Estimating Products and Quotients

Multiplying and dividing decimals can seem a bit overwhelming! Luckily, there are methods and algorithms that break these problems down and make solving for products and quotients easy!

Lets try multiplying decimals and use the estimating products method! We will round each of the numbers to the nearest place value. So, 7.2 can be rounded DOWN to 7 and 37 can be rounded UP to 40. This will make it easier to multiply mentally. 7 x 4 = 28, so we know that 7 x 40 = 280.

Once you have estimated the product, have your child use a calculator to check their answer. When we solve this problem, we get a total of 266.4    Think you could get closer? Have your child try rounding UP and DOWN, to see which gives them the closest estimate possible! Using a calculator is a good way for students to try many different equations and practice rounding to the nearest place value. 

Lets try another!

Here, I chose to round 6.4 UP to 7 and round 45 DOWN to 40. When we solve 7 x 40, we get 280. Try checking your answer on a calculator. We get 288! So this answer is pretty close! If we rounded 6.4 DOWN to 6, and 45 UP to 50, we would have gotten a product of 300.

For more practice, check out the links provided on the side bar!

Now, what about division? How can we estimate quotients to divide decimals?

Lets look at an example:

We can cross off the number 8 that comes after the decimal.

After we do that, we get 28 divided by 9. No we can solve this problem, but the numbers are not compatible. If we try using 27 instead, we can divide evenly to get 3.

Similar to our estimating products algorithm, we are rounding to make solving the problem easier. When we check our estimate, we see that 28.8 divided by 9 = 1.
Our answer was pretty close!

Lets try again:

In this problem, I crossed out the number 9 (after the decimal point) and I chose to change 17 to 18 so that we could divide by 6 evenly. If we check our answer, we get 2.98 

Very close!

Multiplying and Dividing Fractions

This week we will be looking at some helpful approaches for solving multiplication and division problems of fractions.

Common Denominator Approach

There are several different methods we can use to help divide fractions. One of the approaches I find helpful for students, is the Common Denominator Approach. Lets take a look at an example!

Here we can see that the denominators are DIFFERENT. In order to divide them, we must give them both a COMMON denominator. Here, the best denominator (least common denominator) would be 6.

So, you would multiply BOTH of your denominators by a number that would give you 6. Then you would multiply that same number to both your numerators.

 Here, we can multiply 3 by 2 to get 6 as our common denominator. We would also multiply the numerator by 2. This gives us 4/6.

Here, we would multiply our denominator by 3, to get 6. Next, do the same to our numerator, which gives us 3/6.

Now that we have COMMON denominators, we can divide the problem. We know that 6 divided by 6 is just 1, so we can simply divide our numerators. 4 divided by 3 gives you 1 with a remainder of 1, which we put over 3, giving us a final answer of 1 and 1/3. 

Just to make there is no confusion, lets try another example! 

Once again, we must take our denominators that are DIFFERENT and make them COMMON. So, the least common denominator for this problem is 40. 

Remember, we are taking our denominator, and multiplying it by a number that gives us 40. Then, take that number (5) and multiply it by the numerator as well.

Here, we multiply 10 by 4 to get 40. Then, we do the same for the numerator (2 x 4 =8)

Ok! Now we have COMMON DENOMINATORS again! Lets divide!

15 divided by 8 gives you 1, with a remainder of 7. To write this as a fraction, we would place the remainder on top of 8 (7/8). Our final answer is 1 and 7/8.

Multiplying Fractions Using the Area Model 

When we are trying to multiply 2 fractions with different denominators, it sometimes helps to use a visual method to help understand what is really happening in the operation of multiplication. The area model helps students visualize the process of multiplication and understand how fractions look when drawn out. Below is an example of using the area model to multiply two fractions.

The problem reads : 2/5 x 3/4 = 

When we use the area model approach, it helps to draw one of the fractions horizontally and the other  vertically. 2/5 is drawn as 5 horizontal rows, with 2 of them shaded in (to represent two fifths). 3/4 is drawn as 4 vertical rows, with 3 shaded in (to represent three fourths). 

