Class Seven
Properties of Triangle
1. Matchstick Triangles Got a box of matchsticks? Let’s have some fun! - How many matchsticks are in the box? - How long is each matchstick? - How many matchsticks do you need to make a triangle? Three, right? I think you’re spot on: - You need at least three matchsticks! Let’s try thi Grab three matchsticks and make a triangle. This triangle has three sides: 1) Side l1 2) Side l2 3) Side l3 Are all three sides the same length? Measure them with a ruler and write down: - Side 1: ___ cm - Side 2: ___ cm - Side 3: ___ cm Your triangle also has three angles: 1) The angle where l1 and l2 meet – let’s call it Angle A 2) The angle where l1 and l3 meet – let’s call it Angle B 3) The angle where l2 and l3 meet – let’s call it Angle C Are all three angles the same? Use a protractor to measure them and write down: - Angle A: ___ degrees - Angle B: ___ degrees - Angle C: ___ degrees Question: What’s the name of a triangle where all sides are equal and all angles are equal? Now, let’s get creative: Use more than three matchsticks to make a triangle. - Don’t break any matchsticks! - The sides don’t have to be the same length. Each time you make a triangle: - Measure and write down the length of all three sides. - Measure and write down all three angles. Try this out: - How many matchsticks do you need to make an equilateral triangle (all sides equal)? 6? 9? 12? 15? - Can you make a triangle with fewer or more matchsticks than those numbers? - Can you make a triangle where two sides use the same number of matchsticks? - Can you make a triangle where all three sides use different numbers of matchsticks? - Using up to 50 matchsticks, create as many different-sized triangles as you can. - For each triangle, measure and write down the side lengths and angles. - How many matchsticks does it take to make different types of triangles? 2. Triangles in the Alphabet (A to Z) Write all the letters from **A to Z** on a piece of paper. Make sure each letter is the same height – say, 5 cm, or whatever size you like, but keep it consistent! Let’s explore: - Which letters have angles in them? - Which letters already contain a triangle? What kind of triangle is it? - Which letters can form a triangle if you draw one small line? What type of triangle would it be? - Which letters can form a triangle if you draw two small lines? - Rule: The triangle’s sides or height shouldn’t be longer than 5 cm. - Are there any letters that can’t form a triangle no matter what lines you add? 3. Turning Triangles into Quadrilaterals Can you use two triangles to make a quadrilateral (a four-sided shape)? I think there’s a trick to it – a special condition. Can you figure out what it is? Try this: - Take different types of triangles and try joining them. - You can make a quadrilateral only if two triangles have sides of the same length that can be joined together. Questions to explore: - If you join two triangles of the same type, what kind of quadrilateral do you get? - If you join two triangles of different types, what kind of quadrilateral do you get? - Can you use more than two triangles to make a quadrilateral? If yes, which ones?
4. Triangle Property – Sum of Two Sides
Back to matchstick triangles! Got your matchstick box?
You can make a triangle with three matchsticks, and it will be an equilateral triangle (all sides equal).
Fun fact:
- In an equilateral triangle, the sum of the lengths of all three sides is three times the length of one side.
- The sum of any two sides is twice the length of one side.
Big question:
Can we say that for any triangle, the sum of the lengths of any two sides is always greater than the length of the third side?
Test it out:
- Use 25 or more matchsticks to make triangles of different shapes and sizes.
- Don’t break any matchsticks!
- For each triangle, check if the sum of any two sides is greater than the third side to see if this rule holds true.
5. Triangle Property – Sum of Angles
Grab those matchsticks again and make lots of different triangles – big, small, or any shape!
For each triangle:
- Measure the three angles using a protractor.
- Add up the angles. Does the sum always come to 180 degrees?
- Try this with different types of triangles to see if it’s true every time!
These activities are a super fun way to explore triangles, their sides, angles, and properties.
GCD and LCM
Alright, let’s hop on the mathy rollercoaster and make numbers dance with a goofy grin! We’re diving into the wild world of Greatest Common Divisor (GCD) and Least Common Multiple (LCM), and trust me, these number-crunching superheroes are ready to party! Plus, we’ll tackle a fruity puzzle to keep the giggles going.
The Lowdown on GCD and LCM:
GCD: This is the biggest, baddest number that can divide all your chosen numbers without leaving a crumb. It’s like the ultimate cookie-cutter that’s always smaller than (or at most equal to) the smallest number in your gang. Think of it as the math bouncer who says, “You all split up nice and even!”
