Elementary Mathematics

Percents

Percents Notes

percent is blank out of 100 cents, is related to fractions

• using a diagram... you have a diagram of 10 boxes and if you are just demonstrating a percentage, each box represents 10 percent and you fill in the boxes accordingly (if it is half a percent, fill in half the box). If you are demonstrating the percentage of a number, divide the number between the 10 boxes, then fill in the amount of boxes based on the percentage like before, then add the values in each box together. ex: 20% of 40... fill each box with the value 4 because 10 boxes of 4 =40... then fill in 2 boxes... then add 4+4=8
• using mental... use vocal answers only (no writing down), take 10% of the number, and then multiply it by the percentage it is wanting of the number. ex: 30% of 125... 10%= 12.5 and then times it by 3= 37.5

Week 15

Order of Operations

Operations Notes

Order of Operations- how we know what to do and when to do it

G E M/D S/A

(better than PEMDAS, parenthesis does not work for every situation)

• groups is a better way to understand
• multiply/divide and sub/add grouped because either can be first.. must work left to right

ex:

-12-4(2)+5

-12-8+5

-20+5

=-15

Week 14

Main topic

Multiplying Integers

Multiplication rules-(also apply to division)

• same signs= positive answer
• different signs= negative answer

• say as blank groups of blank neg/pos
• use circles and + or - signs

ex: 2X6

two circles with 6 positive signs in each circle= 12

ex: -4(3)= 0-4 (3)

0 take away 4 groups of 3 positives

place as many needed zero pairs with positive and negative signs and take away four groups of 3 positives

= -12

• say as blank groups of blank neg/pos
• use color counters or color tiles (yellow= pos, red= neg)
• neg times neg= positive because we are taking away negative

ex: 2X6

two groups of six positives

place 2 groups of 6 positive color counters

=12

ex: -4(3)= 0-4 (3)

0 take away 4 groups of 3 positives

place as many needed zero pairs with color counters and take away four groups of 3 positives

= -12

Week 13

Subtracting Integers

To solve, you cant use symbols when numbers get bigger..so you can utilize hectors method BUT you need to inverse what you are subtracting into addition in order to use hectors method

ex: -18 - (-15) turns into -18 + (15) then use hectors

• look at the two numbers you have and determine "the bigger pile"... do not reference the signs
• bigger pile has 2 pos/neg symbols under it
• smaller pile has 1 pos/neg symbol under it
• circle one symbol each from the two piles
• the remaining symbol on the outside determines the sign of the answer
• if symbols in bank are the same, you add the two numbers
• if symbols in bank are different, you subtract the two numbers

To show subtracting integers, we still utilize two colors still to represent positive and negative

• you can draw circles, it demonstrates well
• however most efficient way is writing positive symbols and negative symbols ( + and --)

1. most efficient way and introduces symbols to students that they will use in future algorithms

Subtraction:

• the first number represents what we have... write symbol for first number first
• then draw circle around what you are taking away (determined by second number) and use arrow
• if you do not have what needs to be taken away, add zero pairs to be enough pos or neg to take away... then take away an the remaining pos or neg in the zero pair will create zero banks with pos or neg from the first number or be added to the first number
• negative on bottom and positive on top

1. the neg on bottom and pos on top mimics real life ideas (ex: thermometers, arrows, graphs)... students will learn long term if we utilize something they already know

To build subtracting integers, utilize color counter manipulatives

• make sure the manipulative has red on one side and other color on second size
• red side= negatives
• other color (ex: yellow)= positives
• say it as ___ negatives or positives in order for students to better understand integers, easier way to understand rather than negative ____.

