At a Glance...

Before Learning (Warm-Up)

Duration: approximately 30 minutes

Beforehand, place a container (for example, basket, trash can, recycling bin) at the front of the class and mark distances 1 metre, 2 metres, 3 metres, and 4 metres away from this container with masking tape, as shown below.

Explain to students that they will be exploring the concept of percents using the game Aim Straight! Explain that this game involves attempting to throw a small bean bag into a container from a specified distance. Present the chart below (Appendix 6.1) in which successful throws will be recorded as they occur. Invite:

• 5 students to make 1 throw each from the 1 metre distance;
• 10 students to make 1 throw each from the 2 metre distance;
• 20 students to make 1 throw each from the 3 metre distance;
• 25 students to make 1 throw each from the 4 metre distance.

Note: The table below provides a sample of possible results; the subsequent explanations presented are based on these results. However, teachers should always work with actual student results.

Example

Distribute a copy of the chart (Appendix 6.1) to each student so that they can record their results. Fill in the 4th column, Result Represented as a Fraction, together.

Ask students to compare results based on the distance from which the throws are made. Discuss how it can be difficult to compare the results, since the denominators differ from one situation to another. Ask them to write the title "Result Represented as a Fraction (in Hundredths)" in the 5th column and to determine, for each fraction in the 4th column, the equivalent fraction in hundredths.

Note: To determine a fraction equivalent in hundredths, students could use proportional reasoning. For example, 4 successful throws out of 5 throws is equivalent in number to 8 successful throws out of 10 throws (if you double the number of throws made, you double the number of successful throws), to 12 successful throws out of 15 throws made, and so on. We can continue this reasoning until 80 successful throws out of 100 throws (20 times more throws made, so 20 times more successful throws).

Example

Then invite students to share their answers and explain their strategies. Ask questions to prompt the comparison of the fractions obtained and the strategies used. For example:

• How did you obtain the fractions in hundredths?
• Have other students used this strategy?
• Are there any who have used a different strategy?
• What are the benefits of writing the results out of 100?
• What do you think the word "percent" means?
• In what situations have you ever seen a percent? (For example, provincial and federal taxes imposed on purchases, discounted prices…)

Lead students to understand that the results in the 5th column, Result Represented as a Fraction (in Hundredths), can also be expressed as percents (out of 100). Point out, for example, that the fraction \(\frac{{80}}{{100}}\) expressed as a percent is written as 80% and reads as "80 percent". Have students write the title "Result Represented as a Percent" in the 6th column along with the appropriate results.

Example

Lead students to recognize that \(\frac{4}{5}\), \(\frac{80}{100}\) and 80% represent the same result, that is, the success of 4 of the 5 throws. Therefore, \(\frac{4}{5}\) = \(\frac{80}{100}\) = 80%.

Note: It is important to clarify the meaning of a percent. A success score expressed as a percent represents the number of successful throws out of 100 trials. For example, a score of 75% means that if 100 throws are made, 75 throws are successful. However, emphasize that a percent score does not imply that exactly 100 throws were made. Any situation that can be represented by a fraction equivalent to \(\frac{75}{100}\) would be possible, for example, 15 successful throws out of 20 throws made \(\frac{15}{20}\) or 150 successful throws out of 200 throws made \(\frac{150}{200}\).

Then read aloud the fractions expressed in hundredths ( 5th column) and ask the students what other ways they know to represent these numbers. Point out that while they have presented the results of the throws in fractions and percents, they could also have done so in decimals, since these three different forms can represent the same result. Establish the relationships between a percent, a fraction with a denominator of 100, and the corresponding decimal number (for example, \(70\;\% \; = \;\frac{{70}}{{100}}\; = \;0,70\)).

Have students write the title Result as a Decimal Number in the 7th column, then fill in the corresponding results.

Example

Consolidation of Learning

Duration approximately 20 minutes

Ask a few teams to present the strategy they used to complete the table for throws made at a distance of 1 metre, as well as to justify their estimation.

Note: During the mathematical exchange, the choice of teams is very important since it is through the sharing that students will compare their strategies for converting (between a percent, a decimal number and a fraction) with those of other students and to solidify their learning.

