Cornell Note-Taking System Instructions:
- Record: During the lecture, use the note-taking column to record the lecture using telegraphic sentences.
- Questions: As soon after class as possible, formulate questions based onthe notes in the right-hand column. Writing questions helps to clarifymeanings, reveal relationships, establish continuity, and strengthenmemory. Also, the writing of questions sets up a perfect stage for exam-studying later.
- Recite: Cover the note-taking column with a sheet of paper. Then, looking at the questions or cue-words in the question and cue column only, say aloud, in your own words, the answers to the questions, facts, or ideas indicated by the cue-words.
- Reflect: Reflect on the material by asking yourself questions, for example: “What’s the significance of these facts? What principle are they based on? How can I apply them? How do they fit in with what I already know? What’s beyond them?
- Review: Spend at least ten minutes every week reviewing all your previous notes. If you do, you’ll retain a great deal for current use, as well as, for the exam.
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Hi and welcome to course Math: Basic Operations! This is where you will learn to, hopefully, love math!
I say this because this course does not review procedures of the four math operations of addition, subtraction, multiplication, and division. Instead I’ll help you gain confidence with important concepts of these operations, and share some PRO Math Tips that will help you solve any word problem you come across, essentially demystifying the dreaded word problem. So let’s get started!
Let’s start with an overview of some tips that will make you feel like a math pro. First, get into the habit of doing as much mental math as possible. The more math you confidently do in your head, the faster and more accurately you can get to solutions. Second, get really good at rounding and estimating numbers. When you estimate, you can more easily check your answer to make sure it makes sense. Third, practice thinking about the four basic operations conceptually. This can be half the battle in solving word problems, and it is the bulk of the content of this course. And last but not least, get good at the five-step process that I demonstrate here to solve word problems. This has been key to removing the stress of solving word problems for me and made me start to like the challenge.
So what do I mean by mental math? Well, when you feel confident in understanding how the basic operations affect numbers, you can actually do more to solve the word problem in your head before even putting your pencil to the paper. This makes you faster and more accurate in your math overall. I’ll show you how.
How quickly can you multiply 15 and 12? Pretty fast if you take 10 x 10, hold that hundred in your mind and then add 5 x 2. So 110, right? The more math you do, the more shortcuts you will find to increase what you can do in your head. It’s really worth your time.
Understanding the concept of place value, and the names for each relative to the decimal point, is important for avoiding common pitfalls in all four operations. It also helps you estimate faster. I know you estimate numbers regularly in your daily life. For example, if you owe your friend $47, you’re likely to just pay her fifty, right?. In order to estimate that number correctly, you first had to know how to round 47 to the nearest ten. and how did you know it was 50 and not 40? Well, if rounding sometimes trips you up, here’s a helpful rhyme. “5 and above, give it a shove. 4 and below, stay low.” Try it yourself! Round 1,375 to the nearest hundred…did you say “1,400”? Which number was your guide? The 7, because it’s one digit to the right of the place value you were asked to round to. If you feel unsure of rounding, it is worth your time to practice to build your confidence.
And here’s a general concept to keep in mind with any math problem. Ask yourself, “Am I joining values, like when I add or multiply whole numbers?” or “Am I separating values, like when I subtract or divide whole numbers?”. This is important to consider because if you got a much higher or lower answer than you were expecting, maybe you performed the wrong operation. Think – Joining = Bigger (Add or Multiply) and Separate = Smaller (Subtract or Divide) This type of thinking feels like half the battle when to me when I’m solving word problems.
Now I want to share with you a five-step problem-solving plan that has entirely changed my vision of myself as a math person. It’s no magic formula, but a systematic way to break down word problems so you reduce anxiety around solving word problems. Here is a major pitfall that happens folks don’t use a systematic plan to solve word problems – they read it once and then begin taking a stab at the computation. But hold on – take a few minutes to identify the problem, to discover what numbers are important and what operation to use (hint: KEY WORDS), then you make the plan for how to solve it. Guess what? There is no actual math until the 4th step. None! Only planning. So the first 3 steps are about being a good reader! After you actually DO the math in step 4, then you check it!
Step 1 – Identify the problem. Have you ever noticed how the question is always at the end of a word problem? Knowing this, I only pay attention to the numbers in the question on my first reading. I get a general idea of the context and then focus on the question. If I can restate that problem in my own words, noticing the unit,, then I know exactly what I need to do.. Additional tip – I always write the problem with the correct unit. So in this case, I’m being asked to find how much sugar needed, and the sugar is in ounces.
In step 2, identify just the information needed to solve the problem. To do that, I have to read the problem again, but this time paying close attention to all the numbers because there’s often unnecessary information meant to distract. So if you know you need the amount of brown sugar, you notice you can ignore the amount of granulated sugar and vanilla. All you need is 8 oz. but you’re not done in Phase 2 yet. Look for language that signals whether you need to join or separate amounts. Here I see the word doubling, and that tells me my numbers need to get bigger – which is joining. So I’ll need to either add or multiply.
For step 3, you read that you have to DOUBLE the amount of brown sugar you and that you started with 8oz. Now I have all the information I need to write my number sentence. Doubling can mean I either add the amount of brown sugar twice or I can multiply it by 2. So I can use either 8 * 2 equals…or 8+8=… The next step is actually “Do the Math!”, so you see why earlier I said you don’t do any math until step 4? I was serious!
Now step 4. We have our number sentence 8 * 2 equals, so let’s just do the math. The number answer is 16, but the answer isn’t done until you can name the unit. So look back at the problem – since brown sugar is listed in ounces, your complete answer must be 16 oz.
