Learning Story for Grade 7
Number and Computational Fluency
Given that by Grade 6 students have explored whole numbers to billions, in Grade 7 students no longer explore whole numbers. The exception is with extending computational fluency with multiplication and division. By Grade 7 students should be able to recall their multiplication facts and related division facts. For students who do not have such recall, their fact development should be strategy-focused. These same strategies (eg. double-double to multiply by 4) are a big part of what “extending computational fluency” means, as these strategies can be extended to larger whole numbers, decimals, and integers.
In Grade 7 students complete their work with decimal operations that began in Grade 4. They first expand their understanding with multiplying and dividing to include multiplying and dividing by decimal numbers (in Grade 6 they multiplied and divided decimals by whole numbers). They then apply their understanding about order of operations (from Grade 6) to decimal operations.
Speaking of decimals, students in Grade 7 explore more deeply the relationships between decimals and fractions by determining equivalencies for both terminating and repeating decimals, as well as comparing and ordering decimals and fractions. The equivalencies also extend to percents (eg. 1/4 = 0.25 = 25%). In Grade 7 students also applying their learning about percents from Grade 6 to explore financial literacy situations involving sales tax, discounts, and tips.
Grade 7 is the first year that students work with integers. Though in previous curricula, students explored the concept of integers in Grade 6, this is no longer the case. So before doing operations with fractions, students will need to explore the concept of an integer. They then make sense of the operations with integers through concrete (eg. two-sided counters), pictorial (eg. number line), and symbolic representations.
Key Concepts
Operations with Integers
- Throughout unit:
- Extend an understanding of whole number operations to integers
- Explore and make sense of integers concretely, pictorially, and symbolically (eg. 2-sided counters, number line)
- Representing integers
- Comparing and ordering integers
- Adding and subtracting integers
- Multiplying and dividing integers
- Order of operations with integers
- Solve contextual problems involving integers
Operations with Decimals
- Throughout unit:
- Extend an understanding of whole number operations to decimals
- Explore and make sense of decimals concretely, pictorially, and symbolically (eg. base ten blocks, area model)
- Multiplying and dividing decimals
- Order of operations with decimals
Relationships between Fractions, Decimals, and Percents
- Explore the relationships between fractions (terminating and repeating), decimals and percents concretely, pictorially, and symbolically (eg: hundred grid, number line)
- Compare and order fractions and decimals (eg. using equivalencies, benchmarks)
Financial Percents
- Use relationships between percents, decimals and fractions to solve percent problems involving finance (eg: sales tax, discounts, tips)
Key Number Concept 1: Operations with Integers
Overview
In Grade 7, students are introduced to both operations with integers and integer concepts. Before adding, subtracting, multiplying, and dividing integers, students first learn that negative whole numbers exist and have contextual and abstract meaning. Students come to understand that all integers can be placed on number lines that extend to the left of or below zero. Students also build and represent integers using two-sided counters. This model introduces the idea of “zero pairs,” which is helpful when operating with integers. In addition to representing integers, students learn to compare and order pairs or sets of integers; zero becomes a key benchmark.
When exploring and making sense of operations with integers, students see that the fundamental meanings of each operation extend to these “new” numbers. Flexibility is essential. For example, (-6 ) – (-3) can be thought of as removing counters (“takeaway”) whereas (+6 ) – (-3) can be thought of as determining how far apart the two are on a number line (“difference”); (-6) ÷ (+3) can be thought of as distributing six negative counters equally between three groups (“sharing”) whereas (-6) ÷ (-3) can be thought of as counting jumps of -3 required to land on -6 (“measuring”).
In addition to using counters and number lines as helpful representations, a context such as a hot air balloon with sand bags can be helpful. Hot air representing positive integers and sandbags represent negative integers.
Number Sense Foundations:
The following concepts and competencies are foundational in supporting understanding of operations with mixed numbers.:
- Understanding of the fundamental meanings of each of the four operations applied to whole numbers. For example:
- Addition is most commonly thought of as the combining of two (or more) collections into a single collection (e.g. “5 counters and 2 counters make 7 counters”). Addition can also be interpreted as the changing of an original amount (or length) by a given amount (or length) (e.g., “a jump forward of 2 spaces from 5 lands at 7”). Finally addition can be thought of as the sum of the parts that make up a whole.
- Subtraction can be thought of as removing a number (“takeaway”), or it can be thought of as determining how far apart two numbers are (“difference”).
- Multiplication has many meanings, one of which is counting a total of groups of equal size.
- As the inverse of multiplication, one can think of division in the context of groups of a number. In one meaning, it answers how many are in each group (“sharing”). In another meaning, it answers how many groups (“measuring”).
- Computational strategies for adding, subtraction, multiplying, and dividing.
- Familiarity with the facts for the four operations.
- Understanding order of operations.
Progression:
- Understand that the set of integers is composed of zero as well as positive and negative whole numbers; negative integers are counted moving left or down from zero.
- Opposite numbers are the same distance from zero on a number line.
- Understand that (+1) + (-1) = 0 (“zero pair”)
- Opposite numbers form a set of zero pairs.
- Apply zero pairs to represent integers in different ways
- For example, +3 can be 🟡🟡🟡 but it can also be 🟡🟡🟡🟡🟡🔴🔴
- Compare and order a set of integers on a number line.
- Develop an understanding of adding integers by combining both numbers and accounting for zero pairs. For example, for 2 + (-5):
- Using counters:
🟡🟡
🔴🔴🔴🔴🔴 Removing two zero pairs shows the answer is (-3). - Using a number line:
- Using counters:
- Develop an understanding of subtracting integers by applying the meaning of removal or difference. For example:
- Removal:
- 5 – 2 = 3
- (-5) – (-2) = (-3)
- Difference:
- 5 – (-2) = 7
- (–5) – 2 = (-7)
- The difference models can also be thought of as an adding-on In the first one, we need to add 7 to go from (-2) to 5. In the second one, we need to subtract 7 from 2 to get to (-5).
- Removal can also be applied to the second set of problems above through the use of zero pairs.
- 5 – (-2) = 7
- (–5) – 2 = (-7)
- Develop an understanding of multiplying integers.
