Name of Lesson: Watch These Numbers Grow

Topic: Primary Algebraic Thinking Lesson #12

Approximate time: 3 sessions

Gifted Standard and element(s):

G2CG1. Students will reason logically using induction, deduction, and abduction.

a. Explore and develop critical thinking skills through the process of convergent thinking.

d. Solve problems using logical reasoning.

f. Develop verbal and nonverbal communication skills to convey logical reasoning.

Supports CCGPS:

5.OA.3 “Analyze patterns and relationships” fits but the specific element does not—it involves

two patterns and comparing corresponding elements of each, etc.

Writing Gr. 2: 2. Write informative/explanatory texts in which they introduce a topic, use

facts and definitions to develop points, and provide a concluding statement or section.

Language Gr. 2: 1. Demonstrate command of the conventions of standard English grammar

and usage when writing or speaking.

Language Gr. 2: 2. Demonstrate command of the conventions of standard English

capitalization, punctuation, and spelling when writing.

Standards for Mathematical Practice:

3. Construct viable arguments and critique the reasoning of others.

6. Attend to precision.

7. Look for and make use of structure.

Essential Question(s):

How can I use the mathematical meaning of “equivalence” to solve problems?

How can I use variables to solve problems?

*How can I use patterns, rules, and relationships to solve problems?

Lesson Questions: How can I analyze a growing number pattern to determine its rule and to determine future or

missing elements in the pattern?

Lesson Summary: Students will analyze various number patterns, determine the next element or

missing elements, and then determine future elements by using its rule rather than expressing all

intermediate elements. Key idea: There are many strategies for analyzing patterns. These

strategies need to be logically and systematically applied until a solution is reached.

Assessment Description/Performance Task:

Constructed response Informal assessment

Performance task Selected response

Brief Description of Assessment: Teacher will observe groups as they explain various

patterns/rules to each other. The student sheet, “Number Pattern Analysis” has students complete

number sequences and explain their strategies and reasoning independently, both orally and in

writing.

Instructional Methods:

Hook/Activator: Do “Ready to Order” game with pictures that tell a story. (You may need to teach

this game first. One set of pictures is attached. If you have something else, feel free to use it.)

After completing the game, discuss how students determined the order of the pictures. A main

point of this discussion is that one must use all available clues to determine the logical sequence.

In order to do that:

use prior knowledge

look at what is happening in current picture for two purposes: to predict the effect

of the action AND to determine the cause of the action; then decide whether

another picture shows this effect or cause

periodically review the sequence to be sure the order is logical

Say something like, “Now that you have some experience with sequencing pictures, you will use the

same kind of clue-gathering and analysis techniques to determine missing elements in mathematical

sequences. We will be analyzing number sequences to determine the missing or next element, and

then thinking about our strategies for doing this and explaining them to others.” Remind students

of the definitions of the words “analyze” (study the parts of something), “metacognition” (thinking

about one’s thinking), and “explanation” (telling about something so someone else understands).

Teaching Strategy:

Write the following on the board:

1, 2, 3, 4, ______, 6, 7 . . .

Ask, “What number belongs in the blank? How do you know?” Most students will refer to

counting in some way. If no one suggests it, the teacher should lead students to the idea that it

is also “adding 1” to the previous number in the sequence.

Next example: 2, 4, 6, 8, _____, ______, 14, 16 . . .

What numbers belong in these blanks? Have students explain their reasoning. If no one

mentions adding 2 to any element to determine the next element, the teacher should lead them

to it.

Next example: 4, 7, 10, 13, _____, _____, . . .

What numbers belong in these blanks? Have students explain their reasoning. If no one

mentions adding 3 to any element to determine the next element, the teacher should lead them

to it.

Next example: 11, 12, 9, 10, 7, 8, 5, _____, . . .

What number belongs in the blank? Have students explain their reasoning. At some appropriate

point in the discussion, point out that there is not just one straightforward rule for this pattern. If

these have not been introduced by different students lead them to see these ways to explain it:

add 1 to get the next element, then subtract 3 to get the next, then add 1 to get the

next, and continue alternating these two operations

subtract two from the next-to-last listed element to get the next one

Brainstorm strategies to use when solving patterns involving numbers. This may be done

individually or in small groups. Have students record these in the “Numbers” column of the

“Pattern Analysis Tools” chart (begun during previous lesson). Add these strategies to the class

chart. If students have trouble doing this, continue with the next part of the lesson and revisit

after they complete “Number Pattern Analysis”.

Next, the teacher models writing an analysis, possibly using one of the patterns from

the beginning of class or one from “Number Pattern Analysis”. Focus on the critical

thinking skills of metacognition and explanation. This modeling should focus on

explaining one’s thinking in writing, including some strategies that did not work and

their results. This will be important when students complete the assessment. The most

effective way to do this is to show the pattern to the students and then “think out loud”

as you write or type something similar to the example below.

Example for modeling:

I first determined if this (the last example above) was a linear or growing pattern. There is no

repeated pattern unit, so it is a growing pattern. Next, I looked to see if the pattern was getting

larger or getting smaller. In general, it is getting smaller. I asked myself questions like, “Are all

of the numbers the same distance apart?”, “Is skip counting happening?”, “What is similar or

different about the numbers?”, and because it is generally getting smaller, “What is the difference

between each element?” The difference between the first and second elements, 11 and 12, is

“plus one”. The difference between the second and third elements, 12 and 9, is “minus three”.