Subtraction:

• the first number represents what we have... place counters for first number first
• then the second number will be how many of those counters we take away
• if you do not have what needs to be taken away, add zero pairs to be enough pos or neg to take away... then take away an the remaining pos or neg in the zero pair will create zero banks with pos or neg from the first number or be added to the first number
• negative on bottom and positive on top

1. the neg on bottom and pos on top mimics real life ideas (ex: thermometers, arrows, graphs)... students will learn long term if we utilize something they already know

Solving Notes

To solve, you cant use symbols when numbers get bigger..so you can utilize hectors method

• look at the two numbers you have and determine "the bigger pile"... do not reference the signs
• bigger pile has 2 pos/neg symbols under it
• smaller pile has 1 pos/neg symbol under it
• circle one symbol each from the two piles
• the remaining symbol on the outside determines the sign of the answer
• if symbols in bank are the same, you add the two numbers
• if symbols in bank are different, you subtract the two numbers

Week 12

Adding Integers

Showing Notes

To show adding integers, we still utilize two colors still to represent positive and negative

• you can draw circles, it demonstrates well
• however most efficient way is writing positive symbols and negative symbols ( + and --)

1. most efficient way and introduces symbols to students that they will use in future algorithms

ex: show 5 using 9 tiles

draw 5 positive symbols in yellow and create two zero pairs to create a zero bank

total will still= 5 but uses 9 tiles

Addition:

• the first number represents what we have... write symbol for first number first
• then add the second number with symbol
• if you are adding with one sign, use one row
• if you are adding with two signs, use two rows... negative on bottom and positive on top

1. the neg on bottom and pos on top mimics real life ideas (ex: thermometers, arrows, graphs)... students will learn long term if we utilize something they already know

ex: 3 +4

+++ ++++= 7

ex: -4 + (-1)

---- - = -5

ex: 3+ (-5)

+++

_ _ _ _ _

draw zero bank around zero pairs= -2

Building Notes

To build adding integers, utilize color counter manipulatives

• make sure the manipulative has red on one side and other color on second size
• red side= negatives
• other color (ex: yellow)= positives
• say it as ___ negatives or positives in order for students to better understand integers, easier way to understand rather than negative ____.

ex: build 5

place 5 yellow counters

ex: build -2

place 2 red counters

Addition:

• the first number represents what we have... place counters for first number first
• then add the second number with counters
• if you are adding with one sign, use one row
• if you are adding with two signs, use two rows... negative on bottom and positive on top

1. the neg on bottom and pos on top mimics real life ideas (ex: thermometers, arrows, graphs)... students will learn long term if we utilize something they already know

ex: 2+5

place 2 yellow counters then 5 yellow counters= 7

ex: -5 + (-3)

place 5 red counters then 3 red counters= -8

ex: 4 + (-3)

place 4 yellow counters in top row, place 3 red counters below the yellow

draw a box around the zero pairs (one red and one yellow= 0)...the zero bank (the box) represents the number 0 in a understandable way

= 1

Week 11

Solving Fractions

Division Notes

for division, keep change and change

keep the first number, change the symbol, and flip the second fraction

can turn whole number fractions into improper fractions

after this you can multiply the same

simplify then multiply (# of groups and 3 of items)

students need to know math facts in order to factor out the numbers

can turn a number into 1 if it is the same number over itself

for multiplication, simplify then multiply (# of groups and 3 of items)

students need to know math facts in order to factor out the numbers

can turn whole number fractions into improper fractions

can turn a number into 1 if it is the same number over itself

ex: 14/36 x 27/21

2x7x9x3 over 9x4x3x7

left over with 2x1x1x1 over 1x4x1x1

which equals 2/4 or 1/2

Add and Sub Notes

To add and subtract you need to have same size pieces

to do this, use factorization. whatever factor the other fraction is missing multiply it as one unit (x/x)

if there are whole numbers you can just add or sub them, no need to make improper fractions

Week 10

Showing Fractions

Use the area model to demonstrate add, sub, and mult

• model just needs a shape you label as a unit
• rectangle may be easiest to use

• if multiplying a whole number by fraction= answer becomes smaller
• if multiplying fraction by whole number= answer becomes bigger
• demonstrate as groups
• if two fractions are being multiplied, show first fraction in one rectangle and fill in what needs to be filled in then incorporate the second fraction into the rectangle, overlap is the answer

• if same base/size use one box to take away
• if not the same base demonstrate each fraction with one box each and combine together into a third box. the highlighted/ filled in is the answer out of the total
• draw a circle around what you take away and an arrow

• if same base/size use one box to add
• if not the same base demonstrate each fraction with one box each and combine together into a third box. the highlighted/ filled in is the answer out of the total

