Chapter 2, Section 4 2.4 Elementary Education

Section 2.4, p.1

Chapter 2, Section 4 2.4 Elementary Education – Introduction to Fractions Math Topics – Fibonacci Numbers; the Golden Ratio, Figurate Numbers, Successive Differences and Non-Linear Equations; Using Subscripts to Find Patterns

Fibonacci Numbers The Fibonacci numbers were invented in 1202 by Italian mathematician Leonardo of Pisa, who was also known as Fibonacci. But they also appeared over a thousand years before that in Indian Mathematics in Sanskrit poems. The numbers are rich in beautiful patterns, and appear in nature and in art, in spirals and in rectangles.

The Fibonacci numbers begin: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 ….

3+5=8 5+8=13 Each new number is the sum of the previous two. For example, 1+1 =2, 1+2 = 3, 2+3 = 5, 3+5 = 8, etc. We designate F1 to mean the first Fibonacci number, F2 to mean the second, etc. th F7 = 13 means the 7 Fibonacci number is 13.

F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765

Flower petals tend to come in Fibonacci numbers (see for example, britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm)

3 Trillium 5 Texas Yellow Star 8 Clematis 13 Black-eyed Susan

21 Chicory 34 Daisy 55

Section 2.4, p.2

In addition, the number of spirals in the middle of sunflowers, pinecones, artichokes and pineapples tend to be Fibonacci numbers.

You can create a Fibonacci spiral using squares that are the size of Fibonacci numbers.

Example 1 Which sequence is most like a Fibonacci sequence? a. 1, 1, 2, 2, 3, 3 …. b. 1, 1, 1, 3, 3, 3, 9, 9, 9, 27, 27, 27 c. 2, 5, 7, 12, 19, …. Try this out yourself before you go to the next page! Section 2.4, p.3

Example 1, continued Each sequence has some characteristics that are Fibonacci-like. The first sequence, 1, 1, 2, 2, 3, 3 …. starts out with 1, 1, just like the Fibonacci sequence. But then after that, two numbers do not add to equal the next. The next sequence, 1, 1, 1, 3, 3, 3, 9, 9, 9, 27, 27, 27, has an interesting pattern, but you have to add three numbers to get the next and then the number repeats three times -- not the same as the Fibonacci pattern. The third sequence, 2, 5, 7, 12, 19, …. is most like a Fibonacci sequence since you must add two numbers to get the next.

Example 2 Divide consecutive Fibonacci numbers. What pattern do you see in the answers? For example, F4/F3 = 3/2 = 1.5 Try dividing the next pair, F5/F4 = 5/3, and the next pair, F6/F5 = 8/5 , etc. What do you notice? As you divide larger and larger consecutive Fibonacci numbers the result tends to get closer and closer to 1.61… This number is known as the Golden Number, p = = = 1.618033 … This number is similar to pi = π=3.14159…. Both are written 1+ 5 ℎ𝑖𝑖 with Greek√ letters ( and π are letters in the Greek alphabet), both have decimals 𝜑𝜑 2 that continue infinitely, and both are irrational numbers (more on irrational numbers in a later section).𝜑𝜑

The Golden Ratio, or “Divine Proportion” The Greeks thought that things that were in this ratio were more “pleasing to the eye.” This includes the Parthenon, below.

= 1.618 … . = = a

𝑎𝑎 𝜑𝜑 𝑝𝑝ℎ𝑖𝑖 𝑏𝑏 Section 2.4, p.4

Later, Renaissance painters revived the use of the golden ratio. At the right is the Mona Lisa, by Leonardo DaVinci, from around 1503. Try it yourself! Measure the length and width of a rectangle that looks pretty to you. Calculate length divided by width. See if it comes out close to the golden number. Do you think the Greeks were right about what kinds of triangles look the prettiest?

