3rd Grade - Expressions, Equations And Inequalities

Expressions, Equations and Inequalities
Represent relationships of quantities in the form of mathematical expressions, equations, or inequalities.
Algebra and Functions: Making equations and inequalities The ability to recognize that numeric quantities can be expressed using various number combinations and still represent the same value. 

Sample Problems


How do I know where the missing number goes in an equation? The child will be able to explain and use symbols for unknown quantities. (? , x, empty box)


What does an equation look like? The child will recognize equations involving only one operation up to 99. An equation is a mathematical statement that has numbers on either side of an = sign. The numbers on the left side has the same value as the numbers on the right.


What do I do when I see an empty box or question mark in an addition or multiplication problem? The child is able to find missing addition and multiplication facts.


How do I know when and where to put a symbol? The child uses symbols to represent problem situations, which involve unknown quantities.


How can a real-world problem look numerically? The child will be able to formulate problem situations involving one-step equations in one unknown with a whole number solution.


What is a number family? A group of three numbers that is linked together in either multiplication and division or addition and subtraction.

Learning Tips


Ask your child to explain to you what he/she are the same about addition and multiplication. (Both join sets of items into a larger group) How are they different (You can combine sets of different sizes when you add. In multiplication you can only combine sets that are the same size). Ask your child to show you what mathematical symbol should be used if you plan to join sets of different sizes into one group (+) and what mathematical symbol would be used to show that sets of the same size are being grouped together (X).


Give your child about 25 small objects—pennies, buttons, small cubes, pieces of dry pasta, etc. Ask him to show you two groups of items that could be added (2 piles of items; a different number of items in each pile). (In order to continue, don’t use more than half of your counters) Ask your child to count how many are in the two piles if they were joined together. Now ask your child to make another pile that has as many items as the 2 smaller piles together. Are the two piles equal? Discuss what “equal” means and make sure your child knows the mathematical symbol for “equal”. (=)


If you believe that your child does not fully understand the above “equality” principle (children come to this realization at different ages) use the items that were counted out in Tip 2 above. Ask your child to pick up one item from each pile, group them into a pair and set them aside while counting “one.” Continue to pick up pairs, one from each side of the 2 piles, counting as you go. This is called “1 to 1 correspondence” and having an “in your gut” feeling that it is true is essential for a child to be ready to grasp mathematical calculations. (Another test of a child’s readiness to work with these concepts is to verify that your child understands that ten pennies is the same number as 10 horses and that 10 horses are not more just because they are bigger, more spread out, or taller, etc.)


Introduce your child to the Number Family Scale. On the activity sheet that accompanies this lesson some scales are drawn, but you can easily draw your own. In the center of the scale will be two symbols. There will be a plus sign and a minus sign or there will be a multiplication sign accompanied by a division sign. Each of those pairs represent the same function: either items of different sizes are being grouped or separated (addition/subtraction) or groups of items with the same number of items in one group are being combined or divided back into groups of the same size as originally combined. For this activity, we will work with only 3 numbers (two items to combine according to the rules, with the third number representing the result of that operation) although obviously more than 2 items could be added together or multiplied together.

What you want your child to realize is that there are many sets of 3 numbers with a relationship such that if he/she knows any two of that set, and how they are related (whether the unifying operation is addition/subtraction or multiplication/division) then the third number can be figured out. By using the scale graphic your child can easily keep track of how the numbers are related. Since we will work with the same number-facts combinations that your child is learning (addition facts and multiplication facts) you will be reinforcing practice of these combinations.

After you have tried the provided problems, draw scales of your own and make up problems. You want your child to get to the level of competency with manipulating these number facts that he/she can quickly name the missing number. When this happens you know that not only has your child mastered the basic combinations used in arithmetic calculations but also “sees” the three numbers in their special relationship with each other.


Once your child is very comfortable with expressing how these numbers relate to each other, introduce our more formal way of writing down these relationships. We call them “equations” and just as the scale had to balance, our equation has to balance. The operation called for on one side must result in the other number of the relationship appearing on the other side of the equation.

Since we are using facts that your child knows, there is no need to teach additional rules for equations just yet. Instead of using the scale, write a number relationship like this:

4 + 2 = q

Ask your child, if this were our scale, what would go on the side where the box is? (6)

Then write those same numbers in this way:

q =4 + 2

What goes in the box? (6) Does it matter that it is on the other side? (No; 4, 2 and 6 “go together” in addition.

Now, write it this way:

4 + q = 6

What goes in the box now? (2, again because in addition, 4, 2 and 6 “go together.”

Ask your child to write as many number facts, both addition and multiplication as he/she can remember, making up little “problems” for him/herself and filling in the box with the correct number.

If your child knows multiplication facts, the same practice can be initiated, but the operations symbol will be X instead of +.

Extra Help Problems


What is the other number that belongs in this group?

q = 4 + 9 Answer 13


16 = 8 + qAnswer 8


2 + q = 5 Answer 3


7 + q = 15 Answer 8


4 = 3 + qAnswer 1


8 + 7 = qAnswer 15


12 + 2 = qAnswer 14


6 + q = 11 Answer 5


8 + q =12Answer 4


2 + 9 = qAnswer 11


9 + 8 = qAnswer 17


7 + q = 14 Answer 7


(Challenge!) 2 + 3 + q = 4 + 1 + qAnswer varies


2 X 3 = qAnswer 6


4 X q = 12 Answer 3


7 X 7 = qAnswer 49


4 X 9 = qAnswer 36


7 X q = 28 Answer 4


8 X 7 = qAnswer 56


6 X q = 48 Answer 8


273 X 0 = qAnswer: 0 (any number x 0 = 0)


7 x q = 42 Answer 6


6 x 6 = qAnswer 36


1 x 1 = qAnswer 1


(Challenge!) 4 x q = 6 x qAnswer 3, 2


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