Calculos: Mathematics Calculation Game (Kickstarter)

Calculos:  Mathematics Calculation Game  (Kickstarter)

The Card Game

Xavier Wilain created the card game Calculos, which is based on quick, mental mathematics dealing with the four arithmetic operations:  addition, subtraction, multiplication, and division.  There is a mixed version that has all four arithmetic operations.  

Players start with a card and try to find a calculation that matches any of the calculations on the center card.  

For example, if the center card has:

2 × 5

3 × 6

5 × 5

4 × 4

If you have any of the calculations on your card, say 4 × 4, you shout "4 times 4 equals 16" and claim the card.  The object is for find a calculation faster than their opponents.   

Here is a video of the card game and how to play it:

https://www.youtube.com/watch?v=-PEfEVTV4wI

The card game comes in four levels: each level has a different set of calculations on the card, from four to eight.   On the XW_SeriousGames website, each level can be purchased separately or altogether for each operation.

Link:  https://www.xwseriousgames.com/

The Mobile App (Kickstarter)

Wilain is running a Kickstarter campaign to adapt the card game into a mobile app that contains all 20 levels (4 four levels for each arithmetic operation plus one for mixed) with plans to add levels in the future.  Wilain has goal to raise 33'000 CHF (Swiss Francs, which is $35,897.63 in US Dollars on 6/28/2021 - conversion provided by Morningstar).  If successful, the mobile app is set to begin development in October 2021.  The target fund date is July 27, 2021.  

Kickstarter page:

https://www.kickstarter.com/projects/xw-serious-games/calculos/description

Please check this game out, it promises to be a great way to strength our mental calculation skills.  

Eddie

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Casio fx-CG 50 Update and Mental Math: Squaring a Number Ending in 5

Casio fx-CG 50 Update and Mental Math:   Squaring a Number Ending in 5

Casio fx-CG 50 Update

Casio released an update to the operating software for the fx-CG 50, version 3.40.   This update, among other things, expands the number of commands in Casio's Micropython.

There is a new module casioplot module has commands to draw pixels in any color and strings. 

The link to get the updated software is here:  https://edu.casio.com/download/index.php

Click on Graphing Models.

I have not had a chance to try the new drawing commands yet, it's on my list of things to do. 

Mental Math:   Squaring a Number Ending in 5

Squaring a positive integer whose last digit is 5 will result in the answer whose last digits are 25.

Why is this true?

Let n be a positive integer whose last digit is 5.   (n = 15, 25, 35, etc.). 

Let m = n - 5, then n = m + 5.   Note that m is a multiple of 10.

Then:

n^2 = (m + 5)^2
= m^2 + 2 * m * 5 + 25
= m^2 + 10 * m + 25
= (m^2 + 10 * m) + 25
= m * (m + 10) + 25

Since m is a multiple of 10, m^2 and 10 *  m are multiples of 100.  Because of this fact, squaring integers ending in 5 is used in mental math.

Examples

Example 1:  Calculate 45^2.

n = 45 = 40 + 5    (m = 40)
= (40 + 5)^2
= 40 * (40 + 10) + 5^2
= 40 * 50 + 25
= 2000 + 25
= 2025

45^2 = 2025

Example 2:  Calculate 165^2

n = 165 = 160 + 5    (m = 160)
= (160 + 5)^2
= 160 * (160 + 10) + 25
= 160 * 170 + 25
=  27200 + 25
=  27225

Eddie

All original content copyright, © 2011-2020.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

A Look at Some 1st Grade Common Core Problems

A Look at Some 1st Grade Common Core Problems

Common Core… Time to Dive In

My goddaughter is entering first grade this year.  We all heard of Common Core Mathematics.  My introduction to Common Core was an unfortunate one: a Facebook post where a student was marked wrong because a drawing wasn’t done correctly.  However, since Common Core in the United States is a reality and since I anticipate being asked for help with the math homework, I’m going to need to check out what is being taught in first grade. 

I’m going to highlight some types of problems that might be encountered in a first grade class. 