Next, we must combine both units together, to make one big unit that represents a new fraction. If we take 2/5 we get 5 rows across. When we take 3/4s, we add 4 rows going down. This gives us a new unit, with 20 squares. 2/5(yellow) and 3/4(pink) overlap and we now get a new fraction. 6/20 (the 6 orange squares) were made when we combined 2/5(yellow) and 3/4 (pink) together. 

To check your answer, simply multiply your numerators, and your denominators!

Need more practice? Check out the links provided below!! There are some great videos which help explain all operations with fractions and include this method as well!

https://www.youtube.com/watch?v=DJRryNHDyvw

http://www.softschools.com/math/fractions/

I have also included some worksheets for extra practice!! 

Fun with Fractions!

Here are some helpful facts to remember about fractions! I encourage you all to refer to this page when working on fractions with your child. Since it is not always easy to explain these concepts, it is important that we use a method that your child will understand. Your child will have an easier time understanding fractions when they can apply the concept to something tangible, or relatable. They need to be able to visualize HOW and WHY the algorithms work. In class, we have been using manipulatives as well as examples with cookies, pizza, and chocolate bars (who doesn't love those?!) Below are some key facts and examples for improper fractions, equivalent fractions, and converting improper fractions to mixed numbers!

Improper Fractions:

An improper fraction has a TOP number (numerator) that is bigger than the BOTTOM number (denominator). Take a look at this example:
7/4  This means there are 7 parts all together, and each part is a quarter (or 1/4th) of the whole.
                                                                                                                    

Look at the fraction using pizza pies, for instance.
This pizza is sliced into 4 equal parts. For us to represent the fraction 7/4 , we would need to have 7 total slices, which would look like this:              

                                                 

This brings us to the next key fact when learning fractions: mixed fractions.
A mixed fraction is simply converting an improper fraction, into a whole number and a fraction. So, instead of saying 7/4 we could say 1 and 3/4ths.                                                                                           

It is easy to see using the pizza model as a reference,                                                                

because there is 1 whole pie, and another pie with 3 (out of 4) slices.        

How do we convert improper fractions into mixed numbers? Its simple! Just divide the numerator by the denominator. Write the whole number and any remainder you have will go above the denominator. 

So, 7 ÷ 4  = 1 remainder 3 OR 1 and 3/4ths

Try another!

11÷4 = 2 remainder 3 OR 2 3/4ths

Equivalent Fractions:

These are fractions that look different because they use different numbers, but they are equal to one another and represent the SAME quantity.

For example, 6/12 can also be written as 1/2  which we can call reducing or simplifying.          
                                                         

In order to reduce equivalent fractions, we must divide the numerator and denominator by the SAME factor. In this case, that would be 6.
6÷6 =1 and 12÷6 = 2  
We can use the same concept and multiply the numerator and denominator by the same number to make smaller equivalent fractions larger. 1 x 8 = 8 and 2 x 8 = 16. So 1/2 is equal to 8/16 
                                                                                                                                 

Understanding Algorithms

When looking at a new concept or operation for the first time, many students (and parents) become overwhelmed, especially when dealing with 3 or 4 digit numbers! Luckily, there are methods or algorithms designed to make solving these seemingly difficult problems much easier. When explained and modeled the right way, it can change a student's perspective on math! It will help save students a great deal of time and frustration.

