Here are the steps to solve this problem on your whiteboard:
1. Draw axes and label them
2. Write the equation: y = 2x + 1
3. Plot the y-intercept by substituting x = 0. This gives y = 1
4. Plot another point by substituting a value for x, like x = 1. This gives y = 3
5. Draw the line through these two points
6. Write the equation again under the graph
7. Identify the slope as 2 (from the coefficient of x)
8. Identify the y-intercept as 1
Let me know if any part needs more explanation!
Here are the key points about the formula representation of sunflower growth:
The height of sunflower a on any day t is given by the formula h = 3 + 2t.
This formula says that the starting height (y-intercept) is 3 cm, and it grows at a constant rate (slope) of 2 cm per day.
The variables are t (time in days) and h (height in cm). The constants are the y-intercept of 3 and the growth rate (slope) of 2.
So the y-intercept of 3 appears as the first term in the formula, and the slope of 2 appears as the coefficient of t, the time variable.
The document describes a teaching presentation on displaying fractions in diagrams. It provides instructions on inserting or changing numerators and denominators to show equivalent, simplified, improper, and mixed numbers. Users must follow limitations such as entering only whole numbers between 1-10 as denominators to avoid errors in the diagrams. The goal is to demonstrate different fraction types visually through an interactive fraction diagram tool.
1) The document is a mathematics test for key stage 2 levels 3-5. It does not allow calculators and includes questions testing time, numbers, shapes, money, charts, and other math concepts.
2) The test has 45 minutes, with questions worth 1 or 2 marks each. Students are instructed to show working for some questions to possibly earn partial credit.
3) The questions cover a range of math skills including ordering, adding and subtracting numbers, drawing reflections, interpreting data in tables and charts, identifying properties of shapes, and performing division.
The document is a 2011 calendar and newsletter from the Corporation for Supportive Housing, California (CSH). It provides information on upcoming CSH events and developments in supportive housing. It lists details of 4 supportive housing developments that received funding or assistance from CSH, including the location, developer, number of units, population served, and CSH investment. It also provides facts about supportive housing and homelessness.
The document appears to be a survey with results presented in tables. It surveyed 50 people on their satisfaction with ATM machines provided by UBD and the importance of ATM machines. 33 people, or 66%, expressed satisfaction while 17 people, or 34%, expressed dissatisfaction. The top reason for dissatisfaction was limited range of bank accounts supported by the ATMs. A separate question found that 31 people, or 62%, felt ATM machines were very important with 18 people, or 36%, feeling they were of average importance.
The document discusses using a quadratic function f(x) = -(x + discipline)2 + tree planting to model reforesting mountains, where x represents tree planting, the vertex is (discipline, tree planting), and increasing discipline results in fewer trees planted. It also assigns using this function to plan a tree planting activity that applies quadratic functions to environmental issues.
The document contains data from two surveys. The first survey found that 66% of respondents were satisfied with UBD ATM machines while 34% were not. The main reasons for dissatisfaction were limited account range (50%), distance (28%), and limited ATM machines (22%).
The second survey found that most respondents felt ATM machines were very important to UBD - 62% chose this option while 36% chose average and only 2% said not important.
Here are the key points about the formula representation of sunflower growth:
The height of sunflower a on any day t is given by the formula h = 3 + 2t.
This formula says that the starting height (y-intercept) is 3 cm, and it grows at a constant rate (slope) of 2 cm per day.
The variables are t (time in days) and h (height in cm). The constants are the y-intercept of 3 and the growth rate (slope) of 2.
So the y-intercept of 3 appears as the first term in the formula, and the slope of 2 appears as the coefficient of t, the time variable.
The document describes a teaching presentation on displaying fractions in diagrams. It provides instructions on inserting or changing numerators and denominators to show equivalent, simplified, improper, and mixed numbers. Users must follow limitations such as entering only whole numbers between 1-10 as denominators to avoid errors in the diagrams. The goal is to demonstrate different fraction types visually through an interactive fraction diagram tool.
1) The document is a mathematics test for key stage 2 levels 3-5. It does not allow calculators and includes questions testing time, numbers, shapes, money, charts, and other math concepts.
2) The test has 45 minutes, with questions worth 1 or 2 marks each. Students are instructed to show working for some questions to possibly earn partial credit.
3) The questions cover a range of math skills including ordering, adding and subtracting numbers, drawing reflections, interpreting data in tables and charts, identifying properties of shapes, and performing division.
