Bunches of Bananas

What You’ll Need:
Pictures of bunches of bananas

Pose this problem to students:
How many bananas do you think are usually in a bunch?

Example:
I think there are usually about 6 bananas in a bunch.

Habitat Hunt

What You’ll Need:
No materials necessary

What is there only 1 of in your home?
What is there 2 of?
What is there 3 of?
What is there 4 of?

Sample answers:
1 dining room table
2 bathrooms
3 beds
4 sinks

Just Right

What You’ll Need:
Pencil or pen
Paper

Draw cups for Mama Bear, Papa Bear, and Baby Bear that seem the right size for each.

Add Us

What You’ll Need:
Number cube (1–6)

Play in pairs or small groups. Players take turns. Roll the number cube two times to form a two-digit number. Then roll the number cube again for a one-digit number. Add the two numbers. If the sum has a 4 in it, the player gets 1 point. The first player with 10 points wins.

Example: Elio rolls the number cube twice, resulting in a 2 and a 3. His two-digit number is 23. He rolls the number cube one more time, resulting in a 3. Elio adds the two numbers: 23 + 3 = 26. Elio’s sum does not have a 4 in it, so he does not get a point.

Fitri rolls the number cube twice, resulting in a 3 and a 6. Her two-digit number is 36. She rolls the number cube one more time, resulting in a 5. Fitri adds the two numbers: 36 + 5 = 41. The tens digit of Fitri’s sum is a 4. She receives 1 point.

Spirolaterals

What You’ll Need:
Spirolaterals printout

A spirolateral is a complex shape that is made by using a simple set of repeating rules. To make a spirolateral, you start at a point on a grid, move a certain number of spaces, make a turn, move the next number of spaces, and continue the pattern until the shape is closed.
A spirolateral of the form 3–5–10 with quarter turns to the right would look like the sample spirolateral found on the Spirolaterals printout. Create a new spirolateral based on three numbers of your choice and quarter turns. Describe what you notice.

Sample answer: Using the pattern of 3–5–10 with quarter turns, I get four rectangles that have a length of 5 and a width of 3, which matches the two smaller numbers of my three numbers in my pattern. I get a square in the middle of my shape that has a side length of 2, which is the same as the greatest of my three numbers minus the sum of the other two; that is, 10 – (3 + 5) = 2.

Product-Quotient Contest

What You’ll Need:
Product-Quotient Contest cards

Cut out and shuffle the Product-Quotient Contest cards, and arrange them face down in a 4-by-6 array. Then share these instructions with students:
Play with a partner. Take turns turning over two cards. If the cards have the same product or the same quotient, keep the cards. If they don’t match, place them face down again. If you get a division card and a multiplication card, place them face down again and take another turn. Play until all the cards have been matched. The player with the most matched cards wins.

Example: Maria turns over two cards. Her first card is 5 × 320. Her second card is 10 × 120. These cards do not have the same product. Maria places the cards face down again.
Damola turns over two cards. His first card is 130 ÷ 2. His second card is 260 ÷ 4. These cards have the same quotient. The cards match, so Damola keeps the cards.

Pattern Mystery

What You’ll Need:
No materials necessary

What two numbers do you think are missing in this pattern? Why?
1, 3, 6, 11, 18, _____, _____, 59, …

Answer: The missing numbers are 29 and 42. To continue the pattern, you must add the next prime number to the last result.
18 + 11 (the 5th prime number) = 29.
29 + 13 (the 6th prime number) = 42.

Three-Digit Delight

What You’ll Need:
No materials necessary

Choose a three-digit number where the first and last digits differ by at least two. Put the digits in order from greatest to least. Then reverse the digits and subtract the smaller number from the larger number. Reverse the digits in the difference, and add this new number to the original difference.
What do you notice? Why did it happen?

