3rd Grade - Solve Problems With Equations

Solve Problems with Equations
Solve problems involving numeric equations or inequalities.
Algebra and Functions: when to use operational symbols The ability to express a number sentence using standard, recognized symbols.

Sample Problems


What symbols do you see sometimes instead of “x”? (? or an empty box)


What symbol do we give a number when we don’t yet know its value? (often “x”, but any letter can be used to represent an unknown quantity)


How is an inequality solved? If you do a calculation on one side of an equation, you must also do that same operation on the other side. The child will be able to solve 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. Solving the equation often means finding the value of the variables or unknown quantities. For example, x = 5 + 10, x = 15


What do I do with an x in an equation? You want to get the “x” all by itself on one side of the equation. The child will be able to formulate and solve problem situations involving one-step equations in one unknown with a whole number solution.


Solving equations like these is the beginning of understanding what subject? (algebra)

Learning Tips


Review the learning tips and play the games in Lesson 17, which is the lead-in to this lesson.


If your child still struggles with basic number facts (addition and multiplication) provide plenty of manipulatives and try to use the combinations that your child does know in presenting these activities. Brain researchers tell us that we cannot learn two new things at exactly the same time. Children are at a disadvantage if they are trying to master a new concept while learning unfamiliar combinations at the same time. Teach a new concept with familiar facts.


If you remember your own mathematics education, you probably will think that the concepts in this lesson were presented to you when you were in high school. These concepts are included on state achievement tests for third grade; your child is learning this material in school. Studies have shown that most adults in America do not feel confident about their ability to help their child with math. Please try to approach these lessons with confidence that you and your child can enjoy these experiences together. The frame that you put around your image of you and your child’s work together should be that we are going to have fun figuring out how to solve little mysteries, or puzzles, depending on what your child enjoys. Praise your child often for even small steps of mastery. We will be working with larger numbers because by now your child can probably jump immediately to the “answer” if we teach these equation-solution skills with simple combinations. We want children to learn the “everything must be fair” principle when working out these mathematical puzzles.


In this next step we will take your child one step further in understanding how we perform calculations. Now we will introduce concepts that can be used when the needed calculation isn’t as obvious as one of the basic addition or multiplication facts.

Review the simple equations you did in Lesson 17. Now ask your child, “What if you needed to figure out what three numbers go together if the numbers were bigger than those in the relationships that you have memorized? There has to be a way to figure those out too!

There is. This is one way to write a larger addition problem—the way your child has been doing them, most likely:




Your child probably knows that it can be written this way as well.

324+ 135 = q

When children are taught to do problems such as these the emphasis is on getting the calculations right, not on making sure that the child understands that we are working with a number sentence, or equation. All that we are doing now is extending what your child knows about how mathematical thinking is done.

Another way to write this number sentence (equation) is to use an X instead of the box. It’s unfortunate that X is so commonly used, because young children associate it with multiplication so if this is a problem, use the box, but be aware that some tests will use the x.


(Tell your child when we don’t know what the mystery number will be we often give it the temporary name of “x”—The Mystery Name!)

If this problem were one of the number facts your child has learned, he/she would use the relationships he already knows to solve it. But this number set isn’t one of those, so we need to introduce the “Everything must be fair” rule:

What you do to one side (everything to the left or right of the = sign) you must also do to the other side. That’s only fair!

There’s one more rule: if there’s an X or a box in the equation we want to get it all by itself on one side or the other.

(If your child finds this hard to remember, you might use this little story: “We don’t know what’s in the box or who X is yet, do we? We should make him/her stand by himself until we know! This will seem fair to a child, he/she’ll be nodding yes in agreement.)

So, in the example above, to get the box (x) by itself, we need to take the 324 away from that side of the equation. If we take it away on the left side, we must also take it away on the right side (the “fair” rule):

324 – 324 + x = 459 – 324

Since 324-324 is 0, we can leave it out and write:

X = 459 – 324

Your child knows how to do this calculation! He/she can do mental math or write it down in another format, but the answer will be the expected 135.

You can prove to your child that these are “fair rules” by using this process to solve a problem in which he/she knows the relationship between the numbers:

4 + x = 9

Your child knows that in this instance the other number in the relationship is 5, but we will “prove” how it could be obtained if you don’t happen to know it:

Reminder: We want x by himself (or herself!)

Everything must be fair, so to get x alone we must take 4 away from the left side. If we take 4 away from the left side then we must take 4 away from the right side as well:

4 - 4 + x = 9 – 4.

X = 9-4

X = 5.

Allow your child to make up some problems like these and solve them.

(Very important! For now, gently direct your child away from any problems he/she makes up that require him/her to subtract the X or q. Such a problem would look like this: {4 – x=3} because if your child tries to follow the rules we are teaching your child would need to handle negative numbers and that’s really beyond the third grade curriculum! So, no – X or – q in the problems for now!)


Explain to your child that what he/she is learning is the way problems are solved in a VERY grown-up subject called “algebra” and that he/she is very smart to be able to do this while still in the third grade! And it’s true; the principles that your child is learning here about how numbers have a predictable relationship to each other and that relationship can be determined by manipulating the numbers in this particular way is using solid mathematical thinking that will serve your child well in the future. These are not mathematical “tricks” that happen to work, but mathematically-sound reasoning that is worth spending the time needed to teach these concepts.


The same process works for subtraction, as well as multi-digit numbers. Students can use manipulatives to practice adding numbers to balance the equation. For example, x – 8 = 2; 8 can be added to both sides of the equation x – 8 + 8 = 2 +8; x = 10.


Children can check their answers by substituting the value of x into the problem and making sure both sides of the equation balance. For example, 30 + x = 55 can be checked by replacing x with the value the child came up with. 30 + 25 = 55. 30 + 25 = 55 so 55 = 55. Correct.

Extra Help Problems


23 + 17 = q Answer: 40


16 + q = 19 Answer: 3


q + 24 = 40 Answer: 16


54 + 17 = q Answer: 71


3 + q = 39 Answer: 36


q + 16 = 99 Answer: 83


35 + 18 = q Answer: 53


11 + q = 49 Answer: 38


q + 56 = 63 Answer: 7


15 + 19 = q Answer: 34


4 + q = 8 Answer: 4


q + 132 = 179 Answer: 47


323 + 171 = q Answer: 494


316 + q = 518 Answer: 202


q + 224 = 404 Answer: 180


254 + 7 = q Answer: 261


53 + q = 135 Answer: 188


q + 616 = 634 Answer: 18


330 + 11 = q Answer: 341


431 + q = 559 Answer: 128


q + 223 = 639 Answer: 416


215 + 10 = q Answer: 225


74 + q = 89 Answer: 15


q + 49 = 69 Answer: 20


(Challenge) q + q = qAnswer: Any numbers that make a true number sentence.


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