Study Guide

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Kurssin Learn-sivulla tarjottu materiaali. Materiaali pohjautuu osaltaan kirjaan Insinöörin matematiikka, Tuomenlehto ym., 1.-5. painos, 2018. Muutkin painokset käyvät.

Yksilölliset oppimisväylät

Kurssi sisältää lähiopetusta, ja harjoitustehtäviä.

TKI ja työelämäyhteistyö

Ei sisällä

Tentit ja muut määräajat

Kurssin tenttiä saa uusia 2 kertaa.

Opiskelijan työmäärä

Kurssin työmäärä vastaa 3 opintopistettä eli 81 tuntia opiskelua.

Toteutuksen osien kuvaus

Kurssi kertaa tai vahvistaa matemaattisia valmiuksia jatkokursseja varten. Sisältöön kuuluu mm. murtolausekkeet, potenssi- ja juurilausekkeet, 1. ja 2. asteen yhtälöiden ratkaiseminen, funktion käsite ja suorakulmaisen kolmion ratkaiseminen.

Evaluation scale

Approved/Failed

Assessment methods and criteria

Kurssin arviointi perustuu loppukokeen tulokseen.

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Oppikirja: Tuomenlehto ym, Insinöörin matematiikka. EDITA
Kaavakirja: Tekniikan kaavasto. Tammertekniikka tai MAOL-kaavakirja.
Lisäksi kurssin Learn-sivulla ja tunneilla jaettu materiaali.

Yksilölliset oppimisväylät

Työviikkopohjainen oppimisväylä:
Työjärjestyksen mukainen lähiopetus, harjoitustehtävät sekä mahdolliset palautettavat harjoitystyöt.

TKI ja työelämäyhteistyö

Ei ole.

Tentit ja muut määräajat

Opintojakson mahdollisilla välikokeilla tai tentillä. Koko kurssin kattavaa tenttiä saa uusia 2 kertaa.

Opiskelijan työmäärä

Kurssi on kolmen opintopisteen laajuinen, joka vastaa 81 tuntia työskentelyä.

Evaluation scale

Approved/Failed

Assessment methods and criteria

Arvionti perustuu menestykseen välikokeissa tai tentissä.
Mahdollinen harjoitustyö on suoritettava hyväksytysti.

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Evaluation scale

1-5

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Oppikirja: Tuomenlehto ym, Insinöörin matematiikka. EDITA
Kaavakirja: Tekniikan kaavasto. Tammertekniikka tai MAOL-kaavakirja.
Lisäksi kurssin Learn-sivulla ja tunneilla jaettu materiaali.

Yksilölliset oppimisväylät

Työviikkopohjainen oppimisväylä:
Työjärjestyksen mukainen lähiopetus, harjoitustehtävät sekä mahdolliset palautettavat harjoitystyöt.

TKI ja työelämäyhteistyö

Ei ole.

Tentit ja muut määräajat

Opintojakson mahdollisilla välikokeilla tai tentillä. Koko kurssin kattavaa tenttiä saa uusia 2 kertaa.

Opiskelijan työmäärä

Kurssi on kolmen opintopisteen laajuinen, joka vastaa 81tuntia työskentelyä.

Evaluation scale

1-5

Assessment methods and criteria

Arvionti perustuu menestykseen välikokeissa tai tentissä.
Mahdollinen harjoitustyö on suoritettava hyväksytysti.

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Kurssin Learn-sivulla tarjottu materiaali. Materiaali pohjautuu osaltaan kirjaan Insinöörin matematiikka, Tuomenlehto ym., 1.-5. painos, 2018. Muutkin painokset käyvät.

Yksilölliset oppimisväylät

Kurssi sisältää lähiopetusta, ja harjoitustehtäviä.

TKI ja työelämäyhteistyö

Ei sisällä

Tentit ja muut määräajat

Kurssin tenttiä saa uusia 2 kertaa.

Opiskelijan työmäärä

Kurssin työmäärä vastaa 3 opintopistettä eli 81 tuntia opiskelua.

Toteutuksen osien kuvaus

Kurssi kertaa tai vahvistaa matemaattisia valmiuksia jatkokursseja varten. Sisältöön kuuluu mm. murtolausekkeet, potenssi- ja juurilausekkeet, 1. ja 2. asteen yhtälöiden ratkaiseminen, funktion käsite ja suorakulmaisen kolmion ratkaiseminen.

Evaluation scale

Approved/Failed

Assessment methods and criteria

Kurssin arviointi perustuu loppukokeen tulokseen.

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Electronic material as well as web material published on the course's Learn platform. Exercises and their model answers are also given to students on the course's Learn platform and / or Teams channel.

Yksilölliset oppimisväylät

Participation in contact and / or remote learning according to the work schedule. The student completes the assignments, tasks and project work required in the course before the given deadlines and participates in exams or in the final exam at the end of the course if it will be arranged.

TKI ja työelämäyhteistyö

The course does not have RDI or work-life cooperation, since it is a training course for studying mathematics in the university.

Tentit ja muut määräajat

Exercises and assignments will be returned to the Learn platform by the deadline specified there. Possible exam schedule will be agreed upon together with the students.

Opiskelijan työmäärä

A total of 81h, of which about 54hrs is contact learning and 27hrs is independent work.

Evaluation scale

1-5

Assessment methods and criteria

Participation in contact learning, completing course assignments, exams or final exam.

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Oppikirja: Tuomenlehto ym, Insinöörin matematiikka. EDITA
Kaavakirja: Tekniikan kaavasto. Tammertekniikka tai MAOL-kaavakirja.
Lisäksi kurssin Learn-sivulla ja tunneilla jaettu materiaali.

Yksilölliset oppimisväylät

Työviikkopohjainen oppimisväylä:
Työjärjestyksen mukainen lähiopetus, harjoitustehtävät sekä mahdolliset palautettavat harjoitystyöt.

TKI ja työelämäyhteistyö

Ei ole.