The document is a 2011 calendar and newsletter from the Corporation for Supportive Housing, California (CSH). It provides information on upcoming CSH events and developments in supportive housing. It lists details of 4 supportive housing developments that received funding or assistance from CSH, including the location, developer, number of units, population served, and CSH investment. It also provides facts about supportive housing and homelessness.
The document appears to be a survey with results presented in tables. It surveyed 50 people on their satisfaction with ATM machines provided by UBD and the importance of ATM machines. 33 people, or 66%, expressed satisfaction while 17 people, or 34%, expressed dissatisfaction. The top reason for dissatisfaction was limited range of bank accounts supported by the ATMs. A separate question found that 31 people, or 62%, felt ATM machines were very important with 18 people, or 36%, feeling they were of average importance.
The document discusses using a quadratic function f(x) = -(x + discipline)2 + tree planting to model reforesting mountains, where x represents tree planting, the vertex is (discipline, tree planting), and increasing discipline results in fewer trees planted. It also assigns using this function to plan a tree planting activity that applies quadratic functions to environmental issues.
The document contains data from two surveys. The first survey found that 66% of respondents were satisfied with UBD ATM machines while 34% were not. The main reasons for dissatisfaction were limited account range (50%), distance (28%), and limited ATM machines (22%).
The second survey found that most respondents felt ATM machines were very important to UBD - 62% chose this option while 36% chose average and only 2% said not important.
This document discusses various methods of counting and deriving number fields. It begins by describing dot-row counting, pyramid counting, and triangular numbers. It then discusses different methods for deriving the number field, including the copy-down method, coat-hanger method, and algebraic method. The document goes on to describe features of the number field such as conservation of information, negative edges, and applications. It concludes that correct counting involves distinguishing novel from repeated information, and basic exploration of these concepts yields insights into the nature of number fields.
The document contains survey results from 50 people about their satisfaction with ATM machines provided by UBD. 66% of respondents were satisfied, while 34% were not satisfied. The top three reasons for dissatisfaction were limited range of bank accounts supported (50%), distance to ATM locations (28%), and limited number of ATM machines (22%). When asked about the importance of ATM machines, 62% felt they were very important, 36% felt they were average importance, and only 2% felt they were not important.
This document provides instructions for students to create various graphs in Excel using different datasets. Students are asked to make scatter plots, line graphs, pie charts and bar graphs representing fish breathing rates, average temperatures, rocket speeds, lake fish populations, cookie sales and more. They must label axes, title graphs, answer analysis questions and submit their Excel file electronically by the deadline.
This document contains information about three schools with 40 teachers and 600 students. It lists the names of three people - Geri Lorway, Joan Coy, and an AISI Math Coordinator for Division 4. It also mentions Quickdraw and Thinking 101.
The document is a 40-minute, 20-question mathematics test for 5th grade students containing questions on operations with whole numbers, fractions, decimals, word problems, and number lines. The test instructs students to show their work, includes diagrams, and provides the number of marks allocated for each question.
This calendar document provides a summary of each month in 2007 including notable dates, holidays and fun facts. It also includes quick references for designers on symbols, fractions, image formats and sizes. The calendar is from Jupiterimages and features royalty-free stock photos with the 12 Pantone color trends for that year.
Presentatie gehouden door Rob Westerdijk Hoge School Arnhem Nijmegen op Inspiratieontbijt 23-1-13 thema Onderwijs & Ondernemen; hoe kunnen we beter samenwerken? Velden: marketing, logistiek, bedrijfseconomie.
The document discusses the results of a study on the effects of a new drug on memory and cognitive function in older adults. The double-blind study involved 100 participants aged 65-80 who were given either the drug or a placebo daily for 6 months. Researchers found that those who received the drug performed significantly better on memory and problem-solving tests at the end of the study compared to those who received the placebo.
Pemerintah mengumumkan paket stimulus ekonomi baru untuk menyelamatkan bisnis dan pekerjaan. Stimulus ini meliputi insentif pajak, bantuan tunai langsung, dan subsidi upah untuk mendorong pertumbuhan. Langkah ini diharapkan dapat mempercepat pemulihan ekonomi dari resesi akibat pandemi.
This document outlines an algebra topic in a syllabus, describing that students will learn about an unspecified algebra topic and the intended learning outcomes which are that students should be able to do something related to the topic but is not stated.
