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Be sure also to check the Secondary page for general support.
Be sure also to check the Secondary page for general support.
Also check out our Math 8 and 9 Video Series For Parents.
Number: Number represents and describes quantity (how many or how much)
Number: Number represents and describes quantity (how many or how much)
ORDERING RATIONAL NUMBERS
Estimate reasonably
Apply multiple strategies to solve problems in abstract situations
Download activity here.
Part 1:
Have students record where they’d place sets of rational numbers on a picture of an open number line (a/k/a “Clothesline Math”).
Remind students to think not just about the order in which the numbers are placed but also about the relative spacing between them. Ask:
What strategies did you use?
Which numbers were more challenging to “place and space”?
Part 2:
Have students determine what the missing numbers might be.
Encourage them to think about possibilities in term of integers, fractions (proper and improper), and mixed numbers. Ask:
How did you determine the missing number(s)?
What do you notice about the different answers that you came up with?
What’s the same? What’s different?
Is there a relationship?
ADDING RATIONAL NUMBERS
Apply multiple strategies to solve problems in abstract situations
Represent mathematical ideas in pictorial and symbolic forms
Ask your students to compute the sum of two rational numbers.
An example is shown on the right. Ask questions such as:What strategy did you use? Can you think of a different strategy?
Write other pairs of rational numbers that have the same sum as this problem.
Can you write other expressions that have the same sum using 3 or more rational numbers?
OPERATIONS WITH RATIONAL NUMBERS
Demonstrate and apply mathematical understanding through problem solving
Apply multiple strategies to solve problems in abstract situations
GIve your students the expression on the right. Tell them it is made of digit cards filling in spots within the expression. Ask them questions such as:
Without calculating the answer, do you think the answer will be positive or negative?
Calculate the answer and express in lowest terms.
What strategy did you use?Rearrange the digit cards so that:
the answer is the same as the one shown
the denominator is as large as possible (after simplifying)
the denominator is as small as possible (after simplifying)
the answer is positive
Note: If students ask if they can use improper fractions, encourage them to generate solutions only using proper fractions first, then go through and determine how their answers would change.
POWERS
Use logic and patterns to solve puzzles
Develop, demonstrate, and apply mathematical understanding through problem solving
Provide students an image of a branching pattern like the one shown. Ask questions such as:
How is this pattern growing?
How many circles would be in Stage 5? Stage 10? Stage 15?
Which is the first stage that has more than one million circles?
Explore the sum of the circles at each stage (ie. sum of circles in Stages 1 to 3). What pattern do you notice? How many circles will there be altogether after Stage 6?
How many circles would be in Stage 5 if:
Stage 1 had one circle, and each stage tripled
Stage 1 had two circles, and each stage tripled
Create a branching pattern such that one of the stages has 48 circles.
General reflective questions:
How do powers help you to figure out your answers to each question?
What other branching pattern problems can you write and solve?
Note: These problems can be spread across many days instead of all at once.
EXPONENT LAWS
Use reasoning and logic to explore, analyze, and apply mathematical ideas
Provide students with an example of the product of two powers, solved using expanded notation (see below for example). Ask questions such as:
What relationship do you notice between the exponents in the initial question, and the exponent in the answer?
How can you determine the product of any two powers that have the same base?
Explain that the generalization is called an exponent law.
Repeat the process for the quotient of two powers, then for a power of a power (see below for examples).
Ask additional questions such as:
Which exponent law can be used to evalute a number to a power of 0?
For each law, what if the base were the product or quotient of two numbers?
For each law, what if the base was an algebraic term?
Write an expression for each law that results in the answer 26.
Product of Powers
How can you determine the product of any two powers that have the same base?
Quotient of Powers
How can you determine the quotient of any two powers that have the same base?
Power of a Power
How can you determine the power of any power?
ADDING POLYNOMIALS
Connect mathematical concepts to each other
Ask: "How is adding polynomials the same as adding whole numbers? How is it different?"
Encourage students to make as many connections as they can -- pictorial and symbolic, conceptual and procedural.
You may wish to present a particular pair of polynomials and whole numbers, like those below (left).
You may also wish to present pictorial and symbolic representations, like those below (centre and right).
Pose an open question such as "When you add two polynomials, the sum is 3x² + 7x + 6. What could the polynomials be?" Ask:
Can you find possibilities that involve negative coefficients?
Can you find two binomials that add to this trinomial?
SUBTRACTING POLYNOMIALS
Connect mathematical concepts to each other
Ask: "How is subtracting polynomials the same as adding whole numbers? How is it different?"
Encourage students to make as many connections as they can -- pictorial and symbolic, conceptual and procedural.
You may wish to present a particular pair of polynomials and whole numbers, like those below (left).
You may also wish to present pictorial and symbolic representations, like those below (centre and right).
Note: two fundamental meanings of subtraction are removal (or "take away") and comparison (or "difference").
Pose an open question such as "When you subtract two polynomials, the difference is 3x² + 7x + 6. What could the polynomials be?" Ask:
Can you find possibilities that involve negative coefficients?
Can you find two trinomials whose difference is this binomial?
