MTE 280 Investigating Quantity: Number, Operations and Numerical Systems

UnDevCarLo1) Understand the problem2) Develop a plan3) Carry out the plan4) Look back

Before elementary school, kids need to master 1:1 correspondents (which is assigning value to numbers) and cardinality (which is the ability to see how many numbers there are total).Examples for understanding bases: 10 < the number 0 shows we have zero extra numbers^ the number 1 is a ten frame that's been filled

When adding, you follow a similar idea as normal adding.Example: 13 base five + 31 base fiveWhen you add the ones place, you count until you hit five. When you hit five, that becomes a tens place, and vice versa for the tens and hundreds places. Your answer here is 44 base five.Same idea for subtracting.

Algorithm for when you're leaving another base and going to base tenWhen solving without a diagram, you multiple all of the longs together.Example:34 base six converting to base ten:(6)(6)(6) + (4) = 22Answer is 22When we start to see problems with bigger numbers, we do descending exponents.Example: 231 base four converting to base ten:2(4^2) + 3(4^1) + 1(4^0) = 45

In the lattice algorithm you create boxes that help you add together your final answer. This way is great for kids because they can write the number "12" exactly as they know it. They don't need to worry about carry the one.

The first number you see is the number of groups. The second you see is the number of whatever is inside. 2(3) = we have 2 groups of 3 in each

*Post improvement, not success*What to teach first: 1s, 2s, 10s, 5sSecond: 3s, doubles, 9sAt that point they know most of the table. Then, fill in anything that's left.

Base ten blocks method14 x 12, draw the flat, longs, and units. Count what you have Area modelThey can multiply like this forever! LatticeSame as the addition model.

The best way to explain subtraction is to take away. That's easiest for the students to visualize. If showing 6 - 2, literally draw 6 units and circle 2, demonstrating you're taking those away. Then they simply count the remainder.

Red: always the negative Other side: always the positive+ <- we call these together a zero pair-- This allows us to show a student what zero is. Ex: 5 + (-4) = + + + + +-- -- -- -- = 1

Read it like -5 + 9 = "five negatives plus nine positives".Different signs being multiplied: negativeSame signs: positive

Funky ones are how we get rid of factors and simplify our fractions. For example, if we have 24/35, we can pull a 6 out of both of those. Then, we can use our funky ones to get rid of the 6.

When we have a mixed number, we use a backwards C to get rid of that mixed number.Ex:6 2/56x5 + 2 = 32 and then put it over 5Now you have your improper fraction!

Factor trees: they get to start with the numbers they know. They branch out over and over again and stop when you get to a prime number.Upside down division: you start by dividing one of their factors that you know to be prime.LCM: smallest number that is a multiple of both numbersGCF: Most common factor between two numbers

It's always helpful to draw your diagram. If given 24 base six, and you want to go to base ten, you can start by drawing 24 base six. That would look like 2 longs and 4 units. Each long here is worth 6 units. Count every single unit in your diagram. 6 + 6 + 4 = 16. The answer is 16 in base ten.

Same idea as the previous topic. In the problem 13 converting to base eight, draw your diagram. This should have 1 long and 3 units. To convert, count your units until you hit eight. This is your new long, so circle eight units. Then add your remaining units. You should end up with 1 long and 5 units. The answer is 15 base eight.

Lllleeeaving base tenWhen leaving base ten we use upside down division. Example: 46 to base eightWe ask ourselves how many times 8 goes into 46. Our answer is 5 with a remainder of 6, so the final answer is 56 base eight.

A good algorithm reinforces place value and it is repeatable for many different problems (we don't want a "trick" that only works in a specific scenario).Add using expanded form480+364 looks like 400 + 300 + 80 + 60 + 4Left to rightAdding the tens place before the ones placeScratch MethodYou make a scratch every time you hit your ten value when adding. Then you add your scratches to see how many tens values you gained. This eliminates the idea of "carry the one". This way kids also don't have to keep a "running total" in their head. Trade-offTrade around some of the numbers to make the adding easier for you. Only one number here needs to become friendly.Friendly #sBoth numbers need to become friendly numbers. Ex: 28+62 both numbers can be moved around so that they're friendly.

We like this because we're teaching them how to multiply using a form they already know.Ex:27(36) 20 + 7 x30 + 6 <--- then you multiply the number on the bottom with each number on top

Teach kids this when you're teaching them the basics of multiplication, like grouping. Ex: 3(4) = they draw 4 vertical lines intersecting with 3 horizontal lines then count each spot the lines intersect.

The first way to show division. 146 divided by 8. They can ask themselves how many times 8 goes into 146 over and over again. Obviously this is not the most time efficient method, but it works and helps them understand the number on the outside is going into the number on the inside.

"In-n-out" - Use their favorite restaurant to help them remember how to set up division. You have to teach remainders because of testing, but try to rely more on fractions because that makes more sense.

*ideal method*This method helps them understand fraction remainders. You have to learn repeated subtraction first because they need to be able to estimate how much of one number goes into another. Ex:379/4 How many times does 4 go into 3? 0How many times does 4 go into 37? 9 with 1 leftoverHow many times does 4 go into 19? 4 with 3 left over. Answer is 94 and 3/4

Numerators are the number of pieces I have.Denominators signifies the size of my pieces. Terminology to utilize: "the numerator is more or less than half way"

Remember: we aren't changing the value, just changing how it looks.Linear comparison: think lengthSet model: groupArea model: 2 shapesCommon denominators are important because when we are adding or subtracting numbers we are adding their areas. In order to add their areas we gave to be talking about the same size pieces.