# Connecting Whole Number Arithmetic to Algebra

## Description:

When students "hit" algebra, they are usually presented with concepts such as
"balancing"

equations, factoring and expanding polynomials , and linear equations. What
groundwork can be

set by the K-6- curriculum to lead students to see algebra as an extension of
their arithmetic skills?

Can the algebra curriculum bounce off students' previous knowledge of arithmetic
or must the

algebra curriculum start from its own frame of reference? Activities looking at
the operations on

whole numbers, using manipulatives and calculators, will be investigated and
then extended to the

"rules" of algebra. Share your thoughts on the question, "Can there be a
seamless journey from

arithmetic to algebra?"

## Goals:

• Present a connection between the thought processes and algorithms developed
while

studying arithmetic and the thought processes and algorithms shown while
studying

algebra.

• Use word problems, investigations with manipulatives and calculators, and
finally the

written mathematics to present a connected route from arithmetic to algebra.

• Share thoughts on the question, “Can there be a seamless journey from
arithmetic to

algebra?”

## In this Handout:

**Part A:** Word Problems: Natural Problem Solving and Inverse Operations

vs. Balancing Equations

**Part B:** Whole Number Algorithms and a Bit of Algebra!

Use Virtual Base 10 Blocks!

**Part C:** Multiplication Tables to Linear Equations

**Part D: **Using the TI-73 and the TI-Ranger to learn about lines!

## Part A: Word Problems:

Balancing Equations vs . Natural Problem Solving and Inverse Operations

**Addition and Subtraction:**

1. Maggie had 5 cookies. Jamal gave her 3 more

cookies. How many cookies does Maggie have

altogether?

2. Maggie has 5 cookies. How many more cookies

does she need to have 8 cookies altogether?

3. Maggie had some cookies. Jamal gave her 5 more

cookies. Now she has 8 cookies. How many

cookies did Maggie have to start with?

4. Maggie had 8 cookies. She gave some to Jamal.

Now she has 5 cookies left. How many cookies did

Maggie give to Jamal?

**Multiplication and Division:**

1. If three children have two cookies each, how many

cookies are there altogether?

2. If two children have three cookies each, how many

cookies are there altogether?

3. If six cookies are shared among three children, how

many would each child get?

4. If there are six cookies and each child must get three

cookies, how many children can you serve?

**Discussion Question:**

Which should we teach? How do we link the processes?

Balancing Equations |
Natural Problem Solving andInverse Operations |

## Part B: Whole Number Algorithms and a Bit of Algebra!

Using Base Ten Blocks to "See" Algorithms

**Objective:** To look at addition, subtraction, multiplication and
division of whole numbers from a

geometric, "hands-on" perspective and an algorithmic perspective.

**Audience:** This activity is intended for teachers. The activity is
designed to make connections

between the use of manipulatives and the development of algorithms. Parents and
students are

welcomed!

Part 1: Addition - Focus on trading and regrouping.

Part 2: Subtraction - Focus on trading and regrouping.

Part 3: Multiplication - Focus on the distributive property and area models.

Part 4: Division - Focus on the scaffold method and area models.

Part 5: A Bit of Algebra - Focus on the distributive property and area models.

**Part 1: Addition
1. One Type of Addition Algorithm**

2. Try these problems using Base 10 Blocks and the algorithm. Write and draw your work .

**Part 2: Subtraction
1. One Type of Subtraction Algorithm**

2. Try these problems using Base 10 Blocks and the algorithm. Write and draw your work.

**Part 3: Multiplication**

**1. One Type of Multiplication Algorithm** - If you have 23 students in
your class and they

each need 12 straws for a craft project, how many straws do you need to supply?
We

write this as 23 groups of 12 or 23 × 12. Write out how you would solve the
problem.

2. Notice two applications of the **Distributive Property **gives us the
"standard" pieces. Here

you see this in both vertical and horizontal formats. Find the pieces from the
computations

below on the area model. Notice that we are actually finding 12, 23's.

Vertical:

OR

Horizontal:

3. Try these problems using Base 10 Blocks. Draw or printout the area model
you construct.

Write out the details of the algorithm and find the products on your area model.
Notice

that the second problem is multi step. (Why?)

**Part 4: Division**

**1. One Type of Division Algorithm ** - You have 483 sea shells for a
class art project.

Each student needs 21 shells. How many students will be able to make the
project? How

many groups of 21 shells can you form out of 483 objects? We write 483 ÷ 21.
Write

out how you would solve this problem.

2. Find the number of groups of 21 on the

Area Model. Draw in the left most column

with the appropriate “Base 10 blocks.”

3. Next, look at the scaffold method below.

(Is there a correlation to the scaffold “good

guess” method and the Area Model? Does

there have to be a relationship?)

4. Now, you have 483 sea shells for a class art project. There are 21
students in your class.

If you give each student the same number of shells, how many shells will each
student

have? Use the blocks to model this problem. Is it still written 483 ÷ 21?
Discuss.

5. Caution: When you pick problems for illustration with Base 10 blocks, make
sure you

check them out first using the blocks! Sometimes a problem requires that you
break up a

FLAT in order to fill in the area model. This type of problem is not the best
for a first use

of the blocks.