The document discusses introducing algebra through a functions-based approach by focusing on patterns, relationships, and representations rather than procedures. It suggests starting with concrete examples and moving to abstract symbols and equations. A functions approach emphasizes identifying patterns in tables, graphs, and words before learning procedures. This allows students to develop a deeper understanding of algebra as generalized arithmetic and the relationships between different representations.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help boost feelings of calmness, happiness and focus.
Presentatie van de workshop die RCT Rivierenland organiseerde op 17-1-13 om ondernemers te inspireren sneller en beter gebruik te kunnen maken van internet.
We delen dit graag!
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help boost feelings of calmness, happiness and focus.
This document provides an overview of the structure and resources available for the Junior Certificate and Leaving Certificate mathematics syllabi in Ireland. It outlines the sections and topics covered in the syllabi for Probability, Statistics, Geometry, and Trigonometry. It also lists the teaching and learning plans, student materials on CDs, and other online resources available to support teaching and learning for each topic area. These resources are hosted on the Project Maths website and the NCCA website. Sample exam questions and their associated resources are also outlined.
The document discusses using graphs to represent phenomena quantitatively without using formulas. It provides examples of graphs showing motion, depth of water over time, and position over time to describe scenarios. Students are expected to be able to interpret graphs, draw conclusions from them, and describe quantity and change of quantity on a graph. Prior knowledge on graphing coordinates, slope, and representing variables on axes is expected. Matching exercises pair graphs of scenarios with distance-time and depth-over-time graphs.
This document discusses various methods of counting and deriving number fields. It begins by describing dot-row counting, pyramid counting, and triangular numbers. It then discusses different methods for deriving the number field, including the copy-down method, coat-hanger method, and algebraic method. The document goes on to describe features of the number field such as conservation of information, negative edges, and applications. It concludes that correct counting involves distinguishing novel from repeated information, and basic exploration of these concepts yields insights into the nature of number fields.
The document contains survey results from 50 people about their satisfaction with ATM machines provided by UBD. 66% of respondents were satisfied, while 34% were not satisfied. The top three reasons for dissatisfaction were limited range of bank accounts supported (50%), distance to ATM locations (28%), and limited number of ATM machines (22%). When asked about the importance of ATM machines, 62% felt they were very important, 36% felt they were average importance, and only 2% felt they were not important.
This document provides instructions for students to create various graphs in Excel using different datasets. Students are asked to make scatter plots, line graphs, pie charts and bar graphs representing fish breathing rates, average temperatures, rocket speeds, lake fish populations, cookie sales and more. They must label axes, title graphs, answer analysis questions and submit their Excel file electronically by the deadline.
This document contains information about three schools with 40 teachers and 600 students. It lists the names of three people - Geri Lorway, Joan Coy, and an AISI Math Coordinator for Division 4. It also mentions Quickdraw and Thinking 101.
The document is a 40-minute, 20-question mathematics test for 5th grade students containing questions on operations with whole numbers, fractions, decimals, word problems, and number lines. The test instructs students to show their work, includes diagrams, and provides the number of marks allocated for each question.
This calendar document provides a summary of each month in 2007 including notable dates, holidays and fun facts. It also includes quick references for designers on symbols, fractions, image formats and sizes. The calendar is from Jupiterimages and features royalty-free stock photos with the 12 Pantone color trends for that year.
Presentatie gehouden door Rob Westerdijk Hoge School Arnhem Nijmegen op Inspiratieontbijt 23-1-13 thema Onderwijs & Ondernemen; hoe kunnen we beter samenwerken? Velden: marketing, logistiek, bedrijfseconomie.
The document discusses the results of a study on the effects of a new drug on memory and cognitive function in older adults. The double-blind study involved 100 participants aged 65-80 who were given either the drug or a placebo daily for 6 months. Researchers found that those who received the drug performed significantly better on memory and problem-solving tests at the end of the study compared to those who received the placebo.
Pemerintah mengumumkan paket stimulus ekonomi baru untuk menyelamatkan bisnis dan pekerjaan. Stimulus ini meliputi insentif pajak, bantuan tunai langsung, dan subsidi upah untuk mendorong pertumbuhan. Langkah ini diharapkan dapat mempercepat pemulihan ekonomi dari resesi akibat pandemi.
This document outlines an algebra topic in a syllabus, describing that students will learn about an unspecified algebra topic and the intended learning outcomes which are that students should be able to do something related to the topic but is not stated.