Tentit ja muut määräajat

Opintojakso suritetaan kurssitentillä. Koko kurssin kattavaa tenttiä saa uusia 2 kertaa.

Opiskelijan työmäärä

Suomeksi
Kurssi on kolmen opintopisteen laajuinen, joka vastaa 81 tuntia työskentelyä.

Evaluation scale

Approved/Failed

Assessment methods and criteria

Arvionti perustuu menestykseen tentissä.
Mahdollinen harjoitustyö on suoritettava hyväksytysti.

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Learn-alustalta löytyy koko kurssimateriaali: aihepiireittäin jaoteltuna tekstimateriaali,
AC-opetusvideot ja tehtävät.

Yksilölliset oppimisväylät

Miten käsittelet matemaattisia summalausekkeita?
======================================
• Lausekkeen arvo
• Samanmuotoisten termien yhdistäminen
• Sulkujen poistaminen summalausekkeesta
• Summan kertominen ja jakaminen luvulla
• Tekijän erottaminen
======================================
Miten suoritat laskutoimituksia murtoluvuilla?
======================================
• Murtolukujen yhteenlasku
• Murtolukujen vähennyslasku
• Murtolukujen kertolasku
• Murtolukujen jakolasku
======================================
Miten ratkaiset ensimmäisen asteen yhtälön? Miten tulkitset yhtälön kuvaaman ongelman?
======================================
• Yhtälön peruskäsitteitä
• Kaavat sovelluksissa
• Kuvaaja: suora
======================================
Mikä on yhtälöpari? Millaisista ongelmista syntyy yhtälöpareja? Miten se ratkaistaan?
======================================
• ratkaiseminen sijoitusmenettelyllä
• ratkaiseminen yhteenlaskumenettelyllä
• yhtälöparin graafinen tulkinta
======================================
Miten käsittelet potenssilausekkeita? Mitä tarkoittaa negatiivinen potenssi?
======================================
• Neliö ja kuutio
• Eksponenttina nolla
• Eksponenttina negatiivinen luku
• Tulon ja osamäärän potenssi
• Samankantaisten potenssien tulo ja osamäärä
• Potenssin potenssi
======================================
Miten käsitellään juurilausekkeita?
======================================
• Neliöjuuri
• Kuutiojuuri
• Juuret kaavoissa
======================================
Mikä on polynomilauseke? Miten kerrot ja jaat polynomeja keskenään? Mitä ovat binomin neliö ja summan ja erotuksen tulo?
======================================
• Käsitteitä
• Polynomien yhteen- ja vähennyslasku
• Polynomien kertolasku
• Binomin neliö
• Summan ja erotuksen tulo
• Polynomien jakolasku – myös jakokulmaa käyttäen
• Polynomin jakaminen tekijöihin
======================================
Mikä on toisen asteen yhtälö? Miten ratkaistaan erityyppisiä toisen asteen yhtälöitä?
======================================
• Peruskäsitteitä
• Juurtaminen
• Tekijöihin jakaminen yhtälöä ratkaistaessa
• Ratkaisukaava
• Yhtälön juurten reaalisuuden tutkiminen diskriminantin avulla
• Kuvaaja: paraabeli
======================================
Mikä on funktio? Miten piirretään perusfunktioiden kuvaajat (suora ja paraabeli)?
Miten kuvaajia tulkitaan?
======================================
• Funktion käsite
• Ensimmäisen asteen funktion kuvaaja: suora
• Kulmakerroin
• Yhdensuuntaiset suorat
• Nouseva ja laskeva suora
• Toisen asteen funktion kuvaaja: paraabeli
• Nollakohtien ja aukeamissuunnan määrittäminen
======================================
Miten ratkaistaan suorakulmainen kolmio?
======================================
• Pythagoraan lause ja suorakulmaisen kolmion trigonometria
======================================

TKI ja työelämäyhteistyö

Opintojakson toteutukseen ei sisälly TKI- ja työelämäyhteistyötä.

Opiskelijan työmäärä

Oppitunnit 36 h.
Kotitehtävät 15 h.
Learn-materiaaliin perehtyminen 15 h.
Koe ja siihen valmistautuminen 10 h.

Evaluation scale

1-5

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Kurssin Learn-pohjalla jaettu materiaali ja taulukkokirja (esim. MAOL tai tekniikan kaavasto). Learn-materiaali perustuu oppikirjaan, jonka hankkiminen opiskelijoille vapaavalintaista. Kurssilla tarvitset funktiolaskimen ja tietokoneen.

Oppikirja:
Tuomenlehto, Holmlund, Huuskonen, Makkonen ja Surakka. Insinöörin matematiikka, Edita, 2014. ISBN 9789513763527

Yksilölliset oppimisväylät

Työviikkopohjainen oppimisväylä:
Työjärjestyksen mukaiset oppitunnit. Ei läsnäolopakkoa.

Opintoja nopeuttava oppimisväylä:
Suorittaminen mahdollista tenttimällä.

TKI ja työelämäyhteistyö

Ei yhteistyötä

Tentit ja muut määräajat

Viikottaisten kotitehtävien määräajat ja tenttipäivämäärät ilmoitetaan Learn-alustalla kurssin alkaessa.

Kansainvälinen yhteistyö

Ei yhteistyötä

Opiskelijan työmäärä

Opintopisteitä vastaava opiskelijantyömäärä on 81 tuntia, joka koostuu opetukseen osallistumisesta, viikoittaisista kotitehtävistä, sekä tentistä ja tenttiin kertaamisesta.

Further information

Opintojakso kertaa tai vahvistaa matemaattisia valmiuksia jatkokursseja varten.

Evaluation scale

1-5

Assessment methods and criteria

Kurssin arvosana perustuu tentillä ja kotitehtävillä tienattuihin pisteisiin.

Hylättyä tenttiä voi uusia kahdesti.

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Learn-alustalta löytyy koko kurssimateriaali: aihepiireittäin jaoteltuna tekstimateriaali,
AC-opetusvideot ja tehtävät.