LCM: This is the smallest number that all your numbers can divide into without a fuss. It’s the party venue where everyone shows up at the same time, and it’s always bigger than (or at least as big as) the largest number in the crew.
It’s the math DJ syncing everyone’s beats!
How to Find Them: You break those numbers into their prime-factor pieces (like tearing apart a LEGO castle) and put them back together in a special way.
Let’s see how with some fun math!
1. Distribution of fruits
In a wacky village, Bebitai’s the fruit queen, rockin’ a daily haul of 120 chikoos and 180 bananas—that’s a whole lotta fruit-tastic loot!
🍎🍌
Every day, she’s dishing out these juicy treasures to a gang of fruit fans, makin’ sure each one gets the same number of chikoos and bananas, no favoritism allowed!
The twist?
The fruit-hungry crowd gets bigger every single day, like they’re multiplying faster...
If Bebitai’s gotta keep this fruit party poppin’ with no leftovers, how many days can she keep slingin’ chikoos and bananas before the fruit frenzy runs outta steam? 😜
2. fancy flower bouquets
Alright, flower power fans, get ready for a bloomin’ good time!
You’ve got 18 roses and 12 lilies in your garden stash, and you’re itching to make fancy flower bouquets where each one has the same number of roses and lilies.
No flower left behind, okay?
So, how many bouquets can you whip up, and do we need the GCD or the LCM to solve this petal-packed puzzle?
Let’s dive in with a giggle!
The Flower Fiasco:
You’re the floral wizard with 18 roses and 12 lilies, ready to craft bouquets that scream “equal vibes only!” Each bouquet needs the same number of roses and lilies, and you gotta use all the flowers without any sad leftovers wilting in the corner. Let’s figure out the max number of bouquets and which math superhero—GCD or LCM—saves the day!
3. Reducing fraction
Alright, buckle up for a wild ride through fraction-simplifying shenanigans!
If you’ve got a beefy fraction that’s looking way too chubby, like it’s been munching on too many math snacks, here’s how you slim it down to its svelte, simplified self.
Let’s take the chunky fraction 315/549 and give it a makeover!
The Fraction Slim-Down Show:
Step 1: Spot the fraction looking all puffed up: 315/549.
Step 2: Break it down like it’s a dance-off! Factorize those numbers:315 = 3 × 105 = 3 × 3 × 35 = 3 × 3 × 5 × 7
549 = 3 × 183 = 3 × 3 × 61
Step 3: Find the sneaky common stuff!
Both have 3 × 3 = 9 chilling in them. That’s the Greatest Common Divisor (GCD), It’s like the math bouncer that kicks out the extra fluff!
Step 4: Divide both the top and bottom by this GCD (9):
315÷9 / 549÷9
= 35/61
Step 5: Check if 35/61 can slim down more. Since 35 (5 × 7) and 61 (a prime number) have no common factors other than 1, this fraction is now rocking its leanest, meanest form!
The GCD Magic: That 9 we found? It’s the GCD, the ultimate fraction-trimming tool. Whenever you’ve got a big, bloated fraction, hunt for the GCD to chop it down to size.
4. How many schools?
In a village, all the schools are lined up along a single road.
Trees are planted on both sides of the road, with 10 meters between each pair of trees.
On one side of the road, every fourth tree is a fruit tree.
On the other side, every fifth tree is a fruit tree.
Each school is located at a point where there are fruit trees on both sides of the road. If the road is 1.5 kilometers long, how many schools are there in the village?
Since each school is located at a point where there are fruit trees on both sides, and we’ve found 7 such points along the 1,500-meter road, there are 7 schools in the village.
Explain how?
5. When do both rhythms hit their Sam together
Alright, let’s dive into the rhythmic rollercoaster of Indian classical music with a side of silliness! 
We’ve got two big-shot tabla players, one jamming to Trital and the other grooving to Dadra, and they’re both trying to hit that sweet “Sam” (the first beat) at the same time. Let’s figure out when these two beat-buddies sync up their “Dha” moments, with a sprinkle of fun!The Rhythmic Rundown:
6. The Spicy Secret Message
So, I’m chillin’ like a villain on penicillin up on the terrace, yeah? I needed to tip off my sneaky spy pal with a hush-hush note. So, I scribbled down this cryptic gobbledegook: 20 -G n fr ts ymj yjwwfhj 25 And boom, he cracks it like a walnut and figures out I’m loafing on the terrace—I am on the terrace.