Patterns & Relations:
We use patterns to represent identified regularities and to make generalizations
Patterns & Relations:
We use patterns to represent identified regularities and to make generalizations
LINEAR RELATIONS
Explain and justify mathematical ideas and decisions
Share Graphs 5, 8, 21, and 34 from wodb.ca.
Ask students "Which One Doesn’t Belong?"
Challenge students to come up with as many reasons as they can why each graph, equation, or table is different from all the others.
(It’s possible to give a justification for each element in each set being the “odd one out.”)
LINEAR RELATIONS
Use reasoning to analyze mathematical ideas
Share the following four representations of a linear relation.
Ask your learners:
What stays the same and what changes in this relationship? How does this appear in each representation? (or, more formally, Where do you see the constant term and the rate of change in each representation?)
What effect would increasing the starting cost have on each representation? What about decreasing the price per topping?
Sometimes ordering a pizza from Christopher's Pizzeria is less expensive than ordering from Marco's Pizzeria; sometimes it's more expensive. What could the equation for Christopher's Pizzeria be?
SOLVING LINEAR EQUATIONS
Demonstrate and apply mathematical understanding through play and problem solving
Apply multiple strategies to solve problems in abstract situations
Provide students with the open expression on the right, and answer the questions below it. [Note: you may decide whether or not to allow for an improper fraction]
Invite students to share their thinking by responding to questions such as:
What strategies did you use?
What assumptions did you make?
Is there more than one solution? If so, can you find them all?
More problems like this are available on the Open Middle website.
For each of the questions below, use the digits 1 to 9, at most one time each, to create an equation in which:
x has a postive answer
x has a negative integer answer
x has the largest possible answer
x has the least possible answer
LINEAR RELATIONS WITH INDIGENOUS CONNECTIONS
Engage in problem-solving experiences that are connected to place, story, cultural practices, and perspectives relevant to local First Peoples communities
Incorporate First Peoples worldviews and perspectives to make connections to mathematical concepts
Please read this blog post to learn about respectfully connecting cultural practices and perspectives of First Peoples to linear relations.
Cultural context as well as a detailed description of the math task is provided there.
Phyllis Atkins at blessing ceremony for We Are All Connected to This Land
Shape & Space: We can describe, measure, and compare spatial relationships
Shape & Space: We can describe, measure, and compare spatial relationships
SPATIAL PROPORTIONAL REASONING: Scale
Use reasoning to analyze mathematical ideas
Apply multiple strategies to solve problems in contextualized situations
A handout is provided.
Images are shown which have different dimensions than an original image.
Ask your students:
Without making any calculations, which of these images look the same as the original?
Which images ARE an exact enlargement or reduction of the original image? How do you know?
SPATIAL PROPORTIONAL REASONING:
Determining missing values (indirect measurement)
Apply multiple strategies to solve problems in contextualized situations
Connect mathematical concepts to other areas and personal interests
Provide the image on the right.
Ask:
At the same time a 6 foot person casts an 8 foot shadow, one tree casts a 12 foot shadow and another tree casts a 28 foot shadow. What is the height of each tree?
Can you determine the height of each tree using more than one strategy?
Sample responses:
6 ft is 3/4ths of 8 ft, and 3/4ths of 12 ft is 9 ft
12 ft is 1.5 times 8 ft, and 1.5 times 6 ft is 9 ft
If the person were 3 ft, their shadow would be 4 ft; so if a shadow is 12 ft, that's 3 times 4 ft, so the height would be 3 times 3 ft, which is 9 ft.
Why do you think only the shadow lengths were provided instead of giving a tree height and asking for its shadow length?
What is the height of a tree which at this time casts a 17.6 foot shadow?
If you were outside at this time, what would the length of your shadow be?
SPATIAL PROPORTIONAL REASONING: Scale Diagrams
Represent mathematical ideas in pictorial form
Reflect on mathematical thinking
Provide students with grid paper.
Tell them to sketch an original design, then create scaled images using the scales:
1:2
3:1
Ask:
How can you tell whether the scale is saying to draw an enlargement or a reduction?
How did you decide what kind of original design to draw, including its size?
Which parts of your design were the most challenging to enlarge or reduce. Explain why?
How do the areas compare between the three sketches?
Data & Statistics: Analyzing data & graphs enables us to compare and to interpret
Data & Statistics: Analyzing data & graphs enables us to compare and to interpret
MISLEADING GRAPHS
Use reasoning to analyze mathematical ideas
Share the graph on the right. Ask:
How many mistakes did the creator of the graph make? Find as many as you can.
What story does this graph tell? Is this intentional?
What suggestions would you make to the person who made this graph? Can you fix it?
MISLEADING GRAPHS
Use reasoning to analyze mathematical ideas
Explain and justify mathematical decisions
Circle graphs seem to be especially challenging for the media to create.
Share the following graphs with your students. Ask:
What's wrong with this graph?
Is a circle graph appropriate for this data? Why or why not?
What suggestions would you make to the person who made this graph? Can you fix it?
(There are plenty of COVID-19 related misleading graphs out there right now. Like this one! Some students may be very interested in discussing them. Others may not. Please be sensitive to this and use your discretion.)
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