Yksilölliset oppimisväylät

Miten käsittelet matemaattisia summalausekkeita?
======================================
• Lausekkeen arvo
• Samanmuotoisten termien yhdistäminen
• Sulkujen poistaminen summalausekkeesta
• Summan kertominen ja jakaminen luvulla
• Tekijän erottaminen
======================================
Miten suoritat laskutoimituksia murtoluvuilla?
======================================
• Murtolukujen yhteenlasku
• Murtolukujen vähennyslasku
• Murtolukujen kertolasku
• Murtolukujen jakolasku
======================================
Miten ratkaiset ensimmäisen asteen yhtälön? Miten tulkitset yhtälön kuvaaman ongelman?
======================================
• Yhtälön peruskäsitteitä
• Kaavat sovelluksissa
• Kuvaaja: suora
======================================
Mikä on yhtälöpari? Millaisista ongelmista syntyy yhtälöpareja? Miten se ratkaistaan?
======================================
• ratkaiseminen sijoitusmenettelyllä
• ratkaiseminen yhteenlaskumenettelyllä
• yhtälöparin graafinen tulkinta
======================================
Miten käsittelet potenssilausekkeita? Mitä tarkoittaa negatiivinen potenssi?
======================================
• Neliö ja kuutio
• Eksponenttina nolla
• Eksponenttina negatiivinen luku
• Tulon ja osamäärän potenssi
• Samankantaisten potenssien tulo ja osamäärä
• Potenssin potenssi
======================================
Miten käsitellään juurilausekkeita?
======================================
• Neliöjuuri
• Kuutiojuuri
• Juuret kaavoissa
======================================
Mikä on polynomilauseke? Miten kerrot ja jaat polynomeja keskenään? Mitä ovat binomin neliö ja summan ja erotuksen tulo?
======================================
• Käsitteitä
• Polynomien yhteen- ja vähennyslasku
• Polynomien kertolasku
• Binomin neliö
• Summan ja erotuksen tulo
• Polynomien jakolasku – myös jakokulmaa käyttäen
• Polynomin jakaminen tekijöihin
======================================
Mikä on toisen asteen yhtälö? Miten ratkaistaan erityyppisiä toisen asteen yhtälöitä?
======================================
• Peruskäsitteitä
• Juurtaminen
• Tekijöihin jakaminen yhtälöä ratkaistaessa
• Ratkaisukaava
• Yhtälön juurten reaalisuuden tutkiminen diskriminantin avulla
• Kuvaaja: paraabeli
======================================
Mikä on funktio? Miten piirretään perusfunktioiden kuvaajat (suora ja paraabeli)?
Miten kuvaajia tulkitaan?
======================================
• Funktion käsite
• Ensimmäisen asteen funktion kuvaaja: suora
• Kulmakerroin
• Yhdensuuntaiset suorat
• Nouseva ja laskeva suora
• Toisen asteen funktion kuvaaja: paraabeli
• Nollakohtien ja aukeamissuunnan määrittäminen
======================================
Miten ratkaistaan suorakulmainen kolmio?
======================================
• Pythagoraan lause ja suorakulmaisen kolmion trigonometria
======================================

TKI ja työelämäyhteistyö

Opintojakson toteutukseen ei sisälly TKI- ja työelämäyhteistyötä.

Opiskelijan työmäärä

Oppitunnit 36 h.
Kotitehtävät 15 h.
Learn-materiaaliin perehtyminen 15 h.
Koe ja siihen valmistautuminen 10 h.

Evaluation scale

1-5

Assessment methods and criteria

Kurssikoe ja kotitehtävät.

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Kurssin Learn-sivulla tarjottu materiaali. Materiaali pohjautuu osaltaan kirjaan Insinöörin matematiikka, Tuomenlehto ym., 1.-5. painos, 2018. Muutkin painokset käyvät.

Yksilölliset oppimisväylät

Kurssi sisältää lähiopetusta, ja harjoitustehtäviä.

TKI ja työelämäyhteistyö

Ei sisällä

Tentit ja muut määräajat

Kurssin tenttiä saa uusia 2 kertaa.

Opiskelijan työmäärä

Kurssin työmäärä vastaa 3 opintopistettä eli 81 tuntia opiskelua.

Toteutuksen osien kuvaus

Kurssi kertaa tai vahvistaa matemaattisia valmiuksia jatkokursseja varten. Sisältöön kuuluu mm. murtolausekkeet, potenssi- ja juurilausekkeet, 1. ja 2. asteen yhtälöiden ratkaiseminen, funktion käsite ja suorakulmaisen kolmion ratkaiseminen.

Evaluation scale

Approved/Failed

Assessment methods and criteria

Kurssin arviointi perustuu loppukokeen tulokseen.

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Kurssin Learn-sivulla tarjottu materiaali. Materiaali pohjautuu osaltaan kirjaan Insinöörin matematiikka, Tuomenlehto ym., 1.-5. painos, 2018. Muutkin painokset käyvät.

Yksilölliset oppimisväylät

Kurssi sisältää lähiopetusta, ja harjoitustehtäviä.

TKI ja työelämäyhteistyö

Ei sisällä

Tentit ja muut määräajat

Kurssin tenttiä saa uusia 2 kertaa.

Opiskelijan työmäärä

Kurssin työmäärä vastaa 3 opintopistettä eli 81 tuntia opiskelua.

Toteutuksen osien kuvaus

Kurssi kertaa tai vahvistaa matemaattisia valmiuksia jatkokursseja varten. Sisältöön kuuluu mm. murtolausekkeet, potenssi- ja juurilausekkeet, 1. ja 2. asteen yhtälöiden ratkaiseminen, funktion käsite ja suorakulmaisen kolmion ratkaiseminen.

Evaluation scale

Approved/Failed

Assessment methods and criteria

Kurssin arviointi perustuu loppukokeen tulokseen.

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Learn-alustalta löytyy koko kurssimateriaali: aihepiireittäin jaoteltuna tekstimateriaali,
AC-opetusvideot ja tehtävät.