You can also measure from the top of your shoulder to the tips of your fingers and then from the crook of your elbow to the tips of your fingers. The ratio of these two numbers is also often close to 1.618…!

a = 1.618 … . = = 𝑎𝑎 𝜑𝜑 𝑝𝑝ℎ𝑖𝑖 𝑏𝑏

I have tried this in class with students (and me) and it works!! Section 2.4, p.5

Using Subscripts to Find Patterns In this part of our problem solving, we will use the subscripts (the small index numbers) to find patterns.

F12 The small number, 12, is called a subscript. It means the 12th Fibonacci number. When we say that F12 is the sum of the previous two Fibonacci numbers, we can th th th write F12 = F11 + F10, or 144 = 89 + 55, since those are the 12 , 11 and 10 Fibonacci numbers. Notice how 11 is one less than 12, and 10 is 2 less than 12.

We could write F12 = F12-1 + F12-2.

In general, for all Fibonacci numbers, Fn = Fn-1 + Fn-2. To get the general formula using n, we imitate the pattern we see with the numbers themselves.

Example 3 Find the general formula using n for the ratio of two successive Fibonacci numbers, F4/F3 = 3/2, F5/F4 = 5/3, F6/F5 = 8/5 , etc. To see the formula, notice that the bottom number is always one less than the top. That is, = and = 𝐹𝐹4 𝐹𝐹4 𝐹𝐹5 𝐹𝐹5

𝐹𝐹3 𝐹𝐹4−1 𝐹𝐹4 𝐹𝐹5−1 If we imitate this pattern using n, we get . 𝐹𝐹𝑛𝑛 Thus, the formula for the ratio of two successive Fibonacci numbers is Fn/Fn-1 . 𝐹𝐹𝑛𝑛−1 Note: we could have also started with the bottom number, and noticed that the top number is one bigger, so we could also write Fn+1/Fn. There is often more than one correct way to write a formula, depending on what we take n to be.

Example 4 A very interesting pattern emerges when we add any six consecutive Fibonacci numbers. See if you can find the pattern, then write the general form using subscripts. Remember that the Fibonacci numbers are:

F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765

Add the first six: F1 + F2 + F3 + F4 + F5 + F6 = 1+1+2+3+5+8 = 20 Add another six that are consecutive: F3+F4+F5+F6+F7+F8=2+3+5+8+13+21 = 52 Add another six: F5+F6+F7+F8+F9+F10=5+8+13+21+34+55 = 136 Add another six: F6+F7+F8+F9+F10+F11=8+13+21+34+55+89 = 220 Section 2.4, p.6

What is the largest number that you see that divides into each of these answers? Write each answer as a product using that number. Each is divisible by 2, but the largest number that goes into each is 4. We can write each answer as a product of the number 4: F1 + F2 + F3 + F4 + F5 + F6 = 1+1+2+3+5+8 = 20 = 4 × 5 F3+F4+F5+F6+F7+F8=2+3+5+8+13+21 = 52 = 4 × 13 F5+F6+F7+F8+F9+F10=5+8+13+21+34+55 = 136 = 4 × 34 F6+F7+F8+F9+F10+F11=8+13+21+34+55+89 = 220 = 4 × 55

To see even more of the pattern, write which Fibonacci number you have in each case. F1 + F2 + F3 + F4 + F5 + F6 = 4 × 5 = 4 × F5 F3+F4+F5+F6+F7+F8 = 4 × 13 = 4 × F7 F5+F6+F7+F8+F9+F10 = 4 × 34 = 4 × F9 F6+F7+F8+F9+F10+F11 = 4 × 55 = 4 × F10

See if you can now see where that Fibonacci number is, in relation to the ones you just added, then write the general pattern in words and using n. In words: You get four times the fifth Fibonacci number of the ones you added. Using subscripts: Fn + Fn+1 + Fn+2 + Fn+3 + Fn+4 + Fn+5 = 4 × Fn+4

Figurate Numbers The figurate numbers make common figures, or shapes. There are triangular numbers (that make triangles or pyramids), square numbers (that make squares), pentagonal numbers (that make 5-sided shapes called pentagons), and more. You have already seen the square numbers, so we’ll start with those. The square numbers are the perfect squares: 1, 4, 9, 16, 25, etc. We write S1 to indicate the first square number, S2 for the second square number, and so on. The formula to find a 2 square number is Sn = n . When we draw the square numbers, we can represent them as square shapes:

S1 = 1 S2 = 4 S3 = 9 S4 = 16 S5 = 25

2 th Example 5 Use the formula Sn = n to find the 6 square number, and draw the picture. 2 We replace the n in the formula with the number 6, so S6 = 6 = 36. The picture will be a 6 by 6 square.