- When multiplying a positive integer by a negative integer, consider the total of all of the groups. For example 4 x (-3) = (-12)
- When multiplying a negative integer by a positive integer, because of the commutativity of multiplication, the positive number can be considered the number of groups and the above strategies could be used.
- When multiplying a negative integer by a negative integer:
- One can explore a pattern to recognize that the product of two negative integers is positive. For example:
- One can remove groups of negative integers.
For example (-4) x (-3) = 12 - One can also think of multiplying by (-1) as making a number opposite. So (-4) x (-3) = (-1) x 4 x (-3) = (-1) x (-12). The opposite of (-12) is 12, so (-4) x (-3) = 12
- Using arrays on a coordinate grid can also have some meaning for students:
- When multiplying a positive integer by a negative integer, consider the total of all of the groups. For example 4 x (-3) = (-12)
- Develop an understanding of dividing integers.
- One way to do this is to relate it to multiplication. For example, for (-12) ÷ 3, one can think of what to multiply by 3 to get (-12).
- One can also apply the meanings of division. For example:
- (-12) ÷ 3 can be thought of as distributing 12 negative counters equally into 3 groups of (-4).
- (-12) ÷ (-3) can be thought of as how many (-3)s it takes to make (-12)
- Apply the order of operations to integers.
- Throughout above, apply their understanding of operations with integers to solve contextual problems.
- Removal:
Sample Week at a Glance:
Prior to this week, students have explored the concept of an integer using both two-sided counters and a number line. They recognize that opposite numbers are the same distance from zero on the number line. They have developed an understanding of a zero pair, as well as how to compare and order integers.
Focus: Adding Integers using counters, a number line, and symbolically
- Before: Adding numbers using counters and a number line
- Do a Same But Different routine with:
- 6 + 5
- 7 + 4
- Ask them to model the similarities and differences using counters or a number line.
- Counters:
The numbers are different but they both make 11. We can see how the first one connects to the second one by moving one counter from the 5 to the 6 to make 7 and 4. - Number line:
We can see the same thing with the number line.
- Counters:
- Ask what would change if we changed the second to be opposite (negative). Focus on just the second sum, i.e., 7 + (-4).
- Counters:
When we re-arrange the counters we can make 4 zero pairs. So the answer is positive 3. - Number Line:
Instead of moving forward from 7 we do the opposite and go backward to see the answer is positive 3.
Note: In this diagram, an arrow is used to show the first number. You may decide that only showing one arrow, starting from the first number, is a representation that is cleaner and simpler to use.
- Counters:
- Do a Same But Different routine with:
- During: Adding integers using counters, a number line, and symbolically
- Have students work in pairs.
- Provide students with a tool for generating random numbers. For example:
- 2 dice (ideally 10 or more sides). Roll to get the two numbers, then model and solve 3 expressions (pos + neg, neg + pos, neg + neg). Spinners could be used in a similar fashion.
- Deck of cards. Draw 2 cards. Black cards are positive and red cards are negative.
- Ask them to generate several addition questions. For each one:
- Model using counters or a number line, but overall they should use both models at some point.
- Make sense of the answer, and record the process symbolically.
- After:
- Have some pairs share their questions and solutions. Be sure to include the following cases:
- negative + negative, for example:
- positive + negative with a positive answer, for example:
- positive + negative with a negative answer, for example:
- negative + negative, for example:
- Have a class discussion about what things they noticed in general (i.e., generalizations). Sample responses include:
- “If you add two negatives you get a negative answer.”
- “If you add a positive and a negative, you can make zero pairs. If you have more positives, the answer will be positive, and the answer will be negative if you have more negatives.”
- Have students practice with some more questions. Encourage them to try figuring out answers without using models if they can (i.e, answer them symbolically)
- Have students extend their understanding by adding:
- larger numbers, e.g., (-27) + 18
- 3 or more integers, e.g., (-7) + 12 + (-3)
- Have some pairs share their questions and solutions. Be sure to include the following cases:
Focus: Subtracting integers using counters and symbolically
Because using number lines for subtraction is a bit more abstract than using counters, it is advisable to explore using number lines the following day instead of being combined with using counters.
- Before: Meanings of Subtraction
- Write a subtraction equation, such as 7 – 5 = 2
- Have the class turn and talk about how they could model the solution to this equation using counters.
- Have the students share their ideas. Consolidate the ideas into two key strategies for subtraction:
- Take-away (removal): If I take 5 counters away from 7 counters, I have 2 counters left.
- Difference (comparing): 7 counters is 2 more counters than 5 counters.
- Take-away (removal): If I take 5 counters away from 7 counters, I have 2 counters left.
- Discuss which meaning made more sense for this equation. In this case, because 5 was close to 7, the meaning of difference
- During: Subtracting integers using counters
- Have students work in pairs.
- As done the previous day, provide students with a tool for generating random numbers (see above for suggestions).
- Ask them to generate several subtraction questions. For each one:
- Decide whether to use a take-away meaning of subtraction, or a difference meaning, then model the questions using counters. When using counters, a take-away meaning is more likely to make sense.
- Advise them that they may often need to use zero pairs.
- Make sense of the answer, and record the process symbolically.
- Keep a record of their equations and solutions as they will refer to them again the next day.
- As you circulate among the groups, you may need to prompt for the use of zero pairs where appropriate as they may not realize how they can be helpful in these situations.
- After:
- Have some pairs share their questions and solutions. Be sure to include the following cases. The A and B notation is for teacher guidance only.
- A – B with B > A
To take 5 away from 2, we need to add three zero pairs.
- (-A) – B
To take 2 away from (-5), we need to add two zero pairs.
- A – (-B)
Similarly, to take (-2) away from 5, we need to add two zero pairs.
- (-A) – (-B) with A > B
We can just take (-2) from (-5) without any zero pairs.
- (-A) – (-B) with B > A
To take (-5) away from (-2), we need to add three zero pairs.
- A – B with B > A
- All of the above were shown using a take-away meaning of subtraction. It would be good to debrief a couple of examples that use the difference meaning. For example:
The difference between (-2) and (-5) is 3 negatives, so it’s (-3). - Ask students to record any generalizations they have made so far, but do not have a class discussion about them until after they have done the next day’s activity.