Partial Sums is a method designed to teach addition of multi-digit numbers. At first, it might seem more confusing for you than traditional methods for addition, but once you get the hang of it, you will have that "ah-ha" moment! When we use partial sums to add, we are simply breaking down the numbers into parts based on their place value. For example, if we look at the number 45, we are simply breaking it down into two parts : 4(which is in the tens place) and 5(which is in the ones place). We would start by adding the tens column numbers together(the highest place value in the equation) and then the ones. So, when we add two numbers together, such as 45 + 36, it will look like this:

Jump Back Subtraction uses a number line for students to visualize the difference between two numbers. It is particularly helpful when teaching students subtraction because it begins with the larger number and works backward, using addition to find the difference. It begins at the same point on the number line as the problem reads, so for example if you are given 87-34= , you would start at the end of the number line (on 87) and "jump back" to the number 34. As you are moving back on the number line, you count each point until you reach 34, and add them up. Since this algorithm is done for the purpose of visualization, here is an example you may understand better:

Partial Products is similar to the partial sums method, which is used in addition. It is a place value model for multiplication, which is helpful for students to use when understanding multi-digit numbers. Each number is separated into different parts according to their place value. If we look at a problem like 82x3, we know that 8 is in the tens place, and 2 and 3 are in the ones place. A helpful way to teach this method to students is similar to the partial sums method, where we highlight each digit's place value first. When multiplying a 2-digit number by a 1-digit number, we can write out the equation like this:

We break the number 82 down and multiply the tens place (80) by 3, and then the ones place (2) by 3. Finally, these products are added together to make 246.

Partial Quotients looks a lot like long division, however it uses "friendly numbers" to help students. The purpose of using partial quotients is to use mental math in order to solve the problem more quickly. When dividing two numbers with the partial quotients method, you are also using multiplication to help solve the problem. This is a good strategy to use to reinforce multiplication while learning division. In the example below, you can see how we begin with asking how many times 13 can go into 483. Starting with a number like 10 is helpful, since it is easy to multiply by. 13x10 is 130, so we subtract that from 483. Next, we try multiplying 13 by 10 again, and we continue to do so until the number we get is smaller than 130, and then try a smaller number, like 5. We continue this process until we reach the number closest to 13. This method is helpful because it uses multiplication, subtraction, and addition to solve, building upon what students already know. You can see in the model below how we can solve the problem in a row going down.

Technology Time!

Having a hard time getting your child to put down their iPad or tablet? Let's face it, we are living in a world where apps, smartphones, and social media are everywhere. Using technology to help your child learn can be one of the easiest strategies! Technology is incorporated into the classroom on a daily basis, and it really gets students excited to learn. Here are a list of some great programs you can use with your child on the go to help reinforce basic math concepts and practice, practice, practice! Check them out below :)

Splash Math - This app is great for grade levels K-5! It tracks and monitors your child's progress right to your phone, and is categorized by level and mathematical operation! It can also be used right on your computer, so no ipad or tablet is necessary. The games are interactive, eye catching, and provide incentive for children! While it provides great activites for each grade level, 3rd grade seems to provide the most skill topics, especially in multiplication and division.

Mathmateer - Mathmateer is becoming increasingly popular among math apps for children. In this app, children become "engineers" and must complete basic operations to earn money and build their own rocket ships! This app provides activities for all basic math operations, with 3 difficulty levels. It also provides different activities or "missions" everytime, so children are not constantly repeating the same activity. I would recommend this app for focus on addition and multiplication practice, as well as learning to count money.

Operation Math - Operation Math takes children on top-secret missions, where they must solve equations to complete missions successfully. It offers different skill levels and provides over 100 activities for both addition and subtraction practice. Children become engaged quickly, as they must complete their "top-secret" missions in an allotted time frame, which motivates them to think quickly and practice mental math skills.

Math Splat - Students must "splat" the answers to all questions as fast as possible in this highly motivating app. There is an option to save multiple profiles and customize your own difficulty level, which is a great feature when using the app with more than one child! This app allows parents to create and print awards and certificates for their children as they complete different levels and activities. This app is most appropriate for focus on addition and subtraction.

Math Duel - This might just be my favorite app of all! Math Duel is an app with a split screen, which allows two players to participate at once! This helps motivate students and is a great way for you and your child to practice and interact with one another! Rather than sitting down and drilling math facts or doing written homework, parents can sit and play games on this app with their child! It provides differentiation for all levels of ability, and can track both accuracy and speed. This app is a great way to practice multiplication and provides extra focus on division, which often comes more difficult to students.