Yksilölliset oppimisväylät

Miten käsittelet matemaattisia summalausekkeita?
======================================
• Lausekkeen arvo
• Samanmuotoisten termien yhdistäminen
• Sulkujen poistaminen summalausekkeesta
• Summan kertominen ja jakaminen luvulla
• Tekijän erottaminen
======================================
Miten suoritat laskutoimituksia murtoluvuilla?
======================================
• Murtolukujen yhteenlasku
• Murtolukujen vähennyslasku
• Murtolukujen kertolasku
• Murtolukujen jakolasku
======================================
Miten ratkaiset ensimmäisen asteen yhtälön? Miten tulkitset yhtälön kuvaaman ongelman?
======================================
• Yhtälön peruskäsitteitä
• Kaavat sovelluksissa
• Kuvaaja: suora
======================================
Mikä on yhtälöpari? Millaisista ongelmista syntyy yhtälöpareja? Miten se ratkaistaan?
======================================
• ratkaiseminen sijoitusmenettelyllä
• ratkaiseminen yhteenlaskumenettelyllä
• yhtälöparin graafinen tulkinta
======================================
Miten käsittelet potenssilausekkeita? Mitä tarkoittaa negatiivinen potenssi?
======================================
• Neliö ja kuutio
• Eksponenttina nolla
• Eksponenttina negatiivinen luku
• Tulon ja osamäärän potenssi
• Samankantaisten potenssien tulo ja osamäärä
• Potenssin potenssi
======================================
Miten käsitellään juurilausekkeita?
======================================
• Neliöjuuri
• Kuutiojuuri
• Juuret kaavoissa
======================================
Mikä on polynomilauseke? Miten kerrot ja jaat polynomeja keskenään? Mitä ovat binomin neliö ja summan ja erotuksen tulo?
======================================
• Käsitteitä
• Polynomien yhteen- ja vähennyslasku
• Polynomien kertolasku
• Binomin neliö
• Summan ja erotuksen tulo
• Polynomien jakolasku – myös jakokulmaa käyttäen
• Polynomin jakaminen tekijöihin
======================================
Mikä on toisen asteen yhtälö? Miten ratkaistaan erityyppisiä toisen asteen yhtälöitä?
======================================
• Peruskäsitteitä
• Juurtaminen
• Tekijöihin jakaminen yhtälöä ratkaistaessa
• Ratkaisukaava
• Yhtälön juurten reaalisuuden tutkiminen diskriminantin avulla
• Kuvaaja: paraabeli
======================================
Mikä on funktio? Miten piirretään perusfunktioiden kuvaajat (suora ja paraabeli)?
Miten kuvaajia tulkitaan?
======================================
• Funktion käsite
• Ensimmäisen asteen funktion kuvaaja: suora
• Kulmakerroin
• Yhdensuuntaiset suorat
• Nouseva ja laskeva suora
• Toisen asteen funktion kuvaaja: paraabeli
• Nollakohtien ja aukeamissuunnan määrittäminen
======================================
Miten ratkaistaan suorakulmainen kolmio?
======================================
• Pythagoraan lause ja suorakulmaisen kolmion trigonometria
======================================

TKI ja työelämäyhteistyö

Opintojakson toteutukseen ei sisälly TKI- ja työelämäyhteistyötä.

Opiskelijan työmäärä

Oppitunnit 36 h.
Kotitehtävät 15 h.
Learn-materiaaliin perehtyminen 15 h.
Koe ja siihen valmistautuminen 10 h.

Evaluation scale

1-5

Assessment methods and criteria

Kurssikoe ja kotitehtävät.

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Kurssin Learn-pohjalla jaettu materiaali ja taulukkokirja (esim. MAOL). Learn-materiaali perustuu oppikirjaan, jonka hankkiminen opiskelijoille vapaavalintaista. Kurssilla tarvitset laskimen. Funktiolaskin on ominaisuuksiltaan riittävä, mutta graafisen laskimen käyttö on myös sallittua.

Oppikirja:
Tuomenlehto, Holmlund, Huuskonen, Makkonen ja Surakka. Insinöörin matematiikka, Edita, 2014. ISBN 9789513763527

Yksilölliset oppimisväylät

Työviikkopohjainen oppimisväylä:
Työjärjestyksen mukaiset oppitunnit. Ei läsnäolopakkoa.

Opintoja nopeuttava oppimisväylä:
Suorittaminen mahdollista tenttimällä.

TKI ja työelämäyhteistyö

Ei yhteistyötä

Tentit ja muut määräajat

Viikottaisten kotitehtävien määräajat ja tenttipäivämäärät ilmoitetaan Learn-alustalla kurssin alkaessa.

Kansainvälinen yhteistyö

Ei yhteistyötä

Opiskelijan työmäärä

Opintopisteitä vastaava opiskelijantyömäärä on 81 tuntia, joka koostuu opetuksesta, viikoittaisista kotitehtävistä, sekä tentistä ja tenttiin kertaamisesta.

Further information

Opintojakso kertaa tai vahvistaa matemaattisia valmiuksia jatkokursseja varten.

Evaluation scale

Approved/Failed

Assessment methods and criteria

Kurssin arvosana perustuu lopputentin pisteisiin. Hylättyä tenttiä voi uusia kahdesti.

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Lecture notes and calculations.

Yksilölliset oppimisväylät

Final exam.

TKI ja työelämäyhteistyö

This course does not include RDI and work-related cooperation.

Evaluation scale

1-5

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Lecture notes and calculations.

Yksilölliset oppimisväylät

Final exam.

TKI ja työelämäyhteistyö

This course does not include RDI and work-related cooperation.

Evaluation scale

1-5

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Evaluation scale

1-5

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Evaluation scale

1-5

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Kurssin Learn-sivulla tarjottu materiaali. Materiaali pohjautuu osaltaan kirjaan Insinöörin matematiikka, Tuomenlehto ym., 1.-5. painos, 2018.