I want to thank A+ Plus Math Coach, link: http://www.aplusmathcoach.com for posting worksheets that cover Common Core math from grades K-5.  Most of the type of problems I will talk about in today’s blog post is based of these worksheets. 

Math Skills Emphasized

* Place Value, Tens and Ones

* Addition

* Subtraction

* Simple Graphs

* Counting

It would help if the student knows all the addition facts involving 0 to 10.  Yes, that addition table we memorized as kids is still helpful.

For a full list of skills (it’s a long one), click here:  http://www.corestandards.org/Math/Content/1/OA/

Counting Problems

Sample Problem 1:  How many balloons are in the box?

 This is fairly simple, we should count 12 balloons.

Other counting problems have students arranging objects into groups of ten when possible. 

Sample Problem 2:  Rhonda has six pieces of candy and Rita has eight pieces of candy.  How many pieces of candy do they have together?

Illustrated is the candy Rhonda (dark brown pieces) and Rita (tan pieces) have.  One exercise to have the student group the objects in tens when possible.  For this problem, we can have one group of ten pieces while four pieces are left over.  1 ten and 4 ones make 14. 

Adding and Subtracting

Most adding and subtracting problems should be straight forward.  As I mentioned before, it will help greatly if the student knows their addition tables.

Algebra, Without the Symbols

Some problems are presented as they come from first level algebra.  Instead of “Solve for X”, it’s “Solve for the blank space”.

Sample Problem 3:  Fill in the blank:  5 + ___ = 8. 

If the student knows their addition, the student would come up with 3 as the answer.  Sometimes, the problem states “You have 5 units.  How many units do you need to make 8 units?”

Sample Problem 4:  Think of the problem 8 + __ = 10 to solve 10 - ___ = 8.

“You have 8 units.  How many units do you need to make 10 units?”

“You have 10 units, how many units do you need to give away to have 8 units left?”

The answer to both questions is 2.  If someone finds a better way to explain this, please post this in the comments. 

Sample Problem 5:  The following sentence is false:  5 + 7 + 9 = 16.  Remove one of the numbers on the left to make the sentence true.

The goal here is to find the two addends in the sentence that add up to 16.  From the three possibilities:

5 + 7 = 12, no

5 + 9 = 14, no

7 + 9 = 16, yes

Since 7 and 9 are required, the 5 needs to be removed.

Doubles

One concept that may be introduced is the concept of additive doubles.  Simply put, the doubles are:

1 + 1 = 2	2 + 2 = 4	3 + 3 = 6	4 + 4 = 8	5 + 5 = 10
6 + 6 = 12	7 + 7 = 14	8 + 8 = 16	9 + 9 = 18	10 + 10 = 20

How can this come into play?  Some adding problems can be labeled as double plus one and double minus one. 

Sample Problem 6:  7 + 8

This can be seen as a double plus one problem. Note we can break the 8 into 7 + 1, then we have 7 + 7 + 1 (a double addition of 7). 

The thought process:

If 7 + 7 = 14 (double 7 fact)

Since 8 is 1 more than 7, add 1 to 14:

14 + 1 = 15.

An alternate strategy is the doubles minus one strategy. 

Start by recognizing that 8 + 8 = 16 (double 8 fact)

Since 7 is 1 less than 8, subtract 1 from 16:

16 – 1 = 15

This a strategy that facilitates mental math.

For video explanation of doubles, click on this link from Stephanie K:   https://www.youtube.com/watch?v=mbKkasLm5DY or from Bob Kowalec:  https://www.youtube.com/watch?v=elj4aup0wJk 

Graphs and Polls

I saw several problems that would require students to read graphs. 

Sample Problem 7:  Look at the table below, as a classroom of students in Mrs. Roberts said what their favorite toy is.  Mrs. Roberts tallies the results in the box below:

What is the most popular toy?  How many students participated in the poll?  (and similar questions)

This type of problem encourages counting and reading graphic representation of polls.  (Answers:  video game, 3 + 6 + 7 = 16)

Final Remarks

Other problems that I saw included requiring students to mentally add and subtract 10, and compare numbers between 10 and 99 using place value (compare tens digit first, then if they are the same compare the ones digit first). 

