# What Are Friendly Numbers

What Are Friendly Numbers – We begin our unit with subtraction. The goal is to gain a solid understanding of what abstraction is. We hope to develop mental math techniques. This will allow some freedom in problem solving and help develop their creative and critical thinking skills. We will also consider the traditional algorithm.

First we discussed what subtraction means. The concepts of “subtraction” and “difference” between numbers were discussed. But what does this mean? We tested the idea that when doing mental math, we often change one, the other, or both numbers to friendly numbers. But as long as you keep the “distance” between the two numbers the same, you will get the correct answer. So we measured the “difference” and “distance” between the numbers.

## What Are Friendly Numbers

Today’s question:  The school library has 556 books, 112 magazines and 67 DVDs. Students have produced 219 books, 12 magazines and 18 DVDs. How many books are left?

#### Unit 6: Addition And Subtraction Of Fractions

The first thing we did was discuss relevant information and decide whether magazines are considered books or not. We voted that they were NOT considered books. Then the students went to solve the problem.

Another student used the same strategy, but instead of the first 100 jumps, they slowly jumped to a friendly number and then kept going through the friendly numbers, like this:

Starting at a smaller number, you can count forward to a larger number. This is the distance between the numbers and is the answer.

Mr. Wendler was posting the piece on his bulletin board. Its height is 856 cm. Mr. Wendler had a liner of only 325 cm. Mr. How much should Wendler cut?

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Some people solved the question using the “Friendly Front Hop” strategy (different groups were observed to jump different amounts – larger or smaller friendly numbers – depending on what they were comfortable with).

Integration: We focus on the strategy of “backwards” and how it works. Explanation by students, written on the board:

856 – 325 = the subtracted number is divided by its place values ​​(300 + 20 + 5). These numbers are then subtracted from 856 one at a time (856 – 300 = 556 – 20 = 536 – 5 = 531).

Today’s question:  565 students attended the meeting (total number). Of those, 15 were student council members, 35 were on soccer teams, 212 ran cross country, and 285 were members of the choir. The choir was called to the stage to sing. How many students are left in the audience?

First we discussed which numbers in the question are the correct numbers. The students realized that to solve this problem 285 students (in the choir) were subtracted from the number of students (565), so the rest will be the answer.

Some students used the Friendly Front Hop strategy to solve the problem (in these examples one group hopped forward – adding – and the other jumped backwards – subtracting – or whatever):

Some students used the Lexog strategy of jumping backwards (dividing a small number into hundreds, tens and ones – then subtracting each part):

Integration:  Students labeled this strategy as the “grade retention” strategy. The students realized that the numbers can be changed as long as the distance between them is equal. So 2 changed the number 2 in the equation to a friendship number and then changed the number 1 to an even number.

#### Rounding Using Friendly Numbers

Today’s question: Mr. Wendler had 3 X \$100 bills, 5 \$100 bills, 10 bills for \$289.89, a laptop for \$369.49, and the video game for \$73.99. How much money did he have left?

Some students used visual cues to answer the question (although the problem was only solved using whole dollars and the coins were forgotten)  This group displayed all the money as bills and then issued bills one amount at a time (hundreds, tens, and one):

Integration:  Our discussion focused on evaluating our work. Some students had all the right ideas and strategies, but made logical mistakes and got the answers wrong. Students discuss how the function can be checked using back calculations (use addition to check subtraction or vice versa) or by answering the question again using a different mathematical strategy to see if you get the same answer. 1. Play “Go Fish” to add numbers up to ten. For example, if your child is 4, ask for 6.

2. Pig: Alternately roll 2 dice. The player mentally adds the numbers as they roll and can continue rolling, each time adding to their score. If a player chooses to stop playing, they add points to their previous score. BUT, if it becomes a 1, the player scores 0 for that round and it’s the next player’s turn. The game ends when one player reaches 100. Pig can also be played as a draw game. Just start with 100 points and spend your points.

## Adventures In Subtraction Number Talks!

3. Rummy Family Number: Use a deck of 40 cards: Four rows from Ace to Ten. The objective is to create families of three related cards by combination or subtraction. Example: 5, 5 and 10 are a family because 5+5=10 and 10- 5=5. 6, 3 and 9 are family because 6+3=9, 9-6=3 and 9-3=6. Shuffle the deck and deal 6 cards to each player. Place the remaining cards face down in the pile. If you have other card families, set them aside. If you have no clans, you can draw one from the pile and discard yours. You can also throw away what you picked up if you don’t want to. The first player to draw all 6 cards (2 families of events) is the winner. Remember that an ace equals a one.

4. Remove the hold bag: Put 100 of a specific item in the bag. (coins, beans, buttons, etc…) Grab some objects and count them. Use subtraction to find how many items are left in the bag. So if you put 100 items in a bag and take out 20, you would write 100-20=80. Let your partner take turns and whoever leaves the least amount in the bag is the winner.

5. Create a matching game using the egg box, write the numbers inside the egg box. Put two small objects (stones, pens, soft balls) inside the egg box, close and shake. Open to find out where these two things ended up. Add the numbers inside the box where the objects arrived. Now we start the repetition section. There are many replication techniques, varying in effectiveness and complexity.

Another strategy developed by students involves dividing a number into primes or primes before multiplying it. They decided to call this strategy “Separation” or “Friendship Number”.

For 39 x 9, the number 39 is divided into 10, 10, 10 and 9. Then each of these friendly numbers is multiplied by 9 (10×9=90, 10×9=90, 10×9=90 and 9×9 =81). These totals are then added together (90+90+90+81=351).

Another strategy the students came up with was dividing a complex number by its place values ​​(hundreds, tens, and ones). Students then multiply these (best) numbers. They called this strategy as “Splitting” or “Place Value” strategy.

Students discovered another strategy after connecting tables, location, and multiplication. We called this strategy the “Open Array” strategy. It checks for replay by looking at the location. Multiplicative numbers are divided into friendly numbers (32 becomes 30+2 or 10+10+10+2) and multiplied as they would be on the multiplication chart.

Another strategy the students discovered is called the “Rounding” strategy. They first round a number to a friendly number, then multiply, and then add or subtract the numbers they rounded.

## Numbers With Cool Names: Amicable, Sociable, Friendly

Today students made a connection between the traditional (long) multiplication method and the open board strategy. They successfully applied this strategy to a very difficult problem found on the Troubleshooting page of this blog. Check it out.

Today students worked to find the connection between the traditional (long) strategy and the traditional (short) strategy for multiplication. Students then work to solve a more difficult problem.

Today we learned about Napier’s Bones. John Napier, a Scottish mathematician in the early 1600s, developed a unique method of multiplication. It allows students to multiply large numbers without using the “place value operators” required when using the traditional technique. Using a graph (similar to a multiplication graph), the first number is written at the top (one digit per column) and the second number is written to the right (one digit per row). This allows each digit of one number to be multiplied by each digit of a second number (similar to the place value scheme). The answers are written in each box. This method automatically places the numbers diagonally in the correct position value column (other,

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