Yksilölliset oppimisväylät

Kurssi sisältää lähiopetusta, ja harjoitustehtäviä.

TKI ja työelämäyhteistyö

Ei sisällä

Tentit ja muut määräajat

Kurssin tenttiä saa uusia 2 kertaa.

Opiskelijan työmäärä

Kurssin työmäärä vastaa 3 opintopistettä eli 81 tuntia opiskelua.

Toteutuksen osien kuvaus

Kurssi kertaa tai vahvistaa matemaattisia valmiuksia jatkokursseja varten. Sisältöön kuuluu mm. murtolausekkeet, potenssi- ja juurilausekkeet, 1. ja 2. asteen yhtälöiden ratkaiseminen, funktion käsite ja suorakulmaisen kolmion ratkaiseminen.

Evaluation scale

Approved/Failed

Assessment methods and criteria

Kurssin arviointi perustuu loppukokeen tulokseen.

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Kurssin Learn-pohjalla jaettu materiaali ja taulukkokirja (esim. MAOL). Learn-materiaali perustuu oppikirjaan, jonka hankkiminen opiskelijoille vapaavalintaista. Kurssilla tarvitset laskimen. Funktiolaskin on ominaisuuksiltaan riittävä, mutta graafisen laskimen käyttö on myös sallittua.

Oppikirja:
Tuomenlehto, Holmlund, Huuskonen, Makkonen ja Surakka. Insinöörin matematiikka, Edita, 2014. ISBN 9789513763527

Yksilölliset oppimisväylät

Työviikkopohjainen oppimisväylä:
Työjärjestyksen mukaiset oppitunnit. Ei läsnäolopakkoa.

Opintoja nopeuttava oppimisväylä:
Suorittaminen mahdollista tenttimällä.

TKI ja työelämäyhteistyö

Ei yhteistyötä

Tentit ja muut määräajat

Viikottaisten kotitehtävien määräajat ja tenttipäivämäärät ilmoitetaan Learn-alustalla kurssin alkaessa.

Kansainvälinen yhteistyö

Ei yhteistyötä

Opiskelijan työmäärä

Opintopisteitä vastaava opiskelijantyömäärä on 81 tuntia, joka koostuu opetuksesta (22 h), viikottaisista kotitehtävistä (48 h), sekä tentistä ja tenttiin kertaamisesta (11 h).

Further information

Opintojakso kertaa tai vahvistaa matemaattisia valmiuksia jatkokursseja varten.

Evaluation scale

1-5

Assessment methods and criteria

Kurssin arvosana perustuu lopputentin pisteisiin. Hylättyä tenttiä voi uusia kahdesti.

Kotitehtäviä tekemällä on mahdollista tienata lisäpisteitä kurssin loppuarvosanan korottamiseksi.

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Evaluation scale

1-5

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Evaluation scale

1-5

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Lecture notes and calculations.

Yksilölliset oppimisväylät

Final exam.

TKI ja työelämäyhteistyö

This course does not include RDI and work-related cooperation.

Evaluation scale

1-5

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Oppikirja: Tuomenlehto ym, Insinöörin matematiikka. EDITA
Kaavakirja: Tekniikan kaavasto. Tammertekniikka tai MAOL-kaavakirja.
Lisäksi kurssin Learn-sivulla ja tunneilla jaettu materiaali.

Yksilölliset oppimisväylät

Työviikkopohjainen oppimisväylä:
Työjärjestyksen mukainen lähiopetus, harjoitustehtävät sekä mahdolliset palautettavat harjoitystyöt.

TKI ja työelämäyhteistyö

Ei ole.

Tentit ja muut määräajat

Opintojakson mahdollisilla välikokeilla tai tentillä. Koko kurssin kattavaa tenttiä saa uusia 2 kertaa.

Opiskelijan työmäärä

Kurssi on viiden opintopisteen laajuinen, joka vastaa 135 tuntia työskentelyä.

Evaluation scale

1-5

Assessment methods and criteria

Arvionti perustuu menestykseen välikokeissa tai tentissä.
Mahdollinen harjoitustyö on suoritettava hyväksytysti.

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Tuomenlehto ym, Insinöörin matematiikka. EDITA
Kaavakirja: Tekniikan kaavasto. Tammertekniikka tai MAOL-kaavakirja.
Lisäksi kurssin Learn-sivulla ja tunneilla jaettu materiaali.

Yksilölliset oppimisväylät

Työviikkopohjainen oppimisväylä:
Työjärjestyksen mukainen opetus, joka sisältää teoriatunteja ja laskuharjoituksia. Itsenäinen opiskelu ja harjoitustehtävät.

TKI ja työelämäyhteistyö

Ei ole.

Tentit ja muut määräajat

Opintojakson välikokeilla tai tentillä. Koko kurssin kattavaa tenttiä saa uusia 2 kertaa.

Opiskelijan työmäärä

Opintojakso on kolmen opintopisteen laajuinen, joka vastaa 81 tuntia työskentelyä.

Evaluation scale

1-5

Assessment methods and criteria

Kurssin arviointi perustuu loppukokeen tulokseen.

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Tuomenlehto ym, Insinöörin matematiikka. EDITA
Kaavakirja: Tekniikan kaavasto. Tammertekniikka tai MAOL-kaavakirja.
Lisäksi kurssin Learn-sivulla ja tunneilla jaettu materiaali.

Yksilölliset oppimisväylät

Työviikkopohjainen oppimisväylä:
Työjärjestyksen mukainen opetus, joka sisältää teoriatunteja ja laskuharjoituksia. Itsenäinen opiskelu ja harjoitustehtävät.

TKI ja työelämäyhteistyö

Ei ole.

Tentit ja muut määräajat

Opintojakson välikokeilla tai tentillä. Koko kurssin kattavaa tenttiä saa uusia 2 kertaa.

Opiskelijan työmäärä

Opintojakso on kolmen opintopisteen laajuinen, joka vastaa 81 tuntia työskentelyä.