Week 9

Building Add,Sub,Mult Fractions

Multiplication Notes

build the problem and say the problem as blank groups of blank

ex: 1/2 x 4

1) make a group of four

2) bring context: half a group of 4 is red

3) show using color counters 2 red and 2 white

=2

ex: 3/4 x 1/2

1) make two groups of 4 based on the denominators

2) provide context: 3/4 of a half are red

3) show using color counters 3 out of 8 are red

=3/8

Subtraction Notes

place the first fraction on the board and place the fraction you are taking away on the side of the board

take away the same fraction from board and side

if the fractions do not share the same denominator, find a fraction to swap out the fraction on the board that will also match with the fraction on the side of the board

ex: 2/3 circle fractions on board and 1/6 on the side of board

change the 2/3 to 1/6th pieces and take away 1/6

=3/6 = 1/2

Addition Notes

build with one, two, or three colors

build the problem

then put together the two manipulatives

then build with one color ( the fewest amount of tiles the better)

ex:

1/3 + 1/6

⬜⬜▢ (2/3 and 1/6)

▫ ▫ ▫ ▫ ▫ ▫ (sixths)

=5/6

if you are using circle fractions and not tiles make sure the student figures out how many of the color make a whole to find the size of pieces

Week 8

Build Simple/Equivalent Fractions

Building Fractions Notes

Area Model:

• the area of the circle, etc. filled or not filled
• ex: 1/4 of the area of the circle is yellow

Linear Model:

• what in the line is shaded, colored, etc.
• ex: 1/4 of the line is shaded in
• you can use different manipulatives that aren't linear and draw a line through them

Set Model:

• can use any manipulative, or mix of manipulatives/ items
• what out of the set is _____
• ex: two of the set are shaded, or two out of the set are rectangular

To build equivalent fractions with fraction manipulative:

• take any fraction you have and see if you can build it with one color

To build non equivalent fractions with fraction manipulative:

• take any fraction you have and see if you can build it with 2 or more different colors

Fraction Manipulatives

Manipulatives Notes

Different Fraction Manipulatives:

• fraction tiles (everything below the whole is a different fraction, needs context to make sense)
• fraction circles (most simple, but should not be only one used)
• fraction squares (comparing shape below to the one above it and the whole)
• pattern blocks (different color same size rectangles with different size boxes on them, with shading that represents the number of things we have)
• colored counters (doesn't show whole but size of fraction)
• fraction rods (everything below whole is a different fraction, needs context to make sense, the width of the whole is same as the length of the whole

Parts of a fraction:

• number on top= number of pieces/ things we have
• number on bottom is the size of pieces

Week 7

Understanding/Comparing Fractions

Comparing Notes

Looking at two different fractions and determining which is bigger with reasoning

• if the denominator is the same, the one with more amount of pieces is bigger
• if there is a whole number, the one with greater whole number is bigger
• if a fraction has more than half pieces out of its piece size then it is greater than the one less than half
• if you have the same amount of pieces but different size of pieces, the smaller piece size is larger
• if there are none of the above BUT they are both missing the same amount of pieces to fill up the piece size, the smaller the missing pieces the bigger

The more fraction reasoning we can teach students, the more they will understand the size of fractions, how they are less than 1, and the relationships they have

Understanding Notes

What do the parts of the fraction mean?

• numerator: the amount we have, number of pieces
• denominator: the size of the piece

Fractions are tricky for students because this is one of the first times students are working with inverse relationship numbers (the smaller the denominator, the bigger the piece)

When and why do we need to find common denominator?

• when you add and subtract you need the same size pieces
• when you multiply you do not need to find it because the first number acts as the amount of groups and the second number acts as the amount inside each group

Prime Factorization

Factorization Notes

-Finding factors of numbers is important for students to know and develop as they further their math journey

-A prime # is a number that only has two factors, 1 and itself

-non prime # is called a composite, more than two factors

-prime factorization is finding factors of numbers that are only prime numbers

• can do this with a factor tree

ex: 48

/ \

4x12

/\ /\

2x2 3x4

/\

2x2 Factors are: 2x2x2x2x3= 2 to the power of 4, x 3

• OR upside down division

ex:

5 |40

___

2 |8

___

2|4

___

2 Factors are: 5x2x2x2= 2 to the power of 3, x5

• Lowest Common Denominator (LCD)