- Have students practice with some more questions. Encourage them to try figuring out answers without using models if they can (i.e, answer them symbolically)
- Have students extend their understanding by:
- subtracting larger numbers, e.g., (-15) – (-13)
- mixing addition and subtraction, e.g., (-7) + 5 – (-3)
- Have some pairs share their questions and solutions. Be sure to include the following cases. The A and B notation is for teacher guidance only.
Focus: Subtracting integers using a number line, and symbolically
- Before: Do a Clothesline routine (This is about using the difference meaning to subtract integers on a number line.)
- Ask a student to place a zero in the middle of the number line (i.e., clothesline)
- For each of the following problems:
- Ask one student to place one number, and another student to place the second number.
- Invite anyone to shift one of the numbers if they can justify why.
- Write a subtraction expression involving the two numbers.
- Ask the class to think about the difference between the two numbers, and what answer makes sense.
- Explain that when looking at A – B, we are looking for the difference from B to A. In other words, how do we get from B to A?
- Explore these problems:
- 7 – 2
- 2 – 7
- 4 – (-3)
- (-2) – 4
- (-4) – (-6)
- You may wish to try some other ones as well.
- 7 – 2
- During: Subtracting integers using a number line
- Choose a problem from the previous day of the form A – B with B > A. Draw a pictorial representation of the solution using counters. Ask students to turn to a partner to think about how this question could be modeled on a number line. Discuss their ideas. For example:
On a number line, we can start at 2 and then go back 5 to see the answer is (-3).
Note: When adding integers, an arrow was used to show each addend. For subtraction, it may be clearer to have just one arrow that starts from the first number. - Choose another problem from the previous day, this time of the form
A – (-B). Draw a pictorial representation of the solution using counters. Ask students to turn to a partner to think about how this question could be modeled on a number line. Note that this is a challenging concept for students. Discuss their ideas. For example:
Normally when we subtract, we move to the left. However, a negative integer is opposite a positive integer, so we do the opposite. Drawing the arrow underneath the number line is a nice way to indicate subtraction in the opposite direction. - Have students work in the same pairs as the previous day. Ask them to look at their examples using counters and think about how they could model the same problems using a number line. Ask them to record the process symbolically.
- As you circulate among the groups, you may need to help them further on how to make sense of representing subtracting a negative integer.
- Choose a problem from the previous day of the form A – B with B > A. Draw a pictorial representation of the solution using counters. Ask students to turn to a partner to think about how this question could be modeled on a number line. Discuss their ideas. For example:
- After:
- Have some pairs share their questions and solutions. In addition to the 2 cases shared in the Before (A – B with B > A and A – (-B)), be sure to include the following cases.
- (-A) – B
(-5) – 2 = (-7)
- (-A) – (-B)
(-5) – (-2) = (-3)
- (-A) – (-B) with B > A
(-2) – (-5) = 3
- (-A) – B
- Have a class discussion about what things they noticed in general (i.e., generalizations). Sample responses include:
- “If you subtract a positive integer, you move to the left.”
- “If you subtract a negative integer, you move to the right.”
- Have students share their thoughts on which model they prefer, using counters or a number line. Some may prefer counters because they can take action on the operation and number lines are more abstract. Others may prefer number lines. Finally others will be comfortable working doing it symbolically. Ultimately the models are used to make sense of the operation, and to support the development of doing the questions symbolically.
- Have students practice with some more questions. Encourage them to try figuring out answers without using models if they can (i.e, answer them symbolically)
- Have students extend their understanding by:
- subtracting larger numbers, e.g., (-15) – (-13)
- mixing addition and subtraction, e.g., (-7) + 5 – (-3)
- You may wish to discuss other visualizations of adding and subtracting integers. For example:
- Walking on the number line – which direction do you end up walking?
- Have some pairs share their questions and solutions. In addition to the 2 cases shared in the Before (A – B with B > A and A – (-B)), be sure to include the following cases.
Operation | Face direction | Walking direction |
Add a positive | positive | forward |
Add a negative | negative | forward |
Subtract a positive | positive | backward |
Subtract a negative | negative | backward |
- Hot air balloon with sandbags – does the balloon go up or down?
- Add a positive: add more hot air
- Add a negative: add more sandbags
- Subtract a positive: remove hot air
- Subtract a negative: remove sandbags
Focus: Applying whole number strategies to adding integers
- Before: Number Talk with adding whole numbers
- Do a Number Talk with 35 + 28
- Strategies shared may include:
- Adding on:
- Place Value:
30 + 20 = 50
5 + 8 = 13
50 + 13 = 63 - Making a Ten:
- Compensation:
35 + 30 – 2 = 65 – 2 = 63
- Adding on:
- During: Number Talk with adding integers
- Do a Number Talk with (-18) + 23. Encourage students to think about which whole number strategies make sense for this question.
- After:
- Strategies shared may include:
- Adding on (or Making a Zero):
- Place Value:
-10 + 20 = 10
-8 + 3 = -5
10 – 5 = 5 - Making a Ten:
, or - Change to subtraction, then use a subtraction strategy:
(-18) + 23 = 23 – 18 = 23 – 13 – 5 = 10 – 5 = 5
- Adding on (or Making a Zero):
- There may be other strategies shared as well.
- As with any number talk, ask students to reflect on which strategy makes the most sense to them, and discuss what connections they may see between the strategies.
- Have students practice other addition questions involving two-digit integers. Advise them that different questions lend themselves to different strategies, so they should think about a strategy that makes sense to them for each question.
- Strategies shared may include:
Focus: Applying whole number strategies to subtracting integers
- Before: Number Talk with subtracting whole numbers
- Do a Number Talk with 72 – 28
- Strategies shared may include:
- Counting back:
- Counting up:
- Compensation:
72 – 30 + 2 = 42 + 2 = 44 - Same difference:
72 – 28 = 74 – 30 = 44
- Counting back:
- During: Number Talk with subtracting integers
- Do a Number Talk with 24 – (-12). Encourage students to think about which whole number strategies make sense for this question.
- After:
- Strategies shared may include:
- Counting up (making a zero):
- Same difference:
- Change to addition, then use an addition strategy:
24 – (-12) = 24 + 12 = 24 + 10 + 2 = 34 + 2 = 36
- Counting up (making a zero):
- There may be other strategies shared as well. In the above question, because a negative number is being subtracted, strategies which focus on the difference meaning of subtraction make more sense than the take-away meaning.
