# Stage 4 -algebraic techniques - mystery of the bones

## Strategy

Students use the algebraic symbol system to expand and simplify simple algebraic expressions and substitute into algebraic expressions.

## Activities to support the strategy

### Activity 1

Teacher introduces the example of a forensic scientist who uses skeletal remains to infer a person's height from a few bones. Anthropologists have developed formulas to estimate a person's height from the length of long bones such as the humerus and the femur. These formulas change depending on the person's gender and race.

The humerus (single bone extending from the elbow to the shoulder socket) may be used to obtain an indication of a person's height using formulas such as these:

- Female: height = 2.8 x length of the humerus (in mm) + 704 or H = 2.8L + 704
- Male: height = 2.9 x length of the humerus (in mm) + 695 or H = 2.9L + 695

1. Students are given these formulas and they are asked to use the relevant one to estimate their heights (the group may first need to be guided through an example on how to substitute values into the formulas). Students work in pairs to check the accuracy of the formulas by measuring their heights directly to complete the information below. Record:

- Length of their humerus
- Height according to the formula
- Actual height

2. In groups, students apply formulas that estimate the height of a person by using four of the body's long bones to solve an archaeological problem.

### Mystery of the bones

#### 1. The scenario

The remains of four girls have been discovered in a recent archaeological find. There are four bones from each girl left but archaeologists don't know which bones belong together. You have been engaged as the mathematician to work out which bones belong to which girl and approximately how tall the girls were.

#### 2. Background clues

By knowing the length of certain bones in the body, scientists can estimate a person's height. The bones used are the humerus (shoulder to elbow), radius (elbow to wrist), femur (thigh bone), and tibia (knee to ankle).

The formulas for approximate heights (in mm) for males and females are shown in the table below.

Female height | Male height |
---|---|

2.8 x (length of humerus) + 704 | 2.9 x (length of humerus) + 695 |

3.3 x (length of radius) + 800 | 3.3 x (length of radius) + 846 |

2 x (length of femur) + 717 | 1.9 x (length of femur) + 800 |

2.4 x (length of tibia) + 736 | 2.4 x (length of tibia) + 774 |

##### Example

For a tibia belonging to a male and which is 45cm long, we would estimate the height by using the formula:

- H = 2.4L + 774

Converting the length to millimetres first, we substitute L = 450 to obtain:

- H = 2.4 x 450 + 774
- H = 1854mm

Therefore, the tibia belongs to a man who is about 1.85 metres tall.

**3. Your task**

The data table below shows the length of various female bones found at the site. The bones are numbered from 1 to 16 and the measurements are in millimetres. By examining each bone in turn, calculate the approximate height of the female for each bone. Then match the bones to discover which bones go together and find the approximate height of each female.

Number | Bone type | length (mm) |
---|---|---|

1 | Humerus | 235 |

2 | Tibia | 385 |

3 | Femur | 470 |

4 | Tibia | 407 |

5 | Radius | 170 |

6 | Radius | 210 |

7 | Radius | 260 |

8 | Humerus | 280 |

9 | Femur | 320 |

10 | Femur | 495 |

11 | Tibia | 260 |

12 | Humerus | 360 |

13 | Tibia | 315 |

14 | Humerus | 340 |

15 | Radius | 275 |

16 | Femur | 385 |

Bone | Calculation | Approximate height in metres |
---|---|---|

1 | ||

2 | ||

3 | ||

4 | ||

5 | ||

6 | ||

7 | ||

8 | ||

9 | ||

10 | ||

11 | ||

12 | ||

13 | ||

14 | ||

15 | ||

16 |

Mystery of the bones work sheet (PDF 171.72KB)

### Activity 2

1. Pose this problem: Is 6 (2 + 3) = 6 x 2 + 3?

Work through the problem on the board by following these steps.

- Ask: What is 6 (2 + 3) equal to?
- Show how to work through the part in brackets first: 2 + 3 = 5, so 6 (2 + 3) = 6 x 5
- Draw an array of dots to show 6 (2 + 3) = 6 x 5
- Ask: What is 6 x 2 + 3 equal to?
- There are no brackets, so emphasise that the multiplication is done before the addition.
- Draw an array of dots to show 6 x 2 = 12, then + 3

- Remind students of the order of operations when solving number sentences.

2. Have the students investigate problems of the following type in small groups:

Kyle wrote 6 (11 + 7) = 6 x 11 + 7 in his maths book.

**Ask students to:**

- do the calculations to show the statement is incorrect
- draw an array of dots to convince Kyle that his statement is incorrect.