This is not going to be every problem that could be presented in a first grade Common Core math class, but I wanted to get an idea of what is taught in the classrooms.  I don’t know how much different first grade math is from today from when I was in first grade, which is 35 years ago. 

Eddie

All original content copyright, © 2011-2018.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.  Please contact the author if you have questions.

Mental Math and Some Numerical Musings

Mental Math and Some Numerical Musings

Working with mathematics for a real long time, everything from doing this blog, programming, watching a lot of game shows, and being asked about prices after discounts, I picked up a few mental math pointers.  I don’t claim to be a mental math prodigy (those kids that get featured who are, are amazing).

Mentally Adding 9

Mentally adding 9 involves one of two cases:

If the last digit is 0, change it to a 9. 

Example:  1820 + 9.  The last digit now becomes a 9.  Hence: 1820 + 9 = 1829.

If the last digit isn’t a 0, then add 10, then subtract 1. 

Example:  1821 + 9 = 1821 + 10 – 1 = 1831 – 1 = 1830

Example:  1827 + 9 = 1827 + 10 – 1 = 1837 – 1 = 1836

You can do a similar trick with adding 18 and 27.  To add 18:  add 20 then subtract 2.  To add 27:  add 30 then subtract 3. 

Example: Add 18 to 1827, then 27 to the resulting sum. 

(mentally) 1827 + 18 + 27 = 1827 + 20 – 2 + 30 – 3 = 1847 – 2 + 30 – 3 = 1845 + 30 – 3 = 1875 – 3 = 1872

It’s like give to the tens digit, take from the ones digit.

Mentally Multiplying and Dividing by 10

Mentally, it’s a matter of moving the decimal point.  When multiplying by 10, move the decimal right (and fill in a zero if necessary).  Dividing by 10 will cause the decimal point to move to the left. 

Example:  58.238 * 10

Move the decimal point to the right and get the answer:  582.38 

Example:  58.238 ÷ 10

Move the decimal point to the left and get the answer:  5.8238

The 10% Discount

Need to find out the discount when something is 10% off?  Fairly simple, just recognize that 10% is multiplying by 0.1, which is dividing by 10.  Mentally, move one decimal point to the left.

Example:  What is 10% of $38.99?   Multiplying by 10% is the same as dividing by 10, hence move the decimal point to the left, and we get the answer:  $3.899 or rounding to the nearest cent, $3.90. 

The 10% Tip

How about if we add 10% to an amount?  Note that adding 10% is equivalent to multiplying the number by 1.10.  Let n be the number, and:

n + 10% = n * 1.1 = n * 1 + n * 0.1 = n + n ÷ 10

Example:  If a restaurant bill is $32.00 and we needed to find the total cost after adding 10% tip:

32.00 + 10% = 32.00 + 32.00 ÷ 10 = 32.00 + 3.20 = 35.20

Dividing a Number by 5

To mentally divide a number by 5, double the number and divide the result by 10.  Why does this work? 

n ÷ 5 = n * (2 ÷ 10) because the fraction 2/10 is equal to 1/5. 

Example:  Divide 64 by 5. 

Step 1:  Double 64.  Now we have 128.

Step 2:  Divide by 10.  Move the decimal point left.  (think that we starting with 128.0).  The result is 12.8.

Multiplying a Number by 5

To mentally multiply a number by 5, multiply the number by 10 and then half the result.  Observe that for any number n:

n * 5 = n * (10 ÷ 2) = (n * 10) ÷ 2

Example:  Multiply 753 by 5.

(mentally) 753 * 5 =  753 * 10 ÷ 2 = 7530 ÷ 2 = 3765

Dividing Whole Numbers by 3, Will It Divide Evenly?

The way we can tell if a whole number divides by 3 evenly (no remainder, the quotient is also a whole number) is that if the sum of all its digits is also divisible by 3.

Example:  780, 1959, 4839, and 55101 are all divisible by 3.  Why?