- As with any number talk, ask students to reflect on which strategy makes the most sense to them, and discuss what connections they may see between the strategies.
- As students have noticed some generalizations this week when adding and subtracting integers, many integer questions can be expressed as whole number questions. For example:
- (-14) + (-22) is like 14 + 22, but negative
- (-14) + 22 is equivalent to 22 – 14
- 22 – (-14) is equivalent to 22 + 14
- Have students practice other subtraction questions involving two-digit integers. Advise them that different questions lend themselves to different strategies, so they should think about a strategy that makes sense to them for each question.
- Strategies shared may include:
Next week, a good Monday plan would be to use centres to do further practice with adding and subtracting integers. Possible stations could include integer war (with cards), open middle tasks, solving contextual problems, race to 100, and teacher support. Next students would explore multiplying integers using a variety of representations, including noticing patterns. This would be followed by making sense of dividing integers which relates closely to multiplying. Finally students would engage in doing order of operations with integers.
Suggestions for Assessment
By the end of Grade 7, students will be able to apply a variety of strategies and representations to add, subtract, multiply, and divide integers. Rather than remember a set of rules to follow (e.g., two negatives make a positive), they will understand the process of each operation. Finally they will be able to use order of operations with integers.
Suggested Links and Resources
Surrey Schools Videos for Parents Series:
- Representing, Adding, and Subtracting Integers
- Multiplying and Dividing Integers, and Order of Operations
Bucket of Zero
Mathigon: Integer Addition using Algebra Tiles
Make Math Moments: Integers (select Integers as the topic, or scroll down)
Key Number Concept 2: Operations with Decimals
Overview
The learning standard for operations with decimals in Grade 7 mentions all four operations, but one needs to interpret this in light of learning in prior grades. Understanding adding and subtracting decimals was developed in Grades 4 (to hundredths) and 5 (to thousandths). In Grade 7, students need to activate this prior knowledge so that they can add and subtract decimals as part of using the order of operations with decimals. Adding and subtracting decimals is also needed during the process of figuring out products and quotients with decimals.
Multiplying and dividing decimals begins in Grade 6, but to what extent is not clear in the curriculum. A safe expectation is that students have explored these concepts to the same extent as what was done in the prior curriculum (WNCP) which is to multiply and divide decimals by a single-digit whole number. In Grade 7, students will then make sense of multiplying decimals by decimals using a variety of strategies. Finally students will apply the order of operations to decimals.
To make sense of any operation with decimal numbers, students can extend their understanding with whole numbers by using similar strategies. For example, to multiply 0.9 by 0.4, a student may multiply 0.4 by 1, then compensate by removing a tenth of 0.4:.
Using whole numbers and then estimation is another effective strategy. With the above example, a student may use the fact that , then place the decimal using estimation. Estimating by thinking is close to , so the answer should be 0.36 rather than 3.6 or 0.036.
As in prior grades, students should make sense of these concepts at a symbolic level that is developed from concrete and pictorial models. Area models provide a very effective visualization for multiplying and dividing decimals.
Throughout this unit, using contextual problems can be helpful both for developing, extending, and applying these concepts.
Number Sense Foundations:
The following concepts and competencies are foundational in understanding operations with decimal numbers in grade 7:
- Computational strategies with whole numbers, for example decomposition, compensation, and using benchmarks.
- The operations are related to each other (eg. one may use multiplication to figure out a division problem).
- Multiplying and dividing using an area model (using base ten blocks, pictorially)
- Place value and how the place values are related to each other (eg. 100 is ten 10s and 10 is a tenth of 100).
- The concept of a decimal as parts of a whole (tenths, hundredths, thousandths)
- Adding and subtracting decimals to thousandths
- Multiplying and dividing a decimal number by a single-digit whole number
- Estimation strategies (eg: front-end estimation, using benchmarks)
Progression:
- Adding & Subtracting decimals
- Review adding and subtracting decimals, including using computational and mental math strategies
- Multiplying decimals
- Review multiplying a decimal number by a single-digit whole number (pictorially and symbolically), then extend to multiplying by any whole number.
- Explore multiplying a decimal number by 0.1, 0.01, and 0.001
- Develop and apply understanding of multiplying decimal numbers together using:
- Area models
- Whole number products and then estimation
- Computational strategies
- Contextual problems
- Calculators are appropriate to use when multiplying by a number that is more than two digits, (eg. when multiplying by 2.36)
- Dividing decimals
- Connect division to multiplication
- Review dividing a decimal number by a single-digit whole number (pictorially and symbolically), then extend to dividing a whole number by a decimal.
- Recognizing and using different meanings of division is helpful to distinguish these two situations.
- For dividing by a whole number, it may be helpful to think about sharing (dividing into groups of)
- For dividing by a decimal number, it may be helpful to think about measuring (how many times does the decimal go into the whole number)
- Recognizing and using different meanings of division is helpful to distinguish these two situations.
- Explore dividing a decimal number by 0.1, 0.01, and 0.001
- Develop and apply understanding of dividing decimal numbers using:
- Area models – connected to multiplication
- Whole number quotients and then estimation
- Computational strategies
- Contextual problems
- Calculators are appropriate when dividing by a number that is more than one-digit (eg. dividing by 0.27)
- Apply the order of operations using decimal numbers.
Sample Week at a Glance
Prior to this week, students would have reviewed multiplying a decimal number by a single-digit whole number (from Grade 6), and then extended this understanding to multiply a decimal number by larger whole numbers. Students would also have explored multiplying a number (whole number or decimal) by 0.1, 0.01, and 0.001 by exploring the effect this has on place value.
Focus: Multiplying two decimal numbers (tenths)
- Before: Multiplying decimals by 0.5 by connecting it to half.
- What is 0.5 x 0.6? What about 0.5 x 0.9?
- How do these connect to 5 x 6 and 5 x 9?
- During: Explore multiplying tenths with an area model using base ten blocks
- Have students try different choices of decimals (tenths), eg. 0.4 x 0.6
- After: Share and discuss the ideas that emerged. Key ideas include:
- The dimensions of each rectangle represent tenths and the area represents hundredths.