The document discusses introducing algebra through a functions-based approach by focusing on patterns, relationships, and representations rather than procedures. It suggests starting with concrete examples and moving to abstract symbols and equations. A functions approach emphasizes identifying patterns in tables, graphs, and words before learning procedures. This allows students to develop a deeper understanding of algebra as generalized arithmetic and the relationships between different representations.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help boost feelings of calmness, happiness and focus.
Presentatie van de workshop die RCT Rivierenland organiseerde op 17-1-13 om ondernemers te inspireren sneller en beter gebruik te kunnen maken van internet.
We delen dit graag!
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help boost feelings of calmness, happiness and focus.
This document provides an overview of the structure and resources available for the Junior Certificate and Leaving Certificate mathematics syllabi in Ireland. It outlines the sections and topics covered in the syllabi for Probability, Statistics, Geometry, and Trigonometry. It also lists the teaching and learning plans, student materials on CDs, and other online resources available to support teaching and learning for each topic area. These resources are hosted on the Project Maths website and the NCCA website. Sample exam questions and their associated resources are also outlined.
The document discusses using graphs to represent phenomena quantitatively without using formulas. It provides examples of graphs showing motion, depth of water over time, and position over time to describe scenarios. Students are expected to be able to interpret graphs, draw conclusions from them, and describe quantity and change of quantity on a graph. Prior knowledge on graphing coordinates, slope, and representing variables on axes is expected. Matching exercises pair graphs of scenarios with distance-time and depth-over-time graphs.
This document discusses introducing algebra through a functions-based approach focusing on patterns, relationships, and representations rather than procedures. It suggests starting with real-world contexts and moving from concrete to abstract representations like words, tables, graphs, and equations. Students would explore inputs and outputs, rates of change, and the connections between different representations of linear functions before learning algebraic manipulation procedures. The goal is a deeper conceptual understanding of algebra as generalized arithmetic and relationships between quantities.
The document discusses probability and chance. It defines probability as a number between 0 and 1 that indicates how likely something is to occur. It distinguishes between theoretical and experimental probability. Theoretical probability can be calculated without experiments, while experimental probability is determined by performing repeated trials of an experiment and observing outcomes. Examples are provided to illustrate calculating probabilities of events using fractions, decimals, sample spaces, and tally charts.
The document provides examples for adding and subtracting fractions with the same denominator. It shows step-by-step work for solving multiple fraction addition and subtraction problems, with the goal of demonstrating how to combine like fractions by adding or subtracting the numerators and keeping the same denominator. Visual representations are included to illustrate fraction concepts such as what fraction is shaded in a diagram.
This document discusses grouping discrete data and calculating statistics like the mode, mean, and frequency. It provides an example of grouping the number of goals scored in soccer matches to find the mode of 3 goals. Another example calculates the mean shoe size from a survey by tallying shoe sizes into a frequency table and dividing the total by the number of participants.
Okay, let's break this down step-by-step:
* The population is dropping at a rate of 255 people per year
* We want to know how long it will take for the change in population to be 2,040 people
* So we set up an equation: Rate x Time = Change
* Rate is -255 people/year
* Change is -2,040 people
* So the equation is: -255x = -2,040
* Solve for x: x = 2,040/-255 = 8 years
Therefore, it will take 8 years for the change in population to be 2,040 people.
The document provides instructions for students to complete an opener drawing exercise. It instructs students to draw two identical acute triangles, each filling half of the provided box. Students are then asked to draw the perpendicular bisector of each side and the angle bisector of each angle of the triangles. The final instruction asks students to note anything interesting they observe.
The document provides instructions for students to complete an opener drawing exercise. It instructs students to draw two identical acute triangles, each filling half of the provided box. Students are then asked to draw the perpendicular bisector of each side and the angle bisector of each angle of the triangles. The final instruction asks students to note anything interesting they observe.
Sudoku is a logic-based number placement puzzle where the objective is to fill a 9x9 grid so that each column, row, and 3x3 box contains the digits 1-9. The rules are simple and it can be played by people of all ages. To solve a Sudoku puzzle, players use logic and deduction to determine which numbers fit in each empty square based on the numbers already provided in the puzzle.
Sudoku is a logic-based number placement puzzle where the objective is to fill a 9x9 grid so that each column, row, and 3x3 box contains the digits 1-9. The rules are simple and it can be played by people of all ages. To solve a Sudoku puzzle, players use logic and deduction to determine which numbers fit in each empty square based on the numbers already provided in the puzzle.
2. Coloured Blocks Diagram
Is there a pattern to these colours?
