Teaching kids to how to solve math problems is a huge challenge, but often the biggest challenge is knowing where to begin instruction. Without administering some type of pretest, you risk boring your students with problems that are too easy or frustrating them with problems that may seem impossible.

• Math Problem Solving
• Math Problem Solvinguseful Things To Know About You
• Types Of Problem Solving
• Example Of Problem Solving Skills

1. Learn how to solve math problems becomes a crucial aspect in the learning of mathematics. Children, by trial and error, develop their reasoning ability and learn to solve problems. They discover that there is often more than one way to solve a problem and more than one answer possible.
2. Get estimating with this spooky activity that asks your young mathematician to rely on their problem-solving skills to figure out how many items they believe are in spooky Halloween jars!

Before you begin, you need to have some idea of their current problem solving skills. For example:

Math is different than any other subject. It's objective, employs specific procedures, involves symbols and formulas, and it has a vocabulary all of its own. Consequently, the study skills you need to succeed in math are also unique.

• How do they attack different types of problems?
• What strategies do they use?
• Are they functioning below grade level, at grade level, or above?
• If they struggle, is it due to poor computation skills, poor reading skills, or misconceptions about basic math concepts?

The Problem with Problem Solving Pretests

Unfortunately, most math word problem pretests don’t provide enough information to help us answer those questions, let alone know where to begin instruction. Many tests are so challenging that kids who’ve been out of school all summer are likely to give up after making a token effort to solve the first few problems. Also, most tests use a multiple choice format which makes them easy to grade, but not so easy to interpret. Students don’t have a place next to each problem to show their work, so you’re left guessing as to the reason they missed each incorrect answer.

Problem Solving Assessment Made Easy

Since I couldn’t find a math problem solving pretest I liked, I decided to create my own. After a bit of planning, testing, and tweaking, I developed a set of leveled pretests and posttests. I designed it so that there are just 4 word problems per page leaving room for student work. Each test is 4 pages long, but most students will only need a few of the pages because they don’t move to the next level unless they complete 3 of the 4 problems correctly.

The Problem Solving Assessment pack includes a pretest, a posttest, answer keys, a form for recording and analyzing your assessment results, and a scoring guide. The assessment is available in two versions, the American Version which uses customary measurement and the International Version which uses metric measurement. If you’d like to use these assessment with your students, click the link below, fill out the form, and I’ll send them both to you!

Snapshots of Mathematical Thinking

The assessments aren’t multiple choice, but they’re super easy to score. The best part is the insight you’ll gain by observing HOW your students are solving the problems! Sometimes kids will get the right answer, but when you examine their work, you’ll see that they overlooked a method that would have been much simpler. Or perhaps they accidentally got the right answer but their work shows a lack of understanding of essential math concepts. As I examined each test, it almost felt as if I was peeking into the student’s brain to see his or her thought processes at work!

Ultimately, the pretest data provided me with really useful snapshots of student thinking that helped me decide where to get started with my problem solving instruction. Looking at the test results as a whole also gave me an overall picture of my students’ math abilities as a class. An added benefit for me was knowing where to start my students in my Daily Math Puzzlers problem solving program.

If you teach upper elementary students, I think you’ll find these assessments to be informative. Give them a try and let me know how it goes! Good luck with math problem solving this year!

Sometimes we need to solve Inequalities like these:

Words
>	greater than
<	less than
≥	greater than or equal to
≤	less than or equal to

Solving

Our aim is to have x (or whatever the variable is) on its own on the left of the inequality sign:

Something like:	x < 5
or:	y ≥ 11

We call that 'solved'.

Example: x + 2 > 12

Subtract 2 from both sides:

x + 2 − 2 > 12 − 2

Simplify:

x > 10

Solved!

How to Solve

Solving inequalities is very like solving equations .. we do most of the same things ..

.. but we must also pay attention to the direction of the inequality.

Direction: Which way the arrow 'points'

Some things can change the direction!

< becomes >

> becomes <

≤ becomes ≥

≥ becomes ≤

Safe Things To Do

These things do not affect the direction of the inequality:

• Add (or subtract) a number from both sides
• Multiply (or divide) both sides by a positive number
• Simplify a side

Example: 3x < 7+3

We can simplify 7+3 without affecting the inequality:

3x < 10

But these things do change the direction of the inequality ('<' becomes '>' for example):

Multiply (or divide) both sides by a negative number
Swapping left and right hand sides

Example: 2y+7 < 12

When we swap the left and right hand sides, we must also change the direction of the inequality:

12 > 2y+7

Here are the details:

Adding or Subtracting a Value

We can often solve inequalities by adding (or subtracting) a number from both sides (just as in Introduction to Algebra), like this:

Example: x + 3 < 7

If we subtract 3 from both sides, we get:

x + 3 − 3 < 7 − 3

x < 4

And that is our solution: x < 4

In other words, x can be any value less than 4.