- The product of tenths is connected to the product of whole numbers (basic multiplication facts), eg. 4 x 6 = 24, so 0.4 x 0.6 = 0.24 (multiplying tenths produces hundredths)
Focus: Connect multiplying decimals to multiplying whole numbers (2-digits) using an area model (concretely or pictorially)
- Before: Ask students to model (area model using hundred grid or base ten blocks) a product of 2 two-digit whole numbers, eg. 32 x 26.
- During: Using the same model, what if the factors were using tenths, e.g. 3.2 x 2.6?
- Students create and model other questions, recording their process symbolically (pictorially is challenging given the level of detail – a simpler pictorial model will be explored next class).
- After:
- Share and discuss some of the students’ examples.
- Draw connections between the model and its symbolic representation (partial products).
- Discuss how the decimal products compare to whole number products. Ask how estimation may be used to solve these problems. Eg. 3.2 x 2.6 is about 3 x 3 = 9.
32 x 26 = 832, so I need to place the decimal point at 8.32 to be closest to the estimate. - Ask students to create and solve a couple more on their own (their choice, or use dice to generate random numbers). They may decide to use an area model with decimals, or with whole numbers and then estimate. Some may feel ready to do partial products symbolically without modeling first.
Focus: Simplify pictorial model (rectangle model) and connect to symbolic process.
- Before:
- What is the same? What is different?
- If it does not come up when discussing the above, ask how do the areas of these two rectangles compare (without computing them). They should notice that the rectangle on the left has an area that is ten times greater. This example illustrates how they can apply what they’ve learned about multiplying with tenths to multiplying with hundredths.
- During:
- Show the rectangle model below and ask students to figure out the product. Some may need to be prompted to think about what it would look like if built using base ten blocks.
- After:
- Discuss as a class how to fill in each area. Write each product. Then evaluate each product, and then add to determine the overall product of 23.1
- Model this process symbolically. For example:
- Discuss how this process is the same as multiplying whole numbers. And one could multiply 55 by 42 to get 2310, then use estimation (eg. 6 x 4 = 24) to place the decimal place to make 23.1
- Ask students to try two or three more problems. Encourage them to solve them symbolically (either with decimals or using whole numbers and estimation), but they should feel free to use a rectangle diagram if needed.
- Ask students to think about and try a problem such as: 2.6 x 0.48
Focus: Practice to solidify and extend their understanding
- Before: Same but Different
- What is the same? What is different?
- 2 x 0.34 0.52 x 3.4
- During: Math Workshop (Students should bring paper or mini-whiteboards with them. Have base ten blocks available for those who need them. Encourage students to collaborate at each station.)
- What is the same? What is different?
- 3 x 2.1 4.1 x 2.3
- Roll 4 dice. Where would you place the dice to make the greatest product?
- What is the same? What is different?
- This could also be a partner game. Who can get the greatest product?
- Open question: What decimal numbers could you multiply to get as close to 10 as possible?
- Estimation station:
- Given that 63 x 47 = 2961, use estimation to determine:
- 3 x 4.7 63 x 4.7 0.63 x 47 0.63 x 4.7
- Given that 124 x 56 = 6944, what are some products with decimals that you could write?
- Given that 63 x 47 = 2961, use estimation to determine:
- Teacher led small group instruction: multiplying two-digit decimal numbers (tenths or hundredths) pictorially and symbolically
- After: students sharing what they did, and what they learned. Discuss any challenges they faced and what they feel they still need to work on.
Focus: Applying other whole number multiplication strategies to multiply decimals.
- Before: Number Talk
- 18 x 5
- Possible strategies include:
- Double-half: 18 x 5 = 9 x 10 = 90
- Friendly number and compensate: 20 x 5 – 2 x 5 = 100 – 10 = 90
- x5 Strategy (Multiply by 10 and divide by 2):
- Place-value partition: 10 x 5 + 8 x 5 = 50 + 40 = 90
- Use a known fact: 2 x (9 x 5) = 2 x 45 = 90
- During:
- Ask which of these strategies could we use to multiply 1.8 x 0.5? Are there any other strategies we could use?
- Double-half: 1.8 x 0.5 = 0.9 x 1 = 0.9
- Friendly number and compensate: 2 x 0.5 – 0.2 x 0.5 = 1 – 0.1 = 0.9
- Place-value partition: 1 x 0.5 + 0.8 x 0.5 = 0.5 + 0.4 = 0.9
- Ask and discuss: Are there any other strategies we could use?
- 5 is equivalent to a half, and half of 1.8 is 0.9
- Ask which of these strategies could we use to multiply 1.8 x 0.5? Are there any other strategies we could use?
- After:
- Share and discuss their strategies. For example:
- Double-half: 1.8 x 0.5 = 0.9 x 1 = 0.9
- Friendly number and compensate: 2 x 0.5 – 0.2 x 0.5 = 1 – 0.1 = 0.9
- Place-value partition: 1 x 0.5 + 0.8 x 0.5 = 0.5 + 0.4 = 0.9
- 5 is equivalent to a half, and half of 1.8 is 0.9
- Journal: Give them 3 or 4 other problems and ask them to choose which ONE strategy they would choose to figure out each product, and why they chose that strategy.
- Choose problems that lend themselves well to a particular strategy, then vary the strategies.
- The students may include the strategies from earlier in the week as well.
- Share and discuss their strategies. For example:
In the next week, students would next explore some contextual problems involving multiplying decimals (eg. areas of rectangles, an amount times a rate (eg buying vegetables, simple income)). Contexts also provide a good opportunity to multiply larger decimal numbers for which the use of calculators would be appropriate. Students would then begin to explore dividing decimals by connecting both to dividing whole numbers, and to multiplying decimals.
Suggestions for Assessment
By the end of Grade 7, students should be able to think flexibly about which strategy they would use to do operations with decimals, including order of operations. Students should be able to make sense of multiplying and dividing decimals both pictorially (area model) and symbolically (using a variety of symbolic strategies).
Suggested Links and Resources
Surrey Schools Numeracy Support: Video Series for Parents
Number Strings: Life beyond the algorithm: Division of decimals
Key Number Concept 3: Relationships between Fractions, Decimals, and Percents
Overview
Students have been working with fractions and decimals since Grade 4, including the relationship between them in terms of tenths, hundredths, and thousandths. Students have also ordered and compared fractions and decimals, but separately. In Grade 7, students broaden their understanding of the relationships between fractions and decimals in many ways as will be described below. They will also order and compare fractions and decimals together. Students learned a lot about percents in Grade 6, and in Grade 7 students will focus more on the relationship between percents, and fractions and decimals.