Can you use the pattern to predict what
colour will be at a particular point?
How can we investigate if there is a pattern?
3. Table
Block
Colour Black
Position
1 Block Position
2 1 2
3 2 4
4 3 6
5 4 8
6 5 10
Black block is always even.
4. Words: You double the number of the black block
Black Red
Block Position Block Position
1 2 1 1
2 4 2 3
3 6 3 5
4 8 4 7
5 10 5 9
n 2n n 2n – 1
5. Key Outcomes and Words:
identify patterns and describe
different situations using tables,
graphs, words and formulae
predict
generalise in words and symbols
justify
relationship
6. Coloured Blocks Diagram
Is there a pattern to these colours?
Can you use the pattern to predict the next colour?
Ways to investigate the pattern: table
story
general expression
justify
7. Tables
Yellow Black Green
Block Position Block Position Block Position
1 1 1 2 1 3
2 4 2 5 2 6
3 7 3 8 3 9
4 10 4 11 4 12
5 13 5 14 5 15
n 3n – 2 n 3n – 1 n 3n
8.
9. Money Box Problem
Mary Money Box
Start €0
Growth per day €2
We want to investigate the total amount
of money in the money box over time.
Is the growth of the money a pattern?
Can we predict how much money will be in
the box on day 10?
10. Money Box Problem
John Money Box
Start €3
Growth per day €2
Is there a pattern to the growth of this money?
Can we use this pattern to predict how much money
will be in the box at some future time?
How can we investigate if a pattern exists?
11. Table for John’s Money Box
Time/days Money in Box/€
0 3
1 5
2 7
3 9
4 11
5 13
Is there a pattern?
Where is the start value in € and growth per day
in this as seen in the table?
What do you notice about successive outputs ?
12. Money Box Problem
Bernie Money Box
Start €4
€2 on week days
Growth per day
€5 on Weekend days
Is there a constant rate of change here?
14. Identifying variables and constants
Money Box Varying Constant
John
Mary
Bernie
What is varying each Day?
What is constant?
Can you put this into words?
15. Mary John Bernie
Money in Money in Money in
Time/days Time/days Time/days
Box € Box € Box €
0 0 0 3 0* 4
1 2 1 5 1 6
2 4 2 7 2 8
3 6 3 9 3 10
4 8 4 11 4 15
5 10 5 13 5 20
Time on Horizontal Axis
Total Money on Vertical Axis
* Note: in this example day 0 is a Tuesday.
16. Draw a Graph
22
21
20 Mary
19
John
18
17 Bernie
16
Amount of Money Spent
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8 9 10
Days
18. Now, I want you to observe the pattern.
Explain in words & numbers, how to find
the total amount of money in John’s box
after 15 days.
Time/days Money/€ Change
0 3
+2
1 5
+2
2 7
+2 Do this on your
3 9
white board
+2
4 11
+2
5 13
+2
6 15
19. Now, I want you to generalise.
Explain in words & symbols, how to find
the total amount of money in John’s box
after any given day.
Time/days Money/€ Change
0 3
+2
1 5
+2
2 7
+2 Do this on your
3 9
white board
+2
4 11
+2
5 13
+2
6 15
20. Table for Mary’s Money Box
Time/days
Money in
Box € What is the general
0 0 formula for Mary?
1 2 A 0 2D
2 4
3 6
4 8
5 10
21. Jo h n : A 3 2D
M a ry : A 2D 0
O n ly se e in g th e fo rm u la :
C a n yo u re a d th e Jo h n 's sta rt a m o u n t?
C a n yo u re a d M a ry's ra te o f c h a n ge ?
W h a t a re w e b u ild in g u p to ?
y c mx or y mx c
22. Sunflower growth
Sunflower a b c d
Start height/cm 3 6 6 8
Growth per day/cm 2 2 3 2
Is there a pattern to the growth of these sunflowers?
Can we use this pattern to predict height at some future time?
How can we investigate if a pattern exists?
23. Table for Each Sunflower
Time/days Height/cm Change
0
1
2
3
4
5
25. 22
A and B
21
20
19
18
17
16 Sunflower A
15
14
Sunflower B
Height/cm
13
12
11
10
9
8
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8 9 10
Time/Days
26. 22
B and C
21
20
19
18
17
16 Sunflower B
15
14
Sunflower C
Height/cm
13
12
11
10
9
8
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8 9 10
Time/Days
27. 22
C and D
21
20
19
18
17
16 Sunflower C
15
14
Sunflower D
Height/cm
13
12
11
10
9
8
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8 9 10
Time/Days
28. Formula Representation
• Describe in words the height of the sunflower a,
on any day.