-to find LCD, find the blend of the 2 answers from the two given problems (use largest factors), it is the smallest number that two or more denominators can both divide into evenly

ex: 2 to the power of 2 times 3 AND 2 to the power of 4

2 to the power of 4 is divisible by both denominators and 3 is necessary for divisibility of one denominator, so the LCD= 2 to the power of 4 times 3= 48

Divisibility Rules

Divisibility Notes

Divisibility is useful for students to reduce the amount of work they have to do and helps them recognize relationships better between numbers

Divisibility Rules:

2- even numbers

3- sum of digits is divisible by 3

4- last two digits are divisible by 4

5- number ends in a 5 or 0

6- if divisible by 2 AND divisible by 3

7- none

8- last three digits are divisible by 8

9- sum of digits are divisible by 9

10: number ends in 0

Solve Division Alt Algorithims

Upwards Division Notes

Upwards Division

1) expandable

2) efficient

30 not based in math

write the problem as we say it

ex: 372/9= 372

-----

9

work from the bottom up

ex:

9 goes into 37 4 times so add 4 to answer then subtract 36 (9x4) from 37, the remainder 1 goes to 2 so its now 12, then do the same process for 12.

9 goes into 12 once so 1 is added to answer which is now 41, and subtract 9 from 12=3 which is our remainder

so the answer is 41 3/9

Area Model

1) expandable

2) kinda efficient (depends on student ability)

3) based in math

draw rectangle and place dividing number on outside of box on left, start with first value in number we are dividing and place it in first box,

then add divisible number on top of outside of box and multiply it by dividing number

the remainder in that box is added to next place value in next box

repeat process until left with 0 or remainder

add values on the top outside of box along with remainder

-works with what numbers students are strong with

Repeated Subtraction Notes

Repeated Subtraction

1) expandable

2) kinda efficient (depends on student ability)

3) based in math

draw line down side of house and only put numbers in that column

student will pick any number that can go into number being divided

multiply number in side column by number we are dividing with and subtract it from number being divided

repeat this process till we are left with 0 or remainder

then add the values in the column together with the remainder

-works with what numbers students are strong with

Long Division Notes

Long division/ traditional

1) expandable

2) efficient

3) not based in math

number being divided goes inside "house" and number dividing with is outside "house"

-students don't always know which number can go where

-students will make pattern that only small number goes inside house so you need to present problems like 60/10 and 6/12

-the better students are at math facts, the easier long division is

-this method requires more memorization to steps

Showing Division

To show division you create number of groups with smaller numbers, getting into large numbers create groups out of numbers

square= flat, |(line)= long, . (dot)= unit

ex: 25 divided by 4, create four circles (amount of groups) and count off units into each circle till you reach 25 but maintain even amount of units in each circle. so there are 6 units in each group and 1 dot left over which needs to be divided into each group so 1/4 ths. so the answer is 6 1/4

ex: 139 divided by 25, create groups of 25 until you reach 139 or as far as you can to 139, so we created 5 groups of 25 with 14 left which is 14 out of a group of 25 left, so the answer is 5 14/25

Week 6

Building Division

2 ways to look at division:

• the first value is the # of items we have
• but second value can be

1. # of groups we will create or
2. # of things inside each group

with #1 make sure to evenly distribute into each group and if there's a remainder create fractions based on amount of groups you have so if you have five groups and remainder 2 it would be 2/5

with #2 utilize you base blocks and trade off if needed to create as many groups of the second value that you can from the first value, ex: 60 divided by 12 you make as many groups as you can of 12 out if sixty till you run out of base blocks from your group of sixty or till you hit remainders

Solve Multiplication Alt Algorithms

Lattice Notes

Lattice

• draw rectangle and place each number with space in-between each value on top and right side (not using expanded form)
• draw line through rectangle at each space
• add lattice lines and then add the values within each lattice line

1. its expandable
2. its efficient
3. not based in math

Area Model Notes

Area Model

• draw rectangle and place expanded form values of two numbers on top and side of it
• at each addition sign draw a line through the rectangle
• this mimics the model used in base ten blocks where flats are in upper left, etc.

