Estimating vs. calculating sums

Addition and subtraction are fundamental skills that we learn in math. We use them throughout our daily lives for a variety of reasons including: adding up the cost of items or money; measuring to calculate the perimeter; finding the difference in spending, ages or distances, just to name a few. How have you used addition or subtraction in your day-to-day life? If you were to add 67 + 43 without using a calculator, how would you do it? We can estimate or calculate the answer using a variety of strategies. When might you want to estimate a sum versus calculate?

Comparing sums and differences

Before we get started, let’s quickly review the different strategies and approaches to an addition or subtraction number sentence. These strategies help us understand the relationships between numbers. Explore the strategies in the chart provided with a partner, if possible.

Addition strategies

Expanded form (adding from left to right)

In this strategy, you will break each addend apart so that its tens and ones are separated. Once you have done this, you will add your ones together and then add your tens together. Lastly, you will add both totals together to get your final sum. Examine the example in the following image that demonstrate this strategy.

Number line

Create a number line and start with the larger addend. Then, count up by starting with the largest place value. For example, 89 + 34, start at 89, and then add three groups of ten. Lastly, add the ones. Examine the following example representing the number line.

Vertical addition

This is similar to the break apart strategy because we are still focusing on adding the ones and then tens separately. Once you have added each place value separately, add them up together. The following image represents the examples:

In the first example, add 9 to 4 to get 13, then add 80 and 30 to get 110. Then add these amounts together to get 123.

In the second example, add 9 and 3 to get 12, then add 80 and 50 to get 130, then add 300 and 200 to get 500. Then add these amounts together to get 642.

Algorithm

For this method, called the algorithm method, begin by lining up your addends on top of each other. Make sure the place values are lined up so you can add the ones to the ones and the tens to the tens. First, add the digits in the ones column (on the right-hand side). Record the answer under the ones, then continue by adding the tens, the hundreds and so forth.

Examine the following example of the algorithm method:

Subtraction strategies

Base ten drawing

To begin the base ten representation, first, create base ten blocks to represent the larger number. Then, cross off and take away flats, rods or units that represent the smaller number. Then count up the remaining base ten blocks to get your final answer. At times, you may need to break apart rods or flats if smaller units are unavailable. Examine the following base ten image representing this example:

Expanded form

For the expanded form method, line up the numbers in your subtraction sentence, as if you were using the algorithm method. Then, beside each number, represent it in its expanded form. Subtract the ones column, and then subtract the tens column. If the number on top is smaller than the numbers on the bottom, you will need to borrow a group from the next place value over.

In the example represented in the following image, we are expanding the numbers 83 to 80 + 3 and 55 to 50 + 5. However, when we do that, we realize that 5 cannot be taken away from 3. Therefore, we must borrow a group of ten from 80. As a result, we end up representing 83 as 70 + 13 which will make it easier to subtract.

Number line

For this method, create a number line. Start with the larger number on the right-hand side. Then, expand the number that you are subtracting. Hop backwards on the number line by separating that number into tens and ones. The following image represents this action.

Algorithm

For the algorithm method, line up the numbers that you will be subtracting. First, subtract the numbers in the ones column, and then tens, etc. If the number on the bottom is larger, you will need to borrow a group from the next largest column. The following image represents this method:

Task 1: Try it out

Solve the following equations using three different strategies. One of your strategies should be algorithm. Which strategy did you find easiest to use?

26 + 89 =

247 − 189 =

576 + 213 =

56 − 28 =

24 + 55 =

Task 2: Calculating distances

The following map includes a variety of tourist locations throughout Toronto and the GTA. Explore the map with a partner and discuss what the sites are and the distances between. Imagine you are a tour guide and were asked to create an itinerary of the city’s sights. Create 2 itineraries with a minimum of 3 locations for each. One itinerary must use kilometers and the other must use meters. Then compare the distances of some of these locations. How might the differences in their distances impact the number of tourists that visit the various locations each year?

Real world applications

Task 1: Estimating vs. calculating

In your math journal, reflect upon when you would estimate versus when you would calculate the accurate number for something. When is a time you might do both? How does estimating help you to understand the computation of addition and subtraction?

Task 2: Planning a getaway

A great way to experience a little getaway or to enjoy nature in all its beauty is camping or a nature hike. Imagine a hiker planning to spend some time in the outdoors. What sorts of items might they need to purchase to be prepared? Make a list of ten items. You might include things like bug repellent, hiking shoes, or a tent.

Find some flyers or sites online that advertise camping and outdoor materials and add up the totals for these supplies. How much would supplies cost if someone was going camping? What if they were just going for a walk? If their budget is $100, how much is left over? If possible, compare your answer with a friend’s.

Reflection

As you read through these descriptions, which sentence best describes how you are feeling about your understanding of this learning activity? Press the button that is beside this sentence.

Now, record your ideas using a voice recorder, speech-to-text, or writing tool.

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