780:  7 + 8 + 0 = 15; 15 is divisible by 3.  Also, 1 + 5 = 6.  (780 ÷ 3 = 360)

1959: 1 + 9 + 5 + 9 = 24; 2 + 4 = 6.  Divisible by 3.  (1959 ÷ 3 = 653)

4839: 4 + 8 + 3 + 9 = 24.   Divisible by 3.  (4839 ÷ 3 = 1613)

55101: 5 + 5 + 1 + 0 + 1 = 12  (1 + 2 = 3).  Divisible by 3.  (55101 ÷ 3 = 18367)

Dividing by 7

I have not memorized this.  However, something interesting when you divide numbers that are not multiples of 7 happens: 

1/7 = 0.142857142…

2/7 = 0.285714285…

3/7 = 0.428571428…

4/7 = 0.571428571…

5/7 = 0.714285714…

6/7 = 0.857142857…

8/7 = 1.142857142…

9/7 = 1.285714285…

10/7 = 1.428571428…

11/7 = 1.571428571…

12/7 = 1.714285714…

13/7 = 1.857142857…

The decimal portion always follows the pattern 1, 4, 2, 8, 5, 7.  So the next time you divide a whole number by 7 and figure the remainder, you can figure out which part of the pattern to attach if your answer is required as a decimal answer:

If R* = 1; the pattern starts at 1:  142857 142857 142857… (and repeat)

If R = 2; the pattern starts at 2:  2857 142857 142857…

If R = 3; the pattern starts at 4:  42857 142857 142857…

If R = 4; the pattern starts at 5:  57 142857 142857…

If R = 5; the pattern starts at 7:  7 142857 142857…

If R = 6; the pattern starts at 8:  857 142857 142857…

* R: remainder

Example:  1720 ÷ 7.  The division results as 245 with the remainder of 5.  The decimal patter starts at 7, hence 1720 ÷ 7 = 245.7142857142857…

Squaring Any Integer That Ends in 5

Why does squaring every whole number ending in 5 results in the square ending with 25?

Check it out:

5^2 = 25

15^2 = 225

25^2 = 625

35^2 = 1225

45^2 = 2025

55^2 = 3025

65^2 = 4225

…

185^2 = 34225

…

(feel free to use a calculator to check for other numbers)

Let n be a whole number whose last digit is 5.  (n = {5, 15, 25, 35, 45, ... 155 … }).  Then:

n^2

= (n – 5 + 5)^2

Let ϕ = n – 5.   Observe that ϕ is multiple of 10.  (Example:  If n = 25, then ϕ = 25 – 5 = 20)

Then:

n^2

= (ϕ + 5)^2

= ϕ^2 + 10 * ϕ + 25

Note that ϕ^2 and 10*ϕ will be multiples of 100.

The mental trick given when squaring a whole number ending in 5 is:

Step 1: Spilt the number into two parts, separating the last digit 5 from the rest of the number.  Treat the detached as a separate number. 

Step 2:  Square the detached number and the detached number to the result.

Step 3:   “Attach” a 25 to the right side of the result.

Example:  25^2. 

Step 1: “Split and detach” the number:  2 | 5

Step 2:  Square the detached number and add the detached number to the result: 

2^2 + 2 = 6

Step 3:  “Attach” a 25 to the right side of result:  625

Hence:  25^2 = 625

If we use the formula:  n = 25, ϕ = 25 – 5 = 20:

Then 25^2 = 20^2 + 10 * 20 + 25 = 400 + 200 + 25 = 625

Example:  215^2

Step 1:  “Detach”:  21 | 5

Step 2:  Square detached, add the detached to the result:  21^2 + 21 = 441 + 21 = 462

Step 3:  “Attach” a 25 to the right end:  46225

215^2 = 46225

If we use the formula:  n = 215, ϕ = 215 – 5 = 210

Then 215^2 = 210^2 + 10 * 210 + 25 = 44100 + 2100 + 25 = 46225

I hope you find this helpful.  This is some of the math I can do mentally (except I haven’t memorized the 142857 pattern when dividing numbers by 7), it comes with practice and patience.  Of course, it doesn’t hurt to check for accuracy.

Happy August,

Eddie

 All original content copyright, © 2011-2018.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.  Please contact the author if you have questions.