Evaluation scale

1-5

Assessment methods and criteria

Kurssin arviointi perustuu loppukokeen tulokseen.

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Lecture notes and calculations.

Yksilölliset oppimisväylät

Final exam.

TKI ja työelämäyhteistyö

This course does not include RDI and work-related cooperation.

Evaluation scale

1-5

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Kurssin Learn-sivulla tarjottu materiaali. Materiaali pohjautuu osaltaan kirjaan Insinöörin matematiikka, Tuomenlehto ym., 1.-5. painos, 2018.

Tentit ja muut määräajat

Kurssin tenttiä saa uusia 2 kertaa.

Opiskelijan työmäärä

Kurssin työmäärä vastaa 3 opintopistettä eli 81 tuntia opiskelua.

Toteutuksen osien kuvaus

Kurssi kertaa tai vahvistaa matemaattisia valmiuksia jatkokursseja varten. Sisältöön kuuluu mm. murtolausekkeet, potenssi- ja juurilausekkeet, 1. ja 2. asteen yhtälöiden ratkaiseminen, funktion käsite ja suorakulmaisen kolmion ratkaiseminen.

Evaluation scale

Approved/Failed

Assessment methods and criteria

Kurssin arviointi perustuu loppukokeen tulokseen.

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Tuomenlehto ym, Insinöörin matematiikka. EDITA
Kaavakirja: Tekniikan kaavasto. Tammertekniikka tai MAOL-kaavakirja.
Lisäksi kurssin Learn-sivulla ja tunneilla jaettu materiaali.

Yksilölliset oppimisväylät

Työviikkopohjainen oppimisväylä:
Työjärjestyksen mukainen opetus, joka sisältää teoriatunteja ja laskuharjoituksia. Itsenäinen opiskelu ja harjoitustehtävät.

TKI ja työelämäyhteistyö

Ei ole.

Tentit ja muut määräajat

Opintojakson välikokeilla tai tentillä. Koko kurssin kattavaa tenttiä saa uusia 2 kertaa.

Opiskelijan työmäärä

Opintojakso on kolmen opintopisteen laajuinen, joka vastaa 81 tuntia työskentelyä.

Evaluation scale

1-5

Assessment methods and criteria

Kurssin arviointi perustuu loppukokeen tulokseen.

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Lecture notes and calculations.

Yksilölliset oppimisväylät

Final exam.

TKI ja työelämäyhteistyö

This course does not include RDI and work-related cooperation.

Evaluation scale

1-5

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Learn-alustalta löytyy koko kurssimateriaali: aihepiireittäin jaoteltuna tekstimateriaali,
AC-opetusvideot ja tehtävät.

Yksilölliset oppimisväylät

Miten käsittelet matemaattisia summalausekkeita?
======================================
• Lausekkeen arvo
• Samanmuotoisten termien yhdistäminen
• Sulkujen poistaminen summalausekkeesta
• Summan kertominen ja jakaminen luvulla
• Tekijän erottaminen
======================================
Miten suoritat laskutoimituksia murtoluvuilla?
======================================
• Murtolukujen yhteenlasku
• Murtolukujen vähennyslasku
• Murtolukujen kertolasku
• Murtolukujen jakolasku
======================================
Miten ratkaiset ensimmäisen asteen yhtälön? Miten tulkitset yhtälön kuvaaman ongelman?
======================================
• Yhtälön peruskäsitteitä
• Kaavat sovelluksissa
• Kuvaaja: suora
======================================
Mikä on yhtälöpari? Millaisista ongelmista syntyy yhtälöpareja? Miten se ratkaistaan?
======================================
• ratkaiseminen sijoitusmenettelyllä
• ratkaiseminen yhteenlaskumenettelyllä
• yhtälöparin graafinen tulkinta
======================================
Miten käsittelet potenssilausekkeita? Mitä tarkoittaa negatiivinen potenssi?
======================================
• Neliö ja kuutio
• Eksponenttina nolla
• Eksponenttina negatiivinen luku
• Tulon ja osamäärän potenssi
• Samankantaisten potenssien tulo ja osamäärä
• Potenssin potenssi
======================================
Miten käsitellään juurilausekkeita?
======================================
• Neliöjuuri
• Kuutiojuuri
• Juuret kaavoissa
======================================
Mikä on polynomilauseke? Miten kerrot ja jaat polynomeja keskenään? Mitä ovat binomin neliö ja summan ja erotuksen tulo?
======================================
• Käsitteitä
• Polynomien yhteen- ja vähennyslasku
• Polynomien kertolasku
• Binomin neliö
• Summan ja erotuksen tulo
• Polynomien jakolasku – myös jakokulmaa käyttäen
• Polynomin jakaminen tekijöihin
======================================
Mikä on toisen asteen yhtälö? Miten ratkaistaan erityyppisiä toisen asteen yhtälöitä?
======================================
• Peruskäsitteitä
• Juurtaminen
• Tekijöihin jakaminen yhtälöä ratkaistaessa
• Ratkaisukaava
• Yhtälön juurten reaalisuuden tutkiminen diskriminantin avulla
• Kuvaaja: paraabeli
======================================
Mikä on funktio? Miten piirretään perusfunktioiden kuvaajat (suora ja paraabeli)?
Miten kuvaajia tulkitaan?
======================================
• Funktion käsite
• Ensimmäisen asteen funktion kuvaaja: suora
• Kulmakerroin
• Yhdensuuntaiset suorat
• Nouseva ja laskeva suora
• Toisen asteen funktion kuvaaja: paraabeli
• Nollakohtien ja aukeamissuunnan määrittäminen
======================================
Miten ratkaistaan suorakulmainen kolmio?
======================================
• Pythagoraan lause ja suorakulmaisen kolmion trigonometria
======================================

TKI ja työelämäyhteistyö

Opintojakson toteutukseen ei sisälly TKI- ja työelämäyhteistyötä.

