Fractions: Fractions: Computations and Computations and

OperationsOperationsHands-On Experiences in Hands-On Experiences in

support of Student Successsupport of Student Success

Addition of Fractions: Addition of Fractions: Fraction Factory & Pattern Fraction Factory & Pattern

BlocksBlocks• We can join these fractions and rename We can join these fractions and rename

the region using a common fraction the region using a common fraction piece.piece.

• Therefore, Therefore,

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We can rename the sum using a common piece – sixths.

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• Now, change the whole! Assume the Now, change the whole! Assume the whole is:whole is:

Add the Following Using Fraction Factory: Add the Following Using Fraction Factory: Show Your Work!Show Your Work!

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We know that 2 - 1/8 pieces fit into a one fourth piece, therefore:

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We must ask students:

What do we know about the sum of these two fractions?

Will the sum be greater or less than 1?

5/6 is almost one whole; it is 1/6 shy of one whole;

1/3 is greater than 1/6 therefore the sum must be greater than 1!

Try this problem with pattern blocks!

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questions should we be asking our students here?

What about the similarities

between the two pieces?

How many 1/8 pieces

make up the ½ piece?

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What does the common manipulative piece in this example represent?

Repeat the following examples using Repeat the following examples using Pattern BlocksPattern Blocks

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Subtraction of Fractions: Subtraction of Fractions: Fraction Factory & Pattern Fraction Factory & Pattern

BlocksBlocks• When we subtract, we remove a portion of something When we subtract, we remove a portion of something

and name what we have left.and name what we have left.

- We need to remove 1/3 - We need to remove 1/3 of the wholeof the whole from from ½ ½ of the wholeof the whole and name the portion and name the portion (difference) that remains.(difference) that remains.

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-The portion that remains The portion that remains is 1/6 is 1/6

of the indicated whole of the indicated whole (the (the

difference!)difference!)

Subtract Using Fraction Factory & Subtract Using Fraction Factory & Pattern BlocksPattern Blocks

• Complete the following using both FF & Complete the following using both FF & PB for each:PB for each:

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Now make the whole the:

What would the whole look like using Pattern Blocks?

Use the hexagon as the whole

Use patterns blocks to solve and show the

solution…

Multiplication of Fractions: Fraction Multiplication of Fractions: Fraction Factory & Pattern BlocksFactory & Pattern Blocks

• When we multiply, we can think of the When we multiply, we can think of the multiplication sign as making “groups of” an multiplication sign as making “groups of” an amount.amount.

• 3 x 4, we can think of 3 groups of 4.3 x 4, we can think of 3 groups of 4.

- we can think of 1/3 ‘of’ ½- we can think of 1/3 ‘of’ ½

• What do we really know about the product What do we really know about the product here? Do students know what this means?here? Do students know what this means?

• Will 1/3 of ½ be less than 1? Less than ½? Will 1/3 of ½ be less than 1? Less than ½? Less than 1/3?Less than 1/3?

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Since I need to find 1/3 of ½, I showed ½ of the whole, and then partitioned the ½ into thirds.

One of these partitioned parts of ½ is 1/3 of ½ which represents 1/6 of the whole.

Therefore: 1/3 x 1/2 = 1/6 (in relation to the whole)

• Multiply the following using Fraction Multiply the following using Fraction Factory & Pattern Blocks:Factory & Pattern Blocks:

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Half

What does ¾ of the half look like? What is that in relation to our indicated whole?

Multiplication of Fractions: Multiplication of Fractions: Fraction Factory & Pattern BlocksFraction Factory & Pattern Blocks

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Seeing Division of Fractions: Seeing Division of Fractions: Fraction Factory & Pattern BlocksFraction Factory & Pattern Blocks

• When we divide, we ask: “How many groups When we divide, we ask: “How many groups of…?”of…?”

• 12/3 – How many groups of three are in 12?12/3 – How many groups of three are in 12?• Same with fractions:Same with fractions: “How many groups “How many groups

of ¼ are of ¼ are in ½?” in ½?”4

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We can see that there are 2 groups of ¼ . Therefore, ½ divided by ¼ = 2.

Divide Using Fraction Factory & Pattern Divide Using Fraction Factory & Pattern BlocksBlocks

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1 half piece goes into 2/3, plus 1/6 of the

whole.This 1/6 of the whole is

actually 1/3 of the dividend: ½. Therefore,

2/3 divided by ½ = 1 1/3 or 4/3.

Divide Using Fraction Factory & Pattern Divide Using Fraction Factory & Pattern BlocksBlocks

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Only ½ of a 1/3 is in 1/6;

Thus, 1/6 divided by 1/3

= 1/2

There are 1 & ½ groups of

1/6 pieces that are in a ¼

piece

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Will the quotient be < or > 1? How do you know?

Why is 5/12> ¼? How do we know?

