Class work: Final presentations, wrap up integers, tricks with divisibility of numbers
- We will start with three presentations.
- Then we will wrap up integers, and go over any remaining questions. Turn in the multiplication and division worksheets if you haven’t already.
- Look at some story problems: http://www.mathgoodies.com/lessons/vol5/challenge_vol5.html. In particular, we will come up with word problems for multiplication of a positive and a negative number, and for multiplication of two negative numbers.
- We will justify (-1)(-1)=1 in different ways. You can look at these two websites.
- If there is time left, we will talk about divisibility rules and tricks. Here is a nice document that summarizes the divisibility rules. I will also make a handout for it.
Class work: Multiplication and division of integers
Sections covered: 10.2, 10.3
Relevant Common Core Standards:
7.NS.2.Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
- Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
- Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts.
- Come up with rules for adding and subtracting integers. Look, also at Appendix A.8 in your book.
- Finish and turn in subtraction worksheet.
- Here are the lecture notes for today.
- You have now seen three different ways to deal with addition and subtraction of integers: the number line, chip model, and patterns. They are all, in my opinion, preferable to teaching the rules without understanding. I am expecting you to be comfortable with all three for the final. In particular, I really like the chip model. It is able to maintain the idea of subtraction as taking away, which is the context most familiar to students. While solving problems using chips, students should be encouraged to generalize patterns and develop rules. Otherwise, the activity is pointless. Using manipulatives without making connections has little merit.Use the handouts from the packet to use chips to multiply and divide integers.
- Then we will work on the worksheets from your packet about multiplication and division of integers. We will wrap this up on Wednesday.
Final review topics:
- Be able to recognize and extend patterns, and to create formulas for them. This is similar to the work we did at the beginning of the semester.
- Understand what it means to have a place value system and know its properties.
- Know how to use different bases in different contexts, similarly to the problems before and on the first test. For example, know how to count in different bases, how to add and subtract in different bases, and how to convert between bases.
OPERATIONS ON WHOLE NUMBERS, FRACTIONS, AND DECIMALS
- Know contexts for subtraction and division and to make up word problems based on them (take away, missing addend, comparison; partitive, repeated subtraction).
- Be able to use the area model for multiplication, in the context of whole numbers, decimals, and fractions. Understand why the area model is more useful in most contexts than repeated addition.
- Justifying the distributive property using the area model.
- Meaning of operations: being able to recognize when to use which operation in the context of a problem.
- Look at different children’s strategies. Can you justify/explain them mathematically? Which properties of operations and numeration do they use?
- Addition, subtraction, multiplication and division algorithms (for all three types of numbers). Why do they work? Be able to explain each step of the standard algorithms, like we discussed in class. For addition and subtraction, know how to explain the steps using blocks. Why do we find the common denominator when we add fractions? For multiplication, know how the standard algorithm is related to the area model. Why does of mean multiply? For division, know how the standard model is related to division using blocks (this only for whole numbers). Why do we invert and multiply when we divide fractions?
- You do not have to know the CGI problem types.
FRACTIONS, DECIMALS, RATIOS AND PERCENTS
- Understand the meaning of fractions; be able to represent fractions using area, number line, and set models; understand how the size of one whole affects the meaning of a fraction.
- Properties of fractions: know how to show fractions are equivalent, know how to order fractions, know how to estimate sums, differences, products and quotients of fractions using fraction sense. Be able to use diagrams to solve these problems. Look at the diagram problems from worksheets and homework.
- Properties of decimals, especially place value. Know how to represent decimals using manipulatives.
- Ratio and percent problems, similar to the ones from homework and worksheets. There will be no conceptual questions from this area, only problem solving. Make sure you know how to find percent increase or decrease of a price or salary, and other contextual problems.
- Be able to explain the rules for adding, subtracting, multiplying, and dividing integers using chips, number line, patterns, and story problems. In particular, know why the product of two negative numbers is positive, and why subtracting a negative is like adding a positive.
- Be able to create story problems for situations involving integers.
Class work: Addition and subtraction of integers
Sections covered: 10.1, 10.2
Relevant Common Core Standards:
- 6.NS.5. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
- 6.NS.6.Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
- Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.
- Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
- Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
- 6.NS.7.Understand ordering and absolute value of rational numbers.
- Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.
- Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3 oC > –7 oC to express the fact that –3 oC is warmer than –7 oC.
- Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.
- Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.
7.NS.1.Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
- Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.
- Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
- Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
- Apply properties of operations as strategies to add and subtract rational numbers.
- A few more things about ratios, rates, and proportions (from the slides from last time). In particular, we will go over some definitions and properties of percents. What is percent? How do we compute with percents? Here is one form of the general formula: . This formula can be used to find any of the missing pieces. For example, to answer the question 14 is 10% of which number, we solve the following proportion: , so .
- We will talk about integers, and situations where adding, subtracting, multiplying, and dividing integers arises.
- Then we will work on activities from your packet for adding integers using color chips.
- Situations that are especially difficult to create word problems for are multiplying or subtracting two negatives. Here are some “natural” examples. The altitude of the highest point on Earth is 8900m, and the altitude of the lowest point on Earth is -800m. What is the difference in altitudes between the highest and the lowest points? For more examples of “extremes” on Earth, look at this page on Wikipedia. Multiplication and division are harder. Note that negative numbers come up naturally in the context of debt, or any type of decrease.
- I learned something from the book that I hadn’t know before, and that is that the color chips are an adaptation of an ancient Chinese method from 200BCE.
- You can look at subtraction of integers on the National Library of Virtual Manipulatives.
Class work: Test 3.
We will not review for this test, but I will allow you to work in groups if you like, and to use your book and notes.
The test on Wednesday will consist mostly of fraction topics, with a few ratio and percent questions. Know how to use diagrams to solve fraction problems, understand how the algorithms for addition, subtraction, multiplication, and division of fractions work, and be able to solve ratio and percent problems similar to the ones we had had on homework or in class.