The difference between the third and fourth elements, 9 and 10, is “plus one” again. The

difference between the fourth and fifth elements, 10 and 7, is “minus three” again. This looked

like a pattern. It seemed to be alternating between the two operations of adding one to an

element to get the next element and then subtracting three from that element to get the next. I

called this a “plus one, minus three” pattern in my head. I made sure that this pattern worked for

the whole sequence. It did, so the solution to fill in the blank would be “plus one”, or 6. Later I

noticed that you could also say that the number pattern was really two separate growing patterns

combined. One is to skip count down by two beginning at 11. The other is to skip count down by

two beginning at 12. These two skip counting patterns are combined and alternated to form the

entire pattern. That is, 11, 9, 7, … alternates with 12, 10, 8, …

Have students complete the sheet, “Number Pattern Analysis”.

Summary by the Learner: Have students choose a pattern from “Number Pattern Analysis”.

(Students may NOT choose the same pattern about which they already wrote an explanation.)

Each will orally explain to a small group how he/she knew what to write in the blank. The teacher

listens to group discussions, correcting any misconceptions the students do not correct.

Differentiation:

More capable: Have students come up with a minimum of 3 different ways to explain any of the

patterns on “Number Pattern Analysis”. (Example: For number 1, it could be counting by 5’s,

adding 5 to the previous, adding 6 and subtracting 1, etc. If students are REALLY being precise,

they would also name the beginning number.)

Less capable: Suggest specific strategies for completing any sequences the student finds

difficult.

Materials for this Lesson:

picture cards for sequencing (The ones provided are set to “grayscale”. If you want them to be

color, right click on the picture, choose “Format picture”, and then select “automatic” from the

“color” drop-down menu.)

“Ready to Order” dot lines

class set of “Number Pattern Analysis”

students’ “Pattern Analysis Tools” charts from previous lesson

class poster of “Pattern Analysis Tools” from previous lesson

Vocabulary for this Lesson: increasing

decreasing

sequence

analysis

metacognition

explanation

strategies

Name ______________________________________

Pattern Analysis Tools- Suggested answers

Number Shape Word

odd/even

counting by (multiples) increasing by same number

decreasing by same number increasing by a pattern (+1, +2, +3,

etc)

number of sides- increasing or

decreasing position- facing up/down/left/right,

etc. number of shapes in each element

arrangement of shapes in each element (if more than one shape

per element) color or shading of each

shape/element size or shape changes

number of letters

spelling pattern singular/plural

meaning of words (synonym, antonym, etc)

beginning letter alphabetical order

double letters number or position of vowels

ending letter rhyming

CCSD Version Date: May 2011

Name ______________________________

Number Pattern Analysis Use the strategies discussed to analyze each pattern. Then write the missing

element. Be ready to explain how you knew what to write.

1) 5, 10, 15, ______, 25, ______, ______, 40, . . .

2) 21, 19, 18, 16, 15, ______, 12, . . .

3) 122, 128, 134, ______, ______, 152, . . .

4) 48, 24, 12, ______, 3, . . .

5) 51, 40, 29, 18, ______, . . .

6) 2, 4, 8, 16, ______, 64, ______, . . .

7) 31, 35, 33, 37, 35, 39, 37, ______, ______, 43, . . .

8) 1, 8, 15, 22, ______, ______, . . .

9) 65, 50, ______, 40, 45, ______, . . .

10) 207, 208, 210, 213, 217, ______, 228, ______, 243, ______, . . .

11) 9, 11, 15, 17, 21, 23, ______, . . .

12) 1, 2, 5, 10, ______, 26, 37, ______ 65, . . .

13) 850, 800, 810, 760, 770, 720, ______, ______, . . .

14) 100, 90, 79, 67, ______, 40, ______, 9, . . .

15) 1, 2, 3, 6, 7, ______, 15, 30, 31, 62, ______, . . .

16) 11, 2, 16, 7, 21, 3, 26, ______, 31, 4, ______, 9, . . .

CCSD Version Date: May 2011

Choose one of the patterns above and write an explanation of your analysis and

solution. Focus on your thinking, and include any strategies you tried that did not work, as well as those that did.

Re-write your chosen pattern on the line.

___________________________________________________________________

Write the explanation of your analysis and solution below.

CCSD Version Date: May 2011

Name KEY

Number Pattern Analysis Use the strategies discussed to analyze each pattern. Then write the missing

element. Be ready to explain how you knew what to write.

1) 5, 10, 15, __20__, 25, __30__, __35__, 40, . . .

2) 21, 19, 18, 16, 15, __13__, 12, . . .

3) 122, 128, 134, __140__, __146__, 152, . . .

4) 48, 24, 12, __6__, 3, . . .

5) 51, 40, 29, 18, __7__, . . .

6) 2, 4, 8, 16, __32__, 64, __128__, . . .

7) 31, 35, 33, 37, 35, 39, 37, __41__, __39__, 43, . . .

8) 1, 8, 15, 22, __29__, __36__, . . .

9) 65, 50, __55__, 40, 45, __30__, . . .

10) 207, 208, 210, 213, 217, __222__, 228, __235__, 243, __252__, . . .

11) 9, 11, 15, 17, 21, 23, __27__, . . .

12) 1, 2, 5, 10, __17__, 26, 37, __50__ 65, . . .

13) 850, 800, 810, 760, 770, 720, __730__, __680__, . . .

14) 100, 90, 79, 67, __54__, 40, __25__, 9, . . .

15) 1, 2, 3, 6, 7, __14__, 15, 30, 31, 62, __63__, . . .

16) 11, 2, 16, 7, 21, 3, 26, __8__, 31, 4, __36__, 9, . . .

CCSD Version Date: May 2011

CCSD Version Date: May 2011

CCSD Version Date: May 2011

CCSD Version Date: May 2011

CCSD Version Date: May 2011