Given that percents represent hundredths, relating percents and decimals is a natural connection and is a simpler concept than relating fractions to decimals and percents. Relating fractions and decimals begins with connecting decimal place value to fractions of tenths, hundredths, and thousandths, including simplifying such fractions to simplest terms. For example:
The above examples deal with what are called terminating decimals which are decimals that have a specific number of decimal places, and in Grade 7 students would typically go no further than thousandths. The types of fractions (in simplest form) that lead to terminating decimals are the ones which have denominators that can divide into a power of ten (ie. 10, 100, 1000). In other words, this includes any fraction whose denominator is a factor of 1000, though students will mostly focus on fractions whose denominator is a factor of 100 (2, 4, 5, 10, 20, 25, 50, 100).
Because students only typically work with terminating decimals across the grades, it may come as a surprise that most fractions actually do not convert to a terminating decimal. Repeating decimals are decimals which do not terminate, but rather have a core of a digit sequence that repeats. While there are complex methods to convert any repeating decimal to a fraction, more appropriate for Grade 7 is to explore patterns of fractions that have repeating decimals (eg: thirds, sixths, ninths, and ninety-ninths). For example, if , predict . The same pattern could be used to figure out a fraction like because .
Fractions and decimals can be compared using different strategies. For example:
- Using equivalencies (ie. convert all to fractions or all to decimals).
- Using benchmarks (eg. comparing a fraction less than one-half to a decimal that is greater than 0.5).
Number Sense Foundations:
The following concepts and competencies are foundational in supporting understanding of fractions, decimals, and percents in grade 7:
- The concept of a fraction which counts the number of equal parts of a whole
- The concept of a decimal which counts the number of tenths, hundredths, thousandths, etc. of a whole
- The concept of a percent which counts the number of hundreths of a whole
- Equivalent fractions
- Relating mixed numbers and improper fractions
- Different representations of fractions, decimals, and percents (diagrams or area models, grids, number lines)
- Comparing fractions using a variety of strategies (eg. diagrams, common numerators, common denominators, benchmarks)
- Comparing decimals using place value.
Progression:
- Explore the relationships between terminating decimals, and fractions concretely, pictorially, and symbolically (eg: grids, number line)
- Extend the relationship between terminating decimals and fractions to percents (eg. using hundredths grids).
- Explore the relationships between repeating decimals and fractions through pattern exploration.
- Compare and order fractions and decimals (eg. using equivalencies, benchmarks)
Sample Week at a Glance
This sample week describes what could be done during the first week of this unit.
Focus: Activate prior knowledge of fractions and decimals
- Before: Show me ¾ in as many ways as you can.
- For example:
- Concretely (pattern blocks, cuisenaire rods, fraction strips)
- Pictorially (area models, sets, number line)
- Symbolically
- Equivalent fractions concretely, pictorially and symbolically
- For example:
- During: Clothesline
- Place and space cards on a clothesline with different representations (pictorial or symbolic) of fractions and decimals. For example:
- Some cards may end up in the same place (eg. different representations of same fraction or decimal; a decimal that is equivalent to a fraction)
- Begin with benchmarks of 0, ½, and 1 to help with the placing and spacing. You could also extend to 2 or 3 to include some mixed numbers.
- Place and space cards on a clothesline with different representations (pictorial or symbolic) of fractions and decimals. For example:
- After: The After is integrated while the Clothesline routine is taking place. Encourage students to defend their decision about where a specific card belongs. Students can also move a card if they feel it’s needed, or they can create their own (especially if it helps in how to place other cards).
Focus: Relate fractions to terminating decimals
- Before:
- Explore ½
- Shade ½ of a rectangle (top half)
- Divide the rectangle into tenths (this makes a ten-frame)
- How can we express ½ as a decimal?
- How could we record this relationship symbolically?
- Explore ¾ in a similar way, but using a hundred-grid
- Explore ½
- During:
- Explore other fractions on ten-frames and hundred-grids.
For example:- Shade the fraction
- Relate to the equivalent decimal
- Record the relationship symbolically
- Explore other fractions on ten-frames and hundred-grids.
- After:
- Share and discuss both their pictorial and symbolic representations.
- Discuss what general strategies they noticed. For example:
- All of the fractions could be expressed as equivalent fractions using tenths or hundredths in order to convert them to decimals.
- Challenge the class and do a turn & talk to think about how to express ⅛ as a decimal. 8 does not divide evenly into 100, but it does divide evenly into 1000. For example:
- Knowing that ⅛ = 0.125, what about ⅜? ⅝? ⅞?
- Have students create their own practice questions to convert fractions to decimals. They should limit the denominators to numbers which are factors of 100 (ie. 2, 4, 5, 10, 20, 25, 50, 100).
Focus: Relate terminating decimals to fractions
- Before:
- Connecting to prior day’s learning, what if we started with the decimal?
- Explore a decimal whose fraction could be reduced to simplest terms. For example, 0.6
- Shade a ten-frame.
- How can we express 0.6 as a fraction?
- Do you see a fraction we can express in simpler terms?
We can see how
- Explore a decimal like 0.35 in a similar way, but using a hundred-grid.
We can see how
- How could we record this relationship symbolically?
- During:
- Explore other decimals on ten-frames and hundred-grids.
For example: - 3, 0.03, 0.45
- Shade the decimal
- Relate to the equivalent fraction
- Record the relationship symbolically
- Explore other decimals on ten-frames and hundred-grids.
- After:
- Share and discuss both their pictorial and symbolic representations.
- Discuss what general strategies they noticed. For example:
- All of the decimals could be expressed as fractions using tenths or hundredths.
- Use an example of a decimal to discuss expressing a fraction in simplest terms. For example:
- Have students create their own practice questions to convert decimals (tenths and hundredths) to fractions and vice versa. For extension, some students may work with decimals to thousandths.
Focus: Relate fractions and terminating decimals to percents
- Before:
- Review the meaning of percent as how many parts out of 100, eg. 0.56=56%
- Ask students to shade some fractions and decimals on a hundred grid. For example:
- Tell them to express each as a percent.