• Describe in symbols the height of the sunflower
a, on any day.
• Identify the variables and constants in this
formula.
• Where do the y – intercept and the slope of the
graph appear in the formula?
29. Pattern of Growth for 4 Different Sunflowers
A B C D
T H T H T H T H
Pattern Pattern Pattern Pattern
days cm days cm days cm days cm
0 3 3 0 6 6 0 6 6 0 8 8
1 5 3+2 1 8 6+2 1 9 6+3 1 10 8+2
2 7 3+2+2 2 10 6+2+2 2 12 6+3+3 2 12 8+2+2
3 9 3+2+2+2 3 12 6+2+2+2 3 15 6+3+3+3 3 14 8+2+2+2
4 11 3+2+2+2+2 4 14 6+2+2+2+2 4 18 6+3+3+3+3 4 16 8+2+2+2+2
5 13 3+2+2+2+2+… 5 16 6+2+2+2+2+… 5 21 6+3+3+3+3+… 5 18 8+2+2+2+2+…
6 15 3+2+2+2+2+… 6 18 6+2+2+2+2+… 6 24 6+3+3+3+3+… 6 20 8+2+2+2+2+…
Describe in words the height of the sunflower a, on any day.
Describe in symbols the height of the sunflower a, on any day.
Identify the variables and constants in this formula.
Where do the y – intercept and the slope of the graph appear in the formula?
30. Pattern of Growth for 4 Different Sunflowers
A B C D
T H T H T H T H
Pattern Pattern Pattern Pattern
days cm days cm days cm days cm
0 3 3 0 6 6 0 6 6 0 8 8
1 5 3+2 1 8 6+2 1 9 6+3 1 10 8+2
2 7 3+2+2 2 10 6+2+2 2 12 6+3+3 2 12 8+2+2
3 9 3+2+2+2 3 12 6+2+2+2 3 15 6+3+3+3 3 14 8+2+2+2
4 11 3+2+2+2+2 4 14 6+2+2+2+2 4 18 6+3+3+3+3 4 16 8+2+2+2+2
5 13 3+2+2+2+2+… 5 16 6+2+2+2+2+… 5 21 6+3+3+3+3+… 5 18 8+2+2+2+2+…
6 15 3+2+2+2+2+… 6 18 6+2+2+2+2+… 6 24 6+3+3+3+3+… 6 20 8+2+2+2+2+…
h 3 2t h 6 2t h 6 3t h 8 2t
y int ercep t 3 y in t e rce p t 6 y in t e rce p t 6 y int ercep t 8
slop e 2 slop e 2 slop e 3 slop e 2
31. Multi – Representation
Table t/d h/cm
0 3
1 5
2 7
3 9
Graph 4 11
5 13
6 15
Words Height = 3 + 2 times
the number of days
h = 3 +2d Symbols
32. Over to you on your White Boards
1. Draw a rough sketch of the graph: y 2x 1
33. Over to you on your White Boards
2. I start off w ith 6 euro in m y m oney box and put in 3 euro each day.
Draw a rough sketch of the graph.
34. Over to you on your White Boards
3. T he initia l sp e e d of a ca r is 1 0 m / s a nd the ra te a t w hich it incre a se s its
2
sp e e d e ve ry se cond is 2 m / s .
W rite d ow n a line a r la w for the sp e e d of the ca r a fte r t se cond s.
D ra w a gra p h of the la w .
30
28 S o lu tio n :
26 Speed 10 2(n u m b e r o f se c o n d s tra ve llin g)
24 v 10 2t
22 or v 2t 10
20
18
16
14
12
10
8
6
4
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
35. slope 5 slope 4 slope 2
slope 3
slope 1
1
slop e
2
1
slop e
3
1
slop e
4
1
slop e
5
36. All the graphs you a have drawn on your white
boards have been increasing functions. Do this on your
white board
Assess the learning:
Story :
Isabelle has a money box with 20 euro in it. She takes 2
euro out each day to buy sweets in the shop.
Draw a rough graph of how this might look.
Investigate if your graph is close by doing a table.
From your table, what is your rate of change?
Conclusion: Decreasing graph has a negative slope
39. Growing
Squares
Write down a relationship
which defines how many red
squares are required for each
white square.