1. its expandable
2. its efficient
3. based in math

Left to Right Notes

Left-to-Right

• multiply starting with furthest place values with each place value in other number

1. its expandable
2. its efficient
3. based in math

Expanded Notes

Expanded Form

• 23x45= 20+3

x 40+5

1. its expandable
2. not efficient
3. based in math

Traditional Notes

Traditional Algorithm

• right to left solving

1. its expandable
2. its efficient
3. not based in math

Multiplication Automaticity

Automaticity Notes

Multiplication Automaticity

the automatic recognition of answer of 2 numbers working forwards and backwards

Automaticity is important to set students up for success in math in the future

How to teach multiplication table order:

• group 1( learnt by skip counting)

1. start with 1's
2. then 10's
3. then 2's
4. then 5's

• group 2 ( learnt by skip counting)

1. start with 3's
2. then 9's

• how to teach 9's
• in our answers, the 10's digit is 1 less than the number we are multiplying 9 by
• then the 1's is the adding of a value our tens value answer to = nine
• ex: 9x7, 1 less than 7 =6 and 9-6=3, so the answer is 63

1. then doubles (4x4, etc.)

• group 3

1. remaining values that match on both sides, such as 7x6 and 6x7

Other methods to reinforce automaticity

• flash cards: can be great for independent learning but they need to be flash cards that have the groups we created in the multiplication table so you can choose what group they are working on and the frequency of them
• Times table/test: are used for automatic understanding but they can stress students because of the time limit, and we must make sure to utilize the test results correctly. Do not make it a class competition and showing of who didn't do well, etc. Instead make a self progress chart that motivates them to keep trying to do better
• Games: a memory card game called concentration for example with the problems and answers can be beneficial for students to engage in the problems and use their math skills quickly

Build/Show Mutiplication

Show Notes

To show

• with this method still continue to reiterate to your students what the first and second number mean in multiplying
• demonstrate using square as flat, |(line) as long, and . (dot) as unit
• utilize a ray (diagrams laid in rows) to resemble rectangles the most
• ex: ----- . .

----- . . = 2 groups of 12

• for larger numbers you can utilize a diagram where you create a width by length brackets.
• long brackets=10 and shorter brakes= units
• then you draw out the lines for each bracket line and it creates a answer board filled with possible flats, longs, and/or units
• make sure to not close of width and length brackets because then they look like they are part of answer board

Build Notes

What does multiplication mean?

-1st #= # of groups

-2nd #= inside of groups

• better to demonstrate with building rectangles rather than circles and blocks within circles (though it works)

To Build

• start out with placing width value at top and length at side of answer board, then place the needed flats, longs, and/or units to fill up the rectangle on the outside of the answer board
• make sure the values being multiplied are not on the answer board because it is hard to see the answer if they aren't
• fill w/ least amount of base 10 blocks you can
• ex: add flats if applicable, and longs instead of using all units

Week 5

Subtraction Alt Algorithms

Traditional:

working from right-to-left (can be confusing for students who have been learning left to right)

1) its expandable

2)its efficient

3) not based in math

Expanded Form:

expanding each number by its place values (great for beginning learners)

Ex: 46=40 +6

1) its expandable

2)not efficient

3) based in math

Left to right:

subtracting left to right using place values (great for intermediate student learners)

ex: 96-35= (90-30)+(6-5)=61

1) its expandable

2) unsure if efficient

3) based in math

Equal Addends:

measurement between two numbers needing to stay the same but you can add or subtract to make friendly numbers

ex: 64-38

+2 +2= 66-40= 26

1) its expandable

2)its efficient

3) based in math

Build and Show Subtraction

Building Subtraction

we want to give students something concrete to better understand what they are learning, which is why we provide an action phrase for subtraction. For beginners instead of minus, say take away

use base blocks to build subtraction visually

-place first value amount of base blocks on board and place the subtracting value of base blocks on the side of board (NOT ON BOARD, it is just there to help reinforce visually how much we are taking away)

ex: 7-3

-place 7 unit blocks on board and 3 unit blocks off the board

-as you take one unit block from the 3 take one from the board and do this three times

-now you are left with 4 unit blocks on the board

ex:23-8

-for numbers like this start the process and then you will convert or borrow a long to convert it into units so you can take away a total of 8 units. the amount of longs and units left on the board is the answer

*when subtracting large numbers, take away the largest place value first, so take away flats...then longs...then units if applicable

*if you are subtracting with different base using blocks, make sure when you convert from either flat to long or long to units match the base value NOT 10*

Showing Subtraction

utilize shapes and diagrams to mimic base blocks to subtract

square= flat, |(line)= long, and . (dot)= unit

ex: 7-3= . . . . . . . take away . . .