- Note that some may mistake 0.7 as 7%. It is helpful to read 0.7 as seven-tenths to indicate its numeric value, and that it is not the same as seven-hundredths.
- During:
- Create a design in a hundred grid with different labeled regions. Similar to a Fraction Talk, this can be a Fraction/Decimal/Percent talk.
- Ask them to name each region as a fraction, decimal, and a percent.
- For example:
- Depending on your class, you may wish to create a simpler or more complex design. A more complex design could include areas that have a slanted region (eg. triangle, trapezoid).
- After:
- Draw a table to fill out as the class discusses each region.
- For each region, ask students to explain how they knew.
- For the above example, the table could look like:
- Comparing will come later in the unit, but you may wish to ask students to list the regions in order from least to greatest area.
Focus: Practice to solidify and extend their understanding of the relationships between fractions, terminating decimals, and percents
- Before: Which One Doesn’t Belong
- Create or use a Which One Doesn’t Belong involving fractions, decimals, and percents. For example:
- Ask students to defend a choice for why one could not belong. There may be more than one reason for each.
- During: Math Workshop (Students should bring paper or mini-whiteboards with them. Have hundred grids available for those who need them. Encourage students to collaborate at each station.)
- Complete the table
- Provide a table with several rows in which one cell in each row is filled in (a fraction, decimal, or percent). They need to fill in the missing values.
- Roll 2 ten-sided dice (or use cards or spinners). Decide whether to use the two numbers to make a fraction or a decimal. Convert their number to the other forms (ie. fraction, decimal, or percent).
- Create increasing patterns involving decimals. Convert each decimal to a fraction. What do they notice?
- For example: 0.12, 0.24, 0.36, …
- Base Ten Blocks: What decimals/percents can you build using an assortment of 11 base ten blocks? Convert each to a fraction.
- Teacher led small group instruction: relating fractions, terminating decimals, and percents
- Complete the table
- After: students sharing what they did, and what they learned. Discuss any challenges they faced and what they feel they still need to work on.
In the next week, students would explore relating fractions to repeating decimals through pattern exploration (ie. given this fraction is this repeating decimal, what decimal would other fractions with the same denominator have?). They would next explore a variety of strategies for comparing and ordering decimals (terminating and repeating) and fractions.
Suggestions for Assessment
By the end of grade 7, students will be able to relate fractions, terminating decimals (tenths or hundredths, and percents by determining the equivalent form, including expressing fractions in simplest form. Students will also be able to relate certain fractions (denominator of 3, 6, 9, 99, and possibly 11 and 33 as well) to repeating decimals. Students will be able to use a variety of strategies (eg. equivalent forms, benchmarks) to compare and order fractions and decimals.
Suggested Links and Resources
Key Number Concept 4: Financial Percents
Overview
Students first learn about percents in Grade 6 in which they develop an understanding that percents represent a relationship of the number of parts out of 100 in relation to a whole. This relationship can be expressed as . They first solve problems in terms of determining the percent of a number, then move on to solving for a missing percent or a whole. In Grade 7, these same types of problems are explored, but within the context of financial percents.
There are a number of ways that percents are involved in real-life financial contexts. In Grade 7, it is more meaningful to focus on financial percents that relate to their lives. Two such contexts are dining (taxes and tips), and shopping (taxes, discounts).
There are several strategies that are helpful for solving percent problems, including:
- Benchmarks and a double number-line
- Benchmarks and mental math
- Equivalent fractions
- Hundred grid
- Converting the percent to a decimal
In Grade 8, students will explore percents greater than 100, as well as fractional percents.
Number Sense Foundations:
The following concepts and competencies are foundational in understanding financial percents in grade 7:
- Life experiences involving money for dining and shopping
- Percent concepts
- Meaning of percent
- Determining a percent of a given number (eg. what is 40% of 60)
- Determining a whole in a percent problem (eg. 20% of what number is 15?)
- Determining a percent in a percent problem (eg. what percent of 80 is 32?)
- Strategies for solving the above types of problems, including double number-lines, mental math, equivalent fractions, using decimals, and estimation
- Equivalent fractions
- Operations with decimals up to hundredths
Progression:
- Connect to life experiences involving financial percents, eg. Where have you seen percents involved with money in your lives?
- Determine the cost after sales tax (below is a suggested progression):
- Grocery shopping (most food items) are not charged any sales tax
- Only GST of 5% on dining, books, bicycles, and a few others that have PST exemptions
- Only PST of 7% on carbonated sweetened drinks
- This is one of the exceptions that has PST but not GST
- GST and PST on most retail shopping
- Taxes on a mixture of items (eg. going to a grocery store)
- Determine cost when dining after sales tax and tip
- Determine cost of retail item after a percentage discount.
- Integrated above or done after: solve for missing percent (eg. tip) and solve for missing whole (eg. given the total cost, what the cost before taxes and discount)
Sample Week at a Glance
Prior to this week students will have shared their experiences of financial percentages. They would then have explored situations in which there is only 5% GST charged (eg. dining without carbonated beverages, books, bicycles), then only 7% PST (sweetened carbonated beverages).
Focus: Most retail items purchased get charged both GST and PST.
- Before:
- Ask about a retail item with a simple cost. For example, a video game that costs $60.
- How much GST is charged?
- 10% of $60 is $6, so 5% is half of that which is $3.
- How much PST is charged?
- 1% of $60 is $0.60, so 2% is $1.20. 7% = 5% + 2%, so the PST is $3 + $1.20 = $4.20
- What is the cost of the item after taxes?
- $60 + $3 + $4.20 = $67.20
- Ask if there is another way they could figure out the total cost after taxes?
- We could add the GST and PST and just determine 12% for the taxes.
- As a possible extension, note that you can multiply the price of the item by 1.12 to get the total cost. Why does this work?
- How much GST is charged?
- Ask about a retail item with a simple cost. For example, a video game that costs $60.
- During:
- Ask the class to brainstorm 3 to 5 different items that would be charged both GST and PST for sales tax. Ask them only to suggest an item if they have a rough idea about what it would cost. Alternatively, you could suggest a specific store and have the website ready to browse the current prices.
- Ask the students to write down the items with their prices.
- Ask the students to work in small groups to figure out the total cost of all items after taxes.