How’d he do it, you ask? Well, my mate spots the 20, clocks the -G, and spies the 25 at the end. That G ain’t just a letter—it’s code for the Greatest Common Denominator, or as we fancy spies call it, the GCD. Crunching the numbers, 20 and 25 give a GCD of 5 (like, 25 ÷ 20 = 1 with 5 left, then 20 ÷ 5 = 4, no remainder, so 5’s the magic number). The - means “subtract, mate” or “slide left” in spy-speak. We’d already pinky-sworn on this secret handshake of a code.
So, for the jumble n fr ts ymj yjwwfhj, he takes each letter, jogs five steps left in the alphabet (A=0, B=1, ..., Z=25, wrapping around like a dodgy carousel), and voilà—the secret message unravels to I am on the terrace.
Then, my spy pal fires back a message: 20 +G d vh xjhdib njji 25
Same deal: 20 and 25, GCD is 5. But this time, +G means “add 5” or “scoot right” five steps for each letter. Decoding that mishmash gives I am coming soon—like he’s about to crash my terrace party in a jiffy!
Now, can you whip up a secret message like this? Here’s your mission, should you choose to accept it: 33 +G ....... 121 Fill in the blanks with a coded message, keeping it sneaky and slick!
We’ve got two big-shot tabla players, one jamming to Trital and the other grooving to Dadra, and they’re both trying to hit that sweet “Sam” (the first beat) at the same time. Let’s figure out when these two beat-buddies sync up their “Dha” moments, with a sprinkle of fun!The Rhythmic Rundown:- Trital: The king of rhythms with a swagger of 16 matras (beats), split into four groups like a quadruple-decker sandwich (4+4+4+4).
- Bol: Dha Dhin Dhin Dha | Dha Dhin Dhin Dha | Dha Tin Tin Ta | Ta Dhin Dhin Dha
- Sam: The first Dha—the beat that’s like the grand entrance of a rockstar!
- One full cycle of Trital takes 16 matras.
- Dadra: The cool, breezy rhythm with 6 matras, split into two chill groups (3+3), like a quick dance move.
- Bol: Dha Dhin Na | Dha Tin Na
- Sam: Again, the first Dha—the moment everyone’s waiting for!
- One full cycle of Dadra takes 6 matras.
6. The Spicy Secret Message
So, I’m chillin’ like a villain on penicillin up on the terrace, yeah? I needed to tip off my sneaky spy pal with a hush-hush note. So, I scribbled down this cryptic gobbledegook: 20 -G n fr ts ymj yjwwfhj 25 And boom, he cracks it like a walnut and figures out I’m loafing on the terrace—I am on the terrace.
How’d he do it, you ask? Well, my mate spots the 20, clocks the -G, and spies the 25 at the end. That G ain’t just a letter—it’s code for the Greatest Common Denominator, or as we fancy spies call it, the GCD. Crunching the numbers, 20 and 25 give a GCD of 5 (like, 25 ÷ 20 = 1 with 5 left, then 20 ÷ 5 = 4, no remainder, so 5’s the magic number). The - means “subtract, mate” or “slide left” in spy-speak. We’d already pinky-sworn on this secret handshake of a code.
So, for the jumble n fr ts ymj yjwwfhj, he takes each letter, jogs five steps left in the alphabet (A=0, B=1, ..., Z=25, wrapping around like a dodgy carousel), and voilà—the secret message unravels to I am on the terrace.
Then, my spy pal fires back a message: 20 +G d vh xjhdib njji 25
Same deal: 20 and 25, GCD is 5. But this time, +G means “add 5” or “scoot right” five steps for each letter. Decoding that mishmash gives I am coming soon—like he’s about to crash my terrace party in a jiffy!
Now, can you whip up a secret message like this? Here’s your mission, should you choose to accept it: 33 +G ....... 121 Fill in the blanks with a coded message, keeping it sneaky and slick!
Multiplication of Integers
Hey there, Class 7 students! Let’s dive into the fun world of multiplying integers!What Are Integers?
Integers are all the numbers we use: natural numbers (like 1, 2, 3...), zero, and negative numbers (like -1, -2, -3...). All natural numbers are positive, and they’re super common in everyday math. For example, when you calculate the area of a rectangle, you multiply its length and width—both positive numbers!
