Inventory of “Big Ideas”
(Vital Understandings) in Mathematics

1.

Can my students determine how a pattern is created and extend that pattern.

Example Grade K-3:

a.

Join in when you know the pattern:  “Snap, clap, clap, snap, clap, clap, . . .”  Join in on this pattern:  “Snap, clap, clap, stamp, snap, clap, clap, stamp, . . .”  How is this pattern like the first one?  How is it different?

I-1.3

b.

The sticks are arranged like this:

What comes next?  How do you know?  What pattern can you make with the sticks?

Example Grade 3-4:

How would you build the squares that come after these?

I-1.1, 1.3

Example Grade 5-6:

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                     1            3                  6                      10

What are the next 6 triangular numbers?  How many total dots will there be with a side of 10 dots?  What comes next?

Example Grade 7-8:

Describe what is happening in the following patterns. Predict the next number.

I-1.3

2, 3, 5, 7, 11, 13 …

1, 4, 9, 16, 25 …

1, 2, 4, 8, 16 …

25, 23, 21, 19 …

1, 1, 2, 3, 5, 8, 13, 21 …

Could any of these sequences be extended in more than one way? How?

2.

Can my students use a function (rule) to generate a set of ordered pairs?  (A function is a rule where the value of the first quantity determines a corresponding value of the second.)

Example Grade K-3:

What’s my rule?  Tell me an input number.  I will apply a rule to the number and tell you an output number.  What am I doing to each input number?

I-2.3

Example Grade 3-4:

It takes two children to turn a jump rope.  How many children would it take if we wanted turners for 15 jump ropes (with each child turning only one rope)?  

I-2.4

What if you had 50 jump ropes?  Could you figure out how many children you would need?  How could you figure it out?                

Example Grade 5-6:

How can we simplify the following problem and organize the information to help us solve it?

I-2.4

If we tear a sheet of paper in half, then tear each half in half, and continue doing this, how many sheets of paper will we have after 16 tears?

How many pieces of paper would we have after one tear?  Two tears?  Three tears?  Do you see a pattern?  Could we figure out the number of pieces of paper no matter how many tears are made?

Example Grade 7-8:

In your bank you have $24.  Every weekend when you finish your chores, your mom or dad gives you $3.00.  On Saturday, when you go to the movies, you spend $5.00 for your ticket and food.  If you don’t spend any other money during the week, how long will it be until you’re broke?  If your mom or dad lets you borrow, how much will you owe in 6 months?

I-2.4

3.

Can my students recognize the same pattern in different situations?

Example Grade K-3:

One of the patterns that will come up many times is evident in the following examples.

I-1.1

a.

Join in when you know the pattern.

     Jump, jump, clap, jump, jump, clap, . . .

What is another way to show the pattern?

   Circle, circle, square, circle, circle, square, . . .

Another way?

     Tall, tall, short, tall, tall, short, . . .

Another way?

     A, A, B, A, A, B . . .

Write your first name over and over in a 4 x 10 matrix.  Color the first letter of your name.  What pattern do you see?  What other names have the same pattern? Why?  If you put the counting numbers in the matrix and colored in the same squares as your name pattern, what numbers are colored in?  Predict what numbers would be colored in if you continued the pattern.  What if you colored in the last letter of your name?

Example Grade 3-4:

Counting by two’s gives the pattern 2, 4, 6, 8, 10 …

What other situations can we find that give this same pattern?

I-1.1, 1.3

Example Grade 5-6:

One of the patterns that will come up many times is evident in the following examples.

I-1.1

How would you build the squares that come after these?

How would you build the triangles that come after these?

What is the sum of the first two consecutive odd numbers?  The first three?  Four?  Five?  Can you predict what the sum of the first eight consecutive odd numbers will be?

Why are these patterns the same?  Why is this pattern called “square numbers”?

Example Grade 7-8:

If everyone in our classroom were to shake hands with everyone else in the room, how many handshakes would there be?

When you build the triangular numbers, how many counters do you need for each step?

You are setting up a tennis tournament.  If you want every person to play every other person once, how many games of tennis will it take?

What pattern will help you find the sum of all natural numbers from 1 to 100?

Why do all these situations produce the same pattern?