Opiskelijan työmäärä

Oppitunnit 36 h.
Kotitehtävät 15 h.
Learn-materiaaliin perehtyminen 15 h.
Koe ja siihen valmistautuminen 10 h.

Evaluation scale

1-5

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Lecture notes and calculations.

Yksilölliset oppimisväylät

Final exam.

TKI ja työelämäyhteistyö

This course does not include RDI and work-related cooperation.

Evaluation scale

1-5

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Evaluation scale

1-5

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Tuomenlehto ym, Insinöörin matematiikka. EDITA
Kaavakirja: Tekniikan kaavasto. Tammertekniikka tai MAOL-kaavakirja.
Lisäksi kurssin Learn-sivulla ja tunneilla jaettu materiaali.

Yksilölliset oppimisväylät

Työviikkopohjainen oppimisväylä:
Työjärjestyksen mukainen opetus, joka sisältää teoriatunteja ja laskuharjoituksia. Itsenäinen opiskelu ja harjoitustehtävät.

TKI ja työelämäyhteistyö

Ei ole.

Tentit ja muut määräajat

Opintojakson välikokeilla tai tentillä. Koko kurssin kattavaa tenttiä saa uusia 2 kertaa.

Opiskelijan työmäärä

Opintojakso on kolmen opintopisteen laajuinen, joka vastaa 81 tuntia työskentelyä.

Evaluation scale

1-5

Assessment methods and criteria

Kurssin arviointi perustuu loppukokeen tulokseen.

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Learn-alustalta löytyy koko kurssimateriaali: aihepiireittäin jaoteltuna tekstimateriaali,
AC-opetusvideot ja tehtävät.

Yksilölliset oppimisväylät

Miten käsittelet matemaattisia summalausekkeita?
======================================
• Lausekkeen arvo
• Samanmuotoisten termien yhdistäminen
• Sulkujen poistaminen summalausekkeesta
• Summan kertominen ja jakaminen luvulla
• Tekijän erottaminen
======================================
Miten suoritat laskutoimituksia murtoluvuilla?
======================================
• Murtolukujen yhteenlasku
• Murtolukujen vähennyslasku
• Murtolukujen kertolasku
• Murtolukujen jakolasku
======================================
Miten ratkaiset ensimmäisen asteen yhtälön? Miten tulkitset yhtälön kuvaaman ongelman?
======================================
• Yhtälön peruskäsitteitä
• Kaavat sovelluksissa
• Kuvaaja: suora
======================================
Mikä on yhtälöpari? Millaisista ongelmista syntyy yhtälöpareja? Miten se ratkaistaan?
======================================
• ratkaiseminen sijoitusmenettelyllä
• ratkaiseminen yhteenlaskumenettelyllä
• yhtälöparin graafinen tulkinta
======================================
Miten käsittelet potenssilausekkeita? Mitä tarkoittaa negatiivinen potenssi?
======================================
• Neliö ja kuutio
• Eksponenttina nolla
• Eksponenttina negatiivinen luku
• Tulon ja osamäärän potenssi
• Samankantaisten potenssien tulo ja osamäärä
• Potenssin potenssi
======================================
Miten käsitellään juurilausekkeita?
======================================
• Neliöjuuri
• Kuutiojuuri
• Juuret kaavoissa
======================================
Mikä on polynomilauseke? Miten kerrot ja jaat polynomeja keskenään? Mitä ovat binomin neliö ja summan ja erotuksen tulo?
======================================
• Käsitteitä
• Polynomien yhteen- ja vähennyslasku
• Polynomien kertolasku
• Binomin neliö
• Summan ja erotuksen tulo
• Polynomien jakolasku – myös jakokulmaa käyttäen
• Polynomin jakaminen tekijöihin
======================================
Mikä on toisen asteen yhtälö? Miten ratkaistaan erityyppisiä toisen asteen yhtälöitä?
======================================
• Peruskäsitteitä
• Juurtaminen
• Tekijöihin jakaminen yhtälöä ratkaistaessa
• Ratkaisukaava
• Yhtälön juurten reaalisuuden tutkiminen diskriminantin avulla
• Kuvaaja: paraabeli
======================================
Mikä on funktio? Miten piirretään perusfunktioiden kuvaajat (suora ja paraabeli)?
Miten kuvaajia tulkitaan?
======================================
• Funktion käsite
• Ensimmäisen asteen funktion kuvaaja: suora
• Kulmakerroin
• Yhdensuuntaiset suorat
• Nouseva ja laskeva suora
• Toisen asteen funktion kuvaaja: paraabeli
• Nollakohtien ja aukeamissuunnan määrittäminen
======================================
Miten ratkaistaan suorakulmainen kolmio?
======================================
• Pythagoraan lause ja suorakulmaisen kolmion trigonometria
======================================

TKI ja työelämäyhteistyö

Opintojakson toteutukseen ei sisälly TKI- ja työelämäyhteistyötä.

Opiskelijan työmäärä

Oppitunnit 36 h.
Kotitehtävät 15 h.
Learn-materiaaliin perehtyminen 15 h.
Koe ja siihen valmistautuminen 10 h.

Evaluation scale

1-5

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Kurssin Learn-sivulla tarjottu materiaali. Materiaali pohjautuu osaltaan kirjaan Insinöörin matematiikka, Tuomenlehto ym.

Yksilölliset oppimisväylät

Työviikkopohjainen oppimisväylä:
Työjärjestyksen mukainen opetus ja harjoitustehtävät.

TKI ja työelämäyhteistyö

Ei ole.

Tentit ja muut määräajat

Opintojakson tenttiä saa uusia 2 kertaa.

Opiskelijan työmäärä

Opintojakson työmäärä vastaa 3 opintopistettä eli 81 tuntia opiskelua.

Toteutuksen osien kuvaus

Opintojakso kertaa tai vahvistaa matemaattisia valmiuksia jatkokursseja varten. Sisältöön kuuluu mm. murtolausekkeet, potenssi- ja juurilausekkeet, 1. ja 2. asteen yhtälöiden ratkaiseminen, funktion käsite ja suorakulmaisen kolmion ratkaiseminen.