Section 2.4, p.7

Example 6 Find the successive differences for the square numbers.

1 4 9 16 25 36 49

3 5 7 9 11 13

2 2 2 2 2

Notice that now, the successive difference (yellow circles) are not the same each time. We need to make one more row (the blue circles) for the differences to remain the same each time. Algebra note: • A linear equation (which graphs a line) with x1 as the highest powered term, will have one row of successive differences which will remain the same each time – that difference is the slope of the line. • A quadratic equation (which graphs a parabola) with x2 as the highest powered term, will have two rows of successive differences before we get a difference that stays the same each time. • An equation with x3 as the highest powered term, will have three rows of successive differences before the difference stays the same each time. To find the equation, we would probably need to use calculus, using the derivative.

For the triangular numbers, we write T1 to mean the first triangular number, T2 to mean the second triangular number, and so on.

T1=1 T2=3 T3=6 T4=10 T5=?

The first triangular number has 1 block, so T1 = 1; the second triangular number has 3 blocks, so T2 = 3; T3 = 6 and T4 = 10.

Example 7 See if you can draw the correct picture for T5 by copying the picture for T4 and adding one more layer of blocks on the bottom row. How many blocks does it have?

Section 2.4, p.8

Example 7, continued

T5 = 15

( ) The formula for triangular numbers is Tn = . 𝑛𝑛 𝑛𝑛+1 2 Example 8 Use the triangular number formula to find the 6th and 7th triangular numbers. ( ) ( ) th For the 6 triangular number, let n = 6 in the formula. We get T6 = = = 6 6+1 6 7 th = 21. For the 7 triangular number, let n = 7 in the formula. Try it 2yourself2 . The 42answer should be 28. 2

Example 9 Find the successive differences for the triangular numbers and use them to find the 5th, 6th and 7th triangular numbers. Your answers should agree with what you got in the formula, above.

1 3 6 10

2 3 4

1 1 1

Try to get the next differences and the triangular numbers before turning the page.

Section 2.4, p.9

Example 9, continued

1 3 6 10 15 21 28

2 3 4 5 6 7

1 1 1 1 1

Notice that it again takes two rows for the differences to remain constant. This is ( ) consistent with the fact that the triangular number formula, can be 𝑛𝑛 𝑛𝑛+1 nd rewritten as 2 , so the highest power is the 2 power, as expected2 by the fact 𝑛𝑛 +𝑛𝑛 that it took two rows for the difference to remain constant. 2

Fraction Introduction Even young, pre-school aged children understand the fraction , and most understand how to share equal amounts of objects. In Developing1 Effective 2 Fractions Instruction for Kindergarten through 8th Grade, the authors recommend building on this sort of informal understanding by having kindergarten children start with sharing problems that involve whole numbers, like sharing 12 cookies with 3 people (see the picture, at right).

As children get older, such problems can progress to those with fraction results, like sharing 4 pizzas among 8 children.

Section 2.4, p.10

Fractions and Manipulatives Before children formally learn fractions, they can have fun with kitchen math, using measuring cups to help make playdough or food: https://www.youtube.com/watch?v=Yhpiuibv0LY. Children can also play with Tangrams (picture on the right) to make shapes and begin to explore how shapes can fit into a whole.

Tangram shapes and attribute blocks also make great manipulatives to help with early fraction understanding, including creating equivalent fractions which are a key to later understanding how to add, subtract, multiply and divide fractions.

Attribute blocks being washed after playtime. Photo courtesy of Meghan Fitzgerald Raimundo.