Try these by drawing Base 10 Blocks. Solve also using the scaffold method.

**Part 5: Moving to Algebra!**

1. We can use the Base 10 blocks idea to create Base x

blocks or Algebra Tiles. x is unknown !

2. Look at the figure to the right. This is an area

representation of (x + 2)( x + 3). Compare this to

23 ×12. Do you see a similarity?

3. Expand:

(x + 2)( x + 3) =

Find the pieces on the picture to the right.

4. Try these! Draw them out and then expand. Can you see the pieces?

a. x(x+4)

b. (x+1)(2x+3)

c. Extension: Draw and expand (x + 2y + 3)(2x + y).

Hint: You will need y and y^2 blocks

## Part C: Multiplication Tables to Linear Equations

You have no money now. You find out that you are going to get an allowance of
$2 each week.

You would like to buy a __________________ (Calculator) which costs ____________
($20). You will

save your money until you have enough! When will you have enough? How is your
money

growing?

1. With your group, decide how much money you will have week to week. Use the
table and

the graph below to organize your work.

2. If you receive $2 each week you can write that in math symbols as____________________

3. Write the sequence of numbers that describes your money supply week to week.

____________________________________________________________

4. From your graph, write a word that describes the shape of the graph.____________________

5. Write a sentence to explain when you will have enough money to make your
purchase.

Write how you found this answer.

6. Use your work to predict the future! If you don’t spend any money for 14
Weeks, how

many Dollars will you have? Write an explanation of how you found your answer.
Use

words such as add, subtract, multiply, and divide.

**Teacher’s Sheet
How Does Your Money Grow?**

Classroom Organization:

Groups work and present their solution .

(Time saver - use overheads or small white boards for each table or have board

space for each group as they work.)

**Discussions **of Mathematics used in the solutions needs to occur.

Give the suggestion of the use of Tables of Numbers and Graphs as tools.

1. Reinforce multiplication as repeated

addition.

2. If you receive $2 each week you can write that in math symbols as

**+2 , a constant rate of change**

3. Write the sequence of numbers that describes your money supply week to
week.

**2, 4, 6, 8, 10,** This is an Arithmetic Sequence of numbers.

4. From your graph, write a word that describes the shape of the graph. **a
line**

5. Write a sentence to explain when you will have enough money to make your
purchase.

Write how you found this answer.

Encourage students to see the relationship Dollars = 2 × Weeks This is developed
using

Inductive Reasoning.

Use of Division as an Inverse Operation:

This sets the foundation of a function and an inverse function.

Students will also be able to read the answer from the table and the graph.

6. Use your work to predict the future! If you don’t spend any money for 14
Weeks, how

many Dollars will you have? Write an explanation of how you found your answer.
Use

words such as add, subtract, multiply, and divide.

Students can use the table, graph, the calculator to find the solution .
Encourage the

formal writing of Dollars = 2 × 14 = $28. Notice that if students use the table
or the

graph they will have to predict a pattern to use.

Added discussion: Commutative Property of ×

Does it matter?

Dollars = 2 × Weeks

or

Dollars = Weeks × 2

## Part D: Using the TI-73 and the TI-Ranger to Learn about Lines!

**Part I How Fast Do You Need To Go?**

1. Use the Ranger program on your TI-73 to

collect data. A distance-time graph of

constant speed forms a straight line. Draw

your walk on the graph at the left. Write a

description of how you walked a constant

speed._____________________

2. How can you make your line steeper? _______________________________________________________________

3. Repeat the exploration above and walk a

distance-time graph which is steeper than your

first walk. Draw your walk on the graph.

4. Repeat the exploration above and walk a

steeper line. Draw your graph. What did you

have to do to get a steeper
line?_______________________________________________________________

5. Can you walk a line that is vertical (straight up and down)? Why or why not?

_______________________________________________________________

_______________________________________________________________

**Part II How Fast Were You Going? Let's Look at Your Data!**

1. Use the Ranger program on your TI-73 to gather data.

Walk a distance-time graph of constant speed. Draw your

walk on the graph.

2. L_{1} is the time in seconds and L_{2} contains your position from Ranger. Fill
in the following

table with the data

3. The average velocity you walked can be

calculated by

Use the information in the table to answer the

following questions.

4. How far did you walk from 1.1 sec to 1.6 sec?_________

This is your **change in position.**

5. How many seconds did you walk during 1.1 sec to 1.6
sec?_________

This is your **change in time.**

6. Your average velocity during 1.1 sec to 1.6 sec. is found by

7. Find your average velocity during 1.6 sec to 2.1 sec.

8. Find your average velocity during 2.1 sec to 2.6 sec.

9. Find your average velocity during 2.6 sec to 3.1 sec.

10. Did you come close to walking a constant speed? Explain your answer.

________________________________________________________________________________

11. How is the average velocity related to the slope of the distance -time line?

________________________________________________________________________________

12. What is the intercept of your walk ? How do you know?__________________________

13. **Extension:** Use Manual-Fit on the TI-73 to find
the equation that describes your walk.

Compare it to the linear regression line found by using LinReg on the TI-73!

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