.

It is customary to keep the unknowns in the left hand side of the equation. No reason here. It is just some practice that we follow and we need to get used to.

So, let us try to find the weight of an apple. We can safely remove the three (3) weights from either side without disturbing the balance, right? That is also as per the lesson learnt in the previous post that we can subtract the same amount from either side of the balance so as not to disturb the balanced state. Lets see how that looks:

Now, we have learnt recently (previous post) that we can divide the left and right hand side by the same number so as not to disturb the balance.

Finally, we solved the equation. The value of A is 6! So, this is how we arrived at it:

Fraction comparison

This one is traditionally known as a hurdle for teachers, parents and students. When we were kids, we were taught to compare fractions in mechanical way. We were taught LCM/HCF before that. So, it was like 'follow the given process, write exam and show us your score' :-) I hated this approach. While teaching fraction comparison to my kids, I tried to be as natural as possible and tried to bring out the concept from hands-on techniques. Hope you find this useful:

2/5th: 2 out of 5 equal slices

3/5th: 3 out of 5 equal slices

 1. Same denominator fractions: compare 2/5 and 3/5

   * Resolution: Oh, this one is real simple. We have a pizza that is cut into five equal slices. If I eat 2 of these slices and you eat 3 of these slices, who eats more? Very simple! You eat more. So, 3/5 (3 out of 5) is more than 2/5 (2 out of 5). No brainer, right?

2/5th: 2 out of 5 slices

2/7th: 2 out of 7 slices

2. Same numerator fractions: compare 2/5 and 2/7
   * Resolution: This one is slightly trickier than the earlier problem, but not by much and can be solved by sheer intuition and visualization. There are two equal pizzas here. The first pizza is cut into five equal slices. The second pizza is cut into seven equal slices. So naturally, the slices of the first pizza is fatter than those of the second one. If I eat two of the fatter slices and you eat two of the thinner slices, who eats more? Definitely, it's me! Because the number of slices eaten are same, but the sizes of the slices are different. So 2/5 is larger than 2/7
      * So, in general, whenever I cut the same pizza into more number of slices, I reduce the proportions of the slices. This concept is VERY IMPORTANT. I strongly advise parents to get pizzas and cut in front of the kids to get this concept inside their system This will be very helpful throughout the logical part of their education in future :-)

3. Different numerators and different denominators: compare 2/5 and 1/3

 1/3rd: 1 out of the 3 slices shown in different color

• Resolution: This is the challenging one. But something like this may help:
• There are two things for the pizza slices: size of the slices and the number of slices. To compare the slices, we need either the size of the slices (case 1 above) or the number of slices (case 2 above) to be same. Here the problem comes because both the dimensions are different. We need to bring one dimension same for both the pizzas.
• Let us try to make the size of the slices to be same. To so that, we just need to cut both the pizzas into SAME number of slices. The size of each of the slices will automatically become same, right?
• The problem reduces to: how to cut the pizzas to equal number of slices?
• The first pizza was cut into 5 slices and hence could be equally distributed into 5 people.
• The first pizza was cut into 3 slices and hence could be equally distributed into 3 people.
• To make each of the pizza with equal number of slices, we need to ensure than each of them is distributable to 5 as well as 3 people. This is IMPORTANT.
• If the pizza can be equally distributed to 5 as well as 3 people, it should be possible to distribute to 3 groups of 5 people each (or 5 groups of 3 people). That means, it should be possible to distribute the pizza to 15 (= 5 x 3) people.
• What is the easiest way to cut the pizzas to equally distribute to 15 people? Well, it is 15 slices!



The first pizza was cut into 5 slices earlier, now it is  cut into 15 slices

• First pizza, already cut into 5 equal slices: To make it 15 slices, just cut each of the 5 slices into 3 thinner (and equally thin) slices. So each of the 5 fat slices become 3 thinner slices. So 2 out of the fat slices is actually 6 (= 2 x 3) thinner slices! The total number of slices is 15. So our earlier 2/5 is same as 6/15 after further cutting the pizza!

The second pizza was cut into 3 slices earlier, now it is cut into 15 slices. Each of the 3 slices has been cut into 5 thinner slices.