What is a Fraction? In about the third grade, children begin formal instruction in fractions. Fractions can be understood as a part of a whole and at the same time as a particular place on the number line, as a number. For example, can be thought of as 1 part of a whole 1 2 Section 2.4, p.11 that is divided into 2 pieces; it can also be thought of as the number that is half way between 0 and 1 on the number line.

½ 0 ½ 1 2

Half of an inch Half of a pizza Halfway between 0 and 1 One piece out of 2 equal pieces

0 ¼ 1 2 ¼ of an inch

= = ¼ of a pizza 1 1 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑜𝑜𝑜𝑜𝑜𝑜 𝑜𝑜𝑜𝑜 1 𝑡𝑡ℎ𝑒𝑒 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑜𝑜𝑜𝑜𝑜𝑜 𝑜𝑜𝑜𝑜 4 4 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 4 4 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠

0 ¾ 1 2

= ¾ of a pizza ¾ of an inch 3 3 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑜𝑜𝑜𝑜𝑜𝑜 𝑜𝑜𝑜𝑜 = 4 4 𝑒𝑒𝑒𝑒 𝑒𝑒𝑒𝑒𝑒𝑒 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 3 𝑡𝑡ℎ𝑒𝑒 𝑡𝑡ℎ𝑖𝑖𝑖𝑖𝑖𝑖 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑜𝑜𝑜𝑜𝑜𝑜 𝑜𝑜𝑜𝑜 It is also important for children to use many other objects4 to4 𝑒𝑒𝑒𝑒 represent𝑒𝑒𝑒𝑒𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 fractions, not just pizzas and the number line! By using many kinds of objects, students can visualize parts of many different kinds of wholes, and can see equivalent fractions.

Example 10 Using tangrams, if the hexagon is one whole, how can you show halves, thirds and sixths? What about 2/2, 2/3 and 2/6? If a hexagon is one whole, then the trapezoid is ½, since 2 trapezoids fit into a hexagon. = 1 2 2 1

To show 2/2, we use two trapezoids, and we can see that = 1 whole hexagon. 2 2 Section 2.4, p.12

Since three rhombuses (diamonds) fit in a hexagon, one rhombus = 1/3, and we can show by using 2 of them. 2 3 = one = two 2 1 shaded1 shaded 3 3 3 part out of parts out of three equal three equal

parts parts

The hexagon can be divided into 6 triangles, so 1 triangle = and two triangles = 1 2 6 6 Notice that 1 6 makes 1 2 the same 1 6 6 6 shape as . 1 3 Equivalent Fractions Using hexagons, students can start to notice which triangles make the same shapes. For example, 2 triangles = 1 rhombus, so they can see naturally that = . 2 1 Numerically, to get an equivalent fraction, we multiply or divide the top6 and3 bottom of the fraction by the same number. For example, we can divide by 2 on the top 2 ÷ = and bottom to get ÷ . The two fractions and are equivalent6 because 2 2 1 2 1 multiplying or dividing6 2 3by is the same as multiplying6 3 or dividing by 1. 2 2 Notice that the hexagon divides nicely into halves, thirds and sixths, but that it would be hard to show 1/4 or 1/5, since it is hard to divide a hexagon into 5 equal parts or 4 equal parts. Example 11 Using an egg carton as one whole, how can you show , and ? What 1 1 about , and ? What equivalent fractions can you see? What kinds3 of fractions4 2 2 would be hard to show using an egg carton? 3 4

Section 2.4, p.13

To show we can use an egg carton divided into 3 equal parts with 1 part circled 1 or shaded in. 3 To show we can use an egg carton divided into 3 2 equal parts with 2 parts circled or shaded in. 3 1 1 1 Since there are 12 eggs, we get the equivalent 3 3 3 fraction . 8 We also know12 numerically that = because 8 2 when we divide numerator and denominator by 4, 12 3 = ÷ = we get the fraction 2/3: ÷ . 8 8 4 2 × 12 12 4 3 = = We can also think of multiplying by to get : × 2 4 8 2 2 2 8 To show we can use an egg carton3 divided4 into12 4 equal3 3 parts2 12 with 1 part circled 1 or shaded in. 4 To show we can use an egg carton divided into 4 2 equal parts with 2 parts circled or shaded in. 4 1 1 Since there are 12 eggs, we get the equivalent 4 4 fraction . This is also the same as the fraction , which you6 can see very clearly from the picture, 1 12 2 without any calculations. It would be difficult to show a fraction like 1/5, or 1 1/7, since 12 is not divisible by 5 or 7. 1 4 4