1. Where are negative numbers used?
In daily life, where do we need to use negative numbers?
Examples:
1. Money spent
2. Withered flowers
3. Leakage from a tank
4. Floor numbers in a basement
5. Depth below sea level
How many more such examples can you give?
List all of them.
In daily life, where do we need to use negative numbers?
Examples:
1. Money spent
2. Withered flowers
3. Leakage from a tank
4. Floor numbers in a basement
5. Depth below sea level
How many more such examples can you give?
List all of them.
1. Where Do We See Negative Numbers?Negative numbers pop up in real life more than you might think! Here are some examples:
2. When Do We Multiply Negative Numbers?Look at the list of where negative numbers are used. Which of those situations involve multiplication to get an answer? Here are a few:
- Money you’ve spent (like buying snacks!)
- Flowers that have wilted in a garden
- Water leaking from a tank
- Basement floor numbers (like -1, -2 in a building)
- Heights above or below sea level
2. When Do We Multiply Negative Numbers?Look at the list of where negative numbers are used. Which of those situations involve multiplication to get an answer? Here are a few:
- Debt Repayment: If you repay ₹5000 of a loan every month, that’s -5000. For a whole year, you’d calculate 12 × (-5000).
- Opposite Directions: If you’re moving east at +50 km per hour, going west would be -50 km per hour.
- Work Delays: If one person does 100 units of work a day, and 4 people don’t show up, how much work is delayed? That’s 4 × (-100).
3. Multiplication and Signs – Let’s Play with Numbers!Let’s Have Fun with Flowers and Math!
If one vine blooms 50 flowers every day, how many flowers would 8 vines bloom?
You got this!
50 × 8 = 400 flowers. Awesome, right?
Now, imagine 3 of those 8 vines wilt. How does that change the number of flowers?
Since 3 vines are gone, we lose their flowers:
50 × (-3) = -150. That means 150 fewer flowers.
Here’s a new twist: What if each vine starts blooming 40 flowers a day instead of 50? How would that affect the total number of flowers on 8 vines?
Let’s figure it out together!
Your Challenge: Come up with 2 or 3 more examples like this. Maybe think about losing or gaining something fun, like candies or game points!
Your Challenge: Come up with 2 or 3 more examples like this. Maybe think about losing or gaining something fun, like candies or game points!
4. Multiplication of Integers on Graph Paper
Look at the diagram. A vertical and a horizontal line are drawn at the center of the graph paper. Positive numbers are written above the vertical line, and negative numbers below it. Positive numbers are written to the right of the midpoint, and negative numbers to the left. Using this, we can determine the sign of the multiplication of integers.
For example: +3 × +5 = +15
One number is marked on the vertical line and the other on the horizontal line. This forms a rectangle. Check which side of zero the rectangle is formed on to determine the value and sign of the multiplication.
Try various multiplications and verify:
-5 × +6
+8 × -6
-3 × -4
4. Multiplying Integers on Graph Paper – So Cool!Picture a piece of graph paper with a vertical line and a horizontal line meeting in the middle. Above the vertical line, we write positive numbers (+1, +2...), and below it, negative numbers (-1, -2...). To the right of the center, positive numbers; to the left, negative numbers.This setup helps us figure out the sign of a multiplication problem! For example:
+3 × +5 = +15 (positive × positive = positive).You plot one number on the vertical line and the other on the horizontal line. They form a rectangle, and you check which side of zero it’s on to know the sign and value.Try These Out:
+3 × +5 = +15 (positive × positive = positive).You plot one number on the vertical line and the other on the horizontal line. They form a rectangle, and you check which side of zero it’s on to know the sign and value.Try These Out:
- -5 × +6
- +8 × -6
- -3 × -4
5. Dividing Integers on Graph Paper – Flip It!
Division is just multiplication in reverse.
Let’s try 24 ÷ (-6).
Using the same graph paper setup (vertical and horizontal lines, positive and negative numbers), we can figure it out.
Now, let’s solve -24 ÷ (-6).
Start at -6 on the graph and move upward through -6, -12, until you hit -24.
Count how many steps (lines) you cross: +4 steps!
What if you moved downward to +6, +12, up to +24? That’s 4 steps downward, which is -4. But since the problem uses -24, the upward move (+4) is the right one.
Your Turn:
Try solving a few more division problems like this on graph paper.
It’s like a math adventure!
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