After each presentation, encourage other students to add to the mathematical exchange by asking questions and making relevant observations. If necessary, assist them by asking questions such as:

• Is this an effective strategy? Why is it effective?
• How would you explain in your own words the strategy that was just presented?
• How have other students used a strategy like this one?
• How else could results have been determined?
• Is this estimation reasonable? Why or why not?
• Did anyone get the same estimation using different reasoning?
• What other estimations were there?

Then, invite 5 students to throw from the point located 1 metre from the container according to a new instruction: 3 students throw following the constraint and 2 without following it. Determine the percent of success and compare it to the percent of success estimated by the students.

Repeat the same procedure for the other 3 series of throws (Appendices 6.3, 6.4 and 6.5).

Note: Emphasize the meaning of a percent and highlight various conversion strategies:

• converting a fraction to a decimal (for example, \(\frac{2}{5}\; = \;\frac{4}{{10}}\; = \;0.4\) or \( 2\;\div\;5\;=\;0.4\));
• converting a decimal number to a decimal fraction (for example, 0.45 = \(\frac{45}{100}\) 0.7 = \(\frac{7}{10}\) );
• convert a fraction to a percent (for example, \(\frac{3}{5}\; = \frac{6}{{10}}\; = \frac{60}{100} = \;60\;\% \) or \(3\; \div \;5\; = \;0.60\; = \;60\;\% \));
• converting a percent to a decimal fraction (for example, \(5\;\% \; = \;\frac{5}{{100}}\); \(77\;\% \; = \frac{{77}}{{100}}\)).

Differentiated Instruction

The activity can be modified to meet the needs of the students.

Follow-Up at Home

Suggest that students invent a game of skill (for example, knocking over a bowling pin by rolling a ball) and invite family members to participate. Specify the number of trials (5, 10, 20, 25 or 50) to obtain fractions that can be easily converted into percents. Have them compile the results in a table similar to the one used in Warm-Up.

The next day, have them present their game and invite their peers to play. Once they have completed a data table, ask them to compare the percent rate of success results with those obtained at home.

Source: Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 165-175.

Before Learning (Warm-Up)

Duration: 15 minutes

The purpose of this warm-up is to enable students to represent a quantity using a mixed number and an improper fraction. Present the following situation, then complete a table like the one below:

On the menu at a class picnic, there were small pizzas cut into 4 equal pieces. A student ate one piece of pizza. What fraction of a pizza did he eat? (\(\frac{1}{4}\) of a pizza)

He then ate a 2nd piece. What fraction of a pizza did he eat in total? (\(2 \times \frac{1}{4}\) of a pizza or \(\frac{2}{4}\)); students will probably say \(\frac{1}{2}\) a pizza)

Then he ate a third piece. What fraction of a pizza did he eat in total? (\(3 \times \frac{1}{4}\) of a pizza or \(\frac{3}{4}\) of a pizza)

Then he ate a 4th. How much pizza did he eat in total? (4 times \(\frac{1}{4}\) of a pizza, \(\frac{4}{4}\) of a pizza or 1 whole pizza)

Finally, he ate a 5th piece of pizza. How much of a pizza did he eat in total? (5 times \(\frac{1}{4}\) of a pizza, \(\frac{5}{4}\) of a pizza or \(1\frac{1}{4}\) of a pizza)

Point out that in the fractions \(\frac{1}{4}\), \(\frac{2}{4}\) and \(\frac{3}{4}\), the numerator is less than the denominator, so these are called proper fractions. Have students note that it is not essential that the numerator of a fraction be less than the denominator. Explain that a fraction whose numerator is greater than its denominator is called an improper fraction and that it represents a quantity greater than one whole. For example, in this situation, when the student ate 5 pieces of pizza in total, we wrote \(\frac{5}{4}\) of a pizza. The fraction \(\frac{5}{4}\) is pronounced "5 fourths" and it means \(\frac{1}{4}\; + \;\frac{1}{4}\; + \;\frac{1}{4}\; + \;\frac{1}{4}\; + \;\frac{1}{4}\; = \;5\; \times \;\frac{1}{4}\; = \;\frac{5}{4}\). Just as \(\frac{2}{4}\) means 2 pieces of everything separated into quarters, \(\frac{5}{4}\). In the same way as \(\frac{2}{4}\) means 2 pieces of a whole that has been separated into fourths, \(\frac{5}{4}\) means 5 pieces of a whole that has been separated into fourths.