3. Have students explore the following problem: Determine and justify whether a simplified algebraic expression is correct by substituting numbers for letters, does 2x + 3x = 5x when x = 2?

4. Ask students to develop algebraic expressions for their partner to explore, then discuss and check their answers as a class.

5. Students carry out the Modelling seating arrangements activity on maximising seating arrangements by developing simple linear equations to solve the problem.

6. Through substitution, use spreadsheets to investigate formulae:

### Activity 3

As a class, discuss the similiarities and differences in expressions

### Activity 4

1. Pose this problem: Is 6 (2 + 3) = 6 x 2 + 3?

Work through the problem on the board by following these steps.

- Ask: What is 6 (2 + 3) equal to?
- Show how to work through the part in brackets first: 2 + 3 = 5, so 6 (2 + 3) = 6 x 5
- Draw an array of dots to show 6 (2 + 3) = 6 x 5
- Ask: What is 6 x 2 + 3 equal to?
- There are no brackets, so emphasise that the multiplication is done before the addition.
- Draw an array of dots to show 6 x 2 = 12, then + 3

- Remind students of the order of operations when solving number sentences.
- The array shows that 6 (2 + 3) is not equal to 6 x 2 + 3.

2. Have the students investigate problems of the following type in small groups:

Kyle wrote 6 (11 + 7) = 6 x 11 + 7 in his maths book.

**Ask students to:**

- do the calculations to show the statement is incorrect
- draw an array of dots to convince Kyle that his statement is incorrect.

3. Have students explore the following problem: Determine and justify whether a simplified algebraic expression is correct by substituting numbers for letters, - does 2x + 3x = 5x when x = 2?

4. Ask students to develop algebraic expressions for their partner to explore, then discuss and check their answers as a class.

5. Students carry out the Modelling seating arrangements activity, developing simple linear equations to solve the problem.

6. Using substitution in a spread sheet, investigate formulas such as:

### Activity 5

1. As a class, discuss the similarities and differences in these expressions:

No. | 1st term | 2nd term | like/unlike | |
---|---|---|---|---|

1 | 5a | 6a | like | |

2 | 2b | 4b | like | |

3 | 7c | 4c | like | |

4 | 8d | d | like | |

5 | 7e | f | unlike | |

6 | 6g | fg | like | |

7 | 4h | 4h | like | |

8 | 2j | 3k | unlike | |

9 | 3m | 8n | unlike | |

10 | 4p^4 | 7p^4 | like | |

11 | 6q | 2r | unlike | |

12 | 9s | 5s^3 | unlike | |

13 | 9u | 4u | like | |

14 | 4v^2 | 7v | unlike |

Similarities and differences in expressions work sheet (PDF 68.4KB)

Ask students to formulate a definition of like terms.

Like terms are algebraic expressions which have the same variables and powers, the coefficients do not need to match.

2. As a class, determine which terms are like in these expressions:

No | Expression | Like terms |
---|---|---|

1 | 3a + 3b + 7a | 3a, 7a |

2 | 5c + 8d - 4c | 5c, -4c |

Which terms are like? (PDF 68.14KB)

3. As a class, simplify these expressions by first collecting like terms

**Example**: 3a + 3b + 7a

= 3a + 7a + 3b (collect like terms)

= 10a + 3b (simplify)

**Example**: 5c + 8d -4c

= 5c -4c + 8d (collect like terms)

= c + 8d (simplify)

Simplify expressions worksheet (PDF 72.87KB)

4. Lead the class in expanding the grouping symbols and simplifying these expressions:

**Example**: 3 (a + 3b) +7a

3 x a +3 x 3b +7a (expand grouping symbols)

3a + 9b + 7a

3a + 7a + 9b (collect like terms)

10a + 9b (simplify)

Expand grouping symbols and simplify worksheet (PDF 68.66KB)

### Activity 6 – independent

1. Change these relationships from words to symbols

Description | Words | Symbols |
---|---|---|

1 | Start at 13 and add 2 each time | 13, 15, 17, 19 and so on |

Words to symbols worksheet (PDF 59.55KB)

2. Use the cards from the symbols and words sheet to play Concentration.

Symbols and words sheet (PDF 69.41KB)

## References

### Australian curriculum

ACMNA176: Create algebraic expressions and evaluate them by substituting a given value for each variable. ACMNA177: Extend and apply the laws and properties of arithmetic to algebraic terms and expressions.

### NSW syllabus

MA4-8NA: Generalises number properties to operate with algebraic expressions.