Hint: You may need more than one representation to help you!
40. C h a ra cte ristics :
first ch a n ge con sta n t
n te rm
lin e a r gra p h
Are the characteristics of…..
A Linear Relationship
41. Growing Rectangles
Complete the next two rectangles in the above pattern.
There is squared paper in your handbooks.
Is there a relationship between the:
height of the rectangle
length of the rectangle
number of tiles/area of the rectangle?
42. Table
Number of
tiles n in Change of
Height, h Length, l Area = h x l Change
each change
rectangle
1 2 2 2
+4
2 3 6 6 +2
+6
3 4 12 12 +2
+8
4 5 20 20 +2
+10
5 6 30 30 +2
+12
6 7 42 42 +2
+14
7 8 56 56
Investigate the first change and the second change.
What do we notice?
If we let n be the height, write a formula for the area in terms of n,
on your white board.
43. Draw a graph of the table
Number of
tiles n in Change of
Height, h Length, l Area = h x l
each Change change
rectangle
1 2 2 2
2 3 6 6
3 4 12 12
4 5 20 20
5 6 30 30
6 7 42 42
7 8 56 56
What do you observe about the shape of your graph?
44. C h a ra cte ristics :
first ch a n ge va rie s
se con d ch a n ge con sta n t
2
n te rm
cu rve d gra p h
Are the characteristics of…..
A Quadratic Relationship
45. Story: How to ask for Pocket Money
“I only want you to give me pocket money for the month of July.
All I want is for you to give me 2 c on the first day of the month,
double that for the second day, and double that again for the 3rd
day... and so on.
On the first day I will get 2 c, on the 2nd day 4 c, on the 3rd day
8c and so on until the end of the month. That is all I want.”
Is this a good deal for my parents
or is it a good deal for me?
46. Investigate using a Table
Do this on your
Day Money in cent
white board
1 2
2 2x2
3 2x2x2
4
5
6
7
8
9
10
If we let n be the number of days, can we write a formula
for the Amount of Pocket Money?
47. n
F o rm u la : A m o u n t 2
W o rd s : D o u b lin g
49. What if your Dad trebled the amount of
money each day? Trebling
Doubling
Money/Cents
Time/Days
The money would grow even quicker.
50. Lets look at the Changes in a Table
Money in Change of
Days Change
cent change
1 2 +2
2 4 +2
+4
3 8 +4
+8
4 16 +8
+16
5 32 +16
+32
6 64 +32
+64
7 128 +64
+128
8 256 +128
+256
9 512 +256
+512
10 1024
What do you notice about the Change columns……
They develop in a ratio.
51. C h a ra cte ristics :
ch a n ge d e ve lop s in a ra tio
n n
F orm u la : 2 or 3
W ord s: D ou b lin g or T re b lin g
cu rve d gra p h th a t grow s ve ry q u ickly
Are the characteristics of…..
An Exponential Relationship
52. t
F P( 1 i)
Ignoring the principal, the interest rate, and the number of years by setting all these variables equal to "1", and
looking only at the influence of the number of compoundings, we get:
1
ye a rly 1 1 2
2
1
se m i a n n u a lly 1 2 . 25
2
4
1
q u a rte rly 1 2 . 44140625
4
12
1
m o n th ly 1 2 . 61303529022 ...
12
52
1
w e e kly 1 2 . 692596954444 ...
52
365
1
d a ily 1 2 . 71456748202 ...
365
8760
1
h o u rly 1 2 . 71892792154 ...
8760
525600
1
e ve ry m in u te 1 2 . 7182792154 ...
525600
31536000
1
e ve ry se co n d 1 2 . 71828247254 ...
31536000
Can you see a pattern in the table?What is the key to this pattern?Students link between words and table.What is the key to this pattern? Can they see that the unit is 3, it repeats every 3.Green is 3 6 9 12 The first green is 3The second green is 6The third green is 9Can they begin to see 3 x 1st green block is the 3rd position Can they see that 3 x 2nd green block is the 6th positionCan they see that 3x 3rd black block is the 9th position3 is the key Three times every green block gives the position Can they arrive at.....3nIf you know where the green blocks are you can get the one before it to be black every time.Can students move from these words to:The one before green is blackCan they arrive at 3n – 1.And then to 3n - 2
Identify patterns table words graph formulaeDependent (rate of change) independent variableStart amount in the graph / table/ formulae y interceptRate of change of dependent variable in the table graph (slope) formula Linear graphs constant first differences between successive y values (outputs)Parallel lines have the same slope that is the rate of change of y with respect to xIncreasing functions with positive slope
Adds 2 every day is the patternStart is €0The growth of money is seen is seen in the right hand columnEach successive output increases by 2Dependent (rate of change) independent variableStart amount in the graph / table/ formulae y interceptRate of change of dependent variable in the table graph (slope) formula Linear graphs constant first differences between successive y values (outputs)Parallel lines have the same slope that is the rate of change of y with respect to xIncreasing functions with positive slope
What is the difference between Mary & John?
