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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)
ADDING FRACTIONS
Demonstrate and apply mathematical understanding through problem solving
Apply multiple strategies to solve problems in abstract situations
Have students choose at least one of the problems on the right.
For each one, the directions are to use the digits 1-9 at most one time each, to fill in the boxes to make the statement true.
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?
To access each example individually: (A, B, C, D)
Or use this handout.More problems like this are available on the Open Middle website.
RATIOS: Finding Missing Values
Connect mathematical concepts to each other
Reflect on mathematical thinking
Share four approaches (right) to the following problem:
Red paint is mixed with blue paint in the ratio of 8 : 6 to make purple paint. If the paint store used 20 L of red paint, how much blue paint did they use?
Encourage students to connect and reflect by asking questions such as:
What is the same? What's different?
Which approach would you take? Why?
SUBTRACTING FRACTIONS
Demonstrate and apply mathematical understanding through problem solving
Apply multiple strategies to solve problems in abstract situations
Pose the following open question to your students:
The difference between two fractions is Âľ.
What could the two fractions be?
Think of three or more pairs of fractions.
How can you easily create more sets of fractions?
Source: Marian Small
Don Steward's blog, Median, offers several purposeful practice activities that involve strategic thinking about number.
Assign or have students choose one of the puzzles below.
RATIOS: Making Comparisons
Apply multiple strategies to solve problems in contextualized situations
Explain and justify mathematical ideas and decisions
Share the Figure This! challenge below (left) with your students. Ask:
Which tastes juicier? How do you know?
What strategies did you use? Which comparisons were the most challenging?
How else might you have solved this problem? Can you arrive at the same answer but in a different way?
How can you show how you solved the problem? Can you draw a picture?
You may wish to share the image below (right) to encourage less "number-plucking" (i.e., finding a common denominator for 5, 7, 11, and 13) and more intuitive approaches. For example:
By doubling Jerry's Juice, we can quickly see that it is juicier than Jane's Juice. We've made the number of cups of concentrate the same; Jane's Juice has one extra cup of water.
Similarly, doubling Good Grape makes the number of cups of water the same as Grapeade.
Students may also use decimals or percents. Note that these imply a unit ratio. For example:
0.4 in Jerry's Juice represents that four-tenths of ONE can is concentrate.
Source: https://figurethis.nctm.org/challenges/c25/challenge.htm
MULTIPLYING FRACTIONS
Use reasoning and logic to explore and apply mathematical ideas
Communicate mathematic thinking in many ways
Option 1:
Pose an open question such as "When you multiply two fractions, the product is close to ÂĽ. What could the fractions be?" Ask:
How did you know that the product would be close to ÂĽ?
"Two-sevenths is close to one-quarter and nine-tenths is close to one."
"Half of a half is one quarter so I multiplied two-fifths and five-ninths."
"25/100 is one-quarter so I looked for numerators that multiplied to about 25 and denominators that multiplied to about 100."
How can you easily create more pairs from these?
"Just swap the numerators!"
"If I double the first fraction and half the second, the product will be the same."
"Two-fifths is equivalent to four-tenths so..."
Option 2:
The image on the right presents a less open--but more problematic--task.
This problem brings in strategic thinking about number (specifically, divisibility).
See https://nrich.maths.org/peachestoday for a full explanation of the problem!
DIVIDING FRACTIONS
Demonstrate and apply mathematical understanding through problem solving
Apply multiple strategies to solve problems in abstract situations
Pose one of the following open question to your students:
Option 1: The quotient of two fractions is 5/6. What could the two fractions be? (1/2 Ă· 3/5)
Ask:
What strategies did you use? (I thought multiplication. I found two fractions that multiplied to 5/6 and then divided by the reciprocal of the 2nd fraction.)
Once you've found one solution to this problem, how does this help you find more? (You can swap numerators or denominators and the quotient is the same. Also, you can think about 5/6 as 10/12 or 15/18 or ...)
Option 2: Using the digits 1 to 9 at most once, fill in the boxes (see right) to make the smallest quotient that you can.
Ask the same questions as in Option 1.
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
SOLVING LINEAR EQUATIONS
Use logic to solve puzzles
Explain mathematical ideas in many ways
Have students choose a set of SolveMe mobile puzzles.
They can visit the website (or download the iPad app).
Invite students to explain how they solved a puzzle that challenged them.
(If students have not yet learned how to solve linear equations, an intuitive approach is both expected and encouraged! For others, suggest that they consider using variables and equations to determine unknowns.)
SOLVING LINEAR EQUATIONS
Demonstrate and apply mathematical understanding through problem solving
Apply multiple strategies to solve problems in abstract situations
Have students choose at least one of the problems shown.
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.
Using the digits 1 to 9 at most one time each, place a digit in each box to find the greatest (or least) possible values for x.
Using the digits 1 to 9 at most one time each, fill in the boxes to find the largest (or smallest) possible values for x.
Using the digits 1 to 9, at most TWO times each, fill in the boxes to make an equation with no solutions.
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 decreasing the starting cost have on each representation? What about increasing 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?
Note: This task is an adaptation of a task posted earlier under Math 9; the coefficient and constant have been changed from rational to whole numbers. Check out the WODB? sets there, too!
LINEAR RELATIONS
Use reasoning to analyze and apply mathematical ideas
Explain and justify mathematical ideas and decisions
Present the following scenario to your students:
Katherine has $360 in her bank account. Every week she saves $20.