Additional Tip: If the math is complicated it’s recommended you always write all your work, that way if you have to make a correction you can look back to see exactly where the mistake happened and you don’t have to start over.
Step 5 of the 5-step plan is really only to check your answers. Does 16 oz make sense in this context? Sure it does. In this step always check both your number and the unit. if I accidentally said that Susan needed 16 pounds of brown sugar, based on what I know about baking, I need to trust my instinct and go back and check that pounds is the right unit. Discovering that the right unit was actually ounces can save you from a wrong answer. This final check of both number and unit is super important and shouldn’t be skipped.
So now let’s move forward into thinking about the four basic operations very conceptually. I’m not going to teach you how to add, subtract, multiply or divide, but I am going to point out some ways to think about them so that you can avoid some common pitfalls when working with them.
When you recognize these points about addition, you can avoid some common mistakes. Remember that addition is always joining, meaning the end result is larger than the original numbers. When a word problem requires addition, you can add those numbers in any order and get the same result. When adding, you have to line the numbers up according to place value. Here’s what I mean (example)… if you add a digit in the tens place with a digit in the hundreds place, you’re always going to get the wrong answer. When you know addition is the opposite of subtraction, you can check your addition using subtractions and you can better identify vocabulary that tells you to add. Here are a few key vocabulary words, but there are a lot more so have a look at our cheat sheet.
Now let’s practice multiplication. Multiplication is also about joining and when multiplying whole numbers your answer will always be larger than the numbers used. Like in addition, you can multiply numbers in any order, so when working out a word problem don’t stress about which number is written first. When multiplying, you do not have to line the numbers up by place value – just align them to the right margin. And if you know that multiplication is the opposite of division, you can more easily learn the key vocabulary to help you identify it. Here you have someone studying for math and reading. The question is how many minutes each week reading is studied. So I write number of minutes for reading and I finished Step 1. For step 2 II see that reading is studied for 1 hour. 1 hour equals 60 minutes, and I don’t need the number 45 for my answer, so I cross that out so I don’t accidentally use it later. I also see important words like EVERY night and EACH week. These help me know that I have to take the 60 minutes that Regina studies every night, and I know that there are 7 nights in a week. This helps me with Step 3 – write a number sentence. 60 * 7 =…Now I’m in Step 4 and I can finally do the math. My mental math would be 7 * 6 is 42, and add the zero back on for 420. But 420 sounds like a huge amount of time but, so I ask “Does this even make sense?” For step 5, I double checked the unit and see that it’s minutes so I feel confident. Good thing I checked.
When a word problem starts with a total amount and asks you to remove smaller amount that’s separating, so you have to decide if you need to subtract or divide.Here is a common mistake made in subtraction or division word problems: the numbers in the problem have to be put into the number sentence in the correct order. so you have to understand which number is the starting point or the total, and which numbers are being separated from that total. let’s take a look. step one for this problem you notice the contacts you have a couple with some money and they are buying some things. concentrating on the question, I see I need an amount of money left, or remaining to buy an armchair. so I’ll jot down what I understand in my own words…”How much is left over to buy the armchair?” if the unit is dollars. I see they started with a thousand and I know when you buy, that number gets separated from your total. So I’m ready to set up the problem as 1000 – 650 – 29 = .When I know subtraction is the opposite of addition, I can remember to line numbers up according to place value just like in addition. Also I can better recognize key subtraction vocabulary such as reduce, difference, fewer, etc. I see that the couple has $321 left over to buy an armchair. I can quickly estimate that this makes sense by adding 650 + 30 + 300 to get an estimated amount of 980. That’s a good enough ballpark number to match their original $1,000. And did you notice that to check my answer I added? That’s another key reason to understand opposite operations.
For division, notice that you have to separate a total amount into equal amounts. Equal is the key to division, because in subtraction the smaller values don’t have to be equal. It’s important to recognize the order of numbers in a division problem because they can’t be switched, or your answer will be way off.. When you understand that Division and multiplication are opposite operations, you can use one to check your answer in the other. The vocabulary of division is very specific… per, per person or per day, asks you to separate an amount into equal groups. in this problem, we see a yearly amount. The question asked, when stated in my own words, looks like this- How much does Glen earn per week? Step 2 is easy because there’s only one number so the total amount is 45000. But be careful, because that represents yearly earnings, and the question says “per week”. This question assumes you know that there are 52 weeks in a year, and that gives you your second number . Now write your math sentence 45000 divided by 52 =. In my mind, I immediately estimation, so that when I do the math, I know if I’m in the right ballpark. Using mental math, I’ll figure 450 / 50 as 8, so if I put the two zeros I removed back my answer should be around $800 per week. When I solve, I get $865.38. See how I’ve made my step 5 easier by estimating earlier? I’m super confident in my answer. When you get good at knowing the 5 steps, you will collapse some of them when any of the steps is pretty easy, or when your mental math gets stronger.
And that’s it. I hope you now feel a lot more confident in tackling word problems, and that they’ve lost any power they may have had over you. When you remember to think conceptually about math and have confidence in your abilities to perform mental math and estimate, math is easier. When you think about the basic operations in terms of joining or separating, and can quickly recognize key vocabulary, your problem solving skills improve. And when you commit to following the five-step problem solving plan, remembering that the actual computation doesn’t happen until step 4, word problem anxiety becomes a thing of the past. I guarantee you can start to love what is actually really cool about math, and tackle any problem that comes along. Good Luck!
And that’s it for Basic Math Operations. Happy Nursing.