- 5/12 is almost ½ and ½ > ¼! OR ¼ is also 3/12 & 5/12>3/12

Problem SolvingProblem Solving

Teaching Through Problem Teaching Through Problem SolvingSolving

• Mrs. Get Fit teaches Math and Phys-Mrs. Get Fit teaches Math and Phys-ed. To incorporate Math into the ed. To incorporate Math into the Phys-ed class, she divided the class Phys-ed class, she divided the class into eight groups. There are three into eight groups. There are three students in each group. The first students in each group. The first person in the group runs ¼ of a lap of person in the group runs ¼ of a lap of the track, the second person runs 1/6 the track, the second person runs 1/6 of the track, and the third person of the track, and the third person runs 1/3 of a lap of the track. How runs 1/3 of a lap of the track. How many laps of the track are run in total many laps of the track are run in total by all eight teams combined?by all eight teams combined?

Teaching Through Problem Teaching Through Problem SolvingSolving

• Jeff acquired a recipe from his Jeff acquired a recipe from his grandmother which makes 5 dozen grandmother which makes 5 dozen cookies. The recipe requires 2 ½ cups cookies. The recipe requires 2 ½ cups of oatmeal, 2 cups of chocolate chips, 2 of oatmeal, 2 cups of chocolate chips, 2 cups of flour, 2 eggs, 1 cup of butter, 5 cups of flour, 2 eggs, 1 cup of butter, 5 ml of baking powder, and 5 ml of salt. ml of baking powder, and 5 ml of salt.

• A) Adjust the recipe to make 2 dozen A) Adjust the recipe to make 2 dozen cookies.cookies.

• B) Adjust the recipe to make 12 dozen B) Adjust the recipe to make 12 dozen cookies.cookies.

Teaching Through Problem Teaching Through Problem SolvingSolving

• Bill’s Snow Plow can plow the snow off Bill’s Snow Plow can plow the snow off the school’s parking lot in 4 hours. Jane’s the school’s parking lot in 4 hours. Jane’s plowing company can plow the same plowing company can plow the same parking lot in just 3 hours. How long parking lot in just 3 hours. How long would it take Bill and Jane to plow the would it take Bill and Jane to plow the school’s parking lot together?school’s parking lot together?

Think of the math content involved with this problemThink of some “Before” activities that could be used.What would the debrief look/sound like in the classroom after the task was complete?

• Chad made a snack by combining 1/3 of a Chad made a snack by combining 1/3 of a bowl of granola with ¼ of a bowl of bowl of granola with ¼ of a bowl of chopped banana and ½ of a bowl of chopped banana and ½ of a bowl of yoghurt. Did one bowl hold all of the yoghurt. Did one bowl hold all of the ingredients at one time? Explain.ingredients at one time? Explain.

• Neptune completes 1 ½ turns about its axis Neptune completes 1 ½ turns about its axis each day. How many turns does it complete each day. How many turns does it complete in 1 week?in 1 week?

• About 3/4 of the students on the track team About 3/4 of the students on the track team are girls. About ¾ of these girls are in are girls. About ¾ of these girls are in grade 8. What fraction of the students on grade 8. What fraction of the students on the track team are grade 8 girls?the track team are grade 8 girls?

• Mara spent 3/5 of her vacation in British Mara spent 3/5 of her vacation in British Columbia. While in that province, she spent Columbia. While in that province, she spent ½ of her time in Vancouver. What fraction of ½ of her time in Vancouver. What fraction of her vacation did Michaela spend in her vacation did Michaela spend in Vancouver? If her vacation lasted 20 days, Vancouver? If her vacation lasted 20 days, how many days did she spend in Vancouver?how many days did she spend in Vancouver?

• Jackie used to be on the phone 3 ½ times as Jackie used to be on the phone 3 ½ times as much as her brother. Her parents threatened much as her brother. Her parents threatened to take away the phone, so she cut down to to take away the phone, so she cut down to 2/5 of the time she used to be on the phone. 2/5 of the time she used to be on the phone. How many times as much as her brother is How many times as much as her brother is Jackie now on the phone?Jackie now on the phone?

[A][A] A student is sorting into stacks a room A student is sorting into stacks a room full of food donated by the school for the full of food donated by the school for the local food bank. He sorted of it before local food bank. He sorted of it before lunch and then sorted of the remainder lunch and then sorted of the remainder before school ended. What part (fraction) before school ended. What part (fraction) of all the food will be left for him to sort of all the food will be left for him to sort after school?after school?

[B][B]

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Explain the error in Armand’s thinking. Explain the error in Armand’s thinking. Use the manipulatives to support your Use the manipulatives to support your reasoning.reasoning.

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• Let’s explore our current resources:Let’s explore our current resources:

Grade 8 textbook – pg. 57 – 91Grade 8 textbook – pg. 57 – 91

Eating Candies by FractionsEating Candies by Fractions[C] Use manipulatives to solve this problem:[C] Use manipulatives to solve this problem:

Mark ate half of the candies in a bag. Leila ate Mark ate half of the candies in a bag. Leila ate

of what was left. Now there are 11 candies in of what was left. Now there are 11 candies in the bag. How many were in the bag at the start?the bag. How many were in the bag at the start?

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