Ask your learners to express the relationship between the amount of money in her bank account and the number of weeks that she has been saving. Have them express this using a table, graph, and equation.
Ask:
Where do you see the numbers from the scenario in each representation?
What effect would increasing how much Katherine saves per week have on each representation?
What if she had less in her bank account at the beginning? How would that change each representation?
Suppose that Katherine is saving to buy a new smartphone which costs $640.
How many weeks will it take her to save this amount?
Can you solve this using a table, graph, and equation?
Which representation is the most helpful when answering this question? Explain.
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
SURFACE AREA VS VOLUME (Same Volume with different Surface Areas)
Apply mathematical undertanding through play and problem solving
Visualize to explore mathematical concepts
Show the image on the right and ask your students to observe the two rectangular prisms. Ask:
Without doing any calculating, how do the volume and surface area of each compare? (e.g. which is greater?)
Calculate the surface area and volume of each.
Do the calculations confirm what you expected?What other rectangular prisms can you make that have the same volume but different surface areas?
What is the surface area of each of your prisms? What is the greatest surface area that you can create?
This task is also available on this handout.
SURFACE AREA VS VOLUME (Same Surface Area with different Volumes)
Apply mathematical undertanding through play and problem solving
Visualize to explore mathematical concepts
Instructions:
The rectangular prism on the right requires 80 square units to paint all 6 faces.
What other rectangular prisms can you make that also require 80 square units to paint all 6 faces?
What is the volume of each of your prisms? What is the greatest volume that you can create?
This task is also available on this handout.
SURFACE AREA & VOLUME of TRIANGULAR PRISMS
Estimate reasonably
Apply mathematical understanding through problem solving
Show the two triangular prisms on the right.
Ask:
Without calculating, estimate how the volume of one prism compares to the other?
Without calculating, estimate how the surface area of one prism compares to the other?
Calculate the surface area and volume of each.
Do the calculations confirm what you expected?How can the volume of the first prism help you to know the volume of the second prism?
PYTHAGOREAN THEOREM (Discovery)
Develop mathematical understanding through play and inquiry
Visualize to explore mathematical concepts
Have students cut out squares from grid paper of varying sizes, or use this template. They may make more than one square of the same size.
Alternatively you can give them this sheet of sample triangles.Tell them to select 3 squares and form them into a triangle, then ask:
What kind of triangle did you form? (Acute, Obtuse, or Right)
How is the sum of the areas of the smaller two squares related to the area of the largest square?
Tell them to continue exploring with different sets of three triangles. Ask:
Were there any sets of squares that could not make a triangle?
Why do you think this is so?What did you notice was the relationship for every set that produced a right triangle? This is called the Pythagorean Relationship.
You may wish to share some of these Proofs Without Words from Steve Phelps and follow his question prompts.
PYTHAGOREAN THEOREM (Application)
Demonstrate and apply mathematical understanding through problem solving
Explain and justify mathematical ideas and decisions
Present the context of this problem to your students:
As you know, social distancing requires that people maintain a distance of at least 2 m, or 6 feet, from one another.
The image on the right has been making the rounds in math teacher social media circles. (Yes, that's a thing!)
Ask:
What's wrong with this picture?
Can you fix it?
If you'd rather have students analyze distances unrelated to COVID-19, please see "Watson Saves" from YummyMath.
Data & Statistics: Analyzing data & graphs enables us to compare and to interpret
Data & Statistics: Analyzing data & graphs enables us to compare and to interpret
CENTRAL TENDENCY
Estimate reasonably
Explain and justify mathematical ideas and decisions
Ask your students to observe these 3 sets of data:
{1, 1, 3, 3, 4, 6, 6, 6, 6}
{1, 1, 1, 2, 3, 3, 8, 8, 9}
{1, 1, 3, 3, 3, 4, 6, 7, 8}
Ask:
How do they compare in terms of mean, median, and mode? Estimate first before calculating.
Which is the best measure of central tendency for each (mean, median, or mode)?
CENTRAL TENDENCY
Develop, demonstrate, and apply mathematical understanding through problem solving
Reflect on mathematical thinking
Have students choose a few of the problems on the right.
Invite students to reflect on their thinking by responding to questions such as:
What strategies did you use?
What decisions or changes did you have to make as you worked to solve the problem?
What was the most challenging part of the task? Why?
Can you find sets of positive integers that satisfy the following?
Three numbers with mean 3 and mode 2
Three numbers with mean 7 and mode 10
Three numbers with mean 8, median 10, and range 8
Four numbers with mean 7.5, mode 6, and median 7
Four numbers with mean 6, median 6.5, and range 11
Five numbers with mean 4, mode 3, and range 9
Five numbers with mean 4, mode 2, and range 6 (two possible solutions)
Five numbers with mean 7, mode 7, and range 10 (three possible solutions)
Source: https://nrich.maths.org/10995
CENTRAL TENDENCY: Choosing
Use reasoning to analyze and apply mathematical ideas
Explain and justify mathematical decisions
Present the following scenario to your students:
An app developer has created three different apps. Apps are rated on a scale from one to five stars. Each app receives a rating from seven customers. This data is represented on the right.
Ask:
What is the mean, median, and mode of each set of data?
Which of these measures is the best summary of App 1? App 2? App 3? Explain your thinking.
If you had to use just one measure of central tendency and apply it to every app on the store, which would you choose? Defend your decision.
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