-when taking away three dots (units) do not erase but draw a circle around how many you taking away from the 7 dots and draw an arrow

-when you get into large numbers or different bases remember to still convert/borrow ( show this by crossing out the shape) and/or utilize the base value NOT 10

Week 4

Adding Alt Algorithms

Alternative Addition Algorithms

Understanding Traditional vs. Alternative Algorithms

• Traditional Algorithm Issues:Can be confusing for students due to carrying values.
• Example:
• 237 + 185
• Students may struggle with not writing left-to-right.

1. What Makes a Good Algorithm
2. Expandable: Works for all numbers without changes.
3. Efficient: Quick and easy to use.
4. Based on Math Sense: Reinforces mathematical principles rather than memorizing steps.

Alternative Addition Strategies

1. Left-to-Right Addition

• Maintains place value and minimizes rewriting.
• Example:
• 495 + 223
• 400 + 200 = 600
• 90 + 20 = 110
• 5 + 3 = 8
• Total: 618

2. Friendly Numbers (Rounding to Nearest Tens)

• Adjust numbers to make addition easier.
• Example:
• 32 + 58
• Round to 30 + 60 = 90
• Example:
• 364 + 136
• Round to 360 + 140 = 500

3. Trading-Off (Rearrange for Simplicity)

• Adjust numbers by taking what is needed.
• Example:
• 26 + 67
• 23 + 70 = 93
• Example:
• 539 + 285
• 534 + 290 = 824

4. Scratch Method

• Helps students keep track of numbers mentally by adding up to 10 and marking a scratch.
• Example:
• 67 + 73 → Scratch and count groups of 10.

5. Lattice Addition

• Allows numbers to be added in any order.
• Requires vertical alignment.
• Example:
• 426 + 159 using a lattice grid.

5. Expanded Form

• Breaks numbers down into place values for easier understanding.
• Example:
• 36 + 13
• 30 + 6 + 10 + 3
• 40 + 9 = 49
• Example:
• 1368 + 247
• 1000 + 300 + 60 + 8
• 200 + 40 + 7
• Total: 1615

Week 3

Showing Addition

Showing Addition (how do we draw)

□ = flats

| = long

• = unit

4 + 3

• • • • + • • • •

= 7 • • • • • • •

→ make it vertical to match our long instead

5 + 8

• • • • • + • • • • • • • •

Combine into a 10-frame:

| • • •

= 1 long, 3 units = 13

36 + 17

| | | • • • • • • + | • • • • • • •

Combine:

| | | |

• • • • • • + • • • • (from 7) this creates a long and • • • left over

| | | | | • • •

= 5 longs, 3 units = 53

423 + 159

□ □ □ □ | | • • • + □ | | | | | • • • • • • • • •

Combine:

□ □ □ □ □

| | | | | | |

• • • • • • • • • + • (from 3) this creates long and • • left over

□ □ □ □ □ | | | | | | | | • •

= 5 flats, 8 longs, 2 units = 582

286 + 597

□ □ | | | | | | | | • • • • • • + □ □ □ □ □ | | | | | | | | | • • • • • • •

Combine:

□ □ □ □ □ □ □

| | | | | | | | | + | (from 8) this creates a flat and | | | | | | | left over

• • • • • • • + • • • (from 6) this creates a long and • • • left over

□ □ □ □ □ □ □ □ | | | | | | | | • • •

= 8 flats, 8 longs, 3 units = 883

With your students:

Transition from building → drawing to demonstrate quicker and concise steps:

• Colors and drawing help students to see connections of converting and grouping
• Once they see connections, they are ready for algorithms

(Make sure drawings are colored, unlike these!)

Building Addition and Converting Bases

Building Addition with Base Ten Blocks ( notes do now show images of base blocks, use notebook for visuals)

• Using base-ten blocks to demonstrate regrouping:
• Helps students group sizes together to demonstrate place values and add values.