Explain that a quantity greater than the whole can also be represented by a mixed number. Specify that \(1\frac{1}{4}\) is read "1 and 1 fourth", that the number 1 represents the number of wholes and that the fraction \(\frac{1}{4}\) represents the fractional part of an equivalent whole. In this situation, the student has therefore eaten a whole pizza (\(\frac{4}{4}\) of pizza) and \(\frac{1}{4}\) of another pizza of the same size.

Specify that the improper fraction \(\frac{5}{4}\) and the mixed number \(1\frac{1}{4}\) are 2 ways of representing the same quantity. At this stage, it is important to simply present the terminology of these types of numbers without focusing on their connections since students will explore them a little later in this learning situation.

Briefly review the concept of equivalent fractions, pointing out that these are different ways of representing the same quantity. For example, mention that \(\frac{2}{4}\) of sandwich is equivalent to \(\frac{1}{2}\) of sandwich, that \(\frac{1}{3} \) of sandwich is the equivalent of \(\frac{2}{6}\) or \(\frac{4}{{12}}\) of sandwich and show how we can confirm that they represent an equivalent amount.

Present Appendix 5.1A, read the scenario, and explain the task to be performed, that is to establish, using improper fractions and mixed numbers, the quantity that each member of each group received in order to determine whether the distribution of sandwiches was fair.

Ensure that students have understood the task by asking questions such as:

• Who would like to explain the situation in your own words? What happened at the picnic?
• Why might we think each student did not receive the same amount?
• Who can explain the task in their own words?
• What is a mixed number?
• What is an improper fraction?

Consolidation of Learning

Duration: 30 minutes

Invite the selected teams to present their strategies. Ensure that each team identifies the question or statement being presented by number, by reading the text or by writing it on the board. Invite students to intervene and add comments.

Facilitate the discussion by asking questions.

For example

• How can we explain this team's approach?
• Which other teams have you seen who used this or a similar strategy?
• These 2 teams express different answers for the same problem. How is this possible? (Show the equivalence of the various representations: the equivalent fractions as well as the equivalence between the mixed number and the improper fraction.)
• To solve the problem in statement 1, why did they divide by 6?
• How did you go about finding the quantity of Abdul's group even though the number of people in it was not mentioned?
• What do they mean by "we have done the division" when we are talking about a fraction?
• Why do you say that \(1\frac{4}{6}\) is equivalent to \(1\frac{2}{3}\)?
• Some groups speak mainly of sixths while others refer to thirds. What is the difference?
• In order to solve the cantaloupe problem, why are you talking about multiplication, division and remainder?
• What is the difference between Amon's statement ("Each member of the group received 5 thirds of a sandwich.") and Abdul's statement ("It was delicious. There were 8 sandwiches for John, Peter and me.")? [Point out that the 1st statement presents the fraction as part of a whole sandwich \(\frac{5}{3}\) while the 2nd presents a sharing situation (8 sandwiches for 3 people: so 8 ÷ 3 can be represented by \(\frac{8}{3}\)

Encourage students to use clear arguments, precise vocabulary, and logic terminology. Appendix 5.3 provides examples of student reasoning, mathematical arguments, and solution strategies.

Highlight effective strategies for converting an improper fraction into a mixed number and vice versa by establishing connections between semi-concrete representations and the meaning of the numbers that compose them (the wholes, the fractional parts, the division, the remainder ). It is more important for students to understand the meaning of numbers and the operations performed than to memorize the steps to follow.

Differentiated Instruction

The activity can be modified to meet the needs of the students.

Follow-Up at Home

At home, students can use concrete and semi-concrete representations to solve 1 or 2 problems related to mixed numbers and improper fractions.

Examples of problems

• If the rectangle below corresponds to \(\frac{5}{4}\) of a whole, draw the whole.

• Peter covers one fourth of the route planned for bicycle race 7 times during training. The odometer on his bicycle indicates that he has ridden a total of 14 km. How long is the race's route?

• Knowing that on \(1\frac{2}{3}\) of a sheet of stickers there are 35 stickers, how many stickers are there on the whole sheet?
• I have 3 small strips of paper of the same length. When I place 2 of the strips end to end and \(\frac{2}{3}\) of the 3rd, I get a strip of paper measuring 240 cm. How long is one of the small strips?

• After a party, there are 2 whole rectangular cakes and part of a 3rd. How many cakes are left?