Example 7 How many fractions can you show with this bag that has two bags of candy inside it? (Do not cut any objects in half.) How could you show equivalent fractions? What fractions would be difficult to show?

• You can show as 1 candy bag out of 2 and 1 as 3 candies out of 6:2 = . 3 1 • You can show 2 blue6 2candies out of 6 candies, and there are three colors of candies = . 2 1 • You can also show 6 candies is 1 whole6 bag,3 or = 1. 6 • It would be hard to show any fraction out of 5 or6 9 or 10, since the number of things in the bag does not divide evenly by 5, 9 or 10. Section 2.4, p.14

A word about copying Project answers: Please do not let other students copy your project answers! Please think about how you can instead help another student to get the answers so that they really understand the problem. Sometimes students copy from other students and then when they get to the test, they do not pass the test because they don’t really understand. Please don’t let your classmates fail!

Section 2.4 Homework 1. Play the tangrams game, http://www.abcya.com/tangrams.htm. (Note: if it tells you to subscribe or pay money, ignore it.) Do you think it would be more fun to try on paper and with real attribute blocks, or more fun on the computer? How could this help young kids play with the idea of part and whole? How is it different from making fractions with attribute blocks? OR Play the game Chicken Coop Fractions: http://www.echalk.co.uk/tasters/taster2/taster.html. (This is a free preview of the game.) Does the game teach kids about fractions, or is it just to practice what they already know? Would the game help a student get better at understanding fractions? Why or why not? Is it a fun game?

2. Which of the following is like a Fibonacci sequence? Which is not? Why? a. 1, 1, 2, 2, 3, 3, 4, 4, …. b. 2, 2, 4, 8, 32, …. c. 3, 3, 6, 9, 15…

3. Show how you can represent the fraction 3/8: a.) Using a pizza b.) Using a number line or measuring tape c.) Using a bag that has 16 candies in it. What equivalent fraction does this give you? d.) What fractions would be easy to show using a bag of 16 candies? What fractions would be hard to show?

4. Use an egg carton to show ¾. What equivalent fraction does using the egg carton help you to see? Why?

5. Suppose there are special numbers called the Mum numbers. M1 is the first Mum number, M2 is the second, etc. Suppose you see the following pattern: M1 × M2 = M4 M6 × M7 = M9 M7 × M8 = M10 a.) Explain how you would write the pattern in words. b.) Write the general formula using subscripts.

Section 2.4, p.15

6. Suppose there are more special numbers called the Plum numbers. P1 is the first Plum number, P2 is the second, etc. Suppose you see the following pattern: P1 × P2 × P3 = P4 − 1 P2 × P3 × P4 = P5 − 1 P7 × P8 × P9 = P10 − 1 a.) Explain how you would write the pattern in words. b.) Write the general formula using subscripts.

7. A fraction can be reduced using divisibility rules. For example, I can reduce 105 by noticing that both numerator (top) and denominator (bottom) are divisible by 180 5, since the top number ends in 5 and the bottom number ends in 0. So I get: ÷ = ÷ . Next, I might notice that both numerator (top) and denominator (bottom)105 5 21 are divisible by 3, since the digits of 21 add up to 3 (2+1=3) and the 180 5 36 ÷ = . digits of 36 add up to a multiple of 3 (3+6=9). ÷ 21 3 7 Use divisibility rules to reduce these fractions.36 State3 9 the divisibility rules you use without referring to your calculator. a. I know the top and bottom are divisible by ____ and ____ because… 36 b. 60 I know these are divisible by ____ and ____ because… 360

315 8. I am thinking of a number. If I multiply the number by 10, add 8 and then divide the result by 2, I get 29. a. Name two strategies you think might work to solve this problem. b. Solve the problem using either (or both) strategies.

