Fractions: Computations and Operations Hands-On Experiences in support of Student Success

Addition of Fractions: Fraction Factory & Pattern Blocks We can join these fractions and rename the region using a common fraction piece. Therefore, We can rename the sum using a common piece – sixths. When we add, we join things and rename the amount we have. We can use Fraction Factory & Pattern Blocks to join fractions and rename the sum. Now, change the whole! Assume the whole is:

Add the Following Using Fraction Factory: Show Your Work! We know that 2 - 1/8 pieces fit into a one fourth piece, therefore:

Explore 5/6 + 1/3 in more detail: We must ask ourselves and our students what do we know about the sum of these two fractions? For example, will the sum be greater or less than 1? Let’s try and explain why – 5/6 is almost one whole: it is 1/6 shy of a whole; 1/3 is greater than 1/6 therefore the sum must be greater than 1! Try this problem with the pattern blocks (ultimately you will be working with the green triangles!)

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

What does the common manipulative piece in this example represent? What sorts of 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? Eventually, with increased exposure and experience, students will be able to make the connections between ½ and 1/8. They will begin to recognize fraction facts without realizing it – through the repeated use of manipulatives. We know that there are 4/8 in our ½ piece, therefore: 1/8 + ½ is the same thing as 1/8 + 4/8 = 5/8 Therefore:

Repeat the following examples using Pattern Blocks

Subtraction of Fractions: Fraction Factory & Pattern Blocks When we subtract, we remove a portion of something and name what we have left. - We need to remove 1/3 of the whole from ½ of the whole and name the portion (difference) that remains. The portion that remains is 1/6 of the indicated whole (the difference!)

Subtract Using Fraction Factory & Pattern Blocks Complete the following using both FF & PB for each: Now make the whole the: What would the whole look like using Pattern Blocks? 7/8 – ¾ - would have a whole that was two black pattern blocks side-by-side Use the hexagon as the whole Use patterns blocks to solve and show the solution…

Multiplication of Fractions: Fraction Factory & Pattern Blocks When we multiply, we can think of the multiplication sign as making “groups of” an amount. 3 x 4, we can think of 3 groups of 4. - we can think of 1/3 ‘of’ ½ What do we really know about the product here? Do students know what this means? Will 1/3 of ½ be less than 1? Less than ½? Less than 1/3?

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 Factory & Pattern Blocks: Whole Half What does ¾ of the half look like? What is that in relation to our indicated whole?

Multiplication of Fractions: Fraction Factory & Pattern Blocks

Seeing Division of Fractions: Fraction Factory & Pattern Blocks When we divide, we ask: “How many groups of…?” 12/3 – How many groups of three are in 12? Same with fractions: “How many groups of ¼ are in ½?” We can see that there are 2 groups of ¼ . Therefore, ½ divided by ¼ = 2.

Divide Using Fraction Factory & Pattern Blocks 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 Blocks 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 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 Solving

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

Teaching Through Problem Solving Jeff acquired a recipe from his grandmother which makes 5 dozen cookies. The recipe requires 2 ½ cups of oatmeal, 2 cups of chocolate chips, 2 cups of flour, 2 eggs, 1 cup of butter, 5 ml of baking powder, and 5 ml of salt. A) Adjust the recipe to make 2 dozen cookies. B) Adjust the recipe to make 12 dozen cookies.

Teaching Through Problem Solving Bill’s Snow Plow can plow the snow off the school’s parking lot in 4 hours. Jane’s plowing company can plow the same parking lot in just 3 hours. How long would it take Bill and Jane to plow the school’s parking lot together? Think of the math content involved with this problem Think 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 bowl of granola with ¼ of a bowl of chopped banana and ½ of a bowl of yoghurt. Did one bowl hold all of the ingredients at one time? Explain. Neptune completes 1 ½ turns about its axis each day. How many turns does it complete in 1 week? About 3/4 of the students on the track team are girls. About ¾ of these girls are in grade 8. What fraction of the students on the track team are grade 8 girls?

Mara spent 3/5 of her vacation in British Columbia Mara spent 3/5 of her vacation in British Columbia. While in that province, she spent ½ of her time in Vancouver. What fraction of her vacation did Michaela spend in Vancouver? If her vacation lasted 20 days, how many days did she spend in Vancouver? Jackie used to be on the phone 3 ½ times as much as her brother. Her parents threatened to take away the phone, so she cut down to 2/5 of the time she used to be on the phone. How many times as much as her brother is Jackie now on the phone?

[A] A student is sorting into stacks a room full of food donated by the school for the local food bank. He sorted of it before lunch and then sorted of the remainder before school ended. What part (fraction) of all the food will be left for him to sort after school? [B] 1 3 3 4 Explain the error in Armand’s thinking. Use the manipulatives to support your reasoning.

Eating Candies by Fractions Let’s explore our current resources: Grade 8 textbook – pg. 57 – 91 Eating Candies by Fractions [C] Use manipulatives to solve this problem: Mark ate half of the candies in a bag. Leila ate of what was left. Now there are 11 candies in the bag. How many were in the bag at the start?