Answer: You always get a sum of 1089 at the end. The hundreds digit and the ones digit in the difference always add up to 9 and the middle digit is always 9. When you add the difference and its reversed form, you get a ones digit of 9, a tens digit of 8 (since it would have been 18 tens, which is 1 hundred and 8 tens), and a hundreds amount of 10 hundreds (since it would have been 9 hundreds plus 1 additional hundred).

The algebra to explain this, even if you don’t show it to most students, is as follows: Suppose the digits of the numbers in order from greatest to least are a, b, and c. You subtract abc – cba. Notice that if you use a traditional subtraction algorithm, you will need to rewrite the greater number as (a – 1) hundreds + (10 + b – 1) tens + (10 + c) ones. When you subtract the reversed number, you end up with (a – 1 – c) hundreds + 9 tens + (10 + c – a) ones. When you add the reverse of (a – 1 – c) hundreds + 9 tens + (10 + c – a) ones, you end up with 9 hundreds + 18 tens + 9 ones, which is 1089.

Example:
Original number: 512.
521 – 125 = 396.
396 + 693 = 1089.

Right Angle Prediction

What You’ll Need:
Right Angle Prediction printout

Look at the right triangles on the Right Angle Prediction printout. What do you notice? Can you predict the side lengths if the shortest side is 51?

Sample answer:
I noticed the longer sides are consecutive numbers. I noticed that the shortest sides are odd numbers. I also noticed that the shortest side is the square root of the sum of the other two numbers.
If the shortest side length is 51, the other side lengths will be 1300 and 1301.

Decimal-Fraction Contest

What You’ll Need:
Decimal-Fraction Contest cards

Cut out the Decimal-Fraction Contest cards. Take all the fraction cards, shuffle them, and arrange them face down in a 3-by-4 array. Then shuffle all the decimal cards, and arrange them face down in a 3-by-4 array. Then share these instructions with students:
Play in pairs. Take turns turning over two cards, one from each array. If the fraction card and decimal card match, keep the matching cards and take another turn. If the cards don’t match, turn them back over. Play until all the cards have been matched. The player with the most matched cards wins.

Example:
Jaehyun turns over two cards. His first card is the 1.0 card. His second card is the (frac{32}{100}) card. The cards don’t match, so he turns them back over.
Kiera turns over two cards. Her first card is the 0.4 card. Her second card is the (frac{2}{5}) card. The cards match. Kiera keeps the cards and takes another turn.

Four Card Sum

What You’ll Need:
Four Card Sum cards

Play in pairs. Cut out and shuffle the Four Card Sum cards, and place them in a pile face down between the players. The player who did not shuffle can go first. Each player picks up four cards and adds the value of their cards. The player with the greatest sum gets a point. At the end of each round, discard your cards and pick up four new cards. The first player to 10 points wins.

Example:
Miles picks up his four cards. His numbers are –5, 7, 4, and –1. He adds his numbers and gets a sum of 5.
Shaila picks up her four cards. Her numbers are –1, –5, 8, and 6. She adds her numbers and gets a sum of 8.
Shaila’s sum is greater than Miles’s. She receives 1 point.

Percent Precision

What You’ll Need:
Percent Precision printout
Calculator (optional)

Print and cut out the number cards from the Percent Precision printout. Shuffle the cards, and place them face down in a pile. Then give students these instructions:
Play in pairs. Each player picks up four cards and makes a two-digit percent and a two-digit number. Players choose a number so that their percent of that number is close to the two-digit number they created. If their number is between 20 and 40 and their percent of that number is within 2 of their two-digit number, they get a point. Return the cards to the pile, shuffle, and repeat. The first player with 10 points wins.

Example:
Dominique picks up the numbers 3, 2, 4, and 7. She creates 74% and the number 23. She figured that if you took 74% of 31, she’d get about 23. She got a point because she got close to her number.
Jeffrey picks up the numbers 1, 8, 9, and 4. He creates 49% and the number 18. He figured that if he took 49% of 36.7, he’d get about 18. He also gets a point.