• Second pizza, already cut into 3 equal slices: To make it 15 slices, just cut each of the 3 slices into 5 thinner (and equally thin) slices. So each of the 3 fat slices become 5 thinner slices. So 1 out of the fat slices is actually 5 (= 1 x 5) thinner slices! The total number of slices is 15. So our earlier 1/3 is same as 5/15 after further cutting the pizza!

•  Now we have an easy problem in hand! Two equal size pizzas, both cut into 15 equal slices. I eat 6 of the slices (6/15) from the first pizza and you eat 5 of the slices (5/15) from the 2nd pizza. Who eats more? Obviously I eat more! So 6/15 is more than 5/15.
• Our current 6/15 is actually 2/5 in the problem. I mean 6 out of the 15 thin slices is same as 2 out of the 5 fat slices of the same pizza.
• Great! We are done, 2/5 is more than 1/3 :-)
• Let us see if we can arrive at a rule to do this quick:
• STEP 1: Find out the LCM of the denominators. 
• STEP 2: Multiply the numerators and the denominators of the fractions by the relevant numbers to arrive at the equivalent fractions so that the denominators of the fractions are same!
• STEP 3: Just compare the numerators!

Equivalent fractions

In this article, we will understand the concept of Equivalent Fraction from cutting a paper!

Let us start with a piece of paper, a pair of scissors, a pencil and notebook





Figure 1: The 'whole'





The entire piece of paper is what we will call 'one whole' or just 1.
Let us cut this paper into two equal pieces now.

Figure 2: The 'whole' cut into two equal pieces. Each piece is 1/2


Each of the cut pieces is now called as 'one out of two' or 'half' or just 1/2.

Let us cut each of these pieces, i.e halves into two equal pieces. By doing that, the entire piece of paper will be divided into 4equal pieces.




Figure 3: The 'whole' cut into four equal pieces. Each piece is 1/4




Each of the cut pieces is now called as 'one out of four' or 'one fourth' or just 1/4.

Let us now glue two of the smaller pieces together and compare the glued piece with a bigger piece that we had cut initially.


 2 x 1/4 = 2/4  --------- A

 2/4 = 1/2 -------------- B


It is clear from the picture that two of the 'one fourth's added together will be same as one 'half'.

We call 2/4 and 1/2 as Equivalent Fractions.

We could further cut each of the 1/4th pieces to two equal pieces and make 1/8th. The equivalent fractions will look like this:

4 x 1/8 = 4/8  --------- A

2 x 1/4 = 2/4 ---------- B
4/8 = 2/4 = 1/2 ------- C


Extending this further, 16/32, 8/16, 4/8, 2/4 and 1/2 are equivalent fractions.
As a rule, we need to multiply or divide the numerator and denominator of a fraction by the same integer to get its equivalent fraction.

Capacity problems for 3rd grade

1.     If a banana weighs 200 grams, how many will weigh 2 kgs?

2.     If a bottle of medicine holds about 100 milliliters, how many 5 milliliter spoonfuls can be poured from the bottle?

3.     Two rolls of carpet tape measure 35 meters and 50 meters long. What is their total length? 

4.     Tanush went on a walk for charity. For every kilometer he walked he earned Rs 3. He earned Rs 12. How many kilometers did he walk?

5.     A running track is 400 meters long. If you run 8 laps, how many meters will you have run?

6.     I walk 3000m on Saturday. On Sunday I walk twice as far. How far did I walk at the weekend?

7.     The mountain is 700m. I walk halfway and twist my ankle. How far did I get?

8.     I walked a total of 45km over three days. On day one I walked 15km, and on day two I walked 12km. How far did I walk on day three?

9.     I walk 800m in the morning and I walk twice as far in the afternoon. How far did I walk?

10. If 500g of cheese costs Rs 48. How much does 1kg of cheese cost?

11. How many 40cm pieces of string would you need to make a 120cm piece of string?

12.     Drona is 1.35m tall. Sukra is 1.80m tall. How much taller is Drona than Sukra?

13.     A bucket takes 20 liters of water. There is a hole at the bottom of the bucket. Every minute, 200 ml of water leaks out of the bucket. How much time will it take to get the bucket empty by water leaking?