Examples

1. Example: 4 + 3 = 7 units

• Use 4 units to fill the 10 frame.
• Then, add 3 units to the 10 frame.
• Students count the total units out of 10.

1. Example: 6 + 7

• Use 6 units and then 7 units.
• Exchange 10 units for 1 "long" (representing tens), leaving 3 units remaining.

Combining Shapes to Represent Larger Numbers

1. Example: 23 + 42

• Break down the numbers into tens and ones:
• 2 tens (20) + 4 tens (40) = 6 tens (60).
• 3 ones + 2 ones = 5 ones.
• Total: 65

1. Example: 74 + 29

• Break down into tens and ones:
• 7 tens (70) + 2 tens (20) = 9 tens (90).
• 4 ones + 9 ones = 13 ones. Exchange 10 ones for 1 "long," leaving 3 ones.
• Total: 103

Rules for Using Base Ten Blocks

1. 10 units = 1 long (representing tens).
2. 10 longs = 1 flat (representing hundreds).
3. Students group, exchange, then add.

Advanced Examples

1. Example: 247 + 185

• Break down into hundreds, tens, and ones:
• Hundreds: 2 flats (200) + 1 flat (100) = 4 flats (400).
• Tens: 4 longs (40) + 8 longs (80) = 12 longs. Exchange 10 longs for 1 flat, leaving 2 longs.
• Ones: 7 units + 5 units = 12 units. Exchange 10 units for 1 long, leaving 2 units.
• Total: 432

1. Example: 143 + 235

• Break down:
• Hundreds: 1 flat (100) + 2 flats (200) = 3 flats.
• Tens: 4 longs (40) + 3 longs (30) = 7 longs.
• Ones: 3 units + 5 units = 8 units.
• Total: 378

Bases in any other number, not including 10;

ex: In base 6, a Flat is only worth 6 by 6; a Long is only worth 6; a Unit is only worth 1.

Notes for Teaching

• Use clear visual aids to show how units are grouped and exchanged.
• Encourage students to verbalize their thought process while grouping and exchanging.
• Practice with different numbers to reinforce understanding of place value and regrouping concepts.

Week 2

Ploya's Problems

Notes

Ploya's 4-step Problem Solving Process

(method to teach students): 

1) **Understand the problem** 

  - Reread the problem 

  - Explain the problem to someone else 

  - Break down the parts of the problem 

2) **Develop a plan** 

  - Relate the problem to prior knowledge 

  - Identify similarities/differences to other problems 

  - Brainstorm ideas 

3) **Carry out the plan** 

  - Try the plan 

  - Try other plans 

4) **Look back to see if your answer makes sense** 

  - Is it reasonable? 

  - Solve another way to see if you get the same answer 

  - Work backward using your answer 

Example: 

- Two men drive past a farm that has pigs and chickens in a field. 

- One man says, "I see 50 feet in that field," and the other says, "I see 18 animals." 

- How many pigs and chickens are there? 

Possible methods to solve: 

1) Guess & Check

  - Helps students learn how to write equations. 

  Example: 

  - Feet: pig (4 feet) + chicken (2 feet) 

  - Guess: 5 pigs and 13 chickens 

  - Equation: 20+ 26= 46 feet

2) Diagram (needs total 18 animals) 

  - Helps students see and understand visual processes. 

  - Example: use visual representations like circles, dots, or illustrations. 

3) Lists

| Pigs (P) | Chickens (C) | Pig ft (4P) | Chicken ft (2C) | Total | 

|----------|--------------|-------------|-----------------|-------| 

| 1    | 17      | 4      | 34       | 38  | 

| 2    | 16      | 8      | 32       | 40  | 

| 3    | 15      | 12     | 30       | 42  | 

| 4    | 14      | 16     | 28       | 44  | 

Purpose:

- Sets patterns to help students reason. 

- Helps them solve using patterns. 

4) Algorithm

- P= pigs 

- C= chickens 

**Bodies: (P + C = 18) 

**Feet: (4P + 2C = 56) 

Solution: Use two equations to solve. 

Overview and Needed Materials

Materials

Base Ten Blocks, 2-Color Counters and a Fraction Manipulative

Class objective

• what some of the earliest stages of number sense are
• how to apply a problem solving process.

Week 1