9. This is very similar to, but not the same as the previous question. I am thinking of a number. If I multiply the number by 10, add 8 and then divide the result by 2, I get a number between 55 and 60. a. Name one strategy you think might work to solve this problem. b. Solve the problem using the strategy. c. Why are the other strategies you could have used on problem 6 hard to use on this problem?

10. Find the ones digit of 7402. Show your work using problem solving strategies.

th th 11. a. Find S9 and T9, the 9 square and 9 triangular number (p. 4), using the correct formula for each. b. Draw a picture of each.

Section 2.4, p.16

12. a. Find the successive differences for the sequence below.

8 17 32 53 80 ? ?

b. See if you can use the problem solving strategy of “working backwards” to find the next two numbers in the sequence.

Based on the number of rows it took to get the difference to remain the same time, what type of equation is this? (See p. 7)

Section 2.4 Homework Answers 2. Only c is like a Fibonacci sequence, where you have to add the previous two numbers to get the next one. 3. a. 3/8 = 8 equal pieces, shade 3 of them. b. On a number line, make 8 equal sections. 3/8 is the third one.

0 3/8 1

Section 2.4, p.17 c. In a bag with 16 candies, make 8 groups of candies, then take 3 of those groups. This gives you the equivalent fraction = 3 6 8 16

d. Your own words.

4. a. = Since there are 12 eggs in a carton, 3 9 the egg carton can help us see fractions out of 12. 4 12

5. a. The pattern is that the subscript of the answer is two more than what you just multiplied. For example, M6 × M7 = M9 and 9 is 2 more than 7. Or you could say the answer is three more than the one you started with. b. Generally, Mn × Mn+1 = Mn+3 The answer, Mn+3, is two more than what you just multiplied, Mn+1. Or, you could say the answer, Mn+3, is three more than first one, Mn.

6. a. The pattern is that the subscript of the answer is one more than what you just multiplied, and you subtract 1 from that. b. Generally, Pn × Pn+1 × Pn+2 = Pn+3 − 1

÷ = ÷ = = 7. a. ÷ ÷ (You may reduce this in a different order or with different 36 3 12 2 6 3 numbers, the result should be the same.) 60 3 20 2 10 5 Both are divisible by 3 since the digits add up to a multiple of 3; both are divisible by 2 since they end in even numbers. You might also notice that both are divisible ÷ = = by 6 since they are divisible by 2 and by 3: ÷ 36 6 6 3 60 6 10 5 Section 2.4, p.18

÷ ÷ = = b. ÷ ÷ (You may reduce this in a different order, but the result should be 360the 5same.)72 9 8 315 5 63 9 7 Both are divisible by 5, since one ends in zero and the other ends in 5. Both are divisible by 9 because the digits add up to a multiple of 9. 8.a. Working backwards, guess and check, or using algebra would work. b. The number is 5. 9. a. Guess and check. b. The number is 11. It’s hard to use working backwards or algebra because you don’t have a specific ending number. 10. 7 = 7, 7 = 49, 7 = 343, 7 = 2401, 7 = 16807, 7 = 117649, etc. The pattern of ending1 digits2 is 7, 9,3 3, 1, 7, 9,4 3, 1, etc. This5 pattern 6repeats so that every fourth number ends in a 1. Since 400 is a multiple of 4 (is a 4th number), 7400 will end in a 1, 7401 will end in a 7, and 7402 will end in a 9.

2 11. a. S9 = 9 = 81 b. picture: a square made of 81 blocks, 9 on each side ( ) ( ) T9 = = = = 45 Your9 picture9+1 9should10 90 be a pyramid with 45 blocks/dots. 2 2 2

12. a. You should finally end up with the numbers in the bottom row being 6. b. The missing numbers are 113 and 152 c. Quadratic equation (2nd power) since there are two rows.