- After:
- Have the class share and debrief answers and strategies.
- Some may have calculated each tax for each item separately, then added everything together.
- Some may have calculated the total tax for each item, then added everything together.
- Some may have added up the total cost of the items before taxes, then figured out the total tax of that subtotal to add to the subtotal.
- Ask them to choose their own 3 to 5 items that would have GST and PST charged and research the actual prices. Ask them to determine the total cost of all items after taxes.
- Challenge them to figure out a price before taxes (PST & GST) given the total cost after taxes. For example, if the total cost is $27.94, what was the price before taxes? Encourage them to estimate first.
- Have the class share and debrief answers and strategies.
Focus: When purchasing a number of items, the taxes on each item may be different.
- Before:
- Write the headings “No tax”, “PST”, “GST”, and “PST & GST” on the board or collaborative digital document.
- Brainstorm items they may find at a grocery store and list these items under the appropriate heading.
- During:
- Present a shopping list of about 10 items. Include items from various categories of taxation. For example:
- loaf of bread ($3.99)
- bell peppers ($5.99)
- shampoo ($6.99)
- salad dressing ($2.99)
- frozen fish ($12.99)
- laundry detergent ($13.99)
- spaghetti ($2.99)
- toothpaste ($2.99)
- paper towel ($8.99)
- diet coke (20 cans) ($9.99)
- Have a class discussion to estimate what the total shopping bill will be.
- Ask students in groups to figure out what the total of the shopping bill will be.
- Note: due to the conceptual focus on financial literacy and not on number calculations, it is appropriate for students to use a calculator for this task.
- Present a shopping list of about 10 items. Include items from various categories of taxation. For example:
- After:
- Have the class share and debrief answers and strategies.
- For the example above:
- No tax: bread, peppers, salad dressing, fish, spaghetti
Total: $28.95- Note that because the values all end in $0.99, if one were not using a calculator they could round up to the dollar to add, then subtract $0.05
- PST only: diet coke
$9.99 x 7% = $0.70
Some will know that there is also a $0.10 deposit per can = $2
Total: $12.69 - GST only: none of these items
- PST & GST: shampoo, laundry detergent, toothpaste, paper towel
$32.96 x 12% = $3.96
Total: $36.92
TOTAL BILL: $78.56
- No tax: bread, peppers, salad dressing, fish, spaghetti
Focus: When dining, in addition to sales tax one often adds a tip.
- Before:
- Have a class discussion on:
- Why people generally tip at restaurants
- What a reasonable tip amount is (15% is today’s common amount, with more for great service and less for poor service)
- Discuss as a class what a reasonable tip on a $32 restaurant bill is.
- Mental math is a nice strategy for figuring out tips. 10% of a value can be found just by dividing by 10, and then 5% is half of that, so 15% can be figured out fairly quickly. People often round up a tip as well to the nearest dollar, so estimation may come into play as well.
- 15% of $32 = $3.20 + $1.60
- One may tip precisely at $4.80 or may decide to go with $5.
- Tips are usually calculated on the total before taxes.
- Have a class discussion on:
- During:
- Have the class work in pairs to determine a reasonable tip for several scenarios. For example:
- 15% tip on a $63 bill
- 10% tip on a $18 bill
- 18% tip on a $44 bill
- 15% tip on a $23.95 bill
- Have the class work in pairs to determine a reasonable tip for several scenarios. For example:
- After:
- Have the class share and debrief answers and strategies. For the example above:
- 15% = 10% + 5% = $6.30 + $3.15 = $9.45 or $10
- 10% = $1.80 or $2
- 20% = 2 x 10% = 2 x $4.40 = $8.80
18% = 20% – 2% = $8.80 – $0.88 = $7.92 or $8 - 15% = 10% + 5% = about $2.40 + $1.20 = $3.60 or $4
- Ask the class an open problem such as: Someone gave a $5 tip. What could the value of their bill be?
- Present a restaurant bill and ask students to figure out the total amount paid including sales tax and tips. They’ll need to consider:
- 5% GST on the food and non-carbonated beverages
- 7% PST on any carbonated beverages
- How much to tip
- Have the class share and debrief answers and strategies. For the example above:
Focus: When retail items go on sale, sometimes this is advertised as a percentage discount.
- Before: Have a class discussion on different ways they’ve seen things on sale. For example:
- A discounted price, eg. save $10
- A percentage off, eg. save 25%
- A buy-one get-one offer
- During:
- Have students work in pairs to solve randomized problems (using spinners) to determine the sale price after a percentage discount.
- Give each pair of students two spinners and a paper clip. For example:
- Ask them to choose one of their problems and figure out the total cost after sales tax (PST & GST).
- After:
- Have a class discussion about a few of the possible problems, including sharing their solutions and strategies. Discuss also which combinations they found easy, and which they found challenging.
- Use one of the examples to compare two different but related strategies. For example, 30% off of $90:
- 30% of $90 is $27. $90 – $27 = $63
- 30% off means paying 70%. 70% of $90 is $63
- As a possible extension, consider situations involving more than one percent. For example:
- Is a 40% discount followed by a 20% discount the same as a 20% discount followed by a 40% discount?
- Does a 20% discount followed by a 20% increase result in the original price?
Focus: Begin research project on financial percentages.
- Before: Explain that they will be working on a research project to put what they have learned about retail shopping into practice. Elements of the project include:
- They can work in pairs or groups of 3.
- Decide on a reason for the need of the items, eg. hosting a birthday party.
- The class could brainstorm ideas.
- Create a list of the items they need. They have a $100 budget including taxes.
- Go shopping, ie. research the locations and prices for the items they need.
- Determine the total cost of the items on their list, including taxes.
- Decide on how to present their information, eg. poster board.
- Plan an oral presentation of their project.
- During: Observe the class as they begin their plan, prompting with questions as needed.
- After: Have a class discussion about any questions that have emerged so far.
The following week students would spend at least one more day working on their research project, then presenting their research project to the class. They would then do some further purposeful practice to deepen their learning from this unit.
Suggestions for Assessment
At the end of this unit, students will be able to determine which items have tax and how much in order to determine the total cost of a restaurant bill (after taxes and tip) and a shopping trip (after discounts and taxes). They may employ a variety of strategies to do so.