Source: Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 147-157.

Before Learning (Warm-Up)

Duration: approximately 15 minutes

Ask students if they have ever attended a show, play, concert, or large gathering. If so, ask if they remember the approximate number of people in the audience. Discuss with them if they simply guessed the number or if they used an estimation strategy.

Review the words estimation, rounding, more than, less than, about, between, approximate.

Present the following scenario to students:

A theatre company would like to rent a large room to perform their new play. The company's management must determine the price of tickets. After discussion, company members suggested that the price be set at $19.75 per ticket.

Share Appendix 5.2 with students just long enough to given them a sense of the size of the room, but not enough time for them to count, for example, the number of seats per row.

Ask them to write down the number of seats they think the room contains. Later, during the follow-up discussion, they can compare this number with a number they obtained using an estimation strategy. This comparison serves to reaffirm the benefits of knowing and using such strategies, especially when dealing with large numbers.

Note: The number of seats in the room is not specified. It is important in this situation to focus on the strategies for and concept of estimation and not to pay particular attention to the exact number.

Display Appendix 5.1 on the interactive whiteboard or on a large sheet of paper and present the following problem to students.

The theatre company would like to rent the room I just showed you to perform their play. They are budgeting $22 230 for all expenses related to this performance (for example, hall rental, lighting, sound, program). If the company's management accepts the suggestion to set the price of a ticket at $19.75 and if all the tickets are sold, will the company make a profit? If so, approximately what will it be? If not, how much will the loss be?

Ensure that students understand the problem by asking them to explain in their own words the situation and the meaning of the terms profit and loss.

Note: It is important not to direct students too much; they should be allowed to find personal problem-solving and estimation strategies.

Consolidation of Learning

Duration: approximately 30 minutes

First, ask some of the selected teams to take turns presenting the strategy they used to estimate the number of seats the room contains. Encourage students to provide specific justifications using clear and convincing mathematical arguments. For example, "Since the top 3 sections, without the dressing rooms, contain about the same number of seats, we estimated the number of seats in a section and multiplied it by 3."

In discussions, it is important to highlight the different estimation strategies used and compare them by asking questions such as:

• Why does your estimation strategy work?
• Would this strategy work with a different floor plan?
• How have other teams used a similar strategy?
• What other strategies would achieve similar results?
• Which strategy do you think is most effective? Why?
• Are there large differences between the estimations obtained by different teams? (Point out that despite the diversity of strategies used, several teams obtained very similar and equally valid estimations.)

Then ask members of the selected teams to reveal the number of seats they wrote down on a piece of paper when they first saw the floor plan. To encourage all students to think about the difference between the reasonableness of a quantity obtained by guessing and the reasonableness of a quantity obtained by estimating, ask questions such as:

• When you first saw the plan, what number did you write on your piece of paper? Was this number an estimation? (It was not an estimation because the time allotted did not allow for a proper estimation to be obtained.)
• Is a quantity obtained by estimating more reasonable than a quantity obtained by guessing? Why? (Yes, a quantity obtained by estimating is generally more reasonable because it is not random; it is based on a quick, but thoughtful calculation.)

Second, ask the other selected teams to present the calculations they made to determine whether the theatre company will make a profit or a loss. Emphasize the different calculation strategies used and how to check the reasonableness of their answers. Emphasize the importance of rounding numbers to make the calculations easier.

Ask questions such as:

• How did you determine whether the company will make a profit or a loss?
• Why did the teams get different answers? Is this acceptable?
• How do you know your answer is reasonable?
• Is the theatre company likely to experience a greater or lesser profit (or loss) than the amount you have determined?
• What might be a good reason for using rounding in this scenario?

Throughout the discussion, invite other students to respond to each presentation and share their observations. If necessary, ask them questions such as:

• How can we use rounding and estimation to make calculations easier?
• In what type of context is it OK to estimate?
• In what contexts might we round up?
• What would you write about in a math journal (group or personal)?

Ensure that students understand that it is acceptable to estimate and round in contexts where an approximate, not an exact, answer is sought.

To consolidate estimation strategies, take students to a building in their community that can accommodate a large number of people (for example, arena, theatre, church) and ask them to estimate the number of seats. This exercise can also be done using a virtual tour of a theatre or concert hall on the Internet.

Differentiated Instruction

The activity can be modified to meet the needs of the students.