Evaluation scale

Approved/Failed

Assessment methods and criteria

Opintojakson arviointi perustuu loppukokeen tulokseen.

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Luentokalvot ja laskuharjoitukset.

Yksilölliset oppimisväylät

Työviikkopohjainen oppimisväylä:
Osallistut työjärjestyksen mukaisesti opetukseen ja ohjaukseen. Opiskelusi rytmittyvät työjärjestyksen mukaisten ryhmätapaamisten ja itsenäisesti tehtävien oppimistehtävien

TKI ja työelämäyhteistyö

Opintojaksoon ei sisälly TKI- ja työelämäyhteistyötä.

Evaluation scale

1-5

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Luentokalvot ja laskuharjoitukset.

Yksilölliset oppimisväylät

Tentti.

TKI ja työelämäyhteistyö

Opintojaksoon ei sisälly TKI- ja työelämäyhteistyötä.

Evaluation scale

1-5

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Opiskelumateriaali

Learn-alustalta löytyy koko kurssimateriaali: aihepiireittäin jaoteltuna tekstimateriaali,
AC-opetusvideot ja tehtävät.

Yksilölliset oppimisväylät

Miten käsittelet matemaattisia summalausekkeita?
======================================
• Lausekkeen arvo
• Samanmuotoisten termien yhdistäminen
• Sulkujen poistaminen summalausekkeesta
• Summan kertominen ja jakaminen luvulla
• Tekijän erottaminen
======================================
Miten suoritat laskutoimituksia murtoluvuilla?
======================================
• Murtolukujen yhteenlasku
• Murtolukujen vähennyslasku
• Murtolukujen kertolasku
• Murtolukujen jakolasku
======================================
Miten ratkaiset ensimmäisen asteen yhtälön? Miten tulkitset yhtälön kuvaaman ongelman?
======================================
• Yhtälön peruskäsitteitä
• Kaavat sovelluksissa
• Kuvaaja: suora
======================================
Mikä on yhtälöpari? Millaisista ongelmista syntyy yhtälöpareja? Miten se ratkaistaan?
======================================
• ratkaiseminen sijoitusmenettelyllä
• ratkaiseminen yhteenlaskumenettelyllä
• yhtälöparin graafinen tulkinta
======================================
Miten käsittelet potenssilausekkeita? Mitä tarkoittaa negatiivinen potenssi?
======================================
• Neliö ja kuutio
• Eksponenttina nolla
• Eksponenttina negatiivinen luku
• Tulon ja osamäärän potenssi
• Samankantaisten potenssien tulo ja osamäärä
• Potenssin potenssi
======================================
Miten käsitellään juurilausekkeita?
======================================
• Neliöjuuri
• Kuutiojuuri
• Juuret kaavoissa
======================================
Mikä on polynomilauseke? Miten kerrot ja jaat polynomeja keskenään? Mitä ovat binomin neliö ja summan ja erotuksen tulo?
======================================
• Käsitteitä
• Polynomien yhteen- ja vähennyslasku
• Polynomien kertolasku
• Binomin neliö
• Summan ja erotuksen tulo
• Polynomien jakolasku – myös jakokulmaa käyttäen
• Polynomin jakaminen tekijöihin
======================================
Mikä on toisen asteen yhtälö? Miten ratkaistaan erityyppisiä toisen asteen yhtälöitä?
======================================
• Peruskäsitteitä
• Juurtaminen
• Tekijöihin jakaminen yhtälöä ratkaistaessa
• Ratkaisukaava
• Yhtälön juurten reaalisuuden tutkiminen diskriminantin avulla
• Kuvaaja: paraabeli
======================================
Mikä on funktio? Miten piirretään perusfunktioiden kuvaajat (suora ja paraabeli)?
Miten kuvaajia tulkitaan?
======================================
• Funktion käsite
• Ensimmäisen asteen funktion kuvaaja: suora
• Kulmakerroin
• Yhdensuuntaiset suorat
• Nouseva ja laskeva suora
• Toisen asteen funktion kuvaaja: paraabeli
• Nollakohtien ja aukeamissuunnan määrittäminen
======================================
Miten ratkaistaan suorakulmainen kolmio?
======================================
• Pythagoraan lause ja suorakulmaisen kolmion trigonometria
======================================

TKI ja työelämäyhteistyö

Opintojakson toteutukseen ei sisälly TKI- ja työelämäyhteistyötä.

Opiskelijan työmäärä

Oppitunnit 36 h.
Kotitehtävät 15 h.
Learn-materiaaliin perehtyminen 15 h.
Koe ja siihen valmistautuminen 10 h.

Evaluation scale

1-5

Objective

You know how to calculate using different mathematical relations.
You are able to calculate using fraction numbers and relations.
You understand the concept of a function.
You know how to solve different types of first and second degree equations.
You are able to handle the mathematical and the graphical representations of a line.
You can solve systems of linear equations in two variables.
You know how to simplify and calculate with different exponent relations.
You are able to calculate with radical expressions.
You are able to handle the mathematical and the graphical representations of a second degree polynomial function.
You know how to calculate with polynomials and how to factor them.
You know how to solve the right hand triangles.
You are familiar with the basics of calculations related to percentages.

Content

How to simplify mathematical relations?
How to calculate and simplify fraction numbers and relations?
What is the meaning of a function and its graph? What are the zeros? How to draw functions?
How to solve different linear equations?
What is the meaning of a line mathematically speaking? What are the slope and the y-intercept point?
How to solve systems of linear equations in two variables?
How to simplify exponent relations? What are the calculation rules of exponents?
How to simplify radical relations?
How to solve the second-degree polynomial equation?
How to use the information about the discriminant? How to draw the parabola?
How to calculate using polynomials? How to factor polynomials?
How to solve right hand triangles?
How to solve problems related to percentages?

Evaluation scale

1-5