What did we do?

We went from this: To this:

x+3 < 7 x < 4

And that works well for adding and subtracting, because if we add (or subtract) the same amount from both sides, it does not affect the inequality

Example: Alex has more coins than Billy. If both Alex and Billy get three more coins each, Alex will still have more coins than Billy.

What If I Solve It, But 'x' Is On The Right?

No matter, just swap sides, but reverse the sign so it still 'points at' the correct value!

Example: 12 < x + 5

If we subtract 5 from both sides, we get:

12 − 5 < x + 5 − 5

7 < x

That is a solution!

But it is normal to put 'x' on the left hand side ..

.. so let us flip sides (and the inequality sign!):

x > 7

Do you see how the inequality sign still 'points at' the smaller value (7) ?

And that is our solution: x > 7

Note: 'x' can be on the right, but people usually like to see it on the left hand side.

Multiplying or Dividing by a Value

Another thing we do is multiply or divide both sides by a value (just as in Algebra - Multiplying).

But we need to be a bit more careful (as you will see).

Positive Values

Everything is fine if we want to multiply or divide by a positive number:

Example: 3y < 15

If we divide both sides by 3 we get:

3y/3 < 15/3

y < 5

And that is our solution: y < 5

Negative Values

When we multiply or divide by a negative number we must reverse the inequality.

Why?

Well, just look at the number line!

For example, from 3 to 7 is an increase,
but from −3 to −7 is a decrease.

−7 < −3

7 > 3

See how the inequality sign reverses (from < to >) ?

Let us try an example:

Example: −2y < −8

Let us divide both sides by −2 .. and reverse the inequality!

−2y< −8

−2y/−2> −8/−2

y > 4

And that is the correct solution: y > 4

(Note that I reversed the inequality on the same line I divided by the negative number.) Player mfc application.

So, just remember:

When multiplying or dividing by a negative number, reverse the inequality

Multiplying or Dividing by Variables

Here is another (tricky!) example:

Example: bx < 3b

It seems easy just to divide both sides by b, which gives us:

x < 3

Math Problem Solving

.. but wait .. if b is negative we need to reverse the inequality like this:

x > 3

But we don't know if b is positive or negative, so we can't answer this one!

To help you understand, imagine replacing b with 1 or −1 in the example of bx < 3b:

• if b is 1, then the answer is x < 3
• but if b is −1, then we are solving −x < −3, and the answer is x > 3

The answer could be x < 3 or x > 3 and we can't choose because we don't know b.

So:

Do not try dividing by a variable to solve an inequality (unless you know the variable is always positive, or always negative).

A Bigger Example

Example: x−32 < −5

First, let us clear out the '/2' by multiplying both sides by 2.

Because we are multiplying by a positive number, the inequalities will not change.

x−32×2 < −5 ×2

x−3 < −10

Now add 3 to both sides:

x−3 + 3 < −10 + 3

x < −7

And that is our solution: x < −7

Two Inequalities At Once!

How do we solve something with two inequalities at once?

Example:

First, let us clear out the '/3' by multiplying each part by 3.

Because we are multiplying by a positive number, the inequalities don't change:

Math Problem Solvinguseful Things To Know About You

−6 < 6−2x < 12

Now subtract 6 from each part:

−12 < −2x < 6

Now divide each part by 2 (a positive number, so again the inequalities don't change):

−6 < −x < 3

Now multiply each part by −1. Because we are multiplying by a negative number, the inequalities change direction.

6 > x > −3

And that is the solution!

Types Of Problem Solving

But to be neat it is better to have the smaller number on the left, larger on the right. So let us swap them over (and make sure the inequalities point correctly):

−3 < x < 6

Example Of Problem Solving Skills

Summary

• Many simple inequalities can be solved by adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own.
• But these things will change direction of the inequality:
  • Multiplying or dividing both sides by a negative number
  • Swapping left and right hand sides
• Don't multiply or divide by a variable (unless you know it is always positive or always negative)