Adds 2 every day is the patternStart is €4The growth is seen in the right hand columnEach successive output increases by 2
What is the difference between Mary & John?
Adds 2 every day is the patternStart is €4The growth is seen in the right hand columnEach successive output increases by 2
Constant rate of changeWhat varies is the variable. Here we have 2 variables days and amount of moneyWhich variable depends on which? We call one variable the dependent variable and we call the other variable the independent variable.Which is which above? The values which stay the same are constants....constant rate of changeIs there a pattern?Where is the start amount of money in the table?Where is the “rate of change of money per day” in the table
Draw a graph for Mary and JohnWhich variable should go where? Days Amount of Money Independent on X-axis, Dependent on the y-axisX values as inputs and y values as outputsWhat observations can you make?How is each observation seen in the graphHow is each observation seen in the tableAre the amounts ever the same. Explain
Where do you see starting values and growth rates in the tables ? (day 0 values and first differences)W hat do you notice about all the first differences? 2Can students put this into words what they see in the table that is John starts with €4 and each day he gets another 2How would you get what John has on the ??? DayOnly words at ordinaryTotal = Starts + 2 (Number of Days)T = s + 2 dT = 4 + 2d
Adds 2 every day is the patternStart is €0The growth of money is seen is seen in the right hand columnEach successive output increases by 2Dependent (rate of change) independent variableStart amount in the graph / table/ formulae y interceptRate of change of dependent variable in the table graph (slope) formula Linear graphs constant first differences between successive y values (outputs)Parallel lines have the same slope that is the rate of change of y with respect to xIncreasing functions with positive slope
Where is the ‘start height’ for each plant seen in the tables? Where is the amount the plant grows by each day seen in the tables? What do you notice about the first differences between successive outputs for all the tables?For each of the situations and tables (a), (b), (c), and (d) identify 2 values which are staying the same and 2 values which are varying.Situation and TableVaryingStaying the sameWe call the values which vary ‘variables’ and the values which stay the same ‘constants’.We have identified 2 variables. Which variable depends on which? We call one variable the dependent variable and we call the other variable the independent variable.
Where do you see starting values and growth rates in the tables ? (day 0 values and first differences)What do you notice about all the first differences?
Where do you see starting values and growth rates in the tables ? (day 0 values and first differences)What do you notice about all the first differences?
Where do you see starting values and growth rates in the tables ? (day 0 values and first differences)What do you notice about all the first differences?
Graphs are straight lines with constant first differencesThey have a constant rate of change. You can see this in the table – constant first differences between successive outputs for consecutive inputsWhen the left hand column goes up by a constant amount the right hand column goes up by a constant ( not necessarily the same as for the left hand column) amount.The slope is the difference between 2 outputs divided by difference between the corresponding inputs.The bigger the slope – the steeper the graphThe formula has an amount you multiply the independent variable by and an amount you add - the slope and y – intercept respectively.If the graph goes through (0,0) the amount you add is 0
Various Slopes
Line with constant slopeMonica decided to plant a plastic sunflower in the garden whose height was 30 cm. Draw up a table and plot a graph of the height of the sunflower from day 0 to day 5. What shape is the graph? Why is the graph this shape?As the number of days increased what happened to the height of the sunflower?What is the slope of the graph?What formula would describe the height of the sunflower? Line with a negative slope: You have €40 in your money box on Sunday. You spend €5 on your lunch each day for 5 consecutive days (take Monday as day 1). Draw up a table, plot the graph, calculate the slope and write the formula for the situation. If this pattern continues will you ever have €0 in your money box? If so, when? Is sunflower growth a realistic situation for negative slope? As the variable on the x-axis increases what happens to the variable on the y-axis?Can you think of other real life situations which would give rise to linear graphs with negative slopes? (e.g. volume of heating oil in a tank against time in days, assuming constant rate of usage